- Email: [email protected]
Improved efficiency of adaptive robust control by model unfalsification
Automatica 35 (1999) 981}986
Technical Communique
Improved e$ciency of adaptive robust control by model unfalsi"cation1 Hao Xia, SaH ndor M. Veres* School of Electronic and Electrical Engineering, The University of Birmingham, Edgbaston, B15 2TT, UK Received 4 April 1998; revised 1 September 1998; received in "nal form 1 December 1998
Abstract The adaptive robust control scheme introduced by Veres and Sokolov (Automatica, 34, 723}730) provides optimal asymptotic performance 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. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Adaptive control; Robust control; Self-tuning systems; Bounded disturbances; Bounding methods
1. Introduction The 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 relaxing 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 scheme Given a SISO plant, our objective is to present an adaptive controller for the servo problem so that the * Corresponding author. Tel.: 44 121 414 4346; fax: 44 0121 414 4291; email: [email protected]. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Editor Peter Dorato
plant output follows a given reference signal. In this approach approximate linear models will be described by extended model parameters which include model uncertainty with regard to unmodelled dynamics and disturbance (Veres and Sokolov, 1998). The plant model is assumed to be of the form y #a y #2#a y "b u #b u #2 I I\ L I\L I\B I\B\ #b u #P(y )#;(u )#e , (1) K I\K\B I I I where ; and P are bounded linear operators ; : l Pl , P : ROPR and e3 l . The meaning of the di!erent terms can be explained as: P represents parametric perturbations of process dynamics ; represents unmodelled dynamics which is accounted for in the robust control design e represents unstructured additive disturbance. I The unmodelled dynamics is assumed to satisfy a condition ";(u )"4e #u # for all k51 (2) I I with u "[u , u , 2 , u ]2, the parametric perturbation I I an inequality "P(y )"4e #t W # I I
0005-1098/99/$*see front matter 1999 Elsevier Science Ltd. All rights reserved PII: S 0 0 0 5 - 1 0 9 8 ( 9 8 ) 0 0 2 3 4 - 9
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and the additive disturbance #e# 4e (4) for some unknown e '0, e '0, e '0, where [!y , 2 ,!y ]2 if k'q, I\ I\O t W" I [!y , 2 ,!y ]2 if k4q I\ and q5n 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 q\L, B(q\)"b #b q\#2# L b q\K. The notation h"[a , a , 2 , a , b , b , 2 , K L b ]2 3RL>K> is used for the associated parameter vecK tor. To include uncertainty, the extended parameter vector g"[h2, e , e , e ]2 is introduced under each model structure l"(n, m, d)2, 04n4N, 04m4M, 14d4D. Also, a priori "xed compact set MJ LRL>K> of model parameters is given for each model structure l3 S, 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 , k'1. For expository simplicity, linear I controllers of the form
R(q\)(1!q\)u "<(q\)(r !y ) I I I with
(5)
R(q\)"1#r q\#2#r q\LP, LP <(q\)"s #s q\#2#s q\LQ LQ will be used. The nominal closed-loop characteristic polynomial will be denoted by s"AR(1!q\)# q\BBS. Hence the controller will be associated with h rather than g. The performance criterion will be de"ned under a valid plant description and model g" (h2, e , e , e )2 by J(g)"J(u(h)"g*) " : sup lim sup "r !y "w , (6) I I I E I where on the right-hand side the supremum over g is to be understood over all possible perturbations and additive noise allowed by g. w is a weighting function used to I express the importance of keeping the error of tracking small at certain times. The scheme presented is not limited 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(g), 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 introduced 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 parameter
sets will be called unfalsi"ed model sets. For a given model structure l3 S the unfalsi,ed set of extended parameters at time k is de"ned by I FJ"MJ 5 7 HJ , I R RI where MJ is an a priori de"ned set of extended parameter vectors under model structure l and HJ " : +g " y 4g2 > ,5+g " g2 \4y ,. I I I I I Here the regression vectors
> " : [t , #t W # , #u # , 1]2, I I I I W
\ " : [t , !#t # , !#u # , !1]2 I I I I t "[!y , , !y , u , 2 , u ]2 I I\ 2 I\L I\B I\B\K> are used. Then the unfalsi"ed set of models is de"ned as the union F " : 8 FJ , F "M "8 MJ (7) I I JZ1 JZ1 Clearly, each F is the union of a "nite number of I polyhedra. Fig. 1 describes how the modi"ed adaptive scheme works. To any extended parameter vector g"[h2, e , e , e ]2 3M there is a controller u(g) de"ned by Eq. (5). For each model structure l3 S and each sampling time k an estimate gJ is selected from an enI larged set MJMFJ . Then the model structure estimate I I lL is obtained by minimising J. At each time k the controller associated with the estimate gJ( is used, where lL and I gJL minimises J over M . I I
Fig. 1. Block diagram of the modi"ed scheme.
H. Xia, S.M. Veres/Automatica 35 (1999) 981}986
gJ is obtained by constrained optimisation over I a modi"ed unfalsi"ed set M . The exact unfalsi"ed polyI hedron set F can have a large number of vertices and I facets which are computationally demanding to take all into account. Let the update of the unfalsi"ed set in model structure l be I J " M : MJ 5 HJ . I> I I> For each l 3S the feasible set M I J is replaced by a simpler I polyhedron set MJ . Then gJ is obtained by constrained I I> optimisation over MI J (lL is also selected so that the I> I> controller associated with gJL I> can be applied for conI> trol), and to obtain MJ the set MI J is &&relaxed'' by I> I> only retaining its facets around gJ . As the proof below I> shows, the exact form of the relaxation procedure is not that important apart from that a simpler polyhedral set MJ is derived from M I J which satis"es I> I> M I J LMJ . These enlarged polyhedral sets MJ , I> I> I> l3 S will then be used to de"ne a new M " : 8 MJ . (8) I> I> JZ1 The scheme shown in Fig. 1 has the following properties. Theorem 1. ¸et the model structure l3 S be ,xed and use the notations g " : gJ , H " : HJ and M " : MJ . ¸et I I I I I I g*3 MJ be an extended parameter vector that provides a valid description of the plant under any circumstances. ¹hen at any time k: (a) (b) (c) (d)
g*3M . I J(g )4J(g*). I J(g )4J(g ). I I> g 3 K(g*) where K(g*) is an extended parameter I> set de,ned by
K(g*) " : +g"J(g)4J(g*),5M (9) (e) If dist(g , M 5 H )5e then the set ¸ " : I I I> I +g " dist(g, g )5e, j(k; J(g)5J(g ), is non-empty as H I any valid parameter g* is contained in it. (f ) All di+erent g s are separated by at least a distance e. I (g) If K(g*) is compact then there is only a ,nite number of di+erent g s in this procedure. I Proof. (a) As M contains the set of extended parameter I vectors feasible with the data up to time k, any valid description g* of the plant must be contained in it. (b) By de"nition g minimises J(g) over M 5 H I I I> while g* 3M 5 H hence J(g )4J(g*). I I> I (c) By de"nition g is obtained by minimisation of I> J(g) over an area of g where J(g)'J(g ). I (d) This follows from (b) by using k#1 instead of k. (e) As g* 3 M 5 H it follows that dist(g*, g )5e I I> I for any g that has been rejected. If g is rejected then all I I previous g , j(k has been rejected. Also J(g*)5J(g ) H I by part (b).
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(f ) By de"nition g is only changing if dist(g , I I M 5H )5e and then g 3 H , which implies I I> I> I> dist(g , g )5e. I I> (g) By (d) and assumption g*3 M we have that all g 3 K(g*), k'0 where K(g*) is a compact set, therefore I there is only a "nite number of di!erent g , i.e. there is I a k (R such that g "g for k5k . ) I I 3. Convergence of performance In 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 of polytopes. 2. There exists a description of the plant with some suitable model structure l 3 S and extended parameter vector g*"(h*2, e* , e* , e*)2 3MJ , which is unfalsi "ed in all experiments. 3. For every g3M , there is an associated control law u(h), which satis"es J(u(h) " g"(h2, e , e , e )2)(R, g3 M . 4. For some d'0 there is a Lipschitz constant over the a priori set M : ¸(D) " : sup sup d\ JZ1 B" "J(u(h)"(h2, e , e , e)2) ; sup " "4 EJ \EJ B EJ EJ Z+ !J(u(h)"(h2, e , e , e)2)" (10) such that ¸(D)(R. Assumptions 1 and 2 are quite standard and are easy to satisfy. Assumption 3 says that we only consider models for which we have suitable control law when the model is known. Assumption 4 is a rather strict constraint but it does not have to be satis"ed for convergence of control performance, it helps us to evaluate guaranteed asymptotic performance relative to any valid robust model (1). The main result on the modi"ed scheme is stated in the following theorem. Theorem 2. Given a design parameter d'0, d(D, an a priori ,nite set M of polytopes in the extended parameter vector spaces. ¸et g* denote a model of the plant that is always valid. (i) ;nder Assumptions 1}3 the adaptive control scheme described above is convergent in the sense that there exists a ,nite number N such that g "g "(h2 , e, , e, , e, ), k5N and gN *"(h2 , I , , , e,#d, e,#d, e,#d) remains a valid description of the plant for k5N. ¹hen the inequality J(g )4J(g*) ,
(11)
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holds and J(gN *) is an upper bound of the actual asymptotic performance. (ii) If Assumption 4 holds then J(gN *)4J(g*)#d¸(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 performance can be expected. Remark 2. The "rst part of Theorem 2 claims that the asymptotic performance of the system, measured by J, will be bounded by the worst-case performance of gN *"(h2 , e,#d, e,#d, e,#d)2(R. Since the worst , case performance J is continuous in e , e and e , the di!erence between J(g )4J(g*) and J(gN *) can be rather , small if a small d is chosen. Remark 3. From the modelling point of view, both g* and gN * are valid descriptions of the plant, since both of them are unfalsi"ed by the data. As g* is an arbitrary valid description of the plant, it can be better than gN * from the general control performance point of view. The performance di!erence between them is bounded by d¸ if Assumption 4 holds.
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 example Our aim is to show here that the amount of computation 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 considered where the plant order is unknown. The plant will be simulated by the equation 1 B(q\) u# e , y" I A(q\) I A(q\) I
Fig. 2. Reference and output for the simpli"ed and original (Veres and Sokolov, 1998) schemes.
(13)
H. Xia, S.M. Veres/Automatica 35 (1999) 981}986
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Fig. 3. Cost functions and switching between models. l ( * ) and l (#) indicate locations of model updating.
where A(q\)"1#0.8q\!0.4725q\!0.2205q\!0.2q\, B(z\)"1.4q\#0.3q\#0.1q\#o q\, I o and e are pseudo-random with uniform distribution I I 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 simulation which will be indexed by l and l l : y #a y #a y I I\ I\ "b u #b u #P(y )#;(u )#e , I\ I\ I I I (14) l : y #a y #a y #a y I I\ I\ I\ "b u #b u #b u #P(y )#;(u )#e . I\ I\ I\ I I I The model sets represented by l and l are clearly disjoint. The dimensions of the extended parameter vectors will be 7 for model structure l and 9 for model structure l . In each model structure a pole placement controller (placing the poles at 0.5 and 0) will be computed 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 random steps. A large a priori extended parameter set MJ for l is de"ned as an axis aligned box with diagonal vertices [!3,!3,!5,!5, 0, 0, 0]2 and [3, 3, 5, 5, 1, 1, 1]2. An extended parameter set MJ for l is de"ned as an axis aligned box with diagonal vertices [!3, !3, !3, !5, !5, 0, 0, 0]2 and [3, 3, 3, 5, 5, 1, 1, 1]2. Over these large initial sets the control cost function does not satisfy Assumptions 3 and 4, still the scheme works. The asymptotic 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 between 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. 2 show the results as the simulation progressed: the step reference and the output are superimposed on the top graph of Fig. 2. Note that the noise step responses are caused by the high level of dynamical disturbance o and large source noise e is simulated. I I Tuning of the controller was achieved roughly within the "rst 60 sampling 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 for
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each model structure in the top of Fig. 3: the && * '' and &&#'' notation is used for model structures l and l , respectively. The * and &&#'' signs are placed whenever actual model updating happened. It is clear that the model under structure l , which was "nally used for control, was still re"ned up to time period 205. The computation of the "nal estimate takes about 14 000 Flops per sampling period under Matlab 5.2, GBT 6.0 Veres et al. (1993}1998). Using the original method (Veres and Sokolov, 1998) this was about 70 000 Flops per sampling period on average.
References Grace, A., & Branch, M. A. (1996). ¹he Optimisation ¹oolbox,
Kosut, R. L. (1996). Iterative adaptive robust control via uncertainty model unfalsi"cation. Proc. 1996 IFAC =orld Congress, San Francisco, 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. IEEE ¹rans. Automat. Control, AC-37, 942}952. Sokolov, V. F., & Veres, S. M. (1997). Adaptive robust steady-state tracking control. Proc. ACC+97, 4}6 June, Albuquergue, NM. Veres, S. M. et al. (1993}1998). ¹he Geometric Bounding ¹oolbox,
Technical Communique
Improved e$ciency of adaptive robust control by model unfalsi"cation1 Hao Xia, SaH ndor M. Veres* School of Electronic and Electrical Engineering, The University of Birmingham, Edgbaston, B15 2TT, UK Received 4 April 1998; revised 1 September 1998; received in "nal form 1 December 1998
Abstract The adaptive robust control scheme introduced by Veres and Sokolov (Automatica, 34, 723}730) provides optimal asymptotic performance 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. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Adaptive control; Robust control; Self-tuning systems; Bounded disturbances; Bounding methods
1. Introduction The 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 relaxing 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 scheme Given a SISO plant, our objective is to present an adaptive controller for the servo problem so that the * Corresponding author. Tel.: 44 121 414 4346; fax: 44 0121 414 4291; email: [email protected]. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Editor Peter Dorato
plant output follows a given reference signal. In this approach approximate linear models will be described by extended model parameters which include model uncertainty with regard to unmodelled dynamics and disturbance (Veres and Sokolov, 1998). The plant model is assumed to be of the form y #a y #2#a y "b u #b u #2 I I\ L I\L I\B I\B\ #b u #P(y )#;(u )#e , (1) K I\K\B I I I where ; and P are bounded linear operators ; : l Pl , P : ROPR and e3 l . The meaning of the di!erent terms can be explained as: P represents parametric perturbations of process dynamics ; represents unmodelled dynamics which is accounted for in the robust control design e represents unstructured additive disturbance. I The unmodelled dynamics is assumed to satisfy a condition ";(u )"4e #u # for all k51 (2) I I with u "[u , u , 2 , u ]2, the parametric perturbation I I an inequality "P(y )"4e #t W # I I
0005-1098/99/$*see front matter 1999 Elsevier Science Ltd. All rights reserved PII: S 0 0 0 5 - 1 0 9 8 ( 9 8 ) 0 0 2 3 4 - 9
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H. Xia, S.M. Veres/Automatica 35 (1999) 981}986
and the additive disturbance #e# 4e (4) for some unknown e '0, e '0, e '0, where [!y , 2 ,!y ]2 if k'q, I\ I\O t W" I [!y , 2 ,!y ]2 if k4q I\ and q5n 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 q\L, B(q\)"b #b q\#2# L b q\K. The notation h"[a , a , 2 , a , b , b , 2 , K L b ]2 3RL>K> is used for the associated parameter vecK tor. To include uncertainty, the extended parameter vector g"[h2, e , e , e ]2 is introduced under each model structure l"(n, m, d)2, 04n4N, 04m4M, 14d4D. Also, a priori "xed compact set MJ LRL>K> of model parameters is given for each model structure l3 S, 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 , k'1. For expository simplicity, linear I controllers of the form
R(q\)(1!q\)u "<(q\)(r !y ) I I I with
(5)
R(q\)"1#r q\#2#r q\LP, LP <(q\)"s #s q\#2#s q\LQ LQ will be used. The nominal closed-loop characteristic polynomial will be denoted by s"AR(1!q\)# q\BBS. Hence the controller will be associated with h rather than g. The performance criterion will be de"ned under a valid plant description and model g" (h2, e , e , e )2 by J(g)"J(u(h)"g*) " : sup lim sup "r !y "w , (6) I I I E I where on the right-hand side the supremum over g is to be understood over all possible perturbations and additive noise allowed by g. w is a weighting function used to I express the importance of keeping the error of tracking small at certain times. The scheme presented is not limited 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(g), 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 introduced 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 parameter
sets will be called unfalsi"ed model sets. For a given model structure l3 S the unfalsi,ed set of extended parameters at time k is de"ned by I FJ"MJ 5 7 HJ , I R RI where MJ is an a priori de"ned set of extended parameter vectors under model structure l and HJ " : +g " y 4g2 > ,5+g " g2 \4y ,. I I I I I Here the regression vectors
> " : [t , #t W # , #u # , 1]2, I I I I W
\ " : [t , !#t # , !#u # , !1]2 I I I I t "[!y , , !y , u , 2 , u ]2 I I\ 2 I\L I\B I\B\K> are used. Then the unfalsi"ed set of models is de"ned as the union F " : 8 FJ , F "M "8 MJ (7) I I JZ1 JZ1 Clearly, each F is the union of a "nite number of I polyhedra. Fig. 1 describes how the modi"ed adaptive scheme works. To any extended parameter vector g"[h2, e , e , e ]2 3M there is a controller u(g) de"ned by Eq. (5). For each model structure l3 S and each sampling time k an estimate gJ is selected from an enI larged set MJMFJ . Then the model structure estimate I I lL is obtained by minimising J. At each time k the controller associated with the estimate gJ( is used, where lL and I gJL minimises J over M . I I
Fig. 1. Block diagram of the modi"ed scheme.
H. Xia, S.M. Veres/Automatica 35 (1999) 981}986
gJ is obtained by constrained optimisation over I a modi"ed unfalsi"ed set M . The exact unfalsi"ed polyI hedron set F can have a large number of vertices and I facets which are computationally demanding to take all into account. Let the update of the unfalsi"ed set in model structure l be I J " M : MJ 5 HJ . I> I I> For each l 3S the feasible set M I J is replaced by a simpler I polyhedron set MJ . Then gJ is obtained by constrained I I> optimisation over MI J (lL is also selected so that the I> I> controller associated with gJL I> can be applied for conI> trol), and to obtain MJ the set MI J is &&relaxed'' by I> I> only retaining its facets around gJ . As the proof below I> shows, the exact form of the relaxation procedure is not that important apart from that a simpler polyhedral set MJ is derived from M I J which satis"es I> I> M I J LMJ . These enlarged polyhedral sets MJ , I> I> I> l3 S will then be used to de"ne a new M " : 8 MJ . (8) I> I> JZ1 The scheme shown in Fig. 1 has the following properties. Theorem 1. ¸et the model structure l3 S be ,xed and use the notations g " : gJ , H " : HJ and M " : MJ . ¸et I I I I I I g*3 MJ be an extended parameter vector that provides a valid description of the plant under any circumstances. ¹hen at any time k: (a) (b) (c) (d)
g*3M . I J(g )4J(g*). I J(g )4J(g ). I I> g 3 K(g*) where K(g*) is an extended parameter I> set de,ned by
K(g*) " : +g"J(g)4J(g*),5M (9) (e) If dist(g , M 5 H )5e then the set ¸ " : I I I> I +g " dist(g, g )5e, j(k; J(g)5J(g ), is non-empty as H I any valid parameter g* is contained in it. (f ) All di+erent g s are separated by at least a distance e. I (g) If K(g*) is compact then there is only a ,nite number of di+erent g s in this procedure. I Proof. (a) As M contains the set of extended parameter I vectors feasible with the data up to time k, any valid description g* of the plant must be contained in it. (b) By de"nition g minimises J(g) over M 5 H I I I> while g* 3M 5 H hence J(g )4J(g*). I I> I (c) By de"nition g is obtained by minimisation of I> J(g) over an area of g where J(g)'J(g ). I (d) This follows from (b) by using k#1 instead of k. (e) As g* 3 M 5 H it follows that dist(g*, g )5e I I> I for any g that has been rejected. If g is rejected then all I I previous g , j(k has been rejected. Also J(g*)5J(g ) H I by part (b).
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(f ) By de"nition g is only changing if dist(g , I I M 5H )5e and then g 3 H , which implies I I> I> I> dist(g , g )5e. I I> (g) By (d) and assumption g*3 M we have that all g 3 K(g*), k'0 where K(g*) is a compact set, therefore I there is only a "nite number of di!erent g , i.e. there is I a k (R such that g "g for k5k . ) I I 3. Convergence of performance In 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 of polytopes. 2. There exists a description of the plant with some suitable model structure l 3 S and extended parameter vector g*"(h*2, e* , e* , e*)2 3MJ , which is unfalsi "ed in all experiments. 3. For every g3M , there is an associated control law u(h), which satis"es J(u(h) " g"(h2, e , e , e )2)(R, g3 M . 4. For some d'0 there is a Lipschitz constant over the a priori set M : ¸(D) " : sup sup d\ JZ1 B" "J(u(h)"(h2, e , e , e)2) ; sup " "4 EJ \EJ B EJ EJ Z+ !J(u(h)"(h2, e , e , e)2)" (10) such that ¸(D)(R. Assumptions 1 and 2 are quite standard and are easy to satisfy. Assumption 3 says that we only consider models for which we have suitable control law when the model is known. Assumption 4 is a rather strict constraint but it does not have to be satis"ed for convergence of control performance, it helps us to evaluate guaranteed asymptotic performance relative to any valid robust model (1). The main result on the modi"ed scheme is stated in the following theorem. Theorem 2. Given a design parameter d'0, d(D, an a priori ,nite set M of polytopes in the extended parameter vector spaces. ¸et g* denote a model of the plant that is always valid. (i) ;nder Assumptions 1}3 the adaptive control scheme described above is convergent in the sense that there exists a ,nite number N such that g "g "(h2 , e, , e, , e, ), k5N and gN *"(h2 , I , , , e,#d, e,#d, e,#d) remains a valid description of the plant for k5N. ¹hen the inequality J(g )4J(g*) ,
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holds and J(gN *) is an upper bound of the actual asymptotic performance. (ii) If Assumption 4 holds then J(gN *)4J(g*)#d¸(D).
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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 performance can be expected. Remark 2. The "rst part of Theorem 2 claims that the asymptotic performance of the system, measured by J, will be bounded by the worst-case performance of gN *"(h2 , e,#d, e,#d, e,#d)2(R. Since the worst , case performance J is continuous in e , e and e , the di!erence between J(g )4J(g*) and J(gN *) can be rather , small if a small d is chosen. Remark 3. From the modelling point of view, both g* and gN * are valid descriptions of the plant, since both of them are unfalsi"ed by the data. As g* is an arbitrary valid description of the plant, it can be better than gN * from the general control performance point of view. The performance di!erence between them is bounded by d¸ if Assumption 4 holds.
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 example Our aim is to show here that the amount of computation 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 considered where the plant order is unknown. The plant will be simulated by the equation 1 B(q\) u# e , y" I A(q\) I A(q\) I
Fig. 2. Reference and output for the simpli"ed and original (Veres and Sokolov, 1998) schemes.
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Fig. 3. Cost functions and switching between models. l ( * ) and l (#) indicate locations of model updating.
where A(q\)"1#0.8q\!0.4725q\!0.2205q\!0.2q\, B(z\)"1.4q\#0.3q\#0.1q\#o q\, I o and e are pseudo-random with uniform distribution I I 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 simulation which will be indexed by l and l l : y #a y #a y I I\ I\ "b u #b u #P(y )#;(u )#e , I\ I\ I I I (14) l : y #a y #a y #a y I I\ I\ I\ "b u #b u #b u #P(y )#;(u )#e . I\ I\ I\ I I I The model sets represented by l and l are clearly disjoint. The dimensions of the extended parameter vectors will be 7 for model structure l and 9 for model structure l . In each model structure a pole placement controller (placing the poles at 0.5 and 0) will be computed 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 random steps. A large a priori extended parameter set MJ for l is de"ned as an axis aligned box with diagonal vertices [!3,!3,!5,!5, 0, 0, 0]2 and [3, 3, 5, 5, 1, 1, 1]2. An extended parameter set MJ for l is de"ned as an axis aligned box with diagonal vertices [!3, !3, !3, !5, !5, 0, 0, 0]2 and [3, 3, 3, 5, 5, 1, 1, 1]2. Over these large initial sets the control cost function does not satisfy Assumptions 3 and 4, still the scheme works. The asymptotic 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 between 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. 2 show the results as the simulation progressed: the step reference and the output are superimposed on the top graph of Fig. 2. Note that the noise step responses are caused by the high level of dynamical disturbance o and large source noise e is simulated. I I Tuning of the controller was achieved roughly within the "rst 60 sampling 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 for
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each model structure in the top of Fig. 3: the && * '' and &&#'' notation is used for model structures l and l , respectively. The * and &&#'' signs are placed whenever actual model updating happened. It is clear that the model under structure l , which was "nally used for control, was still re"ned up to time period 205. The computation of the "nal estimate takes about 14 000 Flops per sampling period under Matlab 5.2, GBT 6.0 Veres et al. (1993}1998). Using the original method (Veres and Sokolov, 1998) this was about 70 000 Flops per sampling period on average.
References Grace, A., & Branch, M. A. (1996). ¹he Optimisation ¹oolbox,
Kosut, R. L. (1996). Iterative adaptive robust control via uncertainty model unfalsi"cation. Proc. 1996 IFAC =orld Congress, San Francisco, 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. IEEE ¹rans. Automat. Control, AC-37, 942}952. Sokolov, V. F., & Veres, S. M. (1997). Adaptive robust steady-state tracking control. Proc. ACC+97, 4}6 June, Albuquergue, NM. Veres, S. M. et al. (1993}1998). ¹he Geometric Bounding ¹oolbox,