Digital computer applications to process control : proceedings of the 7th IFAC/IFIP/IMACS Conference, Vienna, Austria, 17-20 September 1985

577 Pages·1986·22.51 MB·English

by  Manfred Paul

IFAC PROCEEDINGS SERIES

Editor-in-Chief

JANOS GERTLER, Department of Computer and Electrical Engineering,
George Mason University, Fairfax, Virginia, USA

GERTLER & KEVICZKY (General Editors): A Bridge Between Control Science & Technology
(Ninth Triennial World Congress, in 6 volumes)

Analysis and Synthesis of Control Systems

(1985,

Identification, Adaptive and Stochastic Control

No.

(1985,

/)

No. 2)

Large-scale Systems, Decision-making, Mathematics of Control

Process Industries, Power Systems

(1985, No.

(1985, No. 3)

4)

Manufacturing, Man-Machine Systems, Computers, Components, Traffic Control,

Space Applications (1985, No. 5)

Biomedical Applications, Water Resources, Environment, Energy Systems, Development, Social

Effects, SWIIS, Education (1985, No. 6)

BARKER & YOUNG: Identification and System Parameter Estimation (1985)

NORRIE & TURNER: Automation for Mineral Resource Development

CHRETIEN: Automatic Control in Space (1986, No.

(1985,

(1986, No.

No. 7)

1)

2)

DA CUNHA: Planning and Operation of Electric Energy Systems (1986, No. 3)

VALADARES TAVARES & EVARISTO DA SILVA: Systems Analysis Applied to Water and Related

Land Resources

(1986, No. 4)

LARSEN & HANSEN: Computer Aided Design in Control and Engineering Systems

PAUL: Digital Computer Applications to Process Control (1986,

YANG JIACHI: Control Science & Technology for Development

No.

(1986, No. 5)

6)

(1986, No. 7)

MANCINI, JOHANNSEN & MARTENSSON: Analysis, Design and Evaluation of Man-Machine

Systems (1986, No. 8)

GELLIE, FERRATE & BASANEZ: Robot Control "Syroco '85"

JOHNSON: Modelling and Control of Biotechnological Processes

(1986, No.

(1986,

9)

No. 10)

N OT I CE TO READERS

If your library is not already a standing/continuation order customer or subscriber to this series, may we recommend that you place a
standing/continuation or subscription order to receive irnn1ediately upon publication all new volumes. Should you find that these volumes no longer

serve your needs your order can be cancelled at any time \Vithout notice.

Copies of all previously published volumes are available. A fully descriptive catalogue will be gladly sent on request.

ROBERT MAXWELL

Publisher

IFAC Related T1t/es

BROADBENT & MASUBUCHI: Multilingual Glossary of Automatic Control Technology

EYKHOFF: Trends and Progress in System Identifica1ion

ISERMAN/\:: System Identification Tutorials (Autonwtira Speria/ Issue)
DIGIT AL COMPUTER APPLICATIONS
TO PROCESS CONTROL

Proceedings of the 7th IFACIIFIPIIMACS
Conference, Vienna, Austria, 17-20 September 1985

Edited by

M. PAUL

Technische Universitat Wien, Austria

Published for the

INTERNATIONAL FEDERATION OF AUTOMATIC CONTROL

by

PERGAMON PRESS

OXFORD · NEW YORK · BEIJING · FRANKFURT
S AO PAULO · SYDNEY · TOKYO · TORONTO
U.K. Pergamon Press, Headington Hill Hall, Oxford OX3 OBW, England
U.S.A. Pergamon Press, Maxwell House, Fairview Park, Elmsford, New York 10523, U.S.A.
PEOPLE'S REPUBLIC Pergamon Press, Qianmen Hotel, Beijing, People's Republic of China

OF CHINA

FEDERAL REPUBLIC

Pergamon Press, Hammerweg 6, D-G242 Kronberg, Federal Republic of Germany

OF GERMANY

BRAZIL Pergamon Editora, Rua E�a de Queiros, 346, CEP 04011, Sao Paulo, Brazil
AUSTRALIA Pergamon Press Australia, P.O. Box 544, Potts Point, N.S.W. 2011, Australia
JAPAN Pergamon Press, 8th Floor, Matsuoka Central Building, 1-7-1 Nishishinjuku, Shinjuku-ku,
Tokyo 160, Japan

CANADA

Pergamon Press Canada, Suite 1Cl4, 150 Consumers Road, Willowdale, Ontario M2.J I P9, Canada

Copyright© 1986 IFAC

All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval .1y.1tem or transmitted in

any form or fry any means: electronic, dectrostatir, magnetir tape, rnerhanical, jJlwtoropying, recording or other­

wise, without permi1sion in writing from the cojiyrighl holders.

First edition 1986

British Library Cataloguing in Publication Data

Digital computer applications to prorcss control

Proceedings of the 7th IFAC/IFIP/IMACS

conference, Vienna, Austria, 17-20 September

1985.-(IFAC proceedings series; 1986, no. 6)

l. Pro<.:css control-Data processing

I. Paul, M. II. International Federation of

Automatic Control

670.42'7

Ill. Series

TS 156.8

ISBN 0-08-032554-8

These firoceedings were reproduced by means of the plwto-ojjset process u.1ing the morwsrripts supplied lry the
authors of the difTeren/ papers. The manuscrifils have been ty/Jed wing dif(erent tyjJewriters and ty/Jejaces. The

lay-out.figures and tabfrs of some /Japers did no/ agra cmnpletely ll•ith the standard requirements: romequently

the reproduction does not displa)' complete uniformity. To emure rapid publication thi.1 discrejHlrl<)' rnuld no/ be

changed: nor could the English be rherked completely. Therejine, the readen are asked to excuse any deficiencies

of this JnLhliration whirh may bf due lo !hf abmie mentionnl reasom.

The Editors

Printed in Great Britain by A. Wheaton & Co. Ltd., Exeter
7th IFAC/IFIP/IMACS CONFERENCE ON DIGITAL
APPLICATIONS TO PROCESS CONTROL COMPUTER

Organized by

Austrian Centre for Productivity and Efficiency (OPWZ)

Sponsored by

IFAC Committee on Applications

IFAC Committee on Computers

Co-sponsored by

IFAC Committee on Education

IFAC Economic and Management Systems Committee

IFIP International Federation for Information Processing

IMACS International Association for Mathematics and Computers in Simulation

International Programme Committee

A. Weinmann, Austria (Chairman)

P. Kopacek, Austria

K. J. Astrom, Sweden M. Mansour, Switzerland
D. R. Bristol, USA M. Paul, Austria
A. van Cauwenberghe, Belgium I. Plander, Czechoslovakia
A. A. Concheiro, Mexico I. V. Prangishvili, USSR
G. Davoust, France K. Reinisch, GDR
G. Doolittle, USA G. Schmidt, FRG
K. H. Fasol, FRG V. Strejc, Czechoslovakia
D. Fischer, Austria T. Takamatsu, Japan
C. Foulard, France ]. D. Van Wyk, South Africa
]. Gertler, Hungary E. Welfonder, FRG
R. Isermann, FRG J. H. Westcott, UK

National Organizing Committee

P. Kopacek (Chairman)

]. Hahne! (Secretary)

W. Karner

M. Paul

A. Weinmann
PREFACE

The IFAC/ IFIP / IMACS Conference on "Digital Computer Applications to Process Control" in Vienna resumes a
series which began in Stockholm in 1964 and was continued in Menton (1967 ), Helsinki (1971 ) , Zurich ( 1974 ),
the Hague (1977 ) and Diisseldorf (1980 ) . The aim of this conference is, as of the previous ones, to present,
discuss and summarize recent advances in the application of digital computers to operation and supervision
of industrial processes. Emphasis is based on the realization of modern control principles, including
advanced monitoring and optimization.

Looking at previous Conferences one can observe that there are greater efforts in the area of process
identification and modelling. Adaptive and distributed control have become an essential part of modern
control principles. Software, robotics and data networks become more and more important. Reduction of air
and water pollution caused by industrial processes and at the same time improvement of production quality
require more and more attention.

The papers of the Conference are divided into four groups:

T
A
G

code type no. of contributions
s survey papers
tutorial papers
application oriented papers
papers of general aspects 6
3
3S
40

The six survey papers summarize the trends, developments and state-of-the art of adaptive control, distributed control systems, internal model control and process fault diagnosis. Three tutorial papers
give an introduction into state space control as well as digital control and into digital simulation
methods.

The 7S technical papers are assigned to 17 technical sessions:

code

session

no. of contributions

APPLICATION

Al
A2

Al>
AS
A6
A7

A8

Chemical and Oil Industries I
Water Turbines
Chemical and Oil Industries I I
Energy and P ower Systems I
Energy and Power Systems I I
Robotics and Manufacturing
Cement, Metallurgical Processes
and Traffic
Heating and Climate Systems

6
4
s
6
3
3

s
3

3S

GENERAL ASPECTS

Gl
G2
G3
G4
GS
G6
G7
G8
G9

Adaptive Systems I
Adaptive Systems I I
Control Aspects
Multivariable Systems
Optimization and Reliability
Modelling and Identification I
Modelling and Identification I I
Real Time Software and Languages
Distributed Systems and Data
Networks

7

L,

4
s
3
s
2
5

5

40

Vil
Vlll

Preface

The papers stem from specialists from 21 different countries. It is hoped that their papers will be a good
basis for the Conference and that their achievements may promote further research and development in the
field of digital compu.ter control.

It is a pleasure to thank the members of the International Program Committee for their contributions in
selecting the papers and for their suggestions as well as the members of the National Organizing Committee
for their efforts in organizing the Conference. Furthermore many thanks to the Osterr. Produktivitats- und
Wirtschaftlichkeits-Zentrum for their support in preparation of the Conference as well as the publisher
Pergamon Press in preparing this book.

Sept. 1985

The editor
Copyright© IFAC Digital Computer

Applications to Process Control, Vienna,

Austria, 1985

THEORY AND APPLICATION OF ADAPTIVE
CONTROL

H. U nbehauen

Ruhr University Bochum, Department o/ Electncal Engineering, PO 102118,
D-1630 Bochwn 1, FRG

Abstrac t . Systems which automatically adj ust their controller parameters t o compensate
for changes in the controlled process or its environment are referred to as adaptive
control systems. This survey of adaptive control theory and its applications reviews the
progress during the years 1980 till 1984. Different basic structures of adaptive control
systems, inc luding model reference adaptive controi self-tuning regulators and parameter scheduling control are discussed . It is shown that carefully designed adaptive control systems have been used successfully in a broad variety of application areas.

Keywords. Adaptive control systems; design structures and principles; recursive process
identification; model reference adaptive control; self-tuning controllers; applications.

INTRODUCTION

In adaptive control the controller settings are automatically adj usted in order to achieve good process operation over a wide range of conditions. The
controller adaption is necessary either for poorly
understood processes or to compensate for unanticipated parameter changes of the process due to environmental conditions or unpredictable operating
point changes. Thus adaptive control provides poss:iobilities to control processes with uncertainties,
as e . g . nonlinearities and time-varying parameters.

Although adaptive control strategies have been discussed broadly during the last 30 years it is only
in the last few years that adaptive control has
found real industrial applications. This situation
is based on the one hand on the progress in the development of powerful adaptive control a lgorithms
which have reached today a mature state . On the
other hand, modern microelectronics offers cheap
hardware which allows an easy realisation of adaptive control strategies, leading already to commercially viable solutions.

This paper is intended both to introduce the nonspecialist brief ly to the field of adaptive control
and to evaluate the actual status of this field for
the more specialised control engineer . Therefore ,
the paper i s organized as follows. First a short
c lassification and description of adaptive control
principles is given . Then it is shown that most
adaptive schemes have nearly the same structur e .
The further sections a r e devoted t o a review of recent developments of adaptive control schemes and
of practical applications. This review does not in
any way c laim to be complete , but tries only to discuss the most interesting developments published
during the last four years. Thus this paper is
directly connected to the previous reports of the
author (Unbehauen and Schmid , 1180; Parks et a l ,

1981).

The realization of modern adaptive control schemes
includes a lot of on-line computational operations.
Therefor e , adaptive control algorithms are usually
implemented on digital process computers or microprocessors. Thus, as will be discussed later , most
approaches are based on a discrete system representation .

BASIC STRUCTURES OF ADAPTIVE CONTROL
SYSTEMS

Three main basic control system structures are today relevant to the design of adaptive control systems ( e . g . Unbehauen, 1985)
- model reference adaptive control (MRAC) ,
- sel f-tuning regulators ( STR) ,
- parameter scheduling control (PSC) .
All three schemes have in common a basic feedback
control loop with a process and a controller with
adjustable parameters. All the three adaptive stru�
tures are characterized by automatic adjustment of
the controller parameters to accomodate changes in
the process or its environment ( see Fig . 1).

The adaptive schemes of MRAC and STR are applied to
that c lass of problems where parameter changes .f.
of the process are unknown and cannot be directly
obtained from process measurements. The MRAC-technique uses the reference model to spec ify the desired output behaviour y of the process with respect
to the reference signal w. As the reference model
is a part of the adaptive control system, two control loops have to be defined . While the inner loop
represents the basic control system consisting of
the process and control ler , the parameters of this
controller are adjusted by the outer loop so long
until the model error e *=y-y M becomes small . Thus
the basic ( inner ) closed loop system will achieve
the spec ified model performanc e .

The second structure, the STR-technique , is also
based on an inner c lassical control loop, whereby
the parameters of the controller of this loop are
adjusted by the outer loop, which is composed of an
identification block ( usually a recur sive estimator)
acting on a dec ision block and further on a modif ication block representing the actual adaptation of
the controller parameters. In the second loop the
effect of controller modification is fed back to
the dec ision process through the basic control loop
and the identification process. Thus an adaptive
error forces the adaptation process to achieve the
chosen criterion ( adaptive set point) .

In many control problems the process changes can be
anticipated or inferred from process measurements.
It is then possible to adjust the controller parameters in a predetermined manner as process condi-
2

H. Unbehauen

changes of

_w_....._{ 1--icontro Iler�----+--!
set poin
controller
� parameters
r-a-:d.,-u-j s-tm_e_n_t --,

y

'----l

process parameters are explicitely estimated and the
adaptive controller is designed indirectly on the
basis of the estimated process parameters . Usually
a discrete model is used for the recursive e stimation of the process parameters . Therefore the calculation for the design of this controller has to
be repeated at each sampling interval , whereby the
identification and control ler adaptation are two
different procedures . Without directly identifying
the process parameters it is often possible to identi fy the controller parameters directly . Such an
adaptive control is based on an implicit process
model and i s , therefore, referred to as a direct
(or implicit) adaptive controller. This usually
leads to a s ignificant s implification of the adaptation a lgorithm .

According to the above definitions the MRAC shown
in Fig . l a represents a direct (or implicit) adaptive controller since its parameters and its control
law are directly updated from the s ignals u and y .
The STC shown i n Fig . l b i s thus , however, defined
as an indirect (or explicit) adaptive controller .

controller design

(b)

controller >---·-+---t

£

action

�I

fixed
mapping

_______

I

I
I
I
I
I
I

I

J

©

Fig . 1 . Bas ic schemes of adaptive control (a) MRAC ;
(b) STR ; (c) PSC

tions vary . The decision process thus is reduced to
a f ixed mapping of the process parameters to the
controller parameters, whereby the original decision proces s is a lready realized in the d esign phase
of the adaptive control system , e . g . by a "table
look - up" approach different sets of controller parameters are stored for different operating points of
the plant. This strategy has been originally applied
to the adaptation of controller gain factors and
thus has been referred to as "gain scheduling" . However, in order to be more genera l , this approach
should be defined to as "parameter scheduling contro l" ( PSC). This type of adaptive control structure is wide spread and in vogue today, s ince it
allows one to tune a wide range of controllers using
a manifold of popular on-line process identification
methods . To guarantee a faultnes s operation of systems with the PSC-structure , a good knowledge of
the actual process dynamics i s required . The PSCstrategy represents an open-loop adaptation of the
controller paramete>S of the basic inner loop control
system, because the results of the adaptation of the
controller parameters are not fed back to the adaptation unit and thus cannot be corrected .

Principles of design

MRAC- and STC-schemes are both based on s imultaneous
process identification and control . The operation of
both these adaptive techniques can be clas s if ied into two general groups :

- direct (or implicit) adaptive schemes and
- indirect ( or explicit) adaptive scheme s .

In a n indirect adaptive control system the unknown

Although the STC was originally developed for the
stochastic minimum variance control problem (Astr6m
and Wittenmark , 1 9 7 3 ) many different extensions
have s ince been made . The self-tuning principle had
a lso been successfully applied to adaptive controllers using optimal quadratic cost functions , poleplacement techniques and phase and amplitude margins . Thus the STC-design-principle consists of a
combination of or.e of the above mentioned controller types and a recursive parameter identification
scheme .

The design of MRAC-systems is usually based on the
minimization of the model error e* as shown in Fig .
la . The design problem for MRAC-systems is thus to
determine the structure of the adjustment mechanism
such that the model error e* goes to zero as t-+oo .
This problem had been solved originally by the gradient method . However , this approach does not in general guarantee stability . Therefore, modified adjustment procedures have been proposed using stability theory . In these approaches the adjustment
mechanism has to be determined such that the overall system is globally stable , i . e . all s ignals remain bounded at any time . The problem of proving
g lobal stability in MRAC-systems had been solved
only a few years ago i ndependently by several authors
( e . g . Goodwin et a l , 197 8; Egardt, 1 9 7 9 ; Schmid,
1 9 7 9; Narendra and Lin, 1 9 7 9 ; Morse , 1 980) .

The design principles mentioned here will be described briefly in the following . As adaptive control is based on simultaneous process identification
and control , the problems of on-line parameter e stimation must be dealt with primarily . The process
identi fication and the adaptation mechanisms are
usually both realized by digital process computers .
Therefor e , the corresponding systems are described
in discrete-time form.

Recursive process identification

Most adaptive control algorithms are based on a
linearized process model , which, for a typical singleinput/s ingle-output (SISO) system is given by the
linear difference equation
n
n
( 1 )
y ( k ) = - l a vy M ( k- v ) + l b u ( k-v) .
M
v
v= 1
v =o

For a realistic description of the process model it
i s neces sary to include an additional disturbance
model as in Fig . 2 , where r s ( k ) is a stochastic
noise s igna l , which can be thought of to be generated from a white noise signal E (with normal distribution and zero mean ) by the noise filter trans fer
function
G ( z )
G* ( z ) .
(2)
r
r
Theory and Application of Adaptive Control

E(Z)

U ( z )

Fig . 2 . Complete model structure of the process

It follows using z-transformation from Fig . 2 that

Y ( z ) = yM ( z ) + G ( z ) E ( Z) .
r
By inserting , Eqs . ( 1 ) and ( 2 ) into Eq . ( 3 ) and multiplying by A ( z - 1 ) we obtain
1
- 1
-
( 4 )
A ( z ) Y ( z ) -B ( z ) u ( z ) = G*( z ) E ( Z) = V ( z ) '
r
where V ( z ) is defined as general model error and
- 1
-n
A ( z )
(5)
+ a 1 z
+ . . . + a n z
- 1
- 1
-n
B ( z )
( 6 )
b 0 + b 1 z
+ . . . + bn z
Eq . ( 4) defines an ARMAX-model . Depending on the
selec tion of G; ( z ) all usual model structures are
described by this equation ( Unbehauen, 1 982 , 1 9 85 ) .
E . g . the selection of G; ( z ) = l leads to the least
squares (LS ) technique, which will be used for simplicity but without loss of generality in the following .

Introducing the

�(k) =

3

weighting factor to the matrix P (k+ l ) which can be
obtained by multiplying the r ight hand side of Eq . ( 1 3 )
by the factor 1 /p (Bauer , 1 9 77 ) . A very usual and
effective procedure is to choose a constant weighting factor of 0, 95 :::__ p :::__ 0, 99 , whereby recent measurements are weighted more than older ones . One
draw back of the introduction of the weighting factor may consist in the phenomenon of " estimator
windup " . If the process is operating satisfactorily ,
the exc itation of the process is small, which means
for the expectation

( 1 5 )

Thus according to Eq . ( 1 2 ) q ( k+ l ) =o and from the
modified (weighted) Eq. ( 1 3) follows that

( 1 6 )
P (k )
P- ( k + l ) = .!:._ -
p
grows exponentially , which causes the estimator to
become unstable. If this happens in an adaptive
system, momentary instability of the c losed-loop
system may occur . But the excitation leads again
to an improved estimation followed by improved control .

From this brief discussion it follows, that for a
practical computer realization of identification
algorithms in adaptive control systems the user
should have a lot of operational experience for improvements or compromises .

data vector

T
[ -y (k- 1 ) . . . -y ( k-n ) i u (k- 1 ) . . . u (k-n) ] ,

and the parameter

( 7 )

vector

-
p = [a 1 . . . an ib • . . b ] T
n
1 1
under the assumption b 0=0 (which usually is
led for physical systems ) the output signal
tained directly from Eq . ( 4) as

( 8)

DESIGN OF SELF-TUNING CONTROLLERS (STC)

The original S TC proposed b y Astri:im and Wittenmark
( 1 97 3 ) is based on the stochastic "minimum variance"
( MV ) -controller . The design of the MV-controller is
based on a process model as shown in Fig . 2 with the
transfer functions

fulfilis ob-

( 9 )

The parameter estimation problem is to find a n
mation£ of 12. using the known data vector � ( k )
that the loss function for N measurements
1 T ! .
n +I' N
I = I ( £)
E2 ( k ) = 2 £ (N) £ (N ) = Mm
l
2
k =n + l estisuch

( 1 0)

becomes minimal. The solution of this minimization
problem can be obtained directly by inserting Eq.
( 9 ) into Eq . ( 1 0) by collec ting N pairs of measurements and batch-wise data processing . In adaptive
systems the recursive solution of this problem, however , is prefered .

The recursive estimation of the LS-model is given by
the following equations :

.E_(k+ l )

'.l ( k + l )

!'._(k + l )

E' ( k + l )

.E_ ( k ) + '.l ( k + l ) E' ( k+ l )
- 1
� ( k ) � ( k+ l ) [ ! +�T ( k + l ) � ( k ) � ( k+ l ) ]
T
!'._ ( k ) - '.l ( k + l ) � ( k + l ) � ( k )
T
l
l )
y ( k + -� ( k+ ) .E_ ( k ) .

( 1 1 )

( 1 2 )

( 1 3 )

( 1 4)

For the application of this estimation algorithm a
suitable choice of the initial values .E_(O) and �(O)
must be made. While the choice of .E_ ( O ) is not critical , P (O) should be s elected as a diagonal matrix
with large elements , e . g . 1 04 to 1 05 , which will
cause rapid changes of .E_(k) at the beginning . During
the calculation the values of the diagonal elements
are reduced so that p ( k ) changes only slowly . This
may lead to convergence of parameter s . on the other
hand for slowly varying process parameters and for
large values of k the algorithm may become sluggish .
This can be circumvented e. g . by introducing a

Gr ( z )
where

� z -dA ( z )
1
C ( z - ) -d
--_-- z
1
A ( z )

( 1 7a )

( l 7b )

c z -n
( 1 7 c )
n
The obj ec tive of the MV-controller is to minimize
the variance of the output signal under the assumption that the reference value w=O :
I
( 1 8)
E { y 2 } = E {e2 } � Min .

Substituting Eqs . ( 1 7a , b ) into Eq. ( 3 ) follows
- 1
C (z-- l )
Y ( z ) = � Z-d U ( z ) + -
- -1 - E ( Z)
A ( z )
A ( z )

or
-1
( 1 9 )
B ( z - 1 ) U
C ( z -) d E ( Z)
.
( z ) + --_1 Z
A ( z )
A ( z )
Using the identity
- 1
- 1 )
)
- 1
-d K ( z
C ( z
( 20)
- - _-!- = F ( z ) + z
--_ - 1 -
A ( z )
A ( z )
where
-1
-d + l
( 2 1 a )
F ( z )
+ . . . + fd-l z
- 1
-n+ l
( 2 1 b )
K ( z )
+ . . . + k n- l z
a r e unique polynomials , we obtain from Eq . ( 1 9 )
a fter some manipulation
- 1
- 1
- 1
d
d
- 1
) )
F ( z z
)
K ( z
�i
z Y ( z) =
( z ) +F ( z ) zE ( z ) .
- 1 Y ( z )
( 2 2 )
C ( z ) C ( z )

Applying E q . ( 1 8) t o the predictive form of Eq . ( 2 2 )
leads to
4

H. Unbehauen

where y*(k+dlkl represents the optimal prediction
of y (k+d ) and y ( k+d l k l a prediction error . As y cannot be influenced by the actuating signal u ( k ) the
minimum of Eq. ( 2 3 ) is obtained for
- 1
- 1
- 1
y*(k+d l kl =� - l {� Y ( z) + F ( z ) B ( z ) U ( z )} = o.
1
- 1
C ( z ) C ( z - )
( 24)

Under this condition the control law of the MV-controller directly follwws as
K ( z - l)
U ( z )
( 2 5 )

where the unknown coefficients of the polynomials
F ( z - 1 ) and K ( z - 1 ) are obtained from Eq . ( 2 0) after
multiplication with A ( z - 1 ) and by comparing coefficients of equal powers in z - 1 .

to remove the above mentioned disadvantages of the
c lassical STC is to introduce , as in Fig . 3 the extended process output signal

-1 -1 -d -1 -d
Y ( z ) = P ( z ) Y ( z ) +Q ( z ) z U ( z ) -R ( z ) z W ( z ) ( 33 )
h
1
where P ( z - 1 ) , Q ( z - ) and R ( z - l) represent stable
filter transfer functions . Completely analogous to

Eq . ( 27 ) a control law with the same structure can
be derived as

u ( k ) = - � �(k) � (k ) ,
hho
i n which the extended � and m contain the
mation about the additional fi1'ter transfer
tions and the reference signal w .

The MV-controller discussed above can be easily expanded to become a self-tuning controller (STC ) .
For example, the process parameters could be estimated on-line at every sampling interva l , and can
be used to calculate the parameters of the controller . This would lead to an explicit STC-scheme . However , it is also possible to estimate directly the
controller parameters such that an implicit or direct STC-scheme is obtained . This is very advantageous because the above mentioned comparison of coefficients can be avoided .

Introducing
- 1
F ( z ) B ( z )

where h 0
law
u (k )

( 34 )

inforfunc-

Self -tuning

algorithm

H ( z )

b 0 , into Eq .

z -m-d + l
h 0+h z - 1 + . . . +h
m+d-l
( 26 )
( 2 5 ) , we obtain the control

- �0 �v (k) �v (k) ,

using the estimate
vector
b 0
b 0
the signal vector � of

Fig . 3 . STC-scheme using filtered signals

Self-tuning controllers as described above, in general can be formulated by two laws :
a ) contro l law

( 2 7 )

the controller parameter

[ y (k) . . . y ( k -n+ l) I: u (k- 1 ) . . . u ( k-m-d + l) ]

( 2 8 )

b )

T

( 2 9 )

and

b

0

> b
0 > 0.

( 3 0)

The adaptation law for the controller parameters is
directly obtained from the recursive estimation
scheme similar to Eqs . ( 1 1 ) to ( 1 4 )

�v (k + l)

l
k
= �
v ( ) + S!_ (k+l) v (k+ )

( 3 1 )

using the prediction error
T
v ( k + l) = y ( k + l) - �v (k) � ( k-d + l)

- b 0u ( k -d + l) ( 32 )
wherein S!_ ( k+ l) and !:_ ( k ) are identica l to Eqs . ( 1 2 )
and ( 1 3 ) , and �( k ) i s replaced by � ( k ) .

In this design of the c lassical STC, current estimates of the parameter vector have been accepted
ignoring their uncertainties . This procedure is
usually defined as "certainty-equivalent principle " .
Thus the overall a lgorithm can be considerably simplified . The c lassical STC described above has a
number of disadvantages, e . g . the controller produces relatively large magnitudes of the control
variable u. Furthermore this controller is not directly applicable both to non-minimum phase systems ,
i . e . the case when B ( z - 1 ) has a zero outside the
unit circle , and to servo control problems .

There are many ways in which a STC can be designed
based on the MY-principle . A very general approach

u ( k ) =

-£!( k ) � ( k ) ,

( 35 )

where E_ (k) represents the estimated vector o f
controller parameters and the measurement vector
� ( k ) contains all information about the signals
in the control loop .

adaptation law

p

=--s

(k) = p

=-s

P ( k- 1 )
( k-l) +--s

�

-s (k ) c s (k)

( 36 )

where � ( k ) i s a regression vector obtained from
sensed signals within the control loop and Es ( k )
is the prediction error . For the recursive LSestimator both these variables and the matrix
!'..s ( k ) can be obtained directly from Eqs . ( 1 1 ) to
( 1 4) .

During the last few years considerable progress has
been made in the theoretical treatment of ST-controllers: especially many efforts had been made to
solve the s tability and convergence problems . It is
possible to find sufficient conditions for stabilit�
however , necessary and sufficient conditions are so
far not available for STC . Convergence of the parameter vector Es means that the parameters converge
to the values that would be obtained if the actual
process parameters would exactly be known . As already discussed above there are several possibilities to improve convergence within the parameter
estimation according to Eq . ( 3 6) . Other theoretical
problems are related to the robustness of STC , i . e .
the situation, wherein the assumed process model
structure is incorrect or the process changes its
operational conditions . The theoretical results of
robustness analysis of STC , available today , are
still not satisfactory .

DESIGN OF MODEL-REFERENCE ADAPTIVE CONTROLLERS ( MRAC )

As a lready mentioned above the k ey problem on MRAC