A Comparation Betweeen Diferente Control Laws

8/10/2019 A Comparation Betweeen Diferente Control Laws

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Appendix C

A Com par ison of th e

Self-Tun ing Regulator

o

blstrom and Wit tenm ark w ith

th e Techniques

o

Adapt ive

Inverse Con tro l

The best-known adaptive control methods are based on the self-tuning regulator of Astrom

and Wittenmark. Their

1973

paper

[2]

has had great influence worldwide

in

the field of

adaptive control. Chapter

3

of their book entitled Adaptive Control [13 summ arizes their

work on the self-tuning regulator. Figure

C.

1 is a generic diagram of the self-tuning regu-

lator, based on F ig.

3.1

of Adaptive Con trol.

Process parameters

Design Estimator

Regulator Process

Figure C.l

Conrrol

(Reading, MA: Addison-Wesley, 1989 .

The self-tuning regulator based on Fig.

3.1

of

K.J.

STROM, and B. W I T T E N M A R K , A ~ ~ ~ ~ ~

The system of Fig. C.l is linear and SISO, and

it

works

in

the following way. The

process

or

plant is excited by its input

u

ts output is

y .

This output contains a response

363

Adaptive Inverse Control: Signal Processing App roach Reissue Edition

by Bernard Widrow and Eugene Walach

Copyright 8 the Institute of Electrical and Electronics Engineers


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364

Comparing t he Self-Tuning Regulatorwith Adaptive

Inverse

Control

App. C

u c

to

u

plus plant disturbance. An estimator, receiving both the signal input and signal output

of the plant, estimates the plant parameters. These estimates are fed to an automatic design

algorithm that sets the parameters of the regulator. This regulator could be an input con-

troller, or a controller within the feedback loop, or both. For convenience, we have redrawn

the diagram of Fig. C.

1

in Fig. C.2.

m y

nput

controller

Plant

disturbance

Figure

C.2

An alternative representation of the self-tuning regulator.

In Chapter

8,

we demonstrated that the adaptive disturbance canceler of Fig. 8.1 min-

imizes the plant output disturbance power. In fact, we have shown that no other linear sys-

tem, regardless of its configuration, can reduce the variance of the plant output disturbance

to a level lower than that of Fig.

8.I

Comparing the self-tuning regulator of Fig.

C.2

with

the adaptive disturbance canceler of Fig. 8.1, the question is: Can the feedback controller of

the self-tuning regulator be designed to cancel the plant disturbance as well as the adaptive

disturbance canceler? Another question that arises is: Can an input controller

be

designed

for the self-tuning regulator so that when

it

is cascaded with the plant and its feedback con-

troller, the entire control system will have a transfer function equal to the transfer function of

a selected reference model? It is not obvious that the self-tuning regulator and the adaptive

inverse control system will deliver performances that are equivalent to each other.

C . l

DESIGNING A SELF-TUNING REGULATOR T O BEHAVE L IKE

A N

A DA PTIVE INVERSE CONTROL SYSTEM

To address these issues, we redraw Figs. C.2 and

8.1

as Figs. C.3 and C.4, respectively, in

order to simplify and bring2ut essential features of these block diagrams. For simplicity, we

have drawn Fig. 8.1 with P z ) approximated by P z ) . Also, we included a necessary unit

delay

z-

within

the

feedback loop of the self-tuning regulator that will be necessary only if

there is no delay either in the plant or in the feedback controller. We will assume that

P z )

is stable. If the plant is not really stable, let it be stabilized by a separate feedback stabilizer

and let P z ) represent the stabilized plant. This creates no theoretical problems for the self-

tuning regulator, and

as

demonstrated in Appendix D, creates no theoretical problems for

adaptive inverse control. To compare the two approaches, we need to first show, if possible,

that the transfer function from the plant disturbance injection point to the plant output is


3/6

Sec.

C.l

Designing

a

Self-Tuning Regulator 365

the same for the self-tuning regulator as for the adaptive disturbance canceler. For the self-

tuning regulator of Fig. C.3 his transfer function is

Plant disturbance

W z

Unit delay

C z )

Input

controller

Feedback

controller

Figure C.3

Another representation of the self-tuning regulator.

Plant disturbance

N z )

Input

Plant

output

l- ~--j7+-1

Figure C.4

Another representation of the adaptive plant disturbance canceler

For the adaptive disturbance canceler of Fig.

C.4,

this transfer function is

In order for these transfer functions to be equal, it is necessary that

To

obtain

F C z ) ,

we need

Q z )

and

P z ) .

In practice, both wouldbe readily available

to

a close approximation from adaptive processes already described. So there would be no

problem in getting a good expression for F C z ) .


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66

Comoarina

the

Self Tunina Regulator with AdaDtive Inverse Control

ADD.C

If the feedback controller given by

Eq. C.3)

were used in the self-tuning regulator,

would the resulting system

be

stable? The answ er is yes. Referring to Fig.

C.3,

we may

note that the transfer func tion through the feedb ack loop to the plant output is

This transfer func tion is stable since both

P z )

and

Q z )

are stable.

SO

ar SO good.

To

control the dynam ic response of the sy stem, to make it behave like the dynam ic

response of a selected reference model, we multiply the transfer function 2.4) by ~ C Z )

and set the product equal to

M z ) :

C.5)z ) = ZC Z)

.

I P z) 1 z- P z )

.

Q z ) ) .

Accordingly,

-

M z ) .F C z )

-

Z I P z) Q z )

For the entire self-tuning regulator to be stable, it is necessary that Z C z )be stable. Stability

has already been established for the rest of the system .

M z )

s stable. Since

P z )

and

Q z )

are stable, they would not cancel any unstable poles of F C z ) hat may occur. It is necessary ,

therefore, for

F C z )

o

be

stable in order for

Z C z )

o be stable, although this

is

not sufficient

for stability. I C z ) will

be

unstable if either P z ) , Q z ) , r both are nonminimum-phase.

How would one build a self-tuning regulator if its feedback controller

F C z )

were

unstable? There are two possibilities. One possibility would be to choose an

F C z )

hat is

stable but not optima l for plant disturbance canceling. The other possibility would be to use

the optimal, unstable F C z ) inside the feedback loop and build the input controller ZC(z)

having the ind icated poles and zeros but allowing components of

Z C z )

to be noncausal

as required for stability, The noncausal filter could be realized app roximately with an ap-

propriate delay. The entire system response would be a delayed version of the response of

M z ) .

These difficulties are not encoun tered when the optimal

F C z )

s used and the input

controller

ZC(z)

is stable.

C.2

SOME EXAMPLES

Specific exam ples will help to clarify some of the issues. Suppose that the plant disturbance

is constant, that it is a

Dc

offset or bias. The question is: How w ell do the adaptive dis-

turbance canceler and the se lf-tuning regulator handle this disturbance? For the adaptive

disturbance canceler, the transfer function from the plant disturbance injection point to the

plant outpu t is given by

C.2).

We would like this transfer function to have a va lue of zero

at zero frequency, that is, at z = 1. This is easily accom plished as follows:

1

1

- P 1 ) .

Q 1 )

= 0, or Q 1) =

)

(C.7)


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Sec. C.3

Summary

367

Any form of Q z ) would allow perfect canceling of the constan t disturbance as long as the

value of

Q

at

z

= 1 is the reciprocal of the value of P at z = 1.

So

the

adaptive plant disturbance canceler will perfectly eliminate constant plant dis-

turbance. What will the self-tuning regulator do with this disturbance? Its transfer function

from plant disturbance injection point to plant output point is g iven by

C.1).

This transfer

function shou ld equal zero at zero frequency to eliminate the DC plant disturbance. Accord-

ingly,

Assuming that the plant transfer function is well behaved at

z = 1, i t

is clear that F C z )

must be infinite at z = 1. One way to accomplish this would be to let F C z ) be a digital

integrator,

F C z )

=

(C.9)

1 - - I

giving it a simple pole at

z =

1. T he input controller has the transfer function given by

C.6):

1

C .10)

P z ) and Q z ) are both stable and finite at z = 1. If in addition M z ) has a finite value

at z

= 1, I C z )

will have a pole at

z = 1

and will thereby be unstable making the entire

system unstable. A noncausal realization of I C z ) would not be useful in this case. The

only possibility would be to choose an

F C z )

having its pole slightly inside the unit circle,

sacrificing some d isturbance canceling capability for a stable

I C z ) .

The sam e kind of result would be obtained

if

the plant disturbance were a constant-

amplitude sine wave. The adap tive disturbance canceler would adap t and learn to elimi-

nate it perfectly. The self-tuning regulator would either be unstable or, if stable, would give

somew hat less than optima l disturbance reducing performanc e.

C.3 SUMMARY

The self-tuning regulator has an easier job of disturbance reduction and dynamic control

than adaptive inverse control when the plant is unstable with a pole or poles on the unit circle

or

outside the unit circle. Feedback used by the self-tuning regulator has the capability of

moving the plant poles inside the

unit

circle to stabilize the plant, reduce disturbance, and

control its dynamics. For adaptive inverse control, the first step would be to stabilize the

plant with feedback. The choice of feedback transfer function for stabilization would not be

critical and would not need to be optimized. The only requirement would be to somehow

stabilize the plant. Then adaptive inverse control could be applied

in

the usual manner. A

discussion of initial stabilization for adaptive inverse control systems is given i n Appendix

D.

A difficult case for both approaches occurs when the plant has one

or

more zeros on

the

unit

circle. T he inv erse controller tries to put the

FIR

equivalent of a pole

or

poles on top


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368

Comparing the Self-Tuning Regulator with Adaptive Inverse Control

App. C

of the zeros. The inverse impulse response does not die exponentially but instead persists

forever. An optimal

FIR

inverse filter cannot be constructed. The self-tuning regulator of

Fig. C.3 also canno t cope w ith such a p roblem . Its feedback canno t move around the zeros.

It too would try to choose ZC(z) to put poles on top

of

these zeros, but this would make

ZC(z)

unstable and thereby make the entire system unstable. A remedy that w ould work

for both approaches would be to cascade with the plant a digital filter having poles m atching

the unit circle zeros of the plant and to envelop the cascade within a feedback loop. If, for

exam ple, the plant had a single zero on the

unit

circle at z = 1, the cascade filter would be

a digital integrator. The loop around the cascade would need to be designed to be stable for

adaptive inverse control.

When the plant is nonminimum-phasewith zeros outside the unit circle, we have seen

in Chapters

5,6

and 7 how adaptive inverse control can readily cope w ith such a plant. Deal-

ing with this kind of plant w ith a self-tuning regulator is difficult, much more difficult than

dealing with a m inimum -phase plant. The literature is not clear on how this can be done.

Com paring adaptive inverse control with the self-tuningregulator reveals cases where

one approach is advan tageous, and cases where the other approach is advantageous. Also,

there are m any cases where both approaches give equivalen t performance although the sys-

tem configurations and m ethods of adaptation are totally different.

Bibliography for Appendix C

[ l ] K.J.ASTROM, nd B .

W I T T E N MA R K,daptive

control, 2nd ed. Menlo Park, CA:

[2]

K.J.

ASTROM, nd

B.

W I T T E N M A R K ,

On self-tuningregulators,

Aufomafica ,

Vol. 9,

Addison Wesley, 1995).

No. 2 1973).

Th is situation is not so clear. Refer to simulation examples

in

Chapter 12

A Comparation Betweeen Diferente Control Laws

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