European Journal of Control ():9 # EUCA Delay-Dependent H Control of Uncertain Discrete Delay Systems E. Fridman and U. Shaked Department of Electrical Engineering-Systems Tel-Aviv University Tel-Aviv 998 Israel A delay-dependent solution is given for state-feedback H control of linear discrete-time systems with unknown constant or time-varying delays and with polytopic or norm-bounded uncertainties. Sufficient conditions are obtained for stability and for achieving design specifications which are based on Lyapunov Krasovskii functionals via a descriptor representation of the system. Similarly to the corresponding continuous-time results these conditions provide an efficient tool for analysis and synthesis of linear systems with time delay. The advantage of the new approach is demonstrated via a simple example. Keywords:. Introduction In this paper a H control problem for discrete-time systems with unknown constant or time-varying delays is considered. The control problems for continuous-time delay systems have been extensively investigated in the last decade (see e.g. [9] and the references therein). Delay-independent and less conservative delay-dependent sufficient conditions for H control in terms of Riccati or linear matrix inequalities (LMIs) have been derived by using Lyapunov Krasovskii functionals. Delay-dependent conditions are based on different model transformations. The most recent one a descriptor This work was supported by the C&M Maus Chair at Tel Aviv University. E-mail: emilia@eng.tau.ac.il E-mail: shaked@eng.tau.ac.il representation of the system [] leads to less conservative sufficient conditions. Moreover it allows to treat by Lyapunov Krasovskii approach the important case of time-varying delay no bounds on the derivative of the delay are given []. Note that previously the stability conditions for such systems were derived only via Lyapunov Razumikhin functions. The Razumikhin approach leads to more conservative conditions and it seems to be inapplicable to the case of H control. Less attention has been given to the corresponding results for discrete-time delay systems [8]. Such systems can be transformed into augmented systems without delay but for large delays this augmentation suffers from the curse of dimension. Moreover the augmentation of the system is inappropriate for systems with unknown delays or systems with delays that are time-varying (such systems appear e.g. in the field of communication networks). Delay-dependent conditions for stability and H control have been obtained by Song et al. [] for the case of time-varying delays. The LMI conditions obtained there are conservative since in the case when the upper-bound on the delay is h ¼ these conditions coincide with the well-known delay-independent conditions (see e.g. []). Delay-dependent stability conditions for the case of constant delays have been obtained by Lee and Kwon [8] via a model transformation similar to the one in []. Recently a descriptor approach has been applied to stability analysis of discrete delay systems with normbounded uncertainties []. The advantages of this Received Recommended by
E. Fridman and U. Shaked approach in the continuous-time case are also evident for discrete delay systems. In the present paper we further develop this approach to the H control problem for systems with either polytopic type or norm-bounded uncertainties. Note that the stability conditions of [8] are not affine in the system s matrices and they are thus inapplicable to the case of polytopic type uncertainty. For the cases of constant and time-varying delays we derive bounded real lemmas (BRLs) in terms of LMIs which are affine in the matrices of the systems. Unlike the continuoustime case the derivation of the BRLs in the present paper encounters additional technical difficulties (in comparison with the stability conditions). This is due to the fact that the cross terms that have to be overbounded contain the disturbances. A state-feedback controller that stabilizes the system and satisfies prespecified requirements is found by applying P K iterations. A simple example illustrates the efficiency of the new method... Notation Throughout the paper the superscript T stands for matrix transposition R n denotes the n dimensional Euclidean space with vector norm jj R nm is the set of all n m real matrices and the notation P > for P R nn means that P is symmetric and positive definite. By l we denote the space of sequences fx k g k ¼... with the norm kx k k P ¼ k¼ xt k x k <.. Problem Statement We consider the following discrete-time state-delayed system x kþ ¼ Ax k þ A x k hk þ B w k þ B u k x k ¼ k h k ðabþ z k ¼ Lx k þ Du k x k R n is the state vector w k R q is the disturbance input which is assumed to be of bounded energy u k R p is the control input z k R m is the objective vector h k is a positive number representing the delay h k h and A A B B L and D are constant matrices of appropriate dimensions. For simplicity only we consider the case of a single delay. The results may be easily generalized to the case of multiple delays. It is assumed in the analysis part that A. The eigenvalues of A þ A are all of absolute value less than. We address first the following two analysis problems. Problem. For h k ¼ h that is an unknown number satisfying h h ðþ for fu k g and for a given scalar find whether the system is asymptotically stable and the following holds: J ¼kz k k kw k k < 8 ¼ fw kg for k ¼ h k : ðþ Problem. For all time-varying h k that satisfy () find whether the system () with fu k g is asymptotically stable and () is satisfied for a given scalar. Once solutions are obtained to the above problems the problem of finding a state-feedback control law which stabilizes the system and achieves () for a prescribed will be considered.. Delay-Dependent and Delay-Independent BRLs.. Descriptor Model Transformation Assume A. We consider in this section the case B ¼ D ¼. Denoting y k ¼ x kþ x k ðþ the system () can be represented by the following descriptor form: x kþ y k þ x k ¼ : y k þ Ax k x k þ A x k hk þ B w k Since x k hk ¼ x k P k k y j it follows that Ex kþ ¼ Ax k X k y j þ w k ðþ A k B x ¼ y ¼ðAÞ A h y k ¼ kþ k k ¼ h... and I n A ¼ A þ A I n I n n E ¼ diagfi n g x k ¼ x k y k ðþ : ðþ
Delay-Dependent H Control Thus if for a specific w k x k is a solution of () then fx k y k g y k is defined by () is a solution of () () and vice versa. Denoting P ¼ P P P T and E ¼ diagfi n g ð8abþ P we consider the following Lyapunov Krasovskii functional: V k ¼ V ;k þ V ;k þ V ;k V ;k ¼ x T k P x k ¼ x T k EPEx k < P V ;k ¼ X V ;k ¼ Xk X k m¼ h j¼kþm ð9aþ y T j ½R þ QŠy j < R < Q x T k Sx k < S: ð9b dþ Note that V ;k corresponds to necessary and sufficient conditions for stability of discrete descriptor systems without delay [] V ;k is typical for delay-dependent criteria while V ;k corresponds to delay-independent stability conditions []... The Case of Constant Delay (Problem ) Theorem. Consider the system () with the constant time delay that satisfies () with B ¼ D ¼. If there exist P Z R nn < S R Q R nn and Y R nn X R qn W R qq that satisfy the following LMIs: Z Y W X < Y T R X T Q and ½I ŠP I > ða dþ then () is asymptotically stable and for a prescribed scalar () is satisfied. Proof. We apply the Lyapunov Krasovskii method and require that V kþ V k is strictly negative to guarantee the asymptotic stability of the system and that V kþ V k þ z T k z k! T k! k is strictly negative in order to satisfy (). We obtain that V ;kþ V ;k ¼x T kþ EPEx kþ x T k EPEx k (! ) ¼ x T k A T Xk y T j A T þ! T k B T ( P Ax k X k j þ A y )! k x T k B EPEx k ¼x T k ½ A T PA EPEŠx k þ k þ k þ k þ k ðaþ A is defined in () and k ¼ðx T k xt k h Þ AT P ðx A k x k h Þ k ¼ Xk x T k A T P y A j ðbcþ k ¼! T k B T PAxk þ! T k B T P k ¼ Xk! T k B T P y j A V ;kþ V ;k ¼ hy T k ½R þ QŠy k Xk ¼ x T k h½r x þ QŠ k Xk y T j ½R þ QŠy j V ;kþ V ;k ¼ x T k Sx k x T k h Sx k h ¼ x T S k x k x T k h Sx k h B! k ðdeþ y T j ½R þ QŠy j ðþ ðþ A T PA EPE þ S h½r þ hz þ YI ½ Šþ I Y T A T P Y A T P þ I X T L T þ QŠ A B S þ A T P A T P X T ¼ A B I þ hw þ B T P B I I A¼ ðfþ (e) A I
E. Fridman and U. Shaked and thus Hence V kþ V k ¼ x T k x k x T k h Sx k h Xk y T j ½R þ QŠy j þ k þ k ¼ A T PA EPE þ S h½r : þ QŠ ðþ By [] for any a R n b R n N R nn R R nn Y R nn Z R nn the following holds b T N a b T Z Y N b a Y T N T R a Z Y : ðþ R Y T Applying the latter on the expression we have obtained above for k taking: N¼A h i T P A a ¼ y j and b ¼ x k we have the following: k Xk x T k y T Z Y A T P j A x k y j R Z Y : R Similarly k Xk! T W X ½ BT k yt j ŠP A Q W X : Q! k y j k Xk k Xk y T j Ry j þ hx T k Zx k þ x T k Y A T P A x T k Y A T P A y T j Qy j þ h! T k W! k x k þ! T k X ½ BT ŠP A x k h ðx k x k h Þ: ðþ The asymptotic stability of the system follows from () by []. The performance requirement of () is satisfied if defining k ¼ colfx k x k h! k x k g we require that V kþ V k þ x T k LT Lx k! T k! k T k k < since summation in the latter inequality from k ¼ till k ¼implies (). & Remark. If the LMIs of Theorem are feasible then () is asymptotically stable for h ¼ and thus A holds. The result of Theorem depends on the delay bound h. The corresponding criterion for asymptotic stability which is delay-independent can be readily derived as a special case of Theorem. Choosing Z ¼ W ¼ R ¼ Q ¼ I n Y ¼ X ¼ is a positive scalar and letting tend to zero we obtain the following. Corollary. The system () is asymptotically stable and satisfies () independently of the delay if there exist P R nn and < S R nn that satisfy (d) and the following LMI A T PA EPE þ I SI ½ Š A T P A T P L T A B S þ A ind ¼ T P A T P A B < : I þ B T P B ðþ
Delay-Dependent H Control It follows from (b) that if R and Q are taken to be positive-definite then Z YR Y T W XQ X T and thus there exists a solution to (a d) with R > Q > iff there exists a solution to (ad) Z is replaced by YR Y T. A sufficient BRL condition is thus the following. Lemma. Consider the system () with the constant delay that satisfies () with B ¼ D ¼. This system is asymptotically stable and for a prescribed scalar () is satisfied if there exist P R nn of the structure (8) < S and R R nn and Y R nn that satisfy the following LMIs. ^ < and P > ð8abþ ^ ¼ A T A T B T P A A B T I I L Y þ diagfs P h½r þ QŠ S Igþ þ YT X T Y h h X X hr hq ð8cþ Defining 8 9 I >< A I A I B >= ^J ¼ diag I I mi n I n ð9þ >: >; I it is readily obtained that þ h ^J T^ ^J ¼ A T I A T I B T ½R þ QŠ A T I A T I A T A T T I ¼ A T I I P I A T I I þ diagfs P S Ig B T B T T I I þ ~A T Y þ þ ½YT Aþ½X ~ T ŠŠ X " # I ~A ¼ : ðbcþ A I A I B B T The latter can be used together with Theorem to derive a sufficient condition for stability and prescribed H -norm bound in the case of polytopic uncertainty. Denoting T ¼ ½A A B L Š assuming that Cof j j ¼... Ng namely ¼ XN L T j¼ f j j h A ~ T Y h X hr hq for some f j XN j¼ f j ¼ ðþ the vertices of the polytope are described by j ¼½A ðjþ A ð jþ B ð jþ L ð jþ Š j ¼... N and denoting ðaþ P ð jþ ¼ P " # P Y ð jþ ¼ P T P ð jþ Yð jþ Y h i AA ð jþ ¼ A ð jþ A ð jþ I B ð jþ we obtain the following result. and ða cþ Theorem. Consider the system () with the constant time delay that satisfies () with B ¼ D ¼ and with system matrices that reside in. This system is asymptotically stable and for a prescribed scalar ()
E. Fridman and U. Shaked is satisfied over the entire polytope if there exist P ð jþ R nn of the structure (a) < S ð jþ R Q R nn X ð jþ R qn and Y ð jþ R nn of the structure (b) j ¼... N that satisfy the following set of N LMIs. ~ ðjþ A ðjþt P A ~ ðjþt " # h½r þ QŠ h A ~ ðjþt Y ðjþ h L ðjþt I X ðjþ ¼ P h½r þ QŠ hr hq < j ¼...N ðaþ complements to the second row and column of () with S ¼ one get in the ( ) block of the resulting LMI the element A T diagfp P P PT ga EPE which is never negative-definite. Thus our approach does not give delay-independent solution in the timevarying delay case. Delay-independent stability conditions in the case of time-varying delay were obtained recently in [] via Razumikhin approach. Similarly in the case of polytopic type uncertainty we obtain Theorem. Consider the system () with time-varying delay that satisfies () with B ¼ D ¼ and with ~ ð jþ ¼ A ð jþt I P ½ I I Šþ P T AA ð jþ I þ I I P ð jþ ½ I I ŠþdiagfS ð jþ P S ð jþ Ig I þ A ~ ðjþt Y ð jþ h þ ½I Šþ Y ð jþt A ~ ð jþ i þ X ð jþt ðbþ X ð jþ and ~A ð jþ I ¼ A ð jþ I A ðjþ jþ I Bð : ðcþ.. The Case of Time-Varying Delay (Problem ) Choosing for this case V ;k the required criterion is obtained by substituting S ¼ in (). From Theorem we thus obtain the following: Theorem. Consider the system () with time-varying delay that satisfies () with B ¼ D ¼. This system is asymptotically stable and for a prescribed scalar () is satisfied if there exist P Z R nn W R qq R Q R nn X R qn and Y R nn that satisfy () S ¼. Remark. Necessary conditions for feasibility of () with S ¼ are the following: P < and A is nonsingular. Note that the LMI of Corollary (which corresponds to delay-independent conditions) is never feasible if S ¼. The reason is that applying Schur system matrices that reside in. This system is asymptotically stable and for a prescribed scalar it satisfies () over the entire uncertainty polytope if there exist P ð jþ R nn of the structure (a) Y ð jþ R nn of the structure (b) j ¼... N X ðjþ R qn and R Q R nn that satisfy the LMIs of () S ð jþ ¼. Example (Stability analysis). We consider the system () A ¼ :8 A ¼ : :9 : : B ¼ B ¼ : and ðþ Assuming that h is constant we seek the maximum value of h for which the asymptotic stability of the system is guaranteed. We compare three methods: The criterion of [] Theorem in [8] and Theorem above. It is found that the method of [] does not provide a solution even for h ¼. The maximum
Delay-Dependent H Control value of h achievable by the method of [8] is as a value of h ¼ was obtained by applying Theorem of the present paper. Using augmentation it is found that the system considered is asymptotically stable for all h 8. Allowing h to be time-varying we apply Theorem. We obtain that asymptotic stability is guaranteed for all h 8.. Stabilization and H Control via State-Feedback.. The Case of Polytopic Uncertainty Considering next the case in () B and D are not zero and the uncertainty polytope is given by ¼ ½A A B B L DŠ assuming that Cof j j ¼... ^Ng namely ¼ X^N j¼ f j j for some f j X^N j¼ f j ¼ ðþ the vertices of the polytope are described by h i j ¼ A ðjþ A ðjþ B ðjþ B ðjþ L ðjþ D ðjþ j ¼... ^N: We seek a control law u k ¼ Kx k ðþ which stabilizes the system and achieves a prescribed bound on the H -norm of the closed-loop over. Replacing A ð jþ and L ð jþ in Theorem with A ð jþ þ B ð jþ K and L ð jþ þ D ð jþ K respectively we obtain matrix inequalities which are not affine in the matrices P and K. Similar to the state-feedback design of systems without delay the adjoint of the system () may be considered and the stabilizability and the H -norm requirements can be achieved for the adjoint system restricting the resulting P ½IŠY and R to be proportional to P. This procedure may be quite conservative and it will involve anyhow a search for three scalar parameters. Alternatively a P K iteration method may be used which starting from any stabilizing solution for the required h will minimize iteratively the obtained value of. The stabilizing solution for the required h can be obtained by using a P K iteration starting from h ¼ the stabilizing feedback gain is found while sequentially increasing the value of h. The latter procedure was successfully applied to many examples. We assume that A. The system for h ¼ is stabilizable. The algorithm that achieves for a given h statefeedback controller that asymptotically stabilizes the system over the entire uncertainty polytope is as follows: Algorithm (Finding a stabilizing K). Step : Find a state-feedback gain matrix K that stabilizes the system for h ¼. Here standard stabilization methods can be used []. Set h ¼. Step : Set h ¼ h þ. Replace A ð jþ and L ð jþ in () by A ð jþ þ B ð jþ K and Lð jþ þ D ð jþ K respectively j ¼... N and solve for P i i ¼ Y X R Q S and while minimizing the latter. In the time varying delay case take S ¼ and in the delayindependent problem take R ¼ Q ¼ X ¼ and Y ¼. Step : Substitute the resulting P i i ¼ Y XR QSin the above N LMIs and solve for K and again minimizing. Step : If h ¼ h stop. Otherwise go to Step. Once a stabilizing controller is found we apply the following: Algorithm (Minimizing ) Step : Start with any stabilizing state-feedback gain matrix K that stabilizes the system for example the one derived in Algorithm. Step : Replace A ð jþ and L ð jþ in () by A ð jþ þ B ð jþ K and L ð jþ þ D ð jþ K respectively j ¼... N and solve for P i i ¼ Y X R Q S and while minimizing the latter. In the time varying delay case take S ¼ and in the delay-independent problem take R ¼ and Y ¼. Step : Substitute the resulting P i i ¼ Y X R Q S in the above N LMIs and solve for K and again minimizing. Step : If a prescribed convergence condition on the sequence of previous values of is met (e.g. a decrease of not more than % during iterations) stop. Otherwise go to Step. The latter algorithm converges to some value since the sequence of the obtained values for is nonincreasing and it is bounded from below by zero... The Case of Norm-Bounded Uncertainties In the above we have treated the case the uncertain parameters reside in a polytope. The case
E. Fridman and U. Shaked these parameters are of bounded norm is treated next. We considered () with norm-bounded satisfied if there exist P R nn of the structure (8a) < S R R nn Y R nn and positive scalars d and d that satisfy the following LMI þ d I m H H T ½ I m Šþd I n E T EI n ½ Š M T H dmt < ðþ di di uncertainties namely: x kþ ¼ðAþH k EÞx k þða þ H k E Þx k hk þ B w k þðb þ H k E Þu k x k ¼ k h k z k ¼ðLþH k EÞx k þðdþh k E Þu k ðþ x k R n is the state vector h k is a positive number representing the delay h k h and A A H H E E and E are constant matrices of appropriate dimensions and k R r r and k R r r are timevarying uncertain matrices that satisfy T k k I r T k k I r ð8þ and we address the same stabilization and H control issues that were treated in the previous sections. Replacing in (a) A ð jþ in A ð jþ with A þ H k E A ð jþ with A þ H k E and L ð jþ with L þ H k E the LMI (a) can be written as þm T T k HT M þm T H kmþ½i n Š T E T T k H T ½ I mšþ½ I m Š T H k E½I n Š < ð9aþ is defined as in (a) for A A B and L without superscripts M¼ E E M ¼ P ½ IIŠþ½ IŠY½I I Š P h½r þ QŠ h½ IŠY : ð9bcþ It is well known that the following holds true for any two real matrices and of the appropriate dimensions and for k that satisfies (8) (see e.g. [8]): k þ T T k T d T þ d T ðþ d is some positive scalar. Choosing once ¼M T H and ¼M and then ¼½ I m Š T H and E½I n Š we apply () to (9a) and obtain the following. Theorem. Consider the system () with the constant time delay that satisfies () with B ¼ D ¼ and with k and k that satisfy (8). This system is asymptotically stable and for a prescribed scalar () is is defined as in (a) for the system s matrices without superscripts. The corresponding result for state-feedback control is readily obtained from Theorem A L and E are replaced by A þ B K L þ DK and E þ E K respectively. The resulting LMIs can be solved similarly to the solutions of the LMIs in Sections. and. by applying the above P K iteration... Example : State-Feedback H Control Consider the system of () " A ¼ # : þ g " # " : : A ¼ B ¼ # : þ g " B ¼ # L ¼ ½ Š and D ¼ : ðþ : and the unknown parameter g satisfies jgj :. By [] no solution is found for the nominal system (g ¼ ) with h ¼. Applying the above algorithms to this problem it is found that in the case of constant delay the system is stabilizable for all values of constant delay h and for all admissible values of g. For the case of time-varying delay S ¼ no solution has been found for h ¼ that is feasible for all the uncertain values of g. For constant delay h and for all the admissible values of g a minimum upper-bound on the disturbance attenuation level min ¼ :98 is achieved by the state-feedback with the gain matrix K½:9 98:Š. The question arises what is the difference between the latter minimum value of obtained for the latter K and the upper-bound on the peak value of the Bode magnitude plots of the transfer functions between w and z that are obtained for jgj. It is found that the bound on the peak value is..
Delay-Dependent H Control One of the advantages of the methods proposed in the present paper compared to the one that augments the system in order to incorporate the delayed states in the state vector is that the dimension of the system treated is fixed and does not depend on the delay. In the present example a solution for a constant h ¼ is derived by the method of the present paper without augmenting the system to be of order. The same system was considered in [8] with normbounded uncertainties. Taking there H ¼ : E ¼ E ¼ :I E ¼ H ¼ and g ¼ and considering the stabilization problem only a maximum value of h ¼ was obtained for the case of constant delay. Applying our algorithms we obtain that the system with the above norm-bounded uncertainties is stabilizable for all constant h : For constant h the feedback gain matrix K ¼½ 8:8 :9Š leads to the minimum bound min ¼ 8:: Assuming that the delay is time-varying we substitute S ¼ in Theorem and find that there is a solution to the problem of stabilizing via a state-feedback controller for all h. The resulting gain matrix is K ¼½ : :9Š and the minimum achievable bound on for h ¼ is 9. (for X ¼ and Q ¼ ).. Conclusions Delay-dependent criteria have been derived for statefeedback H control of uncertain discrete-time systems with uncertain constant or time-varying delay. Lyapunov Krasovskii functionals via descriptor model transformation are used. Sufficient conditions for achieving prescribed disturbance attenuation level are derived in terms of LMIs for the case of systems with either polytopic or norm-bounded uncertainties. The delay-independent condition for the case of the constant delay is derived as a special case of our conditions. The H state-feedback controller is obtained by applying P K iterations. The method developed in this paper may be applied in the future to output-feedback H control of discrete delay systems as well as to discrete descriptor systems. References. Boyd S El Ghaoui L Feron E Balakrishnan V. Linear Matrix Inequality in Systems and Control Theory. SIAM Frontier Series 99. Fridman E. New Lyapunov-Krasovskii functionals for stability of linear retarded and neutral type systems. Syst Contr Lett ; : 9 9. Fridman E Shaked U. A Descriptor system approach to H Control of linear time-delay systems. IEEE Trans Autom Control ; ():. Fridman E Shaked U. An LMI approach to stability of discrete delay systems. In: Proceedings of European Control Conference Cambridge. Fridman E Shaked U. Delay-dependent stability and H control: constant and time-varying delays. Int J Control ; (): 8. Kapila V Haddad W. Memoryless H controllers for discrete-time systems with time delay. Automatica 998; (9):. Kolmanovskii V Richard JP. Stability of some linear systems with delays IEEE Trans Autom Control 999; : 98 989 8. Lee YS Kwon WH. Delay-Dependent robust stabilization of uncertain discrete-time state-delayed systems. In: Proceedings of the IFAC Congress Barcelona 9. Li X de Souza C. Criteria for robust stability and stabilization of uncertain linear systems with state delay. Automatica 99; :. Mahmoud MS. Robust H control of discrete systems with uncertain parameters and unknown delays. Automatica ; :. Moon YS Park P Kwon WH Lee YS. Delaydependent robust stabilization of uncertain statedelayed systems. Int J Control ; :. Niculescu SI. Delay effects on stability: A Robust Control Approach Lecture Notes in Control and Information Sciences vol 9. Springer-Verlag London. Song S Kim J Yim C Kim H. H control of discrete-time linear systems with time-varying delays in state. Automatica 999; : 8 9. Verriest E Ivanov A. Robust stability of delaydifference equations. In: Proceedings of the IEEE Conference on Dec. and Control New Orleans 99; 8 9. Xu S Yang C. Stabilization of discrete-time singular systems: a matrix inequalities approach. Automatica 999; :. Zhang J Knopse C Tsiotras P. Stability of time-delay systems: equivalence between Lyapunov and smallgain conditions. IEEE Trans Autom Control ; : 8 8