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, No. /) Identification, Adaptive and Stochastic Control (1985, No. 2) Large-scale Systems, Decision-making, Mathematics of Control (1985, No. 3) Process Industries, Power Systems (1985, No. 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) (1985, No. 7)
NORRIE & TURNER: Automation for Mineral Resource Development (1986, No. 1) CHRETIEN: Automatic Control in Space (1986, No. 2) DA CUNHA: Planning and Operation of Electric Energy Systems (1986, No. 3) V ALADARES 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 (1986, No. 5) PAUL: Digital Computer Applications to Process Control (1986, No. 6) YANG JIACHI: Control Science & Technology for Development (1986, No. 7) MANCINI, JOHANNSEN & MARTENSSON: Analysis, Design and Evaluation of Man-Machine
Systems (1986, No. 8) GELLIE, FERRATE & BASANEZ: Robot Control "Syroco '85" (1986, No. 9) JOHNSON: Modelling and Control of Biotechnological Processes (1986, No. 10)
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Proceedings of the 7th IFACIIFIPIIMACS Conference, Vienna, Austria, 17-20 September 1985
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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 Ill. Series 670.42'7 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 COMPUTER APPLICATIONS TO PROCESS CONTROL
Organized by Austrian Centre for Productivity and Efficiency (OPWZ)
Sponsored by IF AC Committee on Applications IF AC Committee on Computers
Co-sponsored by IF AC Committee on Education IF AC 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) K. J. Astrom, Sweden D. R. Bristol, USA A. van Cauwenberghe, Belgium A. A. Concheiro, Mexico G. Davoust, France G. Doolittle, USA K. H. Fasol, FRG D. Fischer, Austria C. Foulard, France ]. Gertler, Hungary R. Isermann, FRG
National Organizing Committee P. Kopacek (Chairman) ]. Hahne! (Secretary) W. Karner
P. Kopacek, Austria M. Mansour, Switzerland M. Paul, Austria I. Plander, Czechoslovakia I. V. Prangishvili, USSR K. Reinisch, GDR G. Schmidt, FRG V. Strejc, Czechoslovakia T. Takamatsu, Japan ]. D. Van Wyk, South Africa E. Welfonder, FRG J. H. Westcott, UK
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:
code
survey papers tutorial papers application oriented papers papers of general aspects
no. of contributions
6 3
3S 40
The six survey papers summarize the trends, developments and state-of-the art of adaptive control, dis­ tributed 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
A8
session
APPLICATION
Chemical and Oil Industries I Water Turbines Chemical and Oil Industries I I Energy and Power Systems I Energy and Power Systems I I Robotics and Manufacturing Cement, Metallurgical Processes and Traffic Heating and Climate Systems
GENERAL ASPECTS
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
Vil
s 3
5
40
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
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 adjust 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, including model reference adaptive controi self-tuning regulators and parame­ ter scheduling control are discussed . It is shown that carefully designed adaptive con­ trol 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 au­ tomatically adj usted in order to achieve good pro­ cess operation over a wide range of conditions. The controller adaption is necessary either for poorly understood processes or to compensate for unantici­ pated parameter changes of the process due to envi­ ronmental conditions or unpredictable operating point changes. Thus adaptive control provides poss:io­ bilities to control processes with uncertainties, as e . g . nonlinearities and time-varying parameters.
Although adaptive control strategies have been dis­ cussed 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 de­ velopment of powerful adaptive control algorithms which have reached today a mature state . On the other hand, modern microelectronics offers cheap hardware which allows an easy realisation of adap­ tive control strategies, leading already to commer­ cially viable solutions.
This paper is intended both to introduce the non­ specialist 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 classification and description of adaptive control principles is given . Then it is shown that most adaptive schemes have nearly the same structure . The further sections are devoted to a review of re­ cent developments of adaptive control schemes and of practical applications. This review does not in any way claim to be complete , but tries only to dis­ cuss 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 al , 1981).
The realization of modern adaptive control schemes includes a lot of on-line computational operations. Therefore , adaptive control algorithms are usually implemented on digital process computers or micro­ processors. Thus, as will be discussed later , most approaches are based on a discrete system represen­ tation .
BASIC STRUCTURES OF ADAPTIVE CONTROL SYSTEMS
Three main basic control system structures are to­ day relevant to the design of adaptive control sys­ tems ( e . g . Unbehauen, 1985) - model reference adaptive control (MRAC) , - self-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-tech­ nique uses the reference model to specify the de­ sired 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 con­ trol loops have to be defined . While the inner loop represents the basic control system consisting of the process and controller , the parameters of this controller are adjusted by the outer loop so long until the model error e*=y-yM 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 classical 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 recursive estimator) acting on a decision block and further on a modif i­ cation block representing the actual adaptation of the controller parameters. In the second loop the effect of controller modification is fed back to the decision 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 para­ meters in a predetermined manner as process condi-
2 H. Unbehauen
controller parameters
£ action
_______ J ©
y
Fig . 1 . Basic schemes of adaptive control (a) MRAC ; (b ) STR ; (c) PSC
tions vary . The decision process thus is reduced to a fixed mapping of the process parameters to the controller parameters, whereby the original deci­ sion proces s is already realized in the design phase of the adaptive control system , e . g . by a "table look-up" approach different sets of controller para­ meters 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" . How­ ever, in order to be more genera l , this approach should be defined to as "parameter scheduling con­ trol" ( PSC). This type of adaptive control struc­ ture is wide spread and in vogue today, since it allows one to tune a wide range of controllers using a manifold of popular on-line process identification methods . To guarantee a faultness operation of sys­ tems with the PSC-structure , a good knowledge of the actual process dynamics is required . The PSC­ strategy 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 adap­ tation 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 classified in­ to two general groups :
- direct (or implicit) adaptive schemes and - indirect (or explicit) adaptive scheme s .
In a n indirect adaptive control system the unknown
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 estima­ tion of the process parameters . Therefore the cal­ culation for the design of this controller has to be repeated at each sampling interval , whereby the identification and controller adaptation are two different procedures . Without directly identifying the process parameters it is often possible to iden­ tify 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 significant simplification of the adap­ tation algorithm .
According to the above definitions the MRAC shown in Fig . l a represents a direct (or implicit) adap­ tive controller since its parameters and its control law are directly updated from the signals u and y . The STC shown i n Fig . l b i s thus , however, defined as an indirect (or explicit) adaptive controller .
Although the STC was originally developed for the stochastic minimum variance control problem (Astr6m and Wittenmark , 1 9 7 3 ) many different extensions have since been made . The self-tuning principle had also been successfully applied to adaptive control­ lers using optimal quadratic cost functions , pole­ placement techniques and phase and amplitude mar­ gins . Thus the STC-design-principle consists of a combination of or.e of the above mentioned control­ ler 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 gra­ dient method . However , this approach does not in ge­ neral guarantee stability . Therefore, modified ad­ justment procedures have been proposed using stabi­ lity theory . In these approaches the adjustment mechanism has to be determined such that the over­ all system is globally stable , i . e . all signals re­ main bounded at any time . The problem of proving global stability in MRAC-systems had been solved only a few years ago i ndependently by several authors ( e . g . Goodwin et al , 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 de­ scribed briefly in the following . As adaptive con­ trol is based on simultaneous process identification and control , the problems of on-line parameter esti­ mation must be dealt with primarily . The process identification and the adaptation mechanisms are usually both realized by digital process computers . Therefore , 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 single­ input/single-output (SISO) system is given by the linear difference equation
n n yM (k ) = - l avyM (k-v) + l
v=1 v=o b u (k-v) . v ( 1 )
For a realistic description of the process model it is necessary to include an additional disturbance model as in Fig . 2 , where rs ( k ) is a stochastic noise signa l , which can be thought of to be genera­ ted from a white noise signal E (with normal distri­ bution and zero mean ) by the noise filter trans fer function
G ( z ) r G* ( z ) . r (2)
Theory and Application of Adaptive Control 3
E(Z)
Fig . 2 . Complete model structure of the process
It follows using z-transformation from Fig . 2 that
Y ( z ) = y ( z ) + G ( z ) E ( Z) . M r ( 3 )
By inserting , Eqs . ( 1 ) and ( 2 ) into Eq . ( 3 ) and mul­ tiplying by A ( z- 1 ) we obtain
A ( z- 1 ) Y ( z ) -B ( z- 1 ) u ( z ) = G* ( z ) E ( Z) = V ( z ) ' r ( 4)
where V ( z ) is defined as general model error and
- 1 - 1 -n A ( z ) + a 1 z + . . . + a z (5) n - 1 - 1 -n B ( z ) b + b1 z + . . . + b z ( 6 ) 0 n
Eq . ( 4) defines an ARMAX-model . Depending on the selection of G; ( z ) all usual model structures are described by this equation ( Unbehauen, 1 982 , 1 985 ) . E . g . the selection of G; ( z ) = l leads to the least squares (LS ) technique, which will be used for sim­ plicity but without loss of generality in the fol­ lowing .
Introducing the data vector
(k ) = [ -y (k- 1 ) . . . -y ( k-n ) i u (k- 1 ) . . . u (k-n) ] T,
and the parameter vector
- n1 n
under the assumption b0=0 (which usually is led for physical systems ) the output signal tained directly from Eq . ( 4) as
( 7 )
( 8)
( 9 )
The parameter estimation problem is to find a n esti­ mation£ of 12. using the known data vector (k ) such that the loss function for N measurements
I = I ( £) n+N I' 2 1 T ! .
2 l E (k ) = 2 £ (N ) £ (N ) = Mm k=n+ l
( 1 0)
becomes minimal. The solution of this minimization problem can be obtained directly by inserting Eq. ( 9 ) into Eq . ( 1 0) by collecting N pairs of measure­
ments and batch-wise data processing . In adaptive systems the recursive solution of this problem, how­ ever , is prefered .
The recursive estimation of the LS-model is given by the following equations :
.E_ (k+ l )
'.l (k+l )
T - 1 ( k ) (k+ l ) [ ! + ( k+ l ) ( k ) (k+ l ) ]
!'._ ( k ) - '.l (k+ l )T ( k+l ) (k )
y (k+ l ) -T ( k+ l ) .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 criti­ cal , P (O ) should be selected 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
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 , 1 977 ) . A very usual and effective procedure is to choose a constant weight­ ing factor of 0, 95 :::__ p :::__ 0, 99 , whereby recent me­ asurements are weighted more than older ones . One draw back of the introduction of the weighting fac­ tor may consist in the phenomenon of " estimator windup" . If the process is operating satisfactorily , the excitation 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
P (k+ l ) = .!:._ P (k ) - p- ( 16 )
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 con­ trol .
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 im­ provements or compromises .
DESIGN OF SELF-TUNING CONTROLLERS (STC)
The original S TC proposed b y Astri:im and Wittenmark ( 1 973 ) 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
G ( z ) r
1 -
( 1 7 c )
The obj ective of the MV-controller is to minimize the variance of the output signal under the assump­ tion that the reference value w=O :
I
Substituting Eqs . - 1
Y ( z ) = - 1 A ( z ) or
( 1 7a , b ) into Eq. ( 3 )
-d C (z - l ) Z U ( z ) + --- -1 - E ( Z) A ( z )
follows
- 1 -1 B ( z ) U ( z ) + C ( z ) ZdE ( Z) . --_-!- --_-1-
A ( z ) A ( z ) Using the identity
C ( z- 1 ) = F ( z- 1 ) -d K ( z- 1 ) --_-!- + z --_ -1 -
A ( z ) A ( z ) where
-1 F ( z )
- 1 K ( z )
( 1 8)
( 1 9 )
from Eq . ( 1 9 )
- 1 ) Y ( z ) F ( z )iz )
C ( z ) C ( z )
- 1 d ( z ) +F ( z ) zE ( z ) . ( 2 2 )
Applying Eq . ( 1 8) to 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 can­ not 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.
C (z- 1 ) C ( z-1 ) ( 24 )
Under this condition the control law of the MV-con­ troller directly follwws as
U ( z ) K ( z -l) ( 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 coeffi­ cients of equal powers in z- 1 .
The MV-controller discussed above can be easily ex­ panded to become a self-tuning controller (STC ) . For example, the process parameters could be esti­ mated on-line at every sampling interva l , and can be used to calculate the parameters of the control­ ler . This would lead to an explicit STC-scheme . How­ ever , it is also possible to estimate directly the controller parameters such that an implicit or di­ rect STC-scheme is obtained . This is very advanta­ geous because the above mentioned comparison of co­ efficients can be avoided .
Introducing
- 1 - 1 F ( z ) B ( z ) - 1 H ( z ) - 1 -m-d+l h0+h z + . . . +hm+d-lz ( 26 )
where h0 law
using the vector
b 0 v b 0 the signal vector
v [y (k ) . . . y (k-n+l) : u (k- 1 ) . . . u ( k-m-d+l) ] T I
and
( 2 8 )
( 2 9 )
( 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) = v (k ) + S!_ (k+l) v (k+l) ( 3 1 )
using the prediction error
T v (k+l) = y (k+l) - v (k ) (k-d+l) - b0u (k-d+l) ( 32 )
wherein S!_ (k+l) and !:_ ( k ) are identical to Eqs . ( 1 2 ) and ( 1 3 ) , and (k ) i s replaced by (k ) .
In this design of the classical STC, current esti­ mates of the parameter vector have been accepted ignoring their uncertainties . This procedure is usually defined as "certainty-equivalent principle" . Thus the overall algorithm can be considerably sim­ plified . The classical STC described above has a number of disadvantages, e . g . the controller pro­ duces relatively large magnitudes of the control variable u. Furthermore this controller is not di­ rectly 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
to remove the above mentioned disadvantages of the classical STC is to introduce , as in Fig . 3 the ex­ tended process output signal
-1 -1 -d -1 -d Y h( z ) = P (z ) Y ( z ) +Q ( z ) z U ( z ) -R ( z ) z W ( z ) ( 33 )
where P ( z- 1 ) , Q ( z- 1 ) 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
( 34 )
i n which the extended and m contain the infor­ mation about the additional fi1'ter transfer func- tions and the reference signal w .
Self -tuning algorithm
Fig . 3 . STC-scheme using filtered signals
Self-tuning controllers as described above, in gene­ ral can be formulated by two laws :
a ) contro l law
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 .
b ) adaptation law
p ( k ) = p ( k-l) +P (k- 1 ) (k ) c (k ) =--s =-s --s -s s ( 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 LS­ estimator 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-con­ trollers: especially many efforts had been made to solve the stability 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 para­ meter vector Es means that the parameters converge to the values that would be obtained if the actual process parameters would exactly be known . As al­ ready discussed above there are several possibili­ ties 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 CON­ TROLLERS (MRAC )
As already mentioned above the key problem on MRAC-
Theory and Application of Adapt iYe Control 5
systems is that the model error e*=y-yM ( see Fig . la ) becomes small o r even zero . Therefore a n adjustment mechanism has to be determined to solve this prob­ lem. Many methods have been proposed for the solu­ tion of this nontrivial problem . The first attempt is due to Whitaker et al ( 1 958) , who used the gra­ dient method for the continuous adaptation of the controller parameters (MIT-rule) :
dl2_
dt = -an(£)' ( 37 )
where the performance index I (p ) is assumed t o be a function of model error e* ( t ) ,
- e . g . the mean square
model error , which has to be minimized :
2 I
I ( l2_) = f [ e''(t , )2_) ] = e* ( t ,p ) ,; Min . ( 38 )
Inserting E q . ( 38 ) into E q . ( 37 ) and then integrat­ ing this equation gives the adaptation law
t l2_ (t ) = )2_ ( 0) - 2a J e* ( T) ( T, l2_) dT
0 where the sensitivity vector
contains the partial derivations
( 39 )
( 40)
of the process output signal in accordance with the controller parameters pi , and the bar on the inte­ gral signifi es the mean value. The sensitivity func­ tions vi (t , )2_) can easily be generated under the as­ sumption of slowly varying parameter s , from the filtered reference variable w ( t ) as
( 4 1 )
where G and G are the transfer functions o f the controler and5the process respectively . From Eq. ( 4 1 ) it follows that the filter network contains a model of the inner loop, which is defined as sen- ' sitivity mode l . The principal block diagram of the complete MRAC-scheme based on the gradient approach is shown in Fig. 4 . Although the structure of this adaptive scheme is relatively simple to understand ,
w
____ _, model
+
2ae*
Fig . 4 . MRAC-scheme based on the gradient approach
it has one great disadvantage because the overall stability is not guaranteed depending heavily on the selection of the gain parameter a. Therefore, modified adaptation laws had been derived using s tabi lity theory . These rules provide very similar adaptation laws as shown by Eq. ( 39 ) , however , the sensitivity functions are replaced by other ex­ pressions . Such approaches provide for the deter­ mination of the adj ustment mechanism in such a way that the overal l system is globally stable. This means that the plant input and output signals u (t )
and y ( t ) respectively remain bounded for all time and thus either the model error e* or the state error vector * converges to zero . This problem had been solved independently by several authors during the last few years . Ljapunov ' s stability theory and Popov ' s hyperstability method have been extensively applied to MRAC-systems , both with state feedback and output feedback .
The main idea in applying stability theory to MRAC­ systems is to transform the highly non-linear adap­ tive system to the standard form of a non-linear " error system" as shown in Fig . 5, where the model
G (s/z)
Fig . 5. Non-l inear standard "error system"
error e* either in continuous or discrete form re­ presents the output signal of a linear time inva­ riant system, described by the transfer function G , while F contains all non-linear and time-variant subsystems . This structure has the advantage that the stability of the overall MRAC-system can be ob­ tained from the individual properties of the linear and non-linear subsystems , e. g . the linear subsystem must be strictly positive real ( s . p . r . ) . Thus the output error
e* = cT e* ( 4 2 )
or the elements e o f the state error vector
-* [-* -* -*] T = el . . . e£ . . . en ( 43 )
can i n general be represented b y a nonlinear time­ varying differential equation of the form
( 4 4a )
E ( k+ l ) = f 1 [ E ( k ) , R(k), k] , ( 44b)
where E can be replaced either by e* or*, and is either the parameter vector or the corresponding parameter error vector of the adaptive controller , for which the adaptation law
or
t (t ) = ( 0) - f !_2 [ E ( T) , T]dT ( 45a )
0
(k) = ( k- 1 ) - !_2 [ E ( £) , k] ( 45b )
must be developed , such that all signals are uni­ formly bounded and
lim E ( t ) = 0 t-+oo
( 46 )
using a l l available data . The function !_2 in Eq. ( 45 ) can be obtained either by estimation or from filtered process measurements .
In order to obtain a causal control law
u ( t) = T - (t ) (t ) ( 47a)
T ( 47b) u (k) = -(k) R ( k ) ,
which i s linear in the parameters , it is usually necessary to introduce filters for filtering the model error and eventually to augment the model error by adding auxiliary signals . Thus E becomes an "augmented" error signal. To ensure stability the vector ' which in general contains functions
6 H. Unbehauen
of the process input and output signals , must be ge­ nerated to ensure boundedness of u and y and asymp­ totic convergence of E .
Adaptive control laws such as Eqs . ( 45) and ( 47 ) can be derived in a number of different ways , but are not discussed in detail here for the sake of brevi­ ty . However , it should be mentioned that the general adaptation law of Eq . ( 4 5 ) includes in the expres­ sion !_2 a multiplicative connection between the er­ ror E and some regression vector '-.0 which is in ge­ neral represented by functions of u and y similary to !!2MR• see e . g . Unbehauen ( 1 985) . Thus Eq. ( 3 9 ) i s also included i n this law.
The discussion of MRAC-systems shows by comparing Eqs . ( 45 ) and ( 4 7 ) with Eqs . ( 35) and ( 36) that the basic structures of STC and MRAC are nearly the same, although the background of MRAC was the servo con­ trol problem, whereas the STC originally had been designed for the stochastic regulation problem. Both principles are characterized by two feedback loops . However the design principles of these two loops are different .
DEVELOPMENTS IN THEORY DURING 1980- 1984
It is beyound the scope of this paper to describe the development in the theory of adaptive control during the earlier years in detail . Only a few main topics can be treated briefly . For a long time the problem of s tabi lity had not been solved satisfac­ torily . Basic contributions to guarantee global sta­ bility have been published by 1980 by different authors , e . g . Narendra et a l . ( 1 980 a , b ) , Goodwin et a l . ( 1 980) ( see Table 1 ) . Thus it is possible to design adaptive control systems with guaranteed stability properties , Unbehauen ( 198 1 ) . Conditions for the exponential convergence of the adaptation laws of controller parameters have been derived by
Anderson and Johnson ( 1 982 ) . A necessary condition for convergence of the estimated controller parame­ ters is that the process input is persistently ex­ citing . Sin and Goodwin ( 1 982 ) proved the global convergence of a modified rec ursive LS-algorithm . In the case of stochastically disturbed systems the application of the martingale theory provides rea­ sonable results ( Landau 1982c ) .
Robus tness of adaptive control systems means the preservation of stability or boundness properties when ideal conditions are not met ( Ioannu and Koko­ tovic , 1984 ) . The usual assumptions that there are no disturbances and that the plant order is not higher than the model order are very unrealistic . Bounded disturbances and unmodelled dynamics make the basic adaptive scheme unstable. Several propo­ sals had been made to modify, therefore, the adap­ tation laws . The aim is to prevent instability by counteracting the parameter drift through eliminat­ ing the integral action of the adaptation law . If an upper limit of the disturbances is known, the stability of the system can be guaranteed by intro­ ducing a "dead zone" into the adaptation law ( Peter­ son and Narendra 1982 ) . Ioannu ( 1 983a) , Ioannu and Kokotovic ( 1 984 ) introduced the so-called a-modifi­ cation of the adaptation law in order to obtain sta­ bility of the adaptive system under the influence of limited disturbance magnitudes . Within this ap­ proach the usual integral parameter adaptation law, e . g . according to Eq. ( 45) ,
l2_ ( t ) = -!_2 (t ) ( 48a)
or
is replaced by
l2_ (t ) -0.!2_ ( t ) - !.2 ( t ) a > o ( 49a )
.!2_ (k ) a2 ( k- l ) - !.2 (k ) I a I < 1 ( 49b)
This modification should be applied only if the norm of the parameter vector exceeds some a priori defined value II .Ell > N0• In this case simple stabi­ lity is guaranteed for stochastic disturbances and errors for unmodelled high-frequency modes of the proces s . Thus the stability of adaptive control systems using reduced-order models is obtained .
Another idea for robust adaptive control systems , proposed by Narendra and Annaswamy ( 1 984 ) , is based in the sufficient excitation of signals . Rohrs and Shortelle ( 1 984 ) introduce spec ial filters which provide that high-frequency modes of the process are included in damped form in the adaptation law .
The formal extension of single-input/single-output adaptive control systems to multivariab le structures using parameterization issues ( see e .g . Elliot and Wolovich, 1 984 ) includes some difficultiP.s in re­ spect of the dead-time, which can be separated into an input and output portion . Hahn ( 1 98 3 ) introduced a systematic approach for this sepaation .
Whereas most design procedures of adaptive control systems are based on unknown but linear and time invariant systems only a few papers deal with non­ linear and time-varying processes . Goodwin and Teoh ( 1 98 3 ) investigated the convergence of a modified LS-algorithm for time-varying systems , which had been successfully applied to processes with jump parameters and drift parameter s . Mosca and Zappa ( 1 982 ) presented an interesting extension of a STC­ system for processes with variable dead-times , using parallel operating estimation algorithms .
Many contributions have been published on different aspects of adaptive control s tructures. Several papers are devoted to new structures which repre­ sent combinations and modifications of already known adaptation algorithms using appropriate detectors for changeover switching. Other aspects such as
adaptive sampling (de la Sen , 1 984 ) , special design schemes based on quadr•atic cost functions and pole p lacement have been reported . Extensions of the adaptive control principle deal with multi-loop cascaded structures (Gawthrop , 1 984 ) and with the introduction of hybrid adaptive control structures ( see e. g . Narendra et al . , 1 98 3 ) . In principle the hybrid structures consist of a continuous control system combined with a discrete parameter estimation scheme. However in practice these systems are digi­ tally realized by different sampling rates for the control and the adaptation law .
Numerous papers touch upon more general problems for STC- and MRAC-sys tems. Especially the design of non-minimum phase adaptive control system has been of great interest ( e . g . Noth, 1 982; Hahn, 1983; Clarke, 1 984 ) , whereby the introduction of special correction networks parallel to the process provided high advantages for a stable design . Landau et a l . ( 1 983 ) show a variety of possibilities to deal with this problem .
For a broad break-through of adaptive control sys­ tems in industrial appliacations it is necessary to provide simple adaptive control ler structures. Se­ veral proposals and already some industrial solu­ tions are available today . Astr6m and Hagglund ( 1 984 ) describe an interesting solution based on an adaptive PID-controller . Introducing a known non­ linearity and then applying the describing function method the critical gain and frequency of the limit cycles can be obtained on-line. Thus the Ziegler­ Nichols rules for on-line tuning of the controller parameters can directly be applied .
Very little experience is available up to now with adaptive control theory of large scale sys tems s truc­ tures and distributed parameter sys tems. Ioannou and Kokotovic ( 1 983b) show that the <J-modification of the adaptation law can be used also in the case
Theory and Application of Adaptive Control
TABLE 1 Papers dealing with theoretical aspects of adaptive control systems
Survey papers and books : Narendra & Monopoli ( 1 980) , Unbehauen ( 1 980) , Harris & Billings ( 1 98 1 ) , Goodwin & Ramadge ( 1 98 1 ) , Alix et a l . ( 1982 ) , Isermann ( 1 982 ) , Landau ( 1 982b) , Astrom ( 1 983) , Elliot ( 1 983) , Landau ( 1 983) , Goodwin & Sin ( 1 984 ) , Voronov & Rutkovsky ( 1 984 ) , Wittenmark & Astrom ( 1 984 ) .
7
Stabi lity : Astrom ( 1 980) , Gawthrop ( 1 980a ) , Goodwin et al . ( 1980) , Fuchs ( 1 980) , Morse ( 1 980) , Narendra et al . ( 1 980a ) , Narendra & Lin ( 1 980b ) , Lozano & Landau ( 1 98 1 ) , Dugard et al . ( 1 982 ) , Kreiselmeier & Narendra ( 1 982 ) , Landau ( 1 982a) ,de Larminat ( 1982 ) , Kosut ( 1 983a , b ) , Kosut et a l . ( 1 983) , Samson ( 1 983) , Christi ( 1984) Kung & Womack ( 1 984 ) .
Convergence : Goodwin et a l . ( 1 981 ) , Osorio-Cordero & Mayne ( 1 98 1 ) , Johnstone & Anderson ( 1 982 ) , Anderson & Johnson ( 1 982 ) , Dugard et a l . ( 1 982 ) , Landau ( 1 982c ) , Sin & Goodwin ( 1 982 ) , Sternby & Rootzen ( 1 982 ) , Good­ win et a l . ( 1 983) , Boyd & Sastry ( 1 984 ) , Goodwin et al. (1984 ) , Kumar ( 1 984 ) , Moore ( 1 984 ) .
Robus tness : Gawthrop & Lim ( 1 982 ) , Johnson & Goodwin ( 1 982 ) , Kreisselmeier & Narendra ( 1 982 ) , Lim ( 1 982 ) , Peterson & Narendra ( 1 982 ) , Shah & Monopoli ( 1 982) , Bar-Kana & Kaufmann ( 1 983) , Ioannou ( 1 983a) , Ioannou & Kokotovic ( 1 983a) , Praly ( 1 983a , b ) , Chen & Cook ( 1 984 ) , Christi ( 1 984 ) , Fuji et al , ( 1 984 ) , Ioanrou ( 1984 ) , Ioannou & Kokotovic ( 1 984a , b ) , Kokotovic & Riedly ( 1 984 ) , Kosut & Johnson ( 1 984), Krause ( 1 984 ) , Narendra & Annaswamy ( 1984 ) , Rohrs & Shortelle ( 1 984 ) .
Mu ltivariab le sys tems : Goodwin et a l . ( 1 980) , Koivo ( 1 980) , Lu & Yuan ( 1 980) , Bayoumi et a l . ( 1 98 1 ) , Kevicz­ ky & Kumar ( 1 98 1 ) , Koivo et a l . ( 1 98 1 ) , Prager & Wellstead ( 1 981 ) , Wonq & Bayoumi ( 1 981 ) , Elliot & Wolovich ( 1 982 ) , Favier & Hassani ( 1 982 ) , Hahn & Unbehauen ( 1 982 ) , Morris et a l . ( 1 982 ) , Okohawa & Yonezaewa (l'HJ2), Zinober et al . (l'Jl:l:!), Hahn ( l'Jl:l3) , Bar-Kana & Kaufmann ( 1 984 ) , Bezanson& Harris ( 1 984 ) , Dion & Lamare ( 1984 ) , Dugard et a l . (1984a , b ) , Djaferies et a l . ( 1 984 ) , Elliot et a l . ( 1 984 ) , Elliot & Wolo­ vich ( 1 984 ) , Grimble ( 1 984 ) , Lee & Lee ( 1 983 ) , Tsiligiannis & Svoronos ( 1 984 ) .
Nonlinear and time-varying processes : Anbumani et a l . ( 1 98 1 ) , Lachmann & Goedecke ( 1 982 ) , Pajunen ( 1 982 , 1 983) , Anderson & Johnstone ( 1 983) , Goodwin & Teoh ( 1 983) , Mosca & Zappa ( 1 983) , Balestrino et a l . ( 1 984 ) , Kung & Womak ( 1 984 ) , Urwin & Swanick ( 1 984 ) , Xianya & Evans ( 1984 ) .
Contro l s tructures
A. A lgorithms (combinations and modifications) : Goodwin et a l . ( 1 980) , Lozano ( 1 982 ) , Sin & Goodwin ( 1 982 ) , Gupta et a l . ( 1984 ) , Hagglund ( 1 984 ) , Holst & Paulsen ( 1 984 ) , Lam ( 1 984 ) , Milnert ( 1 984 ) , Moore & Bo el ( 1984 ) , Radke & Isermann ( 1 984 ) , Silveira & Doraiswami ( 1 984 ) , Stankovic & Radenkovic ( 1 984 ) .
B. Samp ling prob lems : de la Sen ( 1 984a, b ) , Kanniah et al. ( 1 984a , b ) , Kanniah & Malik ( 1 984 ) .
C. Variab le dead time : Kurz & Goedecke ( 1981) , Fuchs ( 1 98 2 ) , Vogel & Edgar ( 1 982a , b ) , Costin & Buchner ( 1 983) , Mosca & Zappa ( 1 983) , Chien et a l . ( 1 984 ) .
D. Structures based on quadratic performance i ndex : BOhm et a l . ( 1 984 ) , Grimble ( 1 984a , b ) ,Halme & Ahava ( 1 984 ) , Makila ( 1 984 ) .
E. Structures based on pole p lacement: Astrom & Wittenmark ( 1 980) , Tsay & Shieh ( 1 98 1 ) , Clarke ( 1 982 ) , Elliot ( 1 982 ) , Hesketh ( 1 982 ) , McDermott &Mellichamp ( 1 984 ) , Djaferi s et al . ( 1 984 ) .
F. Cascaded s tructures : Gawthrop ( 1 984 ) .
G. Hybrid s tructures : Gawthrop ( 1 98Cb) ,Christi ( 1 982 ) , Christi & Monopoli ( 1 982 ) , Elliot ( 1 982 ) , Narendra et al . ( 1983) .
H. Se lf-tuning structure s : Astrom & Wittenmark ( 1 980) , Allidina & Hughes ( 1 980) , De Keyser & van Cauwen­ berghe ( 1 981 ) , Fjeld & Wilhelm ( 1 98 1 ) , Fortescue et a l . ( 1 98 1 ) , Grimble ( 1 98 1 ) , Radke & Isermann ( 1 984 ) , Warwick ( 1 981 ) , Wellstead & Sanoff ( 1 98 1 ) , Clarke ( 1 982 ) , De Keyser & van Cauwenberghe ( 1 982 ) , Gawthrop ( 1 982a , b ) , Grimble ( 1 982 ) , Noth ( 1 982 ) , Ortega ( 1 982 ) , Wellstead & Zanker ( 1 982 ) , Allidina & Hughes ( 1 983) , Hoopes et a l . ( 1 983) , Toivonen ( 1 983a , b ) , Clarke ( 1 984 ) , Matko & Schumann ( 1 984 ) .
I. Mode l reference structures: Johnson ( 1 980) , Shah & Fisher ( 1 980) , Landau & Lozano ( 1 98 1 ) , Lozano & Lan­ dau ( 1 981 ) , Unbehauen ( 1 98 1 ) , Landau ( 1982 ) , Ambrosino et a l . ( 1 984 ) , Bar-Kana & Kaufmann ( 1 984 ) , Gupta et a l . ( 1 984 ) , Kennedy ( 1 984 ) .
K. Simp le contro l ler s tructures : Glattfelder et a l . ( 1 980) , Wittenmark & Astrom ( 1 980) , Andreiev ( 1 981 ) , Clarke & Gawthrop ( 1 981 ) , Astrom ( 1 982 ) , Banyasz & Keviczky ( 1 982 ) , Gawthrop ( 1 982a) , Bristol ( 1 983) , Cameron & Seborg ( 1 983) , Dexter ( 1 983) , Hawk ( 1 983) , Hetthesy et a l . ( 1983) , Keviczky & Banyasz ( 1 983) , Unbehauen ( 1 983) , Astrom & Hagglund ( 1 984 ) , Halme & Ahava ( 1 984 ) , Nishikawa et al. ( 1 984 ) .
L. Large scale sys tems s tructures : Costin & Buchner ( 1 983) , Ioanmu ( 1 983) , Ioanmu & Kokotovic ( 1 983) .
f\1. Dis tributed parameter systems s tructure s :Balas ( 1 983) , Hulko et al. ( 1 983) .
of decentralized adaptive control . Especially in the field of modern process control these problems of large scale adaptive control structures have to be solved in the future.
APPLICATIONS DURING 1 980- 1 98 1
The following discussion on applications of adaptive control systems does not c laim to be complete. How­ ever the discussion wil l include the most represen­ tative papers published in different fields of appli­ cation . The discussion is directed to Table 2, in which the main applications fields are SlllilIIlarized .
In the classical aerospace field interesting appli­ cations have been made for the adaptive control of large scale structures in space (Balas and Johnson, 1 980) . Because of the broad application of robotic sys tems in production lines these systems demand high positioning accuracy . This can be obtained by adaptive control in robotic manipulators ( Koivo , 1 983; Neumann and Stone, 1983) . Both STC- and MRAC­ systems are broadly applied .
Chemical indus try has become one of those fields, where adaptive control schemes have been introduced most successfully and most widely . Various types of chemical reactors have been equiped with adaptive controllers. In distillation columns multivariable
8 H. U n bchauen
TABLE 2 Papers dealing with applications of adaptive control systems
Survey papers and books : Belanger ( 1 980, 1 982 ) ' Narendra & Monopoli ( 1 980) ' Parks et a l . ( 1 980) , Unbehauen & Schmid ( 1 980) ' Harris & Billings ( 1 98 1 ) ' Unbehauen ( 1 98 1 ) ' de Keyser & van Cauwenberghe ( 1 982 ) ' Azab & Nouh ( 1 98 3 ) ' Bristol ( 1 98 3 ) ' Clough ( 1 983 ) . Seborg et al . ( 1 983 ) .
Air craft and space : Balas & Johnson ( 1 980) ' van den Bosch & Jong kind ( 1 980) ' Kreiselmeier ( 1 980) ' Rynaski ( 1 980) ' Stein ( 1 980) ' Young ( 1 98 1 ) ' Balas ( 1 98 3 ) ' Bar-Kama & Kaufmann ( 1 983 b ) ,Harvey ( 1 98 3 ) .
Robotics : Cao ( 1 980) ' Morris & Neuman ( 1 98 1 ) ' Hondered ( 1 98 3 ) ' Koivo ( 1 98 3 ) ' Neumann & Stone ( 1 98 3 ) ' Neumann & Tourassio ( 1 98 3 ) ' Tomizuka & Horowitz ( 1 983 ) ' Lee & Chung ( 1 984) ' Nicosia & Tome ( 1984) ' Vukobratovic et al . ( 1 984 ) .
Chemical industry
A . Extension: Englander ( 1 98 3 ) .
B. Reactors : Bergmann & Radke ( 1 980) ' Harris et al . ( 1 980) ' Clarke & Gawthrop ( 1 98 1 ) ' Hallager & Jorgen- sen ( 1 98 1 , 1 983 ) ' Yang et al . ( 1 981 ) , Clurett et a l . ( 1 982 ) ' Hodgson & Clarke ( 1 982 ) ' Saxon & Glover ( 1 982 ) ' Kiparissides & Shah ( 1 983 ) ' Koutchoukali et a l . ( 1 983a , b ) , McDermott et a l . ( 1 984 ) .
c. Dis ti l lation: Dahlquist ( 1 980) ' Morris et al . ( 1 98 1 ) ' Chien et a l . ( 1 98 3 ) ' Gerry et al . ( 1 983 ) ' Wiemer et a l . ( 1 983 ) ' Dahhou et . a l . ( 1 984 ) ' Martin-Sanchez & Shah ( 1 984 ) ' Yang & Lee ( 1 984 ) .
D. Evaporation: Bucholt & Kummel ( 1 98 1 ) ' Martin-Sanchez et a l . ( 1 98 1 ) ' Ellis ( 1 982 ) ' Song et al . ( 1 984 ) .
E. PH-neutra lisation: Bergmann & Lachmann ( 1 980) ' Jacobs et a l . ( 1 980) ' Goodwin et al . ( 1 982 ) .
Paper indus try : d ' Hulster et al . ( 1 980) ' Fj eld & Wilhelm ( 1 98 1 ) , de Keyser & van Cauwenberghe ( 1982 ) ' Sikora et a l . ( 1 984 ) .
Therma l processes : Haber et al . ( 1 980) ' Kurz et al . ( 1 980) ' Moden & Nybrant ( 1 980) ' Dahhou et al . ( 1 981, 1983 ) ' Dexter ( 1 981 ) ' Haber et a l . ( 1 98 1 ) ' Naj im et a l . ( 1 982 ) ' Radke ( 1 982 ) , Schumann ( 1 98 2 ) ' Lozano & Bonilla ( 1 98 3 ) .
Cement industry and mineral processes : Westerlund et a l . ( 1 980) ' Westerlund ( 1 98 1 ) ' Rugot & Sauter ( 1982 ) .
Ste e l and meta l lurgical indus try : Desrochers ( 1 98 1 ) ' Yui & Sato et al . ( 1 984 ) .
Power p lants and power sys tems : Bonami & Guth ( 1 980) ' Glattfelder & Schauf elberger ( 1 980) ' Irving et a l . ( 1980a , b ) , Mehra et a l . ( 1 980) , Allidina e t al . ( 1 98 1 ) ' Hamza et al . ( 1 982 ) ' Hahn et a l . ( 1 982 ' 1983 ) ' Amin et al . ( 1 984 ) .
Electromechanical systems : Bonami & Guth ( 1 980) ' Green et a l . ( 1 980) ' Morris & Neumann ( 1 98 1 ) ' Hahn et a l . ( 1 98 2 , 1 983 ) ' Balestrino et a l . ( 1 983 ) ' Brickwede ( 1 9 8 3 ) ' Hondered ( 1 983 ) , Hanus ( 1 983 ) ' Zohdy et a l . ( 1 983 ) .
Position contro l : Claussen ( 1 980) ' Gutmann et a l . ( 1 980) ' Haque & Monopol i ( 1 980) .
Ship steering: van Amerongen ( 1980, 1 981 , 1 982 ) ' Cuong & Parson ( 1 9 8 1 ) ' Fung & Grimble ( 1 98 1 ) ' Mort & Linkens ( 1 98 1 ) ' van Amerongen et a l . ( 1 983 ) ' van Amerongen & Hondered ( 1 983 ) .
Combusting engines and compressors : Wellstead & Zanker ( 1 98 1 ) ' Morris et a l . ( 1 983 ) ' Subbarao & Huntly ( 1983 ) ' Fuj ii et a l . ( 1 984 ) .
Misce l laneous areas : Behar & I nfante ( 1 98 3 ) ' Fjeld et a l . ( 1 983 ) ' Makila & Syrj anen ( 1 983 ) ' Dochain & Bastin ( 1 984 ) ' Kaufmann et a l . ( 1984 ) .
adaptive control systems provide a higher product quality and a considerable reduction of thermal energy ( e . g . Wiemer et a l . 1 98 3 ) . Other applications of adaptive control system in evaporation and pH­ value neutralization processes point out that these techniques are now wel l beyond the theoretical stage in chemical industries.
Interesting examples for practical applications of adaptive control are reported from paper industry, e . g . for the moisture control ( Sikora et a l . 1 984 ) and from the broad field of thermal processes, wherein adaptive systems have been installed success­ fully in phosphate drying ( e . g . Dahhou et al . 1 98 1 , 1 982 ) , i n rotary dryers, glass furnaces (Haber 1980), heating plants (Dexter 1 98 1 ) and air heating sys­ tems.
Only a few applications of adaptive control are in the fields of cement industry and mine ral processes as well as steel indus try and metal lurgical proces­ ses . An interesting example deals with adaptive con­ trol of strip temperature for the continuous anneal­ ing and processing line (Yui and Sato , 1 984 ) .
In power systems a few c learly defined singular problems as e . g . compensation of reactive power ( Zohdy et al . 1 983 ) , control of synchronous gene­ rator ( Hanus 1 983 ) and of turbogenerators (Bonami and Guth , 1 980; Hahn et a l . 1 98 2 , 1 983 ) , power net­ work control (Irving 1 980a , b) , power plant boi ler (Amin et a l . 1 9 84) , nuclear (Mehra et al . 1980 ; Al­ lidina et a l . 1 98 1 ) and hydro power control (Glatt­ felder and Schaufelberger , 1 980) have been solved by adaptive control schemes.
Applications of adaptive control systems have also been reported from the field of e lectromechanical devices and position control such as e . g . a radio telescope (Haque and Monopoli , 1 98 0) . Various papers report on successful implementation of adaptive con­ trol schemes for ship s teering and manoeuvring ( e . g . van Amerongen 1 980, 1 98 1 , 1 982 ) . Also in combus tion engines and compressors adaptive control systems seem to be very advantageous for an economic opera­ tion .
Interesting applications of adaptive systems had been made in misce llaneous areas such as sugar in­ dustry (Behar and Infante , 1 983 ) , r iver regulation (Fjeld et a l . 1 98 3 ) , film thickness control (Makila
Theory and Application of Adaptive Control 9 and Syrj anen, 1 983 ) , control of drug infusion rate (Kaufmann et al . , 1984 ) and bacterial growth (Dochain and Bastin, 1 984 ) .
This very brief discussion shows the surprisingly broad area of applications of adaptive control sys­ tems. Obviously more heuristic ad hoc solutions are becoming rare , whereas most applications are based today on well established approaches of modern adap­ tive control theory. In special cases, however , the theoretical standard approaches have to be slight­ ly modified to overcome special demands of the prob­ lem .
It should also be mentioned that adaptive control schemes can not yet be applied routinely by an in­ experienced engineer . A lot of design specifications including as much as possible 11a-priori11-knowledge about the process must still be regarded . In addi­ tion, various practical aspects for implementation of adaptive control schemes, including e . g . robust­ ness, signal conditioning , parameter tracking , esti­ mator wind-up , reset action, start-up etc . have to be considered by the user ( see e . g . Wittenmark and Astrom, 1 984 ; Goh and Bunn, 1 984 ) .
CONCLUSIONS
Adaptive control theory has reached today a high degree of maturity . A lot of powerful design me­ thods are available now for the experienced control engineer , including also computer-aided design packages for adaptive controllers (Schmid 1985 ) . The numerous applications of adaptive control systems in a broad area of technical fields, discussed in the previous section , indicate that adaptive control can be successfully used in many situations. How­ ever, the inexperienced user still cannot apply a­ daptive control schemes, routinely, because adaptive control structures offer usually a great number of inherent degrees of freedom . In order to make adap­ tive control still more accessible to many control engineers, it will be necessary to reduce the de­ grees of freedom by providing appropriate elements to simplify the tuning of adaptive controllers. For achieving this, further efforts both in practice and theory have to be undertaken.
Acknowledgment. This work was partially supported by research grant Un 25/2 1 from the DFG ( German re­ search foundation ) . This support is gratefully acknowledged .
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