1

Lecture Course in

Adaptive Control

for the 4th

Class of Control Engineering in the

Control and Systems Engineering

Department at the University of

TechnologyCCE-CN445

Prepared By:

Dr. Mohammed Y. Hassan, Ph. D.

Adaptive Control

2

Syllabus

Fourth Year-Control Eng. Theoretical: 2 Hr./ Week

CCE-CN445 (Semester)

1. Introduction and General Aspects: (4 Hrs)

What is Adaptive Control?

Relationship between Non-Adaptive, robust and Adaptive Control.

Performance objectives and design constraints for the control

engineers.

2. Types of Adaptive Control strategies: (2 Hrs)

Gain Scheduling Regulators

Self-Tuning Regulators.

Model Reference Adaptive Control.

3. Gain Scheduling Regulator: (2 Hrs)

4. Self Tuning Regulator: (8 Hrs)

Pole-Assignment by output Feedback.

5. Model Reference Adaptive Control: (6 Hrs)

6. Practical aspects and Implementation: (2 Hrs)

7. Relation to other topics: (6 Hrs)

Expert Systems.

Neuro Controller.

Fuzzy Controller.

Genetic Auto-Tuners.

Total: 30 Hrs

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3

References:

1. “Adaptive Control”

By: K. J. ÅstrÖm and B. Wittenmark.

2. “Self Tuning Systems”

By: P. E. Wellstead and M. B. Zarrop.

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4

ADAPTIVE CONTROL

1-Introduction

1.1-Definitions:

Adaptive controller:

Is a controller that can modify its behavior in response to changes in the

dynamics of the process and the disturbances.

Adaptive system:

Is any physical system that has been designed with an adaptive view

point.

1.2 Historical background:

In the early 1950, there was extensive research on adaptive control, in

connection with the design of autopilots for high performance aircraft. Such

aircraft operate over a wide range of speeds and altitudes. It was found that

ordinary constant-gain, linear feedback control could work well in one

operating condition, but that changed operating conditions led to difficulties.

A more sophisticated regulator, which could work well over a wide range of

operating conditions was therefore needed.

In the 1960s many contributions to control theory were important for the

development of adaptive control. State space and stability theory were

introduced. There were also important results in stochastic control theory.

Dynamic programming introduced by Bellman, increased the understanding of

Adaptive Control

5

adaptive processes. Fundamental contributions were also made by Tsypkin,

who showed that many schemes for learning and adaptive control could be

described in a common framework as recursive equations of a special type.

There were also major developments in system identification and in parameter

estimation.

In the late 1970s and early 1980s correct proofs for stability of adaptive

systems appeared. Rapid and revolutionary progress in microelectronics has

made it possible to implement adaptive regulators simply and cheaply.

Vigorous development of the field is now taking place, both on universities and

industry. Several commercial adaptive regulators based on different ideas are

appearing on the market. A great number of industrial control loops are under

adaptive control. These include a wide range of applications in aerospace,

process control, ship steering, robotics and other industrial control systems.

To works with adaptive control one must have a background in

conventional feedback control and also sampled data systems. The reason for

this is that virtually all adaptive systems are implemented using digital

computers.

However, adaptive control has links in many directions, some of which are

illustrated in figure bellow:

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6

Fig. Links with adaptive control.

There are strong ties to nonlinear system theory, because adaptive systems are

inherently nonlinear.

1.3 Adaptation and tuning:

It is customary to separate the tuning and adaptation problems. In the

tuning problem it is assumed that the process to be controlled has constant but

unknown parameters; in the adaptation problem it is assumed that the

parameters are changing.

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7

Many issues are much easier to handle in the tuning problem. The convergence

problem is to investigate where the parameters converge to their true values.

The corresponding problem is much more difficult in the adaptive case,

because true values are changing.

1.4 Direct and Indirect adaptive controllers:

In general, the adaptive controllers can be divided into two algorithms;

direct and indirect. In direct algorithms, the parameters are updated directly.

If the controller parameters are obtained indirectly via a design procedure, we

use the term indirect algorithms. Sometimes, it is possible to re-

parameterize the process model such that it is possible to use either a direct or

indirect controller.

However, the indirect methods have sometimes been called explicit self-

tuning control, since the process parameters have been estimated. Direct

updating of the regulator parameters has been called implicit self tuning

control.

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8

2-Why adaptive control:

There are many reasons for adaptive control. The key factors are:

a-Variations in process dynamics. Parameters may vary due to nonlinear

actuators, changes in the operating conditions of the process, and non-

satisfactory disturbances acting on the process.

b- Variations in the character of the disturbances.

c- Engineering efficiency.

However, the most common regulators is a feedback controller with

fixed parameters. Through feedback it is possible to decrease the sensitivity to

parameter variations by increasing the loop gain of the system. The main

drawbacks of high-gain controllers are the magnitude of the control signal and

the problem of stability of the closed loop system. If there are bounds on the

uncertainties of the process parameters, it is possible to design robust

controllers by increasing the complexity of the controller. To use this

approach, it is necessary to know the structure of the process fairly accurately

and to have bounds on the variations of the parameters.

In the following, two examples in which parameter variations cause

difficulties when a fixed gain regulator is used:

Example 1: Nonlinear actuators (nonlinear valve):

Many actuators have a nonlinear characteristics which creates

difficulties unless special precautions are taken.

A simple feedback loop with a nonlinear valve is shown in figure below:

Adaptive Control

9

The static valve characteristics be:

u)u(fV4

0u

Linearizing the system around a steady state operating point shows that the

loop gain is proportional to )u(f

. It then follows that the system can perform

well at one operating point and poorly at another. This is illustrated by the step

responses for PI control with k=0.15, Ti=1 and

)1s(

1)s(G

3

.

Fig. Step responses for PI control of the simple flow loop.

Adaptive Control

10

Example 2: Airplane control (Short-period airplane dynamics):

Control of high performance airplane was strong driving force for the

development of adaptive control in 1960s. The dynamics of an airplane depend

on the speed, altitude, angle of attack, …etc. Also, the flexible structure of the

airplane will cause difficulties when designing the control system. In figure

below, the notations for the airplane is given:

The rigid body or the so-called short period dynamics for an experimental F4-

E with canards will be given. The canards make it easier to maneuver the

airplane at the cost of stability. The dynamics can be linearized around

stationary flight conditions (i.e. constant speed and altitude and small angle of

attack α). Let the state variables be normal acceleration (Nz) and pitch rate q=

.

. It is assumed that the dynamics of the servo of the elevons and canards have

the transfer function:

as

a)s(G

with a=14

Let δe be the corresponding state. The model of the airplane has three states

Nz, q, and δe and the linearized equations have the form:

u

a

0

1b

X

a00

23a22a21a

13a12a11a

X.

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11

Assuming the eigen values of the short-period dynamics are λ1 and λ2, it is

seen that the airplane is unstable for subsonic speed and poorly damped for

supersonic speeds as shown in the table below:

mach 0.5 0.9 1.5

Altitude (feet) 5000 35000 35000

a11 -0.9896 -0.667 -0.5162

a12 17.41 18.11 26.96

a13 96.15 84.34 178.9

a21 0.2648 0.08201 -0.6896

a22 -0.8512 -0.6587 -1.225

a23 -11.39 -10.81 -30.38

b1 -97.78 -85.09 -175.6

λ1 -3.07 -1.87 -0.87-43i

λ2 1.23 0.56 -0.87+4.3i

Note: It is essential to know the frequency response at the desired crossover

frequency in order judge whether parameter variation (position of poles) will

have any effect on the closed loop system properties. Step responses are poor

guide; it is much better to look at the frequency responses.

3- Adaptive schemes:

There are three main adaptive schemes:

a-Gain scheduling.

b-Self-tuning regulator (STR).

c-Model-reference adaptive control (MRAC).

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12

a-Gain scheduling:

In many situations it is known how the dynamics of a process change

with the operating conditions of the process. One source for the change in

dynamics may be nonlinearities that are known. It is then possible to change

the parameters of the controller by monitoring the operating conditions of the

process. This idea is called gain scheduling.

Its principle is to reduce the effects of parameter variations by

changing the parameters of the regulator as function of auxiliary variables that

correlate well with these changes in process dynamics. It is a nonlinear

feedback controller of a special type. It has a linear regulator whose

parameters are changed as a function of operating conditions in a

programmed way.

Gain scheduling was used in special cases: such as autopilots for high-

performance air-craft.

The principle:

It is sometimes possible to find auxiliary variable that correlate

well with the changes in process dynamics. It is then to reduce the effects of

parameter variations simply by changing the parameters of the regulator as

functions of auxiliary variables as shown in figure below:

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13

Gain scheduling can thus be viewed as a feedback control system in which the

feedback gains are adjusted using feed forward compensation.

A main problem in the design of systems with gain scheduling is to find

suitable scheduling variables. This is normally done based on knowledge on the

physics of a system. When scheduling variables have been determined, the

regulator parameters are calculated at a number of operating conditions, using

some suitable design method. The regulator is thus tuned or calibrated for

each operating condition. The stability and performance of the system are

typically evaluated by simulation.

Gain scheduling has the advantage that the regulator parameters can be

changed quickly in response to process changes.

However, gain scheduling has two main disadvantages:

1) It is time-consuming to find the correct scheduling which should be

checked by extensive simulation.

2) There is no feedback to compensate for incorrect scheduling.

Design of Gain Scheduling regulators:

It is difficult to give general rules for designing gain scheduling

regulators. The key question is to determine the variables that can be used as

scheduling variables. It is clear that these auxiliary signals must reflect the

operating conditions of the plant. The following general ideas can be useful:

1) Linearization of nonlinear actuators.

2) Gain scheduling based on measurements of auxiliary variables.

3) Time scaling based on production rate.

4) Nonlinear transformation.

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14

Example 1: Nonlinear Actuator:

Consider the system with a nonlinear valve characteristics

discussed previously. The nonlinearity is assumed to be:

u)u(fV4

0u

Let f1 be an approximation of the inverse of the valve characteristics. To

compensate for the nonlinearity; the output of the regulator is fed through this

function before it is applied to the valve. This gives the relation:

))c(f(f)u(fV 1

where c is the output of the PI regulator. The function ))c(f(f 1 should have

less variation in gain the f. If f1 is the exact inverse, then v=c. Assume that

f(u) is approximated by two straight lines: one connecting the point (0,0) and

(1.3,3), and the other connecting (1.3,3) and (2,16) as shown in figure below:

Then:

16C3..............

3C0.............

139.1C0538.0

C433.0)c(f

1

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15

The figure below shows step changes in the reference signal at three different

operating conditions when the approximation of the inverse of the valve

characteristics is used between the regulator and valve.

By improving the inverse it is possible to make the process even more

insensitive to nonlinearity of the valve.

Example 2: Tank System:

Consider a tank where the cross section A varies with height h.

The model is:

h.g.2.aqi)h),h(A(dt

d

Where qi is the input flow and a is the cross section of the outlet pipe. Let qi be

the input and h is the output of the system. The schematic diagram of the tank

is shown in figure below:

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16

The linearized model at an operating point qin and h, is given by the transfer

function:

s)s(G

where : )h(A

1o

and oo

o

oo

o

h).h(A.2

h.g.2.a

h).h(A.2

qin

A good PI control of the tank is given by:

)d)(eTi

1)t(e(K)t(u

where

2K and

2

2Ti

Introducing the expressions for α and β gives the following gain schedule:

o

o

o

h.2

qin)h(A...2K

2oo

o

i

.h).h(A.2

qi.2T

The numerical values are often such that α << 2ζω. The schedule can then be

simplified to: K=2ζωA(h) and Ti=

2

In this case it is sufficient to make the gain proportional to the cross section of

the tank. For ζ=0.5 and ω=1 rad/sec, the gain schedule of the system is:

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17

Cross section A(h), (m2) Gain (K)

0.2 0.2

0.4 0.4

0.6 0.6

0.8 0.8

From the two examples, it was possible to determine the schedules exactly. The

behavior of the closed loop system does not depend on the operating

conditions. In other cases it is possible to obtain only approximate relations for

different operating conditions. The design then has to be repeated for several

operating conditions to create a table. The gain schedule is usually obtained

through simulations of a process model, but it is possible to build up the gain

table online. This might be done by using an auto-tuner or an adaptive

controller. The adaptive system is used to get the controller parameters for

different operating points. The parameters are then stored for later use when

the system returns to the same or a neighboring operating point.

H.w.: Read the other examples discussed in chapter nine.

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18

b- Self-Tuning regulator:

Basic idea:

In an adaptive system, it is assumed that the regulator parameters are adjusted

all the time. This implies that the regulator parameters follow changes in

process. It is difficult to analyze the convergence and stability properties of

such systems. To simplify the problem it can assume that the process has

constant but unknown parameters. When the process is known, the design

procedure specifies a set of desired controller parameters. The adaptive

controller should converge to these parameter values even when the process is

known. A regulator with this property is called Self-Tuning, since it

automatically tunes the controller to the desired performance.

The Self- Tuning Regulator (STR) is based on the idea of separating the

estimation of unknown parameters from the design of the controller. The basic

idea is illustrated in figure bellow:

Fig. Block Diagram of Self Tuning Regulator (STR)

The Unknown parameters are estimated on-line using Recursive estimation

method. The estimated parameters are tested as if they are true. i. e., the

uncertainties of the estimates are not considered.

Adaptive Control

19

Many different estimation schemes can be used, such as:

a) Stochastic approximation.

b) Least squares.

c) Extended and generalized least squares.

d) Instrumental variable.

e) Maximum likelihood.

In the block diagram, the design block represents an on-line solution to the

design problem for a system with unknown parameters. This is called the

underlying design problem. Examples of the design methods can be used are:

a) Minimum variance.

b) Linear quadratic (LQ).

c) Pole placement.

d) Model following.

The design method is chosen depending on the specifications of the closed loop

systems. Different combinations of estimation methods lead to regulators with

different properties.

However, Self Tuning regulators were originally derived for sampled data

systems but continues time and hybrid algorithms have also benn developed.

Moreover, Self Tuning regulators are based on the idea of estimating some

parameters of the process. The most straight forward approach is to estimate

the parameters of the transfer function of the process and the disturbances.

This gives an indirect adaptive algorithm. The regulator parameters are not

updated directly, but rather indirectly via the estimation of the process model.

This type of adaptive control, base com least squares estimation and deadbeat

control.

It is possible to re-parameterize the model in the regulator parameters such

that the regulator parameters can be estimated directly. That is a direct

adaptive control.

Adaptive Control

20

Disadvantages:

There are three main disadvantages:

1- The stability analysis is complicated because the regulator parameters

depend on the estimated parameters in complicated way.

2- The Map from the process parameters to regulator parameters may have

singular points. This happens in design methods based on pole placement.

3- To ensure that the parameters converge to the correct values, it is

necessary that the model structure be correct and that the process input be

persistently exciting.

In order to form a Self tuning or adaptive control system, three forms of

controls can be used:

1- Pole placement (assignment) control.

2- Minimum variance control.

3- Multistage predictive control.

In this lecture, the pole placement control will be taken into consideration.

Pole assignment control:

It is a classical approach to controller specifications. The design of a

feedback controller has two main aims:

1- the first is to modify in some way the dynamic response of a system.

2- The second os to reduce the sensitivity of the system output to

disturbances.

Suppose we have a system with a continuous time transfer function:

)s(us.1

f)s(Y

(1)

where is the time constant of the system and f is the system gain.

The discrete-time domain transfer function is:

Adaptive Control

21

)z(UZ.a1

b.Z)z(Y

1

1

(2)

where

T

ea and f).a1(b (3)

and T is the sampling time.

Common control objective would be to use a feedback controller to alter the

speed of the system response. I.e. the original system might have a certain

speed of response to a sudden change in u(z) associated with time constant ,

but it is required to change this response rate to one associated with time

constant .

In terms of the system model, equation 2 changing the speed of response

corresponds to later the value of a in the denominator in some way assigning it

another value corresponding to faster speed of response.

Also, it is required to ensure that the output y(t) come into correspondence

with reference signal r(t).

To do this, propose the feedback controller structure with the following block

diagram is:

Adaptive Control

22

Where:

u(Z)=-g.y(Z)+h.r(Z) (4)

By combining Eq. (2) and Eq. (4), this gives the closed loop transfer function:

)Z(rZ).g.ba(1

Z.h.b)z(y

1

1

(5)

Notice that the closed loop speed of structure is determined by (a-b.g) instead

of a.

In other words, the open loop system pole Z=a has been assigned (placed) to a

closed loop position Z=a-b.g. Hence, we can specify the location of the poles as

a design parameter.

Suppose we required that closed loop pole at Z=t1 , then:

g.bat1

(6)

or

b

tag 1

(Design equation) (7)

The remaining controller objective is to ensure that steady state (when r(z) is

constant), the output y(z)=r(z).

This is done by selecting the controller parameter h such that the closed loop

transfer function is equal to one at zero frequency. (Zero frequency means

0s or 1Z ). Using equation 5:

Adaptive Control

23

1)g.ba(1

h.b

(8)

or b

)g.ba(1h

. (9)

Equation (9) is the design equation for selecting h so that y(z) tracks r(z) at

steady state conditions.

The key idea of pole assignment is to shift the open loop poles to some desired

set of closed loop poles.

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24

Example:

Design a digital controller for the open loop discrete transfer function of

1Z

2)z(GOL

, so that the digital closed loop transfer function has a zero steady

state error with characteristic equation of 1Z.5.01T

.

Solution:

The open loop transfer function can be re-written as: 1

1

OLZ1

Z2)z(G

The closed loop transfer function is shown below:

)z(rZ).g.21(1

Z.h.2)z(y

1

1

By equating c\cs. equation with T, this gives:

1-2.g=0.5

2g=0.5

g=0.25

For zero steady state error,

5.01

h.21

)z(r

)z(ylim

1Z

2.h=1-0.5

2.h=0.5

h=0.25

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25

The new block diagram of closed loop transfer function is:

and

H.W.:

Obtain T in s-plane if the sampling time is 1 second.

)z(rZ.5.01

Z.5.0)z(y

1

1

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26

General Algorithm of pole assignment:

Consider a system defined by the equation:

A.y(z)=B.u(z)+C.e(z) (10)

Where y(z) is the output, u(z) is the input and e(z) is the disturbance.

Consider a noise-free system model:

A.y(z)=B.u(z) (11)

With a controller of the form:

F.u(z)=-G.y(z)+H.r(z)

Where:

nf

nf

1

.1 Z.f..............Zf1F

ng

ng

1

.1O Z.g..............ZggG

nh

nh

1

.1O Z.h..............ZhhH

Then, the closed loop system equation is, (combining equation 10 and 11):

(F.A+B.G)y(z)=B.H.r(z)

(12)

Fig. Closed loop system with controller.

Now, if the desired closed loop pole set is defined by the roots of:

nt

nt

2

2

1

1 Z.t.......Z.tZ.t1T

)z(rG.BA.F

H.B)z(y

Adaptive Control

27

then the controller coefficients which assign the actual pole set to the desired

set are given by the solution of:

(13)

Equation (13) can be written as a set of simultaneous equations by equating

coefficients of like power of iZ

.If the order nf and ng are chosen correctly,

then the set of simultaneous equations will have a unique solution.

Re-writing equation (13) in terms of the polynomial gives:

)Z.f..............Zf1(nf

nf

1

.1

. )Z.a..............Za1(na

na

1

.1

.+

)Z.b..............Zb(nb

nb

1

.1

. )Z.g..............Zgg(ng

ng

1

.1O

=

nt

nt

2

2

1

1 Z.t.......Z.tZ.t1 (14)

The above equation shows that:

a) Because the largest power of iZ

to occur on the L.H.S are (nf + na) and

(ng + nb), then it is possible to generate, (by comparing coefficients of iZ

), nf + na= ng + nb equations.

b) There are 1+ nf + ng unknown coefficients of the polynomials F and G.

Hence, for a unique solution to equation (13), we require:

nf + na= ng + nb= nf + ng +1

or

Conditions of the design

F.A+B.G=T

nf = nb-1

ng = na -1

nt na+ nb-1

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28

Solution of equation (13)

Equation (13) is called the pole placement identity. It can be solved in a

number of ways. A direct solution has obtained by comparing coefficients of

iZ

leads to a matrix \vector equation with a special banded structure.

Equation (14) becomes:

)1na(

.1.na

3

.1.2

2

.1.1

1

.1

na

na

2

.2

1

.1 Z.fa..............ZfaZfaZfZ.a..............ZaZa1

)nfna(

nfna

)1nf(

nf1

nf

.nf

)2na(

.2.na

3

.2.1

2

.2 Z.f.a......Z.f.aZf...........Zfa....................ZfaZf

.......Z.gb.............ZgbZgbZ.gb.............ZgbZgb)2ng(

.ng.2

3

.1.2

2

o.2

)1ng(

.ng.1

2

.1.1

1

o.1

.

.

.

nt

nt

2

2

1

1

)nbng(

.ngnb

)1nb(

.1nb

nb

o.nb Z.t........Z.tZ.t1Z.gb.............ZgbZgb

By equating coefficients of )nfna(21Z..............Z,Z

or )nbng(Z

, we get:

1o111 tg.bfa power of 1Z

211o22112 tg.bg.bff.aa power of 2Z

.

.

.

and so on.

In matrix form, we get:

ng

1

o

nf

2

1

nbna

2nb

1nb1na

1

121

1

g

.

g

g

f

.

.

.

f

f

.

b..000a...0

...........

...........

b...b0....0

b....ba...a

......1....

...........

.........a.

0...bb0.01a

0...0b0..01

=

0

.

0

0

.

.

.

.

at

at

22

11

(15)

A c b

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29

A.c=b

Where:

c is a vector with controller coefficients.

A is known in linear algebra as Sylvester type matrix.

Example:

Write the identity equation for a system of 3nn ba and 1n t .

Solution:

We have F.A+B.G=T

21nn bf

21nn ag

)ZaZ.aZa1).(Z.fZf1(

3

.3

2

2

1

.1

2

2

1

.1

1

1

2

2.

1

1.o

3

.3

2

2

1

.1 Z.t1)Z.gZ.gg).(ZbZ.bZb(

Sub. in equation (15), to get:

33

2323

12312

121

1

b00a0

bb0aa

bbbaa

0bb1a

00b01

.

2

1

o

2

1

g

g

g

f

f

=

0

0

a

a

at

3

2

11

A c b

The above equation can be solved in the normal way by inverting A to obtain

the vector of controller parameters:

c=A-1

.b (16)

Equation (13) substituted in equation (12) gives the closed loop equation with

desired poles given by T, thus:

)z(rT

H.B)z(y (17)

and this gives the desired stability characteristic.

Adaptive Control

30

However, we still have the problem of ensuring that the output y(z) is equal to

the reference r(z) for constant (or slowly changing) reference signal. The

simplest way is to select:

|B

TH

1Z

(18)

let hH (h is a constant value)

This choice is made so that the closed loop transfer function )T

h.B( will be (1) at

zero frequency and y(z)=r(z) for constant reference signal as required.

An Alternative procedure:

An Alternative procedure is to choose H to give the desired system zeros.

The point of this is that the zeros of a system, which in this case are the roots of

(B.H=0), in fact also influence the transient response shape. The immediate

answer is to cancel the B term by making:

B

1H (19)

But, the B part of a system can be inverse unstable, (the roots of B lie outside

the unit circle), so cancellation must be exact, since any slight error in

cancellation will leave an unstable component in the relationship between r(z)

and y(z).

Time delays:

Any time delay leads to some leading zeros coefficients in B plus possible

non-minimum phase zeros associated with the partial time delay, m. If there

are know to be (k) zero coefficients then it is useful to use the information and

substitute B.ZK for B in the relevant relations. The pole-assignment identity

will now take the form:

TG.B.ZF.AK (20)

Adaptive Control

31

This equation is solved as before except that the K top rows of the B block will

be zero because of the KZ

terms.

Three-term Controller design by pole-assignment:

Three-term or PID controller combines proportional action, integral

action and derivative action and can put into the form:

)z(eZ1

Z.gZ.gg)z(u

1

2

2

1

1o

(21)

The coefficients go, g1 and g2 are related to the proportional derivative and

integral gain setting, (assuming backward shift approximation) with Ts=1

second):

Kp=- g1 – 2. g2

Kd=g2

Ki=go + g1 + g2 (H.W. Check ?)

In order to synthesize exactly the PID controller coefficients, we can assume

the system to be controlled has the following structure:

)z(u.Z.aZ.a1

Z.b)z(y

2

2

1

1

1

1

(22)

and we can assume that a1, a2 and b1 can be estimated by RLS method.

The restriction on the system model form is to ensure that only one set of PID

controller coefficients arises from the design. Combine equations (21) and (22),

we get:

1

1

2

2

1

1o

2

2

1

1

1

2

2

1

1o

1

.1

Z.b).Z.gZ.gg()Z.aZ.a1).(Z1(

)z(r).Z.gZ.gg.(Zb)z(y

(23)

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We can now select the coefficients go, g1 and g2 to give a desired closed loop

performance. Suppose we have selected a desired closed loop the rise time and

damped natural frequency and hence arrived at the corresponding desired

closed loop T polynomial:

2

2

1

1 Z.tZ.t1T (24)

Thus:

2

2

1

1

1

1

2

2

1

1o

2

2

1

1

1Z.tZ.t1Z.b).Z.gZ.gg()Z.aZ.a1).(Z1(

(25)

By equating coefficients of like power of Z, we get:

1

11

ob

)a1(tg

,

1

212

1b

)aa(tg

and

1

2

2b

ag (26)

The steady state matching of y(z) to r(z) for a constant reference signal is

ensured by the factor )Z1(1 in the denominator of equation (23).

Specifically, for zero frequency (Z=1), so that independent of the system or

controller parameters, we have y(z)=r(z) (for constant r(z))

Self-Tuning pole assignment algorithm:

The block diagram below shows the layout of a pole assignment self

tuner in which a recursive estimator is combined with a pole assignment

synthesis rule in order to continuously update the controller coefficients.

Certain performance requirements have fed into the synthesis block. These

consist of the following controller design information:

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Fig. Self-Tuning Pole Assignment System.

a) The desired closed loop pole set, specified by T.

b) The form of the controller e.g. whether it is a servo system, a regulator

or a combination of both.

In addition, further information must be supplied concerning the

configuration requirements of the self-tuner, including:

c) The sampler rate to be used.

d) The degree of the system model polynomials na, nb ….etc.

e) The delay K (for delayed systems) in the system, if known.

Provided with these information, the self –tuning system can be setup to go

through the following cycle of adaptation:

Self-Tuning cycle:

At each sample interval Ts, the following sequence of action is taken:

Step (i) Data capture:

The system output y(KT), reference input r(KT) and any other variables

of importance are measured.

Step (ii) Estimator update:

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The data acquired in (i) is used together with past data and the previous

control signal to update the parameter estimates in a model of the system

using an appropriate recursive estimator.

Step (iii) Controller synthesis:

The updated parameters from (ii) are used in a pole assignment identity

to synthesize the parameters of the desired controller.

Step (iv) Control algorithm:

The controller parameters synthesized in (iii) are used in a controller to

calculate and input the next control signal u(KT).

At the end of the cycle the control computer waits until the end of sample

interval T and then repeats the cycle for interval T+1, and so on.

Steps in the self-tuning cycle have computed sequentially. However,

figure below illustrates this in terms of a timing and sequence diagram. The

total computation time must be less than the sample interval and is

generally assumed to be much less.

Fig. Timing and sequence diagram for self-tuning controller.

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Hint: The conversion from s-plane into z-plane can be done using digital

approximations. Use the following conversions:

Continuous function sdt

dy

Forward difference: ST

1zs

Backward difference: z.T

1zs

S

Central difference: z.T.2

1zs

S

2

Where: sT is the sampling time in seconds.

Tutorials:

Question (1):

Derive the PID controller equation in z-domain using backward

approximation.

Question (2):

Calculate the desired closed loop equation in discrete form for a second order

system with 10n , 7.0 and the sampling time is 0.1 second.

Question (3):

For the following model:

)z(uZ72.0Z7.11

Z2Z)z(y

21

21

Design a digital controller using pole placement that satisfies the following

C\Cs equation: 1Z904.01T

.

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c- Model-Reference Adaptive Controller (MRAC):

The mode-Reference Adaptive Controller is one of the main approaches of

adaptive control. The basic principle is illustrated in figure below:

Fig. Block diagram of RMAC.

The desired performance is expressed in terms of a reference model, which

gives the desired response to a command signal. The system also has an

ordinary feedback loop composed of the process and the regulator. The error

(e) is the difference between the output(s) of the system and the reference

model. The regulator has parameters that are changed, based on the error.

There are thus two loops: an inner loop, which provides the ordinary control

feedback and outer loop, which adjusts the parameters in the inner loop. The

inner loop is assumed faster than the outer loop.

In this adaptive system, two ideas were introduced. First, the performance of

the system is specified by a model. Second, the parameters of the regulator are

adjusted based on the error between the reference model and the system.

There are essentially three basic approaches to analysis and design of MRAC:

1- The gradient approach.

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2- Lyapunov functions.

3- Passivity theory.

For a system with adjustable parameters, the model reference adaptive

methods gives a general approach for adjusting the parameters so that the

closed loop transfer function will be close to a prescribed model. This is called

the Model-following problem.

One important question is how small we can make the error (e)?. This depends

both on the model, the system and the command signal. If it is possible to make

the error equal to zero for all command signals, the perfect model following is

achieved.

Model Following:

The model-Following problem can be solved using pole placement design.

Model –Following is a simple and useful way to formulate and solve a servo

control problem. The basic idea is very simple. Servo performance is specified

indirectly by giving a mathematical model for the desired response. The

specified model can be linear as well as nonlinear. The parameters in the

system are adjusted in order to get the output of the process model (y) as close

as possible to the reference model output (ym) for a given class of input signals.

Optimization methods are thus natural tool in MRAC design.

The Gradient approach:

This is a fundamental idea in the MRAC approach. The parameters

adjustment scheme is usually called MIT rule.

The MIT rule:

Assume that we attempt to change the parameters of the regulator so that the

error between the output(s) of the process and the reference model is driven to

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zero. Let (e) denote the error and () represents the parameters. Introduce the

criterion:

2e

2

1)(J (27)

where J is called the loss function. To make J small, it is possible to change the

parameters in the direction of the negative gradient of (J). i. e.:

(28)

It is assumed that the parameters change much more slowly than the other

variables in the system, then the derivative

ecan be evaluated under the

assumption that is constant. The derivative

eis the sensitivity derivative of

the system. The adjustment rule of equation (28), where

eis the sensitivity

derivative, is commonly referred to as the MIT rule. The choice of loss function

is, of course, arbitrary. If the loss function is chosen as:

e)(J

Thus, the adjustment rule becomes:

)e(signe

dt

d

Equation (28) also applies to the case of many adjustable parameters. The

variable should be interpreted as a vector and

eis as the gradient of the

error with respect to the parameters.

n321i

......

n321i

e......

eeee

Je

.e.J

dt

d

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for 2

e2

1)(J :

n321

e......

eee.e.

dt

d

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Note:

In the MIT rule, it is not necessary to require perfect model-following. The

procedure can be applied to nonlinear systems. The method can also be used to

handle partially known systems.

The MIT rule will perform well if the adaptation gain () is small.

Consequently, it is not possible to give fixed limits that guarantee stability. The

MIT rule can thus give an unstable closed loop system.

Lyapunov’s Second method (1892):

Lyapunov introduced an interesting direct method to investigate the stability

of a solution to a nonlinear differential equation. The key idea is illustrated in

figure below:

Fig. Illustration of Lyaunov’s method for investigating stability.

The equilibrium will be stable if we can find a real function on the state space

whose level curves enclose the equilibrium such that the derivative of the state

variable always towards the interior of the level curves.

To state the results formally, let the differential equation be:

)t,x(fx.

and

f(0,t)=0

where x is a state vector of dimension n. It is assumed that f is such that

solutions exist for all o

tt . The equilibrium point o

t at the origin. This

involves no loss of generality since this can be achieved through a simple

coordinate transformation.

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Lyapunov Stability Theorem:

Let the function RR:V1n

satisfy the conditions:

1. 0)t,0(V for all t R.

2. V is differentiable in X and t.

3. V is positive definite.

4. V.

is a negative definite (or negative Semi-definite).

Then the equilibrium state at the origin is uniformly asymptotically stable.

If )t,x(V as ,x then equilibrium state at the origin is uniformly

asymptotically stable in the large.

Then V is called Lyapunov function.

Note:

a. Positive definiteness of a scalar function: A scalar function V(x) is said to be

positive definite in a region includes the origin if v(x) > 0 for all nonzero

states x and v(0)=0.

b. Negative definiteness scalar function: A scalar function V(x) is said to be

negative definite if –V(x) is positive definite.

c. Positive semi-definiteness of a scalar function: A scalar function V(x) is said

to be positive semi-definite if it is positive at all states except at the origin and

at certain other states where it is zero.

d. Negative Semi-definiteness of a scalar function: A scalar function V(x) is

said to be negative semi-definite if –V(X) is positive semi-definite.

Example:

For the following matrix (P):

Pnn....1Pn

......

......

n2P...22P21P

n1P...12P11P

P

if P11 > 0, 22P21P

12P11P>0, . . . . . . . . . . . . . . . . . . . . .

Pnn.....

......

......

.....21P

n1P...12P11P

> 0

Then P is positive definite.

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Expert Control:

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Neural Networks:

In an expert system, the control law is a logic function that gives the

control action as a function of senor patterns. The function is adaptive in the

sense that it will adjust itself automatically.

The Perceptorn proposed by Rosenbatt (1959) is one way to obtain a learning

function. To describe the perceptorn , let (ui with i=1,2,3,…., n) be inputs and

(yi with i=1,2,3,…, n) be outputs. In the perceptron, the output is formed as:

n

1jjiji

)bu.w(fy , i=1,2,3,…,n

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where wij is the weight, b is the bias and f is the threshold function.

As an example:

0ifx0

0ifx1)x(f

To update the weight, the perceptron uses a very simple idea, which is called

Hebb’s principle. Apply a given pattern to the inputs and clamp the outputs to

the desired response, then increase the weights between nodes that are

simultaneously excited. This principle was formulated in Hebb (1949), in an

attempt to model neuron networks:

)yy).(t(u.)t(wwj

0

jiijij

where 0

jy is the desired response and

jy is the response predicated by the

model. By regarding the weight as a parameter, it becomes clear that updating

formula is identical to gradient method for parameter estimation.

There are several ways to use Neural Network in control systems. In the following,

some of these methods:

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