Output Feedback Guaranteeing Cost Control by Matrix Inequalities for Discrete-Time Delay Systems

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1 Output Feedbac Guaranteeing Cost Control by Matrix Inequalities for Discrete-ime Delay Systems ÉVA GYURKOVICS Budapest University of echnology Economics Institute of Mathematics Műegyetem rp. 3 H-1521 Budapest HUNGARY gye@math.bme.hu IBOR AKÁCS ECOSA Institute for Strategic Research of Economy Society Margit rt. 85 H-124 Budapest HUNGARY tibor.taacs@ecostat.hu Abstract: his paper investigates the construction of guaranteeing cost dynamic output feedbac controller for linear discrete-time delay systems with time-varying parameter uncertainties with exogenous disturbances. A necessary sufficient condition for a controller to have the guaranteeing cost property is given by a nonlinear matrix inequality. As a sufficient condition a linear matrix inequality is derived the solution of which can be used for constructing a control of the desired type. It is also shown that the resulted trajectory is input-to-state stable. A numerical example illustrates the application of the results. Key Words: guaranteeing cost control delay systems discrete-time systems matrix inequalities 1 Introduction Systems with delay occur in several areas of engineering therefore they have always been one of the focuses of control theory. Especially in the past decade a huge amount of papers has been published on the stabilization guaranteeing cost H control problems for uncertain time-delay systems primarily in the continuous-time case but in the discrete-time case as well. o mention but a few of them see e.g. [3] [5] [11] [13] [18] [19] [2] the references therein while a sampled tracing of delay system is presented by [16]. he relatively modest amount of wor devoted to discrete-time delay systems can be explained by the fact that such systems can be transformed into augmented systems without delay. However this approach suffers from the curse dimension if the delay is large inappropriate for systems with unnown or time-varying delays. he present paper deals with the determination of guaranteeing cost output feedbac control for uncertain discrete-time delay systems with given constant delay though a part of the results remains valid also in the case of unnown but bounded delay. We note that to the best of our nowledge time-varying delays are considered in the literature under the assumption that at least the current state is available for measurement memoryless state-feedbac is to be constructed e.g. in [2] [4] [5] [15] [19] [21]. Most papers consider only system uncertainty of either norm-bounded or polytopic type. Similarly to paper [19] we consider both system uncertainty exogenous disturbance but the class of system uncertainty considered here is more general. his is the uncertainty of linear fractional form. Authors are not aware of results on uncertain delay systems with this type of uncertainty. Firstly we formulate a necessary a sufficient condition for a dynamic feedbac to be a guaranteeing cost robust controller. his condition can be transformed into a bilinear matrix inequality BMI. A linear matrix inequality LMI will be shown the solution of which is also a solution of this BMI. In order to set up the LMI the method of the seminal wor [6] will be applied. A further novelty of the present paper is that unlie [13] [19] see remar 2 in [19] the coefficient matrix of the delayed initial states in the cost function bound should not be fixed but it is computed parallel to the weighting matrix of the undelayed initial state the parameters of the dynamic output feedbac controller. It is also shown that the resulted trajectory is input-to-state stable. We note that under a special choice of the weighting matrices of the cost function robust H results can be derived from the results of the present paper. he organization of this paper is as follows: After fixing the problem statement we provide some definitions a preliminary lemma in Section 2. Our main results are stated proved in Section 3. A numerical example is given in Section 4 to illustrate the application of the results finally the conclusions ISSN: Issue 7 Volume 7 July 28

2 are drawn. Stard notation is applied. he transpose of matrix A is denoted by A P > denotes the positive semi- definiteness of P. he minimum maximum eigenvalues of the symmetric matrix P are respectively denoted by λ m P λ M P. Notation w or simply w is used for the infinite vector series {w j } j= while w denotes its truncation to {w τ w τ+1... w } τ is a given positive integer. w denotes the Euclidean norm of the vector w while w w 2 are defined by w = sup w N 1/2 w 2 = w 2. = Notations l 2 l are used for the linear space of infinite vector series with finite norms. Symbol I denotes the identity matrix of appropriate dimension. he notation of time-dependence is omitted if it does not cause any confusion. For the sae of brevity asteris replaces blocs in hypermatrices matrices in expressions that are inferred readily by symmetry. 2 Problem statement preliminaries 2.1 Discrete uncertain time-delay systems Consider the following discrete-time state-delayed uncertain system: x +1 = Ax + A d x τ + Bu +Ew + H x p x Z+ x = φ = 1... τ y = Cx + C d x τ + H y p y q x = A q x + A dq x τ + B q u + G x p x q y = C q x + C dq x τ + G y p y x R n is the state u R m is the control w R s is the exogenous disturbance y R p is the measured output. All the matrices are of appropriate dimensions. he time delay τ is assumed to be a nown constant. he uncertainty appears in the system as p x = x qx p y = y qy the time varying unnown matrices representing the collection of all parametric uncertainties x y satisfy the constraint.. I dot replaces either x or y. It is assumed that the system is well posed i.e. matrices I x G x I y G y are invertible for all admissible realizations of x y. It can easily be shown by the matrix inversion lemma that the inverse of a matrix of type I G exists for all I if only if I G G >. his is also equivalent to I GG >. By substitution we obtain that the uncertain dynamics is x +1 = A + δax + A d + δa d x τ +B + δbu + Ew 1 y = C + δcx + C d + δcx τ 2 δa δad δb = with = H x I x G x 1 x A q A dq B q δc δcd = = H y I y G y 1 y C q C dq. Assign to system 1-2 the objective function Jx u w = Lx u w 3 = Lx u w = x Qx + u Ru w Sw x = {φ τ φ τ+1... φ } is the initial function matrices Q R S are symmetric positive definite. he purpose of this paper is to design a dynamic output feedbac control guaranteeing a certain level of performance for system ISSN: Issue 7 Volume 7 July 28

3 1-3. o this end the output feedbac controller is looed for in the form x +1 =  x + Âd x τ + Ly 4 x = = 1... τ 5 u = K x + K d x τ 6 x R n is the state of the controller the matrices   d L K K d of appropriate dimensions should be determined. he application of the control 4-6 to system 1-2 results in the following closed loop system: z +1 = A + δa z + A d + δa d z τ + with +E w Z + 7 z = ζ = 1... τ z = ζ = A = A d = H = x x φ Z + = 1... τ A B K L C  Ad B K d L C d  d E E = δa = H A q δa d = H A qd Hx LHy = I G 1 Gx G = G y A dq = Aq B A q = q K C q = Adq B dq K d C dq x y. he objective function for the closed-loop system 7 equivalent to the original one under the application of 4-6 can be expressed as J z z w = L z z w 8 = L z z w = z z τ w Sw Q z z τ with Q = Γ K + Q Γ + K d Γ = I R K K d K = K K d = K d. Definition 1 Consider the uncertain system 1-2 with the cost function 3. A controller of the form 4-6 is said to be a guaranteeing cost output feedbac controller for 1-2 with 3 if there exist positive definite symmetric matrices P P d such that for the function inequality V z = z P z + z i P d z i 9 i=1 V z +1 V z + L z z w < 1 holds true for all N for any disturbance sequence w any realization of the uncertainty z = z τ... z z is the solution of 7. Remar 2 Definition 1 is the generalization of that given in [8] inasmuch as it allows the appearance of delayed states in 7 8. It is well-nown that by augmentation system 7 is equivalent to an undelayed discrete-time system with a state space of dimension τ + 1n. hus the above mentioned definition could directly be used to the augmented system. However to avoid the application of a τ +1n τ + 1n matrix as a quadratic cost matrix function V is defined lie a Lyapunov-Krasovsii function. his definition is analogous of those of [1] [14] [17] [22]. Other papers as e.g. [1] [7] accept a definition formulated directly by the objective function. he connections of these two approaches will be discussed below in corollary 6 in remar Input-to-state stability of discrete-time time-delay systems Consider the general discrete-time time-delay system x + 1 = fx x τ w 11 Z + x = φ = 1... τ. ISSN: Issue 7 Volume 7 July 28

4 Definition 3 System 11 is globally input-to-state stable ISS if there exist a KL-function β : R R R a K-function γ such that for each w l each φ = {φ τ... φ} it holds that x; φ w β φ + γ w 12 for each Z + x.; φ w denotes the solution of 11. As it has already been mentioned system 11 is equivalent to an augmented undelayed discrete-time system. his gives the possibility of a straightforward re-formulation of results in [9] on the ISS property of discrete-time systems for delayed discrete-time systems if an ISS-Lyapunov function can be shown for the augmented system. However sometimes this may be difficult. he problem comes from the fact that it is not easy to give an upper estimation for the forward difference of a Lyapunov function cidate along the solution which contains a strictly negative definite term with respect to the whole augmented state. It can be seen that this is the case in the problem under consideration. he difficulty can be overcome if the required estimation holds true with a certain part of the augmented state as it is given in the following lemma. Lemma 4 If there exist a continuous function V : R nτ+1 R two K - functions α 1 α 2 two positive constants α 3 σ such that i α 1 ξ V ξ α 2 ξ ξ R nτ+1 ii V F ξw V ξ α 3 ξ 2 + σ w 2 ξ R nτ+1 w R s ξ = ξ ξ1... ξτ ξi R n F ξw = ξ1... ξτ f ξ τ ξ w then system 11 is ISS. Proof. Let us consider the augmented system applied with an arbitrary initial state ξ R nτ+1 an arbitrary τ + 1-tuple of disturbances {w... wτ}: ξ + 1 = F ξ w = 1... τ. 13 ξ = ξ By property ii we have [V F ξ w V ξ] = = V F ξτ wτ V ξ α 3 ξ 2 + σ w = = Observe that for =... τ the first n elements of ξ denoted by ξ equals to ξ F ξτ wτ = fξ τ ξ w... In this way a mapping... fξ τ τ ξ τ wτ. F : R nτ+1 R sτ+1 R nτ+1 can be defined so that for any ζ R nτ+1 η = η... η τ R sτ+1 the result of the recursion 13 for ξ = ζ w = η is taen Fζ η is defined by Fζ η = F ξτ wτ. From 14 it follows that V Fζ η V ζ α 3 ζ 2 + σ η 2 thus V is an ISS-Lyapunov function in the sense of Definition 3.2 in [9] for the discrete-time system ζ + 1 = Fζ η ζ = ζ. 15 herefore Lemma 3.5 of [9] gives that 15 is ISS. Since ζ j = xτ j τ η. τ + 1 w. the required inequality 12 immediately follows. 3 Main results In this section we propose a guaranteeing cost robust minimax strategy for system 1-2. Firstly a necessary a sufficient condition will be established for the controller 4-6 to be a guaranteeing cost robust minimax strategy for system 1-2. ISSN: Issue 7 Volume 7 July 28

5 3.1 A necessary sufficient condition Set Ĝ = I G G 1 Introduce the following notations: A 1 = A + HĜG A q A d1 = A d + HĜG A dq. Ĝ = I GG 1. heorem 5 he controller 4-6 is a guaranteeing cost output feedbac for 1-2 with 3 if only if there exist positive definite matrices P P d a positive constant ε such that Ψ11 Ψ = <. 16 Ψ 21 Ψ 22 Ψ 11 = Ψ 21 = Ψ 22 = diag Proof. Set P d P P d S A 1 A d1 E P 1 A q A dq H Γ K K d { 1 ε 2 Ĝ 1 ε 2 Ĝ 1 Q 1 R 1}. P = P d P + ΓQΓ + KRK K d = K d RK d P d. By simple substitution from 7 we obtain that 1 can equivalently be written as > V z +1 V z + Lz w = z+1 P z +1 + z P d P z + + z z z τ Q z τ z τ P dz τ w Sw = z z τ w A + δa A d + δa d P E A + δa A d + δa d E + P d P + P d + S ΓQΓ + KRK KRK d K d RK K d RKd z z τ w 17 i.e. the matrix in the square bracets is negative definite. By Schur complement this holds true if only if P K d RK Kd S A + δa A d + δa d E P 1 <. 18 Substituting the definition of δa δa d we obtain that > P K d RK Kd S A A d E P 1 H A q A dq + A q A dq + H. 19 ISSN: Issue 7 Volume 7 July 28

6 Applying Lemma 2.6. of [17] 19 holds if only if there is a positive constant ε such that > 1 ε 2 P K d RK Kd S A A d E P 1 A q A dq + Ĝ A q A dq HĜG A q A dq ε 2 HĜH. 2 By taing into account the definition of A 1 A d1 inequality 2 is obviously equivalent to > P d P P d S A 1 A d1 E P 1 A q Γ K A dq Kd H + 1 Ĝ ε 2 ε 2 Ĝ Q R from which 16 can be obtained applying the Schur complement again. Remar 6 If the delay τ is unnown but constant then 4-6 is not well-defined V depends also on the unnown parameter τ. However if a memoryless dynamic feedbac is applied i.e. Â d = K d = are chosen then the proposed approach is applicable theorem 5 remains valid since 16 doesn t depend on the delay though the above restriction may give more conservative results. Corollary 7 If 4-6 is a guaranteeing cost output feedbac controller for 1-2 with 3 then 7 is ISS sup J z z w V z. 21 w l 2 Proof. Under the condition of the corollary the strict inequality 16 holds true. But then there exists a µ > so that 16 remains valid if µ Γ Γ is added to the right h side of the inequality Γ = I. Let us denote a corresponding modification of L z by L z z w = = z z τ w Sw. Q µi z z τ Going bacward along the equivalent relations from the modified inequality it follows that V z +1 V z + L z z w < 22 holds true for all N for all disturbance sequences w for any realization of the uncertainty. hen 22 involves that the conditions of Lemma 4 are satisfied with α 1 s = min{λ m P λ m P d }s 2 α 2 s = max{λ M P λ M P d }s 2 α 3 = µ σ = λ M S thus the ISS property follows from Lemma 4. On the other h summing up 1 for =... N we have that Since N L z z w V z. = w Sw is finite for any w l 2 = J z z w is finite as well 21 holds true. Remar 8 Since J z z w Jx u w are identical if u is generated by 4-6 corollary 6 shows that the guaranteeing cost output feedbac controller yields a cost bounded by V z = Ṽ x independently from the disturbance w l 2 the uncertainty. We note that one can allow any bounded disturbance sequence to have the ISS property. At the same time in the investigation of the cost function over an infinite horizon one has to restrict the attention to output feedbac controllers 4-6 which ensure that the objective functional is well defined for a class of admissible disturbances in the sense that the cost value belongs to R {+ } { }. For any w l 2 the corresponding cost value J z z w R {+ } { } thus l 2 is a suitable class of admissible disturbances. Remar 9 If τ is unnown then V z in 21 cannot be computed. Nevertheless if < τ τ with given τ we have that V z V z := z P z + z ip d z i i=1 V z is a computable upper bound for the cost function. ISSN: Issue 7 Volume 7 July 28

7 3.2 Derivation of an LMI Matrix inequality 16 is clearly nonlinear. On the basis of the approach of [6] an LMI will be shown the solution of which is also a solution of 16. his means that an LMI can be given for the unnowns K K d L Â Â d P P d which yields a sufficient condition for 4-6 to be a guaranteeing cost output feedbac controller. his is established by the next theorem. o formulate it introduce further notation as follows. Set Π = P 1 P d P 1 partition matrices P P 1 into n n blocs as X M P = M Z Introduce matrices X I F 1 = M WSEAS RANSACIONS on SYSEMS Π = F 1 ΠF 1. Set furthermore P 1 Y N = N W I Y F 2 = N A 1 = A + H x I G x G x 1 G x A q A d1 = A d + H x I G x G x 1 G x A dq B 1 = B + H x I G x G x 1 G x B q C 1 = C + H y I G y G y 1 G y C q C d1 = C d + H y I G y G y 1 G y C dq. K = KN Kd = K d N L = M L 23 Ã = XA 1 Y + XB 1 K + LC1 Y +MÂN 24 Ã d = XA d1 Y + XB 1 Kd + LC d1 Y +MÂdN. 25 heorem 1 Let ε > be a fixed positive constant. If the LMI Ω 11 Ω = Ω 21 Ω 22 Ω 32 Ω 33 < 26 Ω 41 Ω 44 holds for X Y Π K K d L Ã Ãd Ω 11 = Π X I I Y Π Ω 22 = Ω 33 = ε 2 Ĝ 1 Ω 44 = Ω 21 = S E X E XE X I E I Y ε 2 Ĝ 1 Q 1 R 1 ψ 21 Ã ψ 23 Ã d A 1 ψ 32 A d1 ψ 34 A q ψ 42 A dq ψ 44 C q C q Y C dq C dq Y ψ 21 = XA 1 + LC 1 ψ 23 = XA d1 + LC d1 ψ 32 = A 1 Y + B 1 K ψ34 = A d1 Y + B 1 Kd ψ 42 = A q Y + B q K ψ44 = A dq Y + B q Kd Ω 32 = H x X Hx Hy L I Y Ω 41 = K K d then a guaranteeing cost output feedbac controller for 1-2 with 3 can be expressed in the form of 4-6 by solving Proof. Fix ε in 16. Apply the congruence transformation diag{p 1 P 1 I I I I I I } to 16 then multiply the obtained inequality by diag{f 1 F 1 I F 1 I I I I } from the right by its transpose from the left. Let us compute the blocs of Ω. Bloc Ω 11 can immediately be obtained by observing that F 1 P 1 F 1 = X I I Y. 27 Bloc Ω 22 is received from the third to the fifth rows columns of Ψ in 16: S Ω 22 = F1 E F1 P 1 F 1. ε 2 I GG ISSN: Issue 7 Volume 7 July 28

8 Here F 1 E = XE E thus by 27 bloc Ω 22 is verified. Since the last three bloc rows columns are multiplied by diag{i I I} Ω 33 Ω 44 are obviously valid. o compute Ω 21 we observe that P 1 F 1 = F 2. hus Ω 21 = F1 A 1F 2 F1 A d1f 2. A q F 2 A dq F 2 By substitution we obtain that F1 X M A 1 F 2 = I A1 B 1 K LC 1  = analogously ψ21 à A 1 ψ 32 F 1 A d1 F 2 = A q F 2 = A dq F 2 = ψ23 à d I Y N A d1 ψ 34 Aq ψ 42 C q C q Y Adq ψ 44 C dq Y C dq which gives the required expression of Ω 21. Furthermore Ω 32 = H F 1 H F 1 = H x X Hx Hy L thus Ω 32 is also verified. Finally Γ F2 Ω 41 = K K d F 2 I Y = K K d because I F2 = I Y K F2 = K. hus for a fixed ε > 16 follows from the LMI 26. As far as the solvability of is concerned we observe that from 26 it follows that X I > I Y thus I XY is invertible. Because of the definition MN = I XY therefore M N are also invertible they can be determined e.g. by singular value factorization of I XY. hus the system of matrix equations can subsequently be solved for the required matrices K K d L  Âd. Remar 11 Observe that matrices P P d are simultaneously determined here by the LMI 26 fixing ε. Several papers see e.g. [11] proposed an LMI method for systems without exogenous disturbances ε was set to 1 P d was also fixed. Remar 12 We note that in the case of unnown delay the dynamic output feedbac 4-6 had to be considered with Âd = K d =. However in the derivation of 26 it was crucial to introduce à d defined by equation 25 as a new unnown. his equation may not hold if Âd = is fixed. herefore the approach of this paper is not suitable to reduce the construction of a guaranteeing cost dynamic output feedbac controller to the solution of an LMI if the delay is not given. Remar 13 From the proof of heorem 1 one can see that inequality 16 can be transformed into a bilinear one in the variables K K d L P P d ε. However the solution of BMIs requires more sophisticated tools see e.g. [12]. heorem 1 provides a constructive method to find an appropriate control. here are efficient methods software tools to find a feasible solution. Since the main purpose is to find a guaranteed cost it is expedient to assign an objective function to the LMI which assures as low guaranteed cost as possible. Partition Π into n n blocs as follows: Π11 Π12 Π = Π. 12 Π 22 he upper bound of the system s performance is given by aing into account the initial value of x an upper bound U b for the guaranteed cost value ISSN: Issue 7 Volume 7 July 28

9 can be given by the upper left blocs of P P d which are X Π 11 respectively. hus U b = λ M X x 2 + λ M Π 11 x j 2. j=1 o obtain a relatively low value of U b we consider two additional variables ω 1 ω 2 add the LMIs ω 1 I > X ω 2 I > Π 11 to LMI 26 minimize the objective function θω θω 2 with respect to the resulted in new LMI system θ 1 is a given constant. o solve the problem with a fixed nonzero initial state it can be chosen e.g. as θ = x / x x τ. 4 A numerical example Consider the dynamical system 1-3 with the following parameters: let τ = A = A 1.2 d = B = E =.1.1 C = 1 C d = H x = H.2 y =.1 A q = B q = φ = A dq = 1 1 = 1... τ C q =.5.4 C dq =.1.2 G x =.4 G y = consider the following matrices in the performance index: Q =.1I 2 R =.1 S = 1. Solving 26 for ε = 1 with the proposed objective function one obtains  =  d = L = K = K d = ω 1 = ω 2 = P = P d = λ M P = λ M P d = the guaranteed cost with the given initial states is he state responses of above system with x = I y = I are given in figure 1. States Figure 1: State responses of the closed-loop system. 5 Conclusions his paper dealt with the construction of guaranteeing cost output feedbac controller for linear uncertain ISSN: Issue 7 Volume 7 July 28

10 discrete-time time-delay systems. he uncertainty involved time-varying parameter uncertainties of linearfractional form external disturbances. A necessary sufficient condition for a controller to be a guaranteeing cost output feedbac was formulated in terms of a nonlinear matrix inequality. A sufficient condition in terms of a linear matrix inequality was also given which could effectively be solved. A numerical example illustrated the proposed method. References [1] F. Amato M. Mattei A. Pironti: Guaranteeing cost strategies for linear quadratic differential games under uncertain dynamics Automatica pp [2] E. K. Bouas: Discrete-time systems with time-varying time-delay: stability stabilizability Mathematical Problems in Engineering 26 Art ID: pages 26. dvi: 11155/MPE/26/42489 [3] Y. Ch. Chang K.. Lee: Simultaneous Static Output-Feedbac Stabilization for Interval Systems via LMI Approach WSEAS ransactions on Systems 6 27 pp [4] W. H. Chen Z. H. Guan X. Lu: Delaydependent guaranteed cost control for uncertain discrete-time systems with both state input delays Journal of the Franlin Institute pp [5] E. Fridman U. Shaed: Stability guaranteed cost control of uncertain discrete-delay systems International Journal of Control pp [6] P. Gahinet P. Aparian: A linear matrix inequality approach to the H control problem International Journal of Control pp [7] É. Gyurovics. aács: Guaranteeing cost strategies for infinite horizon difference games with uncertain dynamics International Journal of Control pp [8] É. Gyurovics. aács: wo approaches of the construction of guaranteeing cost minimax strategies WSEAS ransactions on Systems 5 26 pp [9] Z.-P. Jiang Y. Wang: Input-to state stability for discrete-time nonlinear systems Automatica pp [1] M. M. Kogan: Robust H suboptimal guaranteed cost state feedbac as solutions to linearquadratic dynamic games under uncertainty International Journal of Control 73 2 pp [11] O. M. Kwon J. H. Par S. M. Lee S. C. Won: LMI optimization approach to observerbased controller design of uncertain time-delay options via delayed feedbac Journal of Optimization heory Applications pp [12] S. Molnár F. Szidarovszy: Introduction to Matrix heory with Application to Business Economics World Scientific Publishing New Jersey London Singapore 22. [13] J. H. Par: On dynamic output feedbac guaranteed cost control of uncertain discrete-time systems: LMI optimization approach Journal of Optimization heory Applications pp [14] I. R. Petersen: Optimal guaranteed cost of discrete-time uncertain linear systems International Journal of Robust Nonlinear Control pp [15] P. Shi E. K. Bouas Y. Shi R. K. Agarwal: Optimal guaranteed cost control of uncertain discrete time-delay systems Journal of Computational Applied Mathematics pp [16] H. Wang Ch. Vasseur A. Chamrov V. Koncar: Sampled tracing for linear delayed plants using piecewise functioning controller WSEAS ransactions on Systems Control 2 27 pp [17] L. Xie: Output feedbac H -control of systems with parameter uncertainty International Journal of Control pp [18] N. Xie G.-Y. ang: Delay-dependent nonfragile guaranteed cost control for nonlinear time-delay systems Nonlinear Analysis pp [19] S. Xu. Chen: Robust H -control for uncertain discrete-time systems with time varying delays via exponential output feedbac controllers Systems Control Letters pp [2] S. Xu I. Lu S. Zhou C. Yang: Design of observers for a class of discrete-time uncertain nonlinear systems with time delay Journal of the Franlin Institute pp [21] L. Yu F. Gao: Optimal guaranteed cost control of discrete-time uncertain systems with both state input delays Journal of the Franlin Institute pp [22] L. Yu F. Gao: Output feedbac guaranteed cost control for uncertain discrete-time systems using linear matrix inequalities Journal of Optimization heory Applications pp ISSN: Issue 7 Volume 7 July 28