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Design and Application of Multivariable Self-Tuning Controllers
Copyright © IFAC Idenlifica lion and Syslem Parameler Estimalion 19115. York, UK, 19115
DESIGN AND APPLICATION OF MULTI V ARIABLE SELF-TUNING CONTROLLERS R. Schumann VDO MeJ3- und Regelteclmik GlIlbH, Hackethalstr. 7, D-3000 Hannover 1, Federal Republic of Germany
Abstract. Various alternatives for the explicit design of multivariable parameteradaptive controllers are outlined and discussed with respect to theoretical properties and application feasability, It is shown that it is possible to derive from the theoretical convergence analysis of the parameter-adaptive control scheme convergence checks which are useful in practical applications and in addition applicability and realizability conditions for the used control algorithm identical to those gained from the application feasability discussion , Keywords.
Multivariable control systems, parameter estimation, adaptive control,
INTRODUCTION Parameter-adaptive controllers, also called selftuning controllers, have found increasing interest in theory and praxis, which is at least partially due to their simple construction : the basic control loop is superimposed by a second information feedback loop in which process model parameters are estimated from process input and output signal measurements in a first step (process identification), and utilized in a second step to calculate the parameters of the control algorithm (controller design), see Fig, 1.
I I
I
.7 (ONTROLLER DESIGN
L ___
,--'-
I l1d W !
lr
I
11
r-
PARAMETER ESTIMATION
I 1-=
---_.- 1--------
(ONTROL ALGORITHM
f-------
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PROCESS
jJ(k)
I
J I
- the properties of the identified process model - the parameter estimation algorithm - the a-priori information to be provided like process model structure, sample time and memory parameter and for the control algorithm on - the tuning facilities - the applicability problem (closed loop stability) - the realizability problem (invertibility problems) - the sample time dependency of the control behaviour
r---------------j
I
will be regarded with respect to theoretical and practical properties : the interest is focus sed for process identification on
The properties of the resulting p.a. control loop are discussed with respect to - the a-priori information to be provided in all - stability and convergence of the p.a. loop - computational efforts
I
~(k'
I
L _________ ___ __ ~
Fig. 1. Parameter-adaptive control loop A survey on the actual state of development and application of parameter-adaptive controllers can be found in ~str8m (1981) or Isermann (1985). It is obvious that the main interest is focus sed on the basic linear singlevariable case and only few authors consider the non linear or multivariable case . In this paper the design of multivariable parameter-adaptive controllers will be regarded. The design will be based on two different linear multivariable models
and the consequences thereof for the selection of appropriate combinations of parameter estimation and control algorithms. It is shown that from practical design considerations and the theoretical convergence analysis supervisory functions can be derived which enable monitoring of the convergence of parameter estimation and stability of the p . a. closed loop and will thus support safe operation. The design components for p.a. state and matrix polynomial controllers have been described in detail in Schumann (1979a, 1979b, 1982), such that their description in the next sections will be restricted to the outline of principal properties as the basis for the foll~wing discussion. PARAMETER-ADAPTIVE STATE CONTROLLERS
- the multi v ariable state space model - the matrix-polynomial model
The basic structure of a p.a. state controller is shown in Fig. 2
For each of the resulting two classes of p.a. controllers the design components, i.e. process identification procedure and control algorithm,
379
R. Schumann
1------
-----l
I I
PARAMETER ESTIMATION
CONTROLLER DESIGN
I I L_
I I I I
:~&..O
A.8.0~!! •. !!,
- the sample time which has to be chosen compromising between the different· internal dynamics of the process to be identified, taking into account the numerical side-conditions for the
applied algorithms, - the model structure parameters which are deter-
minable for the minimal I/O model and the p-canonical I/O model intuitively, -
the memory parameter for the parameter estimation
which controls the weighting of the actual in relation to past measurements.
Parameter estimation is done by the following wellknown algori thm
~kll
I
PROCESS
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I [Q~:~~~~M
I
~. (k) =8. (k-1) + P (k) ';" (k) [Y. (k) - ,;.T(k) 8. (k-1)] -1 -1 ~1 1 ~1 -1
-1
~T(k) P (k-1)
L ____ ____________ ~
1
Fig. 2. Parameter-adaptive state controller
i
with
Process Model
]/A .
-1
P. (k-1) ~. (k)
-1
..::t:1
=
l) ... ,r
(k)
1
(3)
The design of these p. a. controllers is based on the use of the discrete time state space model which is called in the deterministic case RLS (re-
A x(k) + B u(k) + D v(k)
x Ik+1) y(k)
cursive least squares) and in the stochasic case
C x(k) + v(k)
(1)
with dim
[~(k)]
n; dim
[~(k)]
dim [y(k)] dim [.v.(k)]
=
RELS (recursive extended least squares), compare e.g. Isermann (1981); the vectors (k) contain the estimated model parameters , ~. (k) m~asurements of the input and output signals ~nd in the stochastic case also as estimates for the noise signals the
e.
a posteriori errors •T
r
•
ei(k) = Yi(k) - .i (k) ~i (k)
i
=
1, ... ,r (4)
where x is the state vector, u the input, y the
output vector and v the white-noise vector-which is set to zero in the deterministic case.
A principal problem for the design of the p.a. state controller is the identification of a general state space model from input/output measurements as this would require the simultaneous estimation of parameters and states, i. e. the solution of a nonlinear estimation problem. It is,
however, possible to avoid this difficulty by the use of an observable canonical state space model
in which the state vector can be replaced using input and output signal values resulting in the so-called minimal I/O model or p-canonica T/O model, see Ackermann (1972). These I/O mode~s have the principal structure (linear in the parameters
to be estimated) y. (k) = 4.T(k)
-1
-1
e.
-1
The memory parameter A. (k) balances the influence of actual and past mea~urements thus generating a fading memory for A. (k) < 1. In order to ease the choice of the sampl~ time, the implementation of the parameter estimation algorithm should be done in a
numerically optimized form, e.g.
in square
root filter form. The parameters of the associated observable canonic state space model contained in the model matrices ~,
Band D can be calculated straightforwardly from the estimated I/O model parameters, see Schumann (1979a) . For the estimation of the state vector x(k) the elimination of the states for the obser~able canonical state space model can be inverted resulting in a static state reconstruction procedure, which
+ v . (k)
i
1
=
1, ... ,r
(2)
where e. is the vector of the process parameter ·and 4 .(~) contains actual and past input and output ;Ignal values and in the stochastic case also noise values. The model structure parameters for all these models
are directly related to the observability structure of the state space model and can therefore be estimated intuitively : so the structure parameters of the p-canonical I/O model are the orders of the subsystems observable by the single process outputs separately, and the structure parameters of the observable canonic state space model and also the minimal I/O model are determined by the subsystem orders which become observable in addition by adding hypothetically one output after the other for observation of the process. Identification of State Space Model A priori several parameters have to be provided before the parameter estimation can start
avoids possible problems with additional observer dynamics in the p. a. control loop. The estimation of the steady-state constants C enables the introduction of reference values (W(k» in the closed loop by reconstructing the states relatively to the corresponding steady state values of the process input (U (k» and output (= reference) signales which are rel~ed via the steady state constants and the process model parameters. In the stochastic case the noise signals v(k) have to be reconstructed for the control signal calculation using the output prediction of the estimated state space model. State Control Algorithms The principal task for the design of a state control algorithm in the p.a. control loop is the determination of the feedback matrices K and K , which are used for the control signal cal~ulatioX via u(k)
K
-x
(k) x(k) + K (k) v(k) -v
(5 )
Multivariahle Self-;(uning" COl1trollers Several state controllers are outlined in the fol lowinq and discussed with respect to possible use in the p. a. control loop.
381
TABLE 1: Application Properties of Sta te Controllers Con
Linear Quadratic State Controllers. The design of these control algorithms is based on the minimization of n T T I(n) = diEI [x (i) 9. ~(i) + u (i-I) ~ ~(i-ll ]](6) o singlestep controllers (stochastic MVS, deterministic MARll For n=l singlestep controllers result whose parameters are directly calculated from the state space model parameters. The tuning of the co ntrol behaviour is done by selecting the matrices Q and R thus balancing the weights on process inputs, st~tes and/or outputs. For special choices of Q and R (e. g. small R) applicability problems may result and the control behaviour depends strongly upon the chosen sample time. Examples for this kind of controllers are the state space minimum variance controller and also a special decoupling controller.
Single-j step
t
r
o Pole Ass. PA
Multistep
~~::::~ ~:::~~~ ========
e r Decoupling DS
========
Tuning indirect indirect Facilities Sample small strong strong i small Time Dep. Applicabili ty to I ....... . . . ........ ... ···i········ Unstable Progood I good good good cess
Process-wlth-- ------- -------problem. good
Unst. Inverse Computation Effort (DS=1)
good
problem.
4
PARAMETER-ADAPTIVE MATRIX POLYNOMIAL CONTROLLERS The basic structure of a p. a. matrix polynomial controller is shown i n Fig. 3
o multistep controllers (stochastic MARS, determini stic MAR)
,------ ---------- --l
For n - > inf. the steady-state solution of the Matrix-Riccati-equation mu st be calculated with considerable effort, however, this pays as the applicability problems and the influence of the sample time on the control behaviour are reduced considerably thus easing the tuning of these controllers.
I I
Pole Assignment Controller (PA) . Ackermann (1972) proposed the design of a pole placement co ntro ll er from a general controllable state space model on condition that the contro llability indices are known. The design requires the solution of a generalized system of linear equations with considerable computational effort. The tuning of the control behaviour is done by selecting the poles of the closed loop system which influence only the behaviour of the process states and outputs directly whereas the behaviour of the process inputs can only be predicted indirectly from the process model which, however, is unknown a priori in the p. a. case. The need to provide a priori the controllability indices complicates the application of this controller in addition. Decoupling Controller (DS) . The design of this controller is well-known at least since Falb and Wolovich (1967). It can be designed in the discrete ti me caSe straightforward l y from a general state space model, however, requires for realization the invertibility of the so-called decouplability matrix, see Schuma nn (1980). In addition the control algorithm is not applicable to processes with unstable inverse as this would yield unstable process input signals. The control behaviour can be tuned by specifying the decoupled c l osed loop system behaviour, however, only the behaviour of the decoupled process outputs can be infl ue n ced directly whereas the behaviour of the control signaE depends in addition upon the (a priori unknown) process dynamics and can therefore be influenced only indirectly; furthermore, the control behaviour is strongly depending on the sample time. The practical properties of the state controllers outlined above are summarized in Table 1. It is obvious that adventages with respect to tuning simplicity, applicability and realizability must be paid by additional computational effort.
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-
- - - -. - . -
U
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L __ _ ______ ._____ ___ __ .J Fig. 3. Para m.- adaptive matrix polyno mia l controller Process Model The matrix polynomial model can be described in the discrete time case by
~(q-1) Z(k) = ~(q-1) ~(k)
+ .Q(q-1 ) ::(k)
(7)
where again :: is set to zero in the deterministic case. Only an upper limit for the model order m can be derived from the observability structure of the process (i.e. m is less or equal the maximum subsystem order observable by a single process output). The p-canonical I/O model is a special case_9f the matrix polynomial model with a diagonal A(q ) matrix for which the structure parameters (as already mentioned above) can be determined i ntuitiv ely. Identification of Matrix Polynomial Model For the identification of the matrix polynomial model a-priori the following parameters have to be provided -
the sample time, see above
-
the model structure parameter which are determinable intuitively for the matrix polynomial model (upper limit) or the p-canonical model.
Parameter estimation can then be done for the mat rix polynomial mode l or the p-canonical model by the algori thm Eq. (3).
R. Schumann In case of general matrix polynomial model s the recursion of the ~i matrix must only be done once for all subsystems as the measurement vectors
are identical i " this represents a considerable computational advantage. Also for the matrix polynomial model the steady state relations can be identified by steady-state
of these contro llers is restricted to stable processes with stable inversesi
furthermore realizabi-
lity problems may arise for processes with internal deadtimes, see Schumann (1980). As the decoupling controller is in principal a special singlestep controller, its control behaviour depends strongly upon the sample time.
constants C associated with each subsystem thus en-
The practical properties of the matrix polynomial
abling the-calculation of the control signals always related to the process input and output ( = reference) signals.
controllers outlined above are summarized in
In the stochastic case the noise signals v(k) have to be reconstructed for the control signal calcu lation using the output prediction of the estimated matrix polynomial model.
Table 2. In general they require a small computationa l effort, however, their application in a p.a. control loop is complicated by applicability and tuning problems. TABLE 2: Application Properties of Matrix Polynomial Controllers
Matrix Polynomial Control Algorithm
Con Singlestep
The principal task for the design of a matrix polynomial control algorithm in the p. a. control loop is the dete:~inatio~lof the fe:9back matrix polynomials p(q ), Q(q ) and R(q ), which are used for the ~ontrol signal calc~lation via
Q _(q
-1
)
[~(k)
- y(k)
1+
R -1 _(q )
v(k)
t
=~~;;:,,\=~~== Tuning Fadli ties Sample Time Dep. f\pplicabili ty to
r 0 1 Deadbeat MDB
1 e r Decoupling DM
direct
restricted
indirect
strong
considerable
strong
(8)
Several matrix polynomial controllers are outlined in the following and discussed with respect to possible use in the p.a. control loop.
Unstab le Process
Linear Quadratic controllers (determistic single
Computation Effort (DS=l)
step controller Ml;stochastic minimum variance
Process-wttti-Unst. Inverse
problem.
bad
bad
problem.
good
bad
0.1
0.25
0.25
controller MMV). For the matrix polynomial model only the single_ step case will be considered as the
PARAMETER-ADAPTIVE CONTROL LOOP
general case would require the factorization of
Stabi lity a nd Convergence Properties
matrix polynomials with intolerable computational effort. So the design of these controllers will be based on the minimization of the criterion I(1)
= E
[/(1) ~
y(1)
+
~T(O) ~ ~(O) 1
(9)
The parameters of these single ~step co ntroll ers can be determined directly from the process model parameters with small computational effort. The tuning is done by selecting the matrices 5 and R thus balancing the weight of process inputs and-outputs directly. For special choices of § and ~ also in this case applicability problems may result with respect to the application of these controllers to process models with unstable inverse a nd also for unstable process models. Realizability problems may result for process models with internal deadtimes. The control behaviour of these single step controllers is again strongly depending upon the sample time. Matrix Polynomial Deadbeat Controllers (MDB). These controllers are designed to achieve finite time settling of the process inputs and outputs after a step change e . g. of the reference signals. As in the single-variable case the calculation of the co n troller parameters can be done direc~tly from the process model parameters . The tuning, however, is only possible in limits by increasing the finite settling time artificially. As these controllers are cancel li ng the process poles application is restricted to stable processes. The finite timesettling feature causes the control behaviour to depend considerably on the sample time. Matrix Polynomial Decoupling Controllers (DM). These controllers compensate the process dynamics such that the closed loop dynamics are decoupled with respect to the reference signals. The calcu lation of the controller parameters is done directly from the process model parameters, the tuning facilities are restricted to the prescription of the closed loop singlevariable dynamics. As the process dynamics are compensated, application
Following the analysis method of de Larminat (1979. 1980) in Schumann (1982) the convergence properties of the p. a. control loop have been examined. Th e basic approac h is characterized by the analysis of the p. ao control loop under general assumptions about the convergence properties of the parameter estimation method and applicability conditions for the control algorithm. These general assumptions can be used for the selection of appropriate design components and enable the separate analysis of parameter estimation method and control algo rithm.Al ternative approaches as in Egardt (1979) or Goodwin et al. (1980, 1981) regard the parameter -adap tive control loop only for special classes of control algorithms and require for the analysis comp l ete information about the control algorithm. In the following the results of the converge nce analysis are stated for the deterministic case and a linear process with constant parameters which can be transferred to the stochastic case under some additional assumptions.
Convergence Properties of the RLS Method. Under the assumption that the memory parameter A. (k) is appropriately c h osen such that the matrik P. (k) remains positive definite (whi ch is always-the case for A. (k) = 1 » the following convergence proper ties 2an be stated o The a-posteriori-errors are bounded for all k and converge to
o
lim e. (k) k- > inf~
1, ... ,r
o The parameter estimates are bounded for all k and converge to final values, i. e. lim (~i k->inf.
(k)
-
~i
(k
-
1»
=
Q.
i
=
1
J"
•
,r
In case that th e process is excited sufficlent -
ly the parameter estimates may converge even to the true parameters.
M lIll i\'ariablc Sclf-:lUnillg COIlI rollns Applicability and Realizability Conditlon s for the Control Algorithm. It is assumed that for all k the applied control algorithm is applicable to the estimated process model (i.e. yie lds a stable closed loop with the mo del) and is realizab le. This will guarantee the following two properties o for bounded parameter estimates only bounded control signals will be generated o for k-> inf. the closed model loop formed with the estimated process model and the thereof calculated controller is stable. Convergence and Stability of the P.A.Control Loop The convergence analysis is based on the analysis of the closed model loop in which the model signals are replaced by the signals of the real p. a. control loop and correction term based on the a-posteriori error. The following convergence results can be stated o The signals of the p. a. control loop are always bounded and converge to zero (for the regarded steady state). o The p.a. control loop converges to an asymptotically stable system. In case that the process is excited sufficientlv the control behaviour of the p.a. controller may even converge to the behaviour of the true controller. Applications Aspects The Tuning Problem. One basic motivation of the design of self tuning controllenwas the wish to simplify the tuning of control loops during the start-up phase of a proce ss by reduction of the number of parameters to be provided and automation of the tuning procedure in that e.g. the PID parameters are determined automatically. Looking through the table of a-priori parameters which must be provided for a p.a. controller (which is only a typical example for so-called self tuning co ntrollers), i. e. for process identific atio n - the sample time with respect to the process dynamics and numerical propert ies of the parameter estimation algorithm. -
the process model structure parameters with
respect to the model complexity required for a meaningful controller design, - the memory parameter influencing the stability of parameter estimation. and for the control algorithm - possibly additional a-priori knowledge about the process model structure (e.g. for the pole assignment state controller) - the tuning parameter on the basis of rudimentary knowledge about the process model (problematic for all controllers with direct tuning facilities for process outputs only) and possibly complicated by the sample time dependent control behaviour it becomes obvious that without careful selection of the design components of the p.a. controller and additional measures for the reduction of the apriori parameters to be provided the so-called self tuning controllers can only be tuned by specially trained people familiar with terms like process model structure. memory parameter etc. and in general with more tuning effort than a conventional controller. This is especially annoying i n th e singlevariable case if the goal is to simplif y the tuning of PID controllers or even to replace these controllers by self tuners . In the multivariab le case the tuning efforts for the p. a. contro l l e r is at least partially paid as the design and tuning of a conventional multivar iable control system requires a lot more tuning effort and ca n only be done by specialists for whome t he p.a. controller may become a valuable tuning aid. IS VOL I-N
Ho~e ver , i n order to simplify the tuning of p.a . controllers the following measures should be taken
in general o reduction of the necessary a-priori information
by computation, see Schumann et al.
( 1981 ) ,e.g.by
- automatic determination of the optimal model structure parameters
- automatic determination of the optimal sample time o use of sample time independent methods and algorithm. i. e. - numerically improved parameter estimation algorithms - no sample time dependent singlestep or deadbeat controllers o use of controllers with acceptable handling properties i. e. with - direct and transparent tuning facilities balancing directly control and process output signals - no or only few applicability and realizability problems The Stability Problem. In case the p.a. controller is not only used as a tuning aid under supervision of the engineer or technician but for the control of the process with time or load dependend dynamic and static behaviour, the longtime stability problem arises which is influenced by the disturbances acting on the process, the speed of changes in the process dynamics and the appropriate choice of the memory parameter which in this case is the tuning parameter for the balance of adaption velocity versus stability of parameter estimation. As the convergence and stability analysis outlined above is valid for a linear time invariant process nothing can be concluded directly for the stability of the p .a.co ntrol loop in the general case, however, the following measures can be derived in general to supervise stable operation of the p.a. control loop o monitoring of stability of parameter estimation - condition of the matrix ~i (k): The memory parameter A. (k) should be controlled such that P. (k) 'remains well conditioned, compare e~g. Fortescue et al.
(1981)
- convergence of a-posteriori error: The variations of e (k) indicate changes in the linear i process model parameters and can be evaluated e.g. for the implementation of additional mechanisms in the adaptation loop like controlled excitation of the process by an additional test signal, control of the transition of the process model parameters to the controller parameter calculation etc.
o mo nitoring of applicability and realizability c o nditions for the calculated controller; this task is considerably simplified by the choice of controllers which yield in principal no or few problems of that kind like the MAR od PA controller . These and additional supervisory functions must be implemented to ensure longtime stability of the p. a. control loop in the general case. Together with the tuning support functions for the automatic determination of a-priori parameters they
form a third functional lev el , the supervision and coordination level as proposed in Schumann
etal.
(1981),
APPLICATIONS The multivariable self tuning controllers as outlined above have been applied to a number of simulated and real processes.Due to the lack of space
~H4
R. Schumann
in this paper the results of these applications can only be cited and summarized . The application of the self tuning matrix polynomial deadbeat controller and the self tuning linear quadratic state controller to an air-conditioning pilot plant was described in Schumann (1982a). The plant is nonlinear with respect to the bilinear cross-flow type heat exchange process and the water spray diffusion process . It was demonstrated that the applied self tuning controllers could stabilize the process after a short adaptation phase and could achieve satisfactory control performance for time varying air flows at different setpoints for temperature and relative humidity. However, for practical applications, it was concluded that the control performance should be improved especially with respect to the adaption variations of the control signal by measures like predetermination of a rough system model by open-loop identification or on-line modifications of sample time or tuning parameters . In addition in Schumann (1982) the application of the multivariable self tuning controllers to a variety of analog simulated processes was demonstrated including the model of double effect evaporator and the model of a steam generator/superheater process. It was shown that in all applications the self tuning multi variable controller could adapt to the process dynamics very quickly yielding stable control after a short adaption phase.
CONCLUSIONS The design of multivariable p.a. controllers has been based fora long time (as in the singlevariable case) on computational considerations which resulted in the use of singlestep controllers of the minimum variance type or deadbeat controllers, compare e.g. Borisson (1976), Keviczky and Hetthessy (1977), . Koivo (1980). However, the handling of these controllers has been made difficult by applicability and realizability problems, the sample time depending control behaviour and in some case also the limited or indirect tuning facilities available. Today, with improved computational facilities available in form of powerful microcomputers, the handling properties, which are the prerequisite for the use of these controllers in practice, should be improved in general by reduction of the necessary apriori information with computational effort, use of sample time independent methods and algorithms and use of control algorithms with acceptable handling properties . Furthermore, additional on-line supervisory functions for parameter estimation and controller design should be implemented in order to guarantee longtime stability of the p.a. control loop . Only with these additional features p.a. controllers will find their way to widespread practical applications as comfortable tuning devices and selfoptimizing adaptive controllers.
Falb, P. L. and W. A. Wolovich (1967). Decoupling in the design and synthesis of multivarialble control systems. IEEE Trans. Autom. Contr., Vol. AC-12, No. 6, pp. 651 - 659 . Fortescue, T. R., L. S. Kershenbaum and B.E.Ydstie (1981) . Implementation of self tuning regulators with variable forgetting factors. Automatica 17, No. 6, pp.831 - 835. Goodwin, G. C . , P. J . Ramadge and P.E. Caines (1980) . Discrete-time multivariable adaptive control. IEEE Trans . Autom. Contr., Vol. AC-25, No . 3, pp. 449 - 456 . Goodwin, G. C., P. J. Ramadge and P.E.Caines (1981). Discrete-time stochastic adaptive control. SIAM J. Control Optimiz., Vol . 19 . Isermann, R. (1981). Digital Control Systems. Springer-Verlag, Berlin. Isermann, R. (1985). Parameter-adaptive control systems - A review on methods and applications . 6. IFAC Symposium on Identification and System Parameter Estimation, York. Keviczky , L. and J. Hetthessy (1977). Self-tuning minimum variance control of MIMO discrete time systems. Autom . Control Theory & Appl., Vol. 5, Nr. 1, pp . 11 - 17. Koivo, H. N. (1980) . A multivariable self-tuning controller. Automatica, Vol. 16, No. 4, pp. 351 - 366. de Larminat, Ph. (1979). On overall stability of certain adaptive control systems. 5. IFAC Symposium on Identification and System Parameter Estimation, Darmstadt . de Larminat , Ph. (1980). Unconditional stabilizers for nonminimum phase systems . Methods and Applications in Adaptive Control. Lecture Notes in Control and Information Sciences, Nr. 24, Springer-Verlag Berlin. Schumann, R, (1979a). Identification and adaptive control of multivariable stochastic linear systems. 5. IFAC Symposium on Identification and System Parameter Estimation, Darmstadt. Schumann , R. (1979b). Various multivariable computer control algorithms for parameter-adaptive control systems. IFAC Symposium on Computer Aided Design of Control Systems, Zurich. Schumann, R. (1980). Decomposition and decoupling of linear multi variable systems with distributed time delays. 3. lMA Conference on Control Theory, Sheffield. Schumann, R. (1982). Digitale parameteradaptive Mehrgr6Benregelung. - Ein Beitrag zu Entwurf und Analyse. Dissertation, TH Darmstadt, FB 19. Published as PDV-Report KfK-PDV 217, Kernforschungszentrum Karlsruhe. Schumann, R. (1982a). Digital parameter-adaptive control of an air-conditioning plant. Automatica, Vol. 18, No. 5, pp. 569 - 575.
REFERENCES Ackermann, J. (1972) . Abtastregelung. Springer-Verlag, Berlin. ~str5m, K. J .
(1981). Theory and application of adaptive control . Proc. IFAC World Congress, Ky oto, Japan, Pergamon Press, Oxford.
Borisson, U. (1976) . Self tuning regulators for a class o f multivariable s y stems. 4 . IFAC Symposium on Identification and System Parameter Estimation, Tblisi. Egardt, B. (1979). Stability of adaptive controllers. Lecture notes in Control and Information Sciences, No. 2 0, Springer-Verlag, Berlin.
Schumann, R., K. H. Lachmann and R. Isermann (1981). Towards applicability of parameter-adaptive controllers . 8. IFAC Congress, Kyoto.
DESIGN AND APPLICATION OF MULTI V ARIABLE SELF-TUNING CONTROLLERS R. Schumann VDO MeJ3- und Regelteclmik GlIlbH, Hackethalstr. 7, D-3000 Hannover 1, Federal Republic of Germany
Abstract. Various alternatives for the explicit design of multivariable parameteradaptive controllers are outlined and discussed with respect to theoretical properties and application feasability, It is shown that it is possible to derive from the theoretical convergence analysis of the parameter-adaptive control scheme convergence checks which are useful in practical applications and in addition applicability and realizability conditions for the used control algorithm identical to those gained from the application feasability discussion , Keywords.
Multivariable control systems, parameter estimation, adaptive control,
INTRODUCTION Parameter-adaptive controllers, also called selftuning controllers, have found increasing interest in theory and praxis, which is at least partially due to their simple construction : the basic control loop is superimposed by a second information feedback loop in which process model parameters are estimated from process input and output signal measurements in a first step (process identification), and utilized in a second step to calculate the parameters of the control algorithm (controller design), see Fig, 1.
I I
I
.7 (ONTROLLER DESIGN
L ___
,--'-
I l1d W !
lr
I
11
r-
PARAMETER ESTIMATION
I 1-=
---_.- 1--------
(ONTROL ALGORITHM
f-------
I="=-
PROCESS
jJ(k)
I
J I
- the properties of the identified process model - the parameter estimation algorithm - the a-priori information to be provided like process model structure, sample time and memory parameter and for the control algorithm on - the tuning facilities - the applicability problem (closed loop stability) - the realizability problem (invertibility problems) - the sample time dependency of the control behaviour
r---------------j
I
will be regarded with respect to theoretical and practical properties : the interest is focus sed for process identification on
The properties of the resulting p.a. control loop are discussed with respect to - the a-priori information to be provided in all - stability and convergence of the p.a. loop - computational efforts
I
~(k'
I
L _________ ___ __ ~
Fig. 1. Parameter-adaptive control loop A survey on the actual state of development and application of parameter-adaptive controllers can be found in ~str8m (1981) or Isermann (1985). It is obvious that the main interest is focus sed on the basic linear singlevariable case and only few authors consider the non linear or multivariable case . In this paper the design of multivariable parameter-adaptive controllers will be regarded. The design will be based on two different linear multivariable models
and the consequences thereof for the selection of appropriate combinations of parameter estimation and control algorithms. It is shown that from practical design considerations and the theoretical convergence analysis supervisory functions can be derived which enable monitoring of the convergence of parameter estimation and stability of the p . a. closed loop and will thus support safe operation. The design components for p.a. state and matrix polynomial controllers have been described in detail in Schumann (1979a, 1979b, 1982), such that their description in the next sections will be restricted to the outline of principal properties as the basis for the foll~wing discussion. PARAMETER-ADAPTIVE STATE CONTROLLERS
- the multi v ariable state space model - the matrix-polynomial model
The basic structure of a p.a. state controller is shown in Fig. 2
For each of the resulting two classes of p.a. controllers the design components, i.e. process identification procedure and control algorithm,
379
R. Schumann
1------
-----l
I I
PARAMETER ESTIMATION
CONTROLLER DESIGN
I I L_
I I I I
:~&..O
A.8.0~!! •. !!,
- the sample time which has to be chosen compromising between the different· internal dynamics of the process to be identified, taking into account the numerical side-conditions for the
applied algorithms, - the model structure parameters which are deter-
minable for the minimal I/O model and the p-canonical I/O model intuitively, -
the memory parameter for the parameter estimation
which controls the weighting of the actual in relation to past measurements.
Parameter estimation is done by the following wellknown algori thm
~kll
I
PROCESS
!!Ikl
I [Q~:~~~~M
I
~. (k) =8. (k-1) + P (k) ';" (k) [Y. (k) - ,;.T(k) 8. (k-1)] -1 -1 ~1 1 ~1 -1
-1
~T(k) P (k-1)
L ____ ____________ ~
1
Fig. 2. Parameter-adaptive state controller
i
with
Process Model
]/A .
-1
P. (k-1) ~. (k)
-1
..::t:1
=
l) ... ,r
(k)
1
(3)
The design of these p. a. controllers is based on the use of the discrete time state space model which is called in the deterministic case RLS (re-
A x(k) + B u(k) + D v(k)
x Ik+1) y(k)
cursive least squares) and in the stochasic case
C x(k) + v(k)
(1)
with dim
[~(k)]
n; dim
[~(k)]
dim [y(k)] dim [.v.(k)]
=
RELS (recursive extended least squares), compare e.g. Isermann (1981); the vectors (k) contain the estimated model parameters , ~. (k) m~asurements of the input and output signals ~nd in the stochastic case also as estimates for the noise signals the
e.
a posteriori errors •T
r
•
ei(k) = Yi(k) - .i (k) ~i (k)
i
=
1, ... ,r (4)
where x is the state vector, u the input, y the
output vector and v the white-noise vector-which is set to zero in the deterministic case.
A principal problem for the design of the p.a. state controller is the identification of a general state space model from input/output measurements as this would require the simultaneous estimation of parameters and states, i. e. the solution of a nonlinear estimation problem. It is,
however, possible to avoid this difficulty by the use of an observable canonical state space model
in which the state vector can be replaced using input and output signal values resulting in the so-called minimal I/O model or p-canonica T/O model, see Ackermann (1972). These I/O mode~s have the principal structure (linear in the parameters
to be estimated) y. (k) = 4.T(k)
-1
-1
e.
-1
The memory parameter A. (k) balances the influence of actual and past mea~urements thus generating a fading memory for A. (k) < 1. In order to ease the choice of the sampl~ time, the implementation of the parameter estimation algorithm should be done in a
numerically optimized form, e.g.
in square
root filter form. The parameters of the associated observable canonic state space model contained in the model matrices ~,
Band D can be calculated straightforwardly from the estimated I/O model parameters, see Schumann (1979a) . For the estimation of the state vector x(k) the elimination of the states for the obser~able canonical state space model can be inverted resulting in a static state reconstruction procedure, which
+ v . (k)
i
1
=
1, ... ,r
(2)
where e. is the vector of the process parameter ·and 4 .(~) contains actual and past input and output ;Ignal values and in the stochastic case also noise values. The model structure parameters for all these models
are directly related to the observability structure of the state space model and can therefore be estimated intuitively : so the structure parameters of the p-canonical I/O model are the orders of the subsystems observable by the single process outputs separately, and the structure parameters of the observable canonic state space model and also the minimal I/O model are determined by the subsystem orders which become observable in addition by adding hypothetically one output after the other for observation of the process. Identification of State Space Model A priori several parameters have to be provided before the parameter estimation can start
avoids possible problems with additional observer dynamics in the p. a. control loop. The estimation of the steady-state constants C enables the introduction of reference values (W(k» in the closed loop by reconstructing the states relatively to the corresponding steady state values of the process input (U (k» and output (= reference) signales which are rel~ed via the steady state constants and the process model parameters. In the stochastic case the noise signals v(k) have to be reconstructed for the control signal calculation using the output prediction of the estimated state space model. State Control Algorithms The principal task for the design of a state control algorithm in the p.a. control loop is the determination of the feedback matrices K and K , which are used for the control signal cal~ulatioX via u(k)
K
-x
(k) x(k) + K (k) v(k) -v
(5 )
Multivariahle Self-;(uning" COl1trollers Several state controllers are outlined in the fol lowinq and discussed with respect to possible use in the p. a. control loop.
381
TABLE 1: Application Properties of Sta te Controllers Con
Linear Quadratic State Controllers. The design of these control algorithms is based on the minimization of n T T I(n) = diEI [x (i) 9. ~(i) + u (i-I) ~ ~(i-ll ]](6) o singlestep controllers (stochastic MVS, deterministic MARll For n=l singlestep controllers result whose parameters are directly calculated from the state space model parameters. The tuning of the co ntrol behaviour is done by selecting the matrices Q and R thus balancing the weights on process inputs, st~tes and/or outputs. For special choices of Q and R (e. g. small R) applicability problems may result and the control behaviour depends strongly upon the chosen sample time. Examples for this kind of controllers are the state space minimum variance controller and also a special decoupling controller.
Single-j step
t
r
o Pole Ass. PA
Multistep
~~::::~ ~:::~~~ ========
e r Decoupling DS
========
Tuning indirect indirect Facilities Sample small strong strong i small Time Dep. Applicabili ty to I ....... . . . ........ ... ···i········ Unstable Progood I good good good cess
Process-wlth-- ------- -------problem. good
Unst. Inverse Computation Effort (DS=1)
good
problem.
4
PARAMETER-ADAPTIVE MATRIX POLYNOMIAL CONTROLLERS The basic structure of a p. a. matrix polynomial controller is shown i n Fig. 3
o multistep controllers (stochastic MARS, determini stic MAR)
,------ ---------- --l
For n - > inf. the steady-state solution of the Matrix-Riccati-equation mu st be calculated with considerable effort, however, this pays as the applicability problems and the influence of the sample time on the control behaviour are reduced considerably thus easing the tuning of these controllers.
I I
Pole Assignment Controller (PA) . Ackermann (1972) proposed the design of a pole placement co ntro ll er from a general controllable state space model on condition that the contro llability indices are known. The design requires the solution of a generalized system of linear equations with considerable computational effort. The tuning of the control behaviour is done by selecting the poles of the closed loop system which influence only the behaviour of the process states and outputs directly whereas the behaviour of the process inputs can only be predicted indirectly from the process model which, however, is unknown a priori in the p. a. case. The need to provide a priori the controllability indices complicates the application of this controller in addition. Decoupling Controller (DS) . The design of this controller is well-known at least since Falb and Wolovich (1967). It can be designed in the discrete ti me caSe straightforward l y from a general state space model, however, requires for realization the invertibility of the so-called decouplability matrix, see Schuma nn (1980). In addition the control algorithm is not applicable to processes with unstable inverse as this would yield unstable process input signals. The control behaviour can be tuned by specifying the decoupled c l osed loop system behaviour, however, only the behaviour of the decoupled process outputs can be infl ue n ced directly whereas the behaviour of the control signaE depends in addition upon the (a priori unknown) process dynamics and can therefore be influenced only indirectly; furthermore, the control behaviour is strongly depending on the sample time. The practical properties of the state controllers outlined above are summarized in Table 1. It is obvious that adventages with respect to tuning simplicity, applicability and realizability must be paid by additional computational effort.
I I
1 A" ~,IZ ,1o!
L __
,-1
~Ikll
1
I
Jl
j2 CONTROLLER DESIGN
---
---
-PO R
-AIQ·'I,ficq·'l.OCq·'1
I
-~
-'-'-
r- -- - - r-- - - -. - - - - - t- -
shuly
!Jolkl
stole noise
::§p0lkl
r- ~\f'~~"
I
PARAMETER I ;=- ESTIMATION I-- I
-
- - - -. - . -
U
h I I
e-' B i11 e·' g t-~4ao-1='J.!lblk~l
PROCESS
(ONTROL ALGORITHM
LI = ffiikI
I
I b===_.= .. ======~===~~=======dl
L __ _ ______ ._____ ___ __ .J Fig. 3. Para m.- adaptive matrix polyno mia l controller Process Model The matrix polynomial model can be described in the discrete time case by
~(q-1) Z(k) = ~(q-1) ~(k)
+ .Q(q-1 ) ::(k)
(7)
where again :: is set to zero in the deterministic case. Only an upper limit for the model order m can be derived from the observability structure of the process (i.e. m is less or equal the maximum subsystem order observable by a single process output). The p-canonical I/O model is a special case_9f the matrix polynomial model with a diagonal A(q ) matrix for which the structure parameters (as already mentioned above) can be determined i ntuitiv ely. Identification of Matrix Polynomial Model For the identification of the matrix polynomial model a-priori the following parameters have to be provided -
the sample time, see above
-
the model structure parameter which are determinable intuitively for the matrix polynomial model (upper limit) or the p-canonical model.
Parameter estimation can then be done for the mat rix polynomial mode l or the p-canonical model by the algori thm Eq. (3).
R. Schumann In case of general matrix polynomial model s the recursion of the ~i matrix must only be done once for all subsystems as the measurement vectors
are identical i " this represents a considerable computational advantage. Also for the matrix polynomial model the steady state relations can be identified by steady-state
of these contro llers is restricted to stable processes with stable inversesi
furthermore realizabi-
lity problems may arise for processes with internal deadtimes, see Schumann (1980). As the decoupling controller is in principal a special singlestep controller, its control behaviour depends strongly upon the sample time.
constants C associated with each subsystem thus en-
The practical properties of the matrix polynomial
abling the-calculation of the control signals always related to the process input and output ( = reference) signals.
controllers outlined above are summarized in
In the stochastic case the noise signals v(k) have to be reconstructed for the control signal calcu lation using the output prediction of the estimated matrix polynomial model.
Table 2. In general they require a small computationa l effort, however, their application in a p.a. control loop is complicated by applicability and tuning problems. TABLE 2: Application Properties of Matrix Polynomial Controllers
Matrix Polynomial Control Algorithm
Con Singlestep
The principal task for the design of a matrix polynomial control algorithm in the p. a. control loop is the dete:~inatio~lof the fe:9back matrix polynomials p(q ), Q(q ) and R(q ), which are used for the ~ontrol signal calc~lation via
Q _(q
-1
)
[~(k)
- y(k)
1+
R -1 _(q )
v(k)
t
=~~;;:,,\=~~== Tuning Fadli ties Sample Time Dep. f\pplicabili ty to
r 0 1 Deadbeat MDB
1 e r Decoupling DM
direct
restricted
indirect
strong
considerable
strong
(8)
Several matrix polynomial controllers are outlined in the following and discussed with respect to possible use in the p.a. control loop.
Unstab le Process
Linear Quadratic controllers (determistic single
Computation Effort (DS=l)
step controller Ml;stochastic minimum variance
Process-wttti-Unst. Inverse
problem.
bad
bad
problem.
good
bad
0.1
0.25
0.25
controller MMV). For the matrix polynomial model only the single_ step case will be considered as the
PARAMETER-ADAPTIVE CONTROL LOOP
general case would require the factorization of
Stabi lity a nd Convergence Properties
matrix polynomials with intolerable computational effort. So the design of these controllers will be based on the minimization of the criterion I(1)
= E
[/(1) ~
y(1)
+
~T(O) ~ ~(O) 1
(9)
The parameters of these single ~step co ntroll ers can be determined directly from the process model parameters with small computational effort. The tuning is done by selecting the matrices 5 and R thus balancing the weight of process inputs and-outputs directly. For special choices of § and ~ also in this case applicability problems may result with respect to the application of these controllers to process models with unstable inverse a nd also for unstable process models. Realizability problems may result for process models with internal deadtimes. The control behaviour of these single step controllers is again strongly depending upon the sample time. Matrix Polynomial Deadbeat Controllers (MDB). These controllers are designed to achieve finite time settling of the process inputs and outputs after a step change e . g. of the reference signals. As in the single-variable case the calculation of the co n troller parameters can be done direc~tly from the process model parameters . The tuning, however, is only possible in limits by increasing the finite settling time artificially. As these controllers are cancel li ng the process poles application is restricted to stable processes. The finite timesettling feature causes the control behaviour to depend considerably on the sample time. Matrix Polynomial Decoupling Controllers (DM). These controllers compensate the process dynamics such that the closed loop dynamics are decoupled with respect to the reference signals. The calcu lation of the controller parameters is done directly from the process model parameters, the tuning facilities are restricted to the prescription of the closed loop singlevariable dynamics. As the process dynamics are compensated, application
Following the analysis method of de Larminat (1979. 1980) in Schumann (1982) the convergence properties of the p. a. control loop have been examined. Th e basic approac h is characterized by the analysis of the p. ao control loop under general assumptions about the convergence properties of the parameter estimation method and applicability conditions for the control algorithm. These general assumptions can be used for the selection of appropriate design components and enable the separate analysis of parameter estimation method and control algo rithm.Al ternative approaches as in Egardt (1979) or Goodwin et al. (1980, 1981) regard the parameter -adap tive control loop only for special classes of control algorithms and require for the analysis comp l ete information about the control algorithm. In the following the results of the converge nce analysis are stated for the deterministic case and a linear process with constant parameters which can be transferred to the stochastic case under some additional assumptions.
Convergence Properties of the RLS Method. Under the assumption that the memory parameter A. (k) is appropriately c h osen such that the matrik P. (k) remains positive definite (whi ch is always-the case for A. (k) = 1 » the following convergence proper ties 2an be stated o The a-posteriori-errors are bounded for all k and converge to
o
lim e. (k) k- > inf~
1, ... ,r
o The parameter estimates are bounded for all k and converge to final values, i. e. lim (~i k->inf.
(k)
-
~i
(k
-
1»
=
Q.
i
=
1
J"
•
,r
In case that th e process is excited sufficlent -
ly the parameter estimates may converge even to the true parameters.
M lIll i\'ariablc Sclf-:lUnillg COIlI rollns Applicability and Realizability Conditlon s for the Control Algorithm. It is assumed that for all k the applied control algorithm is applicable to the estimated process model (i.e. yie lds a stable closed loop with the mo del) and is realizab le. This will guarantee the following two properties o for bounded parameter estimates only bounded control signals will be generated o for k-> inf. the closed model loop formed with the estimated process model and the thereof calculated controller is stable. Convergence and Stability of the P.A.Control Loop The convergence analysis is based on the analysis of the closed model loop in which the model signals are replaced by the signals of the real p. a. control loop and correction term based on the a-posteriori error. The following convergence results can be stated o The signals of the p. a. control loop are always bounded and converge to zero (for the regarded steady state). o The p.a. control loop converges to an asymptotically stable system. In case that the process is excited sufficientlv the control behaviour of the p.a. controller may even converge to the behaviour of the true controller. Applications Aspects The Tuning Problem. One basic motivation of the design of self tuning controllenwas the wish to simplify the tuning of control loops during the start-up phase of a proce ss by reduction of the number of parameters to be provided and automation of the tuning procedure in that e.g. the PID parameters are determined automatically. Looking through the table of a-priori parameters which must be provided for a p.a. controller (which is only a typical example for so-called self tuning co ntrollers), i. e. for process identific atio n - the sample time with respect to the process dynamics and numerical propert ies of the parameter estimation algorithm. -
the process model structure parameters with
respect to the model complexity required for a meaningful controller design, - the memory parameter influencing the stability of parameter estimation. and for the control algorithm - possibly additional a-priori knowledge about the process model structure (e.g. for the pole assignment state controller) - the tuning parameter on the basis of rudimentary knowledge about the process model (problematic for all controllers with direct tuning facilities for process outputs only) and possibly complicated by the sample time dependent control behaviour it becomes obvious that without careful selection of the design components of the p.a. controller and additional measures for the reduction of the apriori parameters to be provided the so-called self tuning controllers can only be tuned by specially trained people familiar with terms like process model structure. memory parameter etc. and in general with more tuning effort than a conventional controller. This is especially annoying i n th e singlevariable case if the goal is to simplif y the tuning of PID controllers or even to replace these controllers by self tuners . In the multivariab le case the tuning efforts for the p. a. contro l l e r is at least partially paid as the design and tuning of a conventional multivar iable control system requires a lot more tuning effort and ca n only be done by specialists for whome t he p.a. controller may become a valuable tuning aid. IS VOL I-N
Ho~e ver , i n order to simplify the tuning of p.a . controllers the following measures should be taken
in general o reduction of the necessary a-priori information
by computation, see Schumann et al.
( 1981 ) ,e.g.by
- automatic determination of the optimal model structure parameters
- automatic determination of the optimal sample time o use of sample time independent methods and algorithm. i. e. - numerically improved parameter estimation algorithms - no sample time dependent singlestep or deadbeat controllers o use of controllers with acceptable handling properties i. e. with - direct and transparent tuning facilities balancing directly control and process output signals - no or only few applicability and realizability problems The Stability Problem. In case the p.a. controller is not only used as a tuning aid under supervision of the engineer or technician but for the control of the process with time or load dependend dynamic and static behaviour, the longtime stability problem arises which is influenced by the disturbances acting on the process, the speed of changes in the process dynamics and the appropriate choice of the memory parameter which in this case is the tuning parameter for the balance of adaption velocity versus stability of parameter estimation. As the convergence and stability analysis outlined above is valid for a linear time invariant process nothing can be concluded directly for the stability of the p .a.co ntrol loop in the general case, however, the following measures can be derived in general to supervise stable operation of the p.a. control loop o monitoring of stability of parameter estimation - condition of the matrix ~i (k): The memory parameter A. (k) should be controlled such that P. (k) 'remains well conditioned, compare e~g. Fortescue et al.
(1981)
- convergence of a-posteriori error: The variations of e (k) indicate changes in the linear i process model parameters and can be evaluated e.g. for the implementation of additional mechanisms in the adaptation loop like controlled excitation of the process by an additional test signal, control of the transition of the process model parameters to the controller parameter calculation etc.
o mo nitoring of applicability and realizability c o nditions for the calculated controller; this task is considerably simplified by the choice of controllers which yield in principal no or few problems of that kind like the MAR od PA controller . These and additional supervisory functions must be implemented to ensure longtime stability of the p. a. control loop in the general case. Together with the tuning support functions for the automatic determination of a-priori parameters they
form a third functional lev el , the supervision and coordination level as proposed in Schumann
etal.
(1981),
APPLICATIONS The multivariable self tuning controllers as outlined above have been applied to a number of simulated and real processes.Due to the lack of space
~H4
R. Schumann
in this paper the results of these applications can only be cited and summarized . The application of the self tuning matrix polynomial deadbeat controller and the self tuning linear quadratic state controller to an air-conditioning pilot plant was described in Schumann (1982a). The plant is nonlinear with respect to the bilinear cross-flow type heat exchange process and the water spray diffusion process . It was demonstrated that the applied self tuning controllers could stabilize the process after a short adaptation phase and could achieve satisfactory control performance for time varying air flows at different setpoints for temperature and relative humidity. However, for practical applications, it was concluded that the control performance should be improved especially with respect to the adaption variations of the control signal by measures like predetermination of a rough system model by open-loop identification or on-line modifications of sample time or tuning parameters . In addition in Schumann (1982) the application of the multivariable self tuning controllers to a variety of analog simulated processes was demonstrated including the model of double effect evaporator and the model of a steam generator/superheater process. It was shown that in all applications the self tuning multi variable controller could adapt to the process dynamics very quickly yielding stable control after a short adaption phase.
CONCLUSIONS The design of multivariable p.a. controllers has been based fora long time (as in the singlevariable case) on computational considerations which resulted in the use of singlestep controllers of the minimum variance type or deadbeat controllers, compare e.g. Borisson (1976), Keviczky and Hetthessy (1977), . Koivo (1980). However, the handling of these controllers has been made difficult by applicability and realizability problems, the sample time depending control behaviour and in some case also the limited or indirect tuning facilities available. Today, with improved computational facilities available in form of powerful microcomputers, the handling properties, which are the prerequisite for the use of these controllers in practice, should be improved in general by reduction of the necessary apriori information with computational effort, use of sample time independent methods and algorithms and use of control algorithms with acceptable handling properties . Furthermore, additional on-line supervisory functions for parameter estimation and controller design should be implemented in order to guarantee longtime stability of the p.a. control loop . Only with these additional features p.a. controllers will find their way to widespread practical applications as comfortable tuning devices and selfoptimizing adaptive controllers.
Falb, P. L. and W. A. Wolovich (1967). Decoupling in the design and synthesis of multivarialble control systems. IEEE Trans. Autom. Contr., Vol. AC-12, No. 6, pp. 651 - 659 . Fortescue, T. R., L. S. Kershenbaum and B.E.Ydstie (1981) . Implementation of self tuning regulators with variable forgetting factors. Automatica 17, No. 6, pp.831 - 835. Goodwin, G. C . , P. J . Ramadge and P.E. Caines (1980) . Discrete-time multivariable adaptive control. IEEE Trans . Autom. Contr., Vol. AC-25, No . 3, pp. 449 - 456 . Goodwin, G. C., P. J. Ramadge and P.E.Caines (1981). Discrete-time stochastic adaptive control. SIAM J. Control Optimiz., Vol . 19 . Isermann, R. (1981). Digital Control Systems. Springer-Verlag, Berlin. Isermann, R. (1985). Parameter-adaptive control systems - A review on methods and applications . 6. IFAC Symposium on Identification and System Parameter Estimation, York. Keviczky , L. and J. Hetthessy (1977). Self-tuning minimum variance control of MIMO discrete time systems. Autom . Control Theory & Appl., Vol. 5, Nr. 1, pp . 11 - 17. Koivo, H. N. (1980) . A multivariable self-tuning controller. Automatica, Vol. 16, No. 4, pp. 351 - 366. de Larminat, Ph. (1979). On overall stability of certain adaptive control systems. 5. IFAC Symposium on Identification and System Parameter Estimation, Darmstadt . de Larminat , Ph. (1980). Unconditional stabilizers for nonminimum phase systems . Methods and Applications in Adaptive Control. Lecture Notes in Control and Information Sciences, Nr. 24, Springer-Verlag Berlin. Schumann, R, (1979a). Identification and adaptive control of multivariable stochastic linear systems. 5. IFAC Symposium on Identification and System Parameter Estimation, Darmstadt. Schumann , R. (1979b). Various multivariable computer control algorithms for parameter-adaptive control systems. IFAC Symposium on Computer Aided Design of Control Systems, Zurich. Schumann, R. (1980). Decomposition and decoupling of linear multi variable systems with distributed time delays. 3. lMA Conference on Control Theory, Sheffield. Schumann, R. (1982). Digitale parameteradaptive Mehrgr6Benregelung. - Ein Beitrag zu Entwurf und Analyse. Dissertation, TH Darmstadt, FB 19. Published as PDV-Report KfK-PDV 217, Kernforschungszentrum Karlsruhe. Schumann, R. (1982a). Digital parameter-adaptive control of an air-conditioning plant. Automatica, Vol. 18, No. 5, pp. 569 - 575.
REFERENCES Ackermann, J. (1972) . Abtastregelung. Springer-Verlag, Berlin. ~str5m, K. J .
(1981). Theory and application of adaptive control . Proc. IFAC World Congress, Ky oto, Japan, Pergamon Press, Oxford.
Borisson, U. (1976) . Self tuning regulators for a class o f multivariable s y stems. 4 . IFAC Symposium on Identification and System Parameter Estimation, Tblisi. Egardt, B. (1979). Stability of adaptive controllers. Lecture notes in Control and Information Sciences, No. 2 0, Springer-Verlag, Berlin.
Schumann, R., K. H. Lachmann and R. Isermann (1981). Towards applicability of parameter-adaptive controllers . 8. IFAC Congress, Kyoto.