On the Applicability of Adaptive Control

Copnig ht © IF .-\ C .·\da p ti \·e Syste m, ill COlltrol a nd Sign a l Processing. LlI ll d . Swed e n . 19X()

ON THE APPLICABILITY OF ADAPTIVE CONTROL M. M'Saad, M. Duque and I. D. Landau CREeo "5\·.,thll lll es Adaptatifs " (C. ,\ '.R .S.) Laburatoire d'Autolllatiqu f de C rflloble (C.X. R .S. L. A. 288) E.X. S .l. E.C. - LV.P .C. B .P . '; 6 - 38 .;02 Saint Martin D'H nes, Fra nce

Abstract. In the last decade there has been a growing awareness that adaptive control can produce improved performance in real application, mainly as a result of considerable advances in the understanding of adaptive control theory as _11 as in the computer technology. In this paper the authors a1.lll at providing practical iJaplications of recent results in robustness of adaptive control theory . The fundamental deSign features as stability robustness, adaption alertness, offset-free performance, implementation simplicity and control system integrity will be e.phasized. S1.IIIulation studies involving a realiBtic math_tical model of an electrical furnace are given to highlight the applicability of the involved adaptive controllers . Jteywords . Adaptive control I LQ--control Estimated Model Admissibility

I I

1 • IIITROIXJC'l'ION There has been IllUch diBcussion in recent literature on the question of designing controllers that would perform _11 in IIIOre realistic situations. P'Undamental contributions have been made by De Larminat (1986), Lozano Goodwin (1985) , Praly (1983, 1986) and samson (1983) who showed that indirect adaptive algorithmS may lead to a robustly stable behaviour provided that 80lIl8 _11 defined properties of the involved parameter adaptation algorithm (PM) should be satisfied. These xobustness studies contributed, to the authors' belief, to an iJaproved understanding of adaptive control. More specifically the underlying adaptive controllers have shown to be effective in controlling those challenging processes such as non--miniJaum phase and/or unstable ones with time varying and/or nonlinear characteristics (M' 5aad et al (1986» . It is worth noting that this is the case of IIIOSt practical plants particularly when controlled digitally (Aatroaa et a1. (1984». In this paper, the authors will concentrate on those iJDperative aspects that arise when deSigning an adaptive controller I namely (1) Bow to relate the available theoretical results to the practical requi~nts. (2) Bow the ubiquitous nonidealities can be (to a certain extent) safely handled. More specifically the point of view adopted by the authors is that a combination of a control law based on receding-horizon cost function with a robust parameter estimator leada to an applicable adaptive control law. That is the global rebuilt stability will be ensured provided that the two following requi~nts should be satisfied (sa.&on (1983), Praly (1983, 1986».

Generalized Predictive COntrol stability Robustness .

I

Although the fo~r requi~nt would be by an appropriate choice of the distUr"bance model, it is _11 known that the admissibility condition is not necessarily satisfied by the perfo~

available PM. lkNeVer, it is worth noting that the estimated model adlaiss1hility pxobl_ is the exception and not the rule in the practice since it is closely related to a suitable par-*,ri.zation choice I i.e. model order and t1at-delay range. Indeed, i t is iJDperative to check the adaiBs1hility requi~nt to enable the theory work reliably. In other respects, the involved recedinghorizon cost function will be dhoeen to provide a reliable basis for linear quadratic (IQ) or generalized predictive control (Gl'C) approaches. The fo~r is mainly lIIOtivated by the availability of 1JDportant stability rebulltness results (s-on ( 1983» while the latter asserts itself as a potential approach to the adaptive control of cc.plex plants with reduced order .:ldels in realistic situations where the usual techniques _y fail. Along with the rebulltness and safe operation above-stated aspects, the authors will provide a flexible approach for iJIp~nting the considered adaptive control ~s using a cheap OOIIIPUting .PC*8r . SuCh an approach is carried out bearing in mnd those f~ntal engineering features as adaptation alertne8s, offeet-free distrubance rejection, lonq-t:era operation, illple_ntation siJaplicity and start-up procedure. Realistic siD1lation studies involving an electrical furnace Wldel based on physical laws, derived by Electricit. de Prance (EDP), will be given to demonstrate the applicability of the considered adaptive controllers. More particularly the integrity and robustness features of e&dh control law are highlighted. An outline of the paper is as follCIWII. In section II the fOEallation of the pxoblBa i8 addressed with particular 8IIIJhIUIis on the process Wldelling and the PM rebulltness. The applicability of the involved adaptive controllsrs together with the procedure whiCh is used to provide their integrity are diBCullsed in section II. section III deals with ~ concluding ~kB on the adaptive control viability.

( 1) The plant dynaaics can be approximated by a t1at-delay plus a lower-order linear Wldel I i . e. the resulting modelling error normalized by a norm of the input-output data is sufficiently _11 in the~.

(2) The estimated model is adaiBsible with respect to the underlying control law I •• g. the unifora stabilizability in suboptimal t.Q-adapt:ive controllers .

219

~1. ~r Saa d .

220

M. Duque an d I. D. La nd a u

2. PR:lBLDI PORI«1LATION

In this section, the formulation of the involved control probl_ i8 8tated. A reliable and u.eful .odel of the plant to be controlled toqether with the control objective are given . 'ftIe ~~ lying control laws are briefly revi~ . A particular a.phUis is put on the PM robu8tne88. 2. 1. 'ftIe proce88 representation The plant, which i8 to be controlled, i8 to be linearizable. Thu8 it can be, in di8crete tt.e, characterized by a linear parametric .odel . . follOOlB "8~

~

-1

l\(q

) yet) - q

-1

u(t) + v(t)

B(q

(2.1 )

where y( t) and u( t) denote the plant output and input, re8pectively I v( t) i8 a bounded disturbance I d i8 the lllinialal expected value of the DIOdel tillledelay I l\(q-1) and B(q-l) " R(q-1) (the field of pol~l function in the ~ard 8hift operatior q-1) and are of the fora I

(2.6 ) t+ph-l I: (C(q-1 )(y( i )- y *( i » )2+.>.( t )(O(q-1 )u( i»2 i-t

+

where (y*(t» is the reference sequence which can be used to t.prove the control ayat_ tracking dynamics I the polynoa1al C(q-1) may be considered . . the desired closed-loop regulation dynaaics I 0( &- 1) is a transfert function that 8hould be choIIen . . an inverve of a filtered integral oc.penaator (1.e. O(z-1) (1_Z-1)/Iq,(q-1» in order to provide, toqether with the involved .odel structure, a natural integral action in the resulting control law I ).( t) i8 a positive sequence that _y allow a 8.:)Othing tran8ient period I ph is an integer which will be at least greater than the equivalent delay of the plant (1.e. the dead-tt.e augmented by the number of the unstable _roe of the plant (5aJn80n (1982». This performance criterion is _11 ~~ 8tood and has ~ useful interpretations (Cl.arJce et al (1985», (Gawthrop and Lim (1982) , ortega et al (1984» that perlllit to easily choose the deSign parameters . PurthenlOre, simple algebraic ~ipula­ tion8 on the .:)de 1 (2. 1 )-( 2 . 3) lead to the following useful repal._trization of the plant aK)del.

(2.2 )

B(q

-1

) - ho

~

hI q

-1

~.

. .~ b

n

q

A* (q-1 ),,(t) = B* (q- 1 )Au * (t-1)-0 * (q-1 )6y * (t)

-n

where n i8 an upper bound of the degree8 of the A and B polynoaial8. I t i8 worth noting that the leading coefficient8 of the B polynomial 8hould be zero in order to cope with a greater, po88ibly variable , tillle delay and that _ do not need the plant to be minu.a--pha.ae and/or 8table . However, the following aslJUlllPt;ions have to be made. Al. 'ftIe plant is 8tabili8able l\2. n i8 known

(2.3) where the sequence (e(t» is 8ufficiently _11 in a certain sense, i . e. E{e( t » - 0 and P and G are aaywptotically 8table polynoaiala . Such a disturbance DIOde 1 (2 . 3) is .:)re suitable for the industrial proce88 disturbance aK)delling and .:)re adequate from either an .stialation and disturbance points of vi_ (TUffs and ClarJte (1985». More specifically the DIOdel (2.3) is suited to represent the unDOdelled response of the plant under control. The plant DIOdel may be rewritten . . I

-1

,,(t) - C(q

*

)(y(t)-y (t»

G(q-l) e*(t) _ e(t)

*

-1

..

-1

D (q

) - q

~+1

-1

-1

-1

)- A(q

lIe q

)C(q

I

-1

u(t) - Qo(q

..

)u (t)

I

A*(q-l) _ (l~-l)l\(q-l)

)

On( q -1 )C( q -1 )

I

I

* -1 -1-1 ) and C (q ) - C(q )F(q ).

The above parametrization i8 reliable from the control law derivation point of vi_. Indeed, the C08t function (2 . 6) may be restated . . I

J(t,ph ).,,2(t+ph)+

t+ph-1 * 2 I: 6 ( i)+'>'( t)( Au (t» 2 i-t

(2.8 )

That is the control probIe. above 8tated may be vi~ . . a regulation prob~ with respect to the DIOde 1 (2. 7 ). TWo approaChes can be used to realize the control objective, _ l y the IQ and the GPC one8. 2. 3. 'ftIe control law structures

( 2 . 4)

(2.5 )

filtering

(2 . 7)

where

B (q

On the other hand, to retain as much generality as pos8ible, the di8turbance 8equence (v( t » i8 DIOde lled as fo llOOlB .

where "f" denote8 signal G( &-1 )/1"( &-1).

* -1 )e(t) * +C(q

by

2 . 2 . 'ftIe control objective Bearing in .und the involved control law, the coat function to be atiniaized (to a certain extent) is of the fora I

Being liaited in 8pace, _ will not present the underlying control law 8tructureS. Howaver _ would ~i_ that the control objective has been carried out using the LQregulator developed by sa.Bon (1982) and a GPC law largely in8pired by the work of Clarlce et al (1986). Indeed these regulators have been applied to the DIOdel ( 2 • 7 ) in order to provide a natural integration action in the control laws. 'ftIe LQ-requlator structure involves an observer which hrings out an e8tialation of the plant state and a solution of the _11 known Ricatti equation. On the other hand, it is particularly advisable to make the final tt.e ph going to

.\pplicabilitv o f Adapti\e Control infinity fX'Olll both perfonl&nce and caDp.ltational point. of vi_. lk*eVer the follow1..ng assumption. which 18 generally _lilt in industrail enviro~nt. Al. Ay*(t) - 0 should be aide for theoretical coherence sake. It is worth _ntioning that (1) The Riccati equation 18 iterated only once.

starting each tt.a fX'Olll the resultant of the iteration performed at the previous aa.ple instant. That 18 the OOIIIPlting requi~nts are remarkably l:la1ted. (2) The control ayat_ is exponentially stable provided that the assumptions Al-Al should be _tl8fied (Hager and &orowitz (1976». The OPe is fundamentally a long-range approach. Its pith comes fX'Olll the as~ion made when deriving the multi-step-ahead predictors. Such an assumption is closely related to certain control constraint. 1.e. there is a "control horizon" beyond which all control incnt_nts vanish. possibly with 8aII8 use~specified dynaaics. In the involved context. the coat function to be minimized is the prediction of the perfonl&l\ce criterion J(t.ph). given by the equation (2.8). using available data at sample tilDe

predictive

t. _y ph ~ -1 2 J( t. ph/t >- I:
and the control constraint is made via the following as~ion

I

A4. G(q-l) 6(t+j) _ 0

for j

>

ch

( 2.10)

which is an acceptable hypothesiS even if the assumption Al is not rigorously satisfied. i.e. the reference sequence has certain dynaaics. Such an assumption may be interpreted in coat-function terms as placing an infinite weights on control evolution after a certain future time. Ne should emphasized the two following features I (1) The caDp.ltational burden is much reduced when the "control horizon". ch is small enough with respect to the "prediction horizon". ph I e.g. the ch - 0 caae reduces to a scalar computation. ( 2) The OPe robustness is fundamentally due to the degree of freedOlll provided by the choice of the handy design couple (ph.ch). 2 . 4. The adaptive control algorithm ReCent considerable efforts have been at.ed at developing parameter estimators that are robust wi th respect to process control probl_ (i.e. non-lil~arities. load disturbances and unBOdelled. possibly tilDe-varying. dynamics). The key issues to get a robust estimator are suitable data and an adv1aable PM. While the foa.r _y be ensured by data filtering and/or data normalization. the latter requires certain design features that allow its usefulness in adapt.tve control. _ly _ ensuring the unifora boundedness of the par.-ter estimates I _ preventing the adaptation gain fX'Olll going to zero or infinity I _ incorporating a _fe operation procedure which ensures the estimated model admissibility. Indeed. the above cited aspects must be delt with bearing in mind the stability require_nts that have been _11 defined by many researchers as Egardt and s-on (1982). s-on ( 1983) and De Iara1nat (1986). ASC-H·

221

In the involved context the regression model (2.4) 18 the reliable fora for par~r est1lllation. Indeed. the low pass filtering is likely to reduce the fast u~lled dynaaica. protecting therefore the PM fX'Olll large estimated ~ter variations. Moreover the 6 operator allows to obtain ze~an data that iJIprove the n..-rical robustness of the est1lllator. More specifically the dc-levela on the data do not affect the estimation performance. lk*eVer. the .adelling error 18 not necessarily bounded. it 18 then w1ae to noraalize the data in order to reduce these COIIIPOnents. This leads to the follow1..ng "suitable data" model 6yf(t) _ eT 6 .f(t_l) + e(t)

(2.11)

where "-M denotes signals normalization factor is given by (Praly (1983».

whose

1)(t) - l'1)(t-1) +- 1IIIlX(~f(t-1) 6 .f (t-1).1Io) where 0 .. I' < 1 and 110 > o. Notice that the normalization factor is

an input-output data norm. which actually ensures the boundedness of the signal before entering the PM as well as the disturbance tera e( t) of the involved suitable data model. On the other hand. an advisable PM. which is liltely to provide the needed robust parameter estimator. has been proposed in Praly ( 1983). It consists of a regularized noraalized least squares algorithm with parameter projection. say • e*(t) - ~t-1) + ~(t) P(t-1) 6 .f(t-1) y(t) y(t) - 6y (t) - e (t-1).~f(t-1) g(t) ~ l/(1)(t) + ~f(t_1)T P(t-1) ~f(t-1) P(t) > p*(t) = P(t-1) g(t) P(t-1) ~f(t-1) ~f(t_1)T P(t-1) e(t) - 80 + min(1.Rllle*(t) - 80 11 ) (e*(t) - 80)· where eo and R are the center and the radius of the projection sphere S(80.R) I y(t) 18 the est1lllation error. e( t) and P( t) denote the current parameter est1lllate and the adaptation gain. respectively . Notice that the requi~nt P(t) > p*(t) can be realized by any adaptation gain which satisfies the imperative adaptation law condition. namely gO I

..

P(t) .. g1 I

o

< gO < g1 <

Cl)

where the constants gO and g1 are generally cho8en keeping a breast with both the adaptation alertness and the allowable adaptat ion speed. To realize the imperative adaptation law property given above. _ will use a regularized constant trace algorithm that has been shown to per fora well in realistic situation (M'5aad et al (1986». i.e. P(t) - C- 1 (t) p*(t) - U(t) D(t) UT(t) I 0 < C(t) .. 1 where (c( t )} is a forgetting sequence which is OOIIIPlted such that the P( t) _trix trace 18 kept constant I thus the adaptation gain higher bound is ensured. The D(t) and U(t) _trices are provided by the popular 11-0 factorization (8ienl&n ( 1977) ) which is alaoat used as the best fora for n~rical oo.putation of the ildaptation gain _trix. P( t). The regulariaation f_ture 18 obtained by introduciftC) a lower bound. e.g. do. on the e~nts of the D( t) diagonal _trix. 1. e.

That 18 the adaptation gain lower bound 18 ensured. When ilIIpl_nting the PM presented above. two natural questions ar1ae I 1) Bow to get

222

M. I\I'Saad , M. DU411 e a nd I. D. Landall

the involved plant knowledge, l.e. the par_ter projection sphere. 2) Bow to specify the underlying design par_ters. In th18 paper we will focus on the first question, the design parU8ters are generally provided by a "cut and try" procedure . 'ft\e projection sphere 18 .,tivated, _inly, by the "boundedness of the parU8ter estt.ates that .ay be lost becauae of the PAA integral nature . BcNever, t.he latter is \IIOre appropriate to get the plant info~tions as long as they are available. That 18 a procedure which perfor.8 the following stages 18 likely to be reaaonable. ( 1) start with a variable forgetting factor which 18 aayIIIptotically equal to 1 in order to cope with a bad ayatem initialization. ( 2) Use the PAA without par_ter projection as long as the current data is providing info~tion which could improve the ~l. ( 3) Introduce the parameter projection whenever the current info~tion is not likely to ilDprove the ~lling process. Such a procedure requires certain infor_tion ~ure, a silDple _asure can be obtained by the following adaptation gain _trix norm aCt) - c(t)

~fT(t_1)

P(t-1)

~f(t-1)

sia1lation fr..-worlt . In the real-world life the adaptive controller should be able to handle nonlinearitie8, nollllliniJllulll-phase behaviour, 10llddisturbance and u~elled and t.~arying dynaIIIics. A _t~tical .,.,.1 of an electrical furnace, whose 8C~tic diagraa 18 shown in figurf 1, has been used to provide a realistic sia1latior fr..-work . Being non-linear and distributed .xJel, it perlllit8 testing, particularly, the control ayatem robwstne8s against ~elled and t~ varying dynamics. 'ft\e .,re interesting control variable is the load ta.perature which 18 bot} highly non-linear and interacting I The control problem involving the re8istance t8111P8rature vu 8hown to be relatively sillple. 'ft\e ~ipulateC variable is the injected power that was dipped tc lie in the range [O!tW, 17!tW in order to deal witi'. the actuator 8aturation limit8. 3. 2. Implemsntation aspect8 When implementing an adaptive control scheme IllUch attention should be paid to design specifications, the numerical robustness, the longterm operation and the start-up procedure . Design Specifications

Th18 nona has been previously considered by ~y authors for robwstness and stability considerations (ClarJte and Gawthrop (1979), Gawthrop and Lim (1982), ortega et al (1984, 1986». A property of a( t) 18 that it always lies in the range 0 .. a( t) < 1. More specifically a( t) ..aaures how lIIUch the recent info~tion differs from the past one I When aCt) 18 _ller than certain preset level aO' that would usually be easily detenained from Simulation, the current inforaation looks lIIUch the s _ as the past one. In other respects the projection sphere S(90,R) _y be viewed as an adaissible domain for the estt.ated ~el . 90 _y be chosen as the TORUlt:ing parameter vector when the PAA 18 used without a parameter projection. R should be .-11 enough and .ay be provided by preliminary worthwhile identification exper~nts ( Landau et al (1985». Given the parameter estimate sequence (e( t)}, the adaptive control law 18 carried out by replacing the plant ~el parameters e by their estimates e( t). Rafering to S _ n (1983) and Praly (1983, 1986), the involved adaptive control laws allow a stable closed loop behaviour provided that ( 1) there exists a par_ter vector e* such that the no~l1&ed ~elling error A(y(t)~*T ~t-1»/T)(t) should be sufficiently small in the _an. (2) the estt.ates ~l should adaU.ssible. Although the second requi~nt would be perfoD8S by a suitable choice of the d18tu%bance process ~l, l.e. the G and P pol~ls in equation (2.3), it 18 indeed well known that the .,.,.1 adaiasibility requi~nt 18 not necessarily fulfilled by the available PAA. BcNever the adaU.asible problem 18 the exception and not the rule since it arises particularly when the ~el order has not been suitably chosen. Thus it 18 Utperative to introduce a safe operation procedure to check the adaU.asibility condition in one way or another, providing hence the adaptive controller integrity.

There 18 a variety of deSign specificat.ions that would be used to provide an acceptable closed-loop perfo~ce. certain uaeful guidelines of the design parameter choice in the perfo~ function, i.e. C(q-l), Q(q-1), A(t) and y*(t) , are available in the literature (see M' Saad et al (1986) for \IIOre details) and should be used to provide a tighter control. In other respects the deSign parU8ter in the PAA should be chosen bearing in aind the following guidelines I ( 1 ) let the adaptation proceed slowly after the transient period, i.e . the adaptation gain bounds, go and gl' should be as .-11 as possible (2) 5aJIIple slowly enough to reduce the effect of unlllOdelled dynamics. It. is worth recalling, however, that the sampling period is closely related to the desired cloaed-loop dynamics. On the other hand, the PAA has been made able to deterllline by itself a reasonable estt.ate of the projection sphere center I Only its radius has to be chosen. One rule of th\lllll), having proved to be effective in practice, is R - r(g1/90)lI 2 with O
NUBarical AObuatness 3. THE ADAPTIVE COlII'l'R)L APPLICABILITY

Thi8 section discU8se8 the effectivene88 of the involved adaptive controller relating to a wide range of control probl_ u8ing a realistic 8iaJlation fra.eworJt . 3. 1. The controlled plant siallator It. 18 easy to get adaptive control

algoritha

to

perfona

well

under

an

ideal1&ed

In order to illprove the n~rical properties of the involved adaptive controllers, the illplemsntation of both the adaptation gain ( P( t » and the Riccati equation ( R( t » has been carried out through U-O factorization and the GPC control law has been calculated USing Cholesky's technique as the involved _trix 18 aI_ye positive definite provided that the floating-point zero of the computer should be used as lower bound for the sequence (A( t )} .

223

.-\pplicability o f Ad a ptive Co ntro l

Lonq=tera operation Aa mantionnad in the previous section, the integrity of the adaptive controller is . . inly based on P~ requiramants. The most offending ones are the unifora boundednese of the par_ters eet:t.atee and the identified lIIOdel adIIl1ee1bil1ty (i.e. the unifora etabilill&bility i n the caee of L!rapproach and the unifora regularity in the caee of Gl'C-approach) . The former ie performed by the par-..ter projection wile the latter neede ec.e ..,nitoring together with a correction procedure of the ~ter eetu.atee. Aa the admiee1bil1ty requiramant ariaee particularly through wrong par-..tr1zation, a new lIIOdel order ehould be epacified loIhenever the etability property i8 no longer ensured . Por instance it i8 always po881ble to run .ultiple par_ter e8tu.ators in parallel, e&Ch with a corre8ponding order, and use the be8t one in terae of the admi88ibility requiraD8nt together with eufficient _11nee8, in the maan, condition on the noraalized e8timation error and the par_ter e8t1lllate evolution. In particular , thl8 probl_ i8 completely removed i f all the lDOdele in the projection ephere are admi8eible. Indeed, a ba.cIt-up "gain" echeduling controller with a probing eignal have to be deeigned to deal with the admi88ibility problem &8 outlined in Andereon and Johnstone (1985 ) and Janeclti ( 1986). start- up procedure Many approache8 have been used, the better one 18 to etart inclosed-loop with a relay having hysterl8ie, &8 outlined in Aatro. and B&991und (1984), until a reasonable e8timate i8 obtained. The hy8teri8i8 in the relay i8 basically introduced to reduce the dieturbance influence. Furthermore the deeign par_ter (>.( t )} may be choeen &8 t:iJDe--varying sequence according to the allowable adaptation dynamic in order to s.ooth the traneient behaviour during the initial period of the actual adaptive control, l.e . >'(0) exp (-t/z)

i f >.(t) • >.

>.(t) >.

otherwi8e

loIhere >. 18 the de8ired final value of (>.(t)}. 3. 3 . S1a1lation Re8ult8 The re8ulting approach has been experimentally inve8tigated . several 8imulation experimanh involving di8turbed and time-varying linear JDOdels have been performed . They will not be reported in thie paper for our being 11a1ted in 8pace (Bee M'S&ad et al (1986) for more details). ~ver _ will deal with thoae fund_ntal features that can be vi~ as a reliable OClIIIPIlrieon hasis of tha LQ and GPC-approaches. (1) When the plant lIIOdel is known, the LQcontroller is stable for all values of the control _1ghting par_ters, unlike with the GPC loIhere instability . .y re8ult if the prediction and control horizons are not suitably cho. . n . Indeed the latter approach is not yet theoretically understood, m-ver there exists always a couple (ph, ch) wich stabilize8 the control syet_. The prediction horizon, ph, corresponds 1IIOr8 cloaely to the r1ae-ti8e of the plant under test while the ph control horizon, ch, which . .y be inte~retated as the control signal aettl1ng-ti8e, is not &ally to Choose, particularly when the involved plant dynBaics are not _11 dUip8d, i.e . poles on or outside the stability d~in. Thl8 18 in good ~nt with Clarke et al (1985). (2) The control _ighting sequence (>.( t)} can be ...se _11 enough to provide a reference sequence

tracking capability wen the GPC 18 uaed. This 18 not usually the caee with the LQ-OOntroller. ( 3) The involved P~ 18 quite robust against ~ter variations, load dl8turbances and ~ delled dynaaics. Indeed the low-pass filtering and/or data noraal1zation operations are particularly a necessity loIhenever wrong assu.pt;ions about the lIIOdel order &8 _11 as the disturbance .,.sel are

aade. (4) In the adaptive context, the GPC- &8 _11 &8 the L!r-approaches have been found insensitive to under--par_tr1zation. Of particular interest is the robustne8s of the Gl'C to over--par-..tr1zation wich i8 a suitable property in the tu..-varying dynaaic caee. The L!r-approach generally l8ad8 to a rapidly varying control action when the lIIOdel oEder is over-e8t:t.ated . (5) The safe operation conception 18 easier with the LQ-OOntrol law than with the Gl'C-law. Indeed, the latter has not been fully investigated, l.e. no theoretical 8tability re8ult8 are available as far &8 the author8 are aware of . More epacifically the concept of identified lIIOdel ada18sibil1ty 18 not yet defined for the GPC. Neverthele88 i t 18 expected that thi8 concept . .y be at least related to the unifora regularity of the matrix involved in the control law derivation. That ie Gl'C- &8 _11 as L!r-approaches, when coupled with a 8uitable par_ter e8t:t.ator, are 8ufficiently flexible and robust to &Chi_ the adaptive control appl1cability. ~er, there i8 ec.e looaene88 in the perfor.ance definition that prevent8 to easily relate the de8ign par..ters to the closed-loop dynamic re8ponse. Nonethele88 there i8 usually a "cut and try" procedure that allow8 to obtain stable control law and then interactively lIIOdify the de8ign par..tere to 1IIIprove its perfor.ance. The8e re8ult8 motivated comprehensive 81a11ation 8tudie8 including a non-linear distributed .,.sel of an electrical furnace operating over a large range that both the adaptation alertne8s and the robustne88 with re8pect to wrongly ass~ IDOdel order ~ illperative properties. In a fir8t time open-loop lIIOdelling experi8ent8 _re performed, using Par_tric Model Identification pacJtage PIN running on Is.-PC and OOIIP&tible8 ( Land&l1 et al ( 1985 ) ), to detera1ne the .,re suitable par_tr1zation of the involved furnace IDOdel. This led to the foll_ing par~r1zation PAR 2 {d-1, na-1, nA-2}. However, two other paraMtr1zations PAR 1 {d-l, na-A-1} and PAR 2 {d-1, nS-2, nA-3} _re uaed to illustrate the robuatne88 of the Gl'C and LQ approaches against u~lled dynaaica. on the other hand Gl'C and LQ approaches have been oc.pared under the _ conditions . More 8pacifically, the control law de8ign par..tere _re - the aa.pling period T - 25 eeoonds - y*(t) - (.05/(1-.9Sq- l) u*(t) Where (u*(t)} 18 a aet-point sequence that vas initially fu.! at 4720C and vas followed by a equare-vave changing every 200 8aIIIPle8 with a&gnitures 4000C and 545OC. - G(q-1) • p(q-1) • 1 - C(q- 1) _ 1-.8q-1, >.(t) e [0, 1 . 5] l ) - 1--q-1, db - 0, ph e [10, 25] for the Gl'C

- occr

approach

- OCq-1) - (1-q-1)/( 1-.9Sq-1) for the L!r-approadt. Thl8 choice vas nece8sary to reduce the control activity given the all~le range for the sequence (>.(t» •

_ G(q-l) _ 1, p(q-1) _ 1-.9Sq-l - go - . 001, gl - 5, - l' -

. 5, Po - 2 •

ao • •001

224

~1. ~rSa a d.

M. DUqllC and J. D. Landall .

Figures 2 and 3 show the load ta.peratun, the reference sequence and the control signal for the par_trization PAR 2. Notice that the GI'C prov14es a nlatively better reference sequence tracking capability but the intial stage is better with the L\r&PProach. Figure 4 shows how the control signal behaves poorly vhen the I.\radaptive controller is WNId with Q(q-l) _ l-q-l. Figuns 5 and 6 show the control ayat_ performance for the par_trization PAR 1 . In this ~, the ~ltable tracking capability of the GPC was obtained by fixing the costing sequence {~( t to the floati~point zero of the caaputer. s.&11 values of ~(t) led to bad behaviour when the I.Q awroach was used. Figuns 7 and 8 show the control ayat_ behaviour when the par_trization PAR 3 was WNId. Notice the control signal oscillations, with the LQ awroach, that correspond to unstable pole-zero cancellations in the identified .:>del. The GPC provides a better output activity but once again the initial stage is not as smooth and sluggish as is the case with the I.Q approach. TO summarize the parametrization PAR 1 is definitively appropriate for the GPC while the LQ behaviour is IIIOn acceptable for the parametrization PAR 2. Purthermore the par_trization PAR 3 has to be avoided vhen using the LQ approach. 'l'tIat is, the GPC is less sensitive to the underlying para.etrization as outlined in the simulation studies USing linear disturbed .:>delB. Moreover it is worth noting that other experiments, not nported hen, involving greater values of the control horizon did not lead to notable perfonoance illprov_nts.

of adaptive control will, at the authors belief, continue to expand the industrial world when the ubiquitous PlO controllers are .till used.

»

COIICLDSIONS The potential practical prObl-a that has pntvented the adaptive control applicability have been addressed in this paper. More specifically a flexible approach for impl_nting adaptive controllers u.aing a cheap OOIIIPJting ~r, l.e. IBM-PC and oo.patLbles, . has been outlined. Such an awroach ~s its broadness of applicability to those funcS_ntal deSign features as I rObu.at stability, adaptation alertness, free-off. .t Eequlation, t~elay OOIIIP8nsation, safe operation and illpl. . .ntation sillplicity. These intensting deSign features legit:lJllate, in one way or another, the approx:lJllation of the industrial plant behaviour by t~elay plu.a a l<*8r-order linear .:>del on vhidh the advanced control theory is based. Important insights and tools for dealing with the involved approach have been developed by many authors . The lIIOtivation of this paper vu twofold (1) Bow to robustly carry the design of an adaptive controller bearing in IIlind the recent theoretical nsults. (2) Bow to provide the applicability of the underlying adaptive controller. On the other baneS, CClIIIPnhenaive s1mulationa studies have shown that the involved control awroach has actually the design f_tuns stated al:Iave. Mon specifically it has been sucessfully applied to the t811p8rature control of an electrical furnace via a realistic . .t~tical .adsl I i . e. the adaptive controllers have proved to be effective in ntducing both the control and the te.peratun activities. It is however important to ..phaBi&e that a priori identification experiments .hould be done in order to .alut easy the de.ign ~er Choice _ _ 11 _ the supervisory level conception • The conclu.aion frea this work is that the robustnes. studies contributed to an illproved unlSeratanding of adaptive control vhidh is getting better all the tt.e and can already perforll _11 in various expert.ental enviro~nts • Purther.orB _ the aicroprocesaorB are steadily getting f_ter and IIIOre ~rfull, the range of practical applications

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Applicability of Adapti\e Control 1.lI:

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225

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_led

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ua:

Ooel,;u1

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Cc::-.ande adaptatlve

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[6]

lit: 1.111 C: 1·1."('11 lit: 1,1.1\1('11 f:

I

J.oMQ: i Qj:

III:li ,,,I lII:i

I

I

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syst-",

ua:

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noel,; ..1 ,.,.... (lAC) [8]

I

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1980.

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1.

t; f:r~('I) f: 1"" .('1) lrf..k:('1l

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Gawthrop P.J. and I::.W. La (1982). "R0bu8tne•• of aeU-tuning controllera", Proe. tEE, vol. 129, pt.D, no I, pp . 21-29, 1982.

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10Itl1

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rw...o ICI'()

Duturbancee" , rED

".

j:~!i•..Jilll;i

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t; !f.I\1('1l F.1-t·kl'l) Itt" 1.1

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<1DIV:21IIC;

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vol. AC-30, nO 6, pp . 589-592, 1.985.

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I

a

unified