CONTROL OF DISTRIBUTED PARAMETER SYSTEMS A Thesis ...

CONTROL OF DISTRIBUTED PARAMETER SYSTEMS

A Thesis Presented for the Degree of

Doctor of Philosophy

in the Faculty of Engineering University of London

M.J. McCann Elec. Eng. Dept., Imperial College,

July, 1963

London.

i.

ABSTRACT

Many industrial processes are shown to have a common underlying form

involving transport of energy and matter by flow and dispersion or diffusion.

Mathematical models using the Laplace transform and root-locus methods

produce space parameter dependant transfer functions, and explain resonance-

like phenomena characteristic of distributed forcing of flow systems.

Harmonic or functional analysis depends on having eigenfunctions for the

differential operators for the system. The number of sections required for

a lumped parameter model produced by spatial quantization was found (on a

digital computer) to depend on the disturbances being considered and a

parameter characterizing the system. A cheap, simple, special purpose

electronic analogue was developed.

Control design by conventional methods yields a useful standard of

comparison. The absolutely optimal solutions from the calculus of

variations (etc.) are shown to present major computational difficulties

especially when the theory is extended to partial differential and integral

equations. Practical use of sub-optimal control design methods and the

analytical development of a direct feedback controller all depend on having a

state-space of low dimensionality.

A correlation coefficient criterion for instrumentation gives a method

for specifying instrumentation for protection purposes but not for control or

performance measure.

Control based on instantaneous computation on a measure of state is

shown to need only small amounts of instrumentation but sensitivity to

parameter changes has to be taken into account. Spatially distributed

11.

control can deal with disturbances arising anywhere in the system, and

sensitivity to parameter changes is reduced at the cost of greater complexity

The structure of the control scheme and its instrumentation is largely

determined by the spatial location of the measure or measures used for

performance assessment and the relationship between spatial displacement and

time delays in the distributed systems.

ACKNOWLEDGI,EN TS

The author wishes to express his appreciation to the Scholarships

Committee of the University of London for the award of a Post-Graduate

Studentship for the first two years of this research project and to the

Governors of the Imperial College of Science and Technology for an

appointment enabling the work to be continued.

The support of his supervisor, Professor J.H. Westcott, in giving

both direction and considerable freedom was invaluable.

To all the colleagues in the Electrical Engineering Department, who

provided both sounding boards for arguments and innumerable interesting

questions, and to the technical staff for assistance, always available, to

arrange, beg, borrow, or make equipment, grateful thanks are due.

Finally, thanks are due to Miss Jane Knight for typing, from almost

unreadable script, what turned out to be a very long thesis.

iv.


Table of Contents

Chapter I

ABSTRACT

ACKNOWLEDGMENTS

CONTENTS

INTRODUCTION

Page

iii iv

1

1.1 Starting Point: Systems 1

1.2 Starting Point: Control 2

1.3 Problems Classified 3

1.4 Method of Approach 3

PART I - Systems and Models 5

Chapter II DISTRIBUTED PARAMETER SYSTEMS 6

2.1 Introduction 6

2.2 The Basic Structure 7

2.3 Taylor Dispersion 8

2.4 Diffusion of Probability: The Fokker Planck Equation 11

2.5 Tubular Reactor 13

2.6 Distillation Columns 14

2.7 Heat Exchangers 18

2.8 Heat Transfer to Moving Solids 19

2.9 Dimensionless Parameters 22

2.10 Summary and Conclusions 24

Diagrams 25

v. Page

Chapter III TRANSFER FUNCTIONS 26

3.1 Introduction 26

3.2 Operator and Boundary Conditions 26

3.3 The Basic Transfer Function 30

3.4 Physical Realizability and Riemann Surfaces 32

3.5 Frequency Responses 35

3.6 Root Locus Plots 37

3.7 Comparison with Limiting Cases 38

3.8 More Complex Systems 41

3.9 Inversion of the Basic Transform 46


Diagrams 52

Chapter IV LUMPED PARAMETER MODELS 60

4.1 Introduction 60

4.2 Basic Quantized Equation 62

4.3 The Equivalent Stirred Tank Model 64

4.4 How Many Lumps? 68

4.5 Digital Computer Simulation 71

4.6 Conventional Analogue Simulation 72

4.7 Special Purpose Electronic Analogue 74


Diagrams 80

Chapter V 5ARMONIC OR FUNCTIONAL ANALYSIS

5.1 Introduction

102

102

5.2

vi.

The Laplace Transform Applied to

Page

the Space Dimension 103

5.3 General Structure of the Summation 104 Transform

5.4 The Effect of Using Eigenfunctions 107

5.5 Special Functions - not Eigenfunctions 108

5.6 Triangular Interpolator 111


Diagrams 118

PART II - Control 120

Chapter VI CONVENTIONAL CONTROL METHODS 121

6.1 Introduction 121

6.2 Frequency Response Methods 122

6.3 Root Locus Methods 123

6.4 Three Term Controllers 125


Diagrams 132

Chapter VII VARIATIONAL AND ASSOCIATED METHODS 138


7.2 Formulation of the Lumped Parameter Problem 142

7.3 Solution Methods 144

7.4 Goal Seeking Behaviour 153

7.5 Analytical Design of Optimum Feedback Controller 155

7.6 Variational Methods and Partial Differential Equations 159

7 Variational Methods and Integral Equations 162

7.8 Sub-Optimal Policies

7.9 Cost Functions, Liapunov Functions and tric*s

7.10 Special Problems of Distributed Parameter Systems

7.11 Adaptive Controllers

7.12 Summary and Conclusions

Diagrams

181

181

8.2 Non-Unique Trajectories 182

8.3 Correlation Coefficients Criterion for Instrumentation 184

8.4 Trajectories in a Reduced State Space 187

8.5 Comparison with Crude Models 189


Diagrams 193

Chapter IX STATE MEASURE CONTROL 206


9.2 Single Probe Control 207

9.3 Multiprobe Control 215

9.4 Distributed Control Action 218

9.5 Comparison with Conventional Control 226

9.6 Structure and Parameter Optimization 227


Diagrams 231

163

165

173

174

176

180

Chapter VIII INSTRUMENTATION AND A REDUCED STATE SPACE

8.1 Introduction

viii.

Page

Chapter X PROGRESS AND RECOMMENDATIONS 240


10.2 Systems - Flow and Diffusion 240

10.3 Models 242

10.4 Control 244

10.5 Instrumentation 247

10.6 Final Summary and Further Requirements 248

Chapter XI BIBLIOGRAPHY 250


11.2 Distributed Parameter Systems 250

11.3 Frequency Response and Root Locus 260

11.4 Lumped Parameter Models 263

11.5 Harmonic and Functional Analysis 265

11.6 Conventional Control 266

11.7 Variational Methods 268

11.8 Instrujentation and a Reduced State Space 273

11.9 State Measure Control 274

APPENDICES 276

1.

Chapter I

STARTING POINTS : SYSTEMS AND CONTROL

1.1 Starting point : Systems

There are many industrial processes, such as are found in the

chemical engineering fields, which have a distributed nature. They

occupy a continuous region of space and the state of the process at any

point is a function of position in space as well as being a function of

time. For example the temperature of a fluid passing through a heat

exchanger is a function of position in the heat exchanger and of time.

This is a distributed parameter system. A servo-system, for which the

speeds and positions of the parts describe fully its state at any time, is

not a distributed parameter system. Distributed parameter systems may be

described by partial differential equations in space and time variablesi as

opposed to lumped parameter systems described by ordinary differential

equations.

Most examples of this type of system occur in the chemical and

allied industries. For example, heat exchangers mentioned above, chemical

reactors of infinite variety in shapes and sizes, absorbtion and distilla-

tion columns, furnaces and soaking pits.

When considered for control purposes or for any analysis which

involves knowledge of their dynamic characteristicsl each type of system

has usually been considered on its own as a special case. The analysis

of the static patterns of behaviour in these systems has so far dominated

chemical engineering. and dynamic behaviour has only relatively recently

become significant, as faster, less stable manufacturing processes become

2.

generally available or necessary.

There is however no general summary of system analysis for them

from the control engineering point of view.

1.2 Starting point : Control

The frequency response methods based on the Laplace and Fourier

transforms are well established for lumped parameter systems. The

solution of partial differential equations by these transform methods is

also available as a standard technique but the concepts of transfer

functions and the like for distributed parameter systems have received

little attention.

The theoretical methods of control design by the calculus of

variations and the associated approaches through Dynamic Programming and

the Pontryagin method have been treated in considerable mathematical detail

for lumped parameter systems. At the time of initiating this work the

studies had not extended to distributed parameter systems but some theore-

tical material is now becoming available. Considerable work is in progress

to extend the usefulness of these methods by computational developments,

but this is really only for lumped parameter systems.

A considerable amount of practical experience in controlling

industrial processes must not be discounted. The techniques uhioh have

been developed over many years have resulted in useful and efficient

control systems which present few difficulties in operation or maintenance.

Such data as is available is scattered through the literature of the

process industries and deals each time with a specific plant or problem;

certainly there is no general control engineering study of distributed

parameter systems.

3.

1.3 Problems Classified

From the above comments it can be seen that there is a need for a

link between the chemical engineering systems analysis and the control

engineering approaches; . within the control engineering field there is

a marked gap between the advanced theoretical methods and the established

practical techniques. This latter gap is not confined to the study of

distributed parameter systems.

To make progress with the investigation of distributed parameter

systems the first requirement is to establish the basic characteristics to

be found in them. Regardless of the purpose or use of the plant concerned,

only the dynamic behaviour is of interest.

Having extracted the behavioural structure it must be modelled,

mathematically, to provide a basis for computation, simulation, and control

analysis.

The various available techniques of control design must be invest-

igated and extended where necessary either until they become useable or

until they are proved unsuitable.

Complete knowledge of the state of a distributed system cannot be

obtained with a finite number of instrumentss but needs of control schemes

and any other needs must be met with a restricted amount of equipment.

Criteria are required for the assessment of system state.

The problems of modelling, control and instrumentation cannot, of

course be separated, but they provide a classification for investigation

and results.

1.4 Method of Approach

To tackle completely such a wide field would be completely imprac-

4.

ticable, and any treatment general enough to cover virtually all possi-

bilities would be of necessity so hypothetical and theoretical as to be of

little engineering value.

The objective here is to look for pointers to generally applicable

concepts and techniques by dealing with a relatively restricted class of

systems, sometimes reducing to a single example where this shows clearly

the underlying structure of the answers required. Always the ultimate aim

must be to make an assessment of techniques and methods which is of

engineering rather than mathematical standards. The basic criteria of

success are to be; "Could it be made at reasonable cost and will it work

better than anything else?"


Part I

Systems and Models

5.

6.

Chapter II

DISTRIBUTtll PARAMETER SYSTEMS

2.1 Introduction

In many industrial processes, chemicalreactions, material and heat

transfer occur not just at one point but over a considerable region of

space. Take, for example, the tubular reactor where fuel stock enters

and gradually during the passage through the system is transformed into

different material. There is no one number or set of numbers to describe

the whole range of concentrations of the materials inside the reactor, but

at any instant of time properties of concentrations for the materials can

be considered as functions of the distance along the reactor. So it is

with many processes. Some may be distributed in a three dimensional space,

some in two, some essentially only in one. Many of them, like the tubular

reactor involve two transportation processes. These are the direct

movement of material by a flowing stream and the other the overall result

of random motion on a macroscopic or microscopic scale - dispersion or

diffusion. It is only because of these processes of transport of heat or

concentrates that any of these processes take effect. Were it not for

these transfer effects there would be no interaction between effects at

different places, no means whereby the results of chemical reaction could

be utilized, no nuclear reactors or even heat exchangers.

Thus in dealing with distributed parameter systems flow and diffusion

processes are of fundamental importance. Some systems are described here

which exhibit these effects, together with indications of the various

parameters used to describe them; for example, Peclet and Reynolds

numbers. Because the basic problem is that of control the system will

be considered in a structurally simple form, distributed in only one

space dimension, but this does not preclude the extension of the ideas to

more complex systems. Furthermore the systems will be considered as

having constant velocities of flow and constant amounts of dispersion,

though here again generalization is possible, particularly in the sense of

having these parameters as functions of the space dimension, a common

physical situation.

Because of the range of background material (chemical engineering,

automatic control and mathematical literature) there is some clash of

symbolism. The symbolism of the source material has in general been

retained, with suitable explanation, particularly where there is a well

established pattern, such as in the distillation column literature.

2.2 The Basic Structure

The basic process underlying the systems to be considered is that

of a mixture of flow and diffusion. This may be flow of fluid with

dispersion of concentrates or diffusion of probability in the theory of

random walks and the movement of minority carriers in semiconductors.

In these processes there is a behaviour where the fluid flow process

(plug flow),which is supposed to transport concentration and temperature

properties along without modifying themlis blurred and modified so that

as the fluid flows along/ mixing, turbulence or diffusion smooth out the

irregularities in profiles.

The unidirectional fluid flow can be represented by the partial

differential equation;

au = u au at " ax

t is the time variable,

x is the space variable.

where: U is a concentration

V the flow velocity in the direction of the x.

8.

When the dispersion or diffusion term is included this becomes

6U6U a2U - V

at ox + D ax2. where: D is a diffusivity. (2.1)

The following sections deal with specific cases showing how this structure

arises and the relevant values of the parameters V and D.

2.3 Taylor Dispersion of Concentrates in fluid flow

Two cases are to be considered. One case is laminar flow, the

;s other turbulent flow. The name TaylorAused here and elsewhere because

of the work of Sir Geoffrey Taylor published in the Proceedings of the

Royal Society 1953'54. (2.19)

In the case of lcmkor flow,concentrates carried along by the fluid

flow diffuse laterally (radially) through the fluid,while the various layers

of fluid move past each other. The overall effect is that the average

concentration across the fluid flow appears as if it were due to a transport

effect at the average velocity of the fluid (which is assumed to be 4. the

velocity at the centre of the fluid stream for circular pipes) together

with a diffusion effect around the position of a reference particle moving

at this mean velocity. The velocity term, V, in equation 2.1 is the

average velocity of the fluid flow. (For the reference particle moving

at velocity V; x = Ut). The diffusion coefficient D is given by the

formula:- 4a2v2

D - 192.d where: a is pipe radius (2.2)

d is coefficient of molecular diffusion.

9.

In the analysis whereby this result is derived the molecular

diffusion in a longtitudinal direction is neglected because of the relative

times required for it to have any effect in comparison with the radial

diffusion effects. For this, the condition to be met is that:

L a2

2V " (3.8)2d

Part of the explanation for the phenomenon in which the effective

longtitudinal diffusion coefficient is inversely proportioned to the

molecular diffusion coefficient is:"... this means that in the central

part of the pipe, fluid which is free of the dissolved substance passes

into the zone where the concentration is rising. The dissolved

substance is then absorbed until U reaches its maximum value at x = Vt.

The fluid then passes through the region where U decreases with x and

finally leaves this zone, having yielded up the whole of the dissolved

substance which it had acquired." Thus if the coefficient of molecular

diffusion d is small the process of acquiring and losing the concentrate

is protracted and the apparent longtitudinal dWus*ion large.

The experiments described demonstrate the validity of the

theory in predicting the shape of the distributions of concentration

actually measured. And although it is more difficult to assess the

numerical accuracy of the coefficient D, some figures given as estimates

of d from the experiments indicate reasonable agreement.

In the case of turbulent flow the concentrates are dispersed

relative to their mean flow velocity position by the swirling and eddying

of the fluid. Since many industrial flow processes run at Reynolds

numbers high enough to ensure fully developed turbulent flow this is an

where: L is the length of the (2.3) pipe or thereabouts.

10.

important case. Furthermore in view of Reynolds analogy between the

transfer of heat and of momentum and of mass this case is also relevant to

the systems where transfer of heat or momentum is involved.

The main theoretical result from the Taylor paper (2.19) is that

the effective longtitudinal diffusion (or dispersion) coefficient is

D = 10.1 a Vt (2.4)

where and a is the pipe radius.

To is the friction stress on the wall of the pipe due to fluid of

density r .

By writing D = 10.1 a V

with V the mean flow velocity, (2.5)

this can be evaluated in terms of Reynolds number alone;

_ 1 V -r

W Because i - Go and using the usual coefficient for V

0 (2.6)

o = i 6/P V2 pipe friction. r gives (2.7)

i

Y

f'. = J but but W. is a function only of Reynolds

number, R. and is given by the formula

Y i = - 0.04 + 4.00 loqoR + 2.00 lo oW . (2.9)

(See Goldstein 2.13)

The relationship between V, Iand R is shown graphically (p. 454 of

Proc. Royal Society 223A) and fig. 2.1.

Comparison of practical results with the theory showed good

agreement. For example in the equation D = 10.1 a V the coefficient was

found variously as 10.6, 11.6, 100, 12.8 for smooth pipes and 10.5 for a

rough pipe. These were all for straight pipes without end effects but

when the pipes were Lu'ved (e.g. in dia. pipe into 3' dia arch) the

11.

coefficient was 1 or 2 times as big depending on flow velocity. End

effects are also important; discontinuities in pipe diameter,pumps,

corners etc. all tend to increase the effective diffusion

2.4 Diffusion of Probability - The Fokker-Planck Equation

A particle is considered to move backwards and forwards in a

liva, taking steps randomly in either direction (a random walk) but with

a higher probability of moving in one direction than in the other. The

nett result is a drift in one direction but it is also of interest to know

the probability of its being at any particular point as a function of time.

The Fokker-Planck equation which describes the behaviour of this probability

density function for the position of the particle is a partial differential

equation which in its simplest form has the same structure as the equations

for one dimensional fluid flow with turbulence.

The position of the particle may stand for the state of an adaptive

control system, trying to improve performance but never absolutely certain

which way to go, or it may represent an electron moving in a crystal lattice

as discussed below with regard to minority transport in seminoonductors.

A very simple heuristic, derivation of the Fokker-Planck Equation,

which will demonstrate the significance of the coefficients, follows:

(Ref. Feller 2.75)

A particle moves once for each unit of time, L: . The steps

are either +h with probability p, -h with probability q and no movement

at all with probability r. If U(n+1,k)

is the probability of being at

position kh at time (n+1) having started from 0 at time 0, this can

be expressed in terms of the situation at time nt in view of the three

possible ways of reaching the relevant position, either being there and

12.

staying there, coming forward or going back, so that

p Un,k-1 q Un,k+1 + r Un,k (2.10)

for n )* 1 and also for n= 1 if: U oo = 1 and Uo,k = 0 for k = 0

are taken as initial conditions.

The next step is to consider this discrete version approximated by

a differential equation when the step sizes and time periods become smaller a

and smaller. Then Unk corresponds to the integral taken over an interval

of length .h around x = kh at time t = nr. Thus writing Unk as a

function of the continuous variables x and t gives U(kh,u ) = h-1• Uu,k

where U(x,t) is now a probability density function.

Equation 2 10 now becomes

U(x,t+-C) = p U(t,x-h) q U(t,x+h) + r U(x,t) (2.11)

or (noting p+q+r = 1)

U(x,t+t) - U(x,t) = - pIU(x,t) - U(x-h,q+ q{U(x,h,t) -U(x,t)J (2.12)

but this is equivalent to a finite difference version of

(p+q)h2 . a2U(x,t) au (p-q)h au(x,t) U(x,t) = + 67 • (2.13) at • ax

The coefficients can be interpreted thus:

is the average distance moved (forward, the +ve

x direction) per unit time - i.e. the velocity V,

is the mean square displacement per unit time,

equivalent to 2D where D is the effective

diffusion coefficient used in eqn. 2.1.

(p-q) h

(p+q)h2

A more general derivation using more possible step sizes can be used

to give the same results, and, allowing that the probability of each step

13.

is a function of position as well, the equation becomes:-

D(x) . U(x,t)j aU(x at t)

= -ax a rv.(x) . U(x,t) +a

Although this general Fokker-Planck Equation is outside the scope

of the detailed treatment of the control problem considered here some of

the results are applicable for dealing with it.

2.5 Tubular Reactor

The device of interest here is the packed bed tubular reactor in

which fluid flows in the interstices of a random packing of particles which

may be specially shaped ceramic rings or lumps of a material involved in

the reaction. In the common case where the 'bed' is in the form of a long

tube the behaviour of the fluid in so far as it transports material (not at

this stage including terms of reactions) is reasonably well described by

the Taylor Diffusion Equation 2.2.

This effect - axial dispersion in packed beds - has been investigated

practically by Eback and White (2.12) and Liles and Geankoplis (2.16),

among others, who were interested in evaluating the relevant diffusion

coefficients. The results are reported in terms of the dimensionless

parameters, Peclet and Reynolds Numbers. (See Section 2.9 below).

Using the equation

D a2U at ax (177

with the Peclet Number d V

Pe = D (2.14)

and the Reynolds Number d . V p

R

(2.15)

(2.2)

14.

- where; d is the mean particle diameter, V is the interstitia velocity

of the fluid, D is the axial diffusion coefficient (as in equation 2.2),

p is the fluid density, p is viscosity, and V$ is the velocity based on

an empty tube, -

Pe is found to range from 0.3 to 0.8 as R goes from 0.01 to 150

(Eback and White) and from 0.4 to 0.8 as R goes from 2 to 200 (Liles and

Geankoplis). Despite their cave in measurement it is difficult to give

very precise figures. They report that end effects are very important,

though particle shape makes little difference. Their collected results

are reproduced in fig. 2.2.

Other work which takes account of the dispersion effects in packed

bed reactors is that of Amundson, Coste and Rudd (2.7) in which a numerical

analysis of the temperature profiles (etc.) in a reactor in which an

exothermic reaction takes place is carried out by approximating the diffusion

effect by considering the reactor as a suitable number of cascaded stirred

tank reactors. (See also Aris (2.1) for a description of this work and

Chapter 4, here, "Lumped Parameter Models")

2.6 Distillation Columns

In the various theoretical treatments of the transient behaviour of

distillation columns, two basic types are considered, one is the packed

column, the other the plate type, and although these lead to structurally

different equations there is in practice little real difference in their

overall behaviour (Rosenbrock 2.41). These different equations appear as

two first order partial differential equations, of hyperbolic type, one

for each of the two countercurrent streams considered in the packed column

15.;

and one second order, parqbolic partial differential equation for the

continuous variable version of the set of ordinary differential equations,

one for each plate, in the plate column.

The object is to split a fluid mixture of one or more compounds by

making use of their different volatilities. A stream of vapour rises

through the column and a stream of liquid falls. Mass transfer from

liquid to vapour takes place because for the vapour to be in equilibrium

with the adjacent liquid it will (generally) have to have a higher

concentration of the more v)latile components. Thus at any part of the

column the material transported upwards (by the vapour flow) is richer in

the more volatile components. The overall effect is for the top of the 1;qhrer

column to be producing the iittaar components, the bottom producing the

heavier components, with a gradual transition in between. The material

to be refined or separated is fed into the column roughly where the

concentrations inside the column are the same as this fifeed-stock." The

vapour flow is produced by heating ("reboiling") some of the liquid which

has fallen to the bottom and the liquid flow by condensing, and returning

as "refltx", some of the vapour which has risen to the top. The residual

quantities, which must eventually balance the total input, are taken out

as 'bottom' and 'top' products respectively.

The process of interest in this context is the transport of mass

or concentration through the column. Any analysis is bedevilled by the

fact that the relationship between the equilibrium concentrations of

components in vapour and liquid are always non-linear. Furthermore,

because the mixing of liquid and vapour can never be perfect,equilibrium

is not really reached. In the plate type of column this mixing is

16.

achieved by catching the falling liquid in trays and bubbling the rising

vapour through it. In the packed column both liquid and vapour flow

through the interstices between randomly packed ceramic rings (or similar

objects).

These structures lead naturally to the different equations which

represent the behaviour. Consider the plate type of column.

The equation of concentration variation on and around the nth

plate relates the rate of change of concentration to the rate of flow of

the component from below and from above and the loss of the component from

the region, giving the equation 2.16. This analysis is usually done for

the more volatile component in binary separation and for (k-1) components

inak component mixture - all the rest of the mixture must be the other

remaining component.

The equation for the concentration on the nth plate of a column

is thus:-

(2.16) d-t- Hx + h nn nyn l V y n-1 n-1

+ L x n n n+1 n+1 V Y -Lx nn

where Hn and hn are the liquid and vapour 'hold up" capacities at the nth

stage,xn and yn the liquid and vapour concentrations at the nth stage

Ozx

Ln

and Vn the liquid and vapour flow rates: at the nth stage.

Usually hn4< Hand hn is neglected, H often the same for each stage. In

the case where Hn is constant (i.e. not modified by flow rates etc.) and

L, D and V independant of position in the column this reduces to

dxn H dt = Vy

n-1 + Lxn+1 - Vyn - Lxn . (2.17)

The difficulties arise because each yn is a non linear function of

17.

its corresponding xn - the "equilibrium" relationship. For columns with

a very large number of plates the equations can be treated by expanding

Vn-1 yn-1 and L

n+1 xn+1

in Taylor series and keeping terms up to second

order Leqn. 2,17_1 giving the particil differential equation:-

&Hx) = of

(Lx-vy) + T52,(Lx+Vy) where the parameter n (2.18)

has now become the continuous distance variable 1 and H represents the

hold-up per unit length of column. Apart from the non-linear relationships

between x and y the structure is similar to that of equation 2.1:-

8H _ 8H 82u

at - -v ax D

meanings used in that section.

The usual packed column analysis has a partial differential equation

for each stream.

-11(Hv) = a(Lxi - w(x,y) at

a , 7F( hy) = -Vy + w(x,y)

for the liquid stream (2.19)

for the vapour stream. (2.20)

where w(x,y) is the transfer of the component from liquid to vapour unit

length of column,with x and y the relevant liquid and vapour concentrations.

Again, H and h, L and V are liquid and vapour hold-ups and flow rates.

The analytical difficulties now lie in w(x,y).

In practical columns it has been observed that a disturbance in the

feed composition travels up the system to modify the top product and down

again in the reflux, but the disturbance, which may have been quite sudden

and sharp when it went in, comes out considerably blurred.

The parabolic partial differential equation for the plate type

column can describe the blurring in a disturbance but only covers

where the parameters have the (2.1)

18.

propogation in one direction. The pair of hyperbolic partial differential

equations for the packed column can, with suitable boundary conditions

describe the propogation in both directions but wavefronts travel through

just as sharp as they went in. A better answer would seem to lie in the

suggestion of Bowman and Bryant (2.23), where a diffusion or dispersion

term is included in each of the partial differential equations for the

packed column.

The system is thus described by two equations like 2.1 one for each

stream and linked by the transfer terms.

The estimates of the diffusion coefficient depend on either the

effective mixing produced by collection and subsequent mixing in each

section of the plate type column,or for the pecked column, by considering

it as a packed bed tubular reactor and using the available information

for them.

2.7 Heat Exchangers

Designs of heat exchanger are many and various. However they

virtually all involve the transfer of heat between two fluid streams. The

fluids may be similar . (water and water) or quite different liquid sodium

and st eam). They may be moving along, side by side, in separate channels

(e.g. co-current and counter-current heat exchangers) or one may be in a

jacket round the other (e.g. shell and tube type).

The fluid will nearly always be turbulent flow in fully developed

industrial devices although most of the analytical work neglects the effects

of heat transfer by this dispersion in comparison with the transport by

the velocity only. There is evidence to suggest it is rarely valid to

19.

neglect dispersion and diffusion effects. The work of Taylor (See

Section 2.2) shows that for practical systems with discontinuities

and bends in the flow paths dispersion effects will be large. Hougen

and Walsh (3.17) suggest that diffusion is an important part of the process,

having performed tests on a heat exchanger with no fluid flow and got results

similar to those with the device functioning. In a paper on predicting

the dynamics of concentric pipe heat exchangers Mozley (2.55) analyses a

system on the basis of plug flow (no dispersion) but uses an electrical

analogkwhich is actually equivalent to a dispersion only (no flow) system

and gets frequency response results close to the actual measured system

results - though this may be highly fortuitous. In view of the later

results of this work on the theoretical behaviour of flow and diffusion

system it seems that some of the differences between theory and practice

in reported results are explained by including, along with the flow process,

a dispersion term. The equation 2.1.

au _ a2

is (2.1) at - ax + D U . therefore descriptive of a basic process underlying the dynamic behaviour

of heat exchangers.

The numerical data for a heat exchanger, together with the

calculation of the components required for its special purpose electronic

analogue (see Chapter IV) appears in Appendix II.

2.8 Heat Transfer to Moving Solids

With the advent of continuous casting and a general tendency to

make continuous versions of processes which were once treated batchwise

it is of interest to consider the transfer of heat to a moving strip of

20.

solid material, usually metal. Such situations of interest are those in

which a long piece of metal moves slowly through a furnace with the object

of bringing it to the same temperature all through or to some prearranged

profile of temperature. With movement in the direction of x positive at

velocity V and with thermal diffusivity c% , and the other space dimensions

y and Z the equation for the temperature U is

(a2U)

( e

u a2ui

a71' -a7 az?)

It is here assumed that the material is isotropic and moving phase

boundaries are not considered.

To compare this with the previous other systems,it can be noted

that the heat transfer due to diffusion in the direction of motion has to

be considered in comparison with the heat transfer by actually moving the

heated material and the heat transfer in the other (radial) directions

can be considered as distributed forcing applied to the flow and diffusion

(x direction only) system.

The comparable batch process is the soaking pit. In this the

material is static while the temperature is allowed to settle to that of

the environment. Although it is not common practice to do so it is

possible to vary the pit temperature during the time of the soak in order

to speed up the process ("acceleration heating").

The two systems are comparable in that the static material is

subjected to a time dependent environment while the moving material has a

time dependent environment because it goes from a region at one temperature

to a region at 'another. There is thus a correspondence between the time

variable for the static material and the distance variable for the moving

au _ at -

au - v" — + ax (2.21)

21.

material. The difference is that the static material is, at any time,

subject only to environmental conditions then existing, whereas the moving

material, because of diffusion in the direction of motion is also influenced

at any instant of time by the environment both in fromt of and behind it in

space. This is an aspect of the increased dimensionality of the moving

strip problem because the profile can change with time and successive

element of the strip receive different treatments, but because of the

transfer along the strip this canmpffect elements passing through earlier

or later. There is no parallel in the batch process whereby one batch

influences a previous or succeeding batch. The two systems become

equivalent when the transfer by diffusion in the direction of motion is

negligible in comparison with the transfer by motion. This depends on the

velocity of the system as well as the diffusion coefficients, and in two

systems designed to do the same job, one moving slowly through a short

distance and one moving quickly through a long one, the emphasis on the two

processes of transfer will be different.

Two examples with numerical data are treated in Appendix II where the

significance of the various terms is examined for the purpose of simulation.

In any metal being heated the radial heat transfer becomes slower

as the size of the strip or ingot increases. For ingots of the order of

size currently used the calculations show that where the thermal conductivity

is low enough for the radial transfer to be a significant factor the axial

diffusion is negligible and vice-versa. This is justification for treating

the problem of establishing optimal temperature profiles for a furnace to

give even temperature distribution through the ingot by considering the

material as a number of adjacent moving strips with no axial heat transfer.

Re kinematic viscosity dynamic viscosity

velocity x distance velocity x distance x density

22.

It also indicates the validity of neglecting the temperature gradients*

across a turbulent fluid stream when the axial dispersion is significant.

(Section 2.2)

2.9 Dimensionless Parameters

The usual method of assessing the behaviour of fluid systems is in

terms of dimensionless parameters - so that systems of widely differing

sizes and with different fluids can be compared on a normalized scale.

(See Ekert2.61)

The most widely known is the Reynolds number which gives a non-

dimensional assessment of flow velocity. It has the form

(2.22)

The distance term is a representative dimension for the system, e.g.

the bore of a pipe, by which it can be compared in size with similar shaped

systems. The velocity, viscosities and density refer to the fluid

involved.

The Peclet number, associated with diffusivity, is important in

this context and is given by

Pe = velocity x distance

(2.23) diffusivity

Again the same comments apply about the relevant terms.

The Prandtl Number can be considered as relating the Peclet and

Reynolds Numbers thus:-

Pe P

r Re

But if the same distance and (2.24)

velocity terms are used for both Re and P

e then the Prandtl Number is

just the ratio:-

23.

Pr

Kinematic Viscosity (2.25)

Diffusivity

Another important number in heat transfer is the Nusselt Number

which relates the "film transfer coefficient" and "thermal conductivity."

Nusselt No. Film Transfer Coeff. x Distance (2.26)

Thermal Conductivity

To complete the record the dimensions of the terms used above are

recorded

Thermal Diffusivity, 04.= k c-

where k is thermal conductivity e.g. BT'i:•U/hr ft. °F

c is specific heat e.g. BThU/0F . 16

is specific weight

e.g. 16/cu.ft.

Kinematic Viscosity, A = = ea e.g. ft2Aft, sec

where p is density e.g. 11)(mass)/Cu.ft.

d is specific weight e.g. lb(force)/cu.ft.

g is occ,-i.due to gravity e.g. ft/se

fl is dynamic viscosity e.g. lb(force)Secs/ft2

In pipe flow consideration4 which are relevant here, where the

Reynold's number is

R -

Vel. x dia.

Kin. visc.

there is a critical value of around 2,300 for R above which fluid flow

will be turbulent. This covers most industrial processes.

24.


The transfer of heat and mass are, in fluid flow, closely related

effects (viz. Reynold's Analogy). Many processes which include

dispersion and diffusion in fluids as it appears in heat exchangers,

distillation columns and tubular reactors, diffusion of heat in metals and

of carriers in semiconductors, of probability in any random walk process,

all have the same underlying structure - flow and diffusion, which in the

simple one dimensional case can be described by the equation:-

V au D a2U = ► of 8x

ax2

(2.1)

The methods of estimating and measuring the coefficients vary

widely, as does the relative significance of the two terms V and D. One

limiting case occurs for D = 0 at which the whole process is simply a

unidirectional transfer process and the other for V = 0 which leaves the

straightforward diffusion only systelz.

A large proportion of all the industrial distributed parameter

systems are related by a common structure and although much of the

advanced control theory may deal with quite general system structures it

is of great interest to investigate the basic flow and diffusion process

even in its constant coefficients, one space dimension form as treated

here.

36

28

26

at

22

14

12

3

4 S 6

los R

Fig.2.1. V/V* as a function of Reynolds.

number (R), for fluid flow in pipes.

From Taylor (Ref.2.19.)

40

2.0

2

,. 0

• 6

2

4 6

1410 to 60 100 a 0 Co

400

ft

(Particle DiatI(Interstitial vel.) (Axial Diffusivity.)

R. (Particle Dia.)(Interst.vel.)(Density.) (Viscosity)

Fig.2.2. Recorded values of Peolet number (P),

as a function of Reynolds number (R),

for axial diffusion in packed beds.

Taken from Liles and Geankoplis.(Ref.2.&6.)

26.

Chapter III

TRANSFER FUNCTIONS

3.1 Introduction

The method of solving partial differential equations by applying

the Laplace (or related) transforms is well established (3.3, 3.2). The

choice of transform depends on the operators involved in the equation and

the boundary conditions specified for the problem. The usual control

engineering problem is formulated with time as the independent variable.

Transforming the ordinary differential equations leads to a formulation in

terms of frequency as independent variable which is readily interpreted

without necessarily performing the inversion operation. Following this

procedure with partial differential equations in which time is only one of

the independent variables leads to a formulation in terms of frequency in

which the other (space) variable or variables appear as parameters. Once

again the results can readily be interpreted in the frequency domain but

only for a limited class of systems can the process be usefully continued

through to the stage of performing the inversion.

The process is here treated in detail for the basic underlying flow

and diffusion system and then the effects of system modifications are

considered. Finally the inversion process is completed for the basic

system to give formulaefor parameter estimation.

3.2 Operator and Boundary Conditions

The equation to be considered first is that for a flow and diffusion

system with no transfer to the environment except that at entry and exit

27.

to and from the system. The equLtion is

au(x,t) at V au(x,t) + D a

2U(x,t)

ax 8x (3.1)

where U(x,t) is a concentration or temperature, t the time, x the space

variable, V the velocity of flow in the direction of positive x, D is the

effective axial diffusivity.

Suppose that the time taken to travel the length L of the system

at velocity V is C time units. Then:

V = C--

(3.2)

L and if T ° t

, x .,- A

-- E (3.3)

A i D. "t" and a

f 2 then the equation can be (3.4)

; ..;

au = a2U

normalized to OT ax a ;57 (3-5)

where the system has unit length, unit velocity of flow and the diffusivity

is replaced by the inverse of a Peclet number which is based on mean flow

velocity, system length and effective axial diffusivity.

Both the original and normalized versions are used in this chapter.

The boundary conditions to be imposed on the equation are that the

condition of the stream is known at input, i.e. that U(0,t) is known for 5:

all t >0 and that U(4,0) is known as zero. These are not the only set of

boundary conditions that could be applied and the choice doesmot imply

that the others are invalid.

The boundary conditions in the space variables are only an

approximation to a situation where in reality there is a gradual transition

from the realm of validity of one operator to the realm of another.

The symbol g means "is by definition, equal to"

28.

Any process described by partial differential equations must be in an

environment similarly describable. If it is intended to use boundary

conditions which relate values just inside the region of interest to values

just outside themI there is by implication a small transition region and

the boundary conditions must be consistent with the operator which relates

to that small region. If conditions are imposed which specify gradients

and concentrations at some point they must be considered in the light of

the continuity or otherwise of the system.

The conditions

aU = 0

dx x=1 or = o (3.6)

are reasonable when the velocity of flow is zero and the problem is solely

that of thermal diffusion across a non-conducting boundary. If the

velocity of flow is non-zero then a zero gradient precludes the entry or

exit of a disturbance in the concentration or temperature of the flowing

streams for if the temperatures at two points in the fluid stream are to be

different there must be a gradient between them which appears at the point

x=o or x=1 as the fluid moves by. (A possible exception is the unlikely

situation in which a region in which no flow occurs is immediately adjacent

to one in which it does occur - then there arises the problem of having two

different operators at the same point.) Another condition which is used,

for example at the input to a system, is

v(U)0_ = v(U)04. - De) o+ (3.7)

where (U) o- is the state of the input material, just before entry and

au (u)0+ and (7—) are the values of state and gradient just inside the ox 0+

system. (Ref. FAN and AHN 3.14).

29.

This boundary condition can for example be considered as a result of

specifying certain relationships about the behaviour of the system around

x=0.

Consider a small region around x=o, of width ax wherein the state is

U(o). Then allowing for the flow inwards due to fluid velocity and for

the diffusion effectsl the behaviour of the state U(o) is described by the

equation

dU(o) _ dt

- U0_-] D - V (aU) - au 4x Ax ax dx

o+ (3.8)

where the subscripts o- and o+ refer to values just before and just

after the small section, and when it is supposed that the same equation of

behaviour applies to this region as applies to the rest of the system.

Then by specifying that either

du(0) au (—) (3.9) dt ax o-

dU(0) D au, = - — (i the above (3.10) dt Ax ax o-

boundary condition arises. The first of these specifications is

unrealistic in that it eliminates the possibility of changes of state at

x=o. The latter can not be directly explained. This leaves the other

possibility that a different form of behaviour governs the boundary region.

This is consistent with the situation that must exist in changing from say

a large tank with slow moving fluid to a pipe where the flow velocity is

considerable. The problem then requires a detailed specification of the

geometry and behaviour around the input.

In the following analysis a different situation is considered. The

30.

input, x=o, is assumed to be a point in a system where there is no disruption

of behaviour, where the point is really a reference at which U(o,t) is

known. In the same way the output condition U(L,t) is a measure of the

state part way through a process which continues further. Since no flow

and diffusion process can suddenly end with the fluid standing still and

diffusion stopped this is also a reasonable approximation to the case

where conditions change, for example in the feeding of the output stream

of a tubular reactor into a storage tank in which the flow and dispersion

process continues some way into the tank. (It may be noted at this point

that experiments on a digital computer simulation showed that for all

1 practical purposes an extra terminating section of length approximately 5L

or 1 —L was long enough to appear from the point x = L as if it were infinite 2

in extent. (See Chapter 4, Lumped Parameter Models)

Thus the boundary conditions to be used are as above:-

U(o,t) specified, t>,›o

U(x,o) = 0 (or specified), o<x t!zo (3.11)

U(tc,t) = 0 ,

3.3 The Basic Transfer Function

Consider the equation

au _ au a2U - V + D --n at ax axc

with the boundary conditions (as from section 3.2 above)

(3.12) also (2.1 )

U(o,t) specified,

U( ,t) = 0, (3.11)

U(x,o) = 0,

Transform this using the Laplace transform:-

31.

ePt. U(x,t) . dt. (3.13)

0 to give

dU d2U

p U(X r) V -(x

dx D cb&

and transforming the boundary conditions gives:

U(o,p) specified

U(00, r.) = 0

U(x,t=o) = 0 in

The solution of the ordinary differential equationAx leads to

(3.14)

(3.15)

11 + 4Dfd)'

(3.16)

To meet the boundary conditions it would usually be considered essential

that terms in exponentials should have negative real parts so that as

x--P--.a the amplitudes of the responses do not increase indefinitely. This

appears to be a choice between A=o or B=o, and certainly for the case where

r is a positive real number (as in the original approach to the Laplace

transform - Churchill (3.3J ) this choice results in A=o, and B=U(o,p).

Since however the control engineering problems are not restricted to GC

positive it is necessary to investigate the behaviour of the functions for

all values of p and to establish conditions for physical realizability that

can be applied to any system where this choice appears. Furthermore the

condition on the real part of the exponents is not satisfactory and needs

clarification.

Consider the function

Expi [ V + TV2 + 4Dpj) (3.17)

where it is taken that every complex number, such as (V2 + 4Dp) has two

U(x, ) = A.Exp173 V 4Dr .„ ]} B.,42-15 L V - x

32.

square roots. As a function of the Laplace variable p this function is

two-valued and can thus be considered in terms of two Riemann surfaces in

the p -plane. A suitable branch cut can be made and one surface selected

by considering the behaviour of the function

f x ExP 12D [1/ + (r+js)] (3.18)

which is now single valued in the complex variable (r+js), being related

to the variable r, (take p =c4+jw) by

(r+js) = 2 4Dp. (3.19)

The method is to consider the response of the system as a function

of (r+js), selecting that half of the (r+js) plane which corresponds to a

physically realizable system and transforming it into the whole of the

r -plane.

3.4 Physical Realizability and Riemann Surfaces

The function to be investigated is

ExP 2D + )V2 + 4Dp]j c

Replace 11/2 + 4D p by (r+js)

Then if r . , the parameters r,s, and w are related by r2 - s

2 = v2 + 4D ec

and rs = 2wD

In the normalized version of the system's partial differential

equation (See Section 3.2) these basic relationships for r and s become

r2 -s 2 = 1 + 4 ate= 1 + 2. (2a04)

rs = taw, (3.19)

because the velocity and distance parameters are unity and the diffusivity

(3.17)

(3.18)

33.

D is replaced by the parameter a. Thus in the (r + js)-plane the lines

ok= constant and w = constant appear as rectangular hyperbolae. These

are shown in fig. (3 1). Because the transfer function becomes

Exp 7,(1 + r + the amplitude of the (3.20) ,

frequency response is measured logarithmically by (l+r) and the phase

angle is proportional to s. Thus all parts of the (r+js) plane having

r > -1 correspond to a frequency response with amplitude transfer greater

than unity (i.e. (l+r) positive). Also each part of the p-plane is

represented twice on the (r+js)-plane. The criteria of physical

realizability will eliminate half of this (r+js)-plane.

Any transfer function is the ratio of the transforms of the input

and output of a system. Although not usually explicitly stated there is

an implied synchronism in that the ratio is for signals existing in exactly

the same time periods. Because of the transport delay in a process such

as the one being considered herel the output at any time depends mainly on

an input at an earlier time. Thus if an input signal of the form

Exp (+ at). sin(wt), with a- negative, is decaying fast enough the transfer

function, being a ratio of "simultaneous" transforms will indicate a

magnitude of response greater than unity, because the output due to an

early part of the input signal is compared with a later, smaller part.

Furthermore as the spacing between input and output increases this

magnitude ratio will increase so that it is not a good criterion of physical

realizability to assume that the transfer function must always tend to zero

as a finite value as the length of the system tends to infinity. In this

case it cannot be said that (r+1) must be always negative. On the other

hand if the input signal has a positive real part exponent, then the

34.

transfer function must be less than unity in magnitude.

Another aspect of this is that for any input of the form

Exp (+04t) sin (wt), c& negative, a frequency, w, will be

reached at which the apparent increase in magnitude of response due to the

transport delay will be cancelled out by the attenuation due to the disper-

sion effects of the fine alternating profile superimposed on the flowing

stream. This condition is indicated by the locus of unity gain (r=-1).

In general terms this argument yields the criterion that for any given

finite negative the magnitude of response must tend to zerolor at most

a finite value as w tends to infinity.

In any physical, linear, (constant coefficients) system there can

only be one response for any given input. Furthermore the transfer

function associated with one value of p=0/-+ jw must be the complex

conjugate of the transfer function for r= ok-jW, so that the response to

a real input is always real.

Using theaecriterialthe right half of the (r+js)-plane is not

admissible as representing a physically realizable system. Thus (see

thumbnail sketch in fig. 3.1) the region A to the right of the hyperbola

2ao(= o is for positive 04: and magnitude of gain greater than unity - this

is not admissible. In the remainder of the right half place, regions B

and B1 (as also in A) a locus of constant oL leads always to greater values

of r for increasing w. This is not possible. Therefore so that there

should be a value of response defined for every value of p , and so that

conjugate values of p should have the same magnitude of response theplane

must be cut along the r=o axis and the left half plane used to define the

system transfer functions. Observe that the region -14 re.° which gives

35.

transfer functions of increasing magnitudes for increasing length is

admitted.

If this left half plane is transformed back into the p•plane the

region C becomes the right half p-plane, D D1 the left half p-plane, and

the branch cut runs from p = 2 to p = In terms of the original

choice between tak]ng either the positive or the negative square root of,

the term (V2+4Dp), this choice is equivalent to taking the negative square

root for p positive and real. The basic transfer function is thus: (3.21)

u j V Exp 2D 1.-

and all the conventional frequency responses and root locus plots can then

be developed.

3.5 Frequency Responses

Having established the transfer function for all values of p, x,

V and D,the frequency response for all w can be determined by direct

calculation. It is however easier to use the normalized version of the

equation,so that comparisons can be made directly in terms of the

dimensionless parameter "a" - the reciprocal of a Peclet number.

Using the original equation the parameters r and s can be calculated

from: r2-s2

= V2+4D,

rs = 2wD

to give

s = + AI v4 8 Dv2 16D2('-'..2+w2) (V2+4Do&) (3.22)

2 fi.wD and r = ; (3.23)

104 Dv2-+ 161)2(02+w2) - (v2+010.0 . P

0 _ x 2D

36.

Note that the possibility that r and s are themselves imaginary or

complex numbers has been eliminatedi and that for w positive, r and s have

the same sign and vice-versa. Furthermore it has been shown (Section 3.4)

that only r.4::o is required. The transfer function of interest is then of

the form:-

Exp 7i(V+r) . Exp j (ii) , which is in modulus

argument form yielding the conventional frequency responses (for °(= 0):-

-

ti j,A74 + 1617w V2 radians, (3.24)

M = - 8.684 if

2 ,f- coD

decibels.(3.25)

IA A

+ 16D2w2 -

In terms of the normalized system where V becomes unity, the length of

the system is unity, D is replaced by a, and w is measured in terms of

radians per natural time unit these expressions become:

0 - 2a ,t-S. + 4 (2awr 1 radians (3.26)

8.684 (2.w) 2a

L ,f jl + 4 (2aw) 1 decibels. (3.27)

Thus phase and magnitude are parametric on'a'for varying w. Plots of 0

and M are shown in figs. 3.2 and 3.3 for values of'a'from .0625 to 8.0

and of w from 0.10 to 10.

The corresponding behaviour of delay-only system is also shown on

the phase angle plot and the same plots developed for a system having no

flow but only diffusion are shown in figs. (3 6) and (3.-7).

37.

3.6 Root Locus Plots

A conformal transformation of the left half of the (r+js) plane

produces the whole of the p-plane. Each value of r and s yields a line

of constant gain or constant phase angle respectively in the p-plane.

If the p-plane plot is calibrated directly in terms of the magnitudes

of 2ae4and 2awlthe lines for various constant values of r and s can be

readily calculated from the equations:-

r2 - s2 = 1 + 2(2E100 (3.19)

rs = (2aw).

These are manipulated to produce:-

(2aw)2 2(2a00 2 (r2-1) r

2(2atr..) , (2aw/2 2

(l+s ) s2

(3.28)

which yield the relationshipfbetween (28.0) and (2aw) for various values of

r and s respectively.

These constant gain and constant phase angle lines are plotted in

figs. 3.4 and 3.5, for values of (2a09 and (2aw) up to 7.o and 0-7

respectively. Only one quadrant of the p-plane is shown in each case

because only the left half plane is of real interest and this is in any

case symmetrical about the line w=o. For calculating actual phase angles

and amplitude responses the relationships are:-

0 -= 2a radians

M = (l+r) . 8.684 decibels, 2a

which can be considered as phase shift and magnitude change respectively

per unit length of system (using the normalized equations 3.5).

(3.29)

38.

3.7 Comparison with Limiting Cases

The flow and diffusion system has as its two natural limiting cases

the flow only (delay) system and the diffusion only (zero velocity) system.

Because these are structurally simpler and much easier to deal with from the

point of view of engineering calculations,it is of interest to know the

values of system parameters (or particularly the parameter'a) for which each

limiting case becomes a good approximation to the system.

The transfer function for the delay only system is;

Exp [1-4 ] in terms of the original equation/

ExpL-pj in terms of the normalized equation.

As a function of frequency, 0), this has unit magnitude (0410 and a phase

shift proportional to frequency. This characteristic is plotted on the

same chart (fig. 3.3) as the one for the flow and diffusion process.

The system described by the partial differential equation;-

au _ at -

2U D . is a purely diffusion (3.30)

process, and if the same boundary conditions are applied as in the analysis

of flow and diffusion process the transformed equation is A

a2 p U(x,p) = D-- U(x,p) 6)(2

(3.31)

with

U(o,p) specified (as input)

(3.32) U(x,t=o) = 0 (or specified).

The equation cannot be normalized as before because there is no

velocity term involved.

Consistent with making the choice of two possible values for the

transfer function,in the same way as above, the resulting transfer function

39.

is:- Expr-

2D _. ,

„i 4Dp (3.33) —

Once again the analysis can be performed in terms of the parameters r and sy

giving the results:

= -1 ji 1

2 2-- 6 D (co + ) - 4D ,c

(3.34)

-2 S2Dco r = •

•,/,j 16D (co+ a ) - 4Doi-

When the frequency responses are evaluated fort4=o,these become

s = Ea and r = - 2D(o , giving

M = -8.684 2Dc° db (3.35)

= g radians

These results are structurally equivalent to the results that would

obtain by putting V=o in the expressions for the flow and diffusion process.

The same comment cannot be applied to the flow-only process. The difference

is;that while both the flow and diffusion process and the diffusion only

process are described by parabolic equations thedelay-only process has a

hyperbolic structure.

For the purposes of comparing the flow-and-diffusion process with the

diffusion-only onel the frequency responses for the latter are plotted

(fig.. 3.6 and 3.7) for a range of values of D covering the same range as the

values for'at in the first case. In this context the plots for the diffusion-

only case show the effect of neglecting the effects of the flow in the

transfer process.

One result of these comparisons is that at any frequency the flow and

40.

diffusion process exhibits less phase lag than the equivalent delay system.

It also produces more attenuation. (See figs. 3.2 and 3.3). As the

amount of diffusion decreases; the phase shift approaches closer to the delay

only characteristic while the attenuation characteristic develop s a

steeper fall-off which occurs at a higher frequency, so that the nett effect

is that for a < .0) the phase shift is represented to within 15° by the

delay system as far as 0 = 2n (w=27c), and the attenuation is less than 3db

up to w=t. This means that up to a frequency where between half a cycle

and one cycle of input signal corresponds to the mean delay time, the delay-

only model is a fair representation of the flow and diffusion process for

a < 0.03, although a delay-line model may have to be modified slightly to

produce some attenuation at the higher frequencies.

At the other end of the scale ) for a .> 4.0, the flow-and-diffusion

and diffusion-only process give virtually the same characteristics. For

control engineering purposes this leaves the range of 'a' between 0.03 and

4.0 to be represented by other means.

The comparisons on the basis of the root locus or p-plane plots

reveal the regions of equivalence, not only for frequencies along the

jw axis.,but also for non-zero values of 04-.

For "a" large (2aotand taw large) fig. 3.4 shows the lines of constant

gain (r= constant) and constant phase angles= constant) for the flow and

diffusion system. If the whole pattern is displaced 0.5 units (of 2a0(,)

to the right, so that intersections of corresponding r and s lines lie on

the jw axis instead of on the line auk= -0.5 this is the plot for the

diffusion only system. Consequently, for operations in any region in which

this displacement is of small effect , (e.g. for (2a.t + j2awl > 5) the

41.

diffusion only model is adequate. This applies regardless of the value

of'a%and demonstrates that for high frequencies any flow and diffusion

process is more closely allied to the diffusion system, but it must be

appreciated that for engineering purposes this is of little interest for

small'a'because of the large attenuation and phase shift involved.

(Note 0 =2: , (gal) db) = 8.684 (1.41) db)

For low values of 2az.e. and 2aw as shown in fig. 3.5,the lines of

constant gain and constant phase for a pure delay system (T.F: e P) are

the actual cartesian grid lines.The result is that the line r = 0.9,for

example, is to be compared with the line O(= -17-0.9+17 = -0.1. For

12ad +j2aw( < 0.2,the delay line approximation is very close. Whether

this is of interest for engineering purposes depends on the values of a

and w. The implication is that for low frequencies the system behaves

like a delay line, while for high frequencies the diffusion effect is more

important.

3.8 More Complex Systems

So far, the system dealt with has been only the basic flow and

diffusion process. In engineering applications this may form only part

of the whole system. In particular, in heat transfer processes there is

always a transfer to and from adjacent parts of the environment. Consider

the case of fluid flowing through a pipe; Heat may be transferred to the

thermal capacity of the walls and the internal temperature can be

influenced by the state of the environment distributed over the whole

surrounding surface. This brings to light two important aspects: wall

capacity and distributed forcing.

42.

The effects of wall capacity can most easily be described in terms

of the simple flow only system, which will also serve to introduce the

notation. A fluid stream is surrounded by a wall to which heat can be

transferred,but the outer surface of the wall is insulated. The state

of the wall, assumed thin so that radial and axial transfer in the wall

can be neglected, is described by U2(x,t), the state of the fluid stream

by yx,t). Similarly the thermal capacities of wall and stream are C2

and C3 (per unit length) and the conductance between them is G 23 (per unit

length). The system behaviour can be described by the equations:-

8U2 - c

23 (u3 u2) at

2 (3.36)

dU3 .

au G

at -v--- 2 3 ax3 C23 cu u ) . 3

After normalizing the equations to give unit velocity of flow

(as in 3.2) these become:

6U2 8T

B(U3

— u2)

(3.37) 6U3

6U3

6T = —

ax + E(U2 - u3)

Gn Gn where B = (C 3 T) ( E = _a2.7-) (The notation is part of a 2 `3

wider set of symbols covering the more complex cases). -i: is the mean

delay time in the system (as before). Taking the input to the system

to be variations in inlet temperature,U3(o,t),the transfer function to the

outlet is given by:

ED U3(X,p) = U 3 p+B(o,p) Exp -X(p + .

From this the conventional frequency response can be derived:-

(3.38)

43.

m 2

= - 8.684 Ew

db and 0 = -(w + ) radians. (B2+w2

EBw 2 ) B

2+w

(3.39)

Another interpretation which is of interest from the electrical engineering

point of view arises from the equation produced when the parameter U2(x,p)

is eliminated from the transformed equations:-

aU, 2 + pa B) + E(1 ---) U = 0 . ax p+B 3 (3.40)

Comparing this with the equation

au,

d pU3

0 (3.41) x

which results when the wall capacity is isolated, the modification is seen

to be exactly equivalent to replacing the susceptance of the thermal

capacity of the stream by an admittance consisting of the thermal capacity

in parallel with the series combination of thermal capacity of the wall

and the wall to stream conductance.

C

3

is replaced by

C3T

T-c:

The effect of wall capacity is to produce an attenuation of response at

high frequencies in the transfer of disturbances from input to output.

The effect of applying a distributed forcing function to the

system is important for control analysis, because it describes one of the

two basic modes by which system behaviour can be influenced. The

localized disturbance, such as a variation in the input condition, has

appeared above. In the other situation the disturbance takes effect

all over the system. The simplest example is to consider distributed

forcing of a flow only system. The state of the environment is U1(t)'

of the stream U3(x,t),and the conductance between environment and

stream is G13. The relevant equation, in normalized form, is

3 . - G13 (u u ) aT C

3 1 3

when transformed this becomes

44.

(3.42)

--2 (p+C)U3

= U (p) ax G where C = . C

13 15 3

(3.43)

To meet the boundary condition that U3(0't) should be the input condition,

the state of the stream is expressed in terms of transforms by

U3(x'p) = U3(o,p). Exp t (p+C) xi

C.u(p)

(p+C)l - Exp ( -(p+C) (5. 4)

The relevant part here is the transfer function from environment to

output (taken at X = 1):-

74.-eir 1 - Exp I - (p+CPti • (3.45)

This is equivalent to a conventional lag in cascade with a function which

(in root locus terms) has an infinite number of zeros along the line

p= -C occuring at intervals of 27. starting with the one at w = o which

exactly cancels the pole due to 776.7 . The basic pattern on which these

zeros are superimposed is the cartesian grid of the same delay operator.

The important 180° lines travelling in from c.,e= -ooare each caught by a

zero before reaching the 0.=o axis. This results in a conventional

frequency response plot (shown in fig. 3.8) which shows periodic

fluctuations of amplitude response and a phase shift which never exceeds

1800. The frequency response can be calculated from the formula:—

0 = tan-1 ( e sin w ) - tan-1(T)

(1-e Coos w) (3.46)

45.

and: Amplitude = C -

1+e2C -

-0cos w

(c24.2)

(note that this one is not in decibels. See also Hempel 2.53)

The effect of distributed forcing on a moving stream is to produce

a response which at certain frequencies appears to resonate. More

accurately it can be described as a cancellation effect. Fluid entering

the system may for example be heated while passing through the first half

of the system, cooled while in the second halfl and leave having received

very little heating from the environment. This is the situation at the

first dip in the frequency response curve (w = 27). The subsequent humps

and hollows can be explained in the same way. The lag term corresponds

to the effect of the thermal inertia of the stream treated as a whole.

The more general case which exhibits both these effects with+.the

diffusion effect in the fluid stream is described by the equations:

aT

C 2

C cu1 U2)

G (il3 - U2)

2 2

au au 62U, (G

a ) aT3 = 3 + C23 (u

2 U3) x-3 2 ax

(3.47)

Once again the equations have been normalized to give unit flow velocity.

The subscripts 1, 2, 3 refer to environment, wall and fluid respectively.

Apply the Laplace transform to these equations to give:

(p+A+B)U2 = AU1 + BU3 au 3 U

(p+E)U3 ax + a3 + EU2

(3.48)

where A G 2 1: G

23 t G23 2

= C2 c

3 C

(The symbol C has already been used for c13 G13c in - a special case, 3 and D for diffusivity).

46.

Eliminating U2, the state of the wall, which is not of direct interest

gives; a2U

3 au, B -EA a

3 - -- (pE(1 ) ) . U = ax +

p+A+B 3 (p+A+B) Ul 62 (3.49)

The response at X can be written in terms of transfer functions as

(.1 ))

U(X,P) = u3(o,p) Exio 1 +

p+A+B a3

EA r-

+"+- A B U, 1 - X (1 11+4(p+E (1

'.6)."+p(E-)+EA) 2a p+ a3 A+13

(3.50)

Once again the transfer from input to output along the flow

process is modified. This time instead of having F alone inside the

expression /1 + 4ap (as in Section 3.4),the term fr+ Ep+b+A

B appears. A+

This is equivalent to replacing the thermal capacity of the line by the

more complicated structure involving the conductances G32 and G21 and the

capacity C2 as well. The effect of distributed forcing is to produce

a series of zeros in the p-plane,but this time they lie along a line of

constant amplitude for the modified flow and diffusion process, instead

of along a line of constant amplitude for the delay operator, as in the

simple case. Instead of having a single lag due to the thermal inertia

of the fluid stream Ithere is a second order term because of the extra

effect of thermal capacity of the wall.

3.9 Inversion of the Basic Transforms

The transfer function for the flow and diffusion process from

input end to output end is

Exp r ,j v 2 + 4pd1

,J

Exp 2D 21:3"ct5 0,

2 Exp 4Dt

—V2t Exp —Tr s (xV x for t>0.

47.

which can be written as

xV Exp od -x

, Exp p +

Using the standard inverse transform for the factor expj' -kiFt?,(Churchill) R6F33,

the inverse of the whole expression becomes

This can also be put in the form

Ftt3D7 Exdr:1 C 4D

2 (2t 2xv + v2t ,

(3.51)

or 2 3 Dict6 Exp -t - t D (2t-'

It can readily be demonstrated that this is a solution of the partial

differential equation (See Appendix I). This shows the impulse response _2

as a normal distribution (of the form e ) with its mean moving along at

x a velocity v, the whole being modified by the factor r---5 so that

2 Dizt for small values of t and x the distribution is made non-symmetric.

Because of the spatial and temporal distribution of the response,it is

necessary to establish the precise meaning of impulse response in this

context. The impulse (Dirac delta) is an impulse in U(o,t) existing at

the point t=o. Both before and after t=o the state at z=o is zero.

No further specification is made but it precludes the possibility of

material (or heat) diffusing backwards to affect the state at z=o whila

the stream moves on, and it must be pointed out that this is not

necessarily the case in a practical experiment where a small burst of highly

concentrated fluid is injected into the stream at (t=o, x=o). Consider

for example Sir Geoffrey Taylor's Experiments (Chapter 2, Section 3 and

48.

2.19) where this back-mixing was allowed and for solution the equation

was transformed to be solved with respect to a moving reference travelling

along at stream velocity, leavingl by this transformation, only the diffusion

equation.

The result is very similar for large x and ty where the peak of the

response passes any point as if it moved with average velocity,but in

this case, at low x and tithe apparent peak of response)as seem at any

point passesearlier than would be expected. The reason is that;because

the diffusion spreads the concentrated or hot fluid,a high level is

reached at some point before the particles travelling at mean velocity get

there,and by the time they do lthe diffusion has reduced the level at that

point to below the level it reached earlier. Using the above solution

to the equation the time of the peak of response can be calculated for

any point x. (See Appendix III)

The result is

T peak x j [ -3D 4. 9D2 vx 2 2

v x (3.52)

From this it can be seen that for small values of D or large x (or

if the normalized equation is used for small a) the peak response is

practically at T = but as the diffusion becomes more significant the

peak appears earlier. This affect can also be seen in the digital computer

solution of the equation in Chapter 4 and is shown graphically in fig. 4.19.

The other transform of practical interest which can be inverted

analytically is the transform for the response from distributed forcing

to some point x along the line. The transfer function is

-2- [a. - Exp (2D) . Exp if 2D x v u 2+4(p+r)D.V 10+C

49.

(3.53)

which can also be written as

C. Exp xV

p+c (p+c) . Exp E. J

p+C+Y--i 4D

Using the form for the transform of

. x given in

Crank, "Mathematics of Diffusion" (2-0), this can be inverted to give

-Ct V2t x P12-4752)

x JV2t C e - C.Exp (22)Exp (---) ..(exp (- .Erfc 2D . 4D 4D I

- V2

2) . Erfc 4D

V2t exp (..rt.y4D/ 2t).

4D 2 Dt (3.54)

For most engineering purposes this is too complex to be of value. Any

further increase in complexity of the transfer function, even assuming that

it would allow of inversionl would be of little practical interest.


The method of the Laplace transform can be applied to partial

differential equations. In dealing with the flow and diffusion process

of interest here1it produces transfer functions and impulse responses.

These involve terms such as Exp Fp- which are not readily dealt with by

conventional control theory. Root locus and frequency response plots can,

however, be derived. These show that at high frequencies flow and diffusion

systems behave like diffusion only systems and for low frequencies like flow

(delay) only systems. Howeveri for engineering purposes the very high and

very low frequency responses are not of great interest,and the systems can

2,f Dt

50. 10-

be characterized by a parameter, a, the inverse of a Peet number for

the system, so that for 3_4.0.03 the delay process is a fair model and for

a>4.0 the diffusion only process is a good model.

Modifications to the basic process, such as having thermal capacity

in the walls, applying distributed forcing, each produce characteristic

effects which can be interpreted in terms of modifications of the root

locus and frequency response plots, or in terms of changes in the structure

of an equivalent electrical system.

The impulse responses of these systems show considerable complexity

but can, in the simplest case of the basic flow and diffusion process,

yield numerical data about the shape and form of the response which can be

compared directly with measured units for parameter estimation.

The solution of the differential equations produces a need for a

means of establishing a choice between two Reimannsurfaces in the p-plane,

and this results in considering the transfer function not as a relationship

between cause and effect,but as a relationship between two "simultaneous"

transforms. The idea of a passive system in which the magnitude of a

transfer function increases as the input and output ends become further

apart also arisesi and is shown to be feasible for a certain class of input

functions. The selection criteria which result are consistent with the

results based on simpler concepts and the restriction of the Laplace

variable to the positive right half plane.

The boundary conditions that have been used for the problem are

only one possible combination. All the frequency responses and transfer

functions will be different for changed boundary conditions, but any

other choice requires a specification of the terminating environment for

51.

the system which may be difficultl or in many practical problems quite

impossible. The underlying difficulty is that any boundary condition is

an approximation and the real answer lies in having both the system and its

environment described in the same distributed parameter manner.

Fig.3.1. (r+js)-plane representation of flow and diffusion system.

M 4.Oa

i

_

j:"'''''''''''4:

• !

4

-,.'....... •

• 0

4. I

0 14,

I

,--"---.---r----C-----

V '0

4

v 4

/ k.‘• 4./ v.

/

/

/

.

4

r- -.. -

#

M

!' •

4

:

di _ ! 41

, .

ell

1

o oi urk o o at vw m A . m • m m...

4- . o

`11

Fig.3.2. Frequency Response — Magnitude — of flow and

diffusion system subject to diaturbanoa at

input to flow stream. Response in db (M),

relative to input, for range of 'a', as

function of normalized frequency 04).

I •

v.--$:-----11.O

•

7 ,C 2

• . 2

?

2

x if

O

.--.--------------71 • 1 i 4 a-

,4

1 0 4

tO

1/

1.

0!)

-.4

Fig.3.3. Frequency Response — Phase Angle — of floe and

diffusion system. Phase angle relative to input

phase for range of 'al as function of normalised frequenoy (61) •

a

zy,

it

P t tt N

t

6 \ LI

ti

_ 4 3

t: ° 4 . ell

Y * '

? P1

/

4

' ? t.

—'.1Inillig

4

. I

.

r L

41 i .,

.

Fig.3.4. Root Locus or p—plane plot for normalised flow

and diffusion system. Contours of constant phase

and constant magnitude of response for large

values of 2aw and 2a4.

0

N

•

ti

IP 1 \ 'Isms ..4111PITI I ‘41/1001 4.440340i wilL 41 1

4 44 1 idi Fig.3.5. P—plane plot for normalised flow and diffusion

system. Contours of constant gain and phase for

small values of 2a•cand 2a'. Comparable curves for

ideal transfer functiOn and (ncrit) lumped model

shown as broken lines.

8 0 do

0

0 ov

0

O

is

N

3

0

ao

c.r

11 4 O '1

N

a

Fig.3.6. Frequenoy Response - Magnitude - of diffusion

only system, subject to disturbance at one end.

Magnitude (MI) in db relative to input level for

rangeof values of D, as function of normalised

frequency M.

* wi w

......".........,„......................-:

0

0 ww

0 N

2

Is

•41

A

1

on

rwl

3

SI' !!

*16

f

"4

1111

0 In .n a lg. 0

v I. N so ... 2 % i * t I

Fig.3.7. Frequency Response - Phase Angle - of diffusion

only system, subject to disturbance at one end.

Phase angle (0)-1ag, for range of values of D,

as function of normalised frequehoy (0).

6.

0 V

0 Tr

0 M

0 N

e

V

I'

M

N

0 -:.

i

"5'

4fr

P:1)

)

I

N

v, 6

?

P VI 0 .4 0 VI 51 tew rof

vp ‘••• •

Fig.3.8. Frequency Response — Magnitude — of flow

only system subject to distributed forcing.

Magnitude (M) in db relative to disturbance

level, as function of normalised frequency (6.7).

0 u 0 6.

0 .0

0 e

A

0 N

° so

4:1

147

N

..-

,

n

1

+

,,,

4E4 .

IIIIIIIIIIIMIP

41

49 0

II h

u /

) m _ .0 'Zi%\ le g m •••

Fig.3.9. Frequency Response - Phase Angle - for flow

only system subject to distributed forcing.

Phase angle (slag) relative to disturbance

phase angle as function of normalised

frequency (4).

6o,

Chapter IV

LUMPED PARAM7,TER MODELS

4.1 Introduction

The partial differential equation used to describe distributed

parameter systems cannot in general be solved analytically. The forms of

the available analytic solutions depend on the type of the equation.

Analytic solutions to equations of hyperbolic type can be approached through

the methods of characteristics. The systems under discussion here are

generally of parabolic type for which no equivalent theory exists.

For a numerical solution, it is necessary to know the quantization

requirements of a solution found by splitting the relevant regions of the

space of independent variables into discrete elements, and writing a separ-

ate ordinary differential equation for the behaviour in each element. In

a hyperbolic case, quantization of the (two) independent variables can be

arranged in accordance with the behaviour of the characteristics, so that

fairly simple relationships exist between the size of the quantization used

in the time dimension (say) and that of quantization in the space dimension.

This also reveals the regions of interest on the boundaries, and allows

problems to be solved for the behaviour at a particular point or region

using only finite regions of the boundaries. (Ref: Forsythe and Wasow,

4.3). Neither of these two aids is available for parabolic equations,

which with the notable exception of the diffusion equation;

3U _ D. 34-U at -

ax2

61.

receive very little attention in the mathematical literature.

(See for example Lanczos (4.5) and Collatz (4.1), who deal with this and

the Laplacian operator in considerable detail). The basic difficulty

appears to be that while only restricted regions of the space boundaries

can at any time influence the behaviour at a particular point for systems

described by hyperbolic equations, the parabolic equation describes a

situation in which the transfer of effect from every point in the space

(by means of the diffusion) occurs infinitely quickly to every other point,

though the resulting influence may be infinitesimally small. The behaviour

of systems where the particle (or energy transfer) velocity has an upper

bound can be treated; for example, in the analysis of supersonic airflow.

However for most practical purposes the assumption that the maximum particle

velocity is high in comparison with the other (average) velocities is quite

adequate and results in the parabolic equations.

In producing solutions for control engineering problems it is usual

to consider the time as a continuous independant variable and make

quantizations in the space dimension. This is to be compared with the

usual approach in numerical analysis which deals with both quantizations

simultaneously. This approach is tied in with the established methods of

problem solution: i.e. electrical analogs (with time as independant

variable) and Laplace transfer methods. If,when digital computers are

used to solve the partial differential equationsi the quantization in the

space dimension has already been done, the quantization in time can then

be based on producing an adequate solution to a set of ordinary differential

equations, which is a simpler problem.

The basic problem then is to find out how to quantize the space

62.

dimension in the description of a system by a parabolic (in particular a

flow and diffusion) partial differential equation, so that the resulting

set of ordinary differential equations, one for each element or lump of

the space, givelin total a satisfactory description of the system behaviour.

There is another problem and that is to find the extent to which widely

separated parts of the system influence one another, in,particular to find

out how short a model system can be made and still appear from the point

of view of the input to be infinite in length - the problem of the

terminating section. There are situations however when this problem does

not arise and these become apparent as the problem is investigated.

Having found the rules to be used in setting up a mathematical model

there are basically two methods for using it. One involves its use in

digital computation the other in analogue computation. To make a decision

between these two it is necessary to know firstly the costs of solving any

given problem using the two methods, and secondly the minor advantages and

disadvantages of their use. Furthermore the use of lumped parameter

models must be considered in relation to other possible forms and here

again the type of problem is of fundamental importance.

This leads to investigation of special purpose computing techniques

where these might be economically or operationally advantageous, either

relative to one another or relative to other forms of mathematical model.

4.2 Basic Quantized Equation

The system to be investigated is the lumped section model of a

flow and diffusion process described by the normalized equation:-

au_ 8U a2U aT ax a axe (3.5)

-1+20-_.2] dx r+1 x

U dUr Ur-1 Ft• = [1 - (e-

63,

which first appears in Chapter 3, Section 2. The system has unit length

and unit velocity of flow. Consider it to be quantized with n sections,

each of lengthAx. ThenAx = 1 — and the simplest version of the quantized

equation is to take

dUr (U -U r r-1) a

dt Ax 772 (Ur-1+Ur+1-2Ur) (4.1)

Where Ur is the state of the rth section. However, if instead of taking

au J

the simple first order approximation to the gradient -- an estimate is 0x

made which is a weighted average of the gradients just before and after

the section concerned, with the weighting coefficient 0, (c. 8 -..s.1); to choose

the relative significance of the two terms thefollowing equation results:

giving the relationship between the rate of change of state at any point

and the state of the adjacent elements. However another manipulation

reveals the form:

dT Ax

dUr

U ) C rkU r r-1 1 a

+ (-- - a) , (Ur-1 +Ur+1 -2Ur ) (4.4)

which can be seen to have exactly the same form as the first simple

approximation (Eqn 4.1) to the process,where now the term id has been

replaced by ( - 0). Thus, by taking a better approximation to the

original equation, it can be seen that the process of lumping the system

into sections has the effect of introducing some extra dispersion which

can be allowed for by reducing the original coefficient of the dispersion

(or diffusion term). Since the obvious choice for 0 is 0-1 which takes

each measure of the gradient as equally significant it can now be replaced

by this numerical equivalent, having served to trace the source of the

correction to the diffusion

dUr

U _ r-1

term.

I. a 2 Ax

+ r Ax

1 Ax 6X

The equations

[ -2a aX

2 I)

of interest are now:-

Ur+1 [ aj (4.5)

dT A x

and the alternative version

dur (Ur6xr_1) _ k

+21x 6X 2

(Ur-1 +Ur+1 -2Ur ) (4.6) • dT

This introduction of diffusion or dispersion is the reason for the

blurring of wavefronts propagated in digital computer solution of hyperbolic

equations. One cure, as can be seen above, is to make the sections

smaller so that the relative significance of the extra dispersion is made

less and less. Another consequence of this effect is that at some critical

size of section the dispersion term disappears completely because:-

a 1 4x 2

Since this defines a number of sections in the unit length system,

the corresponding critical number of sections can be defined as:

6 1 n crit 2a (4.7)

Itisadmittedthatncrit may not be an integer but this is not important.

This model,made up with the critical number of sections,is worth

investigating because of its very simple structure and its relationships

with other models.

4.3 The Equivalent Stirred Tank Model

Consider instead of a flow and diffusion process a sequence of

little stirred tanks, the output of one being the input to the next. It

is supposed that the materials in each of these little tanks are completely

65.

mixed so that the state is the same throughout the whole of the inside of

any one tank. Furthermore the transfer between each tank is very rapid,

because they are all situated end to end, and there is no "back-mixing"

whereby material in one tank is transported, by dispersion, back into the

previous tank in a direction counter to the direction of flow. It is

intuitively clear that this sequence of tanks is, because of the combination

of flow and mixing (dispersion), a model of the flow and diffusion process.

It is in fact, equivalent to the model with the critical number of sections,

because the equation for the state in the rth tank is:

dU

Cr aT = (Ur Ur-1) q (4.8)

where Cr is the capacity of the rth tank and q the volume rate of flow.

Its structural equivalence to the equation for the ncrit model, (with unit

velocity of flow):-

dUr 41x r r

= (U - U-1 Tr (4.9)

is readily apparent. The model with the critical number of sections

represents the system of flow and dispersion, by the transport between; and

the mixing inside the members of a sequence of stirred tanks. This method

has been used in chemical engineering design to evaluate the static profiles

of temperature and concentrates in tubular reactors, where the diffusion

effects were extremely important, but not easily dealt with if the full

model (Eqn. 4.5) was used,nor satisfactorily treated by a model such as

represented by Eqn. 4.1. (R. Aris (2.1) and Amundson, Coste and Rudd (2.7)).

It is also an extremely useful idea to assist in the analysis of the dynamic

behaviour of these systems.

In the analysis of the basic flow and diffusion process by the

66,

Laplace Transform method two limiting cases were considered where on one

hand the diffusion was so low that the flow-only process was a good model

and the other where a high value of "a" made the diffusion-only model a

good one. The n. . model can also be compared with the root locus crit

representation on the basis of its Laplace transform,which is particularly

simple.

Each section has a transfer function such as

Ur 1 - 1 Ur-1 (p+ r) (4.10)

so that the effect of having ncrit of these in sequence is a transfer

function

Un 1 _

17 (4.11) o ( p+ 413c ) ncrit

Replacing Ax by 2a , (Ax = nd: = 2a) 1 makes the transfer function crit

4. o equivalent to an ncri-t

- cx -th order pole at = 2a jo. (4.12)

If plotted on the same scale diagram as the root locus plots of

Chapter 3 (See figs. 3.4 and 3.5) these ncrit poles are at .2a.04 -1.

The resulting patterns of constant gain and constant phase angle

lines are seen to be sets of concentic circles and radial lines respectively.

They are shown as dotted lines on fig. 3.5. In the left half plane,

inside the zone where)approximately, `tap -11 > i and arg (2ap -1) lies

A A between + T and - T the remarkably close agreement between the parabolas

of the original calculated Laplace transform patterns and the lines and

circles of the approximate ncrit model can be seen.

This representation of the ncrit model also lends itself to comparison

67.

with the Pad e' delay systems used to model the flow-only type of process.

A first order Pade delay approximation has a transfer function;

n

(4.13)

where -V is the delay to be represented by the unit. This corresponds to

a pole of order 7 at p--:,--E„and a zero o± order at p= Along the

jw axis the resulting phase angle is exactly the same as it would be for

a pole of order n at p = - — which would be the representation of the

same amount of delay in an ncrit model of a flow and diffusion process.

The two models differ only in amplitude response, which for the Pade delay

is a constant independent of frequency, and for the ncrit model decreases

with frequency.

In the same way that a second order Pade delay can be made up of

pairs of poles and pairs of zeros suitably locatedlit is possible to consider

a model of the flow and diffusion process built up in the same way. For

examplel instead of having two poles both at the same point as a basic model,

a pair of poles with suitably chosen locations could be used, as a basic

unit. The difficulties are that the resulting system would be less useful

as a model and the choice of the pole pair presents a difficult problem as

the criterion of success in the modelling is much less simple than that for

the pure delay system. Instead of juggling with an arbitrary set of poles

to make a mathematical model, the basic equations can be used (for n / ncrit)

to yield results with any desired degree of accuracy) for which the resulting

pole location will be automatically determined if needed. The problem is

the same as that of finding how many sections are required in the general

lumped model, and the form they have to take.

4.4 How Many Lumps?

1 1 The model based on the critical number of sections (n crit 2a = — ) has

been shown to be a useful model but without precise definition of the

region of validity. It may be necessary to have more sections in some

cases or it may even be possible to use less in others. A criterion for

assessing the validity of the model is required. It is bound to be

arbitrary and must be related to the amount of knowledge available a priori

about the practical systems which are being modelled. Clearly it is of

little advantage to have a system that follows the partial differential

equation to 0.001% if the partial differential equation is only accurate to,

say) 10% as a model of the physical system. Another point of considerable

importance is the use to which the model is to be put. This requires two

factors to be taken into account; firstly the significance or otherwise of

the distributed parameter system as a part of a large system,so that if it wil;k1

dominates it will require accurate representation,if it is only a small part A

it can be treated more approximately; secondly the type of signals with

which it has to deal, because the models become less accurate as the

frequency increases, making the frequency spectrum important.

Take as an example the ncrit model. The frequency responses in

magnitude and phase angle are shown in figs. 4.1 and 4.2. The values for

'd are the same as the values used in the figs. 3.2 and 3.3 for the accurate

transfer function. A direct comparison shows that the phase angle is

accurate to about 5% up to a phase shift of 60n° after which the lumped

model can only produce a limited amount of phase shift as compared to the

(ultimately) infinite amount from the accurate transfer function. In

terms of amplitude response the lumped model produces more response at high

69.

frequencies, less at low frequencies (the accurate curve for a = 0.25 is

shown on fig. 4.1 for comparison) but the differences are extremely

dependant on "a", making an assessment on the basis of frequency response

extremely difficult.

A more satisfactory solution is achieved by measuring the errors in

the response of the lumped model to a signal of known frequency spectrum.

Two input signals lend themselves readily to this, the step input and the

impulse. The first has a spectrum in which the spectral density decreases

with frequency, the latter has a spectral density constant with frequency.

Thus for the same measure of error the impulse response gives a more

stringent criterion of performance.

Both these tests have been applied to a lumped parameter model for a

range of values of tal from 0.02 to 1.0 with the number of sections ranging

from 40 down to 2. (Outside these ranges flow-only and diffusion-only

models are of more interest.)

These experiments were carried out on a digital computer. The

quantization in time was made small enough to make the effect of changing

the time steps to any smaller value negligible (4:0.001 in responses with

max. values 1.0 to 2.0). The time variable was thus effectively continuous.

Because the unit length system could be affected by the back mixing from

the sections beyond x = lithe model was always terminated by an apparently

infinite section from x = 1 onwards.

The overall results of these experiments are shown in figs. 4.3 and

4.4,where the minimum number of sections for any given maximum error is

related to the coefficient "a" for impulse responses and for step responses.

For any specified required accuracytthe ncrit model is better than is

70.

necessary for low values of "a" (high n),and worse for high "a" (low n).

The effect of the wider bandwidth of the impulse signal is to demand a

model with as much as twice as many sections to obtain the same accuracy.

(Compare fig. 4.3 with fig. 4.4)

The experiments to find the number of terminating sections were

carried out to enable the other experiments to proceed satisfactorily.

Although the number of terminating sections for a model should be a

function of both the coefficient "a" and the basic number of sections, the

fact that a model of any given number of sections would be used only for a

restricted range of "a", meant that for practical purposes the length of

the terminating section could be specified in terms of the basic number of

sections only. Fig. 4.5 shows both the regions which were found to be

satisfactory) and the actual terminated sections used in the other

1 experiments. It can be seen that a section of 3 of the length of the

basic model is generally more than adequate.

In establishing the largest satisfactory time steps it was found

that a simple rule based on the time constants of the individual sections

was possible, though the actual rule would depend on the form of step

by step integration used.

Further information on the digital computer programmes, their

construction and operation is contained in Appendix III.

The detailed results of the experiments are the actual impulse and

step responses. These are shown in figs. 4.6 to 4.11 and figs. 4.12 to

4.18 inclusive. Each contains an accurate response together with

samples of less accurate ones. Here the impulse response is seen to have

a peak of response which always appears earlier than the mean delay time.

71.

These times for peak response which can be calculated more precisely from

the formula (3.52) in Chapter 3, are shown graphically in fig. 4.19.

The curves provide a simple means of comparing practical and theoretical

systems on the basis of direct measurement.

4.5 Digital Computer Simulation

The use of digital computers for simulating physical systems by

step by step solution of differential equations has been 7.1pplied here

to investigate the number of sections needed for adequate system representa-

tion, and is an increasingly used technique for solving engineering problems

The type of process to be modelled in investigating the control of

distributed parameter systems always results in having a large set of

simultaneous ordinary differential equations to solve. For example in

the problem treated (Section 4.4) there is an ordinary differential

equation for each section, and this includes both the basic number of

sections and the terminating sections. In a more practical problem this

would be only a partland the immediate environment of the process would

require a similar description. Take for example a heat exchanger with

equations required for the state in corresponding sections of the fluid in

an adjacent stream and in the separating and enclosing walls. With twenty

sections in the basic fluid stream model there would result something like

eighty ordinary differential equations.

This may not present any serious difficulty for making a few runs

to measure, say, step or impulse responses but when the model is to be used

to test a wider range of control schemes, probably varying in structure

as well as in the defined magnitudes of parametersI the time required

72.

becomes several orders of magnitude greater.

The problem is not eased by the fact that as the number of sections

(in the basic model) increases the largest usable time step decreases

(practically in inverse proportion), so that doubt ng the number of sections

also doubles the number of time steps required. The computing time is thus

roughly proportional to n2 for any given length of operating time being

simulated. Since a considerable amount of control engineering work is

done on a qualitative rather than a quantitative basis the need to produce

sufficient printed data 64 for graphical presentations may also make

significant demands on computer time. If this is to be avoided and the

job of performance assessment written in to the computer program, together

with a procedure for making modifications to the control schemes then the

limitations on computer size and program complexity will also become

important.

In general digitalcomputers can provide accurate results for short

runs. The coat of computing is roughly proportional to n2 and to the

simulated time. They are less valuable when producing qualitative

information and not suitable for direct continuous variation of system

parameters from outside, as is sometimes needed for developing control

schemes.

4.6 Conventional Analogue Simulation

The term "conventional" here implies the use of high-gain D.C.

amplifiers and precision feedback and connecting components.

An analogue simulation of the lumped model described above requires

one integrator for each lump. But this is not the total number of high-

73.

gain amplifiers required because each integration process produces an

inversion of signal sense. The number of amplifiers required then depends

on the structure of the model and the sign of the coefficients in the

relevant equations.

Consider the model of the basic flow and diffusion process. For

n n crit

the coefficients in the equations: (Eqn. 4.5)

dUr 1 , a 2a 1 a - + (4.14)

x dt = Ur-1 7 ' x

Ur

- + Ur+1

x 2 x

are all such as to make it possible to use the integrators for each section

in cascade with no inversions (See fig. 4.20a). When however n is

smallerthanncrit the advantages to be gained are offset because one of the

coefficients changes its sign and a structure such as in fig. 4.20b is

required.

The same comments apply to the parts of the analogue computer used

to simulate the environment, and in particular to the connection to be

made between the parts. One notable disadvantage is particularly relevant

to the investigation of heat exchangers and the like. In the investigation

of heat transfer it is very simple to make a lumped resistance-capacity

analogue of a thermal diffusion process. No signal sense inversion occurs

between successive elements in such a model. The result is that if it is

required to represent two parts of a system, one an ordinary thermal

diffusion process and the other a flow and diffusion process the advantage

to be gained by using a cheap, simcle,R.C. analogue for one and a sequence

of analogue integrators for the other; is largely offset by the need to

provide signal inverters at every other integrator,when the two systems

run side by side.

Apart from these difficulties of operation there is also that of

size. While there are available analogue computers sufficiently large to

deal with systems with several hundred separate elements, most commercially

available analogue computers are made in much smaller units, perhaps a

dozen amplifiers or so.

4.7 Special Purpose Electronic Analogue

From the two preceding sections it can be seen that to make useful

engineering application of the knowledge gained so far about lumped

parameter models some means of solving the equations is required which

meets the following requirements. Thinking in terms of an electronic

analogue it must be cheap both to run and in terms of capital equipment, and

requir:e no special techniques or maintenance. It must be compatible

with simple R-C analogues and be usable with conventionable analogue

equipment. It must be capable of producing not only qualitative but also

quantitative information and since the underlying problem is that of

devising control schemes it is a useful advantage if it can be made to run

fast enough for the results of continuous parameter changes to appear as

(almost) simultaneous changes in performance, as for example;in changing

the observed shape of a step response on a CR0 by changing a controller

gain, or measuring the mean response to a white noise disturbance as a

metered function of some other parameter. The digital and conventional

analogue computers can,as discussed above,each do some of these thingsI but

neither of then can do all.

Equation 4.6 provides a design structure for a special purpose

analogue, which is easily produced and which in conjunction with a small

conventional analogue system can meet all the above requirements. The

equation is:- dU

Ax . = 1.(Ur-1 -Ur) dtr - 41(Ur+1+Ur -1 - 2Ur) (4.15)

Consider Ur as the voltage on a capacitor of size/ix. The

equation describes how this voltage depends on the current flowing in from

a source of voltage Ur-1 through a unit conductancel and on the flow of

a current to and from voltages Ur+1 and Ur_i through conductances (2,i 7

as occurs in the simple R-C model of thermal diffusion. The whole

process can thus be modelled by a conventional thermal diffusion type R-C

model (fig. 4.21c) with a "diffusion conductance", (22 - -),together with a

part for producing a current flow;1.(U - U ) into the rth capacitor, r-, r'

without upsetting the voltage on the (r-1)th capacitor. If a simple

unit gain buffer amplifier is used to provide as output,at low impedance,

the voltage Ur-1 from a high impedance input connected across the (r-1)th

capacitor the structure shown in fig. 4.21a results, when this is repeated

for the rest of the capacitors.

Before continuing further it is interesting to compare this structure

with the lumped parameter structures of the ideal transmission line,

shown in fig. 4.21b, and the simple R-C line or thermal diffusion model

which was also the model used by RiAqsdorp and Maarleveld (2.33) to

represent concentration movement in a distillation column, shown in

fig. 4.21c. The ideal transmission line passes signals in both directions

with no dispersion or blurring. The R-C line passes signals equally in

both directions but only by means of the dispersion effect. The

distillation column passes disturbances in concentrations both up and down

(by means of vapour and liquid flows, see Chapter 2, Section 6 ) and

- )

76,.

blurs them out as well. The structure being used here exhibits transfer

by dispersion which is the same in both directions but the flow transfer

is, unlike the other systems, only in one direction, hence the need for

active as well as passive elements.

A number of possibilities present themselves for implementing the

buffer amplifier. The simplest is the simple emitter follower which is

adequate for many engineering purposes. A modified emitter follower which

can be designed to produce a gain closer to unity and with smaller voltage

differences between input (base) and output can be made using two suitably

chosen NPN and PNP transistors and a diode. Using three transistors, all

of the same type (NPN or PNP), amplifiers can be made to meet similar

requirements. (Circuits for buffer amplifiers and some design data are

shown in Appendix IV.)

The overall result is a means of modelling the flow and diffusion

process which is easily and cheaply produced, needs very little maintenance,

is made entirely of standard components, and meets all the other requirements

set out above.

There is a modification to this system which uses inductance to

increase the effective number of sections. The principles are readily

explained in electrical engineering terms but appear to have no counterpart

in the field of numerical analysis.

The underlying idea is to take the second order Pade delay concept

of having pairs of poles (and zeros) and apply this to the simple lumped

model of flow and diffusion by making each section into a second order

LRC circuit. The capacitftor

77.

occupies the same location as before and the inductance is in series with

the R as in fig. 4.22a. The case where the LRC circuit is critically

damped corresponds to a simple doubling of the number of sections in the

representation of the flow process because the transfer function of the'

LRC circuit becomes equivalent to two cascaded lags.

Consider an LRC circuit as in fig. 422b. The two pole locations

can be expressed as functions of the inductance L as it varies from zero

to the critical value (and beyond). The poles appear at

_ R 3

2L 2L

(4.16) 4L

P2 = _ _ 2L R

2C

Expressed as time constants T1

and T2,the same total(T

1 + T2 is always

given by;

T1

+ T2

= RC so the effective delay represented by

the complete system is the same regardless of the value of L. The change

is in the value of 'a' that is represented. Since the objective is to

obtain a model with the highest possible number of sections the case of

greatestinterestisthencrit model, in which the diffusion conductance is

zero, because this represents the smallest value of 'a' that can be modelled

without needing negative conductances or inverters. Using a given model

the introduction of the inductances replaces each of the first order cascade

lags by a second order lag with the same total time constant.

When L is made large enough to make the LRC combination critically damped,

then the effective number of sections is doubled and the inherent dispersion

due to lumping is effectively halved. The details of such a model are

4L R2C

78.

described in Appendix IV, where the use of another modification, that of

making the amplifier gain greater than unity, can be used to model the

effects of unstable reactions for tubular reactors and also provide

alternatives for conventional analogue units in simulating systems.


In solving the partial differential equations associated with

distributed parameter systems it is necessary to reduce the problem to that

of solving a finite set of ordinary differential equations. The same

applies to the problem of making electrical analogue models when it is not

possible to use one process directly as an analogue of another. One method

is to use a lumped-element model. With the exception of the diffusion

equation there is virtually no available treatment of partial differential

equations of the parabolic type from the viewpoint of numerical analysis or

analogue modelling which would enable the quantization to be carried out

satisfactorily solely by reference to the equations and boundary conditions.

Investigating the problem directly by experimental means for the

equation of particular interest here shows that when allowance is made for

the inherent dispersion introduced by quantization in the space domain, the

number of elements required for a satisfactory representation of the system

depends on a normalized diffusivity coefficient, the type of disturbance

signal to be used and the accuracy required. Results are plotted to show

these needs.

Since the accurate representation of these systems makes heavy

demands on conventional analogue or digital computers either in terms of

time, equipment or both, it is necessary to develop special purpose analogue

79.

techniques to overcome these difficulties. An analogue system can be

made from transistors and standard components, which is cheap, compatible

with other simple analoguesl and can be made to operate fast enough for

producing continuous measures of behaviour as functions of parameter

changes. This electronic analogue can be modified to represent twice as

many lumps with the same number of buffer amplifiersland also, though not

discussed in this chapter, to simulate other systems andflow-and-diffusion

processes with reactions.

0

a.

vl I N

-----1;

**.L.

...•'''''''

..-----

-----

w

el

A 9

04 .:, N4.

04

.g—

//..

_.,...-

/

/

) 44 O In 6 v, — A uN * .., .4, , .... o

Soh

0

a

0

‘it

z Fig.4.1. Frequency Response — Magnitude — of Lumped Model

(ncrit

) of flow and diffusion process subject to

disturbance at input of flow stream. Magnitude (M ) in decibels relative to disturbanon level as a

function of normalised frequency (w).

0 0 ,41)N

17M To\ N

ql %...

(4 •

I c II;

.1 ,... ...„,..-

,:,..„,.......,-- ----•--

to

P

-• 1

_________-------------------'1 5:8

/

/ /

/ /

0

0

Fig.4.2. Frequency Response — Phase angle — of Lumped

Model (ncrit) of flow and diffusion process

subject to disturbance at input to flow stream.

Phase angle (0) as function of normalised frequency (0).

0

0 gr

O io

0 N

1-

to

N

to

60 50 /0

30 1

r‘

to

6

3

2

x •01

.•••

sykr..•• 4 •

20

..... ... .....

•

-...... ....._ • ... I.. ... ... ,,

.

--"--.......... -.....„ , 40. 4'.

.... I • 0 \.,,......... \S t ......,„

, , ,

I

• ...,J4)( . • •

•

4 I 0 6

412 .EM •o • '06 'or q .2 .3 .4 .6 -t" 1.1

Fig.4.3. Aocuracy Contours (Approximate Max. Deviation)

for impulse responses of lumped parameter models

of flow and diffusion process, infield of number

of sections (n) and parameter 'a'.

Locations of computer experiments shown by dots.

ez_

.01 .02 • 3 -4 .03 .04 • 6 .1" ko -06 .08' 1. 2 'a

w

D

0

4)

0

30

0

S.

s.NN

C. 1 0

N\

N. •

%..

r

N `4\

N.\ N

N s. 6 N\ N. .......a.

NN '''N.%

.. 4

3 N'''''%

..1);..

N.,,NNN

..)4 N...

..„

2

I

I

Fig.4.4. Accuracy Contours (Approximate Max. Deviation) for

step responses of lumped parameter models of flow

and diffusion systems in field of number of

sections (n) and parameter 'a'.

3

r 6

4-

2

n

0

..... ..... _ ..... ..._ ..... ••• Wan. .1M• •••••• ANNO -••••• . -

Own

i

0 k er

/2

/6 20 24

21?'

71

Fig. 4.5. Numbers of Termiaating Sections (T) used for

Lumped Parameter Models overand above the number

of sections in the model (n). Broken line shows

approximate limit below whioh noticable differences

(>0.1%) ocoured.

Id. t

T 4-'

..---'-' ..----------- ...--"....--...

: fp g /

C4 0

o

\%. -......f. _ h

--.......

ia II a

--- .

N\\.

\ '9

1"

N 0

.- o

-0

CM

.i-

Fig.4.6. Impulse Response of flow and diffusion system.

'a' 110'02. Normalised system. Computed responses. esr

./-

---' .------

,-.- .....---"-

4- .4 ------

; \ N

*." -...70

--..,... --....._,_ o 6 II a

,,...__ -,.

\ I

0

0

41- O

Fig.4.7. Impulse Responseof flow and diffusion system.

'a' 8 0.04, Normalised systemoomputed results. z de

04

0 I-

oo

O

R7

/

/

/

/ /

/

t*:

/

/

/ { Iii

,

O ii v

‘11 z


'a' . 0.1, n

ormalised

system,00mputed responses.

I

e// ...,..L.,

7 /F

N

i

V) n

CD

II

0

,,.....

s'..\..

N

O

a.

.43 O


'a' ... 0•2, normalised system, computed responses. 4.

41'

O

..1)

4.

N

a

(14

----"

4 C

0

m

....,,,..: ....,........

-----'---..

...".---

.............

PO

0.

.4) 4-

N

Fig.4.10. Impulse R

espo

nse o

f flow

and

diffu

sion

system.

'a' u 0'4, normalised system, computed resp

onses.

29

'o

0

4:0

/

dD 0

II 0

tn a F

%..sk

O

cc

Fig.4.11. Impulse response of flow and diffusion system.

'a' = 0.8, normalised system, oomputed responses.

¶0

O

tv M

a

N

IC 11

O

a II

a

Fig.4.12. Step response of flow and diffusion system.

a = 0.02, normalised system, computed responses.

9(

o•

(4

O

i \

W

.0

4..

It

44 4:3 (.°

•9 . ? N n

\

4 1 \

lc

\

\

•t 0

6 11

\ 1

0

co .D

c.f. 0

Fig.4.13. Stepresponse of flow and diffusion system.


.92--

\ tz)

N .

N \

v-

II 0

c)

\

nt O

Fig.4.14.

Step response of flow and diffusion system.


93

c0

O k

c!,

O

N

0

ti

\\ ad

IC \e"

x

N

Col

11 0

0

m

n

an ..n

4- N


a . 0•21 normalised system, computed responses.

O

c0

O

O

O

co N

CO

`

IIa

•r O


a 0.4, normalised system, computed responses.

tO

4-

0

4-

0

N \

'i \

\

40

O II

0

1

0

03 e+1

0

Fig.4

.17. S

tep resp

onse of flow

and d

iffusion

system.

a = 008, norm

alised system, com

puted respon

ses. op

0

1/1 it

0

• gk-

I I 0

0

'0 N

0


a = 1'0, normalised system, computed response.

9-7

GI

'0

•eI ••s .64 • 04 ••r • f • 2 .z •4 .6 •ji Fe a

Fig.4.19. Time (T) for the impulse response of flow and

diffusion system to reaoh its peak value as a

function of 'a'.

/40

.6

0

.7

6

T-

.2

99

+Uro

—w

a) Analogue form when n > ncrit.

b) Analogue form when n < ncrit.

Fig.4.20. Form of conventional analogue computer

simulation of lumped parameter models of

flow and diffusion systems . (Basis for

arrangements is Eqn.4.3.)

T a) Structure of special purpose analogue.

T T T b) Structure of lumped model of transmission

line.

r (3) Structure of lumped model of R—C

transmission line of thermal diffusion

process.

Fig.4.21. Lumped parameter model of flow and

diffusion system with comparable systems.

leo

a) Modified Line.

v..,

b) L-R-C circiut.

Fig.4.22. Modified flow and diffusion (Ncrit)

line with inductances.

102.

Chapter V

HARMONIC OR FUNCTIONAL ANALYSIS

5.1 Introduction

Instead of describing the behaviour of a system by the time

variations of a set of numbers representing the states of the system at a

number of locations spread over the systemt there is the possibility of

representing it by the time variations of a set of numbers,each associated

with one term in a representation of the profile shape by the sum of a

series of functions. This is the harmonic or functional analysis approach.

Usually it involves the use of orthonormal functions, but this is not

necessary. There is a fundamental equivalence between this technique and

the transform methods considered in Chapter 2. The Laplace transform used

there is an integral transform from the t-domain to the p-domain. The

functional analysis requires a summation transform from the x (space)

domain to the domain of a number which characterises each of the terms in

the summation. The result of the operation is a set of O.U's for the

coefficients of the terms in the series.

The structure of the summation transform is not difficult to

establish. The difficulty arises in choosing the series to use. Two

limiting cases can be considered - one is the case of using the Laplace

transform itself in the x-domain to produce an ordinary differential

equation in t with some variable, q say, - the new Laplace variable, as a

parameter, though this is no longer a summation, but an integral transform.

Another is the case where the functions used are so chosen that the lumped

parameter model is produced. The series used must be somewhere between

103.

these limits1 in that it must be a series of functions each existing over

the whole (or a substantial part) of the x-spacel and it must produce a

model of the system capable of representing the unidirectional characteri-

stics of the system. Furthermore the resulting set of ordinary differen-

tial equations for the coefficients of the terms in the series must be a

stable set in the Routh-Hurwitz sense, for any number of terms and any

values of the coefficients in the original partial differential equation.

Another requirement is that the set of functions must be able to meet the

boundary conditions imposed by the problem. It is also an advantage if

the terms used form an orthogonal set,so that the resulting set of first

order ordinary differential equations, with time as the independant

variable, have the simplest possible structure. This avoids the inversion

of a large matrix of coefficients as a pre-requisite to solving or using the

differential equations.

Various sets of orthogonal functions and also a set of non-orthogonal

functions are considered to show the possibilities and disadvantages

together with the relationships between various systems of representation.

5.2 The Laplace Transform applied to the Space Dimension

Making a Laplace transformationf using the space variable on the

partial differential equations for the system,results in an ordinary

differential equation with time as the independent variable. This is of

interest both for its structural properties and also for the numerical

evaluation of terms in summation transformations involving exponentials.

Starting with the equation

6U _6U 2U - V + D 6t ax ax2

(5.1) also (2.2)

104.

where U = U(x,t)

and applying the transformation,

U(q,t) e qt U(x,t) dx , (5.2)

results in the equation

11(q,t) + U(q,t) qV - q2D] = U(x=o,t)(V-qD) + 211(x=o,t .(-D) ax (5.3)

This shows the structure. There is an ordinary differential equation for

U(q,t) in which q appears as a parameter. There is)in ,J.ffect,a different

equation for each value of q, which may be considered as a "space-frequency"

which characterises the forms of profile shapes ultimately to be added

together by the inversion (integral) process to produce the solution to

xxot) the equation. The conditions on U(x=o,t) and aU( are boundary a

conditions which now appear as forcing functions for the differential

equation. The coefficients depend on the parameter q. This same pattern

appears when a summation transform is used,except that instead of integrals

being used for the transformation and inversion, summations are used.

5.3 General Structure of the Summation Transform

The distribution of concentrations or temperatures, U(x,t), in a

distributed parameter system is to be represented by the series sum

U(x,t) aI(t) (x) . i=o

(5.4)

and it is supposed that the process is described by the partial differential

equation:-

8U _ au a2U - V + D at - ax axe

(5.1)

For use in any simulation:the series must necessarily be truncated

at some upper limit (k) to the number of terms. The functions v.(x) are

a.; + D -1V6v.

6x (5.7)

62v.

1) 6x2

105.

the mode shapesl and the problem is to find the ordinary differential

equations satisfied by the coefficients ai(t),which describe the overall

behaviour of the system by specifying the proportions of each mode needed

to reproduce the actual profile, U(x,t), at any particular time. These

ordinary differential equations can be found by substituting the series in

thepartialdifferentialequation,multiplyingbyeachv.(x) in turn and

integrating over the range of interest, which may well be the range of

orthonormality of the functions vi(x).

The equation becomes, after substitution:-

2 ) = 6

--(2a v ) + D — (4:a. v.) at

v i - V ax i i 6x2 1 1

(5.5)

and multiplying by v.(x) and integrating over the range of orthonormality

of vi(x) (say, 0 toop) gives: ...4.0 v.,

6 - V v...f..., a '7: 6v.

I 62v. ( 2. . v .)

i i 6x i 6x2

-si- aj(t) . (I.v.) dx + a.2. dx,

...,0 • (5.6) j = 1... k.

j = 1... k.

At this stage several alternatives can be considered. In equation

5.6 the integrals which are the scalar products of mode shapes and deriva-

tias can be manipulated to give boundary terms and other scalar products.

This is what happens in the derivation of the Laplace transform of a

derivative. One result is, for example,

f. _ i I _

aa.

at = - v .N, a. v. v. 1 + v 4-........., a. (vi --LL)dx +

4.....J

aV4 1 ."

1 1 j i 1 1 aX

1 L Jo 2.

a2v.

a ' (7-71 v.) dx a. oxc. j

.. i6 (5.8)

io6.

where only the first derivative term has been altered, leaving a boundary

4 term and a term which requires the evaluation of the scaler product of the

avi mode shape v. with each of the mode derivatives --" •

ax

The same operation can be performed on the other term, involving

the second derivative, and similar results produced, except that this time

the boundary terms can also include the first derivative. Compare this

with the Laplace transform, Eqn. 5.3, where the same thing happens.

The effects of the boundary condition are included because of the

boundary termsi as in Eqn. 5.8. The environment of the system can also

have an effect in a distributed manner so that it is necessalty to include

its forcing effect in the partial differential equation. Consider the

equation

au aU a2U G

at2 = - V

ax 3 + D 23 C32 (U2 - U3) ax 3 (5. 9 )

which represents the effects of, say, heat transfer from an environment at

temperature U2

to a stream at temperature U3.

(See also Eqn..3.47).

U2

may be a function of both t and x. There is no change in the procedure

needed to deal with this. If the functional dependance of U2 on x is

known or if it is represented by a series such as used for U3

then the

process of multiplying by each v.(x) in turn and integrating yields the

transformed version of U2, which then appears as another forcing term in

the ordinary differential equations.

The behaviour of a distributed parameter system can thus, in

principle, be described by a set of numbers, possibly infinite, which are

coefficients in a series representation of the profile shape in the system.

The coefficients are related by a set of ordinary differential equations

107.

which can be evaluated from the original partial differential equation by

a process like that involved in the standard Laplace transformation or

Fourier analysis. The boundary conditions can appear as forcing functions

in the equations. Although this has been developed around the flow and

diffusion equationIthere is no reason for restricting it to such systems;

and in fact in other cases where the functions, vi(x), can be chosen to

be eigenfunctions of the x-dependant operator, special results apply which

relate the transformation process to the method of solution of partial diff-

erential equations by assuming a separable solution.

5.4 The Effect of using Eigenfunctions

Iflin the transform like approach abovelthe functions vi(x) are not

just any set of orthonormal functions but are related to the x-dependant

operator by being eigenfunctionsIthen the results are simple in structure

and have a separate and independent equation for the behaviour of each mode.

These equations are the same as would arise from a separable solution of

the partial differential equation.

In equation 5.7, if the functions v(x) could be eigenfunctions,

orthogonal and normalized the right hand scalar product would ree-Ice to

the eigenvalue Xi giving the equation

dai(t)

374 = 3 Xj •

This is not possible in this case because the operator;

2 (-V a + D A- ) ox ax2 '

does not have a discrete set of eigenvalues. It does however happen with

the diffusion-only operator:

(5.1o)

108.

2 f 6 1 . ax2

The boundary conditions can appear as forcing terms if the integration '-

parts procedure is carried out first, as in the derivation of the Laplace

transform and in the step immediately prior to equation 5.8.

The use of functions, which, when available, are eigenfunctions for

the x-dependant operator transform the application of the operator to a

simple scalrx multiplication. The result of the analysis is a much simpler

set of ordinary differential equations in which each mode is independent of

the rest. Using this particular form is exactly equivalent to the separable

solution of partial differential equations and, in the case of the diffusion

operator, to Fourier analysis as used in heat transfer. The technique is

widely known and used, (See any general mathematics text - in particular

Lanczos(4.5) and Pipes (5.6).

5.5 Special Functions - not Eigenfunctions

In dealing with the flow and diffusion process the operator

a a2 (-V ax D ;372), together together with the boundary conditions 4-

the function be given at x=o and tends to zero as x tends to infinityl has

no set of discrete eigenvalues. The condition that the application of the

operator should result in a scalar multiplication leads to the equation

2 iD 8x2

V *)1 U (5.11) ax

which has the solution

A Expi551V +,./V2 + 4AD + B Expih- {V 1112 + 444.

which can meet the conditions specified for all >1) >0 and for A = 0 .

Thus any representation of the system by means of the sum of a series of

(5.12)

109.

modes will require special functions which are not eigenfunctions.

The function3to be used have to meet certain criteria before they

can be used. They must, taken together, be able to meet the boUndary

conditions which are applied. They must be capable of representing all

possible behaviour - i.e. they must form a complete set. It is advantageous

if they are orthogonal and normalized but the equations can still be

developed even without these demands being met. If the functions are nearly

orthogonal, that is if their cross scalar products are small in comparison

with their own integral square Valuestthen the resulting set of equations

will have left hand sides (See structure of equation 5.8) which instead of

being solely one term will be dominated by one term but include others as

well. The one overiding criterion for making a useful model of the system

by this method is that the system of ordinary differential equations should

be stable in the Routh-Hurwitz sense for any number of modes (i.e. for

k = 1 ), so that, for example, an analogue simulation using the moael

should be a usable system. There is in the basic formulation of the

process for developing the ordinary differential equations no direct

indication of this stability and it must be investigated in terms of the

functions used.

There is a wide choice of functions available which can be applied.

All the hypergeometric series are available and can be adapted to cover

either the range 0 to 7 for the normalized space variable (See Chapter 3,

Section 2) or by changes of variable can cover the range 0 to o0. Some of

the functions require weighting functions in4integrating process to make

them orthogonal, others do not, but this makes no difference to the approach.

Other functions can be designed either by manipulating and changing the

110.

variables in standard functions or directly from the requirements of the

problem.

Using the procedure outlined abovel the Lasuerre and Legendre

polynomials were applied to model the system on the range 0 to 1 with

special provision for the boundary conditions at x = 1. A special function

was designed by changing the variable in the sin 2nx function to extend the

range (0 to 1) out to (0 toe.Oland putting in a suitable weighting function

in the integration process to make the functions orthonormal. In each case

the result was a system in which the stability depended on the relative

magnitudes of the coefficients of the partial differential equation or in

which the individual modes were each unstable on their own,so that for

k = l‘sayithe system would certainly be unstable—after that (0.1) stability

could depend on coefficients. The details of these investigations are shown

in Appendix V.

One underlying factor is common to all these representations of the

system. They are not related to the differential operator, and thus do

not in any way from a set of functions in which the behaviour of the sye:

can naturally be described. For example, consider the behaviour of a slab

of material co ling down from an even temperature distribution to zero

temperature while its outer surfaces are held at zero temperature. The

temperature profile across the slab can readily be described in terms of a sirNe.

half range Fourier WM series. The various modes, corresponding to

multiples of one half-cycle of the sine wave within the thickness of the .51A1-

decay away to zero as the slab cools. However the higher modes decay more

rapidly so that the profile becomes more and more accurately described by

lower modes, till ultimately the lowest model Le. that corresponding to just

one smooth hump in the temperature profile across the slabiis the only

significant one. No such behaviour can occur with any of the conventional

functions used above to describe the flow and diffusion process. Thus if

the flow and diffusion system starts off with an even distribution of state

all along from x=0 to x=1 and beyoncythen the process of settling down to

zero when the input end is held at zero is a sequence in which the parts

of the system nearest the input reach zero first'and as the fluid flows

alongIthe state at each successive point reaches zero. The representa-

tion of the system by any sort of set of functions which cannot exhibit the

behaviour in which various modes decay away in turn as the system settles

down is unlikely to succeed as a model. To make a model for the flow and

diffusion system in which this sort of thing can happen,it is necessary to

have the various modes arranged so that each deals mainly with one part of

the system. An example which is particularly interesting because it relates

the functional analysis model to the lumped parameter model is the triangu-

lar interpolator described in the next section.

5.6 Triangular Interpolator

The set of functions 0.(x) to be used are a series of triangular

functions centred on x = i.4Ax , i = o n. (See fig. 5.1 (a),(b).)

The heights of the mid-ordinates are unity. The effect of adding them all

together, each scaled to the concentration or temperature at its mid-point,

is to produce a profile which is a linear interpolation between the

ordinates. (fig. 5.1(c)) The series to be used is

al(t). O(x) .

Substitute formally in the differential equation:-

and j being adjacent or the same. This leaves!

at [ai _a. 0j_1 0j dx + a c

dx + a j+1 0. 0. dx +1 ,.,

a at gn'aj-1 3n'aj gri aj+1

2 1

(Chapter 4, Sect. 3) [ aj_i J 3+4

D - 2na. + .na.

= ••• 1 -la + aj +1 2 j-1 I 7

(5.16)

,2

a ai ° = - V a a. 0] + D a. at 0.] • i -- dx{. , i ax2 1 Multiply through by 0j and integrate over the range (- + c'°) in x.

or over (j-l)ax to (j+1$x ;— co pco

at,0ai10. dx = - y2 ai ax 0: 0 . dx + .1. j

a a

:Co Zoo

Taking the first derivative of the 0. functions to be rectangular

functions,and the second derivatives as 8, functions, it will be seen that

the only terms in the summations which remain are those associated with i

a p if ax2

2nci (5.14)

= - v { a . ags 4

j dx + a . 3-1 ax 0 --J 0 dx + ax j aj+1 ax

a0j+1 0j dxj

P 2

a20. 8

D a 0

{ . fa j-1 j+1 2 delj dx + v 2. ---J 0 dx + a0. dx (5.15) ax ax j j+1 ox

And putting in values for the derivatives leaves:

1 x aj In comparison with the lumped section model note that n = [S — and is

here the amplitude at a distance x = j x, to be compared with U., the

amplitude in the jth "tank", in the lumped section model. This involves a

shift of -7 in the x direction if amplitude is supposed measured at the

centre of the section7but this is not important.

113.

The equation for the lumped section model was

a n -V 1 1 —U - U + D - 2U j + (5.17) at j j Ax 2 j+1 2 -1 66.x)` J-L

See also (4.1)

The right hand sides of the two equationb are the same and the left hand

sides have

6U.

at 6a [1 1

1. 3a + 1 -1 j-j ;aj+1

replaced by ye , which would appear

to be a reasonable approximation.

It will be observed that in the triangular function equation the

L.H.S. contains three terms and therefore cannot be used directly as a model.

Three possibilities arise: one, above, to consider the L.H.S. as being a

close approximation to 611 --u

-1 ; secondly to manipulate the equation to extract

at one term only, or to make one term so large in comparison with the others

that it dominates completely/and thirdly/to modify the triangular inter-

polating function so that it is orthogonal to its neighbours.

The values of da --3 can in principle be extracted by inverting the dt

tri-diagonal matrix of their coefficients. However one method of manipula-

tion of the equations involves only scaling and adding them.

In the triangular interpolation equations consider a sequence of left

hand sides of the equations as follows:-

[

2 1 at Taj -3 '

4. 3aj -2 + ri -31 (i)

d [1 2 1 (ii) dt Taj -2 +

rj -1 Via.

d 1 dt 6aj-1 3aj ;"j+1

1, • • • (iii)

dt 4. 2 dt Taj 3aj+1 taj+2i = .00 (iv)

114.

Consider the L.H.S.'d (1), (ii), (iii), (iv) above;by combining

(ii), (iii) and (iv),the k(a.j) term can be made to dominate, in

a d combination with ar(a._ t j 2), dt(a

j+2)

Thus: add; -1-(ii) and -21- (iv) to 2.(iii). The L.H.S. now becomes 2- 2 '

dt 7-12aj-2 + iaj - 12aj+2 [ ]

--- (v)

andthecorrespondingR.H.S.isfoundbyreplacingb.1 2 .by (-- 1 a. 1 + a1 .- 21a.+1 ) -

etc., thus:-

-V

[I

% 1 ] (-a + 2a - le - (-1e+ 2a- 1) Ax 2 1 j+1 2 j+2 ) 2 j_2 j...3. r i

2

(-1a J + 2a j - la J) - 2(-1a. + 2a- la. ) J 2 -2 -1 2 2 J-1 2 j+1

(-1a. + 2a. -1a. ) 2 j j+1 2 j+2

becomes:-

-V 1 1 D [ 1 1 -a. - a. + a. - a. --a. + 3a. - 5a.+ 3 . a x 4 j-2 j-1 j+1 4 3+2 3-1 j aj+1 2 j:2.1 (6x)2 2 J-2 (vi)

Taking the L.H.S. (v) and the R.H.S. (vi) then gives an equation which can

be used as a basis of a model for the system if the L.H.S. (v) is used as an

approximation to

dt(aj) .

daj+3 -- with even smaller coefficients - but the complications involved and dt

the increasing distance between terms lumped together are unlikely to make

it worthwhile.

The triangular interpolating functions (0i) used above do not form

This process can be carried further to eliminate not only the terms aai da. dai 3

adjacent to .3.7-1 in the L.H.S. but also a.3+2 leaving only --v- dt and

115.

an orthogonal set but can be made to do so if some increased irregularity

is accepted in their representation of the profile. Thus functionslP

fig. 5.2, can, with appropriate choice of the parameters of the curves be

made mutually orthogonal.

One simpleformforW.is as shown in fig. 5.2b. Suitable choice

parameterc<inyi -7-(1-xn)(1-m)() ,o 4;x 4= will make n

i+1 do = o. Intheequationfor.replace x by (1 - x) to obtain

' r The integral!

qi+1 dx becomes

(fin

1(1-xn)(1-*Ix)(nx)(1-o4+anx) dx, and solving for the values

J6

of "o0 required to make this zero gives olG= + +

The value (=.- 2 2 - - gives the smoother interpolating function and is to 2

be preferred:-

1.383,

The normalizing factor required for the evaluation of the coefficients in

the equations for the system is given by:-

2

(W. )2dn

3n for either choice of o(.

0 The functions . are made orthogonal at the expense of having an

irregular interpolation function for the profile. The number of sections

required for the model will naturally be approximately the same as needed

for the simpler lumped parameter model (Chapter 4)., to which the linear

interpolation can readily be applied if required.

of the

Om

116.


The method of harmonic or functional analysis can be extended beyond

the normal situation in which the functions are eigenfunctions for the

differential operators involved in system description. This extension

is achieved at the expense of having the equations for the various modes

cross coupled instead of being separate. There is also the further

disadvantage that the resulting set of ordinary differential equations for

the mode amplitudes may not be well behaved in terms of stability, because

the mode shapes bear no relationship to the patterns of behaviour that occur

in the physical system. Another way of describing this unsatisfactory

behaviour is to consider the set of equations for the mode amplitudes ai

forming the vector a.

f • Ji°) The matrix K is found from the partial differential equation by

evaluating the scalar products which appear when applying the "transformation"

process. (See Section 3) When the functions are eigenfunctions the

matrix K has terms only in the leading diagonal, which are the eigenvalues

of the space dependant differential operator. When other mode shapes are

used K has other terms as well, which can be quite large even a long way

from the leading diagonal, and when this happens the results of the

investigations (See Appendix 5) show that the system is probably unsatis-

factory. An example in which the matrix big has only terms near to the

leading diagonal, produced by making the functions bear some relevance to

the natural behaviour of the system, shows satisfactory behaviour.

This same set of functions is also the link between the functional

analysis model and the lumped parameter model becausethe example taken was

Ida

dt

117.

designed to produce linear-interpolation between successive measures of

state at points evenly distributed along the system, and it also produced

equations which were extremely close, in terms of the elements of the

matrix I.K.1 , to the equations for the lumped parameter model.

The lumped parameter model can yield its results in terms of system

state at any point as a direct output, allowing if needed, linear interpola-

tion. This is not the case with a functional analysis model because the

state at any point has to be calculated by evaluating the summation

tai vi(x) L

for the particular value of x required. Any extra computing time or

equipment must be allowed for. The only significant advantage of the

functional analysis model is, given sufficient terms, its smooth representa-

tions of profiles.

The general conclusion is that) except in the case where the space

dependant operator has a suitable set of eigenfunctionsIthere is little to

be gained, over and above the results of using a lumped parameter model, by

using other functions unless they can be designed to be particularly suited

to the job of representing the natural behaviour of the system.

a) Triangular interpolating functions superimposed.

b) Isolated function.

x c) Summed functions give linear interpolation

between specific points.

Fig.5.l. The triangular interpolating function.

HQ'

a) Modified functions superimposed.

b) Isolated function.

X

c) Effect of adding functions to give distorted

version of profile.

Fig.5.2. The modified, orthogonalised interpolating

funetion.


Part II

Control

120.

121.

Chapter VI

CONVENTIONAL CONTROL METHODS

6.1 Introduction

Control methods may be considered conventional because of the

techniques used in analysis or because of the type and use of the necessary

equipment.

Analysis based on frequency response methods and root locus plots is

the normal design method. The practical implementation is achieved with

two-term and three-term controllers.

Valuable and workable control schemes are madet and any alternatives

must be weighed against these.

The usual performance criteria used are that the system should be

stable and give a reasonable (short, small or zero overshoot) response to

step inputs land have small steady state errors. Many industrial processes

are engineered so that they are inherently very stable and have the capacity

to suppress disturbances with little or no control action being taken.

Plant design does not generally depend on having control at all. The

result is often that systems are ultra-stable and sluggish in response.

However, the use of continuously operating processes to replace batch

processesland need to operate plant at higher throughput rates than those

for which they were originally designedlboth require control schemes to

maintain operation. Processes are being developed in which control plays

an integral partland any process which is liable to exhibit unstable

behaviour (Nuclear Reactor) must be held in balance between dying out and

ceasing to operate andthe alternative of catastrophic (usually thermal)

runaway.

122.

For some of these situations conventional techniques may be

inadequate, and more advanced methods required or modifications made to

standard methods to adapt to more difficult situations.

The conventional analysis and control methods are here considered in

application to distributed parameter systems, taking as an example the

common flow and diffusion (dispersion) system which underlies many

industrial operations.

6.2 Frequency Response Methods

In this context frequency response is concerned with Bode and Nyquist

plots, while the full extent of the frequency domain approach is considered

in the next section under Root Locus Methods.

Frequency responses have been developed for a range of structurally

simple flow and diffusion processes, (See Chapter 3), and examples of

these are shown graphically in figs. 3.2 and 3.8.

These frequency responses can be used in the normal way. The

effects of measuring the system output and applying it through a controller

to the input in a feedback loop can be treated as usual. The difficulty is

that this simpleloop structure rarely exists in distributed parameter systems.

Thus;if a disturbance enters by means of the feedstocklit may well be

suppressed by control action applied during transit through the system.

Very rarely will it bo suppressed by injecting another disturbance with the

feedstock, particularly when this may well be too late to do any good. It

is therefore necessary to consider also the frequency responses of structur-

allyup

different arrangements, one for control, one for distftbances. For

example the effect of distributed forcing on a flow-only system is shown in

figs. 3.8 and 3.9.

123.

Logarithmic amplitude plots and phase angle plots can be combined

in the normal way if required and there is no difficulty in combining the

frequency response of a controller with that for the control-action-to-

output response to evaluate overall stability. For example the response

of a system to distributed forcing (figs. 3.8 and 3.9) and eqn. 3.46 show

no phase lag greater than 180° and since this is made up of two components,

one due to the normal lag (max. 90°) for transfer into the stream and an

alternating component due to the relationship between the changing environ-

ment and the flowing stream, which is suppressed by the dispersion process,

the nett effect for a practical process is almost entirely dominated by the

lag and its 900 phase shift. The system can be controlled and be stable

if the controller and instruments do not contribute any phase shift. Even

with another lag due to transfer through the walls (heat exchanger) the

system is stable for quite high loop gainlbut the inclusion of another lag

in the controller makes it unstable very easily. Such a system has been

modelled and its behaviour investigated. (See Section 4 - Three-Term

Controllers)

The difficulties lielnot in using the frequency responses but in

calculating them. If they can be measured either on the plant or a model

then the usual methods apply. (See refs. 3.17, 3.18, 2.54)

6.3 Root Locus Methods

The Root Locus method dealing with the whole of the p-plane (Evans

3.10, 3.11, 3.12) really includes the frequency response method above as a

special case. It has the advantage that a wide range of frequencies are

considered, so that not only is the response to sinusoidal inputs apparent

from the location of the poles (relative to the jw axis)o but so also is the

124.

transient response and the effects of system changes on stability and

performance. The disadvantage is that the root locii have to be calcula-

ted. They can not be measured directly from a model of the system. An

exception to this is the design of systems using electrostatic tanks or

their equivalent for p-plane representation (Westcott, 6.8). Since this

normally requires the transfer functions to be given as rational polynomi-

als it is not directly applicable to distributed parameter systems.

Some examples of calculated p-plane plots are shown in figs. 3.4 and

3.5. These are for the transfer from the input of the flowing stream to

the output. The modification of the plots to represent distributed forcing

requires the transformation from the p-plane to the domain of the related

variable:

E(p+A) P

p+A+B)

or an equivalent operationland the inclusion of extra poles due to the lags

in transfer from the environment. (See equation 3.50) This is too complex

for normal engineering applications.

In the normal root locus methods extra (cascaded) poles and zeros

are readily accounted for)but in a p-plane plot which carries its information,

not as the locations of a finite number of poles and zerost but as the

contours of gain and phase with no locatable singularities,all extra poles

and zeros must be added in by modifying the whole pattern. Thus the

insertion of one pole (due to a controller lag, say) into the plots 3.4 or

3.5 would distort the whole pattern and its effect in amplitude and phase

would have to be added in all over the planes and the usual methods which

allow root locii to be estimated from asymptotes and similar techniques do

125.

not apply.

These difficulties can be partially overcome if instead of the

distributed parameter system a lumped parameter system is used as a model;

In particular the flow and diffusion process can be replaced by the "ncrit

t,

model of Chapter 4 (Section 3). This results in rational polynomial

expressions for the transfer functions and the standard techniques apply.

The p-plane plot for the ncrit model of a flow and diffusion system is

plotted on fig. 3.5. However; the method again becomes impracticable when

distributed forcing has to be considered. The transfer function has to be

evaluated as the summation of the transfer functions through all possible

paths from control input to plant output.

6.4 Three Term Controllers

One of the basic methods of setting up controllers for industrial

processes is to regulate each unit operation or section of a system with

a standard controller. The simplest form is just a proportional controller.

The control actionl which it demands from the actuators of valves and the like,

is just a multiple of the error signal whichit measuresp orl of its input.

More complex devices have a 'reset' or 'integral' action. The objective of

this is to keep on making changes in control demand so long as there is an

error present. Ultimately when the error has disappeared no further changes

are made. This integral or reset demand is added to the proportional demand.

For practical reasons it is not possible to have perfect integrators so the

behaviour is approximated by a (very) long time constant lag, arranged so

that the low frequency gain of the controller is very hi gh. At higher

frequencies the proportional signal dominates.

126.

A further modification is to have a 'derivative' or 'rate' signal

added in so that the controller predicts what is happening and takes

appropriate action. Compare this with the idea of velocity feedback in

servomechanisms. Since perfect differentiation is not possible in

practical systemso the effect is approximated by a lead network (or its

equivalent).

The overall result is a controller which gives high gain with phase

lag at low frequencies for 'reset', and moderate and possibly high gain

with phase advance at medium and high frequencies. The relative values of

the frequencies must be taken in the context of the system under investiga-

tion. The operation of such devices is dealt with in "Handbook of

Automation Computation and Control," by Grabbe, Ramo and Woolridge (6.2).

The "three-term controller", having the three actions proportional,

reset and derivative, can be simulated by a device with the transfer

function:

(1 + pTi) (1 + ppT2) (R3 + R

4)

G(p) = (6.1)

(Ri + R2) (1 +o(pTi) + pT2

where T1 = R1

C1 T

2 R3C3

(6.2)

and R2

= R4

(6.3)

(R1

R2) (R

3 + R

4)

This is the transfer function of the analogue computer device shown

in fig. 6.1(a). The corresponding Bode Plot shown in the form of the

asymptote outline is sketched in fig. 6.1(b).

The effect of applying a three term, and other simple controllers, to

127.

a distributed parameter system is investigated below by means of an analogue

computer simulation. Two slightly different systems are considered.

Both are based on the heat exchanger, the analogue of which is developed in

Appendix II.

Disturbances enter the system in the form of variations in fluid

temperature at input. The objective is to remove these variations from

the output stream. Regulation is achieved by controlling the temperature

in the shell which surrounds the tubes. The difference between the two

systems is in the controller. One has a lag, with a time constant about

one third of the residence time for the flowing fluid, between the control

demand and the change in shell temperature and the other does not. The

system (lag included) is shown in fig. 6.2a and its analogue in 6.2b.

The controller analogue to be used for the system can represent the

range from a simple proportional controller to a full three term controller,

and each type is considered.

Without control at all)the response to disturbances of low frequency

is to transport them through the system almost unchanged. As the frequency

is increased the response decreases. See figs. 6.3a and 6.5a. The control

action to be used to suppress this transfer is applied to the system in a

distributed manner and produces the responses shown in fig. 6.6. It is to

be noted that these responses exhibit a slightly oscillatory nature. This

is the last vestige of the highly oscillatory characteristic associated with

distributed forcing and flow-only systems when it has been modified by

dispersion. (Compare with fig. 3.8) As indicated earlier this is not a

resonance characteristic in the usual electrical engineering sense but is

due to the relationship between the residence time and the period of the

128.

of the external forcing. Thus a control signal applied at 50 c/s to the

analogue used here can be expected to produce a slight increment in the

response of the outputi because each half period corresponds to the transit

time through the system (10 m.secs.). At 100 c/s the response is less

because the transit time corresponds to a whole period andgains in the

first half are partially offset by losses in the second half. (The graphs

have retained the frequency scaling of the analogue, as giving a more

convenient range. 10 m.secs. on the

the original system.)

There is a thermal lag in the

analogue is equivalent to 2 secs. in

transfer from the shell to the tube

walls and from the walls to the moving fluid. On this latter is superim-

posed the effects of the distributed nature of the forcing. Thus the heat

exchanger itself can contribute a maximum of 180° of phase shift with the

possibility of producing more than 180° if there is very little dispersion,

so that the fluctuations in phase angle are large. In this case this does

not happen (See fig. 6.6). When there is another lag in the transfer, due

for example to the time taken to change the pressure and hence the tempera-

ture of the conducing stream in the shall, then the maximum phase shift

increases to 270° with the possibility of producing unstable behaviour.

This also is shown in fig. 6.6.

When a simple proportional controller is applied to this potentially

unstable system themaximum gain that can be applied is restricted by the

requirement for stability. In fig. 6.3(b) the resulting response to input

disturbances shows that at low frequencies the best reduction is around

15db and that near 60 c/s there is a pronounced increase in response. Note

the open loop control system has 180° of phase shift at 65 c/s. The low

129.

frequency response can be improved by including some integral or reset

action (fig. 6.3c) but this does not make any difference at the higher

frequencies.

When the 'rate' or 'derivative' action is available an overall

improvement can be made, not only at the high frequencies but also at low

frequencies, fig. 6.3d. The reason is that with the stabilizing effect of

the rate feedback a higher proportional gain can be used, which would

otherwise make the system unstable, and the corresponding decrease in error

response occurs at all frequencies. The 'rate' term also, in this case,

produces an extra improvement near 40 c/s, and a considerable reduction in

the peak, resonant, response near 60 c/s.

The other system, without the lag in the control loop, is easier to

control and the results of applying a three term controller to this are seen

in fig. 6.5b. (Note that all measurements are made for linear operation.)

The choice of parameters for the three term controller can be

approached either through the frequency response or through the transient

response. The frequency response method deals with the control loop

behaviour and because of the difference between the control path and the

disturbance path through the system this does not give direct information

about the final system behaviour. The frequency response may be difficult

to calculate or measure but the step response is relatively simple to

observe. It gives1 immediately,information on both steady state and resonant

behaviour. This is used in the examples above where the three term control-

lers are set to give maximum useful reset gain, and maximum proportional gain

consistent with stability, for which the derivative term is arranged to give

one overshoot and one small undershoot on the step response. The frequency

130.

response of the controller used to produce the response of fig. 6.3d is

shown in fig. 6.4.


The usual methods of controlling industrial processes are based on

fairly standard techniques and equipment. The basic methods are frequency

response methods; the equipmentithree term controllers.

Frequency responses can be calculated)but not readily enough for most

industrial uses. Systems can be set up without calculation by adjustment

of the standard controllers to give satisfactory transient and steady state

response. The experiments above confirm that this method is the most

suitable for practical implementation.

There is little advantage to be gained from the use of the more

powerful root-locus methods because the two dimensional field of complex

variables involves an excessive amount of calculation andfor distributed

parameter systems is not suitable for predicting the effects of system

changes because the usual poles and zeros do not appear.

In the example investigated above the use of a three term controller

instead of a two term controller produced a significant improvement. The

measurement used for control was also the measurement used for the

assessment of performance and the derivative term corresponds to an

approximation to another term in a more adequate description of the state of

the system. Because of the distributed nature of the process useful

information may be available elsewhere in the system eliminating the need

to try to calculate it by differentiation of the output. This is to be

investigated in later chapters.

Another difficulty is that only a finite gain can be achieved in the

131.

'reset' action,so that there will always have to be some error left over,

uncompensated, even if it is known that there is a sufficient range of

control action to eliminate it completely. This again needs further

investigation.

The conventional control methods provide sound and practical methods

for industrial applications and a basis for comparison with other schemes.

The references in Chapter XI include some reported results of control

investigations for industrial processes (6.9 to 6.14).

a) An analogue computer model of a three term controller.

..h 4' 4' ill; 4 ler 110)

b) Outline of idealised frequency response for above system.

Fig.6.1. Analogue computim version of three term

controller.

132-

W &I/Tr-LOW

,SHOLL

MALI.

PL

siptire,A4 PIRPoSt An/N.040i 1>'. peAqr erte,wimere 6s-ce APA

Lino/T00.2

a) Heat exchanger system to be modelled.

- 7 C one reac

' -4-01- 1 Rift .1. L— — — t— 0- ..• 1 -- --

b) Analogue computer model. (Limiter prevents exoesstve demands on low voltage speoial purpose analogue, allows investigation of saturated controller.

Fig.6.2. Heat exchanger system and its analogue

representation.

133

-----'-

a.UncontrolIed response to input disturbances. b

,Response with proportional control.

o. Response with proportional and integral control.

d.Response with Three—term controller.

e.State measure control, (single probe).

(

\

)

d 41 u e\

•

0 °1

Fig.6.3. Effeots of oontrol with lag in control loop, frequenoy response — magnitude, analogue measured results.

3 k--

0 02

O

07

41)

4.

so

r3c-

0 O to

•

w v, o 1.- * m .1., , ' X . to, -

a

E

Fig.6.4. Frequency response - magnitude - of

• three-term controller used for results

shown in fig.6.3d.

0 0

O

O 0,

O

O

O

0 rt

O

CO

r36

//' /

N\

‘‘.\\ N.N„.

0 .

-....„...„,...

u %....,...., .~

a.Uncontrolled response to input disturbances.

b.Response with three-term controller.

c.Response with state- measure control, (single probe).

•

vs a a IN r Nit ..—... ftv s A Nn 4 V ta M

Fig.6.5. Effects of control without lag in control loop. Frequency response - magnitude - analogue measured results.

0 0

0.1

0

0 41-

0

0 .4

0

le

t•L

O4Z0, O IA 12

cl .. x

( a.Without

b. With

. 64/7/

control

control lag

lag loop

loop

in

in

1

o 1

li • 15, g 41 4: u. M N 1

P

Fig.6.6. Frequency response — Amplitude and Phase — Control

demand to fluid outflow.

0

0

O

0

O fd

0

o

M

438.

Chapter VII

VARIATIONAL AND ASSOCIATED METHODS

7.1 Introduction

In the application of the calculus of variations to control engineer-

ing theory the problem is to make a system perform as well as possible, by

choosing the control action to be applied to it during the time of operation.

The system is described by a set of ordinary differential equations. The

performance is measured by an integral taken over the time of operation.

This integral may, for example, represent the total profit to be made by

running the system with allowance made for the cost of the control action

taken.

This problem can be tackled by at least three approaches which

however are all closely related. The classical calculus of variations deals

with it in the problem of Bolza (Bliss 7.3). A more modern approach which

puts the problem directly in control engineering terms is due to the Russian

school under L.S. Pontryagin (7.31, 7.32). A further method which is

orientated towards machine computation comes from the use of Dynamic

Programming (Bellman 7.16). The literature on these topics is vast, and

mathematical treatments are available for dealing with virtually every type

of engineering problem that may arise, There is however a gap only lightly

covered, between the symbolic mathematical demonstrations of possible tech-

niques and their actual application to realistic engineering problems. The

trouble arises in the difficulties of computation.

The methods of Pontryagin and the classical calculus of variations

both lead to two point boundary value problems. furthermore the number of

139.

variables is double that in the state vector for the system being investi-

gated. In any solution of a two point boundary value problem by analogue

or digital computation it is necessary to choose boundary values at one end

of the range of independent variables so that when the computer integrates

the equation, starting at that end, the solution meets the required

boundary conditions at the other end. This in itself presents significant

search problems when a system with several elements in its state vector is

being investigated. Doubling the number of variables makes it worset and

that is not all. The double set of equations is such that, if the original

system is stable, in the sense that any disturbances would naturally die

away, then the extra part introduced by the variational methods is unstable,

producing ever increasing responses to any initial disturbance. The result

of this is that whole set of equations, for the double number of variables,

which have to be integrated together, exhibits unstable behaviour. A small

change in initial conditions can produce a very large change in final

conditions. If the operating time for the system is long in comparison

with its own natural time constants this effect can make computation

impossible because changes of the same order of magnitude as the rounding

errors in a digital computer made in the initial conditions can produce

changes in final conditions big enough to cause machine overflow. Rever-

sing the direction of integration by running the problem backwards in time

presents no solution; the roles of the two parts of the double size system

are reversed and the whole is unstable when run the other way. The problem

is reduced if the operating time is short in comparison with the natural long

time constants because the computation does not run for enough to get out

of hand. The difficulty is that many processes are expected to run for a

140.

long time, certainly for a good deal longer than the time required for

control demands to have had some effect. Any process which runs continu-

ously, and therefore virtually for infinite time as far as the theory goes,

clearly needs a different approach. The Dynamic Programming method avoids

the difficulties of two point boundary value problems but by treating all

possible events together runs into trouble with the amount of computer

storage space and computing time that is needed. Since the usual Dynamic

Programming solution starts at the end time and runs the problem backwards

there is a clear problem when the end time becomes a very long way off, as,

for example, in a system supposed to run continuously.

These approaches, difficult as they turn out to be, do however yield

useful results. Investigations with simple systems have shown the effects

of changing the performance criterion. In general terms the results show

that since all performance criteria make it inadvisable to operate far

from some optimum condition, explicitly stated or implicit in the performance

criterion, the resulting solutions make the system move close to the optimum

fairly quickly, and unless it is necessary to move away to meet some other

boundary condition the system is made to stay there. As a result performa-

nce criteria of widely different forms will produce similar results if the

same optimum conditions are specified. Another result is that for a certain

restricted, but significant, class of problems the control action to be

taken, which is produced usually as a function of the independant (time)

variable, can also be derived as a direct function of the state of the system.

This means that instead of having a prearranged control action to be applied

the same results can be achieved with a feedback device. The final system

then exhibits the desired goal-seeking behaviour whereby, for any state at

141.

any time, it will continue to keep going on a course which is optimal from

then onwards, always heading for the optimum state.

In the realm of distributed parameter systems there is very little

available control theory. Pontryagin's method has been applied to systems

described by integral equations by Butkoyskii (7.25 to 7.28). The approach

through partial differential equations can be developed, as is shown here,

by means of the classical calculus of variations and the theory pertaining

to the Dirichlet integral. The same or similar techniques can also be

applied to systems described by integral equations. This field of

investigation of partial differential equations also yields some approxi-

thc mation methods for solution,which reverse philosophy of the approach of the

variational methods used in the theory for the control of lumped parameter

systems mentioned above.

Since any answer which is suitable for engineering use will almost

certainly be an approximate answer in some sense or other the so-called

"Sub-Optimal" control methods must be considered. The usual implication

of this term is that the length of time over which the optimization is to

be carried out is reduced. The reduction is like that in a chess game where

instead of considering the consequences of any move right through to the end

of the game the move is assessed on its affect, say, three or four. moves

ahead. This change makes the computation problems considerably less acute.

By considering the reduction of running time in the context of cost functions,

Liapunov functions, metrics in the state space and feedback control the

whole can be fitted into one pattern with variational problems at one end

and control as a function of system state at the other.

142.

7.2 Formulation of the Lumped Parameter Problem

The problem of devising an optimal control policy for a system

described by a set of ordinary differential equations, a lumped parameter

system, is relevant not only to the control of servomechanisms and the like,

but also to the control of distributed p'Arameter systems when it becomes

necessary to replace the partial differential equation, or equations, by a

set of ordinary differential equations.

The system is described by a set of ordinary differential equations

x. = f.(x u t) i = 1.... n. (7.1)

where the terms x , u are used to cover the complete sets of variables in

the description of the system by its state vector (x, an n-vector) and the

control vector (u, an r-vector).

The performance is to be measured by integrating a function (L) of

state and control (and possibly of time) over the period of operation of

the system, denoted here by 0 to T. If the functional which results is the

cost of operation then the problem is to minimize

,T

L(x , u , t) dt . (7.2)

o In any practical system there will be restrictions or limits on the

available range of the control such as

- b k(u3. < + b (7.3)

and similar restrictions on the state variables. These restrictions are

dealt with in the various mathematical treatments by modifications to the

simple formal proofs for the Euler-Lagrange equations or by introducing

extra variables to take up the slack between some variable or function and

143.

its limiting value. (This method is also a key step in linear programming.)

It is not intended to deal in detail with either the proofs or restrictions

here,but solely to consider the solution methods and the results.

The performance criterion is usually defined in a slightly different

way for the Pontryagin approach. The function to be optimized is the value

of one of the variables at the end time, T, say, xo(T). If, however the

behaviour of xo is defined by

xo L(x , u , t) , xo(o) = o

then

0 and the system is exactly the same again. In the Dynamic Programming

method both types of performance measure can be used, depending on

convenience or the type of problem.

The time range of interest can be fixed in one of these ways. It

may be a variable, such as in the "minimum-tithe" problem, where it is

desired to bring a system to some specified state in the shortest possible

time, or may vary depending on the state reached by the system,so that the

end time may be a function of the end state. It may be fixed, as in a

batch process (chemical engineering) for which the period of operation is

predetermined, or any process in which behaviour after a certain time is of

no further interest. Finally,it may be infinite. This means that the

process is expected to operate for everl or)in more practical terms forsuch

a long time that changes in system state can always be achieved in periods

of time which are short in comparison with the operating time. It means

that the operating time is much larger than the system time constants, This

r x (T) =

f , u, t) dt

144.

is the case in many practical engineering problems and leads to some

interesting conclusions. The integrand of the performance integral 7.2

has its own steady state optimum value. Continuous operation in this state

is the best that the system can do. If there are no disturbances after the

initial deviations then the best overall performance will be achieved if the

system takes up the optimum steady state and stays there. This means that

the resulting system must be stable in the Routh-Hurwitz sense,and have as

the point to which it ultimately settles,the steady state optimum. The

infinite time case lays particular emphasis on stability, steady state

optimization and less emphasis on the behaviour, and therefore on the

performance criterion, away from steady state.

7.3 Solution Methods

The problem formulated in the previous section can be dealt with by

the classical calculus of variations and by the methods of Pontryagin and

in a different way by Dynamic Programming.

Since mathematical proofs at any level of complexity and rigour,

abound, only the procedures are treated here.

In the classical calculus of Variations the problem is solved as

follows.

Form the function L' by introducing the Lagrangian multipliers, i,:

L' A

L +)\ - f. ) i i=1 (7.6)

and write down the Euler-Lagrange equations for each variable. The

Euler-Lagrange equations have the form

d taL' I ax -d-t-

ax o (7.7)

T

(7.11) aL 8x.

i=1 axi

145.

where x is one of the variables. These do not result in partial differen-

tial equations because the function L' is expressed algebraically in terms

of the variables so the partial differentiations can be carried through.

Since there are now three sets of variables there are three sets of E - L

equations which are:

d ;Tci 72-1-F

Liz d (aL,) _

1

aL, d ,6L i 511 - kz; =

o i = 1 n.

= 1 n.

j = 1 n.

The variational method also ensures that there are sufficient boundary

values at each end to prevent the problem being run as an initial condition

problem. The overriding criterion is that the summation at the end points

where 8x. is the variation 1

-0

in x., should be zero. This particular

formulation of the problem results in the relationship:-

aL = x . aci 1 (7.12)

Since8x.mustbezerowheneverx.1 is fixed, the boundary condition

requiresX.tobezerowhenx.is free and X.1 to be free when x.1 is fixed. 1

There is a more complex version of the boundary condition for systems with :.1,1"ernt

free end points and varying time of operation - transversality A

condition (Elsgolc 7.5) - but this is not relevant here. When the partial

differential operations are carried out on the function L',the three sets of

146.

Euler-Lagrange equations become:

aL _ axi

of. (x. ox.. --a) 0 = 1 • e • n,

j=1

(7.13)

i = 1 n, (7.14)

aL au. = o j = 1 r. (7.15)

The first set are equations for the "adjoint" system of variables,

the i. The second set are a repeat of the original system equations, and

the third set are conditions for minimizing L with respect to the control,

u. Since this third set givesu as a function of X and possibly x the whole

system has a closed loop structure in which x influences X through the terms

aL (Tc. ), and X influences x through the control u. It is unfortunate that

this closed loop structure does not yield a feedback control scheme because

the adjoint system (X) is unstable if the original (x) system is stable.

It is worth noting that if the total system did not have unstable

characteristics then it would not be possible to meet arbitrary terminal

conditions. For instance a terminal condition far from the stable optimum

condition at a large time, T, could not be met by a system which started near

the optimum and continued to settle down even closer. The process has

produced a two point boundary value problem in double the original number of

variables with a set of equations which show unstable characteristics.

The method of Pontryagin follows similar lines. Bearing in mind that

the performance criterion is xo(T), with the system described by the

equations

• = f.(x 3 u t). = 1 es" n, (7.16)

form the function

n H = pi xi

i=o

The equations to be solved are

147.

(7.17)

• aH pi = ax.

1

ax x. = Pi.

i = o n (7.18)

i = o n. (7.19)

and a further condition: "Minimize H with respect to u."

These sets of equations 7.18 and 7.19 are exactly equivalent to the

Euler-Lagrange equations 7.8 and 7.9 and the "Minimize H" condition corre-

sponds to equation 7.10. The boundary conditions for the Pontryagin

equations have exactly the same form as in the classical problem with the

addition that a boundary condition on pc) arises at T:

po(T) = 1 (7.20)

The two methods both yield the same equations. They can be demon-

strated to be related by a simple transformation (McCann 7.14). Simple

examples of the use of these techniques are given in Appendix VI.

The Dynamic Programming method does not rely on solving equations but

on a simple "principle of optimality"; quoted here from R. Bellman's book

"Dynamic Programming" (7.16),

"An optimal policy has the property that whatever the initial state

and initial decision areithe remaining decision must constitute an optimal

policy with regard to the state resulting from the first decision."

The computational procedure is based on this. Consider the system to

be at a particular end point (t = T), which may be the only allowable end

1484

point in some problems. A cost can be associated with each end point if

more than one is allowable. Now consider the system at an instant of time

a short period immediately prior to the end. For each possible condition

(state) there are a number of possible actions which can be taken over the

next' period. Each action will itself involve some cost. Each action

will result in the system finishing at some end point. Choose the action

which results in the lowest possible total cost allowing for the cost

incurred over the short period and the cost associated with the end point

that is reached. Do this for all possible states. Associate a lowest

total cost of completion with each state. Now consider the system at a

slightly earlier time. Evaluate the cost of moving over the period of time

ahead to a state for which the cost of completion is already known. Choose,

for each state, the action which results in the lowest total cost. This

process is then repeated until the whole time range of interest is covered

or the machine computing and memory space runs out. The whole process is

covered by a functional equation which describes the idea of minimizing the

sum of the cost for completion and the cost of the next small bit of the

trajectory. Once again this can be shown to be exactly equivalent to the

other methods. in particular the equivalence between the dynamic program,-

ing method and the Pontryagin method has been demonstrated by Rozonoer (7.33).

The result of applying Dynamic Programming methods is a set of trajectories

through the state space all heading for the only allowable end point,or for

points as near as can be reached in the available time. Naturally any

restrictions on the control variables or the state variables can readily be

incorporated in the programme for the computer.

The amount of computation required can become quite enormous with

149.

even quite small systems. In the case of the variational methods the problem

of meeting boundary values, even supposing it soluble, involves a search

through a space of as many variables as there are elements in the original

state vector, for if all the values of the elements of x(o) are known then

all the values of X(o) or its equivalent p(o) have to be found to meet any

given condition at T. Each new set of values to be tried requiresa complete

integration of the double size system right through from end to end to test

it. The alternative method of running the whole problem backwards in time

so that the computation starts from the desired end state yields a new

optimal trajectory for each set of values of the adjoint (p or X) system/

and while there is no guarantee that the initial condition to which this

trajectory is found to correspond will be the one of particular interest there

is always the possibility that it could be stored for use when other initial

conditions prevailed.

The amount of computation required for dynamic programming is large

because each possible starting point in an n-space has to be considered at

each stage of the computation. If the ranges of the variables are quantized

into only 10 distinct levels and there are only 3 variables this represents

1000 storage spaces to store the cost function alone for each possible

starting point. If this has to be done for each time step/the storage space

requirements can soon become prohibitive on all except the largest machines

for the simplest problems.

In the calculus of variations the answer which results is a trajectory

through the state space in which the elements of the state vector appear as

functions of time (the independent variable). Likewise the control action

for this optimal trajectory is given as a function of time. There is a new

150.

optimal trajectory for every possible starting point. In the Dynamic

Programming solution the cost function is stored as a function of both time

and state variables. Similarly the corresponding control action is stored

as a function of both. By this means all possible trajectories are covered.

The complexity of the calculus of variations solution to the problem

may depend heavily on the form of performance criterion used, because it

forms a generating function for many of the terms in the differential equa-

tions and itself has to be minimized with respect to the control variables

at every instant of time. In the dynamic programming case it is important

because it has to be evaluated at every point in the state space at every

time instant for every trial in the process of finding the best action at

that point. In view of the influence it has on the solutions to the

mathematical version of the problem the choice is bound to be influenced by

this. The choice will also affect the behaviour of the resulting system/

and since the object is to solve an engineering or economic problem the

choice is important from this point of view.

Before the calculus of variations and the like were used for control

theory certain simple performance criteria had become established. Generally

originating from servomechanism analysis/ they were either specified in terms

of the step responses or frequency responses, based on concepts of second

order systems. The usual measures were; percentage overshoot and rise time

for step responses, maximum amplification factor and desirable ratios between

resonant frequency and damping factor when considering frequency responses

(usually Nyquist plots and root locus plots respectively). All these

measures appear in the standard texts on servomechanism analysis (6.1 to 6.7).

They are all single numbers which are readily measured or estimated, all

151.

intended to characterize the system. When other methods of system design

became available there was a period in which various measures of performance

such as "integral of (product of) time and absolute error" and "integral of

error squared" were put forward and compared. (Westcott 7.40, Shultz and

Rideout 7.39). The only basis of comparison or evalution being the

proximity of the resulting step responses to the responses favoured and

produced by the simpler measures. The introduction of the variational

methods (Gould and Kipiniak, 7.8, Pontryagin, 7.31) made more relevant

criteria, such as profit and costs, available for design, with the big

problem for many processes that there was, and still is, no satisfactory

way or sufficient information to establish financial performance criteria.

Exceptions to this are problems like minimum fuel requirements for rocket

trajectories. The situation is; that while it may be possible to specify

optimum operating conditionsl the means for ascribing a cost to the actual

deviations from this may be much more difficult. Since every performance

criterion used in a variational problem must specify implicitly some steady

state optimum condition,which is that state which optimizes the integrand

in the integral measure (7.2) subject to the condition;

x = f = o , (7.21)

and since many industrial and other processes are expected to run continu-

ously, presumably near this optimum state, the way in which simple,

mathematically usable criteria make systems return to a particular state is

a useful guide to use in setting up mathematical problems to give sensible

engineering answers. The aspect of modifying the mathematical formulation

for ease of computation while retaining the same type of engineering answer

152.

appears in the work of Fuller (7.35, 7.36), and Westcott, Florentin and

Pearson (7.41).

Fuller was comparing the minimum. time solution, which has as its

performance criterion

M ( Error )odt ,

J

with the solutions produced by other criteria of the form

Mx =

( Error )Xdt ,

which includes the minimum mean square error case mentioned above, though

his controller structure was different. His conclusion was,

"More generally, numerical computation shows that the settling time

controller gives a very good approximation to the minimum MX' whatever the

value of X."

His results show the worst deviation between MX computed for the

settling time controller and the best Mx to be around 0.35% for X •= 2.

Westcott, Florentin and Pearson, in investigating various ways of

solving similar problems, of this same type concluded;

"A number of different possible techniques of approximation have been

shown with appropriate examples.

the mathematical setting up of the

problem can often be varied so as to make the computation easier, whilst

still giving a satisfactory physical solution...

The general conclusion is that overall behaviour is not critically

dependent on the performance criterion so long as it ensures that the

larger deviations are costed more heavily than the smaller ones. The inte-

grand of the performance criterion must be a metric in the state space for

153.

the steady state problem. The most simple one to use for mathematical

convenience is a simple quadratic measure. Essentially this is the square

of the distance in the state space, though some directions may be weighted

more heavily than others. Such a performance criterion is represented by

the function:

xT A x (7.22)

where A is a positive definite matrix and the minimum value of the function

occurs at

(7.23)

When solving the variational problems it is necessary to have some

restriction or costing of the control, and a similar function:-

uT B u , (7.24)

may be used. This form of performance measure is often used as an

approximation to any curved surface in the region of a stationary point,

and is therefore particularly useful if the process being optimized is

designed to stay in a small region near the optimum)as is the case for a

continuously operating system. (See Section 7.2).

7.4 Goal-Seeking Behaviour

The results of variational approaches to the optimal trajectory

problems are by their very nature functions of the independant variable

in control engineering this is the time variable. There are, however, two

fundamentally different structures for the family of trajectories which forms

a set of solutions to the problem. These arise from the two Choices Of

starting point for the solution of the boundary value problem.

Consider that the objective is to reach some point x(T) having

154.

started from x(o). If the Euler-Lagrange equations (7.8, 7.9, 7.10) are

integrated (numerically) in the direction of positive t, taking x(o) as

initial conditions and searching through the domain of h(o) or E(o) to

find the right conditions for reaching x(T) then the family of trajectories,

produced in the search fanout in the state space from x(o),as indicated

diagrammatically in fig. 7.1(a). Only one of them, always supposing it

can be found, reaches x(T). The alternative method is to integrate the

equations in the direction of negative t, starting at x(T) and varying R(T)

or A(T) to find the trajectory which reaches x(o). The resulting set of

trajectories now all head for x(T), even though, as before only one meets

both boundary conditions. This structure is illustrated by fig. 7.1(b).

The same structure is produced by the dynamic programming method, because

this too develops the solutions from the end point backwards. Another

version of this behaviour is produced by the infinite time problem in which

the system must ultimately settle down to a steady state optimum regardless

of starting point, but the solution cannot be found by integrating backwards

from T =0. The pattern is illustrated in fig. 7.1(c).

There is an important difference in terms of the use to which they

can be put between the two patterns of behaviour. The family of trajec-

tories which fan out from the end point gives the system which follows them

(in the normal direction of positive time) a built in goal-seeking behaviour.

Regardless of the state of the system or the time left to go before T,

following the unique trajectory through the point in the state space - time

continuum will always result in reaching the desired end point. The

family of solutions radiating from the initial point x(o) are of no value

in this respect. It is thus apparent that if, as is the case in Dynamic

155.

Programming, the results of all the investigations are storedI then the

trajectory to be followed, or the corresponding control action, can be read

off at any time as a function of system states instead of following a defined

sequence or function of time which is fixed once and for all by the initial

conditions. If the problem is perfectly deterministic it makes no differ-

ence in principle which method of generating the control is employed; both

produce the same results. But no practical system is perfectly described

by the equations used and completely free from noise or external unpredicta-

ble disturbances. The advantages thus show up in the engineering case where

the goal-seeking behaviour can be turned to good effect by using the recor-

ded results to define a feedback system. It is still necessary to take

account of the time-left-to-go in defining the optimum controli so that the

information needed for control decisions is the state of the system and the

time. However when the time remaining is infinite, the control action

depends only on system state and although the problem cannot be solved by

integrating backwards from the end time,the inherent stability of the

solution provides a means of establishing the direct relationship between

control and state, as is shown in the following sections.

7.5 Analytical Design of Optimum Feedback Controller

The relationship between the state of a system and the best control

action to be applied to it at any time can be developed analytically for a

restricted class of systems. The system must be continuously operating,

linear, (with constant coefficients in the ordinary differential equations)

and the performance criterion must be the integral of a quadratic function

(7.23, 7.24). This case, though quite restrictive, is of interest because

it represents a commonly used description of a system near its optimum

156.

where the cost function is treated as quadratic and the system linearized.

The steps to be followed are; firstly to establish the relevant

Euler-Lagrange equations; secondly to transform them by a Laplace transfor-

mation which introduces the initial conditions both for x(o) and X(o);and

finally manipulate the Laplace transform solution to give a condition that

must be satisfied relating the initial X(o) to the initial x(o)i in order

that the whole system should be capable of reaching the desired final

optimal steady state.

The process is best illustrated by taking a very simple example.

Let the system be described by the equation

• 1 x = a11

x1 + b1u

and the object of the optimization is to

(7.25)

Minimize, by choosing u1, (C11x12 + d11u1

2) dt (7.26)

0

where c11 and d11 are positive constants.

The modified Lagrangian function (See Section 7.3) is:-

L1 =2 2

cllx1 dllul X1(;c1 a11x1 b1u1).

Write down the Euler-Lagrange equations (See Section 7.3)

2°11x1 all? -

3"1 =

b1u1

+ a11x1 - 1 =

(7.27)

(7.28)

(7.29)

2dilui - Alb/ 0 (7.30)

'Equations 7.28, 7.29 and 7.30 correspond to equations 7.8, 7.9 and

7.10 respectively.

The variable u can be eliminated temporarily, for convenience and

157.

the equations transformed (Laplade) to give

- [x1(0)1

?Li( 0 )1

rp - a11

- f11

x1

- 2c11 P all. 01

(7.31)

where f11

2 b1 2d11

and x1 and Al are functions of P.

The solution for x (p) is

fll X1(°) hl(o) (P+all) x (p) =

(p-all) (174-a11) - 2f11c11

For x1(t) to reach a steady value as t-ip ,

(7.32)

the term,in the

partial fraction expansion for x1(p)which corresponds to an unstable (right

half plane) pole must be zero. The partial fraction expansion depends on

the factorization:-

X1(0))j xi(o) p - - fll x1 (7.33)(o) x1(p) =

and the condition is:

X1 (o )

-all - fll ft37j a112+2flion (7.34)

(This is also equivalent to the condition that in the right hand side of

equation 7.33 the term corresponding to the right half plane pole should be

cancelled out.)

This gives the relationship:-

112+2f11c11.1 + \ 2+2f c

Nall 11 11 [p

f11 al(o) = xl(° ) -all -j 2

all 2f11c11 (7.35)

153.

But f11 X

1(o) is the same as b1u1, having been substituted in equation 7.31

to eliminate the variable ul.

The relationship gives the control in terms of the system state at

the initial time, but this is valid for any instant of time, since any

time can be taken for the start of a process which goes on for ever.

The feedback controller must therefore make

b1u1

=

r-

-a11

-all

2+2f11

c11 (7.36)

and the whole system, feedback included becomes

2 -a11 -ra

11 + 2f11c11I x1

x1 = a11x1 + (7.37)

The structure of this is interesting.,because it can be seen that regardless

of the size or sign of aillits direct and normal effect is suppressed by

the controller and the resulting system always is stable because the term

2

11 + 2f11c11 (7.38)

2 b1 is always positive. (c11 positive, a11 positive, f11 also positive). 2d11

In the more general, multidimensional problem described in Appendix

VII there are just enough conditions produced by this technique to define

the maximum number of linearly independent control variables. Another point

which emerges is that just half the roots of the complete, X and x, system

are stable and half unstable. The procedure adopted above for a first

order system can readily be extended to a second order system and by evalu-

ating the control action in terms of the sums and products of the roots of

the complete system it is possible to evaluate the control action without

having to find the roots first. For higher order systems the whole process

can in principle be taken over into a machine-orientated form though the

2

159.

results are probably of more value for the structure that is demonstrated,as

follows:-'The optimum feedback controller for a linear (constant coefficients)

system, operating continuously, with an integral of error squared and

integral of control squared performance criterion is a linear constant

coefficients arrangement whereLn the control depends solely on system state, end suppresses natural behaviour to ensure that the resulting system is always

stable. (See also Chang :Jen-uei 7.29)

7.6 Variational Methods and Partial Differential Equations.

When a system is described by partial differential equations it is

possible to extend the theory normally associated with the Dirichlet

integral to more general integralsf and to introduce the equations for the

system as constraints in the same way as is done for systems described by

ordinary differential equations. The resulting procedure is thus

practically the same, but instead of ending up with a double size set of

ordinary differential equations the result is a double set of partial

differential equations. The procedure is shown here and the formal

mathematical justification, following the form of Lanczos on ordinary

differential equations, is demonstrated in Appendix VIII. A single

dependent variable is considered here i but the extension to multivariable

systems is immediate.

The system is described by the equations:.

g(7, Ur' UV 3) = (7.40)

where suffices denote partial differentiation; 0 is a control variable;U

is the system state, (using the notation of previous chapters for describing

distributed parameter systems).

160.

The performance criterion is that

L(u, u., ut, a) dxdt (7.40)

is to be minimized.

The solution procedure is as for the ordinary differential equation

systems:

Form the modified Lagrangian

L' = L + Ag (7.41)

Write down the Euler-Lagrange equations for each variable:

First the equation associated with the variable U,

aL.a at,' aL' _ ( ) (-- ) au ax au.2c at aUt

•

0 (7.42)

and similarly for the variable X, which yields once again the original

system equation

aL, _ _ g o (7.43)

and for the control variable 3:-

aL ' _ as -

0 or just to minimize L' with (7.44)

respect to 9 at all points in the domain D of the x,t space.

Once again the variational process yields boundary conditions as

needed but instead of having a summation to be made zero at the two ends of

the range of independent variables the requirement is that

aL 8U dt ' 3,11 dx) 0 (7.45) Oux

where P is the boundary of D and SU is the variation in U along the boundary.

161.

The summation condition has become an integral condition.

The steps above have been shown on the basis of having a system

described by a first order partial differential equation. A formulation

of a second order problem could be given by having a new variable for each

first derivative and thus having more first order equations. In any event

the process is the same,although it may be convenient to use the second

order version of the Euler-Lagrange equation

aL a , aL a al, a2 far., Za2

- —(— ) - + — + )+ a2

(PIJ ) = 0 (7.48) au ax au'lc at aut ax 261J U axat a

xt lat2 Utt xx

Once again, for control engineering purposes, these results are not

as useful as the concepts and ideas that go with them. The equations that

result from this process are no easier to solve than the original equations

used to describe the system. In fact, because of the inherent doubling of

the number of equations, and the difficulties of meeting the boundary

conditions they are more difficult. All the problems associated with the

multidimensional representation of the lumped parameter model would occur

again in increased complexity.

However, the whole philosophy of maximizing an integral by solving

associated differential equations is reversed by techniques used in solving

certain differential equations. Both the Laplace and Poisson equations and

more general equations of similar type arise from the minimization of inte-

grals - for example minimum strain energy for a deflected plate. Instead

of solving the equations to produce the minimum for the integral the Ritz

and Galerkin methods solve the equations by minimizing the integral

(Kantorovich and Koylov 5.3).

The same basic idea, modified to suit the control engineering problem?

162.

in the light of the concept of a continuously operating system being

governed by a feedback controller, leads to an adaptive control system as

is indicated in section 7.11.

7.7 Variational Methods and Integral Equations

Although integral equations are not widely used in engineering

representations of systems they form a means of system description which is

in one to one relationship with the differential equation representation.

For every differential equation description there is an equivalent integral

equation description. (See Refs. 7.21 to 7.24). For each differential

operator and its boundary conditions there is an equivalent Green's Function

used as the kernel in a integral equation description. For example the

ordinary impulse response for a system is a Green's function to be used in

the convolution integral representation of its behaviour. (See Refs. 7.21

to 7.24).

In the same way that the variational techniques can be applied to arphce

systems of differential equations they can also be to integral equations. A

The procedure to be used is exactly the same, and again results in three

groups of equations associated with; the system (state) variables; the

adjoint variables; and the control variables1respectively.

Suppose that the system is described by a set of equations: etl

xi(t) = x1(t) (xn(r) , u1(2) ur(q))dr, i = 1....n (7.49)

0

where the x.1 variables are the state variables, u. the control variables. J

The performance is measured by an integral taken over the range of time

from 0 to t1 such as:

xo = L (r, xi( T) xn(r), u1(Z) ur(z)), d (7.50)

163.

By applying variations to an assumed optimal trajectory, as described in

Appendix IX,Euler type equations are produced in the same way as before

giving:

Xi(t)

)

0

=

=

_ ax.

1

_ aus

X.(t). (Z,t) -x i

x, 11)1a T.

ax . X .(1.).—au (1- -01

dr,

d

i = 1

i = 1

S = 1

n,

n,

r.

(7.51)

(7.52)

(7.33)

[j=1

r t,

f.K*2(t,

0

This time there are no separate boundary conditions, these being included

in the Green's functions, Ki. These equations are all written for one

independent variable but there is no reason why the Green's function itself

should not be dependent on some other variable as well. This is the

situation that occurs when the impulse response of a system is given as a

function of time in the normal way and also depends on another (space)

variable such as the impulse responses produced as inversion5of the Laplace

transforms given at the end of Chapter III. The equivalence between this

approach through the forms of the classical calculus of variations and a

modern approach, through the forms of Pontryaginh principles, produced by

Butkovskii (7.25 to 7.28) is shown in the Appendix IX,together with a summary

of the Butkovskii method for more complex systems.

7.8 Sub-Optimal Policies

The optimal control policy for regulating a system is based on an

assessment made over the whole period of operation. Because of the

164.

difficulties of solving variational or related problems for long periods of

operation it is desirable to reduce the period with some loss of complete-

ness in the solution.

Two measures of time are of interest in discussing sub-optimal

policies. One is the period over which the calculation has been carried

out, the time 0 to T, and the other is the period for which the resulting

calculated best control action is actually followed. This latter may be

less than T but not greater, say, — where r a 1,

Consider a process operated by a computer which can measure system

state at any instant of time, t, and solve variational problems based on

the time left to go, (T - t), taking the system state x(t) as new initial

conditions. At time t = o the computer calculates u(t) for 0 .4: t

and applies u(o) to the system. The control u(t) is applied up till time

t =T If at this stage the measured system state may differ significantly

from the state anticipated from the calculations, then the computer can solve

the new problem based on x(—r) as initial conditions and for a time period of

length (T - r This is also equivalent to using a complete stored set of

results, as from the dynamic programming solution, and looking up a new

solution partway through.

If for one reason or another the period — is much less than T then

there is little advantage in solving the complete problem only to reject

most of it later. A sub-optimal policy may then be used. The best control

action is calculated only for a period, say — where s >1, and this control

action followed for all or part of this time when the process is repeated.

In the limit where the problem becomes one of minimizing only the

integrand of the performance integral (7.2) without taking any regard for

165.

the future behaviour of the system.

The transition of the parameter s from 1 to ,N=)describes the transition

from a complete variational problem to a simple optimization-of-a-function

problem. Various examples of the application of this sub-optimal control

concept show that it produces useful results. (Masanao Aoki 7.42).

The transition of the parameter r from 1 to describes the transition

from a completely pre-scheduled control action to a control action which is

an instantaneous function of system state. Since — must always be less

than or equal to T musing a sub-optimal control policy always makes the

resulting system closer to the feedback system, but does not necessarily

imply that the feedback controller designed on this basis will be the same

as the one evaluated analytically from the solution for s = 1.

This replacement of a problem specified with time as the independent

variable 6y the problem specified with state as the independent variable is

discussed in the next section in the context of methods based on Liapunov

functions and the like.

7.9 Cost Functions, Liapunov Functions and Metrics

In all the above discussions of optimal control the control action

depends on the state and the time left to go. In Section 7.5 the effect

of having infinitely long operating time allowed the analytic design of a

feedback controller based only on system state. However the analysis still

used the time as independent variable. If the problem could be formulated

with system state as independent variable then the control must necessarily

be produced as a function of state only.

The first step is to reformulate the cost function. Instead of using -r L dt

( 7 . 54

166.

the space variables are used to give

Pi do. (7.55)

where M is a function of x and u and do' represents a volume element in the

state space within the domain D.

To maximize this integral with respect to the control variables u

it is only necessary to maximize M(x,u). This is because no account is

taken of the normal dynamics.: of the system in .finking behaviour in one

part of the state space with behaviour in another. To make the system

behaviour relevant two possibilities arise. The first to make the

criterion of performance an integral taken along the trajectory:-

N(x,$) ds (7.56)

where S is the total length of the trajectory, s the distance from the end

point. To solve this problem it is necessary to represent the system

behaviour as parametric on S . Since the elements of the trajectory Ss

are related to the time elements St required to move along them by

Ss = St. ti t (7.57)

this amounts to nothing more than a change of variable from the original

time dependent variational problem. The second alternative is to make the

function M(x,u) into a measure of the cost of completing the trajectory.

Having done this, any improvement due to suitable choice of u automatically

allows for future costs. This principle can be developed into a method for

iterative control improvement.

Let the system be described by a set of ordinary differential equati@ns

x = f(x) (7.58)

167.

before any modifications to it are considered by way of changes in control

action. Any control already built in is included in this description, and

on this basis it is supposed that the system will return to a suitable

operating point without further modification. This is frequently the

situation in practical systems where a working but not necessarily optimum tnoct.f-ed

controller is already built in. Let the system be described by a set of

ordinary differential equations:

= -e (x , u)

(7.59)

where the vector u is the extra control action to be introduced, over and

above any that may already be present.

Let M(x) be the cost of completing the trajectory from x with the

control that is already available, i.e. for the system described by

x = f(x)

(7.6o)

Consider the effect of moving a short distance 5x in the state space.

At the end of the operation there is a change in the value of M(x) given by

8M = grad M. 8x (7.61)

If this step is taken without applying any extra control then

8M = grad M. f(x) 8t (7.62)

where St is the time to complete the small step. But if extra control is

introduced then the trajectory is modified near x and the change in M becomes

8Me, given by:-

8Me = grad M. f e (x,u) St . (7.63)

However this extra control cannot be introduced without cost which

will depend on x and u and be valued, over the period 6t,as

3.68.

C(u , x) St . (7.64)

It is possible that in a system in which control is already available

and usedlcertain choice of u may actually reduce the cost. Whatever the

situation however, if it is possible to modify or introduce control so that

the improvement in the cost of completing the trajectory, represented by

8Me - SM

(7.65)

outweighs the cost of the control, then there is a genuine improvement in

the cost of completing the trajectory from the point x, regardless of what

further improvements may be made later on. Thus the best new control action

u is defined by:—

Minimize, by choosing u, grad M. (4(x,u) - f(x) + C(u,x) (I

(7.66)

The scalar factor St is no longer relevant and the result gives u as a

direct function of x. This u(x) is not necessarily the absolute optimum but

the process can be repeated when this control has been incorporated into the

system description, giving an iterative process for control improvement.

This can be illustrated by a simple example which is also capable of

solution by the calculus of variations, which thus gives an absolute

standard of comparison.

The system is

x = - a x (7.67)

where a is a positive number.

The cost of completing a trajectory is evaluated from the measure

..110

00

where u is the control used, if any. To start with, no control is used and

this gives:

(x2 + c

2) dt (7.68)

169.

2 M(x) = 2a (7.69)

Following the procedure outlined above (the details are given in Appendix X)

the first estimate, u1, of the control action required is

U1 = lac • (7.70)

When this is incorporated into the system and the process repeated,

making due allowance for the costs already incurred with u1

a second

approximation, u2, is found for which

u2 2ac 1 1

2(11-2a2 c)

(7.71)

The same problem treated by the calculus of variations (See also

Section 7.5) gives the control as

j 1 uc x a - a2 - c (7.72)

Taking various values of c ("a" normalized to unity) the rapid

approach of the scaling coefficient in the feedback controller to the

absolute best can be demonstrated; as in the following brief table:

Value of c 0.1 1 10

1st Estimate of coefficient - 5.0 - .5 - .05 2nd Estimate of coefficient - 2.916 - .417 - .049 Absolute best value - 2.316 - .414 - .048

The first estimate of cost is the cost without control, which can

easily be improved on, so that the first estimate of control produces too

much control. When the new cost function is evaluated the standard is

somewhat more accurate and some of this excess control is reduced. The

control is modified at each application so that ultimately the possible

170.

improvements are being weighed against what is very close to the real

(optimal) cost function and little change is produced.

This procedure is satisfactory so long as the function M(x) can

be evaluated, but the difficulty of doing this, though not intractable even

for large systems, is nevertheless significant.

It is when considering methods of evaluating M(x) or approximations

or replacement functions that a wide range of control optimization problems

are seen to be related.

Consider the problem of evaluating M(x) starting at the end point

and working backwards (in time) from there. As the computation proceeds

the evaluated cost of completing any trajectory becomes bigger as the

calculation moves to points more and more distant. M(x) can be recorded

at each point as the calculation proceeds and the starting points of all the

different trajectories expand to fill the state space. If instead of

calculating M(x) for the system without the extra control the best control

is calculated at each stage then the result is the Dynamic Programming

solution to the problem. The whole process requires computation of the

same order as for Dynamic Programming when carried out on a computer. The

advantage lies in the case where the process can be carried through analytic-

ally. Since many practical systems already have some form of control built

in, which can be quite close to optimal, the result of one application of

the iteration process can produce results which are close enough for further

changes to be of no significant value.

A method that can be used avoids the difficulty of evaluating integrals

along unknown trajectories, which is the problem of the calculus of

variations) by using, in effect, successive approximations to the best

171.

trajectory, these approximations being the trajectories followed before

system modification. Suppose however that even this does not make the

problem soluble, because the natural trajectories themselves are too

complex or not known accurately enough. The function M(x) can be

approximated by an integral along an approximate or assumed trajectory.

Some allowance has to be made for the relationship between the assumed

trajectory and the actual,one) and this can be done by taking the velocity L:!,e

vector along the assumed trajectory to be projection of the actual velocity A

vector at any point.

The exact evaluation of the cost of completing a trajectory is

00

L(x, u, t) dt , (7.73)

`t)

where the integration is carried out along the trajectory, which satisfies

the system equations, from start to finish, and gives a value, M(x), of the

integral for each starting point, x . An alternative formulation is:

L(s) 7;7 —0

where again the integration is

• ds (7.74)

carried out along the trajectory but the

s, travelled along the whole problem is expressed in terms of the distance,

trajectory (towards the end point). Si is the total trajectory length and

v(s) the velocity along the trajectory. If an accurate trajectory is

known and followed for the integration then the velocity vector lies along

the trajectory. For an assumed trajectory v(s) is replaced by v(s).cos e,

where 8 is the angle between actual and assumed trajectories at that point.

This angle, or its cosine can be evaluated without producing the complete

solution for the actual trajectory. The integral becomes:

172.

sc'

L(s) v(s) cos e

ds. (7.75)

In the case of a linear system with integral quadratic performance

criterion as considered earlier the replacement of the actual trajectories

by radial lines from the end point (origin) to the starting point, x, leads

to the form

M(x) = (xT Bx) 1 I

2 (xT Ax) • x 1

where the system is described by

• x = A x

(7.76)

(7.77)

and the integrand of the cost function for any point, y, in the state space

is;

L(y) = yT By . (7.78)

The details of this are shown in Appendix XI.

In the very simple first order case this reduces to

2 M(x) =

2a

for the system described by

• x = - a x

(7.79)

(a positive) . (7.80)

This result is the same as the accurate case above (Eqn. 7.67) because the

one dimensional state space only allows of one trajectory form.

In more complex cases however it can be seen that the function M(x)

is a scalar multiple of the distance between the starting point and the

origin.

Because it is an approximate cost function there is no guarantee that

when used for control design it will produce genuine improvements but it is

one stage better than the arbitrarily chosen metric used to modify control

173.

by optimizing a function of state and control which takes no account of

future behaviour. This latter is equivalent to optimizing the integrand

of the cost function at each instant instead of operating on the integral.

It is represented in the scheme of suboptimal policies by the results of

having the period of optimization very short (or zero) in comparison with

the operating time.

Another type of function, used for control optimization, is a

Liapunov function for the system. (See 7.44, 7.43). In the sense that it

is only a metric in the state space it falls in the latter category but

because its use is bound to result in a stable system it is better than that.

The procedure is to modify the system so that the time rate of change of the

Liapunov function is made more negative than it was before. This does not

really allow account to be taken of the extra cost of control which may well

be expressed in quite different units of measure, but no doubts arise as to

system stability.

There are thus two limiting grades of control system optimization

with all shades of approximation between. The best approach takes accurate

account of all behaviour in the period of interest. The simplest methods

only deal with instantaneous behaviour. The intermediate methods

approximate the best either in considering shorter periods for calculation

(approximation in time) or, as in the method discussed above, by considering

simpler behaviour (approximation in state space).

7.10 Special Problems of Distributed Parameter Systems

All the underlying concepts of optimal control schemes go over to

the control of distributed parameter systems but only at the expense of

having systems, of large or of infinite dimensionality, corresponding to the

174.

description of the system by a large number of ordinary differential

equations or a few partial differential equations.

State space concepts are valid in a function space but yield little

of engineering value. The adequate, finite, approximation, for example;

the lumped parameter model of Chapter IV, often yields a state vector far

too large for machine computation of optimal processes, even by the

approximate methods of the previous section.

Even if the results of such an analysis could be produced by

computation the question as to whether the required control scheme could

be engineered would still remain.

The next chapter deals with representation of the system from two

points of view, the engineering aspect of instrumentation and -the optimal

control aspect of the behaviour in a much reduced (and therefore theoretic-

ally inadequate) state space. It appears that the needs of control schemes

and of system observation may conflict.

7.11 Adaptive Controllers

The variational methods of optimal control design have two major

disadvantages: Firstly the size of the computations involved and secondly

the structure of the results.

To find a single trajectory through the state space, between given

end points requires a search for initial or final conditions (depending on

the direction of integration) through an n-dimensional space. (n is the

dimensionality of the state vector). The equations have to be integrated

from end to end for each trial. They are of double dimensionality (2n).

The form of the resultsisbased on time as the independent variable, though

175.

the backward integration process and huge supplies of computer memory make

the transition to giving control as a function of state and time. Similar

quantities of effort are required for dynamic programming though the search

effort is directed to finding the best control at every instant of time for

every possible system state.

In the established theory of partial differential equations the

problem of minimizing a multidimensional integral - such as the general

performance criterion for the control of a distributed parameter system -

leads to a partial differential equation; for example, the Dirichlet

Integral which leads to the Laplace and Poisson equations. But instead of

solving the minimization problem by solving the partial differential

equations, the smoothing effect of the integration operation is employed to

advantage when the reverse process is employed. The integral is minimized

to solve the partial differential equations. (Rig's method and the method

of B.G. Galerkin Kantorovitch and Krylov, 5.3).

The solution is proposed as the sum of a set of functions, the

coefficients in the summation to be chosen to minimize the integral. The

choice of coefficients which does this defines the solution (approximate,

but always improvable) to the partial differential equation.

The philosophy of tackling the problem at the integral rather than

the partial differential equation end can be applied to control engineer-

ing problems, but not in quite the same form as the Ritz-Galerkin method.

It leads, as explained below, to the familiar concept of an adaptive

controller by choice of control. p_e o4 co,s-fe,(

Since the solution to the minimization of the integral action has h

also to meet the constraint imposed by the system equationsl either the setz.:

176.

functions employed must already be solutions to the system equationslor

alternatively they must be generated by a model of the system. The choice

of the set of functions in the Ritz-Galerkin method defines the structure

of the relationship between the independent and dependent variables. In

the same way the structure of the relationship between the indepeneent

variables and the control action can be defined for the control problem.

However the independent variables can be taken to be the system state and

the time (if this latter is relevant))giving a structure which supposes a

feedback system andalso allows for it to be made suitable for engineering

purposes right from the beginning. The control action thus generated is

applied to a model of the system or possibly the system itself and the

resulting behaviour, which naturally satisfies the system equations provides

the response to be measured by the performance integral. The integral

is maximize,lby choice of the parameters which describe u as a function of x.

The process cannot, in generalibe performed analytically but the search in

the space of the variables which define the controller is carried out by

performing an integration for each trial. The amount of computing required

is comparable with that for finding one particular trajectory by the

variational methods and leads directly to a control scheme which can be

used immediately.


The problem of making a system give its performance over some time

period is solved, in principle, by the calculus of variations or a related

method. The theory is well established for lumped parameter systems. It

has been demonstrated to be extendable to distributed parameter systems.

177,

The problem of defining the best control for an nth order system

leads to a two-point boundary value problem requiring a search through an

n-space for suitable conditions. The description of a distributed parameter

system by partial differential equations leads to a double set of equations

with an integral condition on the whole of the boundary of the domain of

independent variables. This boundary condition does not arise when the

problem is formulated in terms of integral equations, but here remains the

difficulty of establishing the necessary Green's functions especially for

systems which are described by parabolic operators for which no adequate

theory exists.

Many industrial processes operate indefinitely or for periods which

are longostein comparison with the natural time constants of the system.

When this happens it is known that control is a function of system state

only. Furthermore,systems operating near an optimum condition can have

their performance functional approximated by the integral of a quadratic

function of state and, in common with any system, have their behaviour

modelled by a linear system. Under these circumstances it has been shown

that the optimal control is a linear function of system state which can be

developed through the use of the calculus of variations, involving the

condition that the resulting system should be stable. The result is a

feedback controller, capable of dealing with any situation and operating

continuously.

This result goes over to distributed parameter systemounder the

same circumstances but even when the system is modelled by a lumped

parameter system the dimensionality may be so high as to preclude solution

of the equations.

178.

To reduce some of the difficulties of variational problems to

manageable proportions sub-optimal policies are used in which the period

over which the optimization is carried out is reduced relative to the

period of operation. The change from optimizing over the whole period of

operation to optimizing over zero time is shown to be equivalent to a

transition from time-dependant to state-dependent control, but this latter

is not necessarily the optimum control.

All the variational problems are framed with time as the independent

variable and naturally result in control being defined as a function of this

variable. The problem can be reformulated with system state as the

independent variable but the need to account for the systems' natural

behaviour linking all points on the same state space trajectory leads back

to a version of the same variational problem as before. If however the

measure of performance at any point in the system state is taken to be the

cost of completing the trajectory from that point, with the system remaining

unmodified then it is shown that an iterative procedure can be developed

which gives an improved control action as a direct function of system state.

The process can be repeated and leads ultimately to the absolutely optimum

condition.

It is in one sense a generalization of the dynamic programming method,

but this method is most suitable for systems which can be described analytic-

ally. Since the necessary cost function may be difficult to produce,

approximations must be considered. This leads to control schemes which are

sub-optimal in the sense thati while they are based on full length operating

periodslare dependent on less accurate state space trajectories. Once again, thett produce control directly as a function of system state,which,however,

179.

cannot be guaranteed to be optimal. The usefulness of this method of

trajectory approximation is that it makes possible the formulation of a

cost function which is directly comparable with the costs of modifications,

unlike the general state-space metric which can only define directions for

trajectories. However for use with distributed parameter systems the

dimensionality of the describing state space must be greatly reduced.

Certain established and satisfactory methods of solving partial

differential equations reverse the philosophy of the variational solution

to control engineering problems. Instead of minimizing an integral by

solving differential equations the equations are solved by minimizing the

integral. Application of this approach to the control problems leads to

adaptive type control design. Enough is known about the best possible

answers to make this worthwhile.

Control is known to be dependent on system state (linearly for a

first approximation to any system). Having set up a controller with this

structure the search for the parameters needed to specify it will be

comparable with or shorter than the corresponding search for a solution in

the calculus of variations.

Associated with this problem is the need for criteria to decia what

information should be available to the controller. There is also the

problem of, finding the structure for the controller, which is

a higher level of adaptive process in which the cost functions involve the

capital cost of the necessary equipment.

The problems of instrumentation and behaviour in a reduced state

space are considered in the next chapter. (Chapter VIII). Chapter IX

deals with the choice of the parameters in the controller, being especially

concerned with the flow and diffusion process considered in Part I.

`X

0 T t a. Fan-out of trajectories from x(0).

o T t b. Fan-in of trajectories to x(T).

t o. Stable trajectories for continuously

operating system, asymptotic to xopt•

Fig.7.1. Structure of resulting trajectories for three

types of solution to control problem.

C!-

181.

Chapter VIII

INSTRUMENTATION AND A REDUCED STATE SPACE

8.1 Introduction

Any decisions taken about operating a process can only be based on

instruments measuring the state of the system. In servomechanisms the

number of relevant parameters is strictly limited, usually to position,

velocity, and possibly acceleration, of the various parts. No such

finite set of data can be specified for a distributed parameter system.

In deciding what data is required, the use to which the data is to

be put is all important. Three distinct aspects can be considered.

Control decisions made either automatically or manually need certain

information. Protection and safety requirements and display information

for operators may be quite different, both in form and quantity. In the

event that the process is to be operated against economic or other perform-

ance Criteriat data is required for costing purposes which may be different

from both the foregoing requirements. In the methods of the calculus of

variations and associated methods, described in the previous chapter, no

distinction is made between the needs of control and of costing and the

safety and display aspect is not relevant.

There are a number of practical problems which can only be considered

in the context of a particular system. Firstly the cost of the instrumen-

tation, often fixed in advance to a small fraction of the total capital cost

of the plant, must always be kept to a minimum. In many processes it is

not possible to insert instruments in arbitrarily chosen locations, either

because it is not possible to build suitable instrumentsleither small enough

or robust enoughlor because it is not possible, having the instrument in

182.

places to provide access for maintenance or even to build in the information

carrying lines, leading to the outside of the plant.

Regardless of particular practical problems theinvestigation of this

chapter is concerned with finding how few instruments need be provided in

any plant to give an adequate description of the state either for display

and safety, or for costing, or for control, 'though this latter can only

be dealt with in the context of the control methods themselves.

The need is always to describe the system by a smaller set of

parameters than is needed for its unique description.

8.2 Non-Unique Trajectories

In the phase plane representation of a second order (lumped parameter)

system the trajectories are unique. That is, for any given point in the

plant there is only one trajectory passing through. Similarly,trajectories

are unique in the state space of higher order systems. For a distributed

parameter system the state space is a function space of infinitely many

dimensions and once again the trajectories are unique. Such a system can

be represented to any required degree of accuracy by a finite number of

parameters (See Chapter IV). However the resulting state space may be of

too high dimensionality to be useful.

This dimensionality can be reduced by the simple process of neglecting

some of the data and considering the behaviour in a reduced state space

having fewer dimensions. In this reduced state space the trajectories are

non-unique. Any point has an infinite number of possible trajectories

through it,corresponding to all the possible values of the neglected

variables. However for practical purposes the range of variation may not

be too great and the reduced state space behaviour may give a fair repre-

183.

sentation of the system for the purposes of protection, control or costing

as required.

Any investigation of these reduced state space trajectories requires

that the range of variation of the neglected variables should be consistent

with normal operating conditions. It is not practicable to consider every

possible trajectory through any point because each point would require a

systematic investigation through the space of neglected variables.

Furthermore because of normal behaviour the distribution of probability of

being in any state would not be uniform over the (state) space of neglected

variables. The distribution depends on the type of disturbances to which

the system is subjected.

The following method allows an accurate mathematical model of a

process to be used to display trajectories in a reduced state space,

starting from suitably chosen random initial conditions.

The system (model) is allowed to run for a time which is long in

comparison with its natural time constants, while subject to disturbances

having the usual statistical structure. Thus at any time the randomly

varying state of the system is subject to the usual probability distributions

The random disturbance is suddenly removed and the system then starts, from

the particular state existing at that time, to follow the state space

trajectories leading to its final settling down. (Unstable systems cannot

be treated because they cannot be operated subject to disturbances). When

these trajectories are recorded in a reduced state space they show non-

unique behaviour,but for any starting point certain directions of movement

have higher probability than othersithus defining a most probable trajectory

for any point.

184.

This method of choosing the initial conditions at random, but subject

to the natural probabilities is equivalent to the Monte-Carlo methods of

system investigation. The circuits for an analogue version of this have

been developed and are shown in fig. 8.1. Examples of the trajectories

which result can be seen in figs. 8.7 and 8.8.

8.3 Correlation Coefficients Criterion for Instrumentation

In setting up instruments to provide information to show the state

of a distributed system there are two limiting cases, albeit somewhat

vaguely defined, which set bounds on the separation between two adjacent

probes.

If two probes into the system are too close together, then both will

provide essentially the same information. If they are too far apart then

there is insufficient data about the variations of state existing between

them.

Correlation coefficients for the variations measured at any two

points show, in a sense, the amount of common information which they carry.

Since the measurements are taken simultaneously from all points when it is

needed to establish the state throughout the system, the correlation at

zero time lag between any two probes, as a function of location and spacing,

is required.

There are basically two methods of finding the necessary coefficients.

Since the objective is to establish instrumentation requirements before

construction the methods must be considered in conjunction with mathematical

models of the system. One method is to measure the correlation directly

while the system is subject to the normal disturbances, the second is to

calculate the coefficient analytically fron the transfer functions or their

the cross correlation of the two measures is:-/-0

Ryly2(7) = f2(u)

J

;(u) Rdd(I -v+u). du dv. fl (u)

185.

inversions. The underlying mathematical treatment is the same in both

cases (Refs. 8.1, 8.2).

Let the input disturbance to a distributed parameter system be

described by d(t). Let the impulse response or weighting function of the

transfer to one point be f1(t) and the transfer to another point be f2(t),

the corresponding disturbances measured at these points being y1(t) and

y2(t).

The object is to evaluate the correlation Ry1y2(t), which is the

expectation of the product of the two signals 71(t) and y2(t) when shifted

relative to each other in time by an amount r. For the purposes of

instrumentation the value of Ry1y2 is required at r=o.

It is shown in Appendix XII that if the input disturbance is white

noise then

JiRyiy2(Z) = fi(u). f2(t+u) du (8.1)

and the power spectrum of thisiPyly2, its Fourier transform,

In the more general case, where the input has an auto-correlation function:

Rdd(T))

03

x. 00

1111111111111111111111111111111111111111111

1111111111110111111111111111111

186.

In dealing with the flow and diffusion system considered earlier,

a digital computer programme was developed to perform the correlation

operation for the white noise case by integrating the cross products of the

impulse responses as they were developed in a step by step integration

process for a set of ordinary differential equations describing a lumped

parameter model of the system. The structure of the programme is shown

in Appendix XIII.

The results are shown in figs. 8.2 to 8.5 where for each value of the

diffusion coefficient "a" the correlation between two measures with a given

spacing is shown as a function of the position of the outer of the two

measures. The results are shown in the form of correlation coefficients

which are:the ratios of the correlation functions for zero lag,to the

product of the root mean square values of the impulse responses themselves.

Identical signals thus have a correlation coefficient of unity (at zero lag)

and completely independant signals a coefficient of zero. Also, to give

some indication of the meaning of these coefficients some sample scatter

diagrams are shown (fig. 8.6), for which the correlation between ordinates

and abscissae for the points are shown.

The results show that as the measuring points move further away from

the input the correlation between measures becomes higher. This means that

for the same correlation between measures the spacing between instruments

can be made greater.

For an example suppose that the system has a = 0.025 and it is

required to have a correlation of 0.7 between the measurements for sucessive

instruments with the white noise input. This is the condition to which

fig. 8.2 corresponds.

187.

Starting with an instrument on the output, x = 1.0 marked as (1)

on the diagram, the spacing required to have a correlation of 0.7 is 0.24 of

the total system length, so that the next probe, (2), is at x = 0.76. The

next spacing of 0.21 gives probe (3) at x = 0.55 and probes (4), (5) and

(6) are at x = 0.38, 0.24 and 0.14 respectively. Another sequence of

probe positions is shown for a correlation coefficient of 0.5.

In either case any further increase in the number of probes would

result in them coming closer together, until ultimately infinitesimal

spacings were demanded because the white noise condition on the input

makes it =correlated with anything else.

Putting a lower limit either on the spacing between probes or on the

distance of the nearest probe from the input point completes the specifica-

tion for instrumentation. The number of positions of the probes can be

developed from a knowledge of system behaviour and a requirement on the

amount of information common to adjacent probes. This is not to be taken

as specifying a means of designing instrumentation for control purposes,

only for protection and display.

8.4 Trajectories in a Reduced State Space

Any group of independent measures of system state can be taken as

the basis for a reduced state space. The objective is to find a set of

measures which provide suitable information for the task in hand (protection,

costing or control) while giving the minimum amount of variation in

behaviour within the reduced state space.

This is most readily interpreted in terms of trajectories., wheee .the

need is for the smallest range of variation of trajectories through any

given point.

188.

For the purposes of demonstration a space of two variables is the

simplest. Using the analogue model of a heat exchanger described in

Appendix II and the analogue Monte-Carlo method for choosing initial

conditions described above (Section 2), a set of trajectories was observed

for each possible combination of any two measures taken from a total

selection of ten equally spaced probe positions.

Two sequences of these sets of trajectories are shown, as examples.

One covers the combination of the second probe (from the input) with all

the others,and the second shows the fifth probe in combination with all

subsequent probes. (Figs. 8.7 and 8.8). The output of the earlier

(near the input) of the two probes is always shown as the vertical deflec- ore

tion and the signal polarities arranged so that the trajectories normally

follow clockwise paths.

Two points emerge from studying these trajectories. For the

trajectories to be consistent or as near as possible to unique the probes

must be close together, e.g. probe 1 and probe 2 or probe 5 and probe 6.

For them to be representative of system behaviour, in that normal operation

results in movement over all the space, the probes have to be apart so that

the outputs are not highly correlated. The two requirements conflict with

each other.

This can be partly avoided by using, instead of direct measures of

system state, linear combinations of measures - weighted averages. The

possibilities are infinite in number, so a few simple weighting functions

are considered. The outputs of the ten equally spaced probes (probe 1

nearest the input, probe 10 the output) are divided into two groups;

1, 3, 5, 7, 9 and 2, 4, 6, 8, 10. These two groups provide the two

189.

signals, one for each axis of the reduced state space. The analogue

computer arrangement for this is shown in fig. 8.9a.

The weighting of the various probes for the summation is shown in

the sequence of little diagrams in fig. 8.9b. Examples of the two-space

trajectories that these produced are shown in fig. 8.10. The ones

corresponding to an even weighting function for each half (Form 1) give

the best trajectories. They are consistent in that there is little cross

over and they cover a large proportion of the state space.

8.5 Comparison with Crude Models

The ultimate step in making a model of a distributed parameter

system for control engineering purposes is to represent the whole system

by a very simple crude model. This may have to be done when the system

forms only a small part of a larger process. This reduction is equivalent

to a reduction of the state space. Using the techniques described above;

the behaviour of such a model can be compared with the more accurate models

to find the differences in behaviour and the possible sources of difficulties

The accurate analogue model used above to investigate other aspects

of control and instrumentation has a mean residence time or delay of

10 m.Secs. Regardless of the other effects,this sort of process may well

have to be represented by, for example, a second order system. The

natural replacement is thus a pair of lag units each having a time constant

of one half the delay time of the original system (5wiSecs). The overall

gain can readily be adjusted to account for the small steady state

attenuation of changes in input level as they pass through the accurate

model. This form of crude model used for the investigation is shown in

fig. 8.15a.

190.

A convenient method for comparing the crude model with the accurate

model is to compare the outputs of the crude model with the measures, or

weighted averages of the measures of the accurate model when both are

responding to the same step or random noise disturbances. The responses

to bursts of random noise show all the necessary information and give a

representative test for all operating conditions. Here the responses of

the two outputs of the crude model are compared with

measures of

the state of the accurate model corresponding to - isolated measures

half way down and at the end of the system and then the weighted

averages shown in fig. 8.9b, (with the exception of the last). The results

of these comparisons, two examples of bursts of noise in each case, are

shown in figs. 8.11 and 8.12 for the first output and the second output of

the crude model respectively.

The first output of the crude model closely resembles that of the

uniformly weighted average over the first half of the accurate model. It

is not nearly so close to the measure halfway down the system which shows a

pronounced delay and considerably less of the high frequency components

present in the input.

The second output is closest to the average over the second half

but the differences are bigger than before. This is because the second

part of the crude model does not receive the same input signal as the second

half of the accurate model.

The most significant difference for control purposes is that between

the final outputs of the two systems. Although adjusted to give the same

amplitude of steady state response the crude model gives a much smaller

response under transient conditions and a smaller delay before input

191.

disturbances start to affect the output. Furthermore there are more

high frequency components present in its output.

A further comparison can be based on the state space trajectories

starting from random initial conditions. The trajectories followed by the

crude model are unique in two space because it is only a second order system.

Examples are shown in fig. 8.13(b). The trajectories they most closely

resemble are those of the evenly weighted averages in fig. 8.10(a).


Any practicable instrumentation of a distributed parameter system,

with a finite number of instruments forany purpose) corresponds to a

representation of the system by a reduced state vector, because the state

space for such a system is really a function space. The more practical

aspect is that of having an adequate representation of the system by fewer

parameters than are necessary to represent it as accurately as calculation

methods allow. Thus in a system which is known to be quite adequately Wne

represented by twenty parameters in its state vector may have/represented

by only five or even two.

A criterion for the location of instruments to ensure that there is

not too much overlap in the information they produce nor too large a chance

of missing significant events can be based on correlation coefficients.

It may prove in practice to form a useful basis for instrumentation design

for protection and display purposes. There is no inherent guarantee that

it gives a suitable arrangement for control needs.

The combination of measures of system state from any two locations

in a system is unlikely to provide an adequate basis for a two-space

representation, unless of course the system can be accurately modelled by a

192.

low order structure. This means that any analysis of behaviour based on

the theoretical optimization techniques described in Chapter VII needs

either a model based on a larger state vector or some other means of making

a compressed measure of state.

This can be done by taking weighted averages of the state over the

whole system instead of isolated measures. The behaviour in the reduced

state space can be made much more consistent at the cost of having to assess

the averages all the time. This same description of system by weighted

averages of state also formsa means for comparing accurate with crude

models and the results can show the significance of the behaviour of the

crude model in terms of its accurate counterpart.

None of these approaches yields any result directly applicable to

control design forwhile a knowledge of system state and the instrumentation

to provide it are based on simultaneous presentation thecontrol design has

to take into account the time at which the information is needed and its

relationship to future behaviour. This aspect is investigated in the next

chapter.

—MAO.

c-a0 ro

SCALE FOR DISPLAY

SQUARE WAVE

GEN 0.514

(-booy)

PVE Aors 'REP.

Willa- NOISE I GEN.

103

SIGNAL GATE CIRCUIT SCALING COMPUTING GFIVERAPON

DISPLAY

Signals from white noise generator are chopped into

bursts by gating circuit and ,fed into input of system.

CR0 display is blanked off except when system it

settling down just after end of noise burst.

Fig.8.1. Circuit for analogue Monte—Carlo method:

providing random initial conditions for state

space trajectories.

C .4

O 0 •f •2

OA", r)

.6

• 2

1 0

• 7

-5

3

9 s pActo‘ . 0.1

vhf ,s.l1

,n11 O.

....18 Q6 .5 / 4 , e;

-0". 03

D 0 . . • . ••• 4

..----7::::"=----°-

e.......0•01 .....

.3 - 4 5 .6 .7 • Cr •9 /. 0

X (00rOrt)

/94'

(Diagram shows sequence of probe position selection

for C=01 and for Cm0.5.)

Fig.8.2. Correlation Coefficients (C) between pairs of

measures of system state, as a function of

spacing between measures and position (X) of

the probe farther from the input end of the

system.

Normalised system,a = o.025.

0.

0.1

[9c

.3

*2

1

0 0 f 2 .3 .4 .5 6 .7 .8 •9 x o

Fig.8.3. Correlation Coefficients (C) between successive measures of system state,

as a function of probe position(X),

for various values of spacing.

Normalised system, a m o.05.

14,46 3 0.‘

o.k

1.0

.9

•8

.7

•6

•5 C

.3 6

.2.

•1

0 1 2 3 1- 5 6 .7 4 .9 10

Fig.8.4. Correlation Coefficient (C) between

succesive measures of system state,

as a function of probe position (X),


Normalised system, a o•l.

1%

.2 .3 •4 •5 -6 7 .8 •te,

X

PO

.9

.7

•6

5

C

.3

2

• 1

0 0 •1

SP \0141.... 0.1

0.%

Fig.8.5. Correlation Coefficients (C) between

successive measures of system state

as a function of probe position (X),


Normalised system, a o•2.

p

cr. •918

•

••

=45.99

c- 40

• Fig.8.6. Sample scatter diagrams for various correl—ation waffle-

-ients (C)

P •

F2

r- I Pl.

I

•

•:. • fit t t4;CICO t 1'1 t r i0114 •

I . I s U.11 in with 0..ch 1- robe

i

, ...

r , • JI • 11 • r

m J 1 n 1 h

', , ) fl .

~-----------------

So — ririn — 004 tts

.50 — f.,rzzr —

."3", —Jo 1616

tic

-

4o- Jo re

/ 3 s 7 O

it

2 96 8 so

I I

ar.rfii— so —

FORM 1

Fotill 2

PoeM 3

53% V.

/NAL)

HEM' c-:PeCm/tA/Cede •A /4h0 4ve (s&e, r )

Ptaii.1 1 1

PI P2 P3 P* P P.‘ P7 P r 1.9 P /0

IM PT

PL

P8

P I 0

Fig.8.9. Analogue Computer circuit and forms of

weighted averages for reduced state space

trajectories shown in fia.8.10.

4

ON-

ReduccH titatti :1-otefi trnjvictorio:; prolucel

with wflitrhtfrii aliiIrnr.7o:i of tly1;tom

C.0rmli of nvra,- i0,0Wh in ri,-.°.0.1

1.— , I .'lai .1 `^/ ^[ (..

INPUT

X,

X 1

a) Analogue arrangement for crude model of flow and diffusion process in heat exchanger. Potentiometer P allows for small loss in steady state transfer.

b) Phase plane trajectories for above model.

Fig.3.13. Crude model of flow and diffusion

process in heat exchanger used for

comparison with accurate model.

206.

Chapter IX

STATE MEASURE CONTROL

9.1 Introduction

The control of distributed parameter systems to economic performance

criteria produces design problems which cannot readily be tackled with

available methods.

The conventional frequency response methods, while applicable if the

system is available and suitable for frequency response measurements, is

not so valuable if it is necessary to predict the transfer functions

analytically. Furthermore the system is always viewed as a single unit

having a certain finite number of transfer functions associated with it.

The distributed nature of the system, and with it the concept of a space

and time dependant performance criterion, is observed.

The calculus of variations yields mathematical equations but these

are not readily solved andfurthermore the form of the answers is not

generally suitable for direct control application.

However, it is apparent that certain advance information is

available about optimum controLbefore any of the problems are solved.

For a system which runs continuously, as do many industrial processes, the

optimum control is known to be a function of system state. Sometimes even

the structure of the optimum controller is known.

It is known that the search problem for finding the best parameters

for producing a control function of system state is comparable with and in

many cases easier than solving the boundary value problems of the calculus

of variations. (See Chapter VII) Furthermore there is precedent in the

207.

established field of mathematical analysis for reversing the philosophy

of the calculus of variations and searching for parameters to minimize an

integral therebysolving differential equations instead of following the

process of the usual control problem whereby the integral is minimized by

solving differential equations.

Special purpose electronic analogues can be produced for distributed

parameter systems which retain the distributed nature and can be used for

any form of control design including problems of the type treated by the

calculus of variations. The same knowledge about the accuracy of

modelling a system can also be applied to digital computer models/but

without the advantages of rapid visual representation of results.

By choosing the time

calculations can be carried

continuously, the effects of

u4S scaling of the electronic analog'', so that

out repetitively and results displayed

control design modifications can be seen at

once. It is on this type of model that the results of this chapter are

based.

Knowing that the optimum control is a function of system state does

not solve the engineering problem of deciding what measure of system state

is necessary. The development of control schemes must thus take account

of practical engineering problems. The starting point used here is to

take only one measurement of system state - a hopelessly inadequate state

vector, it would seem - and see what can be done with it. After that the

complexity of control schemes can be extended to give better control and

more flexible design.

9.2 Single Probe Control

One instrument is to be inserted into a flow and diffusion

20()J.

distributed parameter system and the single measure of state which it

provides is to be used, without any integral or derivative action, to

determine the control action. This is the simplest embodiment of the

general concept of state measure control. One additional advantage is

available over the conventional methods:- the probe can be inserted at any

point.

A performance functional is specified which is an integral or time

averaged measure of any function of state in its general sense of covering

the whole system. In this respect the problem is the same as the one posed

in the calculus of variations. However further restrictions are possible

which relate to the final engineering problems. For example the structure

of the controller can be restricted, in this case to linear operation. An

aspect of control design which is not available by the variational methods

is the possibility of modifying the performance criterion when it is found

that a best solution is obtained which does not specify the final design

precisely enough. Such a case occurs in the Multi-probe example below.

The example for the Single-probe control is the same system as was

used above for the conventional control design case in Chapter VI. The

performance criterion is to be based on the deviations of the output state

from a desired condition and the controller restricted to linear operation.

The problem is thus to choose the location of the probe and the gain

of the controller (no reset or derivative terms allowed) to give best

performance.

This search problem for two parameters can be reduced by one if

another restriction is placed on the overall system behaviour. This is a

useful criterion for continuously operating systems, that there should be no

209.

steady state errors. Meeting this immediately removes one of the difficul-

ties of the conventional control scheme.

The system is shown in fig. 9.1a and the signal flow graph correspon-

ding to it is shown in fig. 9.1b. The transfer functions shown have the

following significance.

G1(x,p) is the transfer function through the fluid and its environ-

ment from the input of the fluid stream U(o,p) to the point x, where the

state is represented by U(x,p).

G2(xp) is the transfer function from the output of the controller

8(p) to point x, and G(p) is the transfer function of instrument and

controller. It is the gain G(o) of this transfer function which is to be

chosen.

From the signal flow graph the overall transfer function from input

to output is

U(x,p) _ U(o,p)

G1(x2,p) G1(xp) G2(x2,p) G(p)

1 - G2(xl,p) G(p)

In the steady state the response must be zerol so that G(o) is defined by!

- G1(xo)

G 1 (x11 o) G2 (x 2 ,o) - G 2 (xl' o) G1 (x2' 0)

Thus G(o), the gain of the controller)is a function of the position of the

probe, xl, and the length of the system, x2.

This formula for the gain as a function of probe position, supposing

x2 fixed, throws further light on the difficulties encountered with the

conventional controllers.

Consider a special case to make the meaning of the formula clearer.

G(o) -

210.

Take G1(x,o) to be identically unity for all x. This means that the system

is lossless and the output disturbance is the same as the input disturbance

in the steady state, when there is no external forcing (0(o) = o).

In the case where there is no feedback from controller to instrument,

G2(x1

) = o then

G2(x2) G(o) = -1.

This expresses the idea that to exactly compensate for the

disturbance the control path gain for the disturbance must be unity and

exactly in opposition to the fluid and environment path gain.

If G2(x1) is non zero so that some of the disturbance is suppressed

before the instrument point then the controller gain must be made bigger to

compensate for this. An alternative argument is that for a given size of

input disturbance a certain size of control action is required to compensate

for it, regardless of the point in the system from which the information is

taken. Thus if the disturbance is measured after it has been reduced by

partial compensation the controller gain must be larger. In the limiting

case in which the disturbance is measured at the output - where it has

presumably been reduced to zero - the gain required must be infinite. At

some point of location for the probe where the gain is still finitel a limit

will be reached because of stability troubles.

G(o) can be calculated from the basic design data or from steady

state measurements regardless of dynamics. Thus in fig. 9.2 possible

curves for G1(x,o) and G2(x,o) are sketched with the curve for G(o) as a

function of the probe location, x.

While increasing demands for controller gain set a limit on the

proximity of the probe to the output end, placing the probe near the input

211.

end makes the whole system more susceptible to parameter changes. (For

example the deposit of fur inside a heat exchanger which modifies its

transfer characteristics).

Let the function 0 represent the steady state transfer factor for

disturbances entering the system. It is thus defined by

0 = G1(x2) 1 - G2(x1) G(o)

and for the correct choice of G(o)

0 = 0.

Now consider the effects of changes in other transfer functions.

a0 G2(x2) G(o)

agi(xl) 1 - G2(x1) G(o)

4 aG1(x2) = 1

a0 - G1(x2) G(o)

aG2(x1)

1 - G2(x1) G(o)

provided in this last case that 0 = o. Finally

80 G1(x1) G(o)

aG2(x2) 1 - G2(x1) G(o)

Then for small changes, denoted by 8G1(x1) etc., in the transfer

functions:

80 -

G(o)

1-G2(x1) 0(o) G2(x2) 8G1(x1)+ 8G1(x2) - G(o)

G1(x1) 8G2(x1) G1(x1) 8G2(x2)

11-G2(x1) G(0).1

Gi(xl) G2(x2) G(o)

212.

Some of these variations will be related to each other. Just suppose

that: xi

= 8G (x . —= 8G1(x1) I 2) x2

and x1

8G2(x1) = 8G2(x2) 2

the implication being that the change in the transfer is proportional to the

length of the system involved. It then follows that

50 = G1(x1) G

1(x2)

G1(x2) 8G1(x2) G2(x2)

15G2(x2) .

2

As the probe position, xl, approaches the output point, x2, the

coefficients become smaller and the sensitivity to parameter changes is

reduced. This applies regardless of the relationships between 8G1(xl)

and 8G1(x2) and between 8G2(x1) and 8G

2(x2) provided only that when x

1

reaches x2 the changes are the same. If a restriction is placed on the

sensitivity to parameter changes then this puts an upper limit on the

distance of the probe from the output.

Subject to these limitations on stability and sensitivity it is

necessary to find the best probe position to give the best performance. An

example of this is shown by the selection of the best probe position for the

analogue heat exchanger of Appendix II, (and Chapters VI and VIII). The

arrangement for these experiments is shown in fig. 9.3a. The instrument

to measure stream temperature could be inserted into the system at any

one of ten equally spaced positions along the fluid flow line. The final

performance criterion for the complete system;with control1was taken to be

the average of the modulus of the output deviation, as measured by a

213.

particular valve voltmeter, when the system was subject to square wave

disturbances. It was found that the same best probe position resulted

from trials with other disturbance forms (sine wave, random white noise)

and any form of performance measure could have been used, the particular

choice being based on the common requirement that the final output of the

process should be as near as possible constant.

As in the experiments to find the effects of conventional control

schemes (Chapter VI), two versions of the system were considered. One had

le an extra lag in the control loop transfer whit's. the other was arranged so

that the 'shell' temperature could be influenced immediately by the demands

of the controller. (See also Chapter VI, Section 4).

The results of these trials are shown graphically in fig. 9.4, and

9.5. The first graph (9. a) shows the gain required to produce zero

steady state error as a function of probe position. The first of the

limitations becomes apparent here because it is not possible to apply

significant gain to give zero steady state error without having the whole

system unstable when the probe is near the output.

The second graph (9.40) shows the two measures of output performance,

one with and one without the lag in the control loop. The two distinct

minima are apparent, one with the probe near x = 0.5, and the other with

the probe near x = 0.3.

These measures of performance can usefully be compared with the

measured disturbance responses of the system shown in fig. 9.5, for the

system without the extra lag. With the probe near the input (x = 0.1,

9.5b) the output transient response shows one single pulse in the opposite

direction to the applied disturbance. When the probe is at x = 0.5, (9.5c)

214.

the transient response is roughly symmetrical, giving a small deviation in

each direction. With the probe inserted further along (x = 0.7, fig. 9.5d)

the predominant deflection is in the same direction as the disturbance and

the beginnings of the oscillatory behaviour are apparent. When the extra

lag was inserted in the control loop similar behaviour was apparent but

masked by the very long settling time associated with the lag.

The important result from this is revealed by the shift of the best

probe position when the lag is introduced. The time constant of the extra

lag (found to be 2.77 milliseconds) corresponds with the mean time delay

between the two best positions (x = 0.2 or 0.3, and x = 0.5) which was

between 3 m.Sec. and 2 m.Sec. Just in the same way that the gain must be

chosen to give zero steady state errors by balancing the effect of the

controller against the disturbance transfer, the best transient response

comes from arranging the time delays in the system so the effect of the

controller reaches the output at the same time as the disturbance.

Fig. 9.5b shows that having the probe tco close to the input produces a

deflection due to the early application of control; fig. 9.5d shows the

effect of taking control action too late and fig. 9.5c shows the reduced

disturbance resulting from a close match of the time required for the

disturbance to reach the output from the instrument point and the time for

the control action to take effect there.

The sensitivity to parameter changes is demonstrated in fig. 9.6,

where the percentage change in output disturbance level, relative to the

uncontrolled disturbance is shown as a function of probe position for a

given percentage change (5%) in the gain of the control loop. As the

probe is placed near to the output the sensitivity decreases. The results

215.

shown are for the steady state error but other performance criteria can be

used, as is done in the 'distributed control' of Section 9.4.

9.3 Multiprobe Control

The next step to be taken after having found the requirements for

single probe control is to investigate the choice of probe positions and

weighting of the resulting measures for computing the control action using

twoand then three probes,and so on.

A preliminary trial showed that there was little extra to be gained

from using two probes,and the further complexity due to organizing not only

a search for weighting factors but also for positions made the process

unnecessarily complicated. For example the transposition of two probes

yields nothing new, only the actual locations used are relevant and the

complications increase with the number of probes. The alternative, which

leads to a search only in the space of one type of parameter, is to insert

probes in every possible position and find the best weighting for each probe

output in the control computation. If the probes are close enough

together this is equivalent, for practical purposes, to finding the best

control as a weighted integral of system state - the natural generalization

of the results produced in Chapter VII where the control was found as a

summation of scalar multiples of the elements of the state vector. This also

has the effect of reducing the overall dimensionality of the problem by a because

factor of two, only a weighting factor has to be found for each probe

instead of both weighting factor and position. The analogue for dealing

with this problem in the case of the heat exchanger is shown in fig. 9..b.

Ten probes are inserted into the system and their measures collected anda

216.

weighted average produced which is the control demand signal, a separate

overall gain factor being available to ensure that the steady state

requirement of zero error is met. This means that only the relative

magnitudes of the weighting factors are important, not the absolute values.

Many techniques are available for finding the optimum values of sets

of parameters, particularly on digital computers (See the references in

Chapter XI, Section 9 on "Hill Climbers".) The analogue compUter versions

are usually less sophisticated because of the difficulties of making either

a large number of logical and sequential decisions or of generating and

then separating the results of a large number of time-orthogonal perturba-

tion signals. However on a high speed analogue this particular problem

can be dealt with by one operator following a type of decision sequence

that might be used in a digital machine. An excellent discussion of the

potentialities of high speed analogues is given by Fisher and McKay (./3).

The results of such an optimizing sequence are shown below. The

procedure adopted was to start with some arbitrary set of weighting factors,

and then vary each factor in turn to give the best performance when the

whole system was adjusted to the zero steady state error condition. When

each factor had been investigated oncelthe procedure was repeated starting

from the resulting set of factors.

217.

Starting Condition: Uniform weighting, P1, P10 set to 30%

Performance Measure at start 0.241

STAGE i STAGE 2

P1

P2

P3

P4

P5

P6

P7

P8

P9

P10

Changes Made

set to

If

It

It

ft

ft

0

0

0

475

100

100

100

0

0

0

Resulting Performance

0.210

.180

.163

.163

.157

.154

.153

.152

.148

.144

P1

P2

1,3

P4

P5

P6

P7

P8

P9

P10

Changes Made

set to

f f

it

,I

"

"

,,

►I

ft

I►

0

0

0

0

100

100

0

0

0

0

Resulting Performance

0.144

.144

.144

.140

.140

.140

.132

.132

.132

.132

At the end of Stage 2 no further measurable improvement could be made with

the system and the only way to obtain a closer decision on the choice of

weighting factors was to introduce another performance criterion such as

the requirement that the average output deviation should be zeroi so that

after the effects of further dispersion and mixing it would disappear 11-Rn

completely. In this case this made the weighting P5 more important7P6,

and the whole multiprobe control system was reduced to the single probe

system found before.

218.

9.4 Distributed Control Action

In the multiprobe control problem treated above and the simpler

single probe control situation which is seen to be closely related the

whole of the control action is applied in the same way to all parts of the

system. A further possibility is available in distributed parameter

systems and that is to have a control action which is a function also of

the space variable. In the multiprobe control situation it was found that

a performance criterion based on one location in space resulted in a similar

ly localized measure being used for the control because of the relationship

between time and spatial displacements. If the performance measure is

based on minimizing deviations notfrom just the desired output end

conditions but is concerned with deviations from a desired state profile all

over the systemla similarly distributed form of instrumentation will be

required. Any practical form of such a distributed space dependant

controller will necessarily require the control to be applied in a finite

number of regions, within which all parts receive the same external forcing.

A further requirement for implementation is that any control region must be

able to provide sufficient effort of forcing to the system to eliminate any

disturbance which enters by the time it reaches the exit from that region.

If this is not possible then it is not possible to maintain a state profile

in the system which will show zero errors elduywhere. In the above single

probe control cases the overall control scheme may well be dominated by a

feedforward effect: the stream of material passes the instrumentation

point and its quality assessed. After this action is taken to eliminate

any existing undesirable characteristics butno final output measures is

made that can in any way be used to modify the material concerned. Once it

219.

has become the output it is too late to do anything about it.

A different situation arises however if many such systems are

cascaded so that the output of one becomes the input of the next. Each

piece of equipment is a feedforward device but to an observer moving with

the material stream it appears that the control action is to repeatedly

sample the material state and then apply correction - a sampled data

feedback controls There is no fundamental reason however for separating

the process into discrete sub-units. The basic underlying scheme is that

thescontrol action at any point is a functional on the state of the system,

and while the functional relationship may apparently be dominated by a

feedforward action from the viewpoint of the static observer as in the

previous multiprobe and singleprobe situations, the overall process is a

feedback process with respect to the moving stream.

The possibility arises of doing the analysis of the feedback system

and from this moving point of view,i\then by a transformation of variables

replacing the 'moving feedback' system by an equivalent static system.

This however can only be done under certain restricted circumstances. The

most important restriction is that there should be no energy or material

storage or 'memory' associated with anything except the moving stream. If

this is not the case then the effects of control action taken on one part

of the moving stream are recorded in the stationary environment so as to

influence another partl and no control policy could be worked out with respect

to a particular part of the stream without involving the past history of all

earlier parts.

In fact the only case which can be dealt with by considering feedback

relative to a moving observer is trivial. In this situation, with no energy

220.

storage outside the stream, control action could be instantaneous and

therefore the profile of control action to be applied would be exactly to

oppose the profile of disturbancel and because the stream moves the control

action profile would move with it. It is just conceivable that this

situation could arise with a temperature profile in a low velocity tubular

reactor where the heat input was controlled electrically by using heating

or cooling elements with low thermal capacity.

The case where ther..: foenergy or material storage in the environment

of the stream must be treated from the static point of view. Here the

formulation of the problem is seen to become the same as the multiprobe

case with the exception that there is now a multiplicity of control actions.

In general terms,control action 0(x,t) can be represented as a function

of space (x) and time (t), and depends on the state of the system every-

where else, at that time.

i

0(x,t) = F(x,xl,t) dx,

o

In the practical case the control action must almost inevitably be achieved

by making a quantization in spacel so that there will be a finite number of

control regions where the control action0i(t), depends on the state of the

system elsewhere. Furthermore since there is a finite number of instruments

the representation becomes:

01(t) = j f(x.,x.,t) .

J where x. is a distance associated with the ith control region and xj with

the location of the jth instrument. The summation is over the whole

numberofinstrimentsandthereis one controlfuriction. 01(t) for each

separate control region.

221.

These integral and summation representations are not the most

general. The problem could, for example, be formulated in terms of

arbitrarily complicated combinations of measured functionals of system state

instead of simple linear summations.

o.bou Applying the approach used to calculate the best multiprobe control

A

scheme for each section willi in principlei produce the best scheme of control

for the system but because of the supposed linear nature of the problem each

control region can be treated separately. The significant difference from

the previous work is that the region which instruments may be inserted is

not necessarily the same as the control region.

Consider the control region shown in fig. 9.7a. A single probe

inserted in the stream controls a small region of the system. A single

probe is considered here because the results of previous work showed that

the improvement due to having a multiplicity of probes was small in so far

as the state at the output of the control region was concerned. In

fig. 9.7b the adjacent control zones are shown and it is apparent that the

instrumentation for one control region may well be inside another. When

this happens it means that instead of being able to treat this as a series

of cascaded systems they are all interlocked. The main difficulty is that

since each region is operating to reduce its output deviations to zero there

may well be errors inside a zone which do not appear at its output. The

instrument for the next section produces measures of effectively non-

existant errors. Compensation is therefore required to make allowance for

the action being taken in a region,when the state at some point in that zone

is being used to derive the control for the next zone.

Once again the criterion for zero steady state errors provides the

222.

basis for setting the gain requirements for each part of the system. The

transient response or some other measure of performance is then modified

by the choice of probe locations.

The topological structure of such a system is shown in fig. 9.8a where

the larger blocks correspond to control regions around the process and the

smaller blocks are the controllers. The lines with arrows show the

directions of information flow. Points 'a'are where there should be zero

(

steady state errors, points1 b the locations of the instruments for each

section inside the previous section and points 'e the summation points for

information about the system state, which comes from points b, and for

information about what control action is being taken in the previous section

which comes from the previous controller. A signal flow graph version of

this is shown in fig. 9.8b and the significance of each transfer term is

shown there.

In the steady state case it is necessary to ensure zero errors at each

point lee. If the system were perfect, having zero error at a' would make

all subsequent errors zero as well. Suppose however that the system is

subject to distributed disturbances and that in the system described by

fig. 9.8 the controller input cr_i is demanding no action because the state

earlier on in the line is satisfactory and ar i is expected to be zero. If

a disturbance dr-1 is injected into the line as shown then the control system

must make the transfer from dr-1

to ar+1 zero,and also, to prevent the

setting up of a steady state error pattern the transfer to crfl must also

be zero. The relevant parts of the signal flow graph (9.8b) are shown as

heavy lines. The two steady state gains g5 and g6 can be chosen:-

2 52 515654 =

and for cr+1 to be zero

gig6g5 g2g1g6 gige3g6 =

These result in the relationships

g3g22

g5 - g2 +

elg4

and g6 g22

glg4

The process of developing such a control scheme in practical terms

is demonstrated by the following investigation.

The analogue of the heat exchanger was modified so that instead of

one shell with a uniform temperature all through theshell was split into

three control regions as shown in fig. 9.9a. The regions 3 and 7 of the

fluid flow and dispersion process arc,• equally influenced by the control

regions on either side.

Each region was operated independently and using the techniques

above (Section 3) it was found that the best control for the control region

3 was achieved with probes in positions 5 and 6, weighted 20, 80 respectively,

and CR2:was best controlled with probes in positions 2 and 3 weighted 85,

15. CR1 was left as a dummy section, no control applied.

Each section operating independently was capable of suppressing the

disturbances presented to it. When both were operated together the system

exhibited over-control and produced a disturbance output opposite to the

input disturbance becausethe region CR3 was acting on a measure of the

error inside CR2 which was actually supressed before the CR3 was reached.

This sort of behaviour could lead to the establishment of a disturbance

pattern in the system.-- a 'sort of feed-forward-spatial-instability:

223.

0

221.

The remaining problem was to find a suitable compensating measure for

the control action being taken in CR2, to feed on the controller for CR3.

Any measure of the controller output for CR2 could be used for compensation,

given a suitable gain in the transfer to the controller for CR3,but the best

overall transient response was achieved, in this case, when the state in the

tube wall was measured alongside position 3. The final form of the control

scheme is shown in fig. 9.9b.

The advantages of this over the single probe or similar system lie

in its insensitivity to parameter changes and its ability to cope with

distributed disturbances. The following figures serve to illustrate the

change in sensitivity. With only one region (CR2) operating, starting with

the correct gain for zero steady state error, the measured performance

(average modulus of error for a square wave input) was increased by 80%

and by 180% for increments of gain of -6.6% and +7.8%. When another

region (CR3) was used as well these same changes in the gain in the CR2

Section produced increments of performance measure of 5.5% and 3.8%

respectively, relative to a much lower level of error.

The sequence of cascaded, interacting control systems forming a

distributed controller for the system is capable of maintaining a defined

profile of state over the system. The steady state operating requirements

for the controller can be evaluated separately in terms of the desired

profile. In this investigation the desired state was taken as zero for all x.

Consider the problem of establishing a specified steady state

distribution of temperature along a moving fluid stream. Apart from the

influence that the exothermic or endothermic character of the interval

processes may have,the fact that there is thermal resistance between the

sources or sinks of heat and the fluid stream, together with stream motion

225.

means that the actual profile of temperature existing in the fluid will

not be the same as the applied profile. For example, consider a shell

and tube heat exchanger where the applied temperature is virtually constant

over all the surface of the tubes while the stream temperature changes from

one end to the other.

The choice of desired profiles in chemical reactors, in the fact of

exothermic or endothermic reactions is dealt with by Aris (2.1) and

Ammundson, Costa and Rudd (2.7). Nuclear reactors and heat exchangers have

their own substantial literature.

An example of the form of the equations, taken from Chapter 3 shows

the conditions to be met.

In the steady state the equations for the shell and tube heat

exchanger are

aU 2 G - V d 0-12I- e72 (U3-U2)

ax

C23G G

(U2-U3) - C12 (U2-U1) = o 2 2

where U1(x) is the shell temperature,

U2(x) the wall temperature and

U3(x) the fluid stream temperature. V is the velocity of flow (positive x)

and D a diffusivity, C2 and C3

thermal capacities and G23 and G12 thermal

conductances. It is required to choose U1(x) to produce a specified

profile U3(x). Once U

3(x) is defined U

1(x) can readily be determined but

some conditions of realizability have to be met.

dU d2U Thus if at some point there is a discontinuity in a7, then dx2

becomes infinite and the function U1(x) could not be produced. This

amounts to trying to maintain sharp kinks in the temperature profile while

0

226.

the dispersion term is trying to smooth them out - naturally the dispersion

will win. The profile must be continuous, with a continuous first

derivative at all points.

Having thus established the steady state conditions Control can be

considered in terms of deviations from this.

9.5 Comparison with Conventional Control

The simplest way of comparing the basic single probe control with the

conventional control schemes is on the basis of frequency response.

In Chapter VI as in this Chapter, two versions of the system (heat

exchanger) have been used for control design. One with a lag between

control demand and shell temperature, the other without.

In the first case the best control action was found to be achieved

with single probe control based on a probe near x = 0.2. In the second

case, with the probe at x = 0.5. The frequency responses of the overall

system to disturbances when these

6'51c.\ figs. *OM 'and, OW 'respectively.

forms of control are applied are shown in

In each case it can be seen that the simple probe control is

distinctly better at low frequencies, which is an important practical

advantage because the cumulative effects of slowly changing or continuous

errors may account for a large proportion of the losses due to ineffective

control. It must be pointed out however that the behaviour of the system

with a three-term controller could be improved in this respect by having

higher reset gainsIbut never quite to the point of complete error elimina-

tion.

The advantages of the single-probe control are better steady state

behaviour and better transient, and frequency, responses. The conventional

227.

control methods have the advantage for many processes of a more accessible

probe location - the actual system output, and less sensitivity of overall

performance to parameter changes.

9.6 Structure and Parameter Optimization

In the investigation of single-probe control the efforts were

directed to finding the best position and the best gains for the controller

by treating it all as one problemland in fact it became a one-parameter

optimization problem. In a similar way the multiprobe problem became that

of choosing a set of weighting factors for the available control measures.

There is however a slight difference between these two problems. The first

was ultimately concerned with finding the best location for an instrument.

The second was concerned with choosing scaling factors for a system whose

structure was already well defined.

The problem of choosing the best probe position is that of structural

optimization, as is that of choosing the best form of calculations to

make on the available information from the plant or process. The other

category) parameter optimization,is that of choosing the best numbers to

use in the calculation. If the distinction is drawn as that between what

an on-line computer can do (parameter optimization) and what it cannot do

(structural and physical changes), then there is a need for two levels of

control system improvement These two appear to be naturally allocated

to the two aspects of control implementation. Structural optimization is

readily performed on a suitable analogue computer, even if the exact values

of the required control parameters are not produced. The final parameter

optimization can be done on the real process where there is no question as

to the validity of the data produced but where changes in structure are

expensive and time consuming.

228.


The starting point for investigating state measure control is the

knowledge that for a continuously operating system the optimum control

action is dependant only on system state. For distributed parameter

systems this puts an excessive demand on instrumentation. Full knowledge

of the state of the system would require infinite amounts of instrumentation

and yet many industrial processes are controlled with very little.

The investigation in Chapter VIII of the instrumentation problem

on its own showed that adequate measures of state may, for many processes,

be obtained with relatively few instruments. This however is no measure

of what is required for control purposeslso the simplest thing is to start

again with the simplest possible arrangement and work up from there,

evaluating the advantages as the control systems become more complex.

This simple starting point is the single probe control system, using

a linear multiple of the measured state to demand the control action. It

degenerates to a one parameter optimization problem - the choice of probe

position.

The next stage, that of choosing the best weighting factors for all

available measures of state again reduces to a single-probe control, but by

way of a more complex parameter optimization problem.

Both these types of control were designed to produce optimal

behaviour against a localized measure of performance. When the performance

was assessed at a number of points it was necessary to have a distributed

control action which was space dependant as well as time dependant, and this

resulted naturally in the state being measured at a number of locations,

including measures of the state of the environment between control action

229.

and the part actually being controlled. In fundamental terms these are

just as much measures of state as the measures of the fluid stream

temperatures that were the object of the control efforts. The requirement

is consistent with the knowledge that if there was no energy storage or

memory in the environment of the flowing stream then the whole problem

could be reduced to considering each element of the flowing stream quite

independantly, and the control action profile would move along, side by

side with the disturbance profile'till the latter was suppressed.

The form of space dependent distributed control for the model used

here was found to be less sensitive to changes in parameters and more

capable of dealing with distributed disturbances than a control scheme with

the same overall capacity but which was only capable of producing one control

action for the whole of the distributed system.

Some minor modifications of normal concepts of feedback also arise

from these investigations. The process of measuring the input to a system

and using this to determine the control action is normally considered as a

'feedforward' control scheme. Similarly,if the output is measured and this

used to define controllit is 'feedback.' In the distributed parameter case

all possible shades between the two can arise and no real distinction be

drawn so long as both are really using measures of system state. Another

aspect of this is that this same classification would depend on the frame

of reference for observing the system. An observer moving with the fluid

stream through a distributed control system such as the one discussed above

sees the state of his environment observed and then corrected and again

observed and then corrected, giving the appearance of a feedback system,

yet to the static observer all that can be seen is a group of feedforward

devices.

230.

The most significant result is concerned with the overall structure

of the control system and its relationship to the form of the performance

criterion. The performance criterion must depend on one or more measures

of the state of the system. Similarly, control action is taken on the

basis of one or more measures of the state of the system, but they are

not generally the same measures. However, once the locations associated

with performance measures are fixed then the locations of the measures

needed for control are also largely determined. The reason is that, in

general terms, the time taken for control action to take effect after the

disturbance has been observed must be matched to the time taken for the

disturbance to reach the point at which performance is assessed. Each and

every location of a measure of state used in a performance criterion thus

defines a point of interest for control instrumentation and so the

controller structure is specified.

Probe at x=x1 Output at x•x2 /NPro

SkiL

WALa

P411113

WA.

Cotitte4

X.. 0 X Xi X

.13t

1,00# CONnt os.i.ta

a) System being studied.

G, (4P)

b) Signal flow graph for system.

Fig.9.1. Structure of single probe control

problem.

1 Sc

.231-

Steady—state amplitude response to external distributed forcing.

Steady—state amplitude response at x to input condition change.

G [

- G,(1-1 «G, 0.1 - 62v)

Controllergain required as function of probe position.

Fig.9.2. Steady state requirements for single probe

control.

a) Analogue arrangement for single probe contra:.

i /0 CfrAIVPIIFC IN f 7.rui14 En, 7.ere./.1.

fliPur

3140d.d.

10 0/1/1vv./ec

INPI/r

CON r4*' I1,

b) Analogue arrangement for Multiprobe control.

23S'

Fig.9.3. Analogue circuits for State measure control.

;34-

Ib

12

to

8

6

4

a 0 • •2 .3 •4 •$ •6 •?

Controller gain (G) for zero steady—state error as a funcIion of probe positiOn (X).

L0%6 I tawrit•u.

ylitm

..4%\,...............

Pie 14.1 •e

5

C.ite• 11e, ...........-'

• •2 .4 •6 .8 v i.a Performance measure (M) as function of probe position(X).

Fig.9.4. Gain requirements and performance of single probe control (Analogue results.)

0.4

M

03

02

01

0

J

4)

INPUT

b) P1

PS

P6

P7

Fig.9.5. Sketches of step responses of complete

system for various controller probe

positions, eaoh system adjusted for zero steady—state error. (No lag in

control loop.)

0

2,31

4

3

E

2

1

0 • 1 •s • 6 .8 x

Fig.9.6. Overall system sensitivity to parameter

as steadj-eLaLe cr in

output level (%-age of uncontrolled

error),(E), for 5% change in controller

loop gain as function of probe position

(X).

INPOr

CON71201, Our/Air

REGION

INS2WWWPir

COAlrat.cER

a) Single Control region applied TO distributed system which exhibits flow(and diffusion)properties

milwr reur

I , •

G G G G

b) Multiple oontrolregions applied to distributed system. (No allowance made for interaction effects.

Fig.9.7. Distributed control schemes.

O

a) Cascaddd control regions — signal flow structure.

b) Signal flow graph. Relevant parts for disturbance correction analysis are shown in heavy line.

Nomenclature: gi tranfer from input of line section to instrument

• in that section. g2 transfer from input of line section to output of

that dectiern. g3 transfer from input of controller to instrument

in that section. transfer from input of controller to output of

g4 line section. g5 transfer from input of controller to next

controller. g6 transfer from instrument in section to next

controller.

23a

Fig.9.8. Structure of Distributed Control Problem.

3

CR I

1 2 3 4 5 6 7 8 9 /0

CRl I I CR2 I I c„

a) Control regions for analogue representation of distributed control. 10 subsections of the system controlled by three regions of which CR1 is a dummy region and CR2 and CR3 are operative.

b) Final form for distributed control. Error measure for CR3 taken from probes 5 and 6, weighted 20% and 80% respectively. Correction for action already being taken in CR2 is measured alongside subsection 3.

239

rnPur

Fig.,9.9• Example of Distributed control

240:

Chapter X

PROGRESS AND RECOMMENDATIONS

10.1 Introduction

The original problems were based on four aspects of systems analysis

and control. They were; to find the underlying structure and characteri-

stics of behaviour of distributed parameter systems, to find out the best

methods of modelling them for computation and simulation, to develop criteria

for instrumentation and to establish the value of the available methods of

control engineering, while developing them and introducing new methods as

necessary.

The treatment of each aspect brought out, not only the results of a

numerical nature and those concerned with techniques but also brought to

light the inadequacies of some concepts,previously developed for lumped

parameter systems,which needed modification and extension to deal with

distributed parameter systems.

All the results are discussed in the context of the various aspects

of interest.

10.2 Systems - Flow and Diffusion

The transfer of heat, of matter, or of both is essential to all the

distributed parameter systems in the chemical industries and in virtually

all continuous processing of raw materials.

Two types of transfer are known; direct transportation in which

particles move along en masse from place to places and diffusion or

dispersion in which the random motion of particles,or of groups of particles,

eventually results in transfer of matter (or energy) from one place to

241.

another.

Even in laminar flow however, the diffusion effect is found to be

significant,and in many industrial processes fluid flow occurs at high

enough Reynolds numbers to give fully developed turbulent flow and the

consequent dispersion effect. Flow cannot occur without the diffusion or

dispersion effectI though diffusion transfer can occur in a static body or

material.

It appears from inconsistencies in other investigations and from the

available records of direct measurements that diffusion or dispersion is

often an important effect in many processes which are usually treated as if

only the flow was significant.

Since many industrial processes show spatial variations of state which

are predominantly functions of only one space dimension with flow in that one

directionia useful structure for the discussion of distributed parameter

systems is a flow and diffusion (or dispersion) process in one space

dimension and the time dimension.

Within this frame of reference it was found possible to characterize

these distributed parameter systems by a single number 11a, - the inverse of

a Peclet number - which could be evaluated from experimental data or

estimated.

For these systems it is necessary to establish clearly the concepts

of "input" and "output" when dealing with the end conditions. Thus if the

input function is defined as the time variation of state at a certain

reference point in the space dimension then it must be appreciated that this

does not necessarily specify the energy or material transfer into the system.

because of the bidirectional transfer associated with diffusion processes,

242.

and similarly for the "output". The real source of difficulty is that a

distributed parameter system can only be described within a similarly

distributed environment. It is not really compatible with lumped parameter

surroundings.

10.3 Models

Three types of mathematical model were considered for the distributed

parameter system, one based on the Laplace and Fourier transform descriptions,

one on a harmonic or functional analysis approach and the other a lumped

section model.

The Laplace transform produces a p-plane or "root-locus" description,

The Fourier transform is concerned with only one line in this same p-plane,

giving the "frequency response". Both give transfer functions which can be

readily developed from the partial differential equations. At this stage

care has to be taken to establish the physical realizability of the transfer

functions producel because the second order equations result in there being

a choice. The criterion of realizability is: "For input signals of an

exponential type having in their transformed form a positive or zero real

partthe resulting transfer function should have a magnitude response which

decreases with increasing spatial separation from the input point."

The transfer functions show the relationship between structural

changes in the system and changes in response. They explain certain

'resonance' phenomena characteristic of distributed flow and diffusion

systems which are not really resonant/ but just exhibit changes in response

when the cycle time of distributed forcing changes relative to the transit

time.

243.

The numerical evaluation of some of the simpler characteristics shows

the relative importance of the diffusion effect. The numerical evaluation

is tedious, though it could be done on a digital computer, and disguises

the distributed nature of the system, leaving it as a single unit with rather

complex characteristics.

The inversion of some of the transforms is possible, to give formulae

which can be used to evaluate physical parameters7but is not possible for

any but the simplest systems.

The use of harmonic and functional analysis models in which the

profiles of state as a function of the space variable are represented by

sums of series of suitable terms is a well established techniquelsuitable for

systems described by differential operators and boundary conditionst which

have convenient eigenfunctions. The type of system considered here does

not fall in this category. The use of other special functions cannot be

guaranteed to produce systems of equations for the modes which are stable in

the Routh-Hurwitz sense, and unless they meet this condition they are not

applicable.

The formal techniques can readily be demonstrated and are equivalent

to a summation transform in the space dimension.

The linear interpolation between two points as a means of profile

representation can by this technique be shown to be equivalent to the

lumped section model.

The lumped section models can readily be established from discrete

versions of the partial differential equations. There are no readily

applicable general rules about the size of sections required for adequate

solution of partial differential equations involving parabolic operators but

244.

for the systems considered here it has been shown that the section size is

dependant on the parameter 'a' and on the frequency spectrum of the distur-

bances to which the system is subject. The numerical evaluation of these

requirements showed that a useful measure is a critical number of sections,

inversely proportional to the parameter 'a' for the system.

Furthermore the model with this critical number of sections has the

same structure as a sequence of little stirred tank reactors, and it is

closely related in terms of its p-plane representation to the Pade delay

and to the system produced by applying linear interpolation to the profiles

of system state.

The lumped section equations can be manipulated to define the

structure of an electrical analogue which can be produced cheaply from

standard components. A high speed version of this was developed and proved

reliable and effective for further studies.

The lumped parameter model retains the distributed nature of the amt.(

system it is suitable for analogue or digital computation with very little

preliminary calculation of coefficients or testing for stability. It can

also be used in analysing non-linear systemsland systems with any physical

shape.

10.4 Control

Three methods of control design were considered; the first based on

established conventional techniques, frequency responses and three term

controllers, the second was the calculus of variations and the like and the

third a hybrid approach using an analogue computer to make a search, as in

an adaptive control scheme, for a solution already in a form suitable for

245.

direct application, but capable of meeting the same form of performance

criteria as in the variational problems.

Using only one measure of state at the output of a system which was

subjected to disturbances at its input, a conventional control scheme was

found to be simple to set up on the basis of transient responses. Its

overall performance depended on the form of disturbance to be corrected.

In general it was incapable of eliminating completely all steady state output

errors even if the system was capable of achieving this.

Design of conventional control schemes on the basis of frequency that'

responses presented no new problems provided the frequency responses were

available, because their calculation could be difficult. Design by root-

locus methods is not practicable for any but the simplest systems because

the necessary modifications and transformations of the p-plane plots become

prohibitively complex, and the frequency dependant response to signals other

than simple continuous sinusoids cannot be measured but has to be calculated.

Using the calculus of variations and invoking a condition of stability

for a system subject to some restricted but useful performance criteria the

optimum controller was evaluated as a feedback device in which the control

was computed as a direct function of system state. This was for lumped

parameter systems, but with such a system of the dimensionality frequently

required to represent a distributed parameter system, the calculations, even

on a large digital computer would become excessive.

The methods of the calculus of variations were extended (formally) to

cover the control of distributed parameter systems. When partial

differential equations are used for system description there results a

boundary condition in the space-time domain of interest, which, apart from

246.

the rest of the problem, makes solution extremely difficult. Representing

the system by integral equations can remove this problem becauseAdifferential

operator and boundary conditions are replaced by a Green's function

representation. The difficulty is that of establishing the Green's function.

This is essentially the same problem as occurs in the harmonic analysis

modelling method and is not generally soluble for parabolic operators.

The computational requirements for solving variational problems

either in the classical form, Pontryagins form or by Dynamic programming,

are enormous for all but the simplest systems and it is apparent that

similar or smaller demands would be made by the search through a multi-

dimensional domain of parameters needed to specify a control scheme directly.

Furthermore there are many precedents for this type of approach to problem

solution and the form of control system which would ultimately be produced,

if the variational problems could be solved, is often known in advance.

- With this knowledge available a high-speed analogue was used to

develop control schemes for a distributed parameter system, by optimizing

the choice of control defining parameters. Starting with the simplest

possible control system it was found that the most important parameter to be

chosen was the position of the instrument used to measure system state.

The location of the instrument was determined by the transport time lag in

the system and the lags in response of the control loop. When the system

was made potentially more complex,little further improvement was found so

long as the measure of performance was confined to an assessment of the state

at one point in the system (usually the output) and the resulting structure

was the same as in the simplest case. Measuring performance by considering

deviations from a desired state profile resulted in the need for distributed

247.

control which was a function of spatial position as well as timel and

consequently required control instrumentation at many more points.

In these applications the concepts of feedback and feedforward need

ammendment. Feedforward is usually considered as using advance information

to predict required control action. Feedback is the use of measured

results or state to demand control action of an error correcting nature.

In distributed parameter systems a measure of state may give advance

information in the feedforward sense as well, and this measure of state which

is fed-back is not necessarily the output or the state measure used in

performance assessment. Another factor is that a control scheme which

appears to be a feedforward scheme in the usual sense from the point of view

of a static observer may be reasonably considered as a feedback scheme from

the viewpoint of the moving stream. For these reasons the term 'state

measure control' seems more appropriate.

In comparison with conventional control(for output point quality),the

state measure control schemes gave beftarresults but other factors such as

sensitivity to parameter changes and the physical difficulties of probe

location must be considered. For distributed control with interconnected'

control zones there is no advantage to be gained from conventional methods

in these respects, because state measure control shows low sensitivity to

parameter changes and the same amount of instrumentation is required in

either case,

10.5 Instrumentation

In the established control theory of lumped parameter systems and in

the methods of the variational calculus the performance of a system is

248.

measured as a function of system state,and this performance measure(or the

elements of the state vector required to define Walso defines the control

action. The problem of instrumentation,just for knowing what is happening,

is not considered.

With distributed parameter systems it is not possible to make a

complete measure of state for any purpose. The best that can be done is to

make a finite number of separately located measures. It has been found that

the best measures of state for control purposes may be quite restricted in

number and have locations determined largely by the locations of the measures

for performance assessment and the relationship between time and space

displacements. Control and performance measure require different instru-

mentation and the former is largely determined by the latter.

For the purposes of protection and safety the need is for a reasonable

certainty that troubles will be located. A means of specifying instrument

locations to give any required degree of correlation between adjacent

measures has been described to give a basis for design.

The problem of giving a limited number of measures of state is that

of reducing the dimensionality of the state space. It was found that a

very much reduced state space could be used to give a visual representation

of average state but this was of little use for control purposes because the

measuras required for this and those needed for control were very different.

10.6 Final Summary and Further Requirements

Many industrial processes have a common distributed structure

involving both flow and dispersion phenomena. For control design purposes

the best way of representing them is generally by means of lumped parameter

249.

models.

Conventional methods of control design by frequency responses are

applicable, but the variational methods are not. The use of high speed

analogue computing has shown that simple control system configurations can

give good results. The most important factor is the relationship between

spatial displacement and time delay and this can be used to good advantage

by careful choice of instrument location for control.

Instrumentation for control is not necessarily the same as that

required for performance assessment and instrumentation for protection

requires different criteria again.

This study has been centred on linear systems with constant coeffici-

ents. Important cases which have not been treated are the non-linear case

such as a tubular reactor in which chemical reactions have widely varying

reaction rates, the non-constant coefficient cases such as systems with

varying velocities of flow and combination of both difficulties such as

distillation columns with control by means of reflux and reboil rates.

Naturally the results of this study suggest possible starting points, for

example, choice of instrument locations as flow rates vary, and non-linear

functions of state measure to give the required amount of correction for

errors, but for these problems no general model is possible and each type

of problem will have to be treated separately.

250.

Chapter XI

BIBLIOGRAPHY

11.1 Introduction

The references listed here are arranged to correspond with the

material content of the chapters. Thus references for Chapter II are found

in Section 2, with numbers such as 2.11, 2.12 etc.

Naturally many references are relevant to more than one aspect of the

subject and these are noted by means of their number, to avoid duplication,

whenever they are required more than once.

Many more references are given than are called for in the text,

particularly for Chapter II which deals with aspects not normally covered

in Control Engineering texts.

Within the lists for each chapter the references are again subdivided

into groups corresponding to subject matterl but these divisions are necess-

arily only a guide. The arrangement after that is in alphabetical order of

authors names.

11.2 Distributed Parameter Systems

These references correspond to the subject matter of Chapter 2. The

list includes references to work outside the immediate scope of text because

there is a need for control engineers to have knowledge of and access to

information about chemical and physical phenomena and plant design which,

while not directly relevant to the control problems, are of interest because

they determine the dynamic behaviour of processes.

The chemical engineering literature is extensive and the main interest

251.

of this list is centred on the dynamic behaviour of processes. Of necessity

there are many references here which are also relevant to later chapters.

The list is in groups corresponding roughly to the main type of

chemical and industrial process of interest.

252.

Chemical Reactors

2.1 Aris, R. / The Optimal Design of Chemical Reactors (A Study in

Dynamic Programming) / B / Vol. 3 of Series "Mathematics in Science

and Engineering" / Academic Press (N.Y., London) 1961.

2.2 Barkelew, C.H. / "Stability of Chemical Reactors" / P / in

"Reaction Kinetics and Unit Operations" / American I. Chem. Engrs.,

Chem. Eng. Progress Symposium Series, No. 25, 1959 (Vol. 55).

2.3 Deans, H.A., Lapidus, L. / "A Computational Method for Predicting

and Correlating the behaviour of fixed bed reactors"; Part 1:

Derivation of Model for non-reactive Systems, Part 2: Extension

to chemically reactive System / P / A.I.Ch.E. Journal. Vol. 6,

No. 4, (Dec. 1960).

2.4 Kermode, R.I., Stephens, W.F. / "Dynamic Behaviour of a Continuous

Stirred Tank Reactor" / P / Can. J. Chem. Eng., Vol 39, No. 2.

(April 1961) pp 81 - 85.

2.5 Kabota, M., Namkoong, S., Akehata, T., Shindo, M. /"Optimum Process

Conditions for a Completely Mixed Multistage Reactor" / P / Can. J.

Chem. Eng., Vol. 39, No. 2. (April 1961) pp 64 - 66.

2.6 Williams, T.J., Otto, R.E. / "A generalized chemical processing

model for investigation of computer control of chemical processes"

/ P / A.I.E.E. Paper CP60 - 119, Winter General Meeting, Feb. 1960.

Diffusion and Dispersion

2.7 Amundson, Coste, Rudd. / P /

/ Can. J. Chem. Eng. Vol. 39, No. 4. (1961).

2.8 Bird, R.B., / "Theory of Diffusion" / Part of book / in 'Advances

in Chemical Engineering, Vol. 1.' Academic Press 1956.

253.

2.9 Clauson, W.A., / "On Unsteady heat transfer in a hollow cylinder

or Sphere" / P / A.I.Ch.E. Progress Symposium Series, Vol. 57, No. 32. (1961).

2.10 Crank, J., / "Mathematics of Diffusion" / B / Clarendon Press,

Oxford, 1956.

2.11 Dankwerts, P.V., / "Continuous Flow Systems" / P / Chem. Eng. Sci.,

Vol. 2. (1953) p. 1 - 13.

2.12 Ebach, White / P / A.I.Ch.E. Journal, Vol. 4, No. 2. (June 1958) pp 161-169.

2.13 Goldstein, S., / "Modern Developments in Fluid Mechanics" / B /

Oxford, Clarendon Press 1938.

2.14 Kramers, Hand, Alberda, G., / "Frequency Response Analysis of

Continuous Flow Systems" / P / Chem. Eng. Sci., Vol. 2. (1953)

pp 173 - 181.

2.15 Levenspiel, O., Smith, W.K., / "Notes on the diffusion-type model

for the longtitudinal Mixing of Fluids in Flow" / P / Chem. Eng.

Sci., Vol. 6. (1957) pp 227 - 233.

2.16 Liles, A.W., Geankoplis, C.J., / "Axial Diffusion of Liquids in

Packed Beds and End Effects" / P / A.I.Ch.E. Journal, Vol. 6, No. 4. (Dec. 1960).

2.17 Mason, H.L., / "Analog Simulation of Zone Melting" / P / Nat. Bur.

Standards, Journal of Research, 65C, No. 2, (April-June 1961)

P. 97.

2.18 Nissan, A.H., Hausen, D., / "Heat and Mass Transfer Transients in

Cylinder Drying" / P / A.I.Ch.E. Journal, Vol. 6, No. 4. (Dec. 1960).

254.

2.19 Taylor, Sir G., / "Dispersion of Soluble Matter in Solvent flowing

Slowly through a tube" / P / Proc. Royal Soc. of London, A219,

(1953) p. 186 - 203.

also / "Dispersion of Matter in Turbulent Flow Through

a Pipe" / Proc. Royal Soc. of London, A223, (1954) pp 446 - 468. also A225 (1954) Addition to paper in A219.

2.20 Vincent, G.C., Hougen, J.D., Driefke, G.E. / "Fluid Mixing in Shell

and Tube Heat Exchangers" / P / Chem. Engineering Progress, Vol. 57, No. 7. (1961) pp 48 - 52.

Distillation Columns

2.21 Acrivos, A., Amundson, N.R., / "Solution of transient stagewise

operations on an analog computer" / Ind. Eng. Chem. Vol. 45, (1953) PP 467 - 471.

2.22 Baker, M.F., Edwards, L.L., Harper, W.T., Witte, M.D., Gerster, J.A.,

/"Experimental transient response of a pilot plant distillation

column" / Paper at A.I.Ch.E. Annual Meeting, Washington, Dec. 1960.

2.23 Bowman, J.R., Briant, R.C., / "Theory of Performance of Packed

Rectifying Columns"/ P / Industrial and Engineering Chemistry,

Vol. 39 (1947) p. 745.

2:24 Chilton, Colburn., / "Continuous Concepts to Replace the McAbe

and Thiele method" / P / Ind. Chem. Eng., Vol. 27, (1935) p. 255.

2.25 Coates, J., Pressburg, B.S., / "How to Make Distillation Calcula-

tions" / P / Chemical Engineering.Pt. I Vol. 68, No. 4. (Feb. 1961) Pt. II Vol. 68, No. 6. (Mar. 1961).

2.26 Davidson, J.F., / "The transient behaviour of plate distillation

columns" / P / Trans. Inst. Chem. Engrs., Vol. 34 (1956) pp 44 - 52.

255.

2.27 Geddes, R.L., /"Progress in Fractional Distillation" / historical

survey paper in 'Reaction Kinetics and Unit operations' / Amer. I.

Ch. Engrs., Chemical Engineering Progress Symposium Series / No. 25,

1959, Vol. 55.

2.28 Houtman, Husain., / "Mathematical treatment of batchwise fraction-

ation / Chem. Eng. Sci., Vol. 5. (1956) p. 178.

2.29 Lamb, D.E., Pigford, R.L., / "Dynamic Characteristics and Analog

Simulation of Distillation Columns" /,Paper at A.I.Ch.E. Annual

Meeting, San Francisco, Dec. 1959.

2.30 McIntyre, R.L., Shelton, R.O., / "Fractionater Design with Automatic

Computing Equipment" / 2 parts,in 'Computer Techniques in Chemical

Engineering', A.I.Ch.E., Progress Symposium Series, No. 21, Vol. 55,

(1959).

2.31 O'Brien, N.G., Franks, R.G.E., / "Development and application of a

general purpose analogue computer circuit to steady state multi-

component distillation calculations" / Paper in 'Computer Techniques

in Chemical Engineering, A.I.Ch.E., Progress Symposium Series, No. 21,

Vol. 55, (1959).

2.32 Rijnsdorp, J.E., / "Computers in Distillation Dynamic Research" /

P / at 'Symposium on Computers for the Chemical Engineer',

University of Birmingham, 28th March 1961.

2.33 Rijnsdorp, J.E., and Maarleveld, A., / "Use of Electrical Analogues

in the Study of the dynamic behaviour and control of distillation

columns" / P / Joint Symposium on Instrumentation and Computation

in process development and plant design, I.Ch.E., S.I.T., Brit.

Comp. Soc., London, May 1959.

2.34 Rose, A., Johnson, C.L., Williams, T.J., / "Transients and Equili-

256.

bration time in Continuous Distillation" / P / Ind. Eng. Chem.

Vol. 48, (1956) pp 1173 - 1179.

2.35 Rose, A., Johnson, C.L., Williams, T.J., / "Stepwise plate-to-plate

computation of batch distillation curves" / P / Ind. Eng. Chem.

Vol. 43 (1951) pp 2459 - 64.

2.36 Rose, A., Johnson, C.L., / "The Theory of unsteady state distillation"

/ P / Chem. Eng. Prog., Vol. 49, (Jan. 1953) pp 15 - 21.

2.37 Rosenbrock, H.H. / "Transient Behaviour of Distillation Columns"

/ P / Brit. Chem. Eng., Vol. 3. (1958) p. 364, P. 432, p. 491.

2.38 Rosenbrock, H.H., Armstrong, W.D., Wilkinson, W.L., / "Distillation

Column Transient Behaviour", Pt. I: Theory, Solution of Equations,

Pt. II: Comparison with Experimental Results / Trans. I. Chem. Eng.

Vol. 35. (1957) No. 5.

2.39 Rosenbrock, H.H., Tavendale, Storey, Challis / "Transient Behaviour

of Multicomponent Distillation Columns" / P / Proceedings IFAC

Congress, Moscow 1960, Published by Butterworths, London 1961.

2.40 Rosenbrock, H.H., /"A theorem of Dynamic Conservation for Distillation'

Trans. Inst. Chem. Engrs. Vol. 38, (1960).

2.41 Rosenbrock, H.H., / "The transient behaviour of distillation Columns

and Heat Exchangers: a Historical and critical review" / P /

European Federation of Chemical Eng. June 1962 and Trans. I.Che.E.

December 1960.

2.42 Schwartz, L.M., / "Dynamic Methods of bubble Plate Analysis" /

M.I.T. Electronic-Systems Lab. Report No. 7793 - R-4. (1959)

2.43 Voetter, H., / "Response of Concentrations in a Distillation Column

257.

to disturbances in the feed Composition" / in 'Plant and Process

Dynamic Characteristics', Butterworths, London 1957, pp 73 - 96.

2.44 Wilkinson, W.L., Armstrong, W.D., / "An approximate method of

predicting composition response of a fractionating column" / Chem.

Eng. Science, (1957) Vol. 7. pp 1 - 7.

2.45 Wilkinson, W.L., Armstrong, W.D., / "An investigation of the

transient response of a Distillation Column"/ in 'Plant and Process

Dynamic Characteristics', Butterworths, London 1957, pp 56 - 72.

Distributed Parameter Systems

2.46 Meredeth, J.F., / Freeman, E.A., /

"The Simulation of Distributed Parameter Systems, with particular

reference to process control problems" / P / I.E.E. Paper 2376 M

(July 1957), Published in Part B of Proc. I.E.E., Vol. 105, (1958).

Heat Exchanger

2.47 Cima, R.N., London, A.L., / "The transient response of a two-fluid

Counterflow Heat Exchanger - the gas turbine regenerator" / Trans.

A.S.M.E., Vol. 8. (July 1958) pp 1169 - 1179.

2.48 Clark, J.A., Arpaci, U.S., Treadwell, K.M., / "Dynamic Response of

Heat Exchangers having Internal Heat Sources" / P / Trans. A.S.M.E.,

Vol. 80, No. 3. (April 1958) pp 612 - 634 (Two Parts).

2.49 Cohen, W.C., Johnson, E.F., / "Dynamic Characteristics of Double

Pipe Heat Exchangers" / P / Ind. Eng. Chem. Vol. 48, No. 6.

(June 1956) pp 1031 - 1034.

2.50 Florentin, J.J., Westcott, J.H., Reswick, J.B., / "Correlation

Analysis of a Heat Exchanger" / Joint Symposium on Instrumentation

and Computation, I.Ch.Engrs., 1959.

258.

2.51 Fricke, L.H., Morris, H.J., Otto, R.E., Williams, T.J., / "Process

Dynamics and Analog Computer Simulation of Shell and tube Heat

Exchangers" / Chem. Eng. Prog. Symposium Series, Vol. 56, 1960,

No. 31.

2.52 Gilmour, C.H., / "Performance of Vapourizers: Heat transfer analysis

of Plant Data" / A.I.Ch.E. Prog. Symp. Series, Vol. 55, No. 29,

p. 67.

2.53 Hempel, A., / "On the Dynamic Behmeibur of Condensing Steam Liquid,

tube and shell heat exchangers" / Report from Chr. Michelsens

Institutt (Bergen) Ref. No. 59, 18/AHe 1.

2.54 Lees, S., Hougen, J.0., / "Pulse testing a model Heat Exchange

Process" / P / 2nd Eng. Chem. Vol. 48, No. 6. (June 1956) pp 1064 -

1068.

2.55 Mozley, J.M., / "Predicting Dynamics of Concentric Pipe Heat

Exchangers" / 2nd Eng. Chem. Vol. 48, No. 6. (June 1956) p. 1035.

2.56 Paynter, H.M., Hainsworth, B.D., Tivy, V.V., / "Dynamic Analysis of

Heat Exchanger Control / I.S.A. Journal, June 1957.

2.57 Rizika, J.W., / "Thermal Lags in Flowing Systems Containing Heat

Capacitors" / Trans. A.S.M.E., Vol. 76 (April 1954) p. 411.

2.58 Rizika, J.W., / "Thermal lags in flowing incompressible fluid Systems

containing Heat Capacitors / P / Trans. A.S.M.E., Vol. 78, No. 7.

(October 1956).

2.59 Taborek, J.J. / "Organization of heat Exchanger Programs on digital

Computers" / P / A.I.Ch.E. Prog. Sym. Series, Vol. 56, No. 30.

(Deals with plant design).

259.

Heat and Mass Transfer

2.61 Eckert, E.R.G., / "Introduction to the transfer of Heat and Mass"

/ B / McGraw-Hill, 1950.

2.62 Harnett, J.P., Irvine, T.F., / "Nusselt Values for Estimating

Turbulent Liquid Metal Heat Transfer in non-circular ducts" / P /

A.I.Ch.E. Journal, Vol. 3, (Sept. 1957) PP 313 - 317.

2.63 McAdams, W.M., / "Heat transmission" / B / McGraw-Hill, 1954.

2.64 Metzner, A.B., Vaughan,R.D., Houghton, G.L., / "Heat transfer to

non-Newtonian Fluids" / P / A.I.Ch.E. Journal, Vol. 3, No. 1.

pp 92 - 100. (March 1957).

2.65 Reilly, P.M., / "Unsteady State Heat Transfer in Stationary Packed

Beds" / P / A.I.Ch.E. Journal, Vol. 3, No. 4. (Dec. 1957)

pp 513 - 516.

2.66 Smith, J.W., Epstein, N., / "Effect of Wall ROUghness on Convective

Heat Transfer in Commercial Pipes" / P / A.I.Ch.E. Journal, Vol. 3,

No. 2. (June 1957) pp 242 - 248.

Nuclear Reactors

2.67 Glasstone, S., Edlund, M.C., / "The Elements of Nuclear Reactor

Theory" / B / MacMillan 1955.

Partial (and Ordinary) Differential Equations

2.68 Bateman, H., / "Partial Differential Equations" / B / Cambridge

University Press, 1932 and 1959.

2.69 Chaundy, T., / "The Differential Calculus" / B / Oxford, Clarendon

Press, 1935.

260.

2.70 Courant, R., / "The Differential and Integral Calculus" / B /

(Two Vols.) Blackie, 1936.

2.71 Forsyth, A.R., / "The Theory of Differential Equations" / B /

Cambridge University Press, 1902.

2.72 Marshall, W.R., Pigford, R.L., / "The Application of Differential

Equations to Chemical Engineering" / University of Delaware, 1947.

2.73 Miller, N., / "Differential Equations" / B / Oxford University

Press, 1935.

2.74 Piaggio, H.T.H., / "Differential Equations" / B / Bell.

Probability

2.75 Feller, W., / "Chance Processes and Fluctuations" / Chap. 6 in

'Modern Mathematics for the Engineer' / 2nd Series / Editor

E.F.Beckenback, McGraw-Hill. 1961.

11.3 Frequency Response and Root Locus

The books and papers listed here have been classified into groups

by subject matter, 'General Theory', 'Root Locus Methods', 'Industrial

Engineering Examples', 'Practical Measurement', and 'Realizability'. A

few references are also given to items in Section 2 where these are also

relevant to this section.

General Theory

3.1 Campbell, G.A., Foster, R.M., / "Fourier Integrals for Practical

Applications" / B / Van Nostrand, N.Y., 1948.

261.

3.2 Carslaw, H.S., and Jeager, J.C., / "Operational Methods in Applied

Mathematics" / B / Oxford University Press, 1941.

3.3 Churchill, R.V., / "Modern Operational Mathematics in Engineering"


3.4 Cutteridge, O.P.D., / "Approximate Transient Response Calculations

using some Special Sets of Polynomials" / P / I.F.A.C. Congress,

Moscow, 1960.

3.5 Papoulis, A., / "The Fourier Integral and its Applications" / B /

McGraw-Hill, 1962.

3.6 Pipes, L.A., / "Applied Mathematics for Engineers and Physicists"

/ B McGraw-Hill, 1958.

3.7 Tsien, H.S., / "Engineering Cybernetics" / B / (See Chap. I)

McGraw-Hill, 1954.

(See also: reference 2.10 (Crank).

Root Locus Methods

3.8 Blackman, P.F., / "Pole-Zero Approach to System Analysis" / Control

Monograph 2, Rowse Muir Publications.

3.9 Blackman, P.F., / "Root-Locus Methods" / Lecture Notes, Electrical

Engineering Department, Imperial College, London.

3.10 Evans, W.R., / "Graphical Analysis of Control Systems" / P / Trans.

A.I.E.E., Vol. 67, 1948, pp 547 - 551.

3.11 Evans, W.R., / "Control System Synthesis by the Root Locus Method"

/ P / Trans. A.I.E.E., Vol. 69, 1950, p. 66.

262.

3.12 Evans, W.R., / "The use of Zeros and Poles for Frequency Response

or Transient Response" / P / Trans. A.S.M.E., Vol. 76, No. 8,

(November 1954) pp 1335 - 1343.

(See also: reference 3.7 (Tsien, Chap. IV).

Industrial Engineering Examples

3.13 Armstrong, W.D., Wood, R.M., / "An Introduction to the theoretical

evaluation of the frequency response of a distillation column to

change in reflux rate" / P / Trans. Inst. Chem. Engrs., Vol. 39,

(1961) pp 80 - 85.

3.14 Fan, Liang-tseng., Ahn, Yong-kee., / "Frequency Response of Tubular

Flow Systems" / P / Joint Automatic Control Conference, New York,

June 1962. Paper 7-2 (A.I.E.E., A.I.Ch.E., I.S.A.).

3.15 Paynter, H.M., / "A new method for evaluating dynamic response of

counterflow and parallel flow heat exchangers" / P / Trans. A.S.M.E.,

Vol. 78, No. 4. (May 1956)

3.16 Trans. A.S.M.E., Vol. 76, No. 8. records papers at "Frequency

Response Symposium" held December 1953. (Includes ref. 3.12 (Evans).

(See also: 2.14 (Kramers, Hand, Alberda), 2.53 (Hempel), 2.58

(Rizika).

Practical Measurement

3.17 Hougen, J.0., Walsh, R.A., / "Pulse Testing Method" / P / Chem. Eng.

Prog. Vol. 57, No. 3. (March 1961) p. 69.

3.18 Seamans, R.C., Blasingame, B.P., Clementson, G.C., / P / J. Aero.

Sci. Vol. 17, 22. (1950).

(See also: 2.54 (Lees and Hougen).

263.

Physical Realizability

3.19 Valley, G.E., Wallman, H., / "Vacuum Tube Amplifiers" / Book, deals

with 'Paley-Weiner' Criterion / McGraw-Hill, 1948.

(See also: 3.5 (Papoulis) and 3.9 (Blackman p. 89).

11.4 Lumped Parameter Models

The background material for this section is largely in the field of

computational methods, in which the step by step integration of partial

differential equations require the equations to be quantized, both in space

and time dimensions. Because of their intractability) parabolic type

equations, with the exception of those based on the Laplacian operator,

receive little attention.

Much the same comment can be made about the relaxation methods which

form another source of information about quantized models.

Some practical work on industrial problems yields results of interest

but in these little attention has been paid to the basis for specifying the

quantization sizes.

For the actual type of analogue developed in the latter part of

Chapter IV almost any standard text on transistor electronics Would prbvide

sufficient background.

Computational Methods

4.1 Collatz, L., / "Numerical Treatment of Differential Equations" / B /

Springer-Verlag, 1959.

4.2 Crandall, S.H., / "Engineering Analysis" B McGraw-Hill, 1956.

264.

4.3 Forsyth, G.E., Wasr_w, / "Finite Difference Methods for Partial

Differential Equations" / B / Wiley, 1960.

4.4 Lance, G.N., / "Numerical Methods for High Speed Computers" / B /

Iliffe, 1960..

4.5 Lanczos, C., / "Linear Differential Operators" / B / Van Nostrand,

1962.

4.6 National Physical Laboratory (Staff Members) / "Modern Computing

Methods" / B / H.M.S.O. (D.S.I.R.), Notes on Applied Science,

No. 16, 1961.

4.7 Ralston, A., Will, H., / "Mathematical Methods for Digital Computers"

/ B / Wiley, 196o.

4.8 Redish, / "Computational Methods" / B / E.U.P., London, 1961.

4.9 Todd, J., / "Survey of Numerical Methods" / B / McGraw-Hill, 1962.

Practical Problems and Models

4.10 Heindlhofer, K., Larsen, B.M., / "An Electrical Analogue of the flow

in a Regenerator System" / P / T.P.1798, Metals Technology, Aug. 1945.

4.11 Hellman, S.K., Habetler, G., Babrov, H., / "Use of Numerical Analysis

in the transient solution of two dimensional Heat Transfer Problem

with Natural and Forced Convection" / P / Trans. A.S.M.E., Vol. 78,

No. 6. (Aug. 1956) pp 1155 - 1161.

4.12 Juhasz, S., / "Hydraulic Analogy for transient Cross-flow Heat

Exchangers" / A.S.M.E. Paper 57-A-125.

265.

4.13 Pasckis, V., Hlinka, J.W., / "Electric Analog Studies of the transient

behaviour of Heat Exchangers" / P / Trans. N.Y. Acad.Sci., Ser. II,

Vol. 19, No. 8. (June 1957) pp 714 - 730.

4.14 Tipler, W., / "An Electrical Analog to the Heat Regenerator" / P /

Proc. of VIIth International Congress for Applies Mechanics, Vol. 3,

1948 (p. 196).

(See also: References 2.51 (Fricke et al), 2.55 (Motley).

Relaxation Methods

4.15 Clauson, W.A., / "On Unsteady State heat transfer in a hollow

Cylinder or Sphere' / P / A.I.Ch.E. Prog. Sym. Series, Vol. 57,

No. 32, 1961.

4.16 Radd, M.E., Tek, M.R., / "Engineering Applications of relaxation

procedures by digital computation " / P / A.I.Ch.E. Journal,

March 1959.

4.17 Shaw, F.S., / "An Introduction to Relaxation Methods" / B / Dover.,

Publications, N.Y., 1953.

4.18 Southwell, R.V., / "Relaxation Methods" / B / Oxford University

Press, 1940.

Transistor Electronics

4.19 Hunter, L.P., / "Handbook of Semiconductor Electronics" / (2nd Ed.)

McGraw-Hill, 1962.

11.5 Harmonic and Functional Analysis

The basic techniques are treated in a number of texts but there is

little material related to parabolic differential operators, except the

Laplace operator.

266.

5.1 Churchill, R.V., / "Fourier Series and Boundary Value Problems"


5.2 Hockney, R.W., Jeffries, T.O., / "The Use of Analog Computers in

Predicting the Space Time Behaviour of Nuclear Reactors" / P /

Proc. I.E.E., Part A, Vol. 109, (1962) p. 131.

5.3 Kantorovitch, L.V., Krylov, V.I., / "Approximate Methods of Higher

Analysis" / B / (Translated by C.D. Benster) / Noordhoff, Groningen,

Netherlands, 1958.

5.4 Kautz, W.H., / "Approximation over a Semi-Infinite Interval" / M.S.

Thesis, M.I.T., 1949.

5.5 Lanczos, C., / "Applied Analysis" / B / Pitman, London, 1957.

5.6 Pipes, L.A., / "Applied Mathematics for Engineers and Physicists" /

B / McGraw-Hill, 1956.

5.7 Whittaker, E., Robinson, G., / "The Calculus of Observations" / B /

Blackie, (Reprint of 4th Ed.) 1952.

(See also: 4.5 (Lanczos), 2.10 (Crank).

11.6 Conventional Control

There is considerable general control engineering literature but

virtually none dealing with the specific problems of control of diStributed

parameter systems. Such coverage as there is for this is to be found in the

chemical engineering literature, dealing with specific control problems.

General

6.1 Ahrendt, W.R., Taplin, J.F., / "Automatic Feedback Control " / B /

McGraw-Hill, 1951.

267.

6.2 Grabbe, E.M., Ramo, S., Woolridge, D.T., /(Editors) / "Handbook of

Automation, Computation and Control" / 3 Vols / Wiley, 1961.

6.3 Murphey, G.J., / "Basic Automatic Control Theory" / B / Van Nostrand,

1957.

6.4 Newton, G.C., Gould, L.A., Kaiser, J.F., / "Analytical Design of

Linear Feedback Controls" / B / Wiley, 1957,

6.5 Seifert, W.W., Steeg, C.W., / "Control Systems Engineering" / B /

McGraw-Hill, 1960.

6.6 Smith, 0.J.M., / "Feedback Control Systems" / B / McGraw-Hill, 1958.

6.7 Truxal, J.G., / "Control System Synthesis" / B / McGraw-Hill, 1955.

6.8 Westcott, J.H., / "The Synthesis of Electrical Networks (with

particular reference to Servomechanisms)" / Ph.D. Thesis, London

University, 1950.

Industrial Applications

6.9 Catheron, A.R., Hainsworth, B.D., / "Dynamics of Liquid Flow Control"

/ P / Ind. Eng. Chem. Vol. 47, pp 2248 - 2249 (1955). (see also 6.14)

6.10 Rademaker, 0., Rijnsdorp, J.E., / "Dynamics and Control of continuous

Distillation Columns" / P / Proc. 5th World Petroleum Congress,

New York, 1959. Section VII, pp 59 - 78.

6.11 Rose, A., Williams, T.J., / "Automatic Control in Continuous

Distillation" / P / Ind. Eng. Chem. Vol. 47, pp 2248 - 2249 (1955).

(see also 6.14).

6.12 Rosenbrock, H.H., / "A Theorem of Dynamic Conservation for Distilla-

268.

tion" / Trans. Inst. Chem. Engrs. Vol. 38, 1960. (Also C.J.B.

Research and Development Report).

6.13 Tivy, V.V., / "Automatic Control of Fractionating Columns" /

Petroleum Refiner, Vol. 27, (1948) pp 603 - 608.

6.14 Williams, T.J., Harnett, R.T., Rose, A., / "Automatic Control in

Continuous Distillation" / Ind. Eng. Chem. (1956) Vol. 48,

pp 1008 - 1019.

6.15 Ziegler, J.G., Nichols, N.B., / "Optimum Settings for Automatic

Controller" / P / Trans. A.S.M.E., vol. 64, (1942) pp 759 - 768.

(See also: 2.56 (Hainsworth et al.), 2.4 (Kermode and Stevens).

11.7 Variational Methods

The literature on the calculus of variations, dynamic programming

and Pontryagin's method is extensive. Many standard texts have been written

on the classical calculus of variations, much of it containing in principle

all that has subsequently been written about control problems but in a

different form.

The sections in this list of references correspond to the above

mentioned headings together with some references to integral equations which

form the background for the development of the calculus of variations applied

to systems described by integral equations. The section about performance

criteria deals with both the earlier servomechanisms aspects and the effects

of changing performance criteria in the variational theory. The references

to Liapunov methods are relevant to the theory of sub-optimal control

systems.

269.

Calculus of Variations and Applications

7.1 Beecher, A.E., / ":Synthesis of Optimum Distillation Controller" /

M.I.T. Electronic Systems Laboratory Report, 7793-R-7, June 1960.

7.2 Bilous, 0., Amundson, N.R., / "Optimum Temperature Gradients in

Tubular Reactors" / P / Chem. Eng. Sci., Vol. 5, (Nos. 1 and 2),

p. 81, p 115.

7.3 Bliss, G.A., / "Lectures on the Calculus of Variations" / B / Univ.

of Chicago Press (Pheonix Science Series) 1961.

7.4 Dreyfus, S.E., / "The Numerical Solution of Variationsl Problems" /

J. Meth. Anal. and Appl., Vol. 5, pp 30 - 45 (1962).

7.5 Elsgole, L.E., / "Calculus of Variations" / B / Pergamon Press Ltd.,

1961.

.7.6 Fox, G., / "Introduction to the Calculus of Variations" / B /

Oxford University Press, 1950.

7.7 Friedland, B., / "The Structure of Optimum Control Systems" / Paper

at 2nd Annual J.A.C.C., Boulder, Colorado, June 1961: A.S.M.E.

Paper -61-JAC-1.

7.8 Gould, L., Kipiniak, W., / "Dynamic Optimization and Control of a

Stirred Tank Chemical Reactor" / Trans. A.I.E.E. Comm. and Electr.

(Jan. 1961), No. 51, pp 734 - 746.

7.9 Hancock, / "Calculus of Variations" / B / University of

Bulletin of Mathematics No. 1, 1904.

7.10 Hildebrande, F.B., / "Methods of Applied Mathematics" / B / Prentice-

Hall, 1952. (See Chapter II).

270.

7.11 Katz, S., / "Best Temperature Profiles in Plug Flow Reactors :

Methods of the Calculus of Variations" / P / Ann. New York Acad.

of Science, 1960, Vol. 84, Art. 12, 441.

7.12 Kipiniak, W., / "Dynamic Optimization and Control: A Variational

Approach" / B / M.I.T. Press and John Wiley, 1961.

7.13 Lanczos, C., / "The Variational Principles of Mechanics" / B /

University of Toronto Press, 1950.

7.14 McCann, M.J., / "Introduction to Variational Methods for Optimal

Control" / P / Trans. Soc. of Inst. Tech., Vol. 13, No. 4, (Dec. 1961) pp 232 - 237.

7.15 Weinstock, R., / "Calculus of Variations" / B / McGraw-Hill, 1952.

Dynamic Programming

7.16 Bellman, R., / "Dynamic Programming" / B / Princeton Univ. Press,

1957.

7.17 Bellman, R., / "Adaptive Control Processes, A Guided Tour" / B /

Princeton Univ. Press, 1961.

7.18 Bellman, R., Dreyfus, S.E., / "Applied Dynamic Programming" / B /

Princeton Univ. Press, 1962.

7.19 Dreyfus, S.E., / "Dynamic Programming and the Calculus of Variations"

/ P / J. Math. Anal. and App., Vol 1, No. 2., (1960) pp 228 - 239.

7.20 Ho, Y.C., / "A Study of the Optimal Control of Dynamic Systems" /

Cruft Lab. Tech. Report, Harvard Univ., No. 335, Office of Naval

Research NU-372-012.

271.

Integral Equations

7.21 Margenau, Murphey, / "Mathematics of Physics and Chemistry" / B /.

7.22 Mikhlin, / "Integral Equations" / B / Pergamon Press.

7.23 Murnagham, F.D., / "Applied Mathematics" /.B / Wiley, 1948.

7.24 Tricomi, / "Integral Equations" / B / Interscience.

(See also: 5.3 (Kantorovich and Krylov), 7.10 (Hildebrand).

(Pontryagin's) Maximum Principle and Applications to Distributed Parameter Systems

7.25 Butkovskii, A.G., Lerner, A.Yo., / "On Optimal Control of Systems

with Distributed Parameters" / Paper intended for presentation

Department of Engineering, Cambridge University, 27th July, 1961.

7.26 Butkovskii, A.G., / "Optimum Processes in Systems with Distributed

Parameters" / Automatika i Telemekhanika, Vol. 22, No. 1. (Jan. 1961).

7.27 Butkovskii, A.G., / "Maximum Principle for Optimum Systems with

Distributed Parameters" / Automatika i Telemekhanika, Vol. 22,

No. 10 (Oct. 1961).

7.28 Butkovskii, A.G., / "Approximate Methods for Solving Problems of

Optimum Control of Distributed Parameter Systems" / Automatika i

Telemekhanika, Vol. 22, No. 12 (Dec. 1961).

7.29 Chang, Jen-Wei / " A Problem in the Synthesis of Optimal Systems

Using Maximum Principle" / Automatika i Telemekhanika, Vol. 22,

No. 10 (Oct. 1961).

7.30 Fuller, A.T., / "Pontryagin's Method for the Optimization of Non-

Linear Control Systems" / Short paper from Dept, of Engineering,

Cambridge University.

272.

7.31 Pontryagin, L.S., / "Optimal Control Processes" / Paper at IIIrd

General Assembly, International Math. Union., St. Andrews, Aug. 1958.

7.32 Pontryagin, L.S., / "Some Mathematical Problems arising in connection

with the Theory of Optimal Systems of Automatic Control" / Proc.

U.S.S.R. Acad, Sci., Vol II, (1957).

7.33 Rozonoer, L.I., / " Sufficient Conditions for Optimality" / Dokl.

Akad. Nauk, U.S.S.R., 127, (1959), 520 - 523. (Russian).

Performance Criteria

7.35 Fuller, A.T., / "Design of Control Systems Containing Saturating

Components" / Ph.D. Thesis / Cambridge University 1959.

7.36 Fuller, A.T., / "Performance Criteria for Control Systems" / J.

Elect. and Control. Vol. 7, pp 456 - 462 (July - Dec. 1959).

7.37 Gibson, IRekasius, McVey, Sridhov, Leedham, / "A Set of Standard

Specifications for Linear Automatic Control Systems" / P / Trans.

A.I.E.E., App. and Ind., No. 54, (May 1961) pp 65 - 77.

7.38 Murphey, G.T., Bold, N.T., / "Optimization based on a Square Error

Criterion with an Arbitrary Weighting Function" / P / Trans. I.R.E.

AC-5, No. 1, (January 1960).

7.39 Schultz, W.C., Rideout, V.C., / "Control System Performance Measures:

Past, Present and Future" / P / Trans. I.R.E. AC-6, No. 1. p. 22.

(January 1961).

7.40 Westcott, J.H., / "Moment of ifirror Squared Criterion for Servo-

mechanisms" / P / Proc. I.E.E., Vol. 101, Part II, No. 83, (Oct. 1954).

273.

7.41 Westcott, Jill., Florentin, J.J., Pearson, J.D., / "Approximation

Methods in Optimal and Adaptive Control" / P / Dept. of Electrical

Engineering, Imperial College, London. (May 1963).

Sub-Optimal Control and Liapunov Methods

7.42 Aoki, Masanao., / "On Optimal and Sub-optimal Policies in the

Choice of Forces for Final Value Systems" / Trans. I.R.E., AC-5,

No. 3, (Aug. 1960) pp 171 - 178.

7.43 Kalman, Bertram, / "Control System Design via the Second Method

of Liapunov" / A.S.M.E. Journal of Basic Engineering, June 1960.

7.44 Liapunov, A.M., / "Problem© Generale de la Stabilite du mouvement" /

Annales de la Faculte des Sciences de Toulouse, Ser. 2, Vol. 9,

(1907), Reprinted as Annals of Mathematics Studies, No. 17,

Princeton, 1946.

11.8 Instrumentation and a Reduced State Space

The mathematical techniques required for the correlation coefficient

criterion can be found in any text on Statistical Theory, for example:

8.1 Lanning, Battin, / "Random Processes in Automatic Control" / B /

McGraw-Hill, 1956.

8.2 Petersen, E.L., / "Statistical Analysis and Optimization of Systems"

/ Wiley, 1961.

The concepts of state space also appear in many texts, for example:

8.3 Fuller, A.T., / "Phase Space in the theory of Optimum Control" / P /

J. Elect. and Control, Vol. 8, (Jan. - June 1960) pp 381 - 400.

and a chapter in 6.5 (Seifert and Steeg).

274.

11.9 State Measure Control

The relevant literature on 'state measurecontrol' in the sense used

here derives from the study of adaptive control on one hand and methods of

finding optimum parameters - hill climbers and non-linear programming -

on the other.

The references to methods of signal flow graphs used to describe

the control systems are also listed here.

Adaptive Control

9.1 Aseltine, Mancine, Sarture, / "Trans. I.R.E. Auto. Control, Dec. 1958.

9.2 Eykhoff, P., / "Adaptive and Optimizing Control Systems" / I.R.E.

Trans. AC-5, No. 2. June 1960. (in Correspondence pp 148 - 151).

9.3 Gibson, J.E., / 'Adaptive Principles" / P / Control Engineering,

August 1960, and October 1960.

9.4 Honeywell Military Products Group - Ho, Y.C., Stone, C.R., Schuck,

0.H., et al, / "Adaptive State Vector Control" / Mineapolis

Honeywell Research Reports: NH MPG 1529-TRI to 1529-TR9 inclusive.

Hill Climbers and Non-Linear Programming

9.5 Arrow, K.J., Hurwicz, L., Uzawa, H., / "Studies in Linear and

Non-Linear Programming" / Stamford University Press, 1958.

9.6 Ho, Y.C., / "Final Value Control Problems and the Method of

Constrained Descent" / Cruft Laboratory, Harvard University,

Tech. Report No. 340.

9.7 Kuhn, H.W., Tucker, A.W., / "Non-Linear Programming" / Proc. 2nd

Berkeley Symposium on Math. Statistics and Probability, (J. Neyman,

Ed.)/ University of California Press, 1951.

275.

9.8 Rosenbrock, H.H., / "An Automatic Method for finding the greatest

or least value of a function" / Computer Journal, (1960) Vol. 3,

No. 3, p. 175.

9.9 Stakhovski, R.I., / "A Multichannel Automatic Optimizer for Solving

Variational Problems" / Automatika i Telemekhanika, Vol. 20, No. 11,

(November 1959).

Signal Flow Graphs

9.10 Hoskins, / "Signal Flow Graph Analysis and Feedback Theory" / Proc.

I.E.E., Vol. 108, Part C, No. 13, (March 1961) p. 12.

9.11 Mason, S.J., / "Feedback Theory - Some Properties of Signal Flow

Graphs" / Proc. I.R.E., (1953) Vol. 41, p. 1144.

9.12 Mason, S.J., / "Feedback Theory - Further Properties of Signal Flow

Graphs" / Proc. I.R.E., (1956) Vol. 44, p. 920.

Analogue Computing

9.13 MacKay, D.M., Fisher, M.2., / "Analogue Computing at Ultra High

Speed" / B / Chapman and Hall, 1962.

Appendix I

VERIFICATION OF SOLUTION TO PARTIAL DIFFERENTIAL EQUATION

Solution of partial differential equation:

au _ - V au D at - ax ax2

Using the Laplace transform method and the conditions on the

boundaries:

x = 0, t ">0 , u(0,t) = 8(t)

t = 0, x 70 , u(x,0) = 0

yields the solution

276,

u(x, t) = x Exp 1 x2 - 2xV +

DIrt3 4D t v2t}j

This can be verified as a solution as follows:

By differentiation

8u at

-s 4 x3 -xV2 2t 4Dt2 4D

Exp . ) . 1 - x2 n. 2Dt 2D

au 1

ax 2/77rt3

and a2u ax2

1 Exp

21.17;tt3

• • .? 2t r, (v v

J - Lb-

"x 2 x 3

t 4Dt2

whence substituting in the equation gives an identity.

The result:-

au _ 1 Exp )

J

xu2

i Drt3 at J1 2t 4Dt2 -

u— z

277.

gives the condition for a stationary point in the observed response at the

point x. Since Ebcp (... can never be zerof the required condition is

thatt

3

+ xv2 4D = 0

4Dtc

+1 9D2

v2.2

Which gives T peak

278.

Appendix II

HEAT TRANSFER EXAMPLES

A2.1 Heat transfer to moving metal strips.

Two examples are considered to show the relative significance of ay;c1 mg

radial and diffusion for heat trammifer)and the design of an electronic

analogue. One deals with a steel strip, the other with an aluminium one.

Example 1. Steel Strip

Steel strip passes through furnace 201 long at 41/hr.

Cross section is circular, 21 diameter.

Physical properties of steel: (All data is approximate)

Density: 490 lb/cu.ft.

Specific heat: 0.15 BTU/°F. lb.

Thermal oond: 30 BTU.ft/sq.ft. °F. hr.

Heat loss to surrounds: 3 BTU/sq. ft. °F. hr. areas

Consider the cross section as two concentric regions of equal aacmd

(and therefore equal thermal capacities).

Each region has thermal capacity 115.4 BTU/°F per foot run.

Distance between centre of inner zone and centre of outer zone =

0.8535'.

Heat transfer between zones = 78.1 BTU/hr. °F per foot run.

Heat transfer through outer surface = 9,86 BTU/hr. °F per foot run.

Zone temperatures are designated thus: u1 outside environment.

u outer zone

u3 - inner zone.

G 12 C2

2-u3)

C3 (u2 — u3)

au2 a a2u V au2G

12 at = ax C

2 ax2

Equation for inner zone

au 8u3

3 = - V a a2u

C2

at

au2 aT

au 3

Normalizing

ax

gives

- au2 +

ax2

,

C3

a2u ax

_ - au

k L2

a /7

ax2

a2u aT ax

(. L2 ax2

(u, - u2) - _21 (u2 u3)

(u2 u3)

279.

Thermal conductances (per foot run) Grs between zone r and zone s.

Thermal capacities (per foot run) Cr in zone r.

Thermal diffusion coefficient a = k = 30 = 0.408 of (0.15)(490)

Equation for outer zone:

where L = length of system, V = velocity of flow, r= L/V

at The value of the coefficient --- is important in determining the L2

number of sections needed to model this flow and diffusion process.

In this case at = 0.408. 5 = .005 (L = 20',2:= 5 hrs) L2 400

whence n crit = 1 = 100 sections. 0.01

280.

A2.1 Example 2. Aluminium strip, calculations and analogue

An aluminium strip 3' wide by 6" deep is considered.

The cross section is considered divided into two regions of equal areas)

the inner region being 2.58' by 0.29',

The common surface area of the regionSis 5.76 sq. ft./per foot run.

The approximate average distances between centre line of inner Zone and

centre line of outer zone is 0.20'.

Physical data for aluminium:

Density: 169 lb/cut.ft.

Specific heat:0.224 BTU/°F. lb.

Thermal cond.; 120 BTU ft./ft2. °F. hr.

Heat transfer between zones:2880 BTU/hr. °F. per foot run.

Thermal capae-Ity:27.9 BTU/°F per foot run.

Heat transfer to environment: ( emmissivity approx. 3%)

0.2 BTU/hr. sq.ft. °F. 74- 1.4 BTU/hr. °F per foot run.

Equations for zones of Al. strip.

Notation: Outer zone suffix 2

Inner zone suffix 3

Environment suffix 0

Temperatures ur in zone r and urs in lumped model section s of

zone r.

Thermal Capacities: Cr of unit length (1 ft.) of zone r

Thermal Conductance: G between zones r and s.

281.

Distances: X denotes distance in original units (ft) measured from

input.

L is total length of system (ft)

x is normalized length =X/L (dimensionless)

Time: t denotes time in original units (hrs)

Z is total (avg) residence time in system (hrs)

T is normalised time = t/ (dimensionless)

Velocity: V is rate of flow (A/hr)

Normalised velocity is unity.

Equations are:

au2 -V au2 • a a2u2 G12 (ul u2) -

G23 (u2 u3)

at r axa7.2 C2

C2

2 au,

-V a u

= + a 1_21 (u2 - u3) at ax

ax2 c3

These equations can be normalised to unity velocity, with the output unit

time and unit distance from the inputI thus: -

2 ( dug - au2 ▪ a7r)), au2 12 C /).(ul u2)

G23 (u2 u3) ,

aT = L2 ax2 ax 2 C2

2 = au c a u,

ax2

2 (▪ ....21G ),(u2 - u3) aT ax c

3 (l,c; )

When expressed as a set of equations for a lumped parameter model by

quantizing in the x coordinate these become

du2r , 1,1J 1 (u r-1 - u r /2 +1(u r-1+ u2r+1- 2u2ar'. = 2 2 -

d T A x 6 x 2. L a x 2

1•• Cop.; u,, I% je

282.

) (uir u ri G23 (u2r rl + (G12 u3 ) 2 / _

C2 2

du r and I = 1 ,(u r-1- u r) + 1 (u r-1 r+1 r) a dT z-i-x 3 3 /5.7 3 + u3 - 2u ---

3 L24, x

/.12.1 ),(u2r u3 r)

\ 3 where Q x is the length of each section in the lumped model

also 4x = 1 where n is the number of sections. n

The electrical system which is to be the analog of this has the

equations (see Fig/12.1)

dV2r

= (V2r - V2r) n + (V2r-1 + V2r+1 - 2V2r) ( .&) n dT1

) k, + (512 k (Vir - V2r ) -(g23 _.2) (2r - V3 r)

k2 k2

1 2 )

and dV

dT' (V r-1 - V r)n + (V + V r+1 - 2V

3r 3 3 3

3 E k3

g .n.

k ) + g23 g

k3 (V2r - V3

r)

gl and g are the conductances used to form the sequence of sections

used to represent the flow and diffusion.

where grs is the total electrical conductance between zones n and s.

kr is an electrical capacity for the full length of one zone:

and -1-1 is the basic time unit of the electrical system

so that T' = VT' where (usually t and "Z are in seconds) and in this

system -ct = -2

283,

11 is again the number of scction.s to be used.

For the electrical system to represent the thermal one the co-

efficients in the normalised equations must be equal (or proportional)

Thus 1 = 4x n

12) g L2 Ax

go-z G 7 = 23 g k2 C

2

5.1 • G23

--c g C3

From these all the electrical components can be specified if g and k3 and

hence -c' are chosen in advance:

Thus g' = (g ) .( a Z L2 / x 12)

and with C2 = C3,

2.0_9 p•23 = g. '1°

C3

g12 = g.

and k2 = k3

Number of sections in model

In this case a q:.* = (2.y) (6 ) = 0.05

L2 Ax 182. Ax /Ix

0.05 1 and for g' to be non-negative 4x ".

281-E.

Thus the largest value of 21x is 0.1 or n = 10.

From the knowledge of the behaviour of lumped models, o ten section

model gives a reasonably satisfactory representation for a -C .05

L2

and so this number of sections can be safely usect.

(Though "accuracy" could be improved slightly havi larger.and

In the model used,take k3

= 1 g.

, 1 and g = 10-4 1; 1 -= 10 K )

then k2 1 pF, each section has a capacity of 0.1 pF.

(10-4 ( 2880 ) . ) 27.9

1 27.9 . 10 K = .0162 Kt

g23- 2880 (6 )

and the actual resistance to be connected between corresponding elements

in the two lines (each of 10 sections)

= 0.162 K

= 162 st-

g12 = 10 1.4 6

1 27.9 . 10 K g12 = 0..4) 6)

= 33.2 K

and the actual resistance to be connected between corresponding elements

in the two lines (each of 10 sections)

= 332 K

The resulting analogue circuit is shown in fig. A2.1.

and g23 =

27.9

au2 G12

at = 2

(ul u2) (G23 )

3 (u3 — u2).

285.

In this case the values of conductances etc. are such that the

two zones would to all intents and purposes act as one,because the con-

ductance between zones is many times greater than that between the outer

zone and the environment. The slow transfer from environment to outer

zone .would dominate the whole system. The problem of producing an even

temperature profile could not arise because of the close coupling between

the zones inside the strip.

A2.2 Heat exchanger analogue

Shell and tube type, fluid flows in 18 copper pipes 6' long and

5/8" x 16 SWG at 3'/sec. (Transit time 2 secs).

Heat transfer coefficients are:

Steam to copper 1000 BTU/hr. °F. sq .ft. = 0.278 BTU/sec.°F. sq.ft.

Copper to fluid 750 BTU/hr. oF. sq .ft. = 0.208 BTU/sec.°F. sq. ft.

Thermal capacity of fluid (water)

2.03 BTU/°F per foot run.

Thermal capacity of copper pipe

0.580 BTU/°F per foot run.

Heat transfer: steam to copper

0.816 BTU/sec. °F per foot run.

Heat transfer: copper to water

0.567 BTU/sec. °F per foot run.

As before)the system equations can be written:-

and au, =

at

D. a2u,

ax2 3

(u2 - u3)

where u3 represents the water temperature

' u2

the wall temperature and ul

286,

the controllable (in this example) steam jacket temperature. For condensing

steam the temperature depends directly on the pressure which can be regula-

ted by standard pneumatic values etc.

The actual value of the parameter D to be used in this analysis can

be estimated from the work of Sir G. Taylor1(2.ID).

He produces the formula

D = (10.1)r ,VV where r is the pipe radius,

Vx a velocity related to the actual average velocity V7 ne relationship

being a function of Reynolds number.

ranges from 103 to 107 the ratio

However while the Reynolds number

ranges from 10 to 30. Furthermore, V

Vx bends, discontinuities in the pipes, which certainly occur in a practical

heat exchanger will all increase the value of the coefficient D. In this

system the pipe radius is 01:266, the flow velocity 3'/sec and taking the

ratio V/Vx r;7.- 10 for a large value of D

D = (10.1) . (.266) . 2- 12 10

= 0.67

and V/Vx = 30 gives D = 0.22

For use in the normalised equations the value

a = D is required where L = length of system.

L2 T:= mean residence time

with L = 6' and V = 3'/sec, t= 2 secs.

= (0.67) . (2) 0.037 for D = .67

a3 36

and a3 = 0.012 for D = 0.22

a (

C3

= 3 a3 8T ax

ax2

u —au a2u G2 (u2 u3)

287.

For a lumped model representation of such systems the critical

number of sections is n = 1 2a

giving n = 13 or 14 for a = .037 (D = .67)

and n = 41 or 42 for a = .012 (D = 10.22).

In the model to be used the effective number of sections will be

taken as 20 but the calculations will be based on a 10 section model because

the effective doubling of the number of sections is achieved by making

each section have a second order transfer function instead of a first

order one. The capacities and conductances are still the same.

When the system equations are normalised they become:,

au2

aT G12

C2 (ul 112) C 23 (u3 - u2) C

2

These equations are quantised into n sections so that the dispersion

effect is represented without having to deal with it separately (ncrit

model), and for an n section model the equations are:

du2r (G12 C ) (u- (G23 T.: (u3r - u2r) I,r u2,r) dT C

2 C , , 2

du3r -1

u3,r-1) dT = (1717

23 7: ,r u 3,r)

C3

Since only ratios of G and C occur all analysis can be done either on foot

run basis or total length basis.

288.

This system is to be modelled by an electrical analogue for which the

corresponding equations are:

dV2r . g

12 . k

3 g23 . (Vg 2r

- V3r) (V

lr - V

2r) - dT k

2 k2

dV3 g23 ) - (V3r - V3r-1)n (V2 - V 3 dT

where the terms g and k, etc., represent conductances and capacitances as

shown in fig. A2.2a.

For the electrical and actual systems to be equivalent the following

relationships have to be met

g12 = (G12 . g C3

g23 (.

:23 C3

k2 C2 k

3 C3

If g is taken as 10-4-Ls. (10 K resistor)

and k is taken as 0.1 µF,

the electronic analogue has a delay time of 10 m.secs. and the other

component values are as shown in the fig. A2.2b. Note that comparisons must

either be based on foot run or full length analysis, and for convenience

in obtaining components the electronic circuit values are not exactly the

calculated ones but the nearest standard sizes. In view of the extremely

approximate nature of the original physical data this is of no importance.

An analogue model using this design was built with buffer amplifiers

289.

of the improved type shown in Appendix 4, Sect. 1, using alternatively

along the system NPN-PNP and PNP-NPN forms to equalise residual steady

state errors.

v3

vs

vi

-.1:3•A•

a) Basic Analogue.

b) Calculated (norit) analogue

Fig.A2.1. Analogue for heat transfer to moving metal strip (Aluminium).

A) Basic analogue.

B) Actual model with doubled effective number of sections.

Fig.A2.2. Heat vachanger Analogue.

290.

Appendix III

DIGITAL COMPUTER PROGRAMME FOR STEP AND IMPULSE RESPONSES

The programme was designed to find out the number of sections

required in lumped parameter modelsi to give adequate representation of

responses. The system represented consisted of a flow and diffusion process

in one space dimension, and the disturbances to which it was subjeCted were

either impulses or step inputs.

The programme was arranged so that one parameter could be used to

specify the system behaviour (the inverse of a Peclet number) and the

number of sections could be specified jand the coefficients and differential

equations were computed automatically.

In operation the programme had the form shown in fig. A3.1.

2 S174 r

I Rifb .DifrA

toomori cotArs.

Foe irQuArtomc

scr 'Pim" AND meinfr

cow/A/no/is

cdvreeveArE ealuics.

(srogs Resmrs)

Peptir our

Refeom44.S.

RecyCh

(err Feb RI. en)

Data Requirements:

1) Diffusion coefficient,a. 2)nNumber of sections 3) Number of terminating sections 4) Choice of step or impulse response 5) Integration step size. 6) Number of steps between each printed

result 7) Number of printed results. 8) Problem identification number.

Output:

1) Problem number 2) Number of sections 3) Value of 'a'. 4) Corresponding values of time

and output response as required.

Fig.A3.1. Programme for investigating numbers of sections requird in lumped parameter models.

291.

Appendix IV

CIRCUITS OF BUFFER AMPLIFIERS AND APPLICATIONS

A4.1 Buffer amplifiers

The amplifiers must have a high input impedance and a low output

impedance with a gain of unity. The input and output voltages are to be

biased to the same level.

The simplest approximation to this is the emitter follower circuit,

and to avoid the change in bias level at each stage the whole system has

to be arranged so that there is sufficient bias current from the supply

lines to correct for it at each stage. The circuit shown in fig. A4.1

represents a sequence of emitter follower stages arranged to represent a

flow and diffusion process with more than the critical number of stages,

so that connections are required between capacitors to represent the

dispersion over and above that represented inherently by the lumping

process.

An improved version of emitter follower circuit which uses both

NPN and PNP transistors is shown in fig. A4.2. In operation,some D.C.

connection must be made to the input terminal, but in the circuits used

here this is always the case. The diode in the circuit has two effects.

The steady state voltage developed across it compensates, at least in part,

for the base-emitter voltage of the first transistor. Because of its Mkt

forward bias resistance the current,flowing through it and through,second

transistor augments the signal voltage developed by the normal emitter

follower actionsof the first transistor, to give a gain closer to unity. The

very high current gain due to the combination of the two transistors results

292.

in a high input impedance. The circuit can be used in either the NPN-PNP

or in the PNP-NPN mode as shown by the two diagrams. The first transistor

must have a low leakage current and if this presents a problem it can be

allevjated by inserting a resistor (RL) into the circuit,as shown by the

broken line in the diagram. (This circuit design is subject to a patent

owned by Rank Cintel Ltd).

A4.2 Modification of simple lumped model for increased number of sections

The transfer function of the single section in a lumped model

corresponds to a first order

G(p) = 1 (1 + pT)

where T represents the fraction of the total delay time of the line repre-

sented by that section, as in Fig. A4.5a.

One section can be used to give a transfer function equivalent to

two simple sections in cascade if instead of a first order R.C. network

an L-R-C arrangements is used, as in fig. A4.5b.

If the value of L is chosen for critical damping and the resistance

and capacitance kept the same then each section becomes equivalent to two

half sections as in fig. A4.5c.

If R = 10 K and C = 0.1 1F as used in the heat exchanger analogue

(Appendix II) then the required value of L is 2.5 H.

This is used in exactly the same way as before, the difference bein5

that the flow and diffusion part of the analogue shows less diffusion.

293.

A4.3 Use of buffer amplifiers with gain greater than unity

To represent the flow and diffusion processi the buffer amplifiers

have to have a gain of unity but if a situation arises in which the con-

centration or temperature being considered is naturally unstable - in that,

for examplei an increase in temperature produces an increase in rate of

change of temperaturel then the coefficients of the quantised version of the

partial differential equation cannot be represented by merely changing the

component sizes in the models shown above.

Consider for example the system represented by the equation

du_ - 8u a a2u a u at ax dx

2

where a is a positive constant

The quantised version of this equation is

dur (u u -1 ,) -1 (u - u ) a ur-1+ ur+1- 2ur +a ur dT = 2

-1 x r r 2 x

r x)2

1u u 1 a- + a ur 2

u r-1+ ur+1 -2u r r 2 x

With an amplifier having a second output terminal giving a gain, ((3),

greater than unity this system can be represented by the analogue circuit

shown in fig, A4.4a ((3 = 1.5). For this circuit the voltage Vr follows

the equation

k dVr = g Vr) gi(Vr..1+ Vr41- 2Vr) g2 ((3 — 1) Vr

dt

The stability of the circuit depends on relationships between the

circuit parameters. Consider for example the case where g1 = 0 ( i.e. the

ncrit model). The equivalent circuit for this is shown in fig. A4.4b.

294.

The transfer function is

V1(p) p k + gi- g2 ((3-1)

The critical case occurs when

gl = g2 (13-1)

and the circuit acts as an integrator. Practical investigation showed

that when such a unit was simulated on an analogue computer the system

could be readily adjusted to give neutral stability. A simple transistor

amplifier version had varying stability depending on the actual voltage

because of the slight variations in gain due to small non-linearities.

The control of reaction systems such as in fig. A4.4a has not been

investigated.

a) Basic emmitter follower.

b) Simple model of flow and diffusion process. RlIC form basic section, R21

C represents transfer by diffusion only, over and above that represented by having lumped instead of continuous model. R2

is ommitted for norit

model.

Fig.A4.1. Electronic analogue of flow and diffusion process.

Fig.A4.2. Two versions of improved emitter follower circuit.

Th

.,2`)ef-c-

g Re.

a) Original arrangement.

rota /kJ w

b) Modified Version. 74, tormedu e"mow4.

R rc 4'i

10 0) 00 46 5/2 ch •

ryz 1

m 7/2

c) Simple equivalent of modified version.

Fig.A4.3. Modification of electrical lumped parameter model.

a) Structure of analogue for system with internal generation.

b) Equivalent circuit for use of amplifier with gain greater than unity.

Fig.A4.4. Amplifiers with greater than unit gain.

I

avn .

VM dx + D

ax a2 Vn.VMdx 0.Vmdx-Gam

U.

dam dT

ax2

0

• 295.

Appendix V

SPECIAL FUNCTIONS APPLIED TO MODELLING FLOW AND DIFFUSION PROCESS

A5.1 Application of Laquerre Polynomials

The system is described by the equation

3u - V 8uD 82u + -I- 0 - Gu A5.1, 8t = 8x 8x

where 0 = 0(x,t) is the input forcing function.

The system operates from t = 0 to t = co and from x = 0 to x =,70

It is proposed to represent the solution of this equation as a

sum of orthonormal functions:-

u(x,t)

ai (t) V.(x) A5.2.

The functions will be chosen sb that

Vn(0) / 0 for all n A5.3.

in order that input conditions may be defined, and so that

Li41.7jVn (x)? = 0 A5.4. j

for physical realizability.

Substitute for u(x,t) in the original equation:-

at a !a n Vn =

ax V a 75:a n n %- V + D a22 !% +0-G a

nVn

ax2

A5.5.

Multiply by um and integrate over the range of orthonormality of

X

A5.6.

LIm(x) = m-1

ml Lr (x) A5.9. r!

296.

. dx + Om - G am

A5.7. where Om is the scalavproduct of 0 and um.

This can be written as

dam = V dt an +

- G am

A5.8.

The Laquerre polynomials can be shown to form an orthonormal set on

the range x = 0 to x = °° when used with the weighting function e-x.

Thus the functions

Vn(x) = 1 e-x/2. Ln (x) from an orthonormal set boc/ t.) n!

Manipulation of the normally quoted recurrence relations for these polynom-

ials Ln(x) shows that they satisfy the relationships:-

r=0

and

L"n (x) = n1 (n-l-r) Lr(x) A5.10. r!

Using these relationships the scale product in the equation A5.8.

can be evaluated, taking due account of the presence of the exponential

terms. The results are

o (' Vn. avm dx -1 n = m.

,/ ax ) 7(n1)2 ••.••••••

ax

( 32V n . Vm dx = 0

= -1

(n1)2

= 0

n m

} m

A5.11.

297.

In the same way

n c m

1

4(n1)2 '

n = m A5.12.

1 n = m+1

(n-1)1 2

1 (n-m),n "7m+1

(m4)2

It can be seen that for n = m the resulting coefficient of the term am

in the R.H.S. of the equation for aam depends on the sum of two terms, at

one negative, the other positive$ whose magnitudes depend on the factors V

and D in the original equation A5.1.

In the very simplest case where only one mode is considered in

representing the system, the stability of the model, in the Routh Hurwitz

case depends on the coefficients of the original equation. This is not

satisfactory.

A5.2 Application of Legendre Polynomials

Instead of representing the system by a set of functions on the

range 0 to .P° , another possibility is the representation of the normalised

298,

system (length = 1) on the range 0 to 1.

Following the same procedure as in the above Section (A5.1) the

equation A5.8. is replaced by

1 m

-V n funum I +

da

dt Jo

--1

u aum n --- ax

dx + DI ( un .0m) dx

o ax2

+ Om - G Vm A5.13.

It will be observed that with this particular organisation of the terms

concerned boundary values are required at both ends of the system, and the

same scalar products are required as before.

If the Legendre polynomials on the range 0 to 1 are used the

orthogonality relationship is

r1 Pm(x) Pn

(x) dx = 0 m n

0 1

(2n+1) m = n

and the scalarproducts are:-

10 Pn. aPm 0 m

ax

0 m)n,n odd and m odd, n even and m even.

2 m >n,n odd and m even, n even and M odd.

The terms such as

cannot readily be evaluated by recurrence

relationships but the first few terms

299.

are given in the following table:

0 1 2 3 4 5 6

0 0 0 12 0 40 0 84

1 o o 0 20 0 56 0

2 0 0 0 0 28 0 72

Two difficulties arise in using this as a model for the system.

There is no way of specifying the boundary conditions at x = I, if the

system is to be considered in any way continuous beyond this point, and

since all the coefficients of the R.H.Sides of the ordinary differential

equations A5.13. are positivelit is only necessary to consider the case

in which all the mode terms Vm(t) are positive,to show that all terms

OV m are positive7 making the model always unstable. This is not

at satisfactory.

A5,3 Application of Specially Designed Function

It is proposed to use the set of functions

sin) 2Tal (1 - e-ax) for n = 1, 2 ... k

together with the function u(0,t) e-ax , so that the whole set

n=k

n=1

u(0,t).e-" + rr [12.7a1(1 - e-") an(t). sin

forms a series approximation to the function u(x,t) which has to satisfy

the partial differential equation (3u_ . v D a2u at

a2u

ox Ox2

300.

together with the boundary conditions

u(0,t) given as a function of t.

u(x,t) tends to zero as x tends to infinity.

The above set of functions is orthonormal on the range (0, 00) with

respect to the weighting function

2a e-ax

with the exception of the term u(0,t)e-ax) which can be treated separately.

Substitute the approximating series into the differential equation

(Note the symbol z is used to stand for L1-e-al ).

a n u(o,t)e-ax + > a(t).sin 2nnz = - V a u(0,t)e-ax+ an(t). at n=1 ax n=1

... sin 21in z + D a2

ax2

k u(0,t)e-ax.+2 an(t).sin 27n z n=1 1

To reduce this to a set of ordinary differential equations for the

coefficients an(t) the whole equation is multiplied by the function

[2a e-ax sin 27: j z. J and integrated over the range (0, cO) for each value of j in turn : j = 1 ... k.

So that each of the equations recorded retains managable proportions

each term will be dealt with separately:

(Note that wherever it arises e-ax is also equivalent to

(1-z) if required, and that in the integration processes the term

can be directly replaced byf4while the limits are changed

from (0,c0 ,to (0,1). )

The first term on the L.H.S. becomes

301.

L (sin 2mj z)

u(0,t). e-ax) .6a e-ax) a

which can be reduced to:

122i2LL2' at

The second term becomes

(sin 2mj z). an(t).sin 27ca z ). 2a e-ax dx

which reduces to

a a,(t) because of the orthogonality of the set of functions. at J.

The first term on the R.H.S. of this equation is.

- V a u(0,t).e ax1 . sin 2nj z. 2a e-ax dx.

ax

and this reduces to

av . u(0,t). nj) The second term on the R.H.S. is

00

— v sin 2nj z. a ax

k

n= )an(t).sin 2iin z . 2a e-ax. dx.

which reduces to n=k

-av n=1 n/j

a n (t) j2-[ n27 + v a.a.(t).

The fourth term on the R.H.S. is

po

D sin 2nj z. ax` ( n=1

a2 an(t). sin 21in z. j.2a e-ax dx.

The third term on the R.H.S. is

302.

+ D

Jo

sin 2nj z. a2

ax2 (u(0,t).e-ax). 2a e-ax dx.

which reduces to

D. a2 u(0,t). 2nj

which can eventually be reduced to

k -- 2 a

2 (n 2+ ji

2, 2n* a2 - D ;75-

n=1

,, an(0 4n +

2 2,2 (j - n i (j2- n2)

n/j

I- + D a.(t) 4 n2j2a2 (2/3 + 1/8n2j2)

J

Collecting all the terms together to make a set of first order

ordinary differential equations for the parameters a.(t) with the form of

equation A5.6. shows that two difficulties arise for practical implementation.

Firstly each equation involves a term in:

au (0,0, at

which is the derivative of the input function, this may not be easily

calculated or generated, even if it exists. Secondly the coefficient of

a.(t) in the R.H.S. of the equation for

da.

dt

303.

involves the sum of two terms, one depending on V, the other on Di as a

scaling factor, and since these have opposite signs the stability of the

resulting model depends on their relative magnitudes. For example, it

would be extremely difficult, if not impossible to set up an analogue

simulation to represent several modes if each mode on its own was unstable.

This is not a satisfactory method for modelling the system.

304.

Appendix VI

SIMPLE EXAMPLES OF TiE USE OF THE CALCULUS OF VARIATIONS

A6.1 First order system, one control variable

A system is described by the equation

X = u

where u,is the control that can be chosen. The objective is to move

between two points, x(0) and x(T), specified at times t = 0 and t = T

while incurring least cost as measured by

T

(x2 + u2) dt.

0

Form the modified Lagrangian

L' = x2 + u2 + X (X - u)

and write down the Euler-Lagrange equations:

2x - n = 0

2u - X = 0

X - u = 0

These simple equations can be solved to give

x = A et B e-t

where A and B can be chosen to meet the boundary conditions. The control

required is given by solving for X

P A6.2 Same problem by Pontriagias method

The problem is to minimize

0

(u2 + x22) dt

given that

x2 = u

and that boundary conditions are specified on x2 at t = 0 and t = T. First

it is necessary to modify slightly the formulation of the problem: Put

X1 = u2 + x2

2

and the problem is now to minimize X1(T).

Set up the Hamiltonian function

H = piki + p2 )12

which is

H = P1°12 x221 4. P2 u

Minimize H with respect to u, which in this case can be achieved by

differentiation with respect to u, to give

u = -p2

2p1

Substitute for th in the Hamiltonian gives

2 2 H = Plx2 - p2 •

The equations of motion are; 2

1 = p2

2 4p

2 + x2 1

8H = p2 X2

= apt 2p1

305.

1

1 4131

and

pl

p2

ax1

-ax ax2

0

-2 p1x2

306.

To solve these equations boundary values are required. There are

four first order equations; two boundary values are given; x2(0) and

x2(T) Two more arise from the solution method: x1(0)= 0 and p

1(T) = 1.

From these, the equations can be considerably reduced. Firstly

pl can be eliminated because;

pl = 1 for all t, 0 e t T,

This leaves two equations to be solved to find the trajectory:-

312 = - P2 2

and p2

-2x2'

which can be combined to give:-

x2 = x2

and the same solution as before.

Appendix VII

DIRECT ANALYSIS OF OPTIMUM LINEAR FEEDBACK CONTROLTRR

A7.1 General case, numbers of variables and roots of equations

Consider a system described by the equations

311 = allxl a12x2 a x *** alrin b 13 3 1 1

X2 = a21x1 + a

22x2 OaDOOODO

ADO0001,

n = anix1 + an2x2 ........ annxn + bnnn A7.1.

There can be no more than n independent control parameters in the u vector

if the system is nth order.

Suppose the performance criterion is to minimize

(C x 1 .... Cnxn ) + (d1u12 .... dnun

2) dt I 12

A7.2. where the period 0 to for the process means that it is to operate

continuously.

The objective is to find the control function u(0) as a direct

function of x(0)? and since this is the start of an infinite period it can

be taken as a general result valid for all finite t, thus giving u (x)

for all t.

Following the methods of the calculus of variations set up the

modified Lagrangian function:-

LI = (C1x12 ... Cnxn2) + (dlul2 + 1 dnun2I X1(1.1 -(allx1+ a12x2—blul"

308.

... +)2 • 1x1+ a22x2...b2u2) ) ••• %n(kn -(anlx1+ an2x2—bnun) )E

A7.3.

Write down the three groups of Euler-Lagrange equations:

Firstly for the u variables; aLl = 0, to give au.

2u1d1 1 = 0 or u = b1 1 1 Tia.1

A7.4. 2undn Xnbn = 0 or u = T bn

n 2dn

Then for the x variables, ail - d = 0, to give ax. dt 8X.

2C1x1 - a11X1 - a21X2 - a31X3 ani.Xn - = 0

04.041 10000

A7.5.

2Cnxn - alnX1 - a2n

annXnn = °

And finally for the X. variables to recover the original system equations:-

blui + anxi + a12x2 ainxn - Xi = 0 A7.6.

bnun anlx1 an2x2 "." annxnn = °

The variables, u, can for convenience be eliminated in favour of the

variables, WI by substituting for them in these system equations.

Furthermore, the coefficients of the X variables which result can be

309.

replaced by a single symbol:

b..2

2d7 is replaced by fi.

This leaves two groups of ordinary differential equations in x and X.

Transform these equations by the Laplace Transform and rearrange them to

give the group of Ln equations:-

(P-all)

-a21

-a

-a12

(p-a22)

-antnl

-a13 -a,

-a23 .... -a2n

(p-ann)

0 • • • f1

0 f2 0 ..

Of n

r x1

x2

Xn

X.1

X2

Xn

=

x (o)

x2(°)

xn(0)

W1 (0)

W2(0)

An(0)

-2Cl

0

0

-2C2

-2C n

(p4-an) a21..

a12 (P4-a22).

(p÷anr,)

A7.7.

This is more conveniently represented by the form

l(p I - A) p I + ATJ I X

A7.8.

where I, F, A, C are all nxn matrices, the matrix I being the unit matrix.

To solve the problem it is necessary that the system should settle

to a stable end point corresponding to the steady state optimum, which

310.

occurs for x = u = 0.

The above set of 2n equations has 2n roots. It will be shown that n

of these are unstable, that isIthey have positive real parts. If the n

coefficients of these roots in the partial fraction expansion of the

solutions for the x variables are made zero then the whole system will

exhibit stable behaviour. Satisfying these n relationshipsgives sufficient

equations to define the values of X(0) in terms of x(0) and hence gives

sufficient information to define u in terms of x for all t.

To show that this 2n x 2n matrix has its roots equally divided

between stable and unstable ones consider

H(p) = p I - A

C p I + AT A7.9.

then

I HT(-p) y = - p I - AT CT

FT -p I + AI A7.10.

Multiply the first n rows of this determinant by -1 and the last n

columns by -1.

(HT(-p)1 p I + AT CT

FT

p I - A

Interchange the first n rows with the last and then the first n columns

with the last:

HT(-0

FT p I - A

CT p I + AT

A7.12.

A7.11.

311.

But F and C are symmetric, and

HT(-p) i = 1H(-p)

Therefore

I11( -p) 1 = p I - A F

p I + AT

But this is also the determinant of H(p), which must therefore be

an even function of p. Therefore the roots of the equations are equally

divided between the positive and negative half planes in the p-domain.

A7.2 Second order numerical example, without solving for roots of

equations

The problem is to devise a controller for a system described by

the equation

d2x dx

dt2 dt X = 1.1) A7.14.

so that it will run continuously and minimize the performance criterion

I (u2 + x2) dt. A7.15.

J O

The first step is to reorganise the equations to the form

X1 x2

x2 -x2 - x1 + u A7.16.

where xi = x.

Form the modified Lagrangian function

= x12 + u2 + Xi (ii - x2) + X2 (X2 + x2 + xi - u)

A7.17.

A7.13.

312.

Set up the Euler-Lagrange equations, eliminate u in favour of X and

transform by the Laplace transform to give:-

11 1

-2x1

OX1

+

+

+

xi 2

x2 +

(p+1)2 +

0;72 +

0x2 +

OT,

o71

II,1

71

+ 0T2

2

- X2

+ (p-1)T2

- 1 7 o

xl(°)

- '2(°)

- X1(0)

- X2(0)

=

=

=

=

0

0

0

A72.

Solving for xi gives

X1 = X1 + (p2-p)x2 2 +1DX2 +x2 +x1(p3+1) 1 2

(p4 + p2 + 2) A7.19.

Whe re the arguments (for time zero) have been dropped as no longer

relevant. Now p4 + p2 + 2 has four roots. Let these be yl and y2 having

negative real parts (the stable ones) and3 and y

4 having positive real

parts (the unstable ones). Note that

1 = - Y

3

and Y2 = - Y4 A7.20.

The numerators in a partial fraction expansion for xl involving these roots

can be evaluated in terms of the rootsand making the numerators of the

terms in y3 and y4 zero gives the conditions for stable behaviour. Thus

to make the coefficient of

1 ( p-.y3)

313.

zero, gives:-

X1 +y32x2 -y3x2 +iy3A2 + x2 + y32 xl + xl = 0

A7.21.

and for the coefficient of 1

(P-Y4 )

x + x + Ne42 xi + xi = - + y42x2 - ylec_ + " 7 '4 2 2 ' 2

A7.22.

Subtracting these two equations, and dividing by the common factor

(Y3- y4) gives:-

(Y3+ Y4) x2 x2 X2 ' „ (Y32 + Y3Y4 '

„ Y42) x1 = 0,

whence:

A7.23.

-2 x2= (Y32 + y42 + y3 y4) xl + (y3 + y4 - 1) x2.

A7.24.

Now the coefficients of x1 and x, in this expression can be evaluated

directly from the coefficients of the equation

1 2)7- 21 op+p !+ 2 = 0,

since y32 and y4

2 are a pair of its roots.

Thus y32 + y4 = -1

A7.25.

Y3 Y4 =

and thus y3+ y4 = j y32 Y42 + 2 y3y4 - 1

3lLE.

The control action u which is required is equal to - %2 which now

becomes

= i 1 - 2/2-- 1 x2 -( W2 - i ) 1 A7.26.

Note that this control has been evaluated without solving for the roots

of the double order system produced by the Euler-Lagrange equations.

Also the form of the control action is seen to consist of two parts. One

part exactly supresses the natural behaviour of the system, since in this

dx case x2 is the same as and x1 is the same as x, and the other part dt

imposes a new, optimal mode of behaviour which is bound to be stable,

315.

Appendix VIII

VARIATIONAL APPROACH TO OPTIMUM CONTROL OF SYSTEMS DESCRIBED

BY PARTIAL DIFFERENTIAL EQUATIONS

Consider a system described by the partial differential equation

G(ux, uy, 8, u, x, y) = 0 A8.1.

where the state is given by u(x,y) and the control by 8(x,y). Suppose

that it is required to operate this system to minimize the integral

I /- I = / Flux, u , u, x, y, e) dx dy. A8.2.

where the domain D is the region of interest for operation and the control

e is to be specified.

Let u(x,y) vary to u(x,y) + a 4(x,y)

where 1(x,y) is continuous and has first and second derivatives as

required. Similarly let 8(x,y) vary to 6(x,y) + a (x,y), where

(x,y) is restricted in the same way.

The'integral I will be stationary for the chosen value of u(x,y)

and the corresponding 8(x,y) if

d (I (u +a: , 8 + a ) da

= 0 A8.3.

_ia=0

Since the .results are to be based on the condition 'a = Of it is

satisfactory Lo consider a-y and al as small quantities, or to neglect

terms in higher powers of a because they will be eliminated when a is made

zero,

316.

Substitute u + a7 and 6 + into the integral I. This results

in:-

). (f(1,4. a Lt. a 2_2 aF cc: aF a? ) dxdy I + , 6 + a ?,. + + . -,15 aux' ax auy ay au / ae

A8.4.

Where terms in a2 etc. have been neglected.

The condition G = 0 must still be satisfied and this gives a relationship

• between 7 and

aG a a -14 aG a LI aG 8G = ( -3Tx. ax auy ay au A2ar)ae ' 0

A8.5.

This must be satisfied at all times and it can be multipled by a factor X

and integrated over the domain D, to be added in to the integral

I(u+a7 , e+aS ) without modifying the value of this latter integral, so

that when the derivative of the integral with respect to a. is evaluated

at a = 0 there results:-

, -1 i rd (I (u+a , e+a ) )1 = aF x aG a -14. aF x aG da aux aux ax au

Y au ay

a=0 D

( () aF .1-

x aG 7? 4-

aF + x aG au / ae ae

dxdy = 0

A8.6.

At this stage the analysis can be based on the modified function

F' = F+XG

which when substituted in the equation gives:-

J (021 LaF' + aF' aF' + dxdy = 0 aux ax au ay au "" • ae

A8.7.

_ a aFI _ a aF' au ax aux ay au y

0 A8.11.

and

Apply Green's formula to the first two terms:-

317.

J.:, 4. al ax au

Y '

.....:i ) ay

cbcdy = 1/ ( a ( j; ax D

1 al ), a ( au ay x '

(1., aF' )) f au

Y

dxdy -

f / Al f ) + a / ffizi t \ aux 53r au

y ) axdY

Y D

A8.8. which becomes:-

aF, dy ax (

27 -.-1 dx) - la 1:1 ), ..,1_,(...E' T ad 1 id ax aux ' ay au Y .

D - Y

A8.9.

7dxcly.

Substituting this back into the equation gives

(..1F' "Li dy — dx J aux i au

Y

al_ 'a( ( 1/ 22)) (au ax auX ay au

r D

(17)_idxdy = 0 A8.10.

For this equation to be satisfied regardless of the choice of 7 and! then

the Euler equations:

OF' ae

= 0

must be satisfied. Also the boundary condition on the whole of the

boundary r of the domain D must be met:

( '

x • 7 dy 7.1 • cbc = o A8.12.

318.

r

The generalized version including more variables to describe state,

more control variables and more independent variables follows exactly the

same formal pattern.

319.

Appendix IX

VARIATIONAL APPROACH TO OPTIMAL CONTROL OF SYSTEMS

DESCRIBED BY INTEGRAL EQUATIONS

Consider the system described by the set of integral equations

,ti

tiC..( ,Z , x1 xn('C), u,(r)...ur(

0 A9.1.

where the state vector x is an n-vector and the control vector u is an

r-vector. The system is to be operated by choosing u(t) so that the

performance integral

t xo 1 L(7; ,xi(r) xn( T), ui( t)...ur(1)

0 A9.2.

is minimized.

Let x. vary to x. 8 x. , i = 1 ... n

Let uj vary to u. 68 u. , j = 1 r

Then on the assumption that x(t) represents the best possible choice

of state in the range OSt1,any allowable small change about it will

make no difference to the performance integral; that is

a xo(t)1 o at 6=0

By using a Taylor expansion for L in the vicinity of x(t),

differentiating with respect to e and then letting 6 tend to zero it

xi(t) =

(: aaXt) 8x1 (t)

n

i=1

r

j=1

320.

is found that

laxo at

e =0

n /aL(t) 8x.(0) r

i=1 j=1

aL(t) 8u.(t)) dt=0 au. • 3

o A9.3.

However this equation alone is not enough to evaluate the best trajectory

because the constraint equations relating the variables x and u have to be

accounted for.

The relationships which must exist between the variations of x and

of u are found by taking the total derivatives of the equations. A9.1. and

equating these to zero.

pt,, 8x.(t) -

' t) 8x .(-0 8K. 811 (17) > ). dr = 0, i=1..n

0( 5

. 1

. j

i=18x. •

s=1 8ui 3 J

A9.4.

Multiply each equation by Xi(t). Sum all equations, integrate over (0,t1)

and add in to the integral in equation A9.3.

,t1 aL(t)

8u.(t)d) au.

i= Ai(t) ( 8xi(t) —

0

aKi (t,t)sx i-1 ax. s=1 -

aKi.su (7)7d-c4t.. aus

s

= 0 A9.5.

Consider the terms like 31 . 8x (z) ? dd dt • 8x. J j j

321.

Reverse the order of integra-bion to give

pt1 f

t1 Xi(t) aKi (t t. ) 8x (t)..1dt

j ax. j

Joio

s(

J --'1•(8x.

J(i) (X. (t)

o 'o

aK i (t,c))}dt d

ax.

Interchange the names of the independent variables:

o o (7) ax. J

t

4/ J ( a.

8K.

1 J 8x.(t) j x.

(1

I (?=, t) )d1". dt A9.6.

Replacing the double integrals in A9.5. by the manipulated versions as

from A9.6. gives

t n

ok j=1

n aL(0811.(t) Xi(t)6xi(t) 3u. j 1=1 i=1 .

( aL(t) 8x3_(0 ax. .)

n n tl aK. 11 oxi(t) / X. (I) ---2; t) a]. x

1=1 j=1

n r

?- 2, Sus(t) 1=1 s=1

aK. x ( *au C) t).? d

s ' d = 0

A9.7.

Collect together terms in Sxi, bus to give

t az(t) 3 8x. I 77 xi(t) - EL

t aK. , )

J 0 i=1 I X.(z) t dil+

1=1 J ax. j 0

322.

aL(t) K. Sus anus) j

j=1

(Z t) )d dt = 0 au

o A9.8.

If then the coefficients of each ox. and 8u. are each (independently)

zero and the system equations A9.1. are also satisfied then the problem is

equivalent to considering the minimization of

t1 . dt) in 2n + r independent variables 0

and produces one equation for each variable as follows:

11 n aK. i ? ax. ax.

--, w(t) = -.21= , ;._ x.(e). --a (-1-,t)( az i = 1 ... n

a. o Z j=1 J 1 I

xi (.0 = „tic

K. (t, 7:, , x, u) ? d .2; , i = 1 ... n , ( 1

0 -

, -r dus

aK. --2 (7-,t)*LT, s = 1 r aus I j

A9.9.

It is interesting to compare this method with the results of applying

Pontriapius method produced by Butkovski (Automatika i Telemeihani-kra„

Vol. 22 No. 1. (Jan. 1961).

In his case the kernels K. were independent of x making the problem

somewhat simpler, but as in the above method, not independent of u. Re-

moving the variable x from the kernels reduces the above set of equations to

Xi(t) = ax.

L'

• • • • (t)x. = tl Ki(t1 E, u) d 1

0

323.

0 = ous

()K./

j au s I A9.10.

Each of the V(t) can be eliminated to give

rt x.(t) = K.(t,c, u) d."(7 i = 1 .. n A9.11.

and 0 = aL aus

rt • n

8K. 5r. aL . (r t))d-z_

t s au ' 0

s = 1 r

A9.12.

Using the notation of this Appendix this solution method resulted in the

condition that the function H / given by:

H = Co t1

Tr ax. -- K. (C t' ' u) d? + C

o L + n

i — CK.

1(t,t,u)

3 . o j=1 i=1 ,

A9.13.

should be minimized with respect to u and Ci = 0, i = 1 n with Co a

positive number. This condition is equivalent to the equations A9.12 and

both methods are subject to the system cauations A9.11.

Using the Pontrigin approach Butkovski has extended this work to

cover systems of greater complexity.

The system is described by the integral equation

Q(p) = [ K(pls„ Q(s), u(s) ) ds 19.14.

where Q(p) is the state at point p in the n space which contains the

324,

domain D. u(s) is the control at s. Q(p) is an n-vector and K is an

n-dimensional L2 kernel.

The performance functional is allowed much wider ranges of forms and other

integral constraints are included by the specification:

Find .:(p) for which the functional Ii = 0, for i=1

and for which the functional IP is minimized.

These functionals are divided into two groups, one group,functionals on

state only, the other dependent on control as well,so that they have the

form:

(Q(p) ) i = 0, 1

1i (Q(p),u(p) ), i = 1+ 1 q A9.15.

To allow even more generality each functional is expressed as a

function of a vector functional)

Il = 0 (z) A9.16.

whore z is a functional (vector) having the form;

z= J F ds. A9.17.

D

The maximum principle then becomes:

.E1 I 0 — i I /dd K I dR 4- {ai az! IF: + tbd grad j , _ . ; 7 • 1111 I '. .1 I I.1 — kW dR

t . - .

• OaQi i-- ''• - D

-1)

A9.18,

should be a maximum with respect to u where [113 has to satisfy the condition

[Id + I arc Q Emi [LI ds = / 14q r14/ ds ✓

Ld . - D

A9.19.

n(s,u) = IS4L3ali

325.

where Lai is the vector [Co, Cl C2 Co and

[Ce+i, Ct+2

is the vector

The subsidiary condition on [ MJ arises from solving a set of integral equations to evaluate the variations in Q(s) as they depend on

each other and on variations in u(p) for all p in D.

The whole process is subject to the feasibility, not only of setting

up the necessary kernel function to represent the system but also of solv-

ing t the resulting integral equations for ;Plias well as the original

system equations.

326.

Appendix X

ITERATIVE CONTROL IMPROVEMENT - DETAILS OF FIRST ORDER EXAMPLE

A system is described by the first order ordinary differential

equation

= - a x = f(x) A1041.

where a is a positive number. The system is stable and the natural

trajectory, starting from x00 is

X = Xoe

-at A10.2.

Suppose that this system is to be operated to give best performance

against the performance criterion

(x2 + c u2) dt

0

A10.3.

where

used,

This cost

jo

u is the control

however, the cost

x2

dt =

is described

F(x) = x2 2a

which may be introduced. Until this control is

of completing a trajectory from x onwards is:

2 -tat xo 2

dt xo

e A10.4. 2a

for any position as

A10.5.

Introduce some control action u, so that the system equation becomes

X = - a x + u = fe(x) A10.5.

To find the local improvement due to introducing u it is necessary

to find u to:

Minimize

Substitute for

; 8F / ax

aF axl '

-7

fe(x) - f(x).1 c u21 A10.7.

fe(x) , f(x) to make the problem:

327.

Minimize 1 3-c- u + c u2

a A10.8.

This mosulit 3irst A.I.. of u as a tunctyon .t x. 1

-x 2ac A10.9.

The system behaviour is now modified for all x to become

+ - x (a 2ac ) A.10.10.

The resulting cost of completing a trajectory is now given by substituting

for x and u, in the complete cost function

(x2 + c u2) dt A10.11.

to give a new value of F(x):

r,, (x) = x2 (1 + 1/4 a2c) A10.12.

2a (1 + 1/2 a2c)

This is less than the cost of completing the trajectory without control.

The next stage is to repeat the process making a change in the control.

The system is

X = -(a + -1- 2ac ) x A10.13.

and is modified to

x -(a + -1- 2ac ) x + v Alo.14.

where V is the extra control. However, the cost of the control, u„ must be

328.

accounted for when the total control cost is assessed)so that the actual

cost of introducing control V is 'the cost of u t and V together less the

cost of ul alone'. The instantaneous cost of adding control V is

therefore

c(u2 , + V2 + 2uiV - u,2 = CV2 Vx a

A10.1.5.

Repeat the minimization processs again, this time it is necessary to

Minimize x a

1 (1+ 4a2c + cV2 -

(1 + 2a2c A10.16.

This is achieved when.

V = x 2ac

1

2(1 + 2a2c) A10.17.

So that the total control is

u2

-x 2ac

1

2(1 + 2a2c)] Alo.18.

The absolutely optimum controller to solve the same problem can be

evaluated from a manipulation of the equations resulting from the variation-

al approach, as shown in Appendix VII and in Chapter VII.

The result of this approach gives the optimum control function as

uor.,t = x ( a. —11 + 1/c )

A10.19.

where for the purposes of establishing a comparison on the basis of one

parameter only the factor a is taken as unity.

The results of the two steps of the iterative scheme are compared

with this in Chapter VII.

329.

Appendix XI

GENERAL APPROXIMATE COST FUNCTION

The approximate cost of completing a trajectory in the state space

is to be evaluated as a function of the starting point for a general linear

system with an integral-quadratic cost function. The assumed trajectory

is the line joining the starting point to the origin (which is the end

point).,and the velocity along this line at any point is the projection of

the actual velocity at that point.

Suppose the system is described by the equations

x = A x A11.1.

The cost function is r

I L (x) dt 1r (xT B x) dt A11.2.

o

To evaluate the cost of completing a trajectory from the point to the

origin, let s be defined as

A11.3.

where x is a point on the line joining z. to the origin.

Then x = s y , and s goes from 0 to 1 as x moves along the

line from the origin outwards.

The projection of the actual velocity in the direction of the line

is

V = yT . x = - sz Az 11Y0 - JIY0

A11.4.

The cost rate L(x) is

330.

L(x) = s2 yT B y All.5.

Then M(z) = /14

L(x) d(x) A11.6.

V(x) o

,1

- s2

J' o

yT B y izi 2. ds. A11.7. s(xTA z)

—a. z B z 1,12. A11.8.

yT A y

Example: First order system

x

- ax

leads to

Ni(y) = - 4. Y2 • Y2

2 A11.10.

2a

This corresponds with the exact case (see Appendix X, where F(x) is

evaluated) because in one-space the approximate and exact trajectories

are bound to correspond.

331.

Appendix X11

CORRELATION COEFFICIENTS AND FUNCTIONS FOR INSTRUMENTATION CRITERIA

The correlation between two measures of system state taken at differ-

ent points in a distributed parameter system can be evaluated analytically

in terms of the transfer functions and weighting functions between the

source of input signal and the two points at which measurements are made.

Suppose that two measurements of state are made, yl(t) and y2(t), at

different pointsl and that both of these responses are due to the same

disturbance function d(t) at the input to the system.

If the weighting functions for the two points are fl(r) and f2('),

respectively) then the responses yi(t) and y2(t) are given by

I -

Y1(t)

J

d(t-u) f1(u) du

co jr. V,

and y2(t) = d(t-V) f2(v) dv

The correlation function between yl and Y2 is

E Yl(t ).Y2 (t+

d( t-u). (u). d( t+ 2= -v ) f (v) du dv

(u)f2(v) E id(t-u).d(t+i- -v)i- du dv

Al2.2. or

R yiy2( 7 ) (u)f2(v) dd(Z.+ u - v) du dv.

Al2.3.

Where Rdd is the autocorrelation function of the disturbance, and R y1y2

332.

the cross correlation of the two measures.

When the input disturbance is white noise then the function Rdd(/;)

reduces to 805) so that

R yly2(T) = f f (1.1)' (F +u) du. 1 2 Al2.4.

Jr

and in the particular case considered for the instrumentation

R yiy2(0) (u) f2(u) du. Al2.5. 1 fl (u)

make this into a correlation coefficient it is normalized by dividing by

1(1/(u) )2du. f(f2(u) )2duj Al2.6.

So that in the event that f1 and f2 are identical the correlation

coefficient is unity.

The computer programme to evaluate these correlation functions for

a particular system is described in Appendix XIII.

The power spectrum associated with the correlation function is

derived by taking the Fourier transform of both sides of the equation for

the correlation function.

The equation is r"

R y1y2( ) = I fl (u). f2 (v) Rdd( ( -C -v) + u) du dv

Al2.7.

f (v ) f1(u) Rdd ( ( -v ) + u) du? dv

_ Al2.8.

f fl

f2(v) q (t -v) dv Al2.9.

333.

where q is some function of Cr -v). Transform both sides of this

equation to give the power spectrum:

P Y1Y2 (jw) = F2(jw) . Q(jw)

Al2.10.

But

Q(jw) =

F1(-jw) . Pdd(jw)

Al2.11.

Therefore

P y1y2(jw) = F1( jw)F2(jw) Pdd(jw)

Al2.12.

and in the special case where d(t) is white noise this becomes

P Y1Y2(jw) = F1(-jw) F2(jw) Al2.13.

334#

Appendix XIII

PROGRAMME FOR COMPUTING CORRELATION BETWEEN SUCCESSIVE

MEASUR7,; OF SYSTEM STATE

The requirement was for a programme to compute the r ponses at

all points in a system to an impulse at the input andthen calculate the

time integral of all the cross products of these responses, for zero time

lag between themland finally to normalize these terms to produce the

correlation functions.

Since only the correlation at zero lag was required the data

storage problem was much reduced by computing the cross products after

each integration stip and forming cumulative totals.

The sequence of operations followed in this programme is shown

in fig. A13.1.

Arreakwre ONE srep

u RDA re

Co Mut A rivE To 7-AL S

NORM* le1

COOPP,O /FLATS

PR/N r Our

ROSuL TS.

1

S TART Data Requirements:

1) Coefficients of differential elns. 2) Length of computer run. 3) Amount of output of impulse resp-

-onse required. bATA

Output:

1) Record of impulse responses(if required).

2) Set of correlation coefficients.

Fig.A13.1. Programme for correlation coefficients between successive measures of system state.

CONTROL OF DISTRIBUTED PARAMETER SYSTEMS A Thesis ...

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