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Supervision and Coordination of Parameter-Adaptive Controllers
Copyright © IFAC Low Cost Automation 1989 Milan, Italy, 1989
SUPERVISION AND COORDINATION OF PARAMETERADAPTIVE CONTROLLERS T. Knapp, R. Isermann Institute of Automatic Control, Technical University of Oarmstadt Landgraf Georg Strasse 4, 0-6100 Oarmstadt, F.R.G.
Abstract. Adaptive controllers have proven to work well as long as all theoretical pre-conditions are fulfilled and the free design parameters are chosen properly. However in practice these pre-conditions may be violated and/or some free design parameters are not chosen properly . 80th result in a lowering of the control performance. Therefore a supervision and coordination level is introduced, monitoring the adaptive controller and taking appropriate actions if malfunctions are detected. The Introduced supervision and coordination level exists altogether of two parts named start - up part and run-time part. The start-up part involves tasks for configuration and implementation of adaptive controllers and the run-time part involves tasks for supervision the adaptive controller in closed-loop operation. A sensible start - up procedure is recommended and an eigenvalue analysis is shown in order to monitor the parameter estimation. To track time-varying process parameters an eigenvalue controlled forgetting factor is introduced. Simulations and experimental results for an a.c. motor are shown, indicating an improvement of the overall control performance. Keywords . supervision and coordination, adaptive control, parameter estimation, eigenvalue analysis, process model mismatch. time-varying processes
Introduction The successful application of parameter-adaptive controllers requires that all assumptions used for the derivation of the parameter-adaptive control principle. for parameter estimation and controller design are met and some design parameters are chosen properly. The closed-loop systems obtained with adaptive controllers are nonlinear . This makes analysis of stability. convergence and control performance difficult. part iculary if there are random disturbances. see Astrom (1983). During the implementation of adaptive controllers the choice of a suitable sample period is an example . The spectrum of possible values for this parameter is bounded by Shannon 's theorem and the demand for good control performance on the one hand and numerical problems as well as limited computer power on the other hand . Another problem is the choice of a suitable process model. of structural parameters (e .g. process model order. process model dead-time) and of some design parameters in connection with parameter estimation and controller design.
Therefore it ons of the are realized coordination stem.
is necessary to monitor all or several functiparameter - adaptive control loop. These tasks in an additional block named supervision and level. which may also include an expert sy-
The first work in this direction was conducted by Clarke and Gawthrop (1981). Schumann et . al . (1981). Fortescue et. al. (1981). Bergmann (1983). Isermann and Lachmann (1985). A sensible use of expert systems in adaptive control is described in Sanoff and Wellstead (1985) . The aim o f this paper is to show. how a supervision level and a coordination level can be designed and what basic elements should be included. In the first part of this paper the parameter-adaptive control scheme is discussed and basic elements f or the supervision level and coordination level are introduced. In the second part of the paper a possible supervis ion and coordination leve l is presented . Finally a few experimental results demonstrate the work ing method o f the presented supervision level and coordination level.
Parameter-Adaptive Control The configuration of the adaptive control system is finished by the selection of a suitable control algorithm . This select ion strongly depends on the requirements spec ified by the plant operator (e .g. rise-time. over-shooting). Furthermore a few theoret ical pre-condit ions must be fulf illed for successful implementation and running of adaptive controllers . When controlling real processes these pre-conditions may be violated or some free design parameters have to be changed, due to expected or unexpected changes in the operating conditions of the controlled process. These violations may result in unacceptable or unstable control behavior of the parameter - adaptive controller .
The principle of parameter-adaptive control used here is called Model Identification Adaptive Control (MIAC) . The structure of this control scheme is shown in Fig . 1. This structure consists altogether o f four levels . The first level is the ordinary digital control loop (lowest levell. The next higher level is the adaptation level and consists of appropriate combinations of recursive parameter estimation techniques and controller design procedures, based on the estimated process parameters (separation principle) Frequently the uncertainties of the parameter estimates are neglected (certainty equivalence principle).
26
T. Knapp, R. Isermann Controller design With regard to the used process model different controllers and control design procedures can be used, based on the estimated process parameters (separation principle). Following controllers have proven to work well and are applicable for many classes of processes : - minimum variance controllers - state controllers - PlO-controllers
IIOt)
There exist several methods to design such controllers as pole-placement, cancellation principle, numerical optimization or approximation methods . For more details see Astrom and Wittenmark (1989), Isermann (1982) and Kofahl (1986).
uOt)
_
Fig. 1: Scheme of parameter-adaptive control
Supervision and coordination Process model description The process can be described by a suitable combination of - linear / non-linear - discrete / continous-time - parametric / non-parametric process models . Before using an adaptive controller some knowledge about the process to control should be gathered in order to apply the most suitable process model. For more details see Lachmann (1983), Isermann and Peter (1989) and Isermann, Jordan and Knapp (1989>'
Parameter estimation
The implementation of only the parameter estimation and the controller design procedure have proven to work well, as long as all necessary pre - conditions are fulfilled and the free design parameters are chosen properly. However these pre-conditions may be violated, leading to a poor or unstable control performance. This drawback requires measures to use adaptive controllers also if pre-conditions are violated. This can be realized in two additional levels called the supervision level and the coordination level (se~ Fig . 1) . The aims of these two levels are to coordinate parameter estimation and controller design, to detect possible malfunctions and to carry out appropriate actions for parameter estimation, controller design and the resulting closed- loop behavior. Also uncertainties of the estimates can be treated in these levels in order to increase the reliability of adaptive controllers.
To obtain information on the process behavior (process parameters §), parameter estimators of the Least-Squares (LS) type are used as they have proven to be robust also in closed-loop .
The supervision level discussed in this contribution involves tasks for :
Using the Least-Squares (LS) parameter estimation algorithm, the estimated parameters !!(k) are updated by
-
monitoring the parameter estimates detecting a process model mismatch decision making, what has changed monitoring the controller design monitoring closed-loop behavior
~(k+1) = ~(k) + y(k+1)[y(k+l) - y(k+1) ] The coordination level involves tasks for : with y(k+1) = JI?(k+1)·ft + v(k+ll
(2)
being the actual measured process output signal perturbed by an unmeasurable discrete white noise signal v(k) and (3)
being the predicted output by the model (one steap ahead prediction) . e(k) = y(k) - y(k)
[1 - y(k+1)·.!I1T(k+1) JE(k)/A
If a malfunction or a violation of pre-conditions is detected by the supervision level, the coordination level has to take suitable actions to guarantee a remaining good control performance. This can be done automatically or by an user supported through an expert system .
Basic Elements for Supervision and Coordination
y(k) is the estimator gain computed recursively with the new incoming data vector .!I1(k+1):
E(k+1) =
performing a start - up procedure switching on/off parameter estimation choosing the most suitable control algorithm decision making, what set of controller parameters will be used
(4)
is the resulting residual or equation error.
y(k+1) = E(k)'JI1{k+1){ A + .!11T(k+1)'E(khl!(k+1)
-
r
1
(5) (6)
A forgetting A (0 < A " 1) is introduced (exponential for getting), to track also slowly varying process parameters . The numerical properties of the basic LS-method can be improved by using square - root filter ing methods or factorization methods, see e.g. Biermann (1977) . Special conditions for closed-loop identification have to be satisfied, see Gustavsson, Ljung and Soderstrom (1977) and Isermann (1982>.
To design a superv is ion level and a coordination level many different approaches are po ssible. The supervision and coordination level should include elements for the supervis',on of the parameter estimation and the controller design. In many cases these functions have been realized by an analys is of the control signals, the equation error or some special matrix . These require several thresholds and limits, resulting from experiments or heuristic knowledge. An overview about such methods is given in Lachmann and Isermann (1985). A drawback is, that a supervision and coord ination level based on only such methods depends on the process to control. In this contribution basic elements for the supervision and coordination level are discussed which do not depend on the process . Further additionai funct ions should be added, depending on the process to control and the knowledge about the process .
SUPERVISION AND COORDINATION OF PARAMETER-ADAPTIVE CONTROLLERS Detection of a process model mismatch
Eigenvalue analysis
To detect a process model mismatch different possibilities are avaible. A favourable method is to analyse the quadratic equation error sum:
The time-variant characteristic equation of the estimator parameter error dynamics is as follows: (11) After a short calculation one has (12)
with j(k-i) being former measured process output signals and y(k - i) being the corresponding predicted process m!(del outputs. based on the estimated process parameters 8(k) at step k .
A discrete-time recursive least-squares parameter estimator of order n (which is the number of parameters to estimate) thus has - (n-1) constant eigenvalues at z=1
(8)
and The parameter M represents the extent of the process model verification . An increasing of the parameter M corresponds to an increasing process model reliability . Decreasing of M leads to a much more sensitive supervision le vel in a straightforward way . A sensible choice of the parameter M is : M = n
(9)
- one time-varying eigenvalue at (13)
which depends on the data JI?(k) and the computed covariance matrix .E(k-1) and lies ins ide the unit circle of the z-plane (0 " z (k) " )..).
with n being the number of parameters to estimate. The decision of process model mismatch is then as follows : Vee > fj, - - > process model mismatch
(10)
/
fj, is a measure for the desired process model accuracy.
-1
"-
/
\
%n(kl-1
•
\"
\
Fig . 2 shows the time history of the equation error sum . Until step k=50 the process behavior does not change. so that the equation error sum V BB only depends on the noise signal. acting on the process output Signal . At step k=50 the process gain K changes (1 - > 1.5) and the equation error sum Vee Pincreases. This increase correspond to a process model mismatch and can easily be de tected by statistical analyse .
1.2
/
"-
-J
J
1
I
~
Fig. 3: eigenvalue zn(k) of the parameter estimator The time-varying eigenvalue shows the following characteristics : excitation
quadratic equation e rror sum
0 .9
~
> 0
(14)
no excitation
- - - - - > )..
(15)
0 .6
0.3
Fig. 4 shows the above results for )..=0.95 .
tt
oJ 0
1.2 16
forgetting fact o r
32 0. 9
Fig. 2: time-history of the equation error sum Vee
((
eigenvalue
Determination of possible process changes 0.3
A change of the process behavior is due to a change of either the static process parameters (process gain or d .c . value) or the dynamic process parameters (time-constants) . In order to increase the reliability of the parameter estimates and to improve the convergence. only such parameters should be estimated. which have changed . Thus require a method to detect possible causes for process change . A change of the static parameters can be detected in steady-state. whereas a change of the dynamiC parameters are detectable only in a transient phase . A method is to estimate the static and dynamic parameters in a separate way. combined with statistic analyse of the corresponding equation error .
Fig . 4 : time-history of the eigenvalue zn(k)
If there is excitation. due to a change of the process input signal u(k) or a change of the d.c . value. acting on the process output signal. the eigenvalue zn(k) tends to zero . If there is no excitation the eigenvalue zn(k) tends to the forgetting factor ).. .
T. Knapp, R. Isermann Variable forgetting factor
Adaptive control of time-varying plants usually requires a forgetting factor A <1 to discount old data and therefore to enable the parameter estimator to follow changing process parameters !!. But this involves compromises. If data are discoA'nted too fast the process model parameter estimates !! will be uncertain, even the true process parameters ~ are constant. If old data are discounted too slow the process parameter estimates ~ of constant process parameters !! will be good, but it will be impossible to track rapid process parameters changes. Fortescue, Kershenbaum and Ydstie (1981) have proposed a method for varying the forgetting factor A automatically. This decreases the probability for bursts but does not eliminate them. Hagglund (1983) has proposed to forget data only in the directions where new information is obtained . Nevertheless, exponential forgetting works well only if the process is properly excited all the time. With regard to the supervision of the parameter estimator (switch off parameter estimation if there is no excitation), the following strategy to influence the forgetting factor is recommended:
(6)
A{k)
with Aa being the value of the forgetting factor when there is excitation (zn{k) ->0) and Ao being the value of the forgetting factor by loss of excitation (Zn{k) - > Ao)' see Fig . 5 . By this equation the forgetting factor is automatically decreased, when there is excitation .
- supervision of the closed-loop stability (5) - decision making as to which set of controller parameters will be used for control (5) - control and monitor'lng of parameter estimation and controller design (C) - stabilization of the control loop in the case of unstable parameter-adaptive controller (C)
These tasks can be supported by an expert system, bringing in the specific knowledge of the plant operator (heu ristic knowledge). Based on this heuristic knowledge and the analytic knowledge a process model mismatch and possible causes are detectible . The detection of possible causes is very interested with regard to parameter es ti mation. If process gain changes then only the static process model parameters have to be estimated and if timeconstants changes, then only the dynamiC process model parameters have to be estimated (part adaptation) .
Start-up part In order to avoid large or possible unacceptable process input and output signals, a pre-identification phase is employed within the coordination level. During this preidentification phase the process is perturbed with a sufficient exciting input signal (e.g. PRB5) to satisfy the identifiability conditions and to obtain a process model as accu rate as possible . Here it is important that the free specification parameters, such as the sampling period To and process model character istics, are chosen properly. This might be a problem , especially for an unknown process. Therefore the following strategy is recommended: - design of free parameters - collect a-priori knowledge - open loop with test signal (e.g. PRB5) - sufficient long identification period - model verification - design of a back - up controller Fig. 6 shows a flow chart of a possible start-up procedu re.
'. Fig.
5: forgetting factor A as a function of the eigenvalue zn
Supervision and Coordination Phllosohy The main aims of the supervision level and the coordination level are to guarantee a sucessful start - up of parameteradaptive controllers, to detect possible malfunctions caused by violations of theoretical pre-conditions and to take appropriate actions if the control performance gets poor . Both the supervision level (5) and the coordination level (C) can be divided into two parts, the start-up part and the run-time part.
M-steps pre- identl f1utlan
Made) veri tiuti an (\I>eps?)
v BacJwp [ontrDlIer design
The start-up part involves tasks for the configuration and implementation of adaptive controllers. These tasks are :
Fig. 6: flow chart of the start-up procedure
- pre-identification procedure with a test signal (e.g. PRB5) (C) - decision making what process model should be used (5) - verification of the process model (5) - decision making which control algorithm should be used, for the design of a back - up controller (5)
By this procedure first of all the free des ign parameters, such as the sampling period To and structural parameters, are des igned and a-priori knowledge is collected. Then a pre-identification is performed, generating a special test signal (e .g . PRB5). Various process models can be applied and the most suitable process model can be cho sen , by checking the identified model behavior against the real process input/output behav ior . This comparison of process output and process model output signal can be aChieved automatically or by an operator in a visual manner on a terminal screen . If no assumed mode l yie lds to a reasonable process model the specification parameters can be changed (e .g. To or structural parameters) and the preidentification phase can be repeated . This procedure can
The run-time part is only active in closed-loop operation and involves task s for: - signal analysis (5) - detection of a process model mismatch (5) - detection of possible causes (5)
SUPERVISION AND COORDINATION OF PARAMETER-ADAPTIVE CONTROLLERS be repeated until a process model is received as accurate as desired
Before closing the parameter - adaptive control loop a robust back - up controller is des igned. depending on the est imated proce ss model . The controller parameters and the reference value for this operating point are stored in order t o sw itch to the back-up cont roller and drive the system to a stable operating point in the case of an un stable parameter - adaptive control loop.
control loop can become unstable. resulting in unreliable process parameter estimates (estimator wind - up) . Therefore parameter estimation has to be switched off by loss of persistent excitation . The results by Egardt (1979) and Peters on and Narendra (1982) indicate. that it is reasonable to estimate only if the absolute value of the process input signal is above a certain level . Therefore the loss of persistent excitation has to be detected. The most favourable method to detect the loss of pers istent excitation is to analyse the eigenvalue zn(k) of the least - square estimator .
If there is a- pr iori knowledge about the process oretical modell'lng. exper iments). this knowledge used to improve parameter estimation . This is t ant fac t. espec ially if continous-time models are
Analysing the time-varying eigenvalue zn{k) parameter estimation can be sw itched o ff. when zn{k) is in the vic inity of the forgetting factor A . Then no additional information about the process is available . Therefore the following equation is used to switch off parameter est imation .
(e .g . the should be an impor used .
zn{k) :t
Run-time part The run-time part is active in closed-loop operation and involves the control and the supervision of both the parameter estimation and controller des ign as a basic part.
E{z~{k)}
-
-> switch off parameter estimation
Fig. 7 shows the flow chart of the parameter estimation supervision .
Supervision and coordination of parameter estimation
The main aim o f supervising the parameter estimation is to ensure that the estimated process model adequately matches the dynamic input/output behavior of the real process . If the process behavior does not change parame ter est imation is not required and should be switched off . A change of the process behavior must be recognized and parameter estimation should be switched on as long as the identifiability condit ions are fulfill e d (persistent excitat ion) and the estimated process mode l does not match the real process . To receive relia ble parameter est imates. it is very important to switch on and off the parameter estimation in the right instant. With regard t o this fact the excitation of the control loop is most important. see Astrom and Bohl in (19661. Here the follow ing strategy is recommended : - detection of a process model mismatch - > monitoring process model quality of process model (Vee )
the estimated
- sw itching on the parameter estimation only if a process model mismatch was detected and whe n there is excitation which is a necessary cond ition for estimation - sw itching off the parameter es timation by loss of exc itation or if a sufficiently accurate proces s model is re ached - > monitor ing eigenvalue and process model quality
s w itch on parameter estimation :
Parame ter estimation should be performed. if a process model mismatch wa s detected and if there is excitation and all neces sary pre-conditions are fulf illed. A very simple and re liable method is to swit ch on the parameter estima tion when there is an external excitation due to change o f the reference var iable w{k) or an additional test signal. If the re is only excitation caused by noise ident ifiability con ditions for parameter est imation in closed - loop have to be proved carefully in order to get re li able parameter est ima tes. The identifi ab ility conditions strongly depend on the used process mo del and the used contr oller . If all precondit ions are fu lfi lled then parameter estimation can be sw it ched on . even if there is or>ly noise . Otherwise there m ust be e xterna l excitation (reference signal or test signal) .
swi tch off parameter estim a tion :
If there is no persistent exc itation the paramete r - adaptive
Fig. 7: f low chart of parameter estimation supervision
Supervision and coordination of controller des ign
In addition to the supervision and coordination of the pa rameter estimation the controller design procedure has to be monitored . Typical violations of pre-conditions for controller de sign are incompatible combinations of process structure and controller structure or cancellations o f process poles or process zeroes near to or outside of the unit-cirle of the z- plane in the discrete - time case . To o ve r come this pos sibl e situation the pro cess model poles and zeroes are calculated for each valid proces s model. and only if all pre - conditions are fulfilled . a new se t of con troller parameters is determined . These tasks are employed w ithin the supervision level .
Before replac ing the ol d controller parameter set by the new one an add it ional control loop an alys is can be performed. This is done by simulating the closed -- loop behavior based on the actual estimated process model combined with the new. the old and the back -up controller parameters.
H
L [w{j)
2
- y{jl]
(17)
j =O
The controller parameters delivering the best control per formance V c then is chosen . It is also poss ible to simulate the process w ith different contr ol algor ithms in order t o obtain the most suitable contr oller for the special process .
T. Knapp, R. Isermann
3()
cess model a back-up controller (PlO-controller) designed and the control loop was closed.
Fig. 8: flow chart of controller design supervision
Supervision of the closed-loop behavior For the supervision of the closed-loop. especially the stability of the closed parameter-adaptive control loop. the behavior of the control loop signals were analysed. If the process input signal is at the maximum or minimum boundary for a given number of control steps (Fig. 9) and the control deviation elk) = w(k) - y(k) is monotonically increasing or oscillatory for a constant reference signal. the parameter-adaptive controller is switched back to the back-up controller and an alarm message is given .
uOt)
",
" '--
.......... .
was
In the following the work ing methods of the supervision level and the coordination level in closed-loop will be shown . Up to step k=20 the process model parameters described the process sufficiently accurate. so that the parameter estimation was switched off (no process model mismatchl. At step k=21 process gain changes. As indicated in Fig. 13 the quadratic equation error sum increases and a process model mismatch was detected by the supervision level. As long as the supervision level ind ic ates a process model mismatch. the parameter estimation was switched on and off (Fig . 11). corresponding to the calculated eigenvalue (Fig . 10l. Based on the eigenvalue the forgetting factor A was calculated. according to Eq . 16 (Fig . 12l. At step k=97 the process model again was as accurate as desired and the parameter estimation was switched off. Only if a process model mismatch was detected aga·,n. the parameter estimation will be switched on . Otherwise parameter estimation will not be performed. Fig . 14 and Fig . 15 shows the resulting process input signal u(k) and the process output signal y(k) and the reference signal w(k) .
Conclusions In this paper procedures have been proposed in order to increase the reliability of adaptive controllers. A start-up procedure was suggested. performing a pre-identification and del ivering the most suitable process model . During this procedure the free design parameters can be optimized. Furthermore strategies to supervise and to coordinate parameter estimation have been presented. which do not depend on the controlled process. For the supervision and coordination of the parameter estimat ion an eigenvalue analyse has proven to work well .
k
Refernces
UR'n ........
L-
.. _
Astrom. K .J . (1983l. Theory and applications of adaptive control - a survey . Automatica . .12. 471-485
.. L- .......... ..
Fig. 9: process input signal u(k) for an unstable control loop behavior
Simulation and experimental results To demonstrate the working methods of the suggested supervision level and the coordination level some simulation and experimental results for an a.c . motor are presented . During the start-up procedure a test signal (PRBS) was generated in open loop and a pre-identification was per formed . The sampling period was chosen to be To=0.5 sec . The process input and output signals were stored and different linear parametric discrete-time process models were applied. Then. the most suitable process model was chosed by an analyse of the quadratic equation error sum (Eq . 7). This could be done automatically or by an operator in a visual manner (table 1) on a terminal screen. d
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Table 1: process model verification
As indicated in table 1 a process model of order m=2 and dead-t ime d=O was selected . Based on the received pro-
Astrom. K .J. and W ittenmark. B . (1989) . Adaptive Control. Addison - Weslev Publish ing Company Astrom . K.J. and Bohlin. T . (1966l. Numerical identification of linear dynamic systems from normal operating records. lD ~ tL Hammond kdJ. Theory cl Self -Adaptive Cont r ol Systems. Plenum Press. New York Bergmann. S. (1983) . Digital parameter -adaptive control with micro-processors (in German) . Doctor-thesis. Fortschr.-Ber. VDI-Z. Reihe § NO.55 Biermann. G.J. (1977). Factorization methods for disc rete sequential estimat ion. Academic Press. New York Clarke. D .W. and Gawthrop . P .J . (1981) . Implementation and application of micro-processor-based self- tuner. Automatica. lZ. 233 Egardt. B. (1979l. Stability of adapt ive controllers . Lecture Notes ill Control and Information Sciences. Springer Verlag. Berlin Fortescue et. al . (1981) Implementation of self-tuning regulators w ith variable forgetting factors . Automatica.
lZ. 831 Gustavsson. I .• Ljung. L . and Soderstrom. T. (1977) Identification o f processes in closed loop - ',dentif,abil',ty and accuracy aspects . Automatica. J]. 59 Ha gglund. T. (1 983) . New estimation techniques for adaptive control. Report Lund ~ cl Techn. Isermann. R. (1982) . Parameter adaptive control algorithms - a tutorial Automatica. ill. 513-528
SUPERVISION AND COORDINATION OF PARAMETER-ADAPTIVE CONTROLLERS Isermann, R., Jordan, M . and Knapp, T . (1989) . Digital adaptive control based on identified parametric and non-parametric models . CIE '89, Anaheim , Cal ifornia Isermann, R. and Lachmann, K .H. (1985) . Parameter-adaptive control with configuration aids and supervision functions. Automatica, Zl, 625-638 Kofahl, R. (1986) . Self-tuning of PlO controllers based on process parameter estimation , Journal 6, 27, NO . 3 Lachmann, K .-H . (1983). Parameter-adaptive control algorithms for special classes of nonlinear processes with non-memory nonlinearities (in German) . Doctor-thesis, Technische Hochschule Darmstadt, Fortschr . Ber . VDI-Z., Reihe ~, NO .66
1 .2
31
Peter, K .-H. and Isermann, R. (1989) . Parameter-adaptive PlO-control based on continous-time process models. IFAC Symposium on Adaptive Control and Signal Processing, Glasgow, Scotland Peterson, B.B . and Narendra, K.s . (1982) . Bounded error adaptive control. IEEE - T rans . Automat. Control., Vol . AC-27 Sanoff, S.P . and Wellstead, P .E . (1985) . Expert identification and control. 7th IFAC Symposium on Identification and System Parameter Estimation, York Schumann et. al . (1981) . Towards applicability of parameter-adaptive control systems . Proc. IFAC-Congress, Kyoto, Pergamon Press, Oxford
threshold quadratic equation error sum
0 .9
0 .6
- eige nvalu e
0 .3 ' 10
1
o . 0 oL--+---:!3--+-----:!: 6-+--:':' 9 -+--:1:;;2~+--1~5--+-+-...,t>
Fig. 10: time-history of the eigenvalue zn
3
2
Fig. 13: time - history of the quadratic equation error sum Vee
3.
et
process input signal
Identification
Fig . 14 : t ime - history of the process output signal u(k)
Fig . 11 : identification
1,J
3" i
p r o c es s output signal A
:\JUVL
f o rgetting f a c t or
c' H) 1 + - - -...........,t >
Fig . 12 : t ime - history of the f orgett ing factor A
Fig . 15 : time-h istory of the process output signal y(k) and the reference signal w(k)
SUPERVISION AND COORDINATION OF PARAMETERADAPTIVE CONTROLLERS T. Knapp, R. Isermann Institute of Automatic Control, Technical University of Oarmstadt Landgraf Georg Strasse 4, 0-6100 Oarmstadt, F.R.G.
Abstract. Adaptive controllers have proven to work well as long as all theoretical pre-conditions are fulfilled and the free design parameters are chosen properly. However in practice these pre-conditions may be violated and/or some free design parameters are not chosen properly . 80th result in a lowering of the control performance. Therefore a supervision and coordination level is introduced, monitoring the adaptive controller and taking appropriate actions if malfunctions are detected. The Introduced supervision and coordination level exists altogether of two parts named start - up part and run-time part. The start-up part involves tasks for configuration and implementation of adaptive controllers and the run-time part involves tasks for supervision the adaptive controller in closed-loop operation. A sensible start - up procedure is recommended and an eigenvalue analysis is shown in order to monitor the parameter estimation. To track time-varying process parameters an eigenvalue controlled forgetting factor is introduced. Simulations and experimental results for an a.c. motor are shown, indicating an improvement of the overall control performance. Keywords . supervision and coordination, adaptive control, parameter estimation, eigenvalue analysis, process model mismatch. time-varying processes
Introduction The successful application of parameter-adaptive controllers requires that all assumptions used for the derivation of the parameter-adaptive control principle. for parameter estimation and controller design are met and some design parameters are chosen properly. The closed-loop systems obtained with adaptive controllers are nonlinear . This makes analysis of stability. convergence and control performance difficult. part iculary if there are random disturbances. see Astrom (1983). During the implementation of adaptive controllers the choice of a suitable sample period is an example . The spectrum of possible values for this parameter is bounded by Shannon 's theorem and the demand for good control performance on the one hand and numerical problems as well as limited computer power on the other hand . Another problem is the choice of a suitable process model. of structural parameters (e .g. process model order. process model dead-time) and of some design parameters in connection with parameter estimation and controller design.
Therefore it ons of the are realized coordination stem.
is necessary to monitor all or several functiparameter - adaptive control loop. These tasks in an additional block named supervision and level. which may also include an expert sy-
The first work in this direction was conducted by Clarke and Gawthrop (1981). Schumann et . al . (1981). Fortescue et. al. (1981). Bergmann (1983). Isermann and Lachmann (1985). A sensible use of expert systems in adaptive control is described in Sanoff and Wellstead (1985) . The aim o f this paper is to show. how a supervision level and a coordination level can be designed and what basic elements should be included. In the first part of this paper the parameter-adaptive control scheme is discussed and basic elements f or the supervision level and coordination level are introduced. In the second part of the paper a possible supervis ion and coordination leve l is presented . Finally a few experimental results demonstrate the work ing method o f the presented supervision level and coordination level.
Parameter-Adaptive Control The configuration of the adaptive control system is finished by the selection of a suitable control algorithm . This select ion strongly depends on the requirements spec ified by the plant operator (e .g. rise-time. over-shooting). Furthermore a few theoret ical pre-condit ions must be fulf illed for successful implementation and running of adaptive controllers . When controlling real processes these pre-conditions may be violated or some free design parameters have to be changed, due to expected or unexpected changes in the operating conditions of the controlled process. These violations may result in unacceptable or unstable control behavior of the parameter - adaptive controller .
The principle of parameter-adaptive control used here is called Model Identification Adaptive Control (MIAC) . The structure of this control scheme is shown in Fig . 1. This structure consists altogether o f four levels . The first level is the ordinary digital control loop (lowest levell. The next higher level is the adaptation level and consists of appropriate combinations of recursive parameter estimation techniques and controller design procedures, based on the estimated process parameters (separation principle) Frequently the uncertainties of the parameter estimates are neglected (certainty equivalence principle).
26
T. Knapp, R. Isermann Controller design With regard to the used process model different controllers and control design procedures can be used, based on the estimated process parameters (separation principle). Following controllers have proven to work well and are applicable for many classes of processes : - minimum variance controllers - state controllers - PlO-controllers
IIOt)
There exist several methods to design such controllers as pole-placement, cancellation principle, numerical optimization or approximation methods . For more details see Astrom and Wittenmark (1989), Isermann (1982) and Kofahl (1986).
uOt)
_
Fig. 1: Scheme of parameter-adaptive control
Supervision and coordination Process model description The process can be described by a suitable combination of - linear / non-linear - discrete / continous-time - parametric / non-parametric process models . Before using an adaptive controller some knowledge about the process to control should be gathered in order to apply the most suitable process model. For more details see Lachmann (1983), Isermann and Peter (1989) and Isermann, Jordan and Knapp (1989>'
Parameter estimation
The implementation of only the parameter estimation and the controller design procedure have proven to work well, as long as all necessary pre - conditions are fulfilled and the free design parameters are chosen properly. However these pre-conditions may be violated, leading to a poor or unstable control performance. This drawback requires measures to use adaptive controllers also if pre-conditions are violated. This can be realized in two additional levels called the supervision level and the coordination level (se~ Fig . 1) . The aims of these two levels are to coordinate parameter estimation and controller design, to detect possible malfunctions and to carry out appropriate actions for parameter estimation, controller design and the resulting closed- loop behavior. Also uncertainties of the estimates can be treated in these levels in order to increase the reliability of adaptive controllers.
To obtain information on the process behavior (process parameters §), parameter estimators of the Least-Squares (LS) type are used as they have proven to be robust also in closed-loop .
The supervision level discussed in this contribution involves tasks for :
Using the Least-Squares (LS) parameter estimation algorithm, the estimated parameters !!(k) are updated by
-
monitoring the parameter estimates detecting a process model mismatch decision making, what has changed monitoring the controller design monitoring closed-loop behavior
~(k+1) = ~(k) + y(k+1)[y(k+l) - y(k+1) ] The coordination level involves tasks for : with y(k+1) = JI?(k+1)·ft + v(k+ll
(2)
being the actual measured process output signal perturbed by an unmeasurable discrete white noise signal v(k) and (3)
being the predicted output by the model (one steap ahead prediction) . e(k) = y(k) - y(k)
[1 - y(k+1)·.!I1T(k+1) JE(k)/A
If a malfunction or a violation of pre-conditions is detected by the supervision level, the coordination level has to take suitable actions to guarantee a remaining good control performance. This can be done automatically or by an user supported through an expert system .
Basic Elements for Supervision and Coordination
y(k) is the estimator gain computed recursively with the new incoming data vector .!I1(k+1):
E(k+1) =
performing a start - up procedure switching on/off parameter estimation choosing the most suitable control algorithm decision making, what set of controller parameters will be used
(4)
is the resulting residual or equation error.
y(k+1) = E(k)'JI1{k+1){ A + .!11T(k+1)'E(khl!(k+1)
-
r
1
(5) (6)
A forgetting A (0 < A " 1) is introduced (exponential for getting), to track also slowly varying process parameters . The numerical properties of the basic LS-method can be improved by using square - root filter ing methods or factorization methods, see e.g. Biermann (1977) . Special conditions for closed-loop identification have to be satisfied, see Gustavsson, Ljung and Soderstrom (1977) and Isermann (1982>.
To design a superv is ion level and a coordination level many different approaches are po ssible. The supervision and coordination level should include elements for the supervis',on of the parameter estimation and the controller design. In many cases these functions have been realized by an analys is of the control signals, the equation error or some special matrix . These require several thresholds and limits, resulting from experiments or heuristic knowledge. An overview about such methods is given in Lachmann and Isermann (1985). A drawback is, that a supervision and coord ination level based on only such methods depends on the process to control. In this contribution basic elements for the supervision and coordination level are discussed which do not depend on the process . Further additionai funct ions should be added, depending on the process to control and the knowledge about the process .
SUPERVISION AND COORDINATION OF PARAMETER-ADAPTIVE CONTROLLERS Detection of a process model mismatch
Eigenvalue analysis
To detect a process model mismatch different possibilities are avaible. A favourable method is to analyse the quadratic equation error sum:
The time-variant characteristic equation of the estimator parameter error dynamics is as follows: (11) After a short calculation one has (12)
with j(k-i) being former measured process output signals and y(k - i) being the corresponding predicted process m!(del outputs. based on the estimated process parameters 8(k) at step k .
A discrete-time recursive least-squares parameter estimator of order n (which is the number of parameters to estimate) thus has - (n-1) constant eigenvalues at z=1
(8)
and The parameter M represents the extent of the process model verification . An increasing of the parameter M corresponds to an increasing process model reliability . Decreasing of M leads to a much more sensitive supervision le vel in a straightforward way . A sensible choice of the parameter M is : M = n
(9)
- one time-varying eigenvalue at (13)
which depends on the data JI?(k) and the computed covariance matrix .E(k-1) and lies ins ide the unit circle of the z-plane (0 " z (k) " )..).
with n being the number of parameters to estimate. The decision of process model mismatch is then as follows : Vee > fj, - - > process model mismatch
(10)
/
fj, is a measure for the desired process model accuracy.
-1
"-
/
\
%n(kl-1
•
\"
\
Fig . 2 shows the time history of the equation error sum . Until step k=50 the process behavior does not change. so that the equation error sum V BB only depends on the noise signal. acting on the process output Signal . At step k=50 the process gain K changes (1 - > 1.5) and the equation error sum Vee Pincreases. This increase correspond to a process model mismatch and can easily be de tected by statistical analyse .
1.2
/
"-
-J
J
1
I
~
Fig. 3: eigenvalue zn(k) of the parameter estimator The time-varying eigenvalue shows the following characteristics : excitation
quadratic equation e rror sum
0 .9
~
> 0
(14)
no excitation
- - - - - > )..
(15)
0 .6
0.3
Fig. 4 shows the above results for )..=0.95 .
tt
oJ 0
1.2 16
forgetting fact o r
32 0. 9
Fig. 2: time-history of the equation error sum Vee
((
eigenvalue
Determination of possible process changes 0.3
A change of the process behavior is due to a change of either the static process parameters (process gain or d .c . value) or the dynamic process parameters (time-constants) . In order to increase the reliability of the parameter estimates and to improve the convergence. only such parameters should be estimated. which have changed . Thus require a method to detect possible causes for process change . A change of the static parameters can be detected in steady-state. whereas a change of the dynamiC parameters are detectable only in a transient phase . A method is to estimate the static and dynamic parameters in a separate way. combined with statistic analyse of the corresponding equation error .
Fig . 4 : time-history of the eigenvalue zn(k)
If there is excitation. due to a change of the process input signal u(k) or a change of the d.c . value. acting on the process output signal. the eigenvalue zn(k) tends to zero . If there is no excitation the eigenvalue zn(k) tends to the forgetting factor ).. .
T. Knapp, R. Isermann Variable forgetting factor
Adaptive control of time-varying plants usually requires a forgetting factor A <1 to discount old data and therefore to enable the parameter estimator to follow changing process parameters !!. But this involves compromises. If data are discoA'nted too fast the process model parameter estimates !! will be uncertain, even the true process parameters ~ are constant. If old data are discounted too slow the process parameter estimates ~ of constant process parameters !! will be good, but it will be impossible to track rapid process parameters changes. Fortescue, Kershenbaum and Ydstie (1981) have proposed a method for varying the forgetting factor A automatically. This decreases the probability for bursts but does not eliminate them. Hagglund (1983) has proposed to forget data only in the directions where new information is obtained . Nevertheless, exponential forgetting works well only if the process is properly excited all the time. With regard to the supervision of the parameter estimator (switch off parameter estimation if there is no excitation), the following strategy to influence the forgetting factor is recommended:
(6)
A{k)
with Aa being the value of the forgetting factor when there is excitation (zn{k) ->0) and Ao being the value of the forgetting factor by loss of excitation (Zn{k) - > Ao)' see Fig . 5 . By this equation the forgetting factor is automatically decreased, when there is excitation .
- supervision of the closed-loop stability (5) - decision making as to which set of controller parameters will be used for control (5) - control and monitor'lng of parameter estimation and controller design (C) - stabilization of the control loop in the case of unstable parameter-adaptive controller (C)
These tasks can be supported by an expert system, bringing in the specific knowledge of the plant operator (heu ristic knowledge). Based on this heuristic knowledge and the analytic knowledge a process model mismatch and possible causes are detectible . The detection of possible causes is very interested with regard to parameter es ti mation. If process gain changes then only the static process model parameters have to be estimated and if timeconstants changes, then only the dynamiC process model parameters have to be estimated (part adaptation) .
Start-up part In order to avoid large or possible unacceptable process input and output signals, a pre-identification phase is employed within the coordination level. During this preidentification phase the process is perturbed with a sufficient exciting input signal (e.g. PRB5) to satisfy the identifiability conditions and to obtain a process model as accu rate as possible . Here it is important that the free specification parameters, such as the sampling period To and process model character istics, are chosen properly. This might be a problem , especially for an unknown process. Therefore the following strategy is recommended: - design of free parameters - collect a-priori knowledge - open loop with test signal (e.g. PRB5) - sufficient long identification period - model verification - design of a back - up controller Fig. 6 shows a flow chart of a possible start-up procedu re.
'. Fig.
5: forgetting factor A as a function of the eigenvalue zn
Supervision and Coordination Phllosohy The main aims of the supervision level and the coordination level are to guarantee a sucessful start - up of parameteradaptive controllers, to detect possible malfunctions caused by violations of theoretical pre-conditions and to take appropriate actions if the control performance gets poor . Both the supervision level (5) and the coordination level (C) can be divided into two parts, the start-up part and the run-time part.
M-steps pre- identl f1utlan
Made) veri tiuti an (\I>eps?)
v BacJwp [ontrDlIer design
The start-up part involves tasks for the configuration and implementation of adaptive controllers. These tasks are :
Fig. 6: flow chart of the start-up procedure
- pre-identification procedure with a test signal (e.g. PRB5) (C) - decision making what process model should be used (5) - verification of the process model (5) - decision making which control algorithm should be used, for the design of a back - up controller (5)
By this procedure first of all the free des ign parameters, such as the sampling period To and structural parameters, are des igned and a-priori knowledge is collected. Then a pre-identification is performed, generating a special test signal (e .g . PRB5). Various process models can be applied and the most suitable process model can be cho sen , by checking the identified model behavior against the real process input/output behav ior . This comparison of process output and process model output signal can be aChieved automatically or by an operator in a visual manner on a terminal screen . If no assumed mode l yie lds to a reasonable process model the specification parameters can be changed (e .g. To or structural parameters) and the preidentification phase can be repeated . This procedure can
The run-time part is only active in closed-loop operation and involves task s for: - signal analysis (5) - detection of a process model mismatch (5) - detection of possible causes (5)
SUPERVISION AND COORDINATION OF PARAMETER-ADAPTIVE CONTROLLERS be repeated until a process model is received as accurate as desired
Before closing the parameter - adaptive control loop a robust back - up controller is des igned. depending on the est imated proce ss model . The controller parameters and the reference value for this operating point are stored in order t o sw itch to the back-up cont roller and drive the system to a stable operating point in the case of an un stable parameter - adaptive control loop.
control loop can become unstable. resulting in unreliable process parameter estimates (estimator wind - up) . Therefore parameter estimation has to be switched off by loss of persistent excitation . The results by Egardt (1979) and Peters on and Narendra (1982) indicate. that it is reasonable to estimate only if the absolute value of the process input signal is above a certain level . Therefore the loss of persistent excitation has to be detected. The most favourable method to detect the loss of pers istent excitation is to analyse the eigenvalue zn(k) of the least - square estimator .
If there is a- pr iori knowledge about the process oretical modell'lng. exper iments). this knowledge used to improve parameter estimation . This is t ant fac t. espec ially if continous-time models are
Analysing the time-varying eigenvalue zn{k) parameter estimation can be sw itched o ff. when zn{k) is in the vic inity of the forgetting factor A . Then no additional information about the process is available . Therefore the following equation is used to switch off parameter est imation .
(e .g . the should be an impor used .
zn{k) :t
Run-time part The run-time part is active in closed-loop operation and involves the control and the supervision of both the parameter estimation and controller des ign as a basic part.
E{z~{k)}
-
-> switch off parameter estimation
Fig. 7 shows the flow chart of the parameter estimation supervision .
Supervision and coordination of parameter estimation
The main aim o f supervising the parameter estimation is to ensure that the estimated process model adequately matches the dynamic input/output behavior of the real process . If the process behavior does not change parame ter est imation is not required and should be switched off . A change of the process behavior must be recognized and parameter estimation should be switched on as long as the identifiability condit ions are fulfill e d (persistent excitat ion) and the estimated process mode l does not match the real process . To receive relia ble parameter est imates. it is very important to switch on and off the parameter estimation in the right instant. With regard t o this fact the excitation of the control loop is most important. see Astrom and Bohl in (19661. Here the follow ing strategy is recommended : - detection of a process model mismatch - > monitoring process model quality of process model (Vee )
the estimated
- sw itching on the parameter estimation only if a process model mismatch was detected and whe n there is excitation which is a necessary cond ition for estimation - sw itching off the parameter es timation by loss of exc itation or if a sufficiently accurate proces s model is re ached - > monitor ing eigenvalue and process model quality
s w itch on parameter estimation :
Parame ter estimation should be performed. if a process model mismatch wa s detected and if there is excitation and all neces sary pre-conditions are fulf illed. A very simple and re liable method is to swit ch on the parameter estima tion when there is an external excitation due to change o f the reference var iable w{k) or an additional test signal. If the re is only excitation caused by noise ident ifiability con ditions for parameter est imation in closed - loop have to be proved carefully in order to get re li able parameter est ima tes. The identifi ab ility conditions strongly depend on the used process mo del and the used contr oller . If all precondit ions are fu lfi lled then parameter estimation can be sw it ched on . even if there is or>ly noise . Otherwise there m ust be e xterna l excitation (reference signal or test signal) .
swi tch off parameter estim a tion :
If there is no persistent exc itation the paramete r - adaptive
Fig. 7: f low chart of parameter estimation supervision
Supervision and coordination of controller des ign
In addition to the supervision and coordination of the pa rameter estimation the controller design procedure has to be monitored . Typical violations of pre-conditions for controller de sign are incompatible combinations of process structure and controller structure or cancellations o f process poles or process zeroes near to or outside of the unit-cirle of the z- plane in the discrete - time case . To o ve r come this pos sibl e situation the pro cess model poles and zeroes are calculated for each valid proces s model. and only if all pre - conditions are fulfilled . a new se t of con troller parameters is determined . These tasks are employed w ithin the supervision level .
Before replac ing the ol d controller parameter set by the new one an add it ional control loop an alys is can be performed. This is done by simulating the closed -- loop behavior based on the actual estimated process model combined with the new. the old and the back -up controller parameters.
H
L [w{j)
2
- y{jl]
(17)
j =O
The controller parameters delivering the best control per formance V c then is chosen . It is also poss ible to simulate the process w ith different contr ol algor ithms in order t o obtain the most suitable contr oller for the special process .
T. Knapp, R. Isermann
3()
cess model a back-up controller (PlO-controller) designed and the control loop was closed.
Fig. 8: flow chart of controller design supervision
Supervision of the closed-loop behavior For the supervision of the closed-loop. especially the stability of the closed parameter-adaptive control loop. the behavior of the control loop signals were analysed. If the process input signal is at the maximum or minimum boundary for a given number of control steps (Fig. 9) and the control deviation elk) = w(k) - y(k) is monotonically increasing or oscillatory for a constant reference signal. the parameter-adaptive controller is switched back to the back-up controller and an alarm message is given .
uOt)
",
" '--
.......... .
was
In the following the work ing methods of the supervision level and the coordination level in closed-loop will be shown . Up to step k=20 the process model parameters described the process sufficiently accurate. so that the parameter estimation was switched off (no process model mismatchl. At step k=21 process gain changes. As indicated in Fig. 13 the quadratic equation error sum increases and a process model mismatch was detected by the supervision level. As long as the supervision level ind ic ates a process model mismatch. the parameter estimation was switched on and off (Fig . 11). corresponding to the calculated eigenvalue (Fig . 10l. Based on the eigenvalue the forgetting factor A was calculated. according to Eq . 16 (Fig . 12l. At step k=97 the process model again was as accurate as desired and the parameter estimation was switched off. Only if a process model mismatch was detected aga·,n. the parameter estimation will be switched on . Otherwise parameter estimation will not be performed. Fig . 14 and Fig . 15 shows the resulting process input signal u(k) and the process output signal y(k) and the reference signal w(k) .
Conclusions In this paper procedures have been proposed in order to increase the reliability of adaptive controllers. A start-up procedure was suggested. performing a pre-identification and del ivering the most suitable process model . During this procedure the free design parameters can be optimized. Furthermore strategies to supervise and to coordinate parameter estimation have been presented. which do not depend on the controlled process. For the supervision and coordination of the parameter estimat ion an eigenvalue analyse has proven to work well .
k
Refernces
UR'n ........
L-
.. _
Astrom. K .J . (1983l. Theory and applications of adaptive control - a survey . Automatica . .12. 471-485
.. L- .......... ..
Fig. 9: process input signal u(k) for an unstable control loop behavior
Simulation and experimental results To demonstrate the working methods of the suggested supervision level and the coordination level some simulation and experimental results for an a.c . motor are presented . During the start-up procedure a test signal (PRBS) was generated in open loop and a pre-identification was per formed . The sampling period was chosen to be To=0.5 sec . The process input and output signals were stored and different linear parametric discrete-time process models were applied. Then. the most suitable process model was chosed by an analyse of the quadratic equation error sum (Eq . 7). This could be done automatically or by an operator in a visual manner (table 1) on a terminal screen. d
K.
C.
2
8
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2
1
8.157
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1.813
-8.551
3.778E-4
1
8.647
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8
1.114
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ft
, J
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Table 1: process model verification
As indicated in table 1 a process model of order m=2 and dead-t ime d=O was selected . Based on the received pro-
Astrom. K .J. and W ittenmark. B . (1989) . Adaptive Control. Addison - Weslev Publish ing Company Astrom . K.J. and Bohlin. T . (1966l. Numerical identification of linear dynamic systems from normal operating records. lD ~ tL Hammond kdJ. Theory cl Self -Adaptive Cont r ol Systems. Plenum Press. New York Bergmann. S. (1983) . Digital parameter -adaptive control with micro-processors (in German) . Doctor-thesis. Fortschr.-Ber. VDI-Z. Reihe § NO.55 Biermann. G.J. (1977). Factorization methods for disc rete sequential estimat ion. Academic Press. New York Clarke. D .W. and Gawthrop . P .J . (1981) . Implementation and application of micro-processor-based self- tuner. Automatica. lZ. 233 Egardt. B. (1979l. Stability of adapt ive controllers . Lecture Notes ill Control and Information Sciences. Springer Verlag. Berlin Fortescue et. al . (1981) Implementation of self-tuning regulators w ith variable forgetting factors . Automatica.
lZ. 831 Gustavsson. I .• Ljung. L . and Soderstrom. T. (1977) Identification o f processes in closed loop - ',dentif,abil',ty and accuracy aspects . Automatica. J]. 59 Ha gglund. T. (1 983) . New estimation techniques for adaptive control. Report Lund ~ cl Techn. Isermann. R. (1982) . Parameter adaptive control algorithms - a tutorial Automatica. ill. 513-528
SUPERVISION AND COORDINATION OF PARAMETER-ADAPTIVE CONTROLLERS Isermann, R., Jordan, M . and Knapp, T . (1989) . Digital adaptive control based on identified parametric and non-parametric models . CIE '89, Anaheim , Cal ifornia Isermann, R. and Lachmann, K .H. (1985) . Parameter-adaptive control with configuration aids and supervision functions. Automatica, Zl, 625-638 Kofahl, R. (1986) . Self-tuning of PlO controllers based on process parameter estimation , Journal 6, 27, NO . 3 Lachmann, K .-H . (1983). Parameter-adaptive control algorithms for special classes of nonlinear processes with non-memory nonlinearities (in German) . Doctor-thesis, Technische Hochschule Darmstadt, Fortschr . Ber . VDI-Z., Reihe ~, NO .66
1 .2
31
Peter, K .-H. and Isermann, R. (1989) . Parameter-adaptive PlO-control based on continous-time process models. IFAC Symposium on Adaptive Control and Signal Processing, Glasgow, Scotland Peterson, B.B . and Narendra, K.s . (1982) . Bounded error adaptive control. IEEE - T rans . Automat. Control., Vol . AC-27 Sanoff, S.P . and Wellstead, P .E . (1985) . Expert identification and control. 7th IFAC Symposium on Identification and System Parameter Estimation, York Schumann et. al . (1981) . Towards applicability of parameter-adaptive control systems . Proc. IFAC-Congress, Kyoto, Pergamon Press, Oxford
threshold quadratic equation error sum
0 .9
0 .6
- eige nvalu e
0 .3 ' 10
1
o . 0 oL--+---:!3--+-----:!: 6-+--:':' 9 -+--:1:;;2~+--1~5--+-+-...,t>
Fig. 10: time-history of the eigenvalue zn
3
2
Fig. 13: time - history of the quadratic equation error sum Vee
3.
et
process input signal
Identification
Fig . 14 : t ime - history of the process output signal u(k)
Fig . 11 : identification
1,J
3" i
p r o c es s output signal A
:\JUVL
f o rgetting f a c t or
c' H) 1 + - - -...........,t >
Fig . 12 : t ime - history of the f orgett ing factor A
Fig . 15 : time-h istory of the process output signal y(k) and the reference signal w(k)