(ebook) Multivariable control, an introduction.pdf

5/25/2018 (ebook) Multivariable control, an introduction.pdf

1

Multivariable Control: An introduction

Dr M.J. Willis

Department of Chemical and Process Engineering

University of Newcastle upon Tyne

email:[email protected]

Written: 4th

November, 1998

Modified: 8th

November, 1999

Aims and Objectives

To introduce the basic concepts of multivariable control (a continuous stirred tank

reactor will be used as the motivating example). To highlight the phenomenon of

loop interactions. To learn how to model multivariable systems using input-output

descriptions. To introduce the relative gain array (RGA) - a tool for the selection of

input-output pairings.

At the end of this section of the course you should be able to select appropriate

manipulated variable - controlled variable pairings to minimise the effect of loop

interactions in multivariable systems. You should know how to formulate and

interpret the RGA.

Plan

Define the terms SISO and MIMO. Introduce MIMO control and loop interaction using a CSTR as the motivating

example.

discuss systems modelling for MIMO systems (the transfer function approach). clarify discussions using a worked example (modelling and control of a mixing

process).

introduce the RGA and discuss how it may be used to select input-outputpairings.
mailto:[email protected]:[email protected]


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Introduction

Processes with only one output being controlled by a single manipulated variable are

classified as single-input single output (SISO) systems. It should be noted however,that most unit operations in chemical engineering have more than one control loop.

In fact, each unit typically requires the control of at least two variables, e.g. product

rate and product quality. There are therefore usually at least two control loops.

Systems with more than one control loop are known as multi-input multi-output

(MIMO) or multivariable systems.

A Continuous Stirred Tank Reactor (CSTR)

A continuous stirred tank reactor (CSTR) is used to convert a reactant (A) to a

product (B). The reaction is liquid phase, first order and exothermic. Perfect mixing is

assumed. A cooling jacket surrounds the reactor to remove the heat of reaction.

EffluentCoolant

CAo, Fo, To

F, CA, T

Tj

A B

Exothermic, first order

Constant Volume

TC

CC

Fig 1 A basic control scheme for a CSTR

In this system variables of interest (from a control engineers perspective) could be,

for example, product composition and temperature of the reacting mass. There will

therefore be a composition control loop as well as a temperature control loop. Feed

to the reactor is often used to manipulate product composition while temperature is

controlled by adding (removing) energy via heating (cooling) coils or jackets. Thisbasic control configuration is demonstrated in Fig (1). 'TC' represents a temperature

controller, the mv for this loop being coolant flowrate to the jacket. 'CC' represents

the composition controller, the mv being reactant feedrate.

Assuming that the control system has been configured in this manner, lets consider

a change in feed flowrate. Perhaps this is necessary to bring composition back to its

desired level. This manipulation of feedflow will also change the temperature of the

reaction mass. Heat removal or addition, on the other-hand, would influence the

rate of reaction and hence composition. This phenomenon, known as loopinteraction, occurs in many processes and must be considered when developing a


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control strategy. If it is not, it may be difficult to run the unit under closed loop

control, creating many operational problems.

Thus, for two loops to work successfully together each loop must know what the

other is doing. Otherwise, in trying to achieve their respective objectives each loopmay act against the interests of the other.

Developing Process Models for Multivariable Systems Analysis

When designing a multivariable control strategy, the process must first be modelled.

This can be achieved either analytically using sets of differential equations to

describe a systems behaviour or empirically, using data obtained from an open loop

step test fitted to an assumed model structure. For the purposes of the control

system design we often use the latter, parameterising the model using 1st

order plustime-delay transfer functions.

Input-Output Multivariable System Models1

For systems with more than one output, input-output models may assume a number

of structural forms. However, discussions will be restricted to the model structure

shown in the diagram below,

G11(s)Loop 1

Loop 2

G21(s)

G12(s)

G22(s)

+

+

v 1

v2

cv 1

cv2

Fig 2 (2 x 2) Multivariable model structure

Here G11(s) is a symbol used to represent the forward path dynamics between mv1

and cv1, while G22(s) describes how cv2 responds after a change in mv2. The

interaction effects are modelled using transfer functions G21(s) and G12(s). G21(s)

describes how cv2changes with respect to a change in mv1while G12(s) describes

how cv1changes with respect to a change in mv2.

1For sake of simplicity during these, and subsequent discussions only (2 x 2) processes will be considered.


4

For the CSTR shown in figure (1) mv1could be the coolant flowrate, while mv2could

be the flowrate of the reactant. The output cv1may be the reactor temperature while

the output cv2would be the effluent concentration.

The mathematical model written in matrix-vector notation

The elements within the blocks of Figure (2) are transfer functions, defining the

relationship between the respective input output pairs. As usual, the following

general transfer function description will be used,

G sk e

spp

s

p

( )=+

1 (1)

where kp is a process gain, p the process time constant and the process timedelay. Note that each of the 4 blocks in Figure (2) will have different parameters that

must be determined.

Referring to fig. 2., on a loop by loop basis, the outputs of the system model are

related to the inputs as follows,

Loop 1: cv1= G11mv1 + G12mv2 (2)

Loop 2: cv2= G21mv1+ G22mv2 (3)

Equations (2) and (3) may be expressed more compactly in matrix-vector notation

as:

cv = G mv (4)

where cv= [cv1, cv2]T; mv= [mv1mv2]

T

and, G

G G

G G=

11 12

21 22

Note that this is a matrix of transfer function elements.

Incorporation of load disturbance terms into the systems model

Thus far, only the major part of the process has been considered. In many

situations, processes are influenced by external factors such as changes in ambient

conditions, changes in the quality of raw materials; changes in the operating

environment and so on. To cater for these effects, load disturbance terms may also


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be incorporated within the model. Incorporation of load disturbance terms in the

model representation leads to the following expression:

cv= Gmv+ Gddv (5)

where [ ]GG

G and dv ,dvd

d

d

=

=

1

21 2

0

0dv

The block diagram representation of this system model is given by,

G11(s)Loop 1

Loop 2

G21(s)

G12(s)

G22(s)

+

+

u1

u2

y1

y2+

Gd1(s)

Gd2(s)

d1

d2

Fig. 3. Incorporating load disturbances into the system model

In other words, disturbances are added to the process output in exactly the same

fashion as considered for single loop systems in process control 1.

Worked Example

Consider the following mixing process,

f1

f2fo

c#

Control objective: regulate foand c#to desired levels


6

Manipulated variables: f1and f2

Assume that the system is a two component system. Let f1be the flowrate (kg/hr) of

stream 1, f2be the flowrate of stream 2 (kg/hr) while fois the total flowrate. Also, if c1

and c2 are the mass fractions of component A in each stream and c#

the massfraction of component A in the mixed stream:

1. What are the material balance equations for this system ?

2. Develop a linear (2 x 2) model representation by linearisation of the material

balance equations around the following steady-state operating point, fo = 100

kg/hr, c#= 60% with c1=80% and c2=20%.

3. If the dynamics of the mixed streams concentration and measurement sensors

and are characterised by the following expressions,

G s f

fm

o

m

o

( )= =1 and G s e c

cm

s m

( ) #= = 2

1(6)

i.e measurement of flow is assumed instantaneous, while the measurement of mass

fraction is assumed to be affected by an analyser delay (which is modelled as a pure

time delay process). Develop the overall linearised dynamic model of the system.

Draw the block diagram representation of the system.

Solution

1) The overall mass balance is given by,

fo= f1+ f2 (7)

The component balance is given by,

foc#=f1c1+ f2c2 (8)

2) We require the mathematical relationship between the two outputs and the two

manipulated variables. Equation (7) provides one expression. Rewriting the material

balance equations (7) and (8) gives the other as,

cf c f c

f f# =

++

1 1 2 2

1 2

(9)

Equation (1) is linear however, equation (3) is not a linear function. To linearise the

relationship between c#and the manipulated variables, fo and f1 it is necessary touse a 1

st order Taylor series expansion of equation (9) around the steady-state


7

operating point (see the Appendix for an explanation about linearisation via the

Taylor series). The basic structure of the linear equation is:

cdc

df f

dc

df f

ss ss

#

#

'

#

'' =

+

1

1

2

2

The following partial differentials are required2

dc

df

f c c

f f

# ( )

( )1

2 1 2

1 2

2=

+and

dc

df

f c c

f f

# ( )

( )2

1 1 2

1 2

2=

+(10)

This gives the linearised model,

cf c c

f f

ff c c

f f

fss ss# ' '' [

( )

( )

] [( )

( )

]=

+

+

+

2 1 2

1 2

2 1

1 1 2

1 2

2 2 (11)

where c# = c

#- c

#ss, f1 = f1- f1ss and f2 = f2- f2ss are deviation variables.

To calculate the steady-state conditions required by equation (12), solve the material

balance. We have two algebraic expressions,

100 = f1+ f2

0.6*100 = 0.8*f1+ 0.2*f2 (13)

Solving for f1and f2gives,

f1= 662/3kg/hr and f2= 33

1/3kg/hr

Therefore,

c# = 2x10-3

f1 + (-4x10-3

)f2 (14)

where c# = c#-0.6, f1 = f1- 662/3 and f2 = f2- 331/3

Using a block diagram to visualise this system model we have,

2To do this differentiation I have used the quotient rule, i.e

d u v

dx

vdu

dx udv

dx

v

( / )= 2


8

f1

f2

fo

c#2

+

+

1

1

2 x 10-3

-4 x 10-3

1

cm

fom

e s

and this may be written as,

f s

c s x e x e

f s

f s

o

m

m

s s

( )

( )

( )

( )

=

1 1

2 10 4 103 31

2

(15)

Summary

So far, these notes have introduced multivariable systems modelling by

consideration of a 2 x 2 (2 input - 2 output) representation. The block diagram

representation is used to clearly illustrate system dynamics and interactions. Once a

mathematical model of the system has been developed, and the presence of

process interactions identified, the next stage of the control design procedure is to

synthesise the control law.

With multivariable systems, where loop interactions exist, configuration of two single

loop PI(D) controllers could cause system instability, or at the very least result in

poor control performance. This can be overcome by,

choosing a manipulated variable - controlled variable pairing so that systeminteractions are minimised. The basic question for a 2 x 2 system is do we want

mv1-cv1, mv2-cv2(i.e. mv1 to control cv1and mv2 to control cv2) or the other way

round ?

the design a multivariable controller that achieves non interacting control.

Design of multivariable controllers will be considered later in the course. The next

section of the notes concentrates on a systematic technique for choosing a

manipulated variable - controlled variable pairing so that system interactions are

minimised.


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The Relative Gain Array (RGA)

One of the most important factors, common to all process control applications, is the

correct (best) pairing of the manipulated and controlled variables. A number of

quantitative techniques are available to assist in the selection process. One of theearliest methods proposed was the Relative Gain Array (RGA), Bristol (1966). The

original technique is based upon the open loop steady state gains of the process

and is relatively simple to interpret2.

Determining relative gains from process experiments

Consider the 2x2 system shown in Figure 2. Suppose mv2remains constant, then a

step change in mv1 of magnitude mv1will produce a change cv1 in output cv1.

Thus, the gain between mv1and cv1when mv2is kept constant is given by:

g11|mv2=

cv

mvmv

1

12 (16)

If instead of keeping mv2 constant, cv2 is now kept constant by closing the loop

between cv2and mv2. A step change in mv1of magnitude mv1will result in anotherchange in cv1. The gain in this case is denoted by:

g11|y2= cvmv

cv1

12

(17)

The gain relationships, equations (16) and (17) may have different values. If

interaction exists, then the change in cv1due to a change in mv1for the two cases

when mv2and cv2are kept constant, will be different.

The ratio: 11=g

g

mv

cv

11

11

2

2

|

|(18)

is a dimensionless value and it defines the relative gain between the output cv1and

the input mv1.

Interpretation of the relative gain

1. ij= 1. There is no interaction with other control loops.

2. ij= 0. Manipulated input, i, has no affect on output,j.

2

As no information about the process dynamics is used a "health-warning" should be attached to the resultsalthough in practice the interpretation is normally valid.


10

3. ij= 0.5. There is a high degree of interaction. The other control loops have the

same effect on the output,j, as the manipulated input, i.

4. 0.5 < ij< 1. There is interaction between the control loops. However, this would

be the preferable pairing as it would minimise interactions.

5. ij > 1. The interaction reduces the effect gain of the control loop. Highercontroller gains are required.

6. ij> 10. The pairing of variables with large RGA elements is undesirable. It can

indicate a system sensitive to small variations in gain and possible problems

applying model based control techniques.

7. ij< 0. Care must be taken with negative RGA elements. Negative off-diagonal

elements indicate that closing the loop will change the sign of the effective gain.

More importantly, negative diagonal elements can indicate integral instability i.e.

the control loop is unstable for any feedback controller.

Elements of the RGA

For the 2 x 2 process, three other relative gain elements can be defined yielding the

matrix,

3=

11 12

21 22

11 11

11 11

1

1

=

Determining relative gains from process models

A computational method is possible if a steady-state model of the system is

available. If this model is given by:

cv1= g11mv1+ g12mv2

cv2= g21mv1+ g22mv2 (19)

then g11|mv2= (cv1/mv1)|mv2= g11 (20)

Eliminating mv2from the steady-state relationships, Eqs. (19) and (20), results in:

cv1= g11mv1+ g12(cv2-g21 mv1)/g22 (21)

from which:

g11|cv2= (cv1/mv1)|cv2= g11-g12 g21/g22

3A property of the RGA is that the rows and columns sum to 1.


11

The relative gain 11is therefore given by: 11=g

g

g

gg g

g

u

y

11

11

11

11

12 21

22

2

2

|

|=

Therefore:11

=1

1 12 21 11 22 ( ) / ( )g g g g(22)

A general calculation procedure for the RGA

Skogestad (1987) demonstrated that the RGA calculation can be expressed in

matrix notation, facilitating computation for systems larger than 2x2.

( )RGA G G T=

.*1

(23)

where, G is the process gain matrix, '.* ' represents Schur (element by element)matrix multiplication, [.]

-1denotes a matrix inverse and [.]

Tis the transpose operator.

The mixing example revisited: calculation of the RGA

For the mixing process the following two control schemes are possible:

Mixer

FTFC

CTCC

F1, x1

F2, x2

F , x

Mixer

CTCC

FTFC

F1, x1

F2, x2

F , x

Now that the steady-state gain matrix has been calculated, it is possible to use the

RGA to determine the "best" manipulated variable control variable pairings.


12

Calculating the RGA in Matlab

To illustrate how the RGA is calculated in Matlab, consider the steady-state gain

matrix of the mixer,

Gx x

=

1 1

2 10 4 103 3

Enter the MATLAB environment and enter the commands preceeded by the

MATLAB prompt >>. The text after the % is intended to tell you what you are doing

(and should not be typed).

% enter the steady-state gain matrix

>>g = [1 1 ;2e-3 -4e-3]

% calculate the RGA4

>> rga = g.*inv(g)

This should give the result,

rga =

0.6667 0.3333

0.3333 0.6667

In other words, the RGA analysis suggests f1should be used to control fo, while f2

should be used to control c#

Exercises

1. Simulate the following transfer function representation of a 2 x 2 process using

simulink

4In Matlab, inv calculates the inverse of a matrix while calculates the transpose of the matrix


13

mv1

mv2

cv1

cv2

+

+

1

10 1s +

1

20 1s+

+0 2

5 1

.

s

0 8

10 1

.

s +

a) discuss the dynamic response characteristics of cv1 and cv2 with respect tochanges in mv1and mv2.

b) Tune two PI(D) controllers the first to control cv1 using mv1 and the second to

control cv2 using mv2. (use the simulink simulation environment). Make critical

comments on the functionality of the system.

Summary

Using a simple process example these notes have:

introduced the concept of systems interaction. demonstrated how multivariable systems can be represented (modelled). introduced the relative gain array (RGA). shown how the RGA may be used to select loop pairings.

If the RGA indicates little interaction within the system then it may be possible to usesingle loop controllers. In some situations this may not be possible in which case it

will be necessary to develop a multivariable control law. The design of multivariable

controllers will be covered in detail in later lectures.


14

Appendix: Linearisation

Given a function,

dxdt

f x= ( )

Expand f(x) using a Taylor series around xo,

f x f xdf

dx x xo x oo( ) ( ) ( ) ( )= + (Ignoring higher order terms)

Control system models are always expressed in terms ofdeviation variables. That

deviation (or difference) being the difference between the actual value and a steady-state value. If xssis the steady-state value of x, then,

dx

dt f x

ss

ss= =( ) 0

and so,

d x x

dt

df

dxx xss

x ssss

( )( ) ( )

=

or, if the deviation variable is defined as x=x-xss,

d x

dt

df

dxxxss

( )( )

'

'=

This concept generalises if there is more than one variable. For instance if there are

two variables we have,

f x x df

dxx

df

dxxx x x xss ss ss ss( , ) ( ) ( ), , , ,,

'

,

'

1 2

1

1

2

21 2 1 2= +

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