Improved efficiency of adaptive robust control by model unfalsification

*Corresponding author.Tel.:441214144346;fax:4401214144291;email:s.m.veres @.This paper was not presented at any IFAC meeting.This paper was recommended for publication in revised form by Editor PeterDoratoAutomatica 35(1999)981}986Technical CommuniqueImproved e $ciency of adaptive robust control by modelunfalsi "cation 1Hao Xia,Sa ndor M.Veres *School of Electronic and Electrical Engineering,The Uni v ersity of Birmingham,Edgbaston,B152TT,UKReceived 4April 1998;revised 1September 1998;received in "nal form 1December 1998AbstractThe adaptive robust control scheme introduced by Veres and Sokolov (Automatica ,34,723}730)provides optimal asymptoticperformance under unmodelled dynamics,unknown disturbance bounds and unknown model orders that make the scheme desirable in practical applications.The amount of numerical computations associated with that scheme is large because of the possible large complexity of the appearing polyhedra.The geometrical details of the polyhedra need large memory space and their updating a lot of computation.This note gives a modi "cation of the scheme so that polyhedron complexity is limited.Asymptotic performance of the scheme is the same as that of the original adaptive scheme in Veres and Sokolov (1998),transient performance can be worse.Simulation illustrates that practical implementation of the scheme is feasible. 1999Elsevier Science Ltd.All rights reserved.Keywords:Adaptive control;Robust control;Self-tuning systems;Bounded disturbances;Bounding methods1.IntroductionThe existence of a globally convergent scheme under unknown model orders,unmodelled dynamics and noise bounds has been proved in Veres and Sokolov (1998).The aim of this note is to modify the scheme so that polyhedral complexity is substantially reduced by relax-ing the polyhedral feasibility sets after each updating step.Simulations show that with the modi "ed scheme it is not impractical in computing e !ort to "nd a suitable controller by using commercial toolboxes (GBT,Veres et al.(1993}1998)and Optimisation,Grace and Branch (1996)).Another improvement upon (Veres and Sokolov,1998)is that a more sophisticated disturbance model is considered here.2.Description of the schemeGiven a SISO plant,our objective is to present anadaptive controller for the servo problem so that theplant output follows a given reference signal.In this approach approximate linear models will be described by extended model parameters which include model uncer-tainty with regard to unmodelled dynamics and distur-bance (Veres and Sokolov,1998).The plant model is assumed to be of the form y I #a y I \ #2#a L y I \L "b u I \B #b u I \B \ #2#b K u I \K \B #P (y I )#;(u I )#e I,(1)where ;and P are bounded linear operators ;:l P l,P :R O P R and e 3l.The meaning of the di !erent termscan be explained as:P represents parametric perturbations of process dynamics;represents unmodelled dynamics which is accounted for in the robust control designe Irepresents unstructured additive disturbance.The unmodelled dynamics is assumed to satisfy a condition";(u I )"4 #u I #for all k 51(2)with u I "[u ,u ,2,u I ]2,the parametric perturbationan inequality "P (y I )"4 # W I #(3)0005-1098/99/$*see front matter 1999Elsevier Science Ltd.All rights reserved PII:S 0005-1098(98)00234-9Fig.1.Block diagram of the modi "ed scheme.and the additive disturbance#e # 4(4)for some unknown '0, '0,'0,whereW I "[!y I \ ,2,!y I \O ]2if k ' ,[!y I \ ,2,!y]2if k 4and 5n is a priori "xed.Let A (q \ )and B (q \ )be the usual polynomials in terms of the backward time-shift operator q \ :A (q \ )"1#a q \ #2#a L q \L ,B (q \ )"b #bq \ #2#b K q \K .The notation "[a ,a ,2,a L ,b ,b,2,b K]23R L>K> is used for the associated parameter vec-tor.To include uncertainty,the extended parameter vec-tor "[ 2, , ,]2is introduced under each modelstructure "(n ,m ,d )2,04n 4N ,04m 4M ,14d 4D .Also,a priori "xed compact set M JL R L>K> of model parameters is given for each model structure 3S ,which is assumed to be the union of a "nite set of polyhedra.The controllers associated with extended parameter vectors will be de "ned as follows.Denote the output reference by r I,k '1.For expository simplicity,linearcontrollers of the formR (q \ )(1!q \ )u I "<(q \ )(r I !y I)(5)withR (q \ )"1#r q \ #2#r L Pq \L P ,<(q \ )"s #s q \ #2#s L Qq \L Qwill be used.The nominal closed-loop characteristic polynomial will be denoted by "AR (1!q \ )#q \B BS .Hence the controller will be associated with rather than .The performance criterion will be de "ned under a valid plant description and model "( 2, , , )2byJ ( )"J (u ( )" *):"sup E lim sup I"r I !y I "w I,(6)where on the right-hand side the supremum over is to be understood over all possible perturbations and addi-tive noise allowed by .w Iis a weighting function used toexpress the importance of keeping the error of tracking small at certain times.The scheme presented is not lim-ited to this class of controllers,it can be extended to other classes of controllers without any di $culty.This note is not concerned with the computation of cost function J ( ),examples of that can be found in Sokolov and Veres (1997)and Veres and Sokolov (1998).First,the set of unfalsi "ed parameter vectors is intro-duced as it was done earlier in parameter bounding (Veres and Norton,1993)for adaptive control.Following the terminology by Safonov and Tsao (1994),Kosut (1996)and Smith and Doyle (1992)the feasible parametersets will be called unfalsi "ed model sets.For a given model structure 3S the unfalsi ,ed set of extended para-meters at time k is de "ned byF J I "M J 5I7R IH J R,where M Jis an a priori de "ned set of extended parametervectors under model structure and H J I :"+ "y I 4 2 >I ,5+ " 2 \I 4y I ,.Here the regression vectors>I :"[ I ,# W I # ,#u I # ,1]2, \I :"[ I ,!# W I # ,!#u I # ,!1]2 I "[!y I \ ,2,!y I \L ,u I \B ,2,u I \B \K>]2are used.Then the unfalsi "ed set of models is de "ned as the unionF I :"8J Z 1F J I ,F "M "8J Z 1M J(7)Clearly,each F Iis the union of a "nite number ofpolyhedra.Fig.1describes how the modi "ed adaptive scheme works.To any extended parameter vector "[ 2, , , ]23Mthere is a controller u ( )de "nedby Eq.(5).For each model structure 3S and each sampling time k an estimate J Iis selected from an en-larged set M J I M F J I.Then the model structure estimateis obtained by minimising J .At each time k the control-ler associated with the estimate J I is used,where andJ I minimises J over M I.982H.Xia,S.M.Veres /Automatica 35(1999)981}986J I is obtained by constrained optimisation over a modi"ed unfalsi"ed set M I.The exact unfalsi"ed poly-hedron set F I can have a large number of vertices and facets which are computationally demanding to take all into account.Let the update of the unfalsi"ed set in model structure beM J I> :"M J I5H J I> .For each 3S the feasible set M J I is replaced by a simpler polyhedron set M J I.Then J I> is obtained by constrained optimisation over M J I> ( I> is also selected so that thecontroller associated with J I>I> can be applied for con-trol),and to obtain M J I> the set M J I> is&&relaxed''by only retaining its facets around J I> .As the proof below shows,the exact form of the relaxation procedure is not that important apart from that a simpler polyhedral set M J I> is derived from M J I> which satis"es M J I> L M J I> .These enlarged polyhedral sets M J I> , 3S will then be used to de"ne a newM I> :"8J Z1M J I> .(8) The scheme shown in Fig.1has the following properties.Theorem1.¸et the model structure 3S be,xed and use the notations I:" J I,H I:"H J I and M I:"M J I.¸et *3M J be an extended parameter vector that provides a valid description of the plant under any circumstances.¹hen at any time k:(a) *3M I.(b)J( I)4J( *).(c)J( I)4J( I> ).(d) I> 3K( *)where K( *)is an extended parameterset de,ned byK( *):"+ "J( )4J( *),5M (9) (e)If dist( I,M I5H I> )5 then the set¸I:"+ "dist( , H)5 ,j(k;J( )5J( I),is non-empty as any valid parameter *is contained in it.(f)All di+erent I s are separated by at least a distance .(g)If K( *)is compact then there is only a,nite number ofdi+erent I s in this procedure.Proof.(a)As M I contains the set of extended parameter vectors feasible with the data up to time k,any valid description *of the plant must be contained in it. (b)By de"nition I minimises J( )over M I5H I> while *3M I5H I> hence J( I)4J( *).(c)By de"nition I> is obtained by minimisation of J( )over an area of where J( )'J( I).(d)This follows from(b)by using k#1instead of k.(e)As *3M I5H I> it follows that dist( *, I)5 for any I that has been rejected.If I is rejected then all previous H,j(k has been rejected.Also J( *)5J( I) by part(b).(f)By de"nition I is only changing if dist( I, M I5H I> )5 and then I> 3H I> ,which implies dist( I, I> )5 .(g)By(d)and assumption *3M we have that all I3K( *),k'0where K( *)is a compact set,therefore there is only a"nite number of di!erent I,i.e.there is a k (R such that I" Ifor k5k .)3.Convergence of performanceIn order to prove the convergence of the modi"ed scheme,the following assumptions will be made.1.The initial set M of is the union of a"nite set ofpolytopes.2.There exists a description of the plant with somesuitable model structure 3S and extended parameter vector *"( *2, * , * , * )23M J ,which is unfalsi-"ed in all experiments.3.For every 3M ,there is an associated control law u( ),which satis"es J(u( )" "( 2, , , )2)(R, 3M .4.For some '0there is a Lipschitz constant over thea priori set M :¸(D):"supJ Z1supB "\;sup"E J\E J"4B E J E J Z+"J(u( )"( 2, , , )2)!J(u( )"( 2,, , )2)"(10) such that¸(D)(R.Assumptions1and2are quite standard and are easy to satisfy.Assumption3says that we only consider mod-els for which we have suitable control law when the model is known.Assumption4is a rather strict con-straint but it does not have to be satis"ed for convergence of control performance,it helps us to evaluate guaran-teed asymptotic performance relative to any valid robust model(1).The main result on the modi"ed scheme is stated in the following theorem.Theorem2.Given a design parameter '0, (D,an a priori,nite set M of polytopes in the extended parameter vector spaces.¸et *denote a model of the plant that is always valid.(i);nder Assumptions1}3the adaptive controlscheme described above is convergent in the sense that there exists a,nite number N such that I" ,"( 2,, , , , , , ),k5N and *"( 2,, , # , , # , , # )remains a valid description of the plant for k5N.¹hen the inequalityJ( ,)4J( *)(11)H.Xia,S.M.Veres/Automatica35(1999)981}986983Fig.2.Reference and output for the simpli "ed and original (Veres and Sokolov,1998)schemes.holds and J ( *)is an upper bound of the actual asymp -totic performance .(ii)If Assumption 4holds thenJ ( *)4J ( *)# ¸(D ).(12)Remark 1.Note that asymptotic performance of this control scheme is the same as that of the original in Veres and Sokolov (1998).However slower convergence of per-formance can be expected.Remark 2.The "rst part of Theorem 2claims that the asymptotic performance of the system,measured by J ,will be bounded by the worst-case performance of *"( 2,, , # , , # , ,# )2(R .Since the worstcase performance J is continuous in , and,thedi !erence between J ( ,)4J ( *)and J ( *)can be rathersmall if a small is chosen.Remark 3.From the modelling point of view,both *and *are valid descriptions of the plant,since both of them are unfalsi "ed by the data.As *is an arbitrary valid description of the plant,it can be better than *from the general control performance point of view.The performance di !erence between them is bounded by ¸if Assumption 4holds.Proof.The proof is similar as in Veres and Sokolov (1998)with the only di !erence appearing in the fact that the unfalsi "ed sets for a given model structure do not form a monotone decreasing set.This relaxation will be compensated by the condition that the optimised cost function minima are required to be monotone increasing which will eliminate the potential problems arising from the non-monotone feasibility sets.)4.A simulation exampleOur aim is to show here that the amount of computa-tion involved does not make the scheme infeasible in practical implementations.Asymptotic performance is the same as in the scheme in Veres and Sokolov (1998).Although the plant simulated will be a fourth-order system,the controller will use a third-and a second-order model with di !erent time delays to approximate the plant.Hence the practically relevant situation is con-sidered where the plant order is unknown.The plant will be simulated by the equation y I "B (q \ )A (q \ )u I #1A (q \ )e I,(13)984H.Xia,S.M.Veres /Automatica 35(1999)981}986Fig.3.Cost functions and switching between models. (*)and (#)indicate locations of model updating.whereA(q\ )"1#0.8q\ !0.4725q\ !0.2205q\ !0.2q\ , B(z\ )"1.4q\ #0.3q\ #0.1q\ # I q\ ,I and e I are pseudo-random with uniform distribution in[!0.025,0.025]and[!0.05,!0.05],respectively. Plant(13)is a fourth-order system system with a small time-varying dynamics and has a relevant unstable pole at q"!1.1683.Two models will be made to&&compete''in the simula-tion which will be indexed by and:y I#a y I\ #a y I\"bu I\ #b u I\ #P(y I)#;(u I)#e I,(14) :y I#a y I\ #a y I\ #a y I\"bu I\ #b u I\ #b u I\ #P(y I)#;(u I)#e I. The model sets represented by and are clearly disjoint.The dimensions of the extended parameter vec-tors will be7for model structure and9for model structure .In each model structure a pole placement controller(placing the poles at0.5and0)will be com-puted and applied on-line based on the current estimate of the extended parameter vector.To make it typical,the reference signal is a square wave altered with some ran-dom steps.A large a priori extended parameter set M J for is de"ned as an axis aligned box with diagonal vertices [!3,!3,!5,!5,0,0,0]2and[3,3,5,5,1,1,1]2.An ex-tended parameter set M J for is de"ned as an axis aligned box with diagonal vertices[!3,!3,!3, !5,!5,0,0,0]2and[3,3,3,5,5,1,1,1]2.Over these large initial sets the control cost function does not satisfy Assumptions3and4,still the scheme works.The asymp-totic parameter estimates may approximate not the true model,but a description of the plant which remains valid during the future runs.There is a subtle di!erence be-tween valid description of a plant against a given class of output references and controllers and a plant description via model(13)which is valid under any circumstances.The graphs in Fig.2show the results as the simulation progressed:the step reference and the output are super-imposed on the top graph of Fig.2.Note that the noise step responses are caused by the high level of dynamical disturbance I and large source noise e I is simulated. Tuning of the controller was achieved roughly within the "rst60sampling period after which the model order used was stabilized as shown in the bottom graph in Fig.3. Evaluation of the optimised cost functions are shown forH.Xia,S.M.Veres/Automatica35(1999)981}986985each model structure in the top of Fig.3:the&&*''and &&#''notation is used for model structures and , respectively.The*and&&#''signs are placed whenever actual model updating happened.It is clear that the model under structure ,which was"nally used for control,was still re"ned up to time period205.The computation of the"nal estimate takes about14000 Flops per sampling period under Matlab5.2,GBT6.0 Veres et al.(1993}1998).Using the original method (Veres and Sokolov,1998)this was about70000Flops per sampling period on average.ReferencesGrace,A.,&Branch,M.A.(1996).¹he Optimisation¹oolbox,<ersion1.5.2.Mathworks Inc.Kosut,R.L.(1996).Iterative adaptive robust control via uncertainty model unfalsi"cation.Proc.1996IFAC=orld Congress,San Fran-cisco,CA.Safonov,M.G.,&Tsao,T.C.(1994).The unfalsi"ed control concept and learning.Proc.33rd Conf.on Decision and Control (pp.2819}2824).Smith,R.S.,&Doyle,J.C.(1992).Model validation:A connection between robust control and identi"cation.IEEE¹rans.Automat.Control,AC-37,942}952.Sokolov,V.F.,&Veres,S.M.(1997).Adaptive robust steady-state tracking control.Proc.ACC+97,4}6June,Albuquergue,NM. 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