H 2 and H 1 cost estimates for time-invariant uncertain

Transcription

1 INT. J. CONTROL, 00, VOL. 75, NO. 9, ±79 Extended H and H systems norm characterizations and controller parametrizations for discrete-time M. C. DE OLIVEIRAy*, J. C. GEROMELy and J. BERNUSSOUz This paper presents new synthesis procedures for discrete-time linear systems. It is based on a recently developed stability condition which contains as particular cases both the celebrated Lyapunov theorem for precisely known systems and the quadratic stability condition for systems with uncertain parameters. These new synthesis conditions have some nice properties: (a) they can be expressed in terms of LMI (linear matrix inequalities) and (b) the optimization variables associated with the controller parameters are independent of the symmetric matrix that de nes a quadratic Lyapunov function used to test stability. This second feature is important for several reasons. First, structural constraints, as those appearing in the decentralized and static output-feedback control design, can be addressed less conservatively. Second, parameter dependent Lyapunov function can be considered with a very positive impact on the design of robust H and H 1 control problems. Third, the design of controller with mixed objectives (also gain-scheduled controllers) can be addressed without employing a unique Lyapunov matrix to test all objectives (scheduled operation points). The theory is illustrated by several numerical examples. 1. Introduction Using Lyapunov stability theory, a symmetric instrumental matrix variable can be used to check the stability of a linear system. For instance, consider the linear time-invariant discrete-time system x k 1 ˆ Ax k Bw k z k ˆ Cx k Dw k 1 where the state vector x n and all other vectors and matrices have appropriate dimensions. The system (1)± () is asymptotically stable if, and only if, there exists a symmetric matrix P ˆ P T n n such that the LMI P PA T AP P > 0 is feasible. Very recently this stability test has been extended by adding to the above LMI one more instrumental variable (de Oliveira et al a). It has been shown that the system (1)±() is asymptotically stable if, and only if, there exist a symmetric matrix P ˆ P T n n and a general matrix G n n such that the LMI Received 1 June 001. * Author for correspondence. DT/FEEC/UNICAMP Av Albert Einstein 00 Campinas, SP, Brazil mauricio@ mechanics.ucsd.edu y LAC-DT/School of Electrical and Computer Engineering, UNICAMP, CP 101, , Campinas, SP, Brazil. z LAAS-CNRS, 7, Avenue du Colonel Roche, 1077, Cedex, Toulouse, France. P AG GA T G G T P > 0 is feasible. While the introduction of this extra variable provides little advantage in the analysis of precisely known systems (de Oliveira et al a) has shown that this extra degree of freedom can be used to build a parameter dependent Lyapunov function (Feron et al. 199) that is able to prove the stability of time-invariant uncertain systems where the uncertainty is con ned to a bounded and convex polytope. For this class of uncertain systems, the extended stability condition contains the quadratic stability condition (Barmish 1985) as a particular case (de Oliveira et al a). It has also been anticipated in de Oliveira et al. (1999 a) that the new extended stability condition might also provide advantage in the context of synthesis of linear controllers. This constitutes the subject of this paper. This objective will be accomplished according to the following route. In }, the analysis condition provided in de Oliveira et al. (1999 a) is generalized to provide extended LMI characterizations for H and H 1 norm computations (Colaneri et al. 1997). As in de Oliveira et al. (1999 a), these extended analysis conditions feature a symmetric variable coming from Lyapunov theory as well as an extra instrumental variable. Consequently, they inherit all the nice properties from the extended stability test (de Oliveira et al a). For instance, they can be used to provide less conservative (parameter dependent) guaranteed H and H 1 cost estimates for time-invariant uncertain systems. Section concerned with the derivation of controller parameterizations to cope with the extended analysis conditions developed in the rst part. In the context of the standard Lyapunov analysis conditions, the exist- International Journal of Control ISSN 000±7179 print/issn 1±580 online # 00 Taylor & Francis Ltd DOI: /

2 Extended H and H 1 norm characterizations 7 ence of state-feedback and dynamic output-feedbac k controllers that are able to keep the H and H 1 norm of the closed-loop system below a prespeci ed level can be checked by solving an LMI optimization problem. Although these analysis conditions are not simultaneously convex (ane) on the instrumental Lyapunov variable and the controller parameters, linearizing change-of-variable s are known that are able to transform these synthesis problems into LMI optimization problems (see Bernussou et al. (1989) for statefeedback and Scherer et al. (1997) and Masubuchi et al. (1998) for dynamic output-feedback). The contribution of this paper is to show the existence of similar linearizing change-of-variable s that are able to render control synthesis problems formulated with the new extended analysis conditions into LMI optimization problems. A limitation of the standard synthesis parametrizations (Bernussou et al. 1989, Scherer et al. 1997, Masubuchi et al. 1998), is that the controller parameters are functions of the instrumental Lyapunov variable. In the extended synthesis parametrization developed here, that dependence no longer exists. Section is dedicated to show how this feature can bring advantage in the following situations.. In the synthesis of controllers with structure, such as decentralization (Siljak 1978). In Geromel et al. (199), de Oliveira et al. (000) and Geromel et al. (1999 a) decentralization has been obtained by imposing a block-diagonal structure on both the controller optimization parameters and the Lyapunov instrumental variable. With the extended conditions, a block-diagonal structure on the controller optimization parameters can be imposed without imposing unnecessary (conservative) constraints on the Lyapunov matrix. The same holds true for the structural approach to the synthesis of static output-feedbac k problem developed in Geromel et al. (199).. In the synthesis of robust controllers for timeinvariant uncertain systems. State-feedback robust controllers that are less conservative than the controllers developed in Geromel et al. (1991), which are based on the quadratic stability concept, can be developed with the new extended synthesis conditions for parameter dependent Lyapunov functions. The approach can be extended to cope with dynamic output-feedbac k robust synthesis in the context of de Oliveira et al. (000) and Geromel et al. (1999 a).. In the synthesis of multi-objective controllers. To obtain computationally ecient solution to control problems with multiple objectives, the usual approach is to force all Lyapunov matrices used to test the several design speci cations to be the same. This constraint is the core of the Lyapunov shaping paradigm (Scherer et al. 1997, Khargonekar and Rotea 1991), and constitutes an important source of conservatism. The extended synthesis conditions releases some of these constraints, and let several Lyapunov matrices be simultaneously considered in the multi-objective synthesis. While the new extended synthesis controller parametrizations release the constraints on the Lyapunov instrumental variable, this is obtained at the expense of imposing conservative constraints on the extra instrumental variable. Although this still imposed some degree of conservatism on the results, the number of free parameters in the extended synthesis is signi cantly increased as compared with the available techniques. Furthermore, it is shown that the results obtained with the standard parametrizations are always encompassed by the new extended formulations. Numerical examples spread throughout } help to illustrate the signi cant reduction in the conservatism that can be obtained using the proposed extension techniques. Several other control problems could also bene t from the extended analysis and synthesis conditions as, for instance, the design of gain-scheduled controllers, the design of model predictive controllers (MPC) and the design of robust lters. The use of extended controller parameterizations on these problems are currently under investigation and are not addressed in this paper due to the lack of space. An extended lter parametrization is investigated in Geromel et al. (1999 b). Part of the results contained in this paper have preliminarily appeared in short form in the conference paper (de Oliveira et al b). The notation used throughout the paper is fairly standard. Capital letters denote matrices and small letters denote vectors. Scalars are denoted with small Greek letters. For matrices or vectors T indicates transposition. For symmetric matrices, X > 0 0 indicates that X is positive de nite (semipositive de nite). In some partitioned symmetric matrices, the symbol T generically denotes each of its symmetric blocks. A real and square matrix X is said to be Schur stable if all its eigenvalues lie inside the unitary circle.. Extended LMI characterization of the H and H speci cations Consider the discrete-time time-invariant linear system (1)±(). The symbol H wz ± denotes the transfer function from the input w to the output z. The following lemmas are well known results that completely characterize the H and H 1 norm constraints through LMI.

3 8 M. C. de Oliveira et al. Lemma 1 (H norm): Assuming that D ˆ 0, the inequality kh wz ± k < holds, if, and only if, there exists symmetric matrices P and W such that W CP trace W < ; > 0; PC T P P AP B 5 PA T P 0 7 B T 0 I is feasible. Lemma (H 1 norm): The inequality kh wz ± k 1 < holds if, and only if, there exists a symmetric matrix P such that P AP B 0 PA T P 0 PC T B T 0 I D T 7 is feasible. 0 CP D I These results are now considered standard and proofs may be easily found in the literature. As expected, both conditions require that matrix A be Schur stable since the fundamental Lyapunov inequality () appears as one of their diagonal blocks. It has been shown in de Oliveira et al. (1999 a) that it is possible to extend the Lyapunov inequality () with the introduction of an additional instrumental matrix variable in the form (). This property is generalized in the next theorem to cope with the H norm calculation. Theorem 1 (Extended H norm): Assuming that D ˆ 0, the inequality kh wz ± k < holds if, and only if, there exists a matrix G and symmetric matrices P and W such that W CG trace W < ; > 0; GC T G G T P P AG B 7 GA T G G T P 0 7 B T 0 I is feasible. Proof: (Necessity) Choose G ˆ G T ˆ P in order to recover Lemma 1. (Suciency) Assume that the inequalities (7) are feasible. Hence G G T > P > 0. Note that this implies that G is non-singular. Since P is positive de nite the inequality P G T P 1 P G 0 holds. Therefore establishing G T P 1 G G G T P which yields W CG GC T G T P 1 G > 0; P AG B GA T G T P 1 7 G 0 B T 0 I Recalling that G is non-singular and multiplying the rst of these inequality on the right by T :ˆ diag I; G 1 PŠ and on the left by T T, the second inequality in (5) is recovered. The third inequality in (5) follows from the multiplication of the second inequality above by T :ˆ diag I; G 1 P; IŠ on the left and by T T on the right which concludes this proof. & Following the same steps it is possible to extend the H 1 norm calculation given in Lemma as below. Theorem (Extended H 1 norm): The inequality kh wz ± k 1 < holds if, and only if, there exist a matrix G and a symmetric matrix P such that P AG B 0 GA T G G T P 0 G T C T B T 0 I D T 7 8 is feasible. 0 CG D I Proof: (Necessity) Choose G ˆ G T ˆ P. (Suciency) Follow the same steps as in the proof of Theorem 1 so as to obtain P AG B 0 GA T G T P 1 G 0 G T C T B T 0 I D T 7 0 CG D I which recovers () if multiplied on the right by T ˆ diag I; G 1 P; I; IŠ and on the left by T T. & At this point it is interesting to put in evidence some important features of the given conditions. As in the standard case, a stability test can be drawn by extracting the diagonal block () from the inequalities (7) and (8). As in de Oliveira et al. (1999 a), the extra variable G is a full general matrix, i.e. it does not present any structural constraint such as symmetry. Also note the separation between the Lyapunov instrumental variable P and the dynamic system matrices A, B, C and D. This property and anity with respect to P, A, B, C and D provides a way to derive guaranteed H and H 1 cost LMI conditions based on parameter dependent Lyapunov functions (Feron et al. 199). For that sake, assume that the matrices that characterize the dynamic system (1)±() are uncertain but belong to a convex and bounded set. This set is such that matrix M de ned by M ˆ A B C D 9

4 belongs to a convex bounded polyhedron F. That is, each uncertain matrix in this domain may be written as an unknown convex combination of N given extreme matrices M 1 ;... ; M N such that F :ˆ M ¹ : M ¹ ˆ XN iˆ1 XN ¹ i M i ; ¹ i 0; iˆ1 ¹ i ˆ 1 10 The following theorems characterize guaranteed H and H 1 costs that can be computed using LMI optimization. Theorem (Extended guaranteed H cost): If there exist symmetric matrices W i ; P i ; i ˆ 1;... ; N; and a matrix G such that W i C i G trace W i < ; > 0; GC T i G G T P i P i A i G B i GA T i G G T P i 0 B T i 0 I 7 11 hold for all i ˆ 1;... ; N; where the matrices A i, B i, C i and D i de ne the extreme matrices M i ; i ˆ 1;... ; N; then the inequality kh wz ± k < holds for all matrices M in the domain F. Theorem (Extended guaranteed H 1 cost): If there exist symmetric matrices P i ; i ˆ 1;... ; N; and a matrix G such that P i A i G B i 0 GA T i G G T P i 0 G T C T i B T i 0 I D T 7 > 0; i ˆ 1;... ; N i 5 0 C i G D i I 1 where the matrices A i, B i, C i and D i de ne the extreme matrices M i ; i ˆ 1;... ; N; then the inequality kh wz ± k 1 < holds for all matrices M in the domain F. These theorems can be proved with the introduction of the ane parameter dependent Lyapunov matrix de ned as the convex combination of the uncertain parameters P ¹ :ˆ XN iˆ1 ¹ i P i 1 The key point is that the inequalities (11)±(1) are ane in the extreme matrices M i and in the variables W i and P i ; i ˆ 1;... ; N (see also de Oliveira et al a). For instance, to prove Theorem assume that all inequalities in (1) hold. Multiply each of these inequality by ¹ i > 0 and sum them up to obtain Extended H and H 1 norm characterizations 9 P ¹ A ¹ G B ¹ 0 GA ¹ T G G T P ¹ 0 G T C ¹ T B ¹ T 0 I D ¹ T 7 0 C ¹ G D ¹ I In this form, Theorem can now be used to show that Theorem is true since the above inequality holds for every ¹ such that M F. A similar argument can be used to prove Theorem. As in de Oliveira et al. (1999 a), a very important feature of these results, is that they contain the guaranteed H and H 1 costs provided by quadratic stability (Geromel et al. 1995) as a particular case. This fact can be stated as the following corollaries. Corollary 1 (Quadratic guaranteed H cost): Whenever there exist symmetric matrices P, W such that W C i P trace W < ; > 0; PC T i P P A i P B i PA T i P 0 7 ; i ˆ 1;... ; N B T i 0 I 1 where the matrices A i, B i, C i and D i de ne the extreme matrices M i ; i ˆ 1;... ; N; then Theorem also holds. Corollary (Quadratic guaranteed H 1 cost): Whenever there exists a symmetric matrix P such that P A i P B i 0 PA T i P 0 P T C T i B T i 0 I D T 7 > 0; i ˆ 1;... ; N 15 i 5 0 C i P D i I where the matrices A i, B i, C i and D i de ne the extreme matrices M i ; i ˆ 1;... ; N; then Theorem also holds. Indeed, if there exists a single matrix P that satis es all inequalities in Theorems or, then it is clear that the choice of variables G ˆ G T ˆ P i ˆ P; i ˆ 1;... ; N must also be feasible for the inequalities given in these theorems. This property is fundamental since it ensures that the new guarantee d will always provide better cost estimates than the standard quadratic guaranteed costs.. Extended controller parametrizatio n Consider the following discrete-time time-invariant linear system

5 70 M. C. de Oliveira et al. x k 1 ˆ Ax k B w w k B u u k z k ˆ C z x k D zw w k D zu u k y k ˆ C y x k D yw w k where the state vector x n and all other matrices and vectors have appropriate dimensions. The connection of this system with the linear controllers to be de ned will always provide linear systems with closed-loop statespace representation ~x k 1 ˆ A~x k Bw k z k ˆ C~x k Dw k 19 0 Unfortunately, the substitution of these closed-loop matrices (which are typically ane functions of the controller parameters) into the previously introduced analysis conditions generate products between the controller parameters and the instrumental variables P and G which are very hard to handle. In the context of the standard Lyapunov analysis conditions (Lemmas 1 and ), some non-linear transformation s are known to provide controller parametrizations that reduce important controller synthesis problems into LMI optimization problems. These non-linear transformation s typically maps the set of controller parameters plus the Lyapunov matrix P into a set of auxiliary variables so that no products between the Lyapunov matrix P and the transforme d controller optimization parameters occur. The purpose of this section is to show how these parametrizations can be generalized so that they can be used with the extended analysis conditions discussed in }. First, as anticipated in de Oliveira et al. (1999 a), it is shown that the change-of-variable s introduced in Bernussou et al. (1989) can be used to provide solution to state-feedback problems. Second, the linearizing change-of-variable s introduced in Scherer et al. (1997) and Masubuchi et al. (1998) is modi ed to provide a new framework for dynamic output-feedbac k synthesis problems. It is shown that with the introduction of some more independent variables it is possible to cast full order (the controller has the same order as the plant) dynamic output-feedbac k extended control problems as LMI optimization problems..1. State-feedbac k problem Throughout this section it is the assumed that the state vector x k is available for feedback. Moreover, the state information is not corrupted by the input w k. These assumptions are standard and can be enforced on the measurement equation (18) by assigning to the matrices C y and D yw the values C y ˆ I and D yw ˆ 0. The following linear static state-feedback control law u k ˆ Kx k 1 is sought. This feedback structure produces a system in the form (19)±(0) where ~x k ˆ x k and the closedloop matrices are given by A :ˆ A B u K; B :ˆ B w C :ˆ C z D zu K; D :ˆ D zw The main contribution of this section is to show that the non-linear transformation (change-of-variables ) X :ˆ G; L :ˆ KG; P :ˆ P is able to reduce the non-linear conditions obtained after replacing ()±() into the inequalities of Theorems 1± into LMI on the synthesis variables X, L and P. For instance, the following synthesis versions of Theorems 1 and are obtained using (). Theorem 5 (H state-feedback): There exists a controller in the form 1 such that the inequality kh wz ± k < holds if, and only if, the LMI W C z X D zu L trace W < ; > 0; T X X T P P AX B u L B w 5 T X X T P 0 7 T T I hold, where the matrices X and L and the symmetric matrices P and W are the variables. Theorem (H 1 state-feedback): There exists a controller in the form 1 such that the inequality kh wx ± k 1 < holds if, and only if, the LMI P AX B u L B w 0 T X X T P 0 X T Cz T L T D T zu T T I D T 7 zw T T T I holds, where the matrices X and L and the symmetric matrix P are the variables. The above inequalities are promptly obtained from () after substituting the closed loop matrices ()±() into Theorems 1 and. The non-linear transformatio n () maps the variables (K, G, P) into the synthesis variables (L, X, P), which provides necessity to Theorem 5 and. On the other hand, the fact that X X T > P > 0 implies that the matrix X is non-singular so that the original matrices (K, G, P) can always be recovered form a feasible set of synthesis variables (L, X, P) using K ˆ LX 1 ; G ˆ X; P ˆ P 7 Consequently, suciency follows immediately. The nice form of these synthesis LMI come from two facts: (a) in the analysis inequalities developed in } the

6 Extended H and H 1 norm characterizations 71 dynamic system matrices always appear in products with the instrumental variable G; (b) a single instrumental variable G is used to test all inequalities involved in the H and H 1 norm characterization. These facts do not trivially come from the stability conditions introduced in de Oliveira et al. (1999 a) and should not be overlooked. These properties are also responsible for the nice form of the synthesis conditions under dynamic output-feedbac k to be presented in the next section. Note that the synthesis variables (X, L) that are used to recover the controller gain K do not depend directly on the Lyapunov matrix P (P) nor on the dynamic system matrices. These facts will be further explored in the forthcoming sections. The standard state-feedback parametrization (Bernussou et al. 1989) is recovered by imposing on Theorems 5 and the additional constraint X ˆ X T ˆ P... Dynamic output-feedbac k problem Dynamic output-feedbac k control is addressed with respect to the full order linear dynamic controller x c k 1 ˆ A c x c k B c y k u k ˆ C c x c k D c y k 8 9 where it is assumed that the controller state x c n. The connection of this controller with the system (1)±(18) provides a linear system in the form (19)±(0) where ~x k :ˆ x k x c k 0 and the closed-loop matrices " A :ˆ A B # ud c C y B u C c ; B :ˆ Bw B u D c D yw B c C y A c B c D yw 1 C :ˆ C z D zu D c C y D zu C c Š; D :ˆ D zw D zu D c D yw Š As in the state-feedback case, a suitable non-linear transformation (change-of-variables ) is sought so as to reduce the non-linear conditions obtained after replacing (1)±() into the inequalities of Theorems 1± into LMI on some additional synthesis variables. Under dynamic output-feedbac k this transformatio n is somewhat more involved. The starting point is the work (Scherer et al. 1997, Masubuchi et al. 1998). First de ne the matrices K, G and G 1 partitioned into blocks as K :ˆ Ac B c C c D c ; G :ˆ X? U? ; G 1 :ˆ Y T? V T? where `? denote blocks in these matrices with no importance for the derivations to be presented in the sequel. Also introduce the transformation matrix T :ˆ I Y T 0 V T that has already been used in Scherer et al. (1997) and Masubuchi et al. (1998). The main contribution of this section is to show that the non-linear transformatio n (change-of-variables) Q F L R :ˆ V YB u 0 I P J K U 0 C y X I Y 0 A X 0Š 5 J T H :ˆ T T PT S :ˆ YX VU 7 has linearizing properties with respect to the dynamic output-feedbac k synthesis for the extended analysis conditions. The linearizing property of this transformatio n relies on the following identities T T AGT ˆ AX B ul A B u RC y Q YA FC y ; T T B ˆ Bw B u RD yw YB w FD yw CGT ˆ C z X D zu L C z D zu RC y Š; D ˆ D zw D zu RD yw 8 9 The main point here is that the matrix T, originally de ned in Scherer et al. (1997) and Masubuchi et al. (1998) with respect to the partitioned blocks of the symmetric Lyapunov matrix P, keeps its linearizing properties even if the symmetric matrix P is replaced by the non-symmetric instrumental matrix G. Note however that this modi cation also calls for the introduction of the non-linear transformatio n involving the extra variable S, de ned in (7), which is not required in Scherer et al. (1997) and Masubuchi et al. (1998). This extra transformation along with the additional variables () are required to take care of the linearization of terms in the form T T G G T P T ˆ X X T I S T I S Y Y T P J J T H 0 that have been introduced with the extended analysis conditions. LMI conditions for H and H 1 output-feedback synthesis can be derived by using the change-of-variables (5)±(7) along with the transformation matrix

7 7 M. C. de Oliveira et al. T, de ned in (). The following theorems can be obtained. Theorem 7 (H output-feedback): There exists a controller in the form 8 ± 9 such that the inequality kh wz ± k < holds if, and only if, the LMI trace W < 1 W C z X D zu L C z D zu RC y T X X T P I S T 7 J T T Y Y T H P J AX B u L A B u RC y B w B u RD yw T H Q YA FC y YB w FD yw T T X X T P I S T J 0 T T T Y Y T H T T T T I D zw D zu RD yw ˆ 0 > 0 hold, where the matrices X, L, Y, F, Q, R, S, J and the symmetric matrices P, H and W are the variables. Theorem 8 (H 1 output-feedback): There exists a controller in the form 8 ± 9 such that the inequality kh wz ± k 1 < holds if, and only if, the LMI P J AX B u L A B u RC y B w B u RD yw 0 T H Q YA FC y YB w FD yw 0 T T X X T P I S T J 0 X T C T z L T D T zu T T T Y Y T H 0 Cz T Cy T R T D T zu T T T T I D T zw D T ywr T D T 7 zu 5 T T T T T I > 0 5 hold, where the matrices X, L, Y, F, Q, R, S, J and the symmetric matrices P and H are the variables. The inequalities in these theorems have been obtained by substituting for the closed-loop matrices (1)±() and appropriately applying congruence transformations based on the transformation matrix T to the extended analysis conditions de ned in }. Then linearity is recovered by using the change-of-variable s (5)± (7) and the identities (8)±(0). For instance, inequality (5) comes from (8) multiplied on the right by matrix T :ˆ diag T ; T ; I; IŠ and on the left by T T. Note that T de nes a congruence transformatio n and that the inertia of the matrix inequality being transformed is not altered if T is a full rank matrix. Notice also that T is full rank whenever T is full rank, Or, in other words, T is full rank whenever matrix V is full rank. Following an argument similar to the one in Gahinet (199), when dealing with strict inequalities the matrix V can be assumed to be full rank without loss of generality since, if necessary, a small perturbation can be used to recover the full rank property. This congruence transformation on (8) produces T T PT T T AGT T T B 0 T T GA T T T T G G T P T 0 T T G T C T B T T 0 I D T 7 0 CGT D I from where inequality (5) follows by using the identities (8)±(9). Similar manipulations hold for the inequalities (1)±(). These tools provide the arguments required to show necessity of the results in Theorems 7 and 8. Suciency is recovered with a constructive argument. First note that a controller can be obtained from the synthesis variables by inverting the changeof-variables (5)±(7) as K ˆ V 1 V 1 YB u 0 I Q YAX L F R U 1 0 C y Xu 1 I 7 The condition for the inversion of these matrices is the non-singularity of matrices U and V. Considering again the inequality (5), note that any feasible solution must satisfy X X T I S T I S Y Y T > P J J T H > 0 8 which implies that X and Y are non-singular. Moreover, multiplying the left-hand side of (8) by T T ˆ X T IŠ on the left and by T on the right one obtains S YX X 1 X T S YX T < 0 9 which implies that (S YX) is also non-singular, hence, there exist matrices V and U both non-singular satisfying (7). Therefore, for any non-singular matrices U and V such that UV ˆ S YX, the formula (7) provides feasible controller matrices from a set of feasible synthesis variables. It is important to stress that although the recipe to obtain the inequalities given in Theorems 7 and 8 follows from Scherer et al. (1997) and Masubuchi et al. (1998), the constructive issues that have been addressed above do not appear in these papers, where the use of a symmetric Lyapunov matrix to parametrize the controller simplify the analysis. Indeed, the parametrization provided in Scherer et al. (1997) and Masubuchi et al. (1998) can be recovered as a particular case by imposing the additional linear constraints

8 Extended H and H 1 norm characterizations 7 X ˆ X T ˆ P; S ˆ J ˆ I; Y ˆ Y T ˆ H 50 which come from G ˆ G T ˆ P. Note that with (50) the non-singularity of matrices U and V is immediately guarantee d since, in this case, UV YX ˆ I and the matrices X and Y are symmetric and positive de nite. As in the state-feedback case, the most important feature of this new parametrization is that it does not depend on any of the Lyapunov matrices P, J or H.. Impact of the extended controller parametrization on control synthesis Several control design problems can be expressed in terms of LMI. The reader is referred to Colaneri et al. (1997) for more details. Many design problems with practical importance require the controller to have structural constraints, such as decentralization. Yet, since LMI controller design problems are usually solved in terms of auxiliary synthesis variables, which are related to the controller parameters via a non-linear transformation (change-of-variables), structural controller constraints remain dicult to handle. Typically, constraints are imposed on the auxiliary synthesis variables so as to achieve the desired controller structure. In the context of the standard Lyapunov based controller parametrization (Bernussou et al. 1989, Scherer et al. 1997, Masubuchi et al. 1998), where the controller is a function of the Lyapunov matrix, these constraints usually leads to unnecessary (conservative) constraints on the Lyapunov matrix. The fact that the extended controller parametrization developed in } does not directly depend on the Lyapunov matrix let one impose structural constraints on the control synthesis variables which are less conservative. The implications of this fact are explored in the sequel, where some important control problems that can bene t from the extended analysis conditions and the extended controller parametrization are discussed..1. Decentralized control Decentralization is a very common structural controller constraint and is typically required (for practical reasons) on the design of large scale system (Siljak 1978). On the context of state-feedback of large interconnected systems, decentralization can be seen as the design of decentralized (decoupled) control laws u i k ˆ K i x i k ; i ˆ 1;... ; N; where each input is allowed to feedback only local state information. In other words, state-feedback decentralization can be equivalently expressed as the computation of a state-feedback control law (1) where the feedback gain matrix is blockdiagonal, that is, K ˆ diag K 1 ;... ; K N Š. In the light of the state-feedback controller parametrization (), a block-diagonal gain K is obtained whenever LX 1 is block-diagonal. This imposes a non-linear constraint on the synthesis variables X and L that is very hard to handle. Instead, more tractable constraints seems to be L ˆ diag L 1 ;... ; L N Š; X ˆ diag X 1 ;... ; X N Š; i ˆ 1;... ; N 51 The above constraints are linear constraint on the matrices L and X (the o-diagonal components should be zero) and can be easily incorporated in Theorems 5 and in order to provide H and H 1 decentralized state-feedback controllers. Note that in the presence of the additional constraint (51), the value of in Theorems 5 and should be seen as upper bounds (guaranteed costs) for the H and H 1 norms of the closed-loop transfer function. If a feasible solution to the synthesis inequalities with (51) is obtained, then the decentralized state-feedbac k gains are obtained as K i ˆ L i X 1 i ; i ˆ 1;... ; N 5 The above should be compared with Geromel et al. (199) that uses the same idea but is based on the standard controller parametrization (Bernussou et al. 1989). In Geromel et al. (199), where X ˆ X T ˆ P, the constraint (51) imposes the much stronger constraint that the Lyapunov matrix should also be block-diagonal. In contrast with Geromel et al. (199), the extended controller parametrizatio n lets the matrix P remain unconstrained. The above approach can be extended to cope with dynamic output-feedbac k control in the context of de Oliveira et al. (000) and Geromel et al. (1999 a). In de Oliveira et al. (000) dynamic output-feedbac k controllers are designed on the basis of an ad-hoc separation principle. An iterative procedure is given in Geromel et al. (1999 a). A complete description of these procedures is omitted for brevity. The improvements obtained with the extended parametrization in both state and dynamic output-feedbac k synthesis are illustrated in the following example. Example 1: Consider the time-invariant discrete-time linear system de ned by matrices 0:8189 0:08 0:0900 0:081 0:5 1:00 0:01 0:00 A ˆ ; 0:055 0:010 0:7901 0: :1918 0:10 0:10 0:80 0:005 0:00 0:1001 0:0100 B u ˆ ; 0:000 0: :0051 0:09

9 7 M. C. de Oliveira et al. B w ˆ 0: : ; 0: : C z ˆ ; D zu ˆ This example is borrowed from de Oliveira et al. (000), where a continuous-time decentralized output-feedbac k controller is obtained via an ad-hoc separation procedure. The above discrete-time system matrices have been obtained assuming a zero-order sample-and-hold at the inputs with a sampling period equal to 0.1. The rst objective is to design a decentralized statefeedback controller. For state-feedbac k design, the matrices C y and D yw are set to C y ˆ I and D yw ˆ 0. The desired decentralized state-feedbac k controller should feedback the rst two states through the rst input and the last two states through the second input. The performance objective is to minimize a guaranteed H cost of the transfer function from the input w to the output z. The procedure (Geromel et al. 199) based on the standard Lyapunov controller parametrization provides K 1 ˆ 1:891 K ˆ 0:051 kh wz ± k < ˆ 0:1 1:00Š; 0:00Š; In order to obtain this controller the Lyapunov matrix P had to be constrained to be block-diagonal. On the other hand, using Theorem 5 with (51) only requires the auxiliary matrix X, but not P, to be block-diagonal. This result yields the controller K 1 ˆ 1: K ˆ 0:111 kh wz ± k < ˆ 0: 1:918Š; 0:017Š; This controller represents an improvement in the guaranteed H cost of 0% with respect to the one obtained from Geromel et al. (199). The second design is a decentralized and strictly proper dynamic output-feedbac k controller that uses the measurement matrices C y ˆ ; D yw ˆ The rst controller should feedback the rst measured output to the rst control input and the second controller should feedback the second output to the second control input. Using as a starting point the decentralized state-feedback controller obtained from Geromel et al. (199), a decentralized dynamic output-feedbac k has been designed using the separation procedure proposed in de Oliveira et al. (000). This controller is given by K 1 ± ˆ K ± ˆ :0 ± 0:7 ± 1:18± 0:0 ; 0:08 ± 0:70 ± 0:0 ± 0:1 ; kh wz ± k < ˆ 5:5 The same separation procedure appropriately modi ed to cope with Theorem 7 provides the extended controller K 1 ± ˆ K ± ˆ 1:9 ± 0:7 ± 1:7± 0:51 ; 0:1 ± 0:9 ± 0:8± 0:1 ; kh wz ± k < ˆ 18:7 The extended controller parametrization brings an improvement on the guaranteed H cost of 59%... Static output-feedbac k A structural approach to the synthesis of static output-feedback problem has been developed in Geromel et al. (199). In order to solve static output-feedbac k problems involving stability and H and H 1 norms it is assumed that the matrices C y and D yw in the measurement equation (18) are C y ˆ I 0Š and D yw ˆ 0. That is, the output vector y k is available for feedback and the input w k does not corrupt its measurement. Note that the particular structure of the output matrix C y can be enforced without loss of generality by a simple yet not unique similarity transformation. From the extended parametrization of state-feedback controllers (), a static output-feedbac k gain can be obtained by imposing on matrices L and X the structure L :ˆ L out 0Š; X :ˆ Xout 0 X 1 X 5 This structure can be incorporated in Theorems 5 and by imposing additional linear constraints on L and X. From (7) a static state-feedback gain is obtained as K ˆ K out 0Š ˆ L out X 1 out 0Š 5 Hence the state-feedback gain matrix exhibits the desired output-feedbac k structure so as to make u k ˆ K out y k a static output-feedback control. The above result should be compared with Geromel et al. (199). In this work, the static output-feedbac k structure (5) has been obtained imposing a block diagonal constraint on both L and the Lyapunov matrix

10 Extended H and H 1 norm characterizations 75 X ˆ X T ˆ P. The extended parametrization clearly relax that constraint in two directions. First, the Lyapunov matrix P remains unconstrained. Second, exploring the fact that the auxiliary matrix X is not symmetric, the block-diagona l constraint on matrix X can be relaxed to the one shown in (5), where the blocks X 1 and X are free. Example : In this example a static output-feedbac k controller is designed for the same system given in Example 1 for the noise-free output measurements C y ˆ ; D yw ˆ 0 The procedure (Geromel et al. 199) provides the controller K out ˆ 1:879 0:5597 0:89 0:18 ; kh wz ± k < ˆ 0:5 which minimizes a guaranteed H cost from the input w to the output z. Theorem 5 with (5) produces K out ˆ 1:15 0:110 0:70 0:1757 ; kh wz ± k < ˆ 0:18 The extended controller parametrization provided the signi cant improvement of 8% on the guaranteed H costs... Robust control The robust stability results of de Oliveira et al. (1999 a) have been generalized to cope with the extended conditions for H and H 1 norm robust analysis given in } (see Theorems and ). With respect to state-feedback controller synthesis, Theorems 5 and can be also generalized to cope with robust H and H 1 state-feedback synthesis. By analogy with (9), a matrix M should be de ned as M :ˆ A B w B u C z D zw D zu 55 This de nition and Theorems and provide the following modi ed version of Theorems 5 and, which are presented without proof for brevity. Theorem 9 (Robust H state-feedback): There exists a controller in the form 1 such that the inequality kh wz ± k < for all M F if the LMI W i C z i X D zu i L trace W i < ; > 0; T X X T P i P A i X B u i L B w i 5 T X X T P i 0 7 T T I hold for all i ˆ 1;... ; N; where the matrices A i, B w i, B u i, C z i and D zu i de ne the extreme matrices M i ; i ˆ 1;... ; N; and the matrices X and L and the symmetric matrices P i ; i ˆ 1;... ; N are the variables. Theorem 10 (Robust H 1 state-feedback): There exists a controller in the form 1 such that the inequality kh wz ± k 1 < for all M F if the LMI P i A i X B u i L B w i 0 T X X T P i 0 X T C z T i LT D zu T i > 0 T T I D zw T 7 i 5 T T T I i ˆ 1;... ; N 57 hold, where the matrices A i, B w i, B u i, C z i, D zw i and D zu i de ne the extreme matrices M i, i ˆ 1;... ; N; and the matrices X and L and the symmetric matrices P i ; i ˆ 1;... ; N are the variables. Notice that the above synthesis results rely on the fact that the state-feedback controller parametrization () does not depend on system matrices A i, B w, B u, C z and D zu which, in the robust synthesis case, are not precisely known. For this reason, Theorems 9 and 10 do not have an exact analogue for the extended outputfeedback controller parametrization (5). However, as in the decentralized control problem, extended robust H and H 1 dynamic output-feedback can be solved in the context of de Oliveira et al. (000) and Geromel et al. (1999 a). Example : The robust control problem to be solved in this example is a reliable control problem (Veillette et al. 199). First, a (centralized) robust state-feedback controller is sought so as to stabilize the same system given in Example 1 in three dierent scenarios: (a) the nominal plant, (b) outage of the rst actuator, (c) outage of the second actuator. Furthermore, it is desired to minimize a guaranteed H cost of the transfer function from the input w to the output z. Three sets of extreme matrices corresponding to these scenarios have been generated and the standard quadratic robust controller (Geromel et al. 1991) has been obtained K ˆ 0:57 0:700 0:589 0:190 0:590 0:980 0:07 0:1 ; kh wz ± k < ˆ 0:7 Theorem 9 provides the extended robust controller K ˆ 0:51 0:78 0:5 0:551 0:9 0:990 0:510 0:170 ; kh wz ± k < ˆ 0:0

11 7 M. C. de Oliveira et al. The improvement provided by the extended robust synthesis procedure has been of 10%. Then, using the same data as in Example 1, a (centralized) robust and strictly proper dynamic output-feedbac k controller has been designed using the ad-hoc separation procedure (de Oliveira et al. 000). In the dynamic output-feedbac k design ve scenarios have been considered: (a) the nominal plant, (b) outage of the rst actuator, (c) outage of the second actuator, (d) outage of the rst sensor, (e) outage of the second sensor. In this case, the procedure (de Oliveira et al. 000) initialized with the robust state-feedback controller based in Geromel et al. (1991) produces the robust controller A c C c B c D c ˆ 0:77 0:08 0:0 0:011 0:019 0:0 0:00 0:801 0:111 0:11 0:09 0:071 0:178 0:15 0:778 0:58 0:0 0:08 0:88 0:105 0:1 0:850 0:009 0:0 7 1:9 1:757 0:70 1: :15 :0 0:90 0: kh wz ± k < ˆ 8:0 The less conservative extended controller synthesis, using as a starting point the robust state-feedback controller obtained by Theorem 9, produces the robust controller A c C c B c D c ˆ 0:5 0:09 0:058 0:00 0:0 0:0 0:00 0:81 0:070 0:171 0:101 0:071 0: 0:091 0:85 0:8 0:100 0:07 0:08 0:079 0:099 0:87 0:085 0: :7 1:1 0:588 0: :09 1:57 1:088 0:1 0 0 kh wz ± k < ˆ :8 The extended synthesis provide the signi cant improvement in performance of 5%... Multi-objective control In this section, the fact that the extended controller parametrizations do not depend on the Lyapunov matrices used in the H and H 1 constraints is explored in the design of controllers with multiple objectives, as the ones in Scherer et al. (1997) and Khargonekar and Rotea (1991). The multi-objective problem to be dealt with in this section is de ned as the problem of determining a controller for the plant (1)±(18) such that several closed-loop H and H 1 speci cations are met. It is assumed that these speci cations are imposed on closed-loop transfer functions of the form H wi z i ± :ˆ L i H wz ± R i where the matrices L i and R i are responsible for selecting the desired input/output channels. From the dynamic matrices of system (1)±(18), a state-space realization of the closed loop system H wz ± i is obtained by appropriately replacing the matrices B w, C z, D yw, D zu and D zw by B w i :ˆ B w R i C z i :ˆ L i C z 58 D yw i :ˆ D zu R i D zw i :ˆ L i D zw R i D zu i :ˆ L i D zu 59 in the closed-loop matrices ()±() or (1)±(). In this form, closed-loop performance and robustness may be ensured by constraining the H and H 1 norms of the transfer functions associated to the pairs of signals w i :ˆ R i w z i :ˆ L i z 0 For instance, suppose that a controller that meets N closed-loop speci cations (constraints) is sought. Applying Theorems 1 and to the N closed-loop transfer functions, one has to deal with two kinds of variables:. The controller parameters.. Instrumental matrices G i, P i, i ˆ 1;... N. Using the results of } it is possible to linearize each one of the N constraints by choosing an appropriate controller parametrization. However, once the actual controller matrices are obtained as a function of the instrumental matrices G i, and in order to keep the problem tractable, it is necessary to impose the additional constraint G i ˆ G; i ˆ 1;... ; N 1 These constraints ensure that the controller considered in each speci cation (inequality) will be the same. By doing this the multi-objective synthesis problem is kept as an LMI optimization problem, which may be eciently solved. This approach should be compared with the Lyapunov shaping paradigm (Scherer et al. 1997). In this work, the more conservative constraints P i ˆ P ˆ G ˆ G T ; i ˆ 1;... ; N are imposed. By analogy, the extended approach proposed here is denoted G shaping paradigm. An interesting feature of

12 Extended H and H 1 norm characterizations 77 the G shaping paradigm is that it contains the Lyapunov shaping paradigm. That is, if a controller satisfying the control speci cations with the constraints () is available, so is an extended controller satisfying the less restrictive constraints (1). Hence, all advantage s associated with multi-objective control based on the Lyapunov shaping paradigm (see Scherer et al for details) are automatically inherited by the G shaping. Notice that although some conservativeness is still present due to the constraint (1), some degrees of freedom that are vital to the multi-objective design are kept in the design problem. For instance, an independent Lyapunov matrix P i is available at each constraint. Hence, all degrees of freedom provided by these extra Lyapunov matrices may be explored in the search for a controller that jointly satis es the performance speci cations. These features will be illustrated in the next example. Example : Consider the uncertain system x k 1 ˆ 1 1= :5 y k ˆ 0 1 0Šx k 7 5 x k u k where i; i ˆ 1; ; are the uncertain parameters. By setting these uncertain parameters to zero the nominal matrices A, B u and C y are found. The goal is to design a strictly proper output-feedbac k controller which is robust to such uncertainties and meets a closed-loop performance index, to be de ned latter. Robustness is addressed by bounding the H 1 closed-loop norms of the transfer functions of some appropriately de ned extra inputs and outputs that `pull out the uncertain parameters. More speci cally, the following inputs and outputs are added to the above system and outputs 1 x k 1 ˆ Ax k B u u k w 1 k w k w x k 0 y k ˆ C y x k w k w y k 0 z 1 k ˆ 1 0 0Šx k z k ˆ u k z k ˆ 0 1 0Šx k z ˆ x k u k so that the robustness requirements can be translated as to keep kh wi z i ± k 1 < ; i ˆ 1; ; where de nes the robustness level. For a given, performance is obtained by minimizing the guaranteed H cost such that kh w z ± k <, where H w z ± is the transfer function from the input w k :ˆ wx k w y k to the output z k. By gathering the appropriate LMI for each H and H 1 constraint this control problem can be cast as a multichannel and multi-objective control synthesis problem that can be eciently solved by the G shaping paradigm. The same problem is solved by the Lyapunov shaping paradigm (Scherer et al. 1997) for comparison. Starting from some arbitrarily large value of (robustness level) the value of (performanc e level) is minimized. By progressively reducing the value of the trade-o curves given in gure 1 are obtained. In this gure the optimal values obtained with the G shaping are given by the solid lines while the dotted lines represent the performance obtained with the Lyapunov shaping. The minimum values of the upperbound to kh wi z i ± k 1 ; i ˆ 1; ; ; achieved by the G shaping and the Lyapunov shaping are denoted by G and P, respectively. The scalars G and P denote the actual values of associated to these upperbounds. These gures reveal an interesting pattern: the G shaping provided less conservative upperbounds as well as actual performances that are much better than the one obtained with the Lyapunov shaping. Although the global minimum value of is not available, the optimal (non-robust) H controller can be easily calculated and provides :ˆ :8 and :ˆ 1:7. Hence, for a robustness level > it is expected that the H performance of an (optimal) multi-objective controller should not exceed. A similar behaviour has been obtained by the (suboptimal) controller obtained by the G shaping paradigm. It is remarkable that the G shaping paradigm has been able to generate a controller that matches the optimal value of the H at a nite (and reasonably small) value of. Due to the use of a single Lyapunov matrix in all inequalities, the Lyapunov shaping paradigm can only generate controller with such performance asymptotically for in nite values of.

13 78 M. C. de Oliveira et al. Figure 1. Actual H 1 and H performances. 5. Conclusion A recently introduced extended stability condition for discrete-time time-invariant linear systems have been generalized to cope with H and H 1 norm analysis. The extension process consists in adding an additional instrumental variable other than the usual symmetric matrix obtained through standard Lyapunov methods. These new conditions provide less conservative guaranteed H and H 1 costs for uncertain systems making use of ane parameter depended Lyapunov functions. All analysis computations can be performed by solving LMI optimization problems. The extended analysis conditions introduced in this paper have been generated so as to make possible the development of adequate controller parametrizations for control synthesis. It has been shown that several state-feedback and full order dynamic output-feedbac k control problem involving the extended conditions can be solved as LMI optimization problems by employing a linearizing change-of-variables. These new parametrizations have the interesting property that the controller matrices are not functions of the Lyapunov matrix. This fact has been used to provide less conservative solutions to several H and H 1 control problems in the literature. Namely, decentralized state-feedback and dynamic-outpu t feedback control, structural static-output feedback control, state-feedback and dynamic-outpu t feedback robust control with parameter Lyapunov functions, and statefeedback and dynamic-outpu t feedback multi-objective control involving H and H 1 performance speci cations. In all these problems the controllers have been solved from an LMI optimization problem. It has been also shown that the new procedures always contain the equivalent results available in the literature to date. Several numerical examples have been solved in order to illustrate the improvement in performance obtained with the new techniques. Acknowledgments This work has been supported in part by grants from `FundacË aä o de Amparo aá Pesquisa do Estado de SaÄ o Paulo ± FAPESP and `Conselho Nacional de Desenvolvimento Cientõ co e Tecnolo gico ± CNPq ± Brazil. References Barmish, B. R., 1985, Necessary and sucient conditions for quadratic stabilizability of an uncertain system. Journal of Optimization Theory and Application,, 99±08. Bernussou, J., Geromel, J. C., and Peres, P. L. D., 1989, A linear programming oriented procedure for quadratic stabilization of uncertain systems. Systems and Control Letters, 1, 5±7. Colaneri, P., Geromel, J. C., and Locatelli, A., 1997, Control Theory and Design: An RH ±RH 1 Viewpoint (San Diego, CA: Academic Press). de Oliveira, M. C., Bernussou, J., and Geromel, J. C., 1999 a, A new discrete-time robust stability condition. Systems and Control Letters, 7, 1±5. de Oliveira, M. C., Geromel, J. C., and Bernussou, J., 1999 b, An LMI optimization approach to multiobjective controller design for discrete-time systems. Proceedings of the 8th IEEE Conference on Decision and Control, Phoenix, AZ, USA, pp. 11±1. de Oliveira, M. C., Geromel, J. C., and Bernussou, J., 000, Design of decentralized dynamic output feedback controllers via a separation procedure. International Journal of Control, 7, 71±81. Feron, E., Apkarian, P., and Gahinet, P., 199, Analysis and synthesis of robust control systems via parameterdependent Lyapunov functions. IEEE Transactions on Automatic Control, 1, 101±10. Gahinet, P., 199, Explicit controller formulas for LMI-based H 1 synthesis. Automatica,, 1007±101.