Observer-Based Control of Discrete-Time LPV Systems With Uncertain Parameters

Transcription

1 230 IEEE TRASACTIOS O AUTOMATIC COTROL, VOL. 55, O. 9, SEPTEMBER 200 Observer-Based Control of Discrete-Time LPV Systems With Uncertain Parameters W. P. Maurice H. Heemels, Jamal Daafouz, and Gilles Millerioux Abstract In this note, linear matrix inequality-based design conditions are presented for observer-based controllers that stabilize discrete-time linear parameter-varying systems in the situation where the parameters are not exactly known, but are only available with a finite accuracy. The presented framework allows to make tradeoffs between the admissible lel of parameter uncertainty on the one hand and the transient performance on the other. In addition, the lel of parameter uncertainty can be maximized while still guaranteeing closed-loop stability. Index Terms Linear matrix inequalities (LMIs), linear parameter-varying (LPV) systems, output feedback and observers, robust control, separation principle. I. ITRODUCTIO Linear parameter-varying (LPV) systems and controllers have receivedconsiderableattentionfromthecontrolcommunityinrecentyears [2] [4], [6], [5] [7]. When LPV controllers are implemented in practice two important properties need to be satisfied. First of all, the controller needs to be output-based, as in practice it is rarely the case that the full state variable is available for feedback. Secondly, the controller must be robust with respect to some degree of mismatch between the available and the true parameters as the real parameters are not always known exactly, although this is often assumed in the literature on LPV systems. This note will address the design of stabilizing controllers for discrete-time LPV systems that satisfy these two properties. In [2] and [2], the continuous-time version of this problem was considered, but, unfortunately, only conditions in terms of bilinear matrix inequalities (BMIs) were presented. Only recently a solution was given in [8] using convex programming techniques. In the discrete-time case, output-based control design for LPV systems for which the measured parameters do not exactly fit the real ones is at present an open problem. In [4], it is shown that an observer that is asymptotically recovering the state when the parameters are exactly measured, is input-to-state stable (ISS) [0], [8] with respect to mismatch between the true and the available parameters. Hower, [4] does not study the observer synthesis nor the output-based stabilization problem. These two important problems will be solved in this note. Closely related to LPV systems are switched linear (SL) systems and piecewise affine (PWA) systems, which can be perceived as a subclass of LPV systems in which the parameters only take a finite number of values. Observer-based control design for SL systems has been considered in [5] under the assumption of having exact knowledge of the parameter values. In case of unknown parameters, [] proposes design Manuscript received July 29, 2009; rised December 20, 2009; accepted May 05, 200. Date of publication May 24, 200; date of current version September 09, 200. This work was supported in part by the AR project ArHyCo, ARPEGE, number AR-2008 SEGI 004, and the European Committee through the EU-FP7 project MOBY-DIC (248858). Recommended by Associate Editor F. Wu. M. Heemels is with the Hybrid and etworked Systems Group, Department of Mechanical Engineering, Eindhoven University of Technology, 5600 MB Eindhoven, The etherlands ( M.Heemels@tue.nl). J. Daafouz is with CRA, UMR CRS 7039, ancy University, 5456 Vandoeuvre Cedex, France ( jamal.daafouz@ensem.inpl-nancy.fr). G. Millerioux is with CRA, UMR CRS 7039, ancy University, ESSTI, 5459 Vandoeuvre-Les-ancy, France ( gilles.millerioux@esstin.uhpnancy.fr). Digital Object Identifier 0.09/TAC conditions for observers that include an estimation procedure for the parameters. In [], [9] observers and observer-based controllers were designed for PWA systems based on linear matrix inequalities (LMIs). In this case the parameters are also unknown as they depend on the state variable that has to be estimated. Hower, as for SL and PWA systems the number of parameter values is finite, these results are not applicable to general LPV systems. This note provides a solution to the open problem of output-based controller design for discrete-time LPV systems with uncertain parameters. The main contributions are LMI-based conditions for the separate design of state observers and input-to-state stabilizing state feedbacks for discrete-time LPV systems. We prove that the resulting closed-loop system is globally exponentially stable for some lel of mismatch between the true parameters and the available ones. The flexibility in our framework allows to make tradeoffs between the lel of mismatch and the transient performance of the closed loop in terms of the decay factor. Moreover, the lel of parameter uncertainty can be maximized while still guaranteeing closed-loop stability. OTATIO AD BASIC DEFIITIOS, 0, and are the field of real numbers, the set of nonnegative reals and the set of nonnegative integers, respectively. The i-th entry of a real vector x is denoted by x i (subscripts are pused for denoting discrete-time dependence). We denote by kxk = x T x the Euclidean norm of x in n, where M T denotes the transpose of a vector or matrix M, and by kxk its infinity norm given by max i jx i j.for a sequence fv k g k2 with v k 2 n, we denote its supremum norm sup k2 kv k k by kvk. For a matrix M 2 n2m we denote its spectral norm max (M T M ) by kmk, where max (M T M ) denotes the largest eigenvalue of M T M. When a matrix P is positive definite (including symmetry), we write P 0. If it is positive semi-definite, we use P 0. Similarly, for (semi-)negative definiteness we write and.by0and we denote the zero and the identity matrix of appropriate dimensions. A function ' : 0! 0 belongs to class K if it is continuous, strictly increasing and '(0) = 0 and to class K if additionally '(s)!as s!. A function : 0 2 0! 0 belongs to class KL if for each fixed k 2 0, (;k) 2Kand for each fixed s 2 0, (s; ) is decreasing and lim k! (s; k) = 0. Consider now the discrete-time nonlinear systems and x k+ = G(x k ;! k ) () x k+ = G v(x k ;v k ;! k ) (2) where x k 2 n is the state, v k 2 d is an unknown disturbance input and! k 2 d is an uncertainty parameter at discrete time k 2. G : n 2 d! n and G v : n 2 d 2 d! n are arbitrary nonlinear functions. We assume that! k 2, k 2 for some set d. Definition : [0], [8] The system () with uncertainty set is called globally asymptotically stable (GAS), if there exists a KL-function such that, for each x 0 2 n and all f! k g k2 with! k 2, k 2, it holds that the corresponding state trajectory satisfies kx k k (kx 0 k;k) for all k 2.If can be taken of the form (s; k) =ds k for some d 0 and 0 <the system () with uncertainty set is called globally exponentially stable (GES). The system (2) with uncertainty set is said to be ISS with respect to v if there exist a KL-function and a K-function such that, for each x 0 2 n, all fv k g k2 and all f! k g k2 with! k 2, k 2, it holds for all k 2 that kx k k(kx 0 k;k)+(kvk ): /$ IEEE

2 IEEE TRASACTIOS O AUTOMATIC COTROL, VOL. 55, O. 9, SEPTEMBER We call a decay factor for () and the function an ISS gain of (2). ext we state sufficient conditions for ISS using so-called ISS Lyapunov functions. The proofs are omitted for shortness, but can be based on [0], [3] by adopting parameter-dependent Lyapunov functions. Theorem : Let d ;d 2 2 0, let a; b; c; 2 >0 with c b and let (s) := as, 2 (s) := bs, 3 (s) := cs and 2K. Furthermore, let V : n 2 d! 0 be a function such that (kxk) V (x;!) 2 (kxk) V(G v (x; v;! );! 2 )0V(x;! )0 3 (kxk)+(kvk) (3a) (3b) for all x 2 n, all v 2 d and!,!,! 2 2. Then system (2) with uncertainty set is ISS with respect to v. In case (3a) and V (G(x;! );! 2) 0 V (x;! ) 0 3(kxk) hold for all x and!,!,! 2 2, then system () with uncertainty set is GES with decay factor 0 (c=b) 2 [0; ). A function V that satisfies (3) is called an ISS Lyapunov function. II. PROBLEM STATEMET We consider discrete-time LPV systems given by x k+ = A( k )x k + Bu k y k = Cx k + Du k (4a) (4b) with x k 2 n, y k 2 m and u k 2 r the state, output, and control input at discrete time k 2, k 2 L is a time-varying parameter and A() 2 n2n for each, B 2 n2r, C 2 m2n, D 2 m2r. The parameter k, k 2 lies in some set 2 L and we assume that A : 2! n2n can be written in the polytopic form A() = i= i ()A i for certain continuous functions i :2! and matrices A i 2 n2n, i =;...;. In addition we assume that the mapping :2! given by := ( ;...; ) > is such that (2) S with S = f 2 j i 0; i =;...; and i= i =g. Hence, A() lies for each 2 2 in the convex hull CofA ;...;A g. In this note, we focus on the situation where the true (time-varying) parameter k is not available, but only an estimated parameter ^ k 2 2 fulfilling k k 0 ^ k k is known, where is some nonnegative constant indicating the uncertainty lel. Problem : Design an observer-based controller ^x k+ = A(^ k )^x k + Bu k + L(^ k )(y k 0 ^y k ) ^y k = C ^x k + Du k u k = K(^ k )^x k (5a) (5b) (5c) with L(^ k )= i= i (^ k )L i and K(^ k )= i= i (^ k )K i by appropriately choosing the gains L i and K i, i = ;...; such that the closed-loop system (4) (5) is GAS when the uncertainty satisfies k k 0 ^ k k and ^ k 2 2 for all k 2. III. OBSERVER DESIG We first focus on the estimation of the state x k using a polytopic observer of the form ^x k+ = A(^ k )^x k + Bu k + L(^ k )(y k 0 ^y k ) ^y k = C ^x k + Du k ; where ^ k 2 2 and possibly k 6=^ k. The estimation error e k :=x k 0 ^x k is governed by (6) e k+ = A e (^ k )e k + v k (7) with A e(^ k ):= i= i (^ k ) ~ A i, where ~ A i = A i 0 L ic and v k =(A( k ) 0 A(^ k ))x k : (8) =:A( ;^ ) Sometimes we write k := ( k ) and ^ k := (^ k ) for shortness. Theorem 2: Assume that there exist symmetric matrices P i 2 n2n, matrices G i 2 n2n, F i 2 n2m, i = ;...; and a scalar satisfying for all i; j =;...; the LMIs G T i + G i 0 P j 0 G ia i 0 F ic G i 0 0 A T i G T i 0 C T Fi T P i 0 G T i (9) then the error dynamics (7) with uncertainty set 2 for ^ and L i = G 0 i F i is ISS with respect to v with ISS gain (s) = s, s 2 0. Moreover, V e(e k ; ^ k )=ek T ( ^ i i= kp i)e k is an ISS Lyapunov function that satisfies for all ^ k, ^ k+ 2S, e k 2 n, v k 2 n V e (e k+ ; ^ k+ )0V e (e k ; ^ k ) 0ke k k 2 + kv k k 2 ke k k 2 V e(e k ; ^ k ) ke k k 2 : (0a) (0b) Proof: The feasibility of the LMIs (9) for all i; j = ;...; implies that P i 0 and G T i + G i 0 P j G i G T i 0 () are satisfied for all i; j =;...;. From () it follows that P i and G T i + G i 0 P j 0 G ig T i 0 for all i; j =;...;. Since (P 0=2 j G T i 0 P =2 j ) T (P 0=2 j G T i 0 P =2 j ) 0 implies G i P 0 j G T i G i + G T i 0 P j (2) it follows now that G i Pj 0 G T i 0 G i G T i and thus P j, j = ;...;, because G i is invertible. Invertibility of G i follows from G T i + G i P j as this leads for G i x =0that x T P j x 0 and thus x =0. As such, we have (0b) for all e k 2 n and all ^ k 2S. To prove (0a), note that feasibility of the LMIs (9) gives together with (2) for all i; j the LMI G i Pj 0 G T i 0 G i A i 0 F i C G i 0 0 A T i G T i 0 C T Fi T P i 0 G T i 0 0 This is equivalent for all i; j to and ij 9 ij T ij 0 with ij = 9 ij = G ipj P j 0 P j (A i 0 L i C) P j 0 0 (A i 0 L ic) T P j P i 0 P j 0 0 0: (3) : (4) Hence, we have that for all i; j 9 ij 0. For shortness, we write ^ k i = i (^ k ) and k i = i ( k ). Multiplying 9 ij 0 by ^ k i and summing, The LMIs (9) imply that G is invertible for each i =;...; as is shown in the proof.

3 232 IEEE TRASACTIOS O AUTOMATIC COTROL, VOL. 55, O. 9, SEPTEMBER 200 multiplying by ^ j k+ and summing, and using the Schur lemma yield P k 0 A T e(^ k ) P k+ 0 0 A e (^ k ) 0 0 with P k := P (^ k ) := ^ i i= kp i and P k+ := P (^ k+ ) = ^ j Pj. ote that we used ^ j= k+ k, ^ k+ 2S(due to ^ k, ^ k+ 2 2). Hence, for all e k 2 n and v k 2 n ( ek T vk T ) M e k 0 with v k M = 0+P k 0A e(^ k ) T P k+ A e(^ k ) 0A e(^ k ) T P k+ 0P k+ A e (^ k ) 0P k+ + : This implies for all e k and all v k that (A e (^ k )e k + v k ) T P k+ (A e (^ k )e k + v k ) 0 e T k P k e k 0e T k e k + v T k v k : This can be rewritten as (0a). We could base ourselves now on Theorem to obtain ISS, but we proceed here to explicitly compute the ISS gain. From (0a) and (0b), one has V e (e k+ ; ^ k+ ) 0 V e (e k ; ^ k )+ kv k k 2 : (5) Applying (5) repetitively leads to V e(e k ; ^ k ) 0 0 k V e (e 0 ; ^ 0 )+ k0 l=0 k V e (e 0 ; ^ 0 )+ 2 kvk 2 : 0 k0l0 kv l k 2 Finally, by using again (0b), taking the square root, we obtain the inequality ke k k p 0 k=2 ke 0 k + kvk: (6) This inequality shows ISS with respect to v with linear ISS gain (s) = s, s 2 0. In case the conditions of Theorem 2 hold, the polytopic observer (6) guarantees GES of the error dynamics (7) in the nominal case where k = ^ k for all k 2 (as then v k =0, k 2 ). In case k 6= ^ k, ISS as in (6) guarantees only a steady-state estimation error e that is smaller than sup k2 kx k k with := supfa(; ^) jk 0 ^k g. Hence, a kind of steady state relative error can be obtained in the sense that lim sup k! ke k k lim sup k! kx k k as was used also in [] in the context of observer design for discontinuous PWA systems. Remark : ote that the normalization of certain constants in (0) to is without loss of generality as any ISS Lyapunov function V e of the quadratic type as specified in the theorem for (7) can be multiplied by a sufficiently large positive constant to satisfy (0). As mentioned, if the hypotheses of Theorem 2 are satisfied, the polytopic observer (6) guarantees GES of the error dynamics in the nominal case ( k =^ k for all k 2 ). Actually, the observer satisfies the matrix inequalities (A i 0 L i C) T Pj ~ (A i 0 L i C) 0 P ~ i 0; i;j=;...; and ~P i 0; i=;...; (7) which are both necessary and sufficient conditions for the existence of a parameter-dependent quadratic Lyapunov function proving GES of the estimation error dynamics in the nominal case (^ k = k ) [7], [4]. Interestingly, the nominal conditions in (7) also guarantee that the hypotheses of Theorem 2 are satisfied (as will be shown in Theorem 3 below). This shows the nonconservatism of the LMIs (9) as the existence of a nominal observer for the exact LPV system, with a parameter-dependent quadratic Lyapunov function proving GES of the error dynamics, is sufficient for (9) to hold. This also shows that any GES observer for the exact LPV system has some degree of robustness. Theorem 3: [9] If there exist ~ Pi and L i, i = ;...; such that (7) holds, then there are symmetric matrices P i and matrices F i, G i, i =;...; and a scalar satisfying for all i; j =;...; the LMIs (9). IV. STATE FEEDBACK DESIG We now focus on the design of a state feedback for (4a) using an estimated state given by u k = K(^ k )^x k = K(^ k )(x k 0 e k ) (8) with K(^ k )= i= i (^ k )K i and e k the estimation error. This results in the closed loop x k+ = A x (^ k )x k + v k 0 BK(^ k )e k (9) with, as before, v k given by (8) and A x (^ k )= i= i (^ k )(A i + BK i ) : Again, we sometimes write ^ k i = i (^ k ) and k i = i ( k ).Wenow study ISS of (9). Theorem 4: Assume that there exist symmetric matrices Y i 2 n2n, matrices Z i 2 m2n, i = ;...; and scalars xv; xe; with >0and xv satisfying for i; j =;...; the LMI conditions Y i 0 0 Y ia T i + Zi T B T Y i 0 xv xe 0Zi T B T 0 A i Y i + BZ i 0BZ i Y j 0 Y i and for i =;...; Y i A 0 (20a) (20b) then the closed-loop system (9) with uncertainty set 2 for ^ and K i = Z iyi 0, i =;...;, is ISS with respect to e and v and V x(x k ; ^ k )= x T k ^ i i= ks i x k with S i = Yi 0, i =;...;, is an ISS Lyapunov function that satisfies for all ^ k, ^ k+ 2S, all x k 2 n, all e k 2 n and all v k 2 n V x(x k+ ; ^ k+ ) 0 V x(x k ; ^ k ) 0kx k k 2 + xv kv k k xe ke k k 2 and kx k k 2 V x(x k ; ^ k ) xvkx k k 2 : (2a) (2b) Proof: Assume that the LMIs in (20) are feasible and define S i := Y 0 i. Premultiply the LMIs in (20a) by T = diag(s i; ;S i; ; ),

4 IEEE TRASACTIOS O AUTOMATIC COTROL, VOL. 55, O. 9, SEPTEMBER postmultiply it by T T = T and apply the Schur lemma to arrive for i; j =;...; at S i (A i + BK i) T 0 xv xe Si 2 0Ki T B T A i + BK i 0BK i S 0 j 0: (22) Multiply (22) by ^ k, i sum for i =;...; (note that ^ i i= k =, since ^ k 2 2) and use the Schur lemma pivoting again around the south east block to obtain for j =;...; T () T () T 0S j ^ ka i BKi 0S j + xv () T i= T 3 i= ^ i kbk i T S j T 33 0 (23) where T = 0( ^ i i= ka BKi ) T S j ( ^ i i= ka BKi )+ ^ i i= ks i 0, T 3 = ( ^ i i= kbk i ) T S j ( ^ i i= ka BKi ), and T 33 = 0( ^ i i= kbk i ) T S j ( ^ i i= kbk i ) + ^ i xe i= ksi 2. Using that Si 2 02, i = ;...; due to the second LMIs in (20), multiplying (23) by ^ j k+ and summing for j = ;...; (note that ^ j j= k+ =, since ^ k+ 2 2) leads, in a similar way as we obtained (0a) in the proof of Theorem 2, to V x (x k+ ; ^ k+ ) 0 V x (x k ; ^ k ) 0kx k k 2 + xvkv k k xeke k k 2 : (24) As (22) implies S i and S i xv, we have for all x k 2 n and k 2Sthat (2b) holds. From Theorem it follows now that the closed-loop system is ISS with respect to v and e. The following corollary applies when the full state x k is known (i.e., e k =0for all k 2 ). Corollary : Let the hypotheses of Theorem 4 be satisfied. Then the LPV system consisting of (4a) and the state feedback u k = K(^ k )x k with uncertainty set 2 for and K i = Z iy 0 i, i =;...; is GES for all uncertainties satisfying ka(; ^)k ; when <= xv. Proof: From (2a) with e k = 0, k 2, and v k = A( k ; ^ k )x k, it follows that V x (x k+ ; ^ k+ ) 0 V x (x k ; ^ k ) 0( 0 xv )kx k k 2 : (25) Together with (2b) this proves GES on the basis of Theorem. An analogous result to Theorem 3 can also be shown for the state feedback design. In particular, a nominal state feedback u k = K( k )x k with K( k )= i= i ( k )K i (i.e., without estimation error (e k = 0, k 2 ) and exact knowledge of the parameters, k = ^ k, k 2 ) coupled to the LPV system (4a) is GES if there are K i, S ~ i, i =;...; such that (A i + BK i) T Sj(A ~ i + BK i) 0 S ~ i 0; i;j=;...; ~S i 0; i=;...;: (26) Clearly, a state feedback (8) that renders (9) ISS (proved by parameter-dependent quadratic ISS Lyapunov functions) certainly satisfies (26). Interestingly, the converse also holds in the sense that a nominally stabilizing state feedback for (4a) satisfying (26) has some robustness properties in the sense that (2) holds for some V x and en stronger, the LMIs in (20) are feasible. This clearly indicates the nonconservatism of the derived LMIs in Theorem 4. Hower, note that (26) does not allow any minimization of the ISS gains, while the results of Theorem 4 do. Theorem 5: [9] Suppose that there exist K i, ~ S i, i =;...; such that (26) is satisfied. Then there are symmetric matrices Y i and matrices Z i, i =;...; and scalars xv; xe; with > 0 and xv satisfying the LMIs (20) for i; j =;...;. V. OBSERVER-BASED COTROL DESIG ext, we will show that the separate design of the observer as in Section IV and a state feedback as in Section V leads to a stabilizing output-based controller for some nontrivial lel of uncertainty := supfka(; ^)k j k 0 ^k g. The closed-loop system is given by x k+ e k+ = A( k)+bk(^ k ) 0BK(^ k ) A( k ) 0 A(^ k ) A(^ k )0L(^ k )C x k e k : (27) Theorem 6: Let an observer (6) that satisfies the hypotheses of Theorem 2 and a state feedback law that satisfies the hypotheses of Theorem 4 be given. Then for any maxf 0 = ; 0 = xv g"< and any 0 < ( 0 ( 0 ") )= 02 xe the closed-loop system (27) is GES with decay factor equal to p " for all uncertainties satisfying ka(; ^)k := ( 0 ( 0 ") xv) + xv : Proof: Consider the candidate Lyapunov function V (x k ;e k ; ^ k ) := V x(x k ; ^ k )+V e(e k ; ^ k ) for the closed-loop system (27) with > 0. From (0) and (2) and noting that v k =A( k ; ^ k )x k with we have that = supfka(; ^)k jk 0 ^k g V (x k ;e k ; ^ k ; ^ k+ ) (0 + xv )kx k k 2 0 ( 0 02 xe )ke k k 2 ; (28) where V (x k ;e k ; ^ k ; ^ k+ ) := V (x k+ ;e k+ ; ^ k+ ) 0 V (x k ;e k ; ^ k ) with (xk+;e T k+) T T as in (27). To obtain GES with decay factor " it suffices to guarantee p V (x k+ ;e k+ ; ^ k+ ) "V (x k ;e k ; ^ k ) as V can be bounded by quadratic K functions (s) =as 2 and 2 (s) =bs 2 as in (3a) in the norm k(xk T ;ek T ) T k. To obtain this inequality, it is sufficient to have V (x k ;e k ; ^ k ; ^ k+ ) 0( 0 ")( xv kx k k 2 + ke k k 2 ) (29) because V (x k ;e k ; ^ k ) xvkx k k 2 + ke k k 2 : Due to (28), the inequality (29) holds when (i) 0 xv ( 0 ") xv and (ii) 0 02 xe ( 0 "). Obviously, under the hypotheses of the theorem these conditions are true, which completes the proof. It is of interest to find the Lyapunov function V that provides the largest robustness in terms of. To maximize the value for 2 (for a fixed value of the decay factor p ") it is clear that we have to maximize f () := =( + xv ): Since df ()=d = =( + xv) 2 0; the maximum is obtained for the largest allowable value of, which is ( 0 ( 0 ") )= 02 xe and thus the maximum of is (") = ( 0 [ 0 "] )( 0 [ 0 "] xv ) 02 xe +(0 [ 0 "] ) xv : (30) Hence, we obtained the following corollary. Corollary 2: Let an observer (6) that satisfies the hypotheses of Theorem 2 and a state feedback law that satisfies the hypotheses of Theorem 4 be given. Then for any maxf 0 = ; 0 = xvg "< the closed-loop system (27) is GES with decay factor equal to p " for all uncertainties satisfying ka(; ^)k (") with (") as in (30).

5 234 IEEE TRASACTIOS O AUTOMATIC COTROL, VOL. 55, O. 9, SEPTEMBER 200 Suppose we now would like to find the value of " such that the admissible uncertainty lel (") is maximal. Since it can be inspected that d 2 (")=d" > 0 for any maxf 0 = ; 0 = xv g"<, maximizing robustness requires maximizing (actually taking supremum of) " and thus taking it close to. This yields that the maximal value of can become arbitrarily close to () = 02 xe + xv (3) while still guaranteeing stability. Hence, for maximizing robustness in terms of maximizing ("), we should maximize " meaning that the performance in terms of the decay factor p " is worst. As such, we encountered a classical tradeoff between robustness and performance. The reasoning above maximizes robustness for fixed values of xv, and xe. Since we have determined the maximum () as in (3) given these s, we can now optimize robustness by appropriately selecting the gains L i and K i, i =;...;. From (3), it is clear that we have to minimize 02 xe + xv to get the maximal value for the uncertainty lel (just below) () = =( 02 xe + xv), while still guaranteeing GES (for decay factor just below ). This gives rise to the following procedure to get maximal robustness in the mismatch between the scheduling parameter ^ k and the actual one k as reflected in, while still guaranteeing GES. Design Procedure: Step : Minimize subject to (9) for i; j =;...;. This gives the minimum 3 and the corresponding observer gains L i, i =;...;. Step 2: Given 3 as in Step. Fix >0and minimize the expression 02 xe 3 + xv subject to the LMIs given in (20). This results in the feedback gains K i, i =;...;. The optimization problems in Step and 2 are convex problems as we are minimizing linear costs subject to LMI constraints. Step 2 might en be extended by performing a line search in and applying the above procedure repetitively. Once, the minimal value 302 xe xv is found, one can on the basis of Theorem 6 and (30) still make tradeoffs between transient performance in terms of the decay factor p " and robustness in terms of ("). Letting " increase from maxf 0 = ; 0 = xvg (maximal performance, minimal robustness) to (minimal performance, maximal robustness), tradeoff curves between performance and robustness are obtained, as was already indicated in Corollary 2. Fig.. Line search. Fig. 2. Tradeoff performance/robustness. VI. ILLUSTRATIVE EXAMPLE Consider the LPV system (4) with ; B = 0:25 0 A( k )= 0 0: :6 + k C =[02]; D =0 and k 2 [0 ; 0:5], k 2. In this case, we can take the functions () = (0:5 0 )=0:5 and 2 () = =0:5 with A = A(0) and A 2 = A(0:5). The observer is designed using Theorem 2 along with the optimization problem in Step of the design procedure presented in the prious section. The optimal solution is given by 3 = 5:8277 with observer gains L = [00: :00 0:3870 ] T and L 2 = [00: :00 0:7094 ] T : With this optimal observer and the associated slope of the linear ISS gain, 3 a line search involving > 0 is performed in order to minimize the cost J = 02 xe 3 + xv subject to the LMIs given in (20) for all i; j (Step 2). Fig. shows the minimum of J for each fixed, which is the smallest 0 for 3 =0:2986 yielding 3 xe =0:2663 and 3 xv =3:9284 and corresponds to the controller gains K =[00: :24 0 0:2387], K 2 =[0: : :648]. As a consequence, the maximum lel of uncertainty is 3 max = xe xv =0:786: Hence, for A( k ; ^ k ) = j k 0 ^ k j < 0:786 GES of the closed-loop system (27) is guaranteed (with a decay factor close to ). Letting " increase from maxf 0 =; 3 0 =xvg 3 to leads to the tradeoff curves between performance in terms of the decay factor p " and robustness to uncertainty A( k ; ^ k ) in terms of as depicted in Fig. 2. VII. COCLUSIO In this note, the design of robustly stabilizing output-based feedback controllers is considered for discrete-time LPV systems in which the scheduling parameters are only known up to a given precision. The

6 IEEE TRASACTIOS O AUTOMATIC COTROL, VOL. 55, O. 9, SEPTEMBER output-based controllers are obtained using a separate design of the observer and the state feedback and we showed that the interconnection of the LPV plant, observer and state feedback leads to a globally exponentially stable closed-loop system for certain lels of mismatch between estimated and true parameters. The nonconservatism of our approach is demonstrated by showing that well known conditions for nominally stabilizing observers and feedbacks (i.e., without mismatch between true and available parameters) imply our LMI-based conditions. The flexibility in the framework allows to construct the controller that guarantees global exponential stability for the largest lel of parameter uncertainty and to make tradeoffs between transient performance in terms of decay factors and robustness with respect to parameter uncertainty. REFERECES [] A. Alessandri, M. Baglietto, and G. Battistelli, Luenberger observers for switching discrete-time linear systems, Int. J. Control, vol. 80, no. 2, pp , [2] P. Apkarian and P. Gahinet, A convex characterization of gain-scheduled H infinity controllers, IEEE Trans. Autom. Control, vol. 40, no. 5, pp , May 995. [3] G. Becker and A. Packard, Robust performance of linear parametrically varying systems using parametrically-dependent linear feedback, Syst. Control Lett., vol. 23, pp , 994. [4] F. Blanchini and S. Miani, Stabilization of LPV systems: State feedback, state estimation, and duality, SIAM J. Control Optimiz., vol. 42, pp , [5] F. Blanchini, S. Miani, and F. Mesquine, A separation principle for linear switching systems and parametrization of all stabilizing controllers, IEEE Trans. Autom. Control, vol. 54, no. 2, pp , Feb [6] G. Chesi, A. Garulli, A. Tesi, and A. Vicino, Homogeneous Polynomial Forms for Robustness Analysis of Uncertain Systems. ew York: Springer, [7] J. Daafouz and J. 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Pavlov, Tracking and synchronisation for a class of PWA systems, Automatica, vol. 44, no., pp , Adaptive Tracking for Stochastic onlinear Systems With Markovian Switching Zhao Jing Wu, Jun Yang, and Peng Shi Abstract The problem of the adaptive tracking for a class of stochastic nonlinear systems with stationary Markovian switching is considered in this note. An Ito formula is proposed for stochastic integral equations with an integral about martingale measure. An adaptive backstepping controller is designed such that the closed-loop system has a unique solution that is globally bounded in probability and -norm of the tracking error converges to an arbitrarily small neighborhood of zero. A simulation example demonstrates the efficiency of the proposed scheme. Index Terms Backstepping, Markovian switching, nonlinear stochastic systems. I. ITRODUCTIO Recently, there have been increasing attentions doted to stochastic hybrid systems, in which the stochastic continuous dynamics (states) are intertwined with the stochastic discrete ents (modes). The recent work [] and [2] provided a framework of theory of stochastic differential equations with Markovian switching for the following general system: dx(t) =f (x(t);t;r(t))dt + g(x(t);t;r(t))dw (t) () where x(t) 2 n is the state of system and W (t) is an m-dimensional independent standard Wiener process (Brownian motion). The underlying complete filtration space is taken to be the quartet (; F; F t ;P) with F t satisfying the usual conditions. r(t) is a right-continuous Markov process on the probability space taking values in a finite mode space S = f; 2;...;g with generator 0=( pq ) 2 given by P pq (t) =P fr(t + s) =qjr(s) =pg = pqt + o(t); if p 6= q (2) + ppt + o(t); if p = q for any s; t 0 where pq > 0 is the transition rate from mode p to mode q if p 6= q while pp = 0 q=;q6=p pq. The Markovian process r(t) is independent of the Brownian motion W (t). Functions f : n S! n and g : n S! n2m are locally Lipschitz in x 2 n for all t 0. As in [3], let pq(p 6= q) be consecutive, left closed, right open intervals of the real line each having the length pq such that 2 =[0; 2 ); 3 =[ 2 ; ) and so on. Define a function h : S 2! by h(p; ) = q 0 p; if 2 pq 0; otherwise Manuscript received July 2, 2009; rised January 07, 200; accepted May 4, 200. First published May 24, 200; current version published September 09, 200. This work was supported by the ational atural Science Foundation of China ( , , , ), the China Postdoctoral Science Foundation ( ), the Shandong Postdoctoral Science Foundation ( ), and the Engineering and Physical Sciences Research Council, UK (EP/F02995). Recommended by Associate Editor J.-F. Zhang. Z. J. Wu and J. Yang are with the School of Mathematics and Informational Science, Yantai University, Yantai, Shandong Province , China ( wuzhaojing00@88.com). P. Shi is with the Department of Computing and Mathematical Sciences, University of Glamorgan, Pontypridd, CF37 DL, U.K., and the School of Engineering and Science, Victoria University, Melbourne, Vic. 800, Australia ( pshi@glam.ac.uk). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier 0.09/TAC (3) /$ IEEE