Stability and guaranteed cost control of uncertain discrete delay systems

Transcription

1 International Journal of Control Vol. 8, No., 1 March 5, 5 Stability and guaranteed cost control of uncertain discrete delay systems E. FRIDMAN* and U. SHAKED Department of Electrical Engineering-Systems, Tel-Aviv University, Tel-Aviv 998, Israel (Received March ; revised 5 November ; accepted January 5) Robust stability and the guaranteed cost control problem are considered for discrete-time systems with time-varying delays from given intervals. A new construction of Lyapunov Krasovskii functionals (LKFs), which has been recently introduced in the continuous-time case, is applied. To a nominal LKF, which is appropriate to the system with nominal delays, terms are added that correspond to the system with the perturbed delays and that vanish when the delay perturbations approach zero. The nominal LKF is chosen in the form of the descriptor type and is applied either to the original or to the augmented system. The delayindependent result is derived via the Razumikhin approach. Guaranteed cost state-feedback control is designed. The advantage of the new tests is demonstrated via illustrative examples. 1. Introduction During the last decade, a considerable amount of attention has been payed to stability and control of continuous-time linear systems with delays (see e.g. Li and de Souza 199, Kolmanovskii and Richard 1999, Fridman 1,, Niculescu 1, Fridman and Shaked, and the references therein). Delayindependent and, less conservative, delay-dependent sufficient stability conditions in terms of Riccati or linear matrix inequalities (LMIs) have been derived by using Lyapunov Krasovskii functionals or Lyapunov Razumikhin functions. Delay-dependent conditions are based on different model transformations. The most recent one, a descriptor representation of the system (Fridman 1), minimizes the overdesign that stems from the model transformation used. The conservatism that stems from the bounding of the cross-terms in the derivation of the derivative of the Lyapunov Krasovskii functional has also been significantly reduced in the past few years. An important result that improves the standard bounding technique of, for *Corresponding author. emilia@tau.ac.il exmaple, Li and de Souza (199) has been proposed in Moon et al. (1). Less attention has been drawn to the corresponding results for discrete-time delay systems (Verriest and Ivanov 1995, Kapila and Haddad 1998, Song et al. 1999, Mahmoud, Lee and Kwom, Chen et al., Gao et al. ). This is mainly due to the fact that such systems can be transformed into augmented systems without delay. This augmentation of the system is, however, inappropriate for systems with unknown delays or systems with time-varying delays (such systems appear, for example, in the field of communication networks). For the case of constant small delay from ½, Š the delay-dependent conditions were derived in Lee and Kwon (), Gao et al. () and Chen et al. () by applying the discrete counterparts of the method developed in Moon et al. (1) and of the descriptor approach of Fridman and Shaked () correspondingly. There is a difference between the Lyapunov functions V for the descriptor discrete-time system Exðk þ 1Þ ¼AxðkÞ, E ¼ diagfi,g and the continuoustime system E _xðtþ ¼AxðtÞ. Thus, in the discretetime V ¼ x T EPEx, P ¼ P T is a full matrix (Xu and Yang 1999), while in the continuous-time V ¼ x T EPx with P of block-triangular structure International Journal of Control ISSN 19 print/issn 1 58 online ß 5 Taylor & Francis Group Ltd DOI: 1.18/151

2 E. Fridman and U. Shaked (Takaba et al. 1995). The method of Chen et al. () allows the treatment of the discrete-time case in a continuous-time manner with block triangular P. The case of small time-varying delay has been studied in Fridman and Shaked (5) via a discrete descriptor Lyapunov function. The case of uncertain non-small time-varying delay, the nominal delay value is non-zero and constant, has been recently considered in Xu and Chen (). A Lyapunov function has been used there with a nominal part that corresponds to delay-independent stability of the nominal system (i.e. of the systems with a nominal value of the delay). Thus, the necessary condition for the feasibility of the LMIs derived in Xu and Chen () for stability is the delayindependent stability of the nominal system, which is very restrictive. For continuous-time systems with uncertain nonsmall delay a new construction of the LKF has been introduced recently in Fridman (). To a nominal LKF, which is appropriate to the nominal system (with nominal delays), terms are added which correspond to the perturbed system and which vanish when the delay perturbations approach zero. In the present paper we apply such a construction of the LKF to the discrete-time systems with timevarying non-small delay, the descriptor type nominal LKF is applied first to the original system. Further, we augment the system to one with uncertain delay in a segment, starting from zero, and apply the conditions via descriptor nominal LKF to this augmented system. Such an augmentation is impossible in the continuous-time case, the complete LKF should be used (see, e.g. Kharitonov and Zhabko ), which leads to complicated conditions. The augmented system approach essentially improves the results obtained by the direct application of the descriptor nominal LKF (see, Examples 1 and ). The trade-off is in the higher-dimensional LMIs that are obtained, which require more computational efforts. Moreover, the state-feedback via augmentation depends on the current and the delayed states, while in the direct descriptor approach a memoryless statefeedback is obtained. To derive the reduced-order conditions we apply the descriptor model transformation of the augmented system and the discrete descriptor Lyapunov function of the form V ¼ x T EPEx. New delay-independent robust stability conditions are derived in the case of time-varying delay, that are based on the Razumikhin approach. Guaranteed cost state-feedback control is designed via descriptor nominal LKF. Examples are given which show that our conditions are less conservative than those that have appeared in the literature.. Robust stability.1 Problem formulation We consider the following unforced discrete-time statedelayed system xðk þ 1Þ¼ðAþHðkÞEÞxðkÞþð þ HðkÞE 1 Þxðk ðkþþ, xðkþ¼ðkþ, h k ð1þ xðkþ R n is the state vector, (k) is a positive number representing the delay ðkþ ¼ h þ ðkþ with the nominal constant value h > and a time-varying perturbation ðkþ½ 1, Š, h 1,. The matrices A,, H, E and E 1 are constant and ðkþr r 1r is a time-varying uncertain matrix satisfying the following inequality T ðkþðkþ I: For simplicity only we consider the single delay. The results are easily extended to systems with multiple delay.. Lyapunov Krasovskii method for discrete systems with delays Denoting yðkþ ¼xðk þ 1Þ xðkþ and taking into account that xðk ðkþþ ¼ xðk hþ k h 1 X j¼k h ðkþ we represent (1) in the descriptor form xðk þ 1Þ 5 ¼ yð jþ yðkþþxðkþ yðkþþða þ HE IÞxðkÞ þð þ HE 1 Þxðk hþ P k h 1 j¼k h ðkþ ð þ HE 1 ÞyðkÞ ðþ ðþ 1 C A 5, ðþ 9 xðþ ¼ðÞ, = yðþ ¼ðAþHE IÞðÞþð þ HE 1 Þð ðþþ, ; yðkþ ¼ðk þ 1Þ ðkþ, k ¼ h,..., 1: ð5þ Thus, if x(k) is a solution of (1), then fxðkþ, yðkþg, yðkþ is defined by (), is a solution of () and (5) and vice versa.

3 Uncertain discrete delay systems Lemma 1: If there exist positive numbers, and a continuous functional VðkÞ ¼Vðxðk hþ,..., xðkþ, yðk h Þ,..., yðk 1ÞÞ such that 9 VðkÞmaxf max j½k h,kš jxðjþj, max j½k h,k 1Š jyðjþj g = VðkÞ¼ Vðk þ 1Þ VðkÞ jxðkþj ; ða, bþ for x(k) and y(k) satisfying (), then (1) is asymptotically stable. Proof: From (b) it follows that X k j¼ ðvð j þ 1Þ Vð jþþ ¼ Vðk þ 1Þ VðÞ Xk j¼ jxð jþj : Therefore, for x(k)andy(k) satisfying () we have due to (a) jxðkþj Xk jx j j 1 VðÞ max max jxð j½ h j¼,š jþj, max jy jj, 8k : ðþ j½ h, 1Š Let x(k) be a solution of (1) and y(k) be defined by (), then fxðkþ, yðkþg satisfies (), (5) and thus (). Equation () implies that jxðkþj is small enough for small enough kk ¼ max j½ h,š P j jj. Moreover, 1 j¼ jxð jþj <1 and, hence, jxð jþj! for j!1. œ We suggest to construct the LKF for () in the form of V a ðkþ ¼ VðkÞ ¼V n ðkþþv a ðkþ X 1 1 m¼ Xk 1 j¼kþm h ð8þ yð jþ T R a yðjþ, < R a ð9þ and V n is a nominal Lyapunov function which corresponds to (), with ðkþ ¼ and H ¼. We intend to construct V n in the form of descriptor type (see, e.g. Chen et al. ) and to apply it either to the original system or to the augmented one.. Robust stability via descriptor type nominal LKF The nominal LKF (which corresponds to () with ðkþ ¼, H ¼ ) is given by (see, e.g. Chen et al. ) V n ðkþ ¼x T ðkþp 1 xðkþþ X 1 þ Xk 1 j¼k h m¼ h j¼kþm yð jþ T Ryð jþ xð jþ T Sxð jþ, P 1 >, R>, S >: ð1þ The nominal system is asymptotically stable if there exist nn matrices <P 1, P, P, S, Y, Z 1, Z, Z, R such that the following LMIs are feasible G n ¼ C n þ hz P T Y T 5 <, S R P ¼ P 1 P P Y ¼½Y 1 Y Š, Z ¼ Z 1 Z, i ¼ 1, Z C n ¼ P T I I T þ P A I I A I I þ S þ Y þ Y T, hr þ P 1 Y Z ð11a, bþ ð1a dþ We obtain the following lemma. Lemma : Equation (1) with is asymptotically stable for h 1 ðkþ h þ if there exist n n matrices < P 1, P, P, S, Y 1, Y, R and R a > that satisfy the LMI C P T Y T P T hy T G 1 ¼ S < R a 5 hr ð1þ ¼ maxf 1, g, Y and C n are given by (1) and C ¼ C n þ : ð1þ ð 1 þ ÞR a

4 8 E. Fridman and U. Shaked Proof: We find when VðkÞ is strictly negative. The difference V n ðkþ along the trajectories of the nominal system satisfies the inequality (Chen et al. ) V n ðkþ T ðkþg n ðkþ G n is given by (11a) and ðkþ ¼colfxðkÞ, yðkþ, xðk hþg ð15þ ð1þ provided (11b) is satisfied. Note that along the trajectories of () x T ðk þ 1ÞP 1 xðk þ 1Þ x T ðkþp 1 xðkþ ¼ x T ðkþp 1 yðkþþy T ðkþp 1 yðkþ " ¼ x T ðkþp yðkþ # T þ y T ðkþp 1 yðkþ yðkþ ¼ x T ðkþp T yðkþþða IÞxðkÞþ xðk hþ þ y T ðkþp 1 yðkþþðkþ xðkþ ¼colfxðkÞ, yðkþg ðkþ ¼ x T ðkþp T k h 1 X j¼k h ðkþ yð jþ ð1þ while along the trajectories of the nominal system with ðkþ h (1) is obtained with ðkþ. Therefore V n along the trajectories of the perturbed system satisfies the inequality From (8), (9), (18), (19) and Schur complements formula we find, provided (11b) is satisfied, the following VðkÞ 1 ðkþ T ðg 1 þ diagfhz,,,gþ 1 ðkþ ðþ 1 ðkþ ¼colfðkÞ, yðkþ,g. From (11b) it follows that Z Y T R 1 Y. Choosing therefore in () Z ¼ Y T R 1 Y and applying Schur complements formula it is obtained that (1) implies VðkÞ < and the asymptotic stability of (1). œ In the case of norm-bounded uncertainties (i.e. ¼ ) we replace A and in Lemma by A þ HE and þ HE 1, respectively. Applying the bounding (Xie 199) ðkþ þ T T ðkþ T 1 T þ T ð1þ is a positive number and T ¼ ½H T P H T P Š and ¼½E E 1 E 1 Š, we obtain by Schur complements that VðkÞ < along the trajectories of () if the following LMI holds P T H E T P T H G 1 E1 T 5 E1 T I I 5 < : ðþ 5 We have V n ðkþ T ðkþg n ðkþþðkþ: k h 1 X ðkþ x T ðkþp T R 1 a j¼k h ðkþ A ½ AT 1 ŠP xðkþ 1 k h 1 X þ y T ð jþr a yð jþ j¼k h ðkþ x T ðkþp T R 1 ia ½ AT 1 ŠP xðkþ þ k hþ X 1 1 j¼k h y T ð jþr a yð jþ: ð18þ ð19þ We have thus proved the following Theorem 1: Consider (1), h 1 ðkþ h þ. This system is asymptotically stable if there exist n n matrices < P 1, P, P, S, Y 1, Y, R, R a and a scalar that satisfy (), ¼ maxf 1, g.. Augmentation and descriptor nominal LKF In the case when the non-delayed system is not asymptotically stable or h 1 is not large, we represent (1) in the form of the augmented system ðk þ 1Þ ¼ ðaþ HðkÞEÞðkÞ þð þhðkþe 1 Þðk 1 ðkþþ ðþ

5 Uncertain discrete delay systems 9 xðk h þ 1 Þ I n ::: xðk h þ 1 1Þ ðkþ¼ ::: 5, A¼ ::: ::: ::: ::: ::: ::: I n 5 xðkþ ::: A ::: ::: ::: ::: ::: ¼ ::: 5, H¼ ::: 5, ::: H E¼½ EŠ, E 1 ¼½E 1 Š: ðþ Note that for 1 ¼, the nominal system (), ðkþ and, has no delay and the nominal exact Lyapunov function V n ðkþ ¼ T ðkþp 1 ðkþ should be used. This is different from the continuous case, the exact (complete) LKF has a complicated form and leads to complicated robust stability conditions (Kharitonov and Zhabko ). In the general case of 1 we apply Theorem 1 to (), h ¼ 1, and obtain the following: Theorem : Consider (1), h 1 ðkþ h þ. This system is asymptotically stable if there exist ðh 1 þ 1Þn ðh 1 þ 1Þn matrices < P 1, P, P, S, Y 1, Y, R, R a and scalars i >, i ¼, 1 that satisfy () with ¼ maxf 1, g and h ¼ 1, A,, E, E 1 and H should be changed correspondingly to A,, E, E 1 and H. Remark 1: The augmentation of the system till some h < h 1 with T ðkþ ¼½x T ðk h Þ,..., x T ðkþš can also be applied to obtain less restrictive conditions than those obtained by the descriptor approach. Such augmented system is of lower-order than () and has delay h h þ ðkþ. Here Theorem 1 should be applied with h substituted by h h..5 Augmentation and discrete descriptor Lyapunov function We consider 1 ¼ and ¼. To reduce the size and the number of the decision variables by the previous augmented method, we consider h 1 and the state vector ¼½ 1... hþ1 Š T given by (). Defining yðkþ ¼ xðk þ 1 hþ xðk hþ ¼ ðkþ 1 ðkþ and representing (1) in the form we obtain the descriptor form E ðk þ 1Þ¼A d ðkþþ yð jþ E ¼ diagfi ðhþ1þn, nn g, ðkþ¼ ðkþ yðkþ I n ::: I n I n ::: ::: ::: ::: ::: ::: ::: ::: A d ¼, ¼ : ::: I n ::: A 5 5 I n I n ::: I n ða eþ We construct the LKF for () in the form of VðkÞ ¼V n ðkþþv a ðkþ, V a ðkþ ¼ X 1 Xk 1 m¼ j¼kþm yð jþ T R a yð jþ, < R a ðþ and V n is a nominal Lyapunov function which corresponds to (a), with ðkþ ¼ V n ¼ T ðkþepe ðkþ, P ¼ P T, EPE : ð8þ We have " A T # V n ðkþ ¼ T d ðkþ P A d ðkþ T ðkþepe ðkþ h T ðkþ ¼ T ðkþ and A T 1 P k 1 i yt ð jþ V a ðkþ ¼ yt ðkþr a yðkþ y T ð jþr a yð jþ j¼k yt ðkþr a yðkþ ðkþ By Cauchy Schwartz inequality y T ð jþr a yð jþ: ð9þ xðk þ 1Þ ¼AxðkÞþ xðk hþ yð jþ ð5þ ðkþ y T ð jþr a yð jþ y T ð jþr a!! yð jþ :

6 E. Fridman and U. Shaked Hence V a ðkþ yt ðkþr a yðkþ y T ð jþ! R a Finally we find that VðkÞ ¼V n ðkþþv a ðkþ T ðkþg d ðkþ G d ¼ AT d PA d EPE þ R a! yð jþ : ðþ A T d P R a þa T 1 P 5ð1Þ Therefore, G d < implies asymptotic stability of (1). We proved the following. Lemma : Consider (1),, 1 h ðkþ h þ. This system is asymptotically stable if there exist a ðh þ Þn ðhþþn matrix P ¼ P T, such that ½I ðhþ1þn ŠP½I ðhþ1þn Š T >, and a n n matrix R a that lead to G d <, G d, A d and are given by (1), (d) and (e), correspondingly. The condition of Lemma can also be written as A T P I T þip T! AþIP I T A T C ¼ þdiagf P 1, R P 1 a, R a g 5 < ðþ P 1A P1 " # A¼ I ðhþ1þn Ad..., I¼ I n Ad and we substituted in (1) the structure of P ¼ P 1 P P T P and applied the Schur complements formula. In the case A and are replaced by A þ H ðkþe and þ HðkÞE 1, respectively, we require that C þ IP P 1 C þ 1 IP P 1 HðkÞEþ E T ðkþ T H T P I T P 1 Š HH T P T I T P 1 þ E T E < for a positive scalar, H is defined in () and E ¼½E 1 E E 1 Š: The latter leads to the following. T Theorem : Consider (1), 1 h ðkþ h þ. This system is asymptotically stable if there exist: ðh þ 1Þn ðh þ 1Þn matrix < P 1, ðh þ 1Þn n matrix P, n n matrix P, n n matrix < R a and a positive scalar that satisfy the following LMI C IP T P 1 H E T I I 5 < ðþ C is defined in (). Remark : Considering 1 > and combining V n of the form of discrete descriptor LKF (i.e. V n of (1), the first term should be changed to x T ðkþepexðkþ with V a of () may lead to further improvement of the results by Lemma and Theorem. We next consider the case of the small delay ðkþ ½, Š with h ¼, ¼ and representing (1) in the descriptor form E xðk þ 1Þ¼ ~ A xðkþþ ~ yð jþ, E ¼ diagfi n, nn g xðkþ¼ xðkþ, A¼ ~ I n I n yðkþ þ A I n I n ~ ¼ : ða eþ By applying the above derivations to () we obtain a new stability criterion: Corollary 1: Consider (1),, ðkþ. This system is asymptotically stable if there exist n n matrix P ¼ P T, satisfying ½I n ŠP½I n Š T > and n n matrix R a such that G d <, G d is given by (1) and A d and should be substituted by A ~ and ~ correspondingly.. Delay-independent conditions in the case of time-varying delays As in the continuous-time situation, this case is treated adopting the Lyapunov Razumikhin approach (see Zhang and Chen 1). Lemma : Consider the system (1), ðkþ, with time-varying delay. This system is asymptotically stable if there exist < P R nn and scalars ð, 1Þ and q > 1 that satisfy the LMI G ind ¼ AT PA P A T P A T 1 PA < : 1 ðð1 Þ=qÞP ð5þ

7 Uncertain discrete delay systems 1 Proof: Choosing the Lyapunov Razumikhin function VðkÞ ¼xðkÞ T PxðkÞ and assuming that for some q >1 we find Vðk iþ qvðkþ, h i 1, k Vðk þ 1Þ VðkÞ¼ðxðkÞ T A T þ xðk ðkþþa T 1 ÞPðAxðkÞ þ xðk ðkþþ xðkþ T PxðkÞ ¼ xðkþ T ða T PA PÞxðkÞ þ x T ðk ðkþþa T 1 PAxðkÞ þ x T ðk ðkþþa T 1 Pxðk ðkþþ ð1 ÞxðkÞ T PxðkÞ xðkþ T x T xðkþ ðk ðkþþ G ind xðk ðkþþ and thus due to (5) Vðk þ 1Þ VðkÞ <, which implies the asymptotic stability of (1) (see Zhang and Chen 1998). œ By Schur complements (5) is equivalent to P A T P O ¼ ðð1 Þ=qÞP A T 1 P 5 < : ðþ P We replace A with A þ HðkÞE and with þ HðkÞE 1 and obtain, applying Lemma to the uncertain system (1), that the stability of the system is guaranteed if the following inequality holds O þ ET T H T P E1 T 5 E T ½E E 1 Š O þ " E1 T 5 þ " 1 5 < : PHH T P ðþ In the latter inequality we used the bounding (1) ¼ " and E T T ¼ 5, T ¼ E T H T 1 5: P We proved the following. Theorem : Consider the system (1) with (k) that satisfies (). This system is asymptotically stable for all delays (k) if there exist P ¼ P T R nn, ð, 1Þ, q > 1 and " > that satisfy the LMI P þ "E T E "E T E 1 A T P ðð1 Þ=qÞP þ "E1 TE 1 A T 1 P P PH 5 < : "I ð8þ. Examples Example 1: We consider the system (1) A ¼ :8, A :9 1 ¼ :1 and H ¼ : :1 :1 ð9þ 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 Song et al. (1999), Theorem 1 in Lee and Kwon () and Theorem 1 above. It is found that the method of Song et al. (1999) does not provide a solution even for h ¼ 1. The maximum value of h, achievable by the method of Lee and Kwon (), is 1, as a value of h ¼ 1 was obtained by applying Chen et al. (). Using augmentation it is found that the system considered is asymptotically stable for all h 18. The criterion of Lemma did not provide a solution, so that no delay-independent solution has been found. Allowing to be time-varying we apply Lemma, h ¼ 1 ¼ 1 and ¼. We obtain thus that asymptotic stability is guaranteed for all ðkþ 8. The same result is obtained by Corollary 1 via discrete descriptor Lyapunov function. Choosing h ¼ 8, 1 ¼ ¼ ; h ¼ 1, 1 ¼ ¼ and h ¼ 11, 1 ¼ 1, ¼ we verified that conditions of Lemma are feasible. Hence the system is asymptotically stable for all (k) from the following intervals: ½, 1Š, ½5, 11Š, ½8, 1Š and ½1, 1Š. Note that conditions of Xu and Chen () are not feasible even for ðkþ 1. By augmentation via the discrete descriptor Lyapunov function we verify that the conditions of Lemma are feasible for h ¼, ¼ ; h ¼ 5, ¼ ; h ¼, ¼ 5 and for h ¼ 9, ¼ and thus the stability intervals are larger for h : ½, 1Š, ½5, 11Š, ½, 1Š and ½9, 1Š. The augmented approach via descriptor LKF of Lemma leads to the same stability intervals as Lemma, but needs essentially more time for computation. Next considering the case the system parameters are uncertain, with A and given in (9) and with H ¼ :1 :, E ¼ I and E 1 ¼ :5I,

8 E. Fridman and U. Shaked we apply Theorem 1 for h ¼, and 5 and verify that the system (1) is asymptotically stable for all (k) that satisfy (19) and for (k) from the following segments: ½, Š, ½, 5Š and ½5, Š. By the augmented system approach via descriptor LKF, we find that the conditions of Theorem are feasible for h ¼, 1 ¼ 1, ¼ and for h ¼ 5, 1 ¼ 1, and ¼ 1. Thus the stability intervals, starting from non-zero values, are larger ½, 5Š and ½, Š. By augmentation via discrete descriptor approach we find the following intervals: ½,5Š and ½, Š. The augmented system approach improves the results, but it takes essentially more time for computations due to high dimensional LMIs. The conditions by Theorem need less time for verification than those by Theorem. Example (Wu and Hong 199): We consider the system (1) A ¼ :5, A :5 : 1 ¼ : and H ¼ : In the case of constant delay, this system is delayindependently stable by the conditions of Wu and Hong (199). In the case of time-varying delay, by conditions of Song et al. (1999) the system is asymptotically stable for <ðkþ. By Theorem, it is verified that also in the case of time-varying delay the system is delayindependently stable. This is achieved by taking ¼.5 and q ¼ 1:1.. Guaranteed cost control Extending the description of (1) to include a control input uðkþ R m, we consider the system xðk þ 1Þ ¼ðA þ HðkÞEÞxðkÞþð þ HðkÞE 1 Þ xðk ðkþþ þ ðb þ HðkÞE ÞuðkÞ, xðkþ ¼ðkÞ, h k ðþ xðkþ R n, (k), A,, H, E, E 1 and (k) are as in (1) and () and B and E are constant matrices of the appropriate dimensions. We also consider the cost function J ¼ X1 j¼ z T ðkþzðkþ the objective vector zðkþ R p is defined by zðkþ ¼LxðkÞþDuðkÞ ð1þ ðþ for matrices L and D of the appropriate dimensions. A control law uðkþ ¼KxðkÞ ðþ is sought that for a given ðkþ, h 1 ðkþ h þ leads to a minimum guaranteed cost for J(), namely, JðÞ for the delay described in (1) and for all that satisfy ()..1 Guaranteed cost via descriptor nominal LKF Denoting FðkÞ ¼Vðk þ 1Þ VðkÞþz T ðkþzðkþ ðþ V(k) is defined in (8) and (9), it is obtained, similarly to the proof of Lemma, that in the case H ¼ and uðkþ the following holds FðkÞ T ðkþg z ðkþ ðkþ ¼colfxðkÞ, yðkþ, xðk hþ, yðkþ, zðkþg and Requiring that G z ¼ G 1 G z < ð5þ L T 5: ðþ I p ðþ we take the sum of the two sides of (5), from to N, and obtain that and thus X N k¼ z T ðkþzðkþ VðÞ VðN þ 1Þ VðÞ J VðÞ ¼ T ðþp 1 ðþþ X 1 X 1 m¼ h j¼m Rðð j þ 1Þ ð jþþ þ X 1 þ X 1 1 X 1 m¼ j¼m h ðð T ð j þ 1Þ ð T ð j þ 1Þ ð jþþ j¼ h T ð jþþr a ðð j þ 1Þ ð jþþ: T ð jþsð jþ ð8þ ð9þ Admitting the control law () the following result is thus obtained.

9 Uncertain discrete delay systems Lemma 5: Consider the system () H ¼ and the cost function (1). The control law () stabilizes the system and achieves a prescribed guaranteed cost <, namely J, if there exit n n matrices < P 1, P, P, S, Y 1, Y, R and R a and a m n matrix K that satisfy the following two inequalities ^G 1 ^G z ¼ VðÞ L T þ K T D T 5 < ð5aþ I p ð5bþ ^C P T Y T P T hy T ^G 1 ¼ S < ð5cþ R a 5 hr ^C ¼ P T I I T þ P A þ BK I I A þ BK I I þ S þ Y þ Y T hr þ P 1 þð 1 þ ÞR a ð5d Þ and V() is given in (9) and P has the structure of (1a). The inequality (5a) is non-linear in P and K. In order to obtain a LMI we consider the case Y ¼ " A T 1 P, for some tuning parameter ". Realizing that the second block on the diagonal of ^C in ^G 1 is P P T þ hr þ P 1 þð 1 þ ÞR a it is found that P is invertible. Denoting P 1 ¼ Q ¼ Q 1 Q Q we obtain the following. Lemma : Consider the system () H ¼ and the cost function (1). The control law () stabilizes the system and achieves a prescribed guaranteed cost < if for some tuning scalar parameter " there exit n n matrices Q 1, Q, Q, S, R, R a, M R, M Ra and M S, a scalar M Q, and a m n matrix Y that satisfy the following six inequalities and Q 1 L ^G T þ Y T D T k 5 < ð51aþ I p M Q T ðþ M R I n >, > ð51b, cþ Q 1 R M Ra I n M S I n >, > ð51d, eþ R a S JðÞ¼M Q þ X 1 þ X 1 j¼ h X 1 m¼ h j¼m T ðj þ1þ T ðjþ MR ððj þ1þ ðjþþ T ðjþm S ðjþþ X 1 1 M Ra ððj þ1þ ðjþþ X 1 m¼ j¼m h T ðj þ1þ T ðjþ ð51fþ C k ð1 "Þ S R A a "h R 1 ^G k ¼ S R a 5 h R and ð51gþ! Q þ Q T Q 1 þ Q Q T þq 1 ða T þ "A T 1 Þþ Y T B T Q Q T C k ¼ Q 1 hq T Q T ð 1 þ ÞQ T hq T Q T ð 1 þ ÞQ T S h R : ð51hþ Q 1 5 ð 1 þ Þ R a

10 E. Fridman and U. Shaked If a solution to the above exists, the feedback gain that achieves the guaranteed cost is given by K ¼ YQ 1 1 : ð5þ Proof: We first multiply (5a) by diagfq T,,,g and diagfq,,, g, from the left and the right, respectively. Using the fact that: " Q # T Q ¼ QT hr Q Q and hr Q T " Q S # " T Q ¼ Q # 1 S Q 1 and denoting Y ¼ KQ 1, R ¼ R 1, R a ¼ R 1 a, S ¼ S 1, the requirement of (51a) follows applying Schur complements arguments. The requirement (51b) follows from the fact that M Q, M R, M R_a and M S are upperbounds on Q 1, R, R a and S, respectively, and thus VðÞ < JðÞ. œ Lemma 5 addresses the case H ¼. Its result can be readily extended to the case H ¼. Replacing A, and B in the LMI of Lemma 5 by A þ HE, þ HE 1 and B þ HE, respectively, and applying Schur complements arguments we readily obtain the following result for the case with normbounded uncertainties. Theorem 5: Consider the system () and the cost function (1). The control law () stabilizes the system and achieves a prescribed guaranteed cost < if for some tuning scalar parameter " there exit n n matrices Q 1, Q, Q, S, R, R a, M Q, M R, M Ra and M S, amn matrix Y and a scalar that satisfy (51b f) and! Q 1 ðe þ "E 1 Þ T Q 1 L T H þy T E T þy T D T ^G k ð1 "Þ SE 5 1 T R a E1 T < 5 5 "h RE1 T I I 5 I ð5þ ^G k is defined in (51g). If a solution to the latter inequalities exists, the state-feedback gain K that achieved the guaranteed cost is given by (5).. Guaranteed cost via augmentation and descriptor nominal LKF The stability result of x. can be extended to solve the guaranteed cost problem. However, dissimilar to the solution of the stability verification problem there, when attempting to obtain a state-feedback control law that stabilizes the system and achieves a given bound on its cost, using the augmented system of (), the problem becomes one of finding a static outputfeedback controller. The latter task is known to be non-convex. If, on the other hand, a state-feedback control law of the form uðkþ ¼ Xh 1 K j xðk jþ j¼ ð5þ is sought the problem can be solved using the augmented system representation of (). We have ðk þ 1Þ ¼ðAþHðkÞEÞðkÞ þð þhðkþe 1 Þðk 1 ðkþþ þ BuðkÞ T ðkþ ¼ T ðk h þ 1 Þ T ðk h þ 1 þ 1Þ T ðkþ ¼ T ðkþ, 1 k zðkþ ¼LðkÞþDuðkÞ B¼½... B T Š T, L¼½,..., LŠ and uðkþ ¼½K h 1 ::: K ŠðkÞ: ð55þ Then Theorem implies the following result. Theorem : Consider the system () and the cost function (1). The control law (5) stabilizes the system and achieves a prescribed guaranteed cost < if for some tuning scalar parameter " there exist nðh 1 þ 1Þ nðh 1 þ 1Þ matrices Q 1, Q, Q, S, R, R a, M Q,M R M R, M Ra, M S, amnðh 1 þ 1Þ matrix Y and scalars and 1 that satisfy (51b e),(5a, b) with ^G k and JðÞ given by (51g) and (51f), h, A,, E, E 1, H, L and should be substituted by 1, A,, E, E 1, H, L and correspondingly. If a solution to the above inequalities exists, the state-feedback gain K ¼½K h 1 ::: K Š that achieves the guaranteed cost is given by (5). Remark : Guaranteed cost control via the augmented discrete descriptor approach may be designed by an iterative method similarly to Fridman and Shaked (5) and will not be considered here. Razumikhin approach seems to be inapplicable to guaranteed cost and to H 1 control.

11 Uncertain discrete delay systems 5.. Example We consider the problem of Chen et al. (). Given the system () with 1:1 :5 :1 A ¼, ¼, B ¼ 1: :1 1 H ¼ I, E ¼ E 1 ¼ :8I, E ¼ p L ¼ ffiffiffiffiffiffi I p :1, D ¼ ffiffiffiffiffiffi e k :1, ð kþ ¼ : 5 1 For h ¼, 1 ¼ ¼ a state-feedback control law () is sought that stabilizes the system and achieves a minimum cost bound. In Chen et al. () such a feedback control has been found which achieves the cost bound of ¼ 1:8. Applying the result of Theorem 5 for " ¼ 1 we obtain a cost bound of ¼ :19 for K ¼ ½: 1:1Š. For a smaller uncertainty, H ¼ I, E ¼ E 1 ¼ :I and E ¼, application of Theorem 5 yields a cost bound of ¼ :5 for K ¼ ½:1 1:Š: For this uncertainty we also consider the case h ¼, 1 ¼ and ¼ 1. For xð Þ ¼½e Š T, the application of Theorem 5 yields a minimum cost bound of ¼ :18 for K ¼ ½: 1:5Š. Applying the exact LKF result of Theorem to the system with the latter uncertainty ( H is :I, h ¼, 1 ¼ and ¼ 1) we obtain ¼ :891, for k ¼ [ ]. The difference between the latter result and the corresponding result that was obtained via descriptor V n is accentuated in the case H ¼. Then, the descriptor based result of Theorem yields a bound of ¼ :1 compared to ¼ :8 obtained by the exact LKF result of Theorem.. Conclusions Delay-dependent and delay-independent criteria have been derived for determining the asymptotic stability of discrete-time systems with uncertain delay and norm-bounded uncertainties. The delay is assumed to be time-varying either from a given interval or unbounded. In the first case the Lyapunov Krasovskii method is applied via descriptor model transformation, while the second (delay-independent) case is treated by Laypunov Razumikhin technique. The Lyapunov Krasovskii method is applied to guaranteed cost control. For the first time, augmentation is applied to the case of time-varying delay in order to reduce the nominal value of the delay which appears in the augmented system. Such an augmentation leads to less conservative conditions. However, the resulting LMIs possess highdimensional decision variables which require longer computational time. Another possibility for reducing the conservatism is to apply the discrete-time counterpart of the complete LKF that corresponds to the necessary and sufficient conditions for the stability of the nominal system and which was applied to robust stability of continuoustime systems by Kharitonov and Zhabko (). The latter LMIs may have computational advantages over the high-order ones that are based on the augmentation. This issue is currently under study. Acknowledgement This work was supported by the KAMEA fund of Israel and by the C&M Maus Chair at Tel Aviv University. References W.-H. Chen, Z.-H. Guan and X. Lu, Delay-dependent guaranteed cost control for uncertain discrete-time systems with delay, IEE Proccedings of Control Theory Applications,, 15, 1 1. E. Fridman, New Lyapunov Krasovskii functionals for stability of linear retarded and neutral type systems, Systems and Control Letters, 1,, E. Fridman, Stability of linear functional differential equations: A new Lyapunov technique, in Proceedings of MTNS, Leuven, July. E. Fridman and U. Shaked, An improved stabilization method for linear systems with time-delay, IEEE Transactions on Automation Control,,, E. Fridman and U. Shaked, Delay-dependent H 1 control of uncertain discrete delay systems, European Journal of Control, 11, no. 1, 5. H. Gao, J. Lam, C. Wang and S. Xu, H 1 model reduction for discrete time-delay systems: delay-independent and dependent approaches, International Journal of Control,,, 1 5. V. Kapila and W. Haddad, Memoryless H 1 controllers for discretetime systems with time delay, Automatica, 1998,, V. Kharitonov and A. Zhabko, Lyapunov Krasovskii approach to the robust stability analysis of time-delay systems, Automatica,, 9, 15. V. Kolmanovskii and J.-P. Richard, Stability of some linear systems with delays, IEEE Transactions on Automatic Control, 1999,, Y.S. Lee and W.H. Kwon, Delay-Dependent robust stabilization of uncertain discrete-time state-delayed systems, in Proceedings of the 15th IFAC Congress on Automation and Control, Barcelona, Spain,. X. Li and C. de Souza, Criteria for robust stability and stabilization of uncertain linear systems with state delay, Automatica, 199,, M. Mahmoud, Robust H 1 control of discrete systems with uncertain parameters and unknown delays, Automatica,,, 5.

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