Adaptive Robust Control under Unknown Plant Orders

Adaptive Robust Control under Unknown Plant Orders

PII: S0005–1098(98)00007–7 Automatica, Vol. 34, No. 6, pp. 723—730, 1998 ( 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 00...

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PII: S0005–1098(98)00007–7

Automatica, Vol. 34, No. 6, pp. 723—730, 1998 ( 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0005-1098/98 $19.00#0.00

Brief Paper

Adaptive Robust Control under Unknown Plant Orders* S. M. VERES- and V. F. SOKOLOV‡° Key Words—Robust adaptive control; unmodelled dynamics; parameter bounding; self-tuning control.

model order, the time delay and the disturbances are bounded or stochastic. Stability robustness for known nominal plant under parametric uncertainty and norm bounded perturbation has been studied by Chapellat et al. (1990) and Dahleh and Khammash (1993). Dahleh and Dahleh (1990) proposed a weakly stable l -adaptive scheme for the case of output disturbances. 1 Equivalence of stability robustness and performance robustness under time varying and nonlinear perturbations has been discussed by Khammash and Pearson (1991), Dahleh and Dahleh (1990) and Dahleh and Khammash (1993). The question to be answered in this paper is: can similar robust performance be achieved by an adaptive scheme for an unknown plant to the one that can be achieved for a known plant with given uncertainty bounds under known model orders and time delay? No knowledge will be assumed on the bounds of the perturbations and disturbances, and only upper bounds will be supposed to be known for the model orders and time delay. It will be shown in this paper that there is an adaptive scheme, which is not only stable under unknown model orders, but also achieves an asymptotic robust performance arbitrarily near to that for a known plant. The assumptions under which this scheme exists are general, there is no requirement on the knowledge of minimum phase property or the sign of the high-frequency gain (Goodwin, 1991; Middleton et al., 1988; Goodwin et al., 1990). The size of the finite memory perturbations is not assumed to be known in advance either. With regard to the plant parameters, a technical assumption will be that for each model order and time delay, there is a prior compact set of parameter vectors known. In practice, this is not a serious limitation as the prior compact parameter sets can be selected to be arbitrarily large. Concerning the robust controller for a known plant (see Khammash, 1994, 1995; Khammash and Pearson, 1991, 1993; Sokolov, 1997; for various results on this topic), it will be assumed that with a valid model of the plant its asymptotic performance is acceptable to us. Robust regulation (Sokolov, 1996) and tracking performance (Sokolov and Veres, 1997), which can be placed arbitrarily near to that achievable for a known plant, have recently been developed for the adaptive case. Switching control was suggested for the case of unknown plant orders by Middleton et al. (1988), where parallel running estimators were used to avoid for instance pole-zero cancellation in the estimated model. Unmodelled dynamics was handled and asymptotic tracking was proved. For parameter estimation least squares was used with dead zone and control was based on pole-placement and internal model principle. The techniques used in the present paper are very different although the problem addressed is similar. Multiple model schemes have been presented by Narendra and Balakrishnan (1994). They suggested a stable strategy of switching between MRAC schemes with different initial conditions in continuous time. Although a search algorithms was proposed for the best controller, the approach was direct control without explicit listing of model structures for the plant, and therefore their approach is different from the indirect control scheme considered in this paper. Pickhardt and Unbehauen (1996) suggested a multi-model approach to adaptive control. Rules of switching between model structures was based on a multi-step prediction error criterion and the parameter estimates were updated by recursive least squares in each model order. The most relevant difference with

Abstract—A robust indirect adaptive controller under additive and parametric disturbances is introduced in this paper. The orders and time delay of the plant are assumed to be unknown apart from their known upper bounds. It is also assumed that a robust linear controller of the plant can be defined for known parameters, model orders and time delay. The robust adaptive controller switches between model structures depending on an overall criterion of performance. It is not only proved that the adaptive control scheme is stable but that its asymptotic performance can be set arbitrarily near to the performance achievable for a known plant. Finally, a simulation illustrates the performance of the scheme. ( 1998 Elsevier Science Ltd. All rights reserved. Notations N Rn q, q~1 #, &, $, ( n, m k N, M, K S 0 w h I, J l ,l = 1 DD . DD p D.D E.E

44

set of positive integers—discrete time domain n dimensional real Euclidean space forward and backward time-shift operators compact subsets in parameter space model orders model time delay upper bounds of model orders and time delay a priori known set of model structures additive disturbance in plant model perturbation of plant model model parameter vector asymptotic performance criteria spaces of real scalar valued sequences over N with supremum norm and absolute value sum norms, respectively l -norm of vectors and sequences defined by p 1@p E(x , x , 2)E " + Dx Dp , p 51 1 2 p i i for vectors this is a short notation for the l -norm = as this occurs most frequently in this paper, for scalars D . D is also the modulus or absolute value. the steady state norm is defined as ExE :" 44 lim sup Dx D for a sequence x"(x , x , x , 2). t 1 2 3 t?=

A

B

1. Introduction Global stability of adaptive regulation of discrete-time LTI systems has been the topic of numerous results since the early 1980s (Goodwin et al., 1981; Lozano and Goodwin, 1985; Kreisselmeier and Smith, 1986; Giri et al., 1989; Lozano and Zhao, 1994). Most of these results assume exact knowledge of the * Received 28 January 1997; revised 29 October 1997. This paper was recommended for publication in revised form by the Associate Editor D. W. Clarke under the direction of Editor C. C. Hang. This paper was presented at SYSID’97, 8—11 July 1997, Fukuoka, Japan. Corresponding author S. M. Veres Tel. 44-0 121-414 4346; Fax 44 0 121 414 4291; E-mail [email protected]. - School of Electronic and Electrical Engineering, The University of Birmingham, Edgbaston, B15 2TT, UK. ‡ Department of Mathematics, Syktyvkar State University, 55, Oktyabrskii prospect, Syktyvkar, 167001, Russia. ° The work of this author has been supported by RFFR Grant no. 96-01-00459 and by the University of Birmingham, UK. 723

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Brief Papers

respect to our paper is the statistical handling of signals and that no proofs are provided on performance guarantees. A time-delay adaptation technique for adaptive control was proposed by Ba´nya´sz and Keviczky (1994), where step-response equivalent and modified bilinear transformations between continuous and discrete-time models were used to adapt the time delay in the discrete-time model. Model perturbations and effects of noise have not been explicitly handled, unlike the approach presented here. Structure selection in the parameter bounding context has been analysed by Veres and Norton (1991). Although the parameter bounding setting was similar to the one here, consistency conditions for the model structure were not proved under closed-loop adaptation. The approach presented here can easily be related to recent results on ‘‘identification for control’’ by Gevers (1995). Closedloop identification is taking place in the scheme presented as the controller is applied while robust modelling is performed by computing feasible polyhedral sets of the extended parameter vector. Similar approaches have been taken by Kosut et al. (1992), Veres and Norton (1993) and Sokolov (1996), which is generalised here to unknown model orders and different assumptions on the parametric perturbations. Weyer et al. (1994) discuss the limitations of robust adaptive pole placement under similar, l -norm bounded perturbations, and some of their = results are used in this paper. Note that the scheme presented in this paper is not limited to a single type of controller an therefore leaves the door open for the application of l -optimal 1 techniques. In the control scheme that model structure will be selected which ‘‘promises’’ the best performance out of the total set of feasible models up to a given time instant. It is proved that this ‘‘optimistic’’ strategy will lead to achieving the best-possible asymptotic performance. Some distant parallelism can be discovered with the work of Buesher and Kumar (1993) on learning systems. It will not be proved, however, that the model orders and parameters converge to some ‘‘true’’ ones. Indeed, the concept of a ‘‘true’’ model order and parameters is not crucial here, since the perturbation and disturbance bounds are not assumed to be known a priori: it can happen, in principle, that several model orders and parameters can be considered as ‘‘true’’. It will be seen that best asymptotic control performance might be possible to achieve without convergence to a true model, which situation has some parallels with the well-known results of As stro¨m and Wittenmark (1973) on convergence of self-tuning regulators. Finally, the robust adaptive control scheme presented has the following property: it only updates the model set when the ‘‘optimistic’’ model is falsified and triggers such an action. There is, however, no active probing or caution designed into the scheme as in dual control (see e.g. As stro¨m and Wittenmark, 1971, or Tse and Bar-Shalom, 1973), and asymptotic performance is achieved rather passively, using a feedback mechanism from performance to model updating. Section 2 sets the problem and Section 3 discusses the feasibility sets of parameters and model structure. Section 4 presents the adaptive algorithm and Section 5 illustrates it by a simulation example. Section 6 gives a discussion of the results and potentials for applications. 2. Assumptions and problem statement The plant is assumed to be describable by the equation a(q~1)y(t)"q~kb(q~1)u(t)#d(t), t3N

(1)

with unknown initial conditions, where y(t), u(t) and d(t) are scalar output, control input and disturbance, k is an unknown time delay satisfying 14k4K where K'0 is a priori known, and a(q~1)"1#a q~1#2#a q~n, 1 n b(q~1)"b #b q~1#2#b q~m, 0 1 m where the polynomial orders n50 and m50 are unknown except that upper bounds 04n4N and 04m4M are known. Denote the a priori known set of model structure vectors by S:"M(n, m, k)D04n4N, 04m4M, 14k4KN.

For each model structure vector (n, m, k)3S assume that a compact polyhedral set #(n, m, k)LRn`m`1 of parameter vectors h"(a , 2 , a , b , b , 2 , b )T is a priori known, which is not 1 n 0 1 m necessarily convex or connected. It will be assumed that the real plant can be described by a parameter vector from #1" : Z #(n, m, k). (n,m,k)|S The equation error d(t) is assumed to be the sum of disturbances d(t):"0 (t)#w(t),

(2)

where 0 (t) and w(t) will be called the additive disturbance and parametric perturbations, respectively. Prior information about the additive disturbance is that o "sup : D0 (t)D(R t

(3)

and the value of o is unknown. Prior information about the parametric disturbance w is that Dw(t)D4dDt(t!1)D,

t3N

(4)

holds with some d'0 unknown and t(t!1):"(!y(t!1), 2 , !y(t!n),u(t!k), 2 , u(t!k!m))T. The assumed range of model orders with n4N, m4M and k4K allows for a better performance in case of unknown model orders. As it happens in practice, the model structure, and especially the time delay are frequently unknown. The aim of the present paper will be to propose a class of adaptive control algorithms which not only stabilise the plant under the above weak prior knowledge, but give asymptotic performance guarantees as well. The performance of the adaptive controllers will be measured by the performance index lim sup Dy(t)D#rDu(t)D t?=

(5)

with a fixed r'0. For brevity, the vector of model structure will also be denoted by a single letter l"(n, m, k). It will be assumed that for any h3#(l), l3S, a regulator is defined with zero initial conditions and satisfying a (q~1, h)u(t)"b (q~1, h)y(t) l l which ensures a robust performance

(6)

J (h, d, o)" sup sup lim sup Dy(t)D#rDu(t)D, d'0, o'0, l D0 D4o w tPR where w runs through all sequences satisfying equation (4). J (h, d, o) can actually be characterised as follows. Define l I (h, d)" sup sup sup Dy(t)D#rDu(t)D, h3#(l), l3S, l D0 D41 w tPR The following result holds (Sokolov, 1996). ¸emma 2.1. Assume that the control system (1) and (6) is asymptotically stable without disturbances. Let *(h)"sup Md D I (h, d)(RN, h3#(l), l3S. l Then the function J (h, d, o) is positive for every d3[0, *(h)), l h3#1, o'0 and J (h, d, o)"PI (h, d), o(R. l l Later on in the on-line algorithm in equations (14) and (15) it will be possible to use an upper bound JM (h, d, o) of J (h,d,o) (or l l an upper bound IM (h, d) of I (h, d)) instead of the exact value of l l J (h,d,o), which will result in asymptotically achieving the upper l bound JM (h,d,o) instead of J (h, d, o) in the adaptive scheme. In l l practice, the use of upper bounds might be needed because of the possible high computational complexity of evaluating the smallest upper bound. Assumption 1. For each l3S there is a compact prior parameter set #1(l)LM(h, d)Dh3#(l), 04d4*(h)!2cN known with c'0

Brief Papers known. There is a description of the plant by some h3#(l), l3S, 0(o(R, 0(d4*(h)!2c, so that equations (1)—(4) hold and (h, d)3#1(l). An illustration of the performance upper bound JM (h, d, o) can l be obtained, for instance, by the results of Weyer et al. (1994). Using a pole-placement controller of the form (6) with given desired closed-loop characteristic polynomial a(q~1)a(q~1)!q~k b(q~1) b(q~1)"s(q~1), the closed-loop response from the disturbances to the input/ output is s(q~1)y(t)"a(q~1)d(t),

s(q~1)u(t)"b(q~1)d(t).

This implies

K

a(q~1) EyE 4 44 s(q~1)

K

EdE , 44

K

b(q~1) EuE 4 44 s(q~1)

K

EdE 44 1 1 and by our assumption on the disturbance it follows that

(7)

EyE 4Ea/sE (o#d max (EyE ,EuE )), 44 1 44 44 EuE 4Eb/sE (o#d max (EyE ,EuE )) . 44 1 44 44 Taking the maximum of both sides it follows that m "max (EyE ,EuE ) satisfies xy 44 44 m 4max (Ea/sE , Eb/sE ) (o#dm ) xy 1 1 xy and if d max(Eb/sE ,Ea/sE )(1 1 1 then using equation (7) and EdE 4o#d max (Ea/sE , 44 1 Eb/sE )EdE it follows that 2 44 Ea/sE 1 o, EyE 4 44 1!d max (Ea/sE , Eb/sE ) 1 1 Eb/sE 1 EuE 4 o, J (h, d, o) (8) 44 1!d max (Ea/sE ,Eb/sE ) l 1 1 Ea/sE #rEb/sE 1 1 4JM (h, d, o):"oIM (h, d):"o . 1!d max (Ea/sE ,Eb/sE ) 1 1 To evaluate the asymptotic performance the following assumption will be made which is not necessary for achieving good performance but helps making comparisons with the performance achievable by using any valid model of the plant. Assumption 2. There is a Lipshitz constant ¸'0 such that D IM (h, d )!IM (h, d )D4¸Dd !d D, l 1 l 2 1 2 0(d , d (*(h)!c, h3#(l), l3S, 1 2 0(d , d (*(h)!c, h3#(l), l3S 1 2 holds uniformly over all model structures and parameter sets, where c'0 is fixed a priori as in Assumption 1. 3. Feasibility sets of parameters and model structure Define the extended vector of model parameters of the plant by m"(hT, d, o, n, m, k)T. Part of this vector will be denoted by m "(hT, d, o)T and l"(n, m,k), so that m"(mT ,l)T gives the l l whole extended parameter vector. This section will clarify non-identifiability of the model structure within finite time and the concept of most parsimonious model structure. It will be clear that based on finite number of i/o samples the model parameter m cannot be uniquely determined due to the perturbations w(t) of the dynamics and the additive disturbance 0 (t) for which no upper bounds are known. To study the problem in general the class of admissible control laws C will be assumed to be the set of all recursive continuous functions u "f (yt, ut~1) where yt"(y ,y , 2 , y ) and t t 1 2 t ut"(u , u , 2 , u ). 1 2 t Definition 3.1. Given a control law u3C, a vector m"(hT,d, o, n, m, k)T will be said to be consistent with the observed data yt, ut~1 if d4*(h)!2c and there are disturbances w(s), 0 (s), s3[0, t], such that Dw (s) D4d Dt(s!1)D,

D0 (s) D4o, s4t,

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where the observed data satisfy equations (1) and (2) with h3#(n,m,k) and (n,m,k)3S. Denote the set of extended parameter vectors m consistent with the data yt, ut~1 by &t(u), indicating that the control law u has also an effect on the formation of this set. Given a control law u3C, the parameter set consistent with all the data is defined by = &=(u)" Y &t(u) t/1 which is not empty due to Assumption 1. The set &=(u) is unbounded in the direction of the o component and the model is non-identifiable in finite time. &=(u) is a theoretical limit and a set too broad for our purpose. Therefore, the following definitions are made for t(R. Let the set of additive disturbance bounds consistent with yt, ut~1 be defined by R" : MoD&h& d& l : (h, d, o, l)T3&t(u)N. t Let o " : inf R . For o5o define t 6t N t d (o):"inf D (o), D (o):"Md D &h&l:(h, d, o, l)T3&t(u)N. t t 6t For given o'o and d5d (o) let the set of model structures 6 t be defined by 6 t data yt, ut~1 consistent with the S (o, d):"Ml3S D &h:(h, d, o, l)T3&t(u)N . t The consistent set of auto-regressive (AR-) orders will be defined by SAR (o, d):"Mn D &m&k: (n, m, k)3S (o, d)N. t t Let n (o, d):" inf Mn D (n, m, k)3SAR (o, d)N, t4R. t N t For each n3SAR (o, d), o3R define t m (o, d, n):"min SMA (o, d, n), t N t SMA (o, d, n):"MmD&k:(n,m,)3S (o, d)N, t4R t t and finally

(9)

k1 (o, d, n)"max Mk D (n,m (o, d, n), k)3S (o, d)N, t4R. t t N t The set of parsimonious model structures, with the AR-order n varying, can then be defined by PAR (o, d)"M(n, m (o, d, n), k1 (o, d, n)) D n3SAR (o, d)N. t t t N t Finally, the most parsimonious model structure is defined by v (o, d)"(n (o, d), m (o, d, n (o, d)), k1 (o, d, n (o, d))), t t N t N t N t N t which is clearly o-dependent by the definition of its components. The non-identifiability of the model structure and parameters can be expressed by the following properties: Proposition 3.1. Assume that u3C. Then the following statements hold: (1) sup R "R, t4R. t (2) The set of feasible pairs (o, d) is not a singleton, i.e. cardM(o, d)D&h, &l3S: (h, d, o, l)3&=(u)N'1. (3) If #(l) is a polytope then for any o5o and d5d (o) the set t 6 6 t of consistent parameters (t (o,d)"Mh3#(l) D (hT, d, o,l)T3&t(u)N, l3S. l is a polytope in Rn`m`1. (4) n (o, d), t'0 is a monotone increasing function of t and N t n (o, d)"lim n (o, d) t N = t?= N with n (o, d) as defined in equation (9). N = n5n (o, d), m (o, d, n), t'0, is a monotone in(5) For any N = of t and N t creasing function m (o, d, n)"lim m (o, d, n). t N t?= N

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(6) For any n5n (o, d), k1 (o, d, n), t'0, is a monotone decreast = ing function Nof t and k1 (o, d, n)"lim k1 (o, d, n). t = t?= (7) For a fixed o, d also lim (n , m (n ), k1 (n ))"(n , m (n ), t?= N t N t N t t N t N = N = N = k1 (n )) holds. N = sufficiently (8) For large o'0 the limits is K (n , m (n ), k1 (n ))"(0, 0, 1). = = = = = N N N N These statements show that the model structure cannot be uniquely determined on the basis of input—output measurements because of the non-identifiability of the (o, d) parameters. Asymptotically, there is a most parsimonious model structure for each pair (o, d). Although it might be intuitively appealing to choose the simplest feasible model in control design, the next section will suggest an adaptive control algorithm where the model structure used will be selected on the basis of a control performance criterion. Hence, the approach taken will be control oriented modelling on line. 4. Adaptive control under unknown model orders An outline of the adaptive control scheme is shown in Fig. 1. The main parts of the on-line procedure are: (i) conditional updating of the polyhedron feasibility sets of extended parameter vectors for each model structure, (ii) the selection of an overall estimate of the model structure and extended parameter vector, (iii) application of the controller associated with the overall estimate. Next, each part will be described in detail. A set estimation algorithm will be applied to compute feasible parameter sets of m "(hT, d, o)T and an associated sequence of l estimates m (t)"(hT(t), d(t), o(t))T. l Initialisation: Let the initial feasible parameter set be defined by $(0)"Z $ (0), l l|S where $ (0) "# : 1 (l)][0, R) is a polyhedral extended parameter l set for each l3S. Let I1 (h, d)5I (h, d) be a known upper bound l l of the asymptotic criterion function. Define for each l3S a para-

meter estimate by m (0):"arg min J (h, d, o) . (10) l l m| $l(0) Let e'0 be a fixed design parameter of the adaptive algorithm. For each model order v3S a sequence of polyhedra $ (t), t'0, l will be computed by an estimation algorithm as follows. Estimation algorithm: At each time instant t'0 and for each l"(n, m, k)3S compute s (t#1)"sign(y(t#1)!tT(t)h (t)), l l l u (t)"[s (t)tT(t),Et (t)E ,1]T, (11) l l l l = t (t)"[!y(t!1)2!y(t!n), u(t!k), 2 , u(t!k!m)]T l and define ) (t#1)"MqDqT/ (t)5y(t#1)s (t#1)N, l l l $ (t), if m (t)T/ (t)5y(t#1)s (t#1)!eD/ (t)D l l l l $ (t#1)" l l (12) $ (t)W) (t#1) otherwise, l l and let

G

G

m (t) if $ (t#1)"$ (t), l l l m (t#1)" arg min I1 (h, d)o otherwise, l l N m| l (t`1)

(13)

where in the case of multiple minima any of those is selected. Denote the set of parameter estimates for all model structures by E(t#1):"M(mT (t#1),l)Dl3SN. l The overall estimate of model orders and parameter vectors will be defined by mª (t#1):"(mK (t#1), lL (t#1)):"arg min IM l (h, d)o, (14) lK (t`1) m|E(t`1) where in case of multiple minima the one with the smallest DlD:"nM#m!k is selected. (This way in some sense the most parsimonious model will be selected, i.e. the one with the smallest n, m and largest k, in order of priority as listed.) Application of the controller: The controller computations will then be based on m (t#1), lL "lL (t#1) using the linear controller lK (t`1) a (q~1, m (t#1)) u(t#1)"b (q~1,m (t#1))y(t#1). (15) lK lK lK lK ¹heorem 4.1. Under Assumptions 1 the solutions of the closed loop system (1), (6) and (10)—(15), for any e'0 such that e(c, satisfy the inequality sup sup lim sup Dy(t)D#rDu(t)D4IM (h3, d3#e) (o8#e) l v w t?= for some (h3 , d3, o3 ) 3$ 8 (0) and l8 3S, with I1 8 (h3 , d3)o84IM J (h, d)o so l l l that (h3, d3#e,o8#e) is a valid description of the plant. Under Assumption 2 a further inequality IM (h3, d3#e) (o8#e)4 l IM (h, d)o#Ke holds with regard to any valid description (h, d, o) l of the plant, where K"supq 3#(l),l3S IM (q,*(q)!c)#¸o#¸e. l Proof. Let e be such that 0(e(c. First, we prove that for each l3S updating in equation (12) can only happen a finite number of times and therefore the model structure as well as the parameter estimates remain unchanged after a finite time. Assume that at time t#1 the set $ (t#1) has been updated. Then l m (t)T/ (t)(y(t#1)s (t#1)!eD/ (t)D, l l l l mT/ (t)5y(t#1)s (t#1) ∀m3$ (t#1), l l l by the definition of the estimate and condition of updating. These two inequalities clearly imply that (m!m (t))T/ (t)'eD/ (t)D, ∀m3$ (t#1) l l l l and by Ho¨lder’s inequality

Fig. 1. Block diagram of the adaptive control scheme.

Em!m (t)E D/ (t)D5D(m!m (t))T/ (t)D'eD/ (t)D, l 1 l l l l ∀m3$ (t#1), l

Brief Papers i.e. Em!m (t)E 'e, ∀m3$ (t#1) if D/ (t)D"E/ (t)E O0. This l 1 l l l = means that the l -ball with centre m (t) and radius e does not 1 l intersect with $ (t#1). This will also imply that the l -ball with l 1 centres m (t) and radius e/2 will all be disjoint. l The first aim will be to prove that the number of updatings equation (12) is finite for each l3S. This can be proved in three steps: (i) proving that the overall estimates mK (t) converge in finite time (ii) proving global stability of the scheme (iii) concluding that convergence is in finite time for each l. Part (i) of the proof is possible by contradictio ad absurdum. Assuming that the overall estimates mK (t) do not converge in finite time implies by the disjoint e/2 balls that o (t)PR as tPR l for all l3S. Considering that the overall estimates oL (t) are at any time equal to one of o (t), l3S, they satisfies the inequality l oL (t)4IM (h (t),d (t))o (t)4IM (h,d) o l l l l with the ‘‘true’’ parameters on the right-hand side, which then contradicts oL (t)"o (t)PR for all l3S. l To prove part (ii) it is enough to note that outputs and control inputs remain bounded since no updating happens after the final estimate of the model structure l is reached, meaning that the plant satisfies Dy(t)!hT (R)t (t!1)D l l "s (t)(y(t)!hT(R)t (t!1)) l l l 4eD/ (t!1)D#d (R)Dt (t!1)D#o (R) l l l l "e max (Dt (t!1)D, 1)#d (R)Dt (t!1)D#o (R) l l l l 4e Dt (t!1)D#e#d (R)Dt (t!1)D#o (R) l l l l "(e#d (R))Dt (t!1)D#e#o (R) l l l which by the initialisation, Assumption 1 and e(c is then stabilised by the robust controller associated with m (t)" l m (R)"(h (R) d (R) o (R))T for sufficiently large t, and hence l l l l the input—output sequence is bounded. Finally, boundedness of the input/output implies that the sequence o (t) remains l bounded for each l3S. Since the l -balls with centres m (t) and 1 l radius e/2 will all be disjoint, this implies that the estimates converge in finite time for each l3S. For brevity introduce the notations h3 " : hl8 (R) (R),

d3"dlˆ (R)(R),

o8"olˆ (R)(R), l8"lˆ (R).

Then the regulator al8 (q~1, h3)u(t)"b L (q~1, h3 )y(t) l gives an asymptotic performance sup sup lim sup Dy(t)D#rD(u(t)D4IM lJ (h3, d3#e) (o8#e) 0 w tPR 4IM lJ (h3 , d3)o8#e¸oJ #eIM lJ (h3, d3)#e2¸ 4IM lJ (h3, d3)o8#e[IM lJ (h3 , d3 )#¸oJ #e¸] 4IM lJ (h3 , d3)o8#eK4IM lJ (h, d)o#eK, where m"(hT d o l)T represents the plant under control, i.e. a ‘‘true’’ parameter vector. Here the Lipshitz condition of Assumption 2 was possible to use since d3"dª (R)4*(hª (R)!2c holds and e(c by the assumptions. IM lJ (h3, d3)o84IM lJ (h, d)o follows by the definition of the estimates for the model order and parameter vectors in equaitons (13) and (14), completing the proof of Theorem 4.1. K 5 A simulation example To keep the example simple the plant was simulated by the model y(t)#ay(t!1)"bu(t!2)#w(t)#0 (t), t55, t4150, where 0 (t) is a pseudo-random additive disturbance uniformly distributed in [!o, o] and w(t) is a pseudo-random

727

quantity uniformly distributed in [!d* , d* ] where t t * "max (Dy(t!1)D, Du(t!2)D). t Three model structures will be used in the scheme, denoted by l , l and l 1 2 3 l : y(t)#a y(t!1)"b u(t!1)#w(t)#0 (t), l "(1,0,1), 1 1 1 1 l : y(t)#a y(t!1)"b u(t!2)#w(t)#0 (t), l "(1,0,2), 2 2 2 2 l : y(t)#a y(t!1)"b u(t!3)#w(t)#0 (t), l "(1,0,3), 3 3 3 3 A priori sets of the parameter vectors h "(a , b )T, 1 1 1 h "(a , b )T, h "(a , b )T will be defined by #(l )" 2 2 2 3 3 3 1 #(l )"#(l )"convex hull of M[!1.5, 0.1], [!1.5, 10], 2 3 [1.5, 0.1], [1.5, 10]NXconvex hull of M[!1.5,!0.1], [!1.5,!10], [1.5,!0.1], [1.5,!10]N, the second of which contains the ‘‘true’’ unstable parameter vector h0"(1.1,1.5)T used to generate 2 the plant under model structure l . This way parameter b is 2 allowed to have a large a priori range over [!10, 10] with the exception of the interval [!0.1, 0.1] around b"0 where controllability is lost. Parameter a is assumed to fall into the a priori interval [!1.5, 1.5]. The standard dead-beat control schemes a u(t)" y(t) for l"(1, 0, 1), b a2 u(t)"au (t!1)! y(t) for l"(1, 0, 2), b a3 u(t)"au(t!1)!a2u(t!2)# y(t) for l"(1, 0, 3) b are used. The measure of asymptotic robust performance for these controllers for h"(a, b)T is J i (h, d, o)"IM lJ (h, d)o, i"1, 2, 3, l for which upper bounds IM lJ (h, d)5I (h, d) can be defined as l 1#rDaD/DbD if (d#c) max (1,DaD/DbD)(1, IM (h, d)" 1!d max (1,Da/bD) (1,0,1) R otherwise,

G

IM (1,0,2)

G

if (d#c) max (1#DaD, (1#ra2/DbD) (h, d)" 1!d max (1#DaD, a2/DbD) a2/DbD)(1, R

G

IM (h, d)" (1,0,3)

otherwise,

(1#rDaD3/DbD) 1!d max (1#DaD#a2, DaD3/bD) if (d#c) max (1#DaD#a2,DaD3/DbD)(1, R otherwise

The simulation example shown in Figs 2—4 is for real noise bounds o"0.05, d"0.05 and design parameters c"0.05, r"0.01, e"0.001. The initial values of i/o are u(2)"u(3)"u(4)"0 and y(4)"0 and simulation started at t"5. In Fig. 2 the output is regulated around 0 and the number of adaptations (i.e. polyhedron updatings) is displayed for each of l , l , l . 1 2 3 The time-delay estimates settle for the correct k"2 in Fig. 4. In Fig. 3 the final value of the parameter estimates is h"(a, b)T"(1.0901, 1.4602)T which is not far from the ‘‘true’’ h0"(1.1, 1.5)T. The final estimates of the bounds are 2 (d, o)"(0.0000 0.0209) which are consistent with the data but they might increase or decrease during a longer runtime. In this simulation optimisation of the control criterion over #1(l ), 1 #1 (l ), #1 (l ) was simplified to optimisation over the finite set of 2 3 those vertices m "(h d o)T of the polyhedral sets which satl isfythe condition I (h, d#2c)(R with c"0.05. In spite of l this simplification the algorithm performed well.

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Brief Papers

Fig. 4. Performance criteria Jl , i"1, 2, 3 and estimates of the i time delay k.

Fig. 2. I/O and number of updatings of the control law. The number of adaptations was quickly reaching 8 for k"1,3 while it was settled at 4 for k"2.

DIM (h, d )!IM (h, d )D (1,0,2) 1 (1,0,2) 2 (1#ra2/DbD) max (1#DaD, a2/DbD)D d !d D 1 2 " 1!d max (1#DaD, a2/DbD)) (1!d max (1#DaD, a2/DbD)) 1 2 if (d #c) max (1#DaD, a2/DbD)(1, i"1, i DIM (h, d )!IM (h, d )D" (1,0,3) 1 (1,0,3) 2 (1#rDaD3/DbD) max (1#DaD#a2, DaD3/DbD)Dd !d D 1 2 1!d max (1#DaD#a2, DaD3/bD)) (1!d max (1#DaD#a2, DaD3/DbD)) 1 2 if (d #c) max (1#DaD#a2,DaD3/DbD)(1, i"1 i and hence 1 ¸:" max c2

G

(1#rDaD/DbD) (1#ra2/DbD) sup , sup max (1,DaD/DbD) max(1#DaD,a2/DbD) # # (a,b) | (l1) (a,b) | (l2)

H

(1#rDaD3/DbD) sup max (1#DaD#aN 2,DaD3/DbD) # (a,b) | (l3) 1

Fig. 3. Estimates of the extended parameter vector m 2. l

It is not difficult to evaluate the Lipshitz constant ¸ valid over all three model structures. DIM (h, d )!IM (h, d )D (1,0,1) 1 (1,0,1) 2 (1#rDaD/DbD) max (1, Da/bD)Dd !d D 1 2 " (1!d max (1,Da/bD)) (1!d max (1,Da/bD)) 1 2 if (d #c) max (1,DaD/DbD)(1, i"1,2, i

is a suitable choice for ¸ to satisfy Assumption. 2. Under the a priori parameter sets of the example this gives ¸"(1#r)/c2"404 as a suitable Lipshitz constant. Note that K still depends on the unknown o. To illustrate the size of K, if an upper bound o "0.02 were known for o then K could .!9 be bound from above by K "(1#r)/c#¸o #¸e" .!9 .!9 20.2#404]0.02#404]0.001"28.12 and hence Ke"0.028 in Theorem 4.1. If there is no good upper bound known for o then the asymptotic robust performance will still be bounded by sup sup lim sup Dy(t)D#rDu(t)D4IM lJ (hI , d3#e) (oJ #e), v w t?= where h3 , d3 , oJ , lJ are the final estimates with the property that IM lJ (h3, d3) o84IM lJ (h, d)o, the latter indicating the ideal robust performance for a known plant. 6. Discussion and potential of implementations 6.1. Comments on convergence of the model estimates. The tuning of a regulation problem considered in this paper is most relevant for an unstable nearly time invariant plant with some

Brief Papers perturbations of its nominal dynamics, as this was described in equations (1)—(4). In such a situation the excitation, which helps estimation of the plant, can only originate from the disturbances d(t). This excitation will not necessarily cause the estimator in the adaptive control scheme to converge to the ‘‘true’’ model represented by h, d, o, l in Assumption 1. The estimator will instead converge to another model represented by h3, d3 , oJ , l8 which is a good enough description of the plant to ensure a performance near to what could have been achieved by knowing h, d, o, l. 6.2. Comment on the a priori parameter sets. Large polytope sets #(l) have to be defined for each l3S so that for any (h,d)3#1(l) a stabilising controller exists. If a h allows for a stabilising pole-placement controller, then it follows by continuity that there is a small enough d'0 such that the perturbed system represented by (h,d) is also stabilised by the same controller. Surfaces where h does not allow for a stabilising pole-placement controller have to be cut out from #1(l) which can make it non-convex. In this case #1 (l) should be the union of a finite set of convex polytopes. A software package to handle polytopes and polyhedra of higher dimensions is now available as a MATLAB toolbox (Veres et al., 1996). In practice, some very large upper bounds are always available on the plant to allow for an acceptable definition of #1 (l), the less knowledge is available a priori, the more complicated the structure of #1 (l) will have to be in order to ensure good performance. 6.3. Comment on Assumption 1. Closer scrutiny of the meaning of the ‘‘true’’ h,d,o,l quickly reveals that, however complicated the plant dynamics may be, for any model structure l3S there will be be a (h,d)3#1(l) and some o'0 which describe the plant, as the o can always be increased to such a degree. Some of these descriptions might be very poor but valid if the model structure is very unsuitable for the plant. Hence the priorities of a practical application are to have the compact sets #1(l) and the upper bounds N, M, K of the model order as large as processing power allows that. Then the adaptive control algorithm presented in this paper will achieve nearly the best performance that can be achieved relative to a given S, #1(l), l3S. Weak computational power will therefore not guarantee good control performance and strong computational power will imply better performance. It is not possible to make a statement on what M, N and K is reasonable to assume in applications, as this not only varies form field to field but a complex system does not necessarily imply a complex controller for the reasons described above. 6.4. Comment on Assumption 2. Note that Assumption 2 is not needed for good asymptotic performance of the scheme but helps to compare performance with what could have been achieved with any given valid model of the plant. The calculations after the simulation example give an illustration of how the Lipshitz constant can be evaluated. For higher-order models computation of the Liptshitz constant requires the maximisation of nonlinear functions over a set of convex polytopes. Note also, that there is no such strict relationship between the Lipshitz constant being large and therefore asymptotic performance would be bad. The region where the Lipshitz constant is high tends to fall far from the area where the estimates occur for little perturbed systems. 6.5. Comment on the computational demand of the adaptive scheme. The simulation in the previous section gives an illustration of the amount of computation the algorithm requires. In the example polyhedra had to be updated in 4D. Parallel processing of polytope updating is now possible (Veres, 1992) and limited complexity algorithms can also be used. Optimisation of the control criterion over sets of polytopes also requires substantial computing power. Large amount of computing is involved in the adaptive algorithm presented, which becomes less and less of a disadvantage with the advance parallel-processor technology. Analysis of numerical techniques is beyond the scope of this paper.

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7. Conclusion A robust indirect adaptive controller based on ‘‘set-membership’’ modelling techniques has been introduced which can achieve an asymptotic performance under unknown model order arbitrarily near to that achievable for a known plant. No assumption has been made on a priori known bounds of the additive disturbance or on accurate knowledge of the coefficients of the ‘‘parametric’’ model perturbations. Model orders have been selected on-line by choosing the one with the most promising performance. Future research might be concerned with reducing the computational complexity of the scheme by (i) limited complexity polytope updating (ii) using parallel processors and by (iii) exploring the properties of the control criterion over convex polytopes of plant parameters in order to speed up on-line optimisation. References As stro¨m, K. J. and B. Wittenmark (1971). Problems of identification and control. J Math. Anal. Appl., 34, 90—113. As stro¨m, K. J. and B. Wittenmark (1973). On self-tuning regulators, Automatica, 9, 195—199. Ba´nya´sz, Cs. and L. Keviczky (1994). A new indirect adaptive time-delay estimation method. In ¹rans. S½SID’94, Copenhagen, pp. 549—554. Buesher, K. L. and P. R. Kumar (1993). Selecting model complexity in learning problems. In ¹rans. 32nd CDC, San Antonio, pp. 3527—3533. Chapellat, H., M. Dahleh and S. P. Bhattacharyya. (1990). Robust stability under structured and unstructured perturbations. IEEE ¹rans. Automat. Control, AC-35, 1100—1108. Dahleh, M. and M. A. Dahleh (1990). Optimal rejection of persistent and bounded disturbances: continuity properties and adaptation. IEEE ¹rans. Automat. Control, AC-35, 687—696. Dahleh, M. A. and M. H. Khammash (1993). Control design for plants with structured uncertainty. Automatica, 29, 37—56. Goodwin, G. C., B. Ninness, P. Cockerell and Salgado (1990). Illustration of an integrated approach to adaptive control. Int. J. Adaptive Control Signal Process., 4, 149—162. Gevers, M., (1995). Identification for control. In Preprints 5th IFAC Symp Adaptive Systems in Control and Signal Processing, Budapest, pp. 1—12. Giri, F., J. M. Dion, M. M’Saad and L. Dugard (1989). A globally convergent pole-placement indirect adaptive controller. IEEE ¹rans. Automat. Control, AC-34, 353—356. Goodwin, G. C. (1991). Can we identify adaptive control? In Proc. ECC’91, Grenoble, pp. 1714—1725. Goodwin, G. C., P. J. Ramadge and P. E. Caines (1981). Discrete time stochastic adaptive control. SIAM J. Control Optim., 19, 829—853. Khammash, M. H. (1994). Robust performance bounds for systems with time-varying uncertainty, In Proc. of the 33rd Conference on Decision and Control, Lake Buena Vista, FL, pp. 28—33. Khammash, M. H. (1995). Robust steady-state tracking. IEEE ¹rans. Automat. Control, AC-40, 1872—1880. Khammash, M. H. and J. B. Pearson (1991). Performance robustness of discrete-time systems with structured uncertainty. IEEE ¹rans. Automat. Control, AC-36, 398—412. Khammash, M. H. and J. B. Pearson (1993). Analysis and design for robust performance with structured uncertainty. Systems Control ¸ett., 20, 179—187. Kosut, R. L., M. K. Lau and S. P. Boyd (1992). Set-membership identification of systems with parametric and nonparametric uncertainty. IEEE ¹rans. Automat. Control, AC-37, 929—941. Kreisselmeier, G. and M. C. Smith (1986). Stable adaptive regulation of arbitrary nth order plants. IEEE ¹rans. Automat. Control, AC-31, 299—305. Lozano, L. R. and G. C. Goodwin (1985). A globally convergent adaptive pole placement algorithm without a persistancy of excitation requirement. IEEE ¹rans. Automat Control, AC30, 795—798. Lozano, R. and X. H. Zhao (1994). Adaptive pole-placement without excitation probing signals. IEEE ¹rans. Automat. Control, AC-39, 47—58 Middleton, R. H., G. C. Goodwin, D. J. Hill and D. Q. Mayne (1988). Design issues in adaptive control. IEEE ¹rans Automat. Control, AC-33, 50—58.

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Van den Hof, P. and R. J. P. Schrama (1994). Identification and control closed loop issues. In Preprints 10th IFAC/IFORS Symp. on System Identification, Copenhagen, pp. 1—14. Veres, S. M. (1992). Limited complexity and parallel implementation of polytope updating. In Proc. American Control Conf., 24—26 June, 1992, Chicago, pp. 1061—1062. Veres, S. M. and J. P. Norton (1991). Structure selection for bounded parameter models: consistency conditions and selection criterion. IEEE ¹rans. Automat. Control, AC-36, 474—481. Veres, S. M. and J. P. Norton (1993). Predictive self-tuning control by parameter bounding and worst-case design. Automatica, 29, 911—928. Veres, S. M., D. S. Wall, S. Hermsmeyer, I. Valyi, A. V. Kuntsevich and S. Sheng (1996). ¹he Geometric Bounding ¹oolbox for MA¹¸AB. Software licensed by the University of Birmingham, Birmingham. Weyer, E., I. M. Y. Mareels and J. W. Polderman (1994). Limitations of robust adaptive pole placement control. IEEE ¹rans. Automat. Control, 39, 1665—1671.