Research Article Delay-Dependent H Control for Networked Control Systems with Large Delays

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1 Mathematical Problems in Engineering Article ID pages Research Article Delay-Dependent H Control for etworked Control Systems with Large Delays Yilin Wang Hamid Reza Karimi 2 and Zhengrong Xiang School of Automation anjing University of Science and Technology anjing 294 China 2 Department of Engineering Faculty of Engineering and Science University of Agder 4898 Grimstad orway Correspondence should be addressed to Zhengrong Xiang; xiangzr@mail.njust.edu.cn Received 22 January 23; Accepted 5 March 23 Academic Editor: Xiaojie Su Copyright 23 Yilin Wang et al. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. We consider the problems of robust stability and H control for a class of networked control systems with long-time delays. Firstly a nonlinear discrete time model with mode-dependent time delays is proposed by converting the uncertainty of time delay into the uncertainty of parameter matrices. We consider a probabilistic case the system is switched among different subsystems and the probability of each subsystem being active is defined as its occurrence probability. For a switched system with a known subsystem occurrence probabilities we give a stochastic stability criterion in terms of linear matrix inequalities (LMIs). Then we extend the results to a more practical case the subsystem occurrence probabilities are uncertain. Finally a simulation example is presented to show the efficacy of the proposed method.. Introduction etworked control systems (CSs) are distributed systems in which communication between sensors actuators and controllers is supported by a shared real-time network. Compared with conventional point-to-point system connection this new networked based control scheme reduces system wiring low cost high reliability information sharing and remote control [ 2. evertheless the introduction of communication network also brings some new problems and challenges such as time-delay packet dropout quantization and band-limited channel [3 which all might be potential sources to poor performance even instability. etwork-induced delay is one of the main problems in CS [2 3 and has attracted much research interest. Compared with the constant delays the random or time-varying ones are more difficult to be dealt with especially when the delay is larger than one sampling period (long-time delays). Some important methods on dealing with time delay which were investigated [4 have shown that the input-output technique is an effective way to deal with time delay; the delay partitioning approach is also advanced to deal with the time-varying delay [5 6. Reference [7 applies for the dynamic output feedback controller design of discrete-time systems with time-varying delays. Sliding mode control (SMC) is an effective robust control strategy to deal with the time-delay systems [8 9. In this paper the uncertainties of the delays are transformed into those of the system models in the uncertain system approach. As has been mentioned above it is difficult to deal with the CS with long time-varying or random delays and one aspect of the difficulties lies in providing an appropriate modeling method for such CSs. Since the delay may be larger than one sampling period more than one control signals may arrive at the actuator during one sampling interval. Moreover the numbers of the arriving control signals vary over different sampling intervals; thus the dynamic model of the overall closed-loop CS varies from sampling period to sampling period. Reference [2 deals with the problem of stability and stabilization controller design for CS with long timevarying delays. The CS with time-varying delays is modeled as a discrete-time switched system with multiple state delays and with both stable subsystems and unstable subsystems [5.

2 2 Mathematical Problems in Engineering In most models CSs are modeled as Markov jump linear systems (MJLSs). In an MJLS the subsystem occurrence probability is called the stationary probability distribution of the Markovian states. According to the known transition matrix for an MJLS we can readily infer the subsystem occurrence probability for each of the involved subsystems [ Reference [25 focuses on studying the MJLS with partly unknown transition probabilities due to the complexity of network. Then the MJLS can be converted to a switched system with known subsystem occurrence probabilities. Compared with the MJLS model the advantage of the switched system model in this paper mainly lies in that one does not need to identify the transition probability from each mode to another. Practically there is an uncertainty on the information of the subsystem occurrence probability. However most of the current works seldomly consider the uncertain environment ofthecsandthefeatureofstochasticlongtime-delay CSs. With the motivation of the above reasons it is natural to consider the stability and controller design problems for CS with stochastic long delays and less conservative results can be achieved by incorporating the available information of the occurrence probabilities even when there are uncertainties on the information. In this paper we are interested in investigating the problems of the robust stochastic stability and H performance analysis for a class of CSs with a continuous-time nonlinear plant and the delay larger than the sampling period maybeanarbitraryvalueinafiniteinterval;adiscretetime stochastic switched CS model is proposed. A modedependent state feedback controller is designed by using a cone complementary linearization approach to ensure that closed-loop system is stochastically stable and achieves the disturbance attenuation level. The remainder of the paper is organized as follows. In Section 2 we model the CS as a discrete-time switched system by considering the subsystem occurrence probabilities. Further the definition of the stochastic stability is introduced. The stochastic stability for switched systems with known and unknown subsystem occurrence probabilities is analyzed by the LMIs technology and state feedback controller is designed to stabilize the CS in Section 3. Simulation results are given in Section 4 to verify the proposed scheme. Finally Section5 concludes the paper. otation. Throughout the paper the superscripts and T stand for the inverse and transpose of a matrix respectively; R n denotes the n-dimensional Euclidean space and the notation P>means that P is real symmetric positive definite matrix. E{x} is theexpectationof thestochastic variable x. I and represent identity matrix and zero matrix with appropriate dimensions in different place. In symmetric block matrices or complex matrix expressions we use an asterisk to represent a term that is induced by symmetry and diag{ } stands for a block diagonal matrix. refers to the Euclidean norm for vectors and induced 2-norm for matrix. λ min (Q) and λ max (Q) are defined as the minimum and maximum eigenvalue of Q respectively. The set of all positive integers is represented by Z Model for etworked Control System The plant is a continuous-time system described by x (t) =A p x (t) +B p u (t) +f(x t) +H p w (t) z (t) =Cx(t) x(t) R n u(t) R m z(t) R q are the state vector control input vector and controlled output vector respectively w(t) R d is the exogenous disturbance signal belonging to L 2 [ ). A p B p H p andc are known real matrices with appropriate dimensions. f:ω [t ) R n (Ω R n ) is the nonlinear function vector f( t )=. f satisfies the local Lipschitz condition that is f(x t) f(x 2 t) 2 α x x 2 2 x x 2 Ω R n t [t ) α >is a known constant. In the considered CS time delays exist in both channels from sensor to controller and from controller to actuator. Sensor-to-controller delay and controller-to-actuator delay are denoted by τ sc and τ ca respectively. The assumptions in the above CS are as follows. () (2) () The discrete-time state-feedback controller and the actuator are event-driven and the sensor is timedriven with sampling period T. (2) The network-induced delay τ k τ sc k +τca k satisfies T τ k <Tduring the kth sampling period is a given positive number and 2. (3) In order to improve the real-time ability of CS the data packet will be discarded and be held at previous valueoncethetimedelayofthisdatapacketislonger than T. Define the indicator function with...and ξ (k) =[ξ 2 (k)...ξ l (k)...ξ (k) T (3) ξ l (k) = { the lth subsystem is active { the lth subsystem is not active. { Remark. At any time instant k ifthelongdelayτ k [(i )T it) thenwecangetξ i = ξ j =j=i i j = 2...and i=2 ξ i = ;thatiswecaninstantaneously identify which subsystem is active at any time instant k so we can define the switching rule as follows σ(k) : Z + = {2...} l={23...}. (4)

3 Mathematical Problems in Engineering 3 Then the system () canbewrittenintodiscretetime modelduringtheinterval[kt (k + )T) k is a nonnegative integer we get x (k+) =Ax(k) + ξ l [B l (τ k )u(k l+) +B l (τ k )u(k l) + f (x k) +Hw(k) z (k) =Cx(k) A = e ApT B l (τ k ) = lt τ k e Aps dsb p B l (τ k ) = T e Aps dsb lt τ k p H= T eaps dsh p and T f (x k) = e Aps dsf (x k). (6) According to Assumption 2 τ k [TT) is stochastic variable; therefore B l (τ k ) and B l (τ k ) arealsostochastic matrices. Define a scalar τ k and the parameters in (6) could be transformed as follows: lt τ k B l (τ k )= e Aps dsb p lt τ k = T B l (τ k )= e Aps dsb p lt τ k T = lt τ k e Aps dsb p +e A p(lt τ k ) τ k τ k e Aps dsb p e Aps dsb p e A p(lt τ k ) τ k τ k e Aps dsb p. For convenience of the following proof we can define Γ l = lt τ k e Aps dsb p Γ l = T e Aps dsb lt τ p G l =e A p(lt τ k ) k F(τ k )= τ k τ k e Aps ds ande=b p F T (τ k )F(τ k ) I l l. Then we get (5) (7) B l (τ k )=Γ l +G l F(τ k )E (8) B l (τ k )=Γ l G l F(τ k )E. The discrete-time state feedback controller is u (k) =K(k) x (k) (9) x(k) R n and u(k) R m are the value of x(t) and u(t) at the sampling instant kt respectively and state feedback gain K(k) corresponds to the subsystem at the sampling instantkt to be considered. Consider the plant input: (k) u (k)={ u u () if u (k) and x (k) is successfully transmitted if u (k) or x (k) is lost during transmission. () Then the resulting closed-loop CS is shown as follows: x (k+) =Ax(k) + ξ l [B l (τ k )K l x (k l+) +B l (τ k )K l x (k l) + f (x k) +Hw(k). Let x(k) = [ x(k) x() ()leadsto x (k+) = A x (k) + + f (x k) + Hw (k) z (k) = C x (k) ξ l B l K l x (k l+) A =[ A I K l =[ K l K l f (x k) f (x k) =[ H =[ H C =[C B l B l =[ B l =[Γ l Γ l +[G l F(τ k )[E E = Γ l + G l F(τ k ) E. From (2) (6) the nonlinear f(x k) satisfies () (2) (3) f T (x k) f (x k) = f T (x k) f (x k) x T (k) U T U x (k) (4) with U=[ α λ max(π)i is a known constant positive-definite matrix Π=( T eaps ds) T T eaps ds. ow for each subsystem we define the subsystem occurrence probability β l. Subsequently we have a priori information about ξ l foralll l.asdiscussedpreviouslythe probability of ξ l =is represented by β l thatis Prob {ξ l =}=β l Prob {ξ l =} = β l. (5) We consider two cases: () the value of β l is precisely known; (2) β l is subject to an uncertainty. A general form for ξ l is depicted as follows Prob {ξ l =}=β l [β l β 2l (6) β l and β 2l are two constants satisfying β l β 2l.

4 4 Mathematical Problems in Engineering Lemma 2 (see [26). The stochastic stability in discrete-time implies the stochastic stability in continuous time. Lemma 3 (see [27 (Schur complement)). For a given matrix S = [ S S 2 S S T 2 S S 22 are square matrixes then the 22 following conditions are equivalent: () S<; (2) S < S 22 S T 2 S S 2 <; (3) S 22 < S S 2 S 22 ST 2 <. Lemma 4 (see [28). Let D E and F be matrices with appropriate dimensions. If F T F Ithenforanyscalarε> one has DFE + E T F T D T εdd T +ε E T E. (7) Lemma 5 (see [29). Assume that i l = {23...}and is a given positive number and 2thenforanypositivedefinite matrix P R n n onehas ( ) ( s=k + x T (s) Px (s) x T (s))p( x (s)) k Z +. (8) Definition 6 (see [3). The system (2)withw(k) is said to be stochastically stable if for every finite x = x()andthe following inequality holds: E{ k= x (k) 2 x }<. (9) Definition 7 (see [3). The closed-loop system (2)is robustly stable with H performance γ if there exists a state feed-back controller u(k) = K(k)x(k) and the following conditions are satisfied. (a) The closed-loop system (2) with ω(k) is stochastically stable. (b) Under the zero-initial condition it holds that k= E{z T (k) z (k)} <γ 2 E{w T (k) w (k)} w(k) L 2 [ ). k= 3. Main Results (2) 3.. Stochastic Stability Analysis. With Lemma 2the stability of system () can be converted into the stability of system (2). Then a sufficient condition for stochastic stability of system (2)withw(k) = isgiveninthefollowingtheorems Stability with the Known Occurrence Probability Theorem 8. Suppose that the occurrence probability for each subsystem is known. The networked control system (2) is stochastically stable if there exist positive definite matrices P i R 2n 2n Q R 2n 2n R R 2n 2n and S i R 2n 2n i lwith appropriate dimensions satisfying the following LMI: Φ Φ 2 Φ 3 [ Φ 22 < ij l (2) [ Φ 33 Ω i Si T Si T Q d Φ = Q d [ Q I [ R β 2 A T P j β i A T P j β A T P j β 2 ( B 2 K 2 ) T P j Φ 2 = β i ( B i K i ) T P j β ( B K ) T P j [ β 2 P j β i P j β P j [ β 2 ( A I) T R β i ( A I) T R β ( A I) T R β 2 ( B 2 K 2 ) T R Φ 3 = β i ( B i K i ) T R. β ( B K ) T R [ β 2 R β i R β R [ Φ 22 = diag {P j...p j...p j } Φ 33 = diag {R...R...R} 2 ( ) Ω i = P i +S i +S T i +U T U+( ) Q. (22) Proof. Construct Lyapunov function candidates for closedloop system (2) as follows: V ( x (k)) =V ( x (k)) +V 2 ( x (k)) + ( ) V 3 ( x (k)) (23)

5 Mathematical Problems in Engineering 5 V ( x (k)) = x T (k) P σ(k) x (k) V 2 ( x (k)) = V 3 ( x (k)) = θ= θ= s=k+θ+ s=k+θ+ x T (s) Q x (s) e T (s) Re (s). (24) P σ(k) > Q>andR>are matrices to be determined. The new variable e(k) satisfies the following equation: e (k) = x (k+) x (k). (25) Thenwe can furtherwrite (25) in the equivalent descriptor form e (k) = ( A I) x (k) + ξ l B l K l x (k l+) + f (x k). (26) Assume that the ith and jth modes are active at times k and k+respectively.thatisτ k [(i )T it) and τ k+ [(j )T jt) for any i j l.alongthesolutionofthesystem (2)withw(k) = and using Lemma 5wehave ΔV ( x (k)) =E[V ( x (k+)) V ( x (k)) =E[ x T (k+) P j x (k+) x T (k) P i x (k) =E[ A x (k) + ΔV 2 ( x (k)) P j [ A x (k) + x T (k) P i x (k) ξ l B l K l x (k l+) + f (x k) ξ l B l K l x (k l+) + f (x k) +2 x T (k) S T i [ x (k) x (k i+) =E(V 2 ( x (k+))) V 2 ( x (k)) = θ= x T (k) Q x (k) ( ) x T (k) Q x (k) s=k + s=k + x T (s) Q x (s) x T (s) Q x (s) T e (s) ΔV 3 ( x (k)) =E(V 3 ( x (k+))) V 3 ( x (k)) = θ= e T (k) Re (k) ( ) e T (k) Re (k) ( s=k + e T (s))r( e T (s) Re (s) e (s)). (27) Denote V(k) = e(s) foralli land combine (27) with (4) then we can obtain ΔV ( x (k)) =ΔV ( x (k)) +ΔV 2 ( x (k)) + ( ) ΔV 3 ( x (k)) ΔV ( x (k)) +ΔV 2 ( x (k)) + ( ) ΔV 3 ( x (k)) + x T (k) U T U x (k) f T (x k) f (x k) =η T (k) Φ (i j) η (k) (28) η T (k) = [ x T (k) x T (k ) x T (k i+) x T (k +) f T (x k) V T (k)and with Φ (i j) =Φ + Φ 2l =[( A I) Φ l =[ A l l β l Φ T l P j Φ l + ( ) 2 β l Φ T 2l R Φ 2l (29) B l K l B l K l l l I I ijl l. (3) By Schur complement lemma the inequality (2) guarantees Φ(i j) <. Thus with the above relations the inequality (28) can be rewritten as follows: ΔV ( x (k)) = E {V ( x (k+) σ(k+) = j)} V( x (k) σ(k) =i) λ min ( Φ (i j)) x (k) 2 = μ x (k) 2. From the previous inequalities we can obtain E{V( x (k+)) σ(k+) =j} V( x () σ()) μe{ k k= x (k) 2 } (3) (32)

6 6 Mathematical Problems in Engineering which implies that Proof. By Schur complement lemma (34) implies E { k= x (k) 2 } E {V ( x ())}. (33) μ Φ + β 2l Φ T l P j Φ l + ( ) 2 β 2l Φ T 2l R Φ 2l <. (36) Therefore by Definition 6 it can be obtained that the closed-loop system (2) is stochastically stable.the proof is completed. ote that for positive definite matrices P j and Rand β l β l β 2l Stability with Uncertain Active Probabilities. In Section 3.. we studied the stochastic stability for the switched networked control system with known subsystem occurrence probabilities. As a matter of fact there is an uncertainty on the information of the subsystem occurrence probability. ext the stochastic stability criterion of the closed-loop system (2) with uncertain subsystem occurrence probabilities is given in the following theorem. Theorem 9. Supposethattherangeofoccurrenceprobability for each subsystem is known. The networked control system (2) is stochastically stable if there exist positive definite matrices P i R 2n 2n Q R 2n 2n R R 2n 2n and S i R 2n 2n i l with appropriate dimensions satisfying the following LMI: β l Φ T l P j Φ l β l Φ T 2l R Φ 2l β 2l Φ T l P j Φ l β 2l Φ T 2l R Φ 2l j = l. Thus if the LMI (34)holdsthenweobtain Φ + (37) β l Φ T l P j Φ l + ( ) 2 β l Φ T 2l R Φ 2l <. (38) Φ Φ 2 Φ 3 [ Φ 22 < ij l (34) [ Φ 33 According to Theorem 8 the discrete-time stochastic switched system (2) with uncertain subsystem occurrence probabilities is stochastically stable. This completes the proof. Φ 2 = [ [ Φ 3 = [ [ β 22 A T P j β 2i A T P j β 2 A T P j β 22 ( B 2 K 2 ) T P j β 2i ( B i K i ) T P j β 2 ( B K ) T P j β 22 P j β 2i P j β 2 P j β 22 ( A I) T R β 2i ( A I) T R β 2 ( A I) T R β 22 ( B 2 K 2 ) T R β 2i ( B i K i ) T R β 2 ( B K ) T R β 22 R β 2i R β 2 R. (35) 3.2. H Controller Synthesis with Uncertain Active Probabilities. This section is devoted to synthesizing a controller given in the form u(k) = K i x(k) foralli l that guarantees the closed-loop system is robustly stable with the noise attenuation level γ. Theorem. Suppose that the range of occurrence probability for each subsystem is known. The closed-loop system (2) is stochastically stable and achieves the given disturbance attenuation performance if there exist constants ε i > and positive definite matrices P i R 2n 2n Q R 2n 2n R R 2n 2n X i R 2n 2n Y R 2n 2n and S i R 2n 2n i l with appropriate dimensions and feedback gain matrix K i satisfying matrix inequalities: Θ Θ 2 Θ 3 Θ 4 Θ 22 [ Θ 33 < [ Θ 44 P j X j =I RY=I ij l (39)

7 Mathematical Problems in Engineering 7 i Si T Si T Q d.... Q Θ = d Q I. [ R [ γ 2 I β 22 A T β 2i A T β 2 A T β 22 ( Γ 2 K 2 ) T β 2i ( Γ i K i ) T Θ 2 = β 2 ( Γ K ) T β 22 I β 2i I β 2 I [ [ β 22 H T β 2i H T β 2 H T β 22 ( A I) T β 2i ( A I) T β 2 ( A I) T β 22 ( Γ 2 K 2 ) T. β 2i ( Γ i K i ) T Θ 3 = β 2 ( Γ K ) T β 22 I β 2i I β 2 I [ [ β 22 H T β 2i H T β 2 H T β 22 ( E K 2 ) T β Θ 4 = 2i ( E K i ) T. β 2 ( E K ) T [ [ Θ 22 = diag {δ 2...δ i...δ } δ i = X j +ε i G i G T i Θ 33 = diag {θ 2...θ i...θ } θ i = ( ) 2 Y+ε G i i G T i Θ 44 = diag { ε 2 I... ε i I... ε I} i = P i +S i +S T i + ( ) Q+U T U+ C T C. (4) Proof. From Theorem 8 the closed-loop system (2) with w(k) = is stochastically stable. Assume that the ith and jth modes are activated at instants k and k+respectively.fornonzerow(k) using thesamelyapunovfunctioncandidatesasintheorem 8we have ΔV ( x (k)) =E[V ( x (k+)) V ( x (k)) =E[ x T (k+) P j x (k+) x T (k) P i x (k) =E[( A x (k) + ΔV 2 ( x (k)) + f (x k) + Hw (k) ) P j ( A x (k) + x T (k) P i x (k) ξ l B l K l x (k l+) T ξ l B l K l x (k l+) + f (x k) + Hw (k) ) +2 x T (k) S T i [ x (k) x (k i+) =E(V 2 ( x (k+))) V 2 ( x (k)) = θ= x T (k) Q x (k) ( ) x T (k) Q x (k) ΔV 3 ( x (k)) s=k + s=k + =E(V 3 ( x (k+))) V 3 ( x (k)) = θ= e T (k) Re (k) ( ) e T (k) Re (k) ( s=k + e T (s))r( x T (s) Q x (s) x T (s) Q x (s) e T (s) Re (s) e (s)). e (s) (4)

8 8 Mathematical Problems in Engineering From inequations (4) and combining the nonlinear condition (4) we have From Theorem 8 the following inequality can be obtained: ΔV ( x (k)) =ΔV ( x (k)) +ΔV( x (k)) 2 + ( ) ΔV 3 ( x (k)) ΔV ( x (k)) +ΔV 2 ( x (k)) 2 + ( ) ΔV 3 ( x (k)) + x T (k) U T U x (k) f T (x k) f (x k) +E{z T (k) z (k)} γ 2 E{w T (k) w (k)} E{z T (k) z (k)}+γ 2 E{w T (k) w (k)} ΔV ( x (k)) =E{V ( x (k+))} V( x (k)) E{z T (k) z (k)}+γ 2 E{w T (k) w (k)} + min {λ min ( Θ (i j))} x (k) 2 E{z T (k) z (k)}+γ 2 E{w T (k) w (k)}. (47) =ζ T (k) Θ (i j) ζ (k) E{z T (k) z (k)} +γ 2 E{w T (k) w (k)} (42) ζ T (k) = [ x T (k) x T () x T (k i+) x T (k +) f T (x k) V T (k) w(k)and with Θ (i j) = Θ + Θ l =[ A l Θ 2l =[( A I) β l Θ T l P j Θ l + ( ) 2 β l Θ T 2l R Θ 2l l l. l B l K l B l K l l l I H I H. (43) (44) Using the condition (3)and by Lemmas 3 and 4we can obtain φ l =[ +l+ φ T l FT (τ k )φ l + ε φ l =[ l G T l φ T l F(τ k )φ l l φ T l φ l + ε l φ T l φ l β l E K l G T l 3 l+ l l. (45) (46) Using the similar analysis methods as in Theorem 9 the inequality (39)guaranteesΘ(i j) <. Taking expectation and summing up from k=to on both sides of inequality (47) it can be obtained that the above inequality (47)isequivalentto E [V ( x ( )) V( x ()) which implies that k= k= E{z T (k) z (k)} +γ 2 E{w T (k) w (k)} k= (48) E{z T (k) z (k)} γ 2 E{w T (k) w (k)}. (49) Therefore the closed-loop system (2) is stochastically stable with disturbance attenuation level γ.this completes the proof. Remark. It should be pointed out that the sufficient conditions proposed in Theorem are not standard LMI condition anymore. The subsystem occurrence probabilities are not coupled with the Lyapunov weighting matrix P j and R. In this paper it is suggested to use the cone complementarity linearization (CCL) algorithm in [32 and a nonlinear constraint can be converted to a linear optimization problem with a rank constraint. Remark 2. The CCL algorithm has been used to solve the nonconvex feasibility problems by formulating them into some sequential optimization problems subject to LMI constraints and the work in [33 has used the CCL algorithm to solve the model reduction problem recently. Remark 3. In Theorems 9 and theupperboundofoccurrence probability for each subsystem is used to establish the sufficient conditions. In order to reduce the computation complexity we can employ some constraints of the subsystem occurrence probability to narrow the range of the upper bound. Then by using the CCL algorithm we can obtain the upper bound of subsystem occurrence probability. k=

9 Mathematical Problems in Engineering 9 4. umerical Example In this section we present an example to illustrate the effectiveness of the proposed approach. Consider system () with parameters as follows: x (t) =[ x(t) +[ u(t) +[.4x sin x.4x 2 cos x 2 +[.2. w(t) z (t) =[ x(t). (5) Here thenonlinear function f(x t) satisfies sector condition (2) and we can obtain that α =.4. Choose the sampling period T =.s and suppose. τ k <.3s sowecan know that the upper bound of the long time delays is =3 and the CS can be modeled with two subsystems the corresponding system matrices are obtained as follows: A=[ Γ 2 =[ Γ 2 =[ Γ 3 =[ Γ 3 =[ G =[ G 3 =[ E=[ H=[.9 U=diag {.2.2 }..3 (5) State Switching mode x x 2 Time (s) Figure : State trajectories of the closed-loop system Time (s) Figure 2: The stochastic switching signal. By using iterative algorithm-ccl and solving the constraint conditions in Theorem wecanobtaintheupper bounds of the occurrence probability for the first subsystem and the second subsystem that are β 22 =.4366 and β 23 =.735respectively. Then the controller can be obtained as follows: K 2 =[ (52) K 3 =[ Then the range of the subsystem occurrence probability is β 2 [ and β 3 [ Assume that the actual subsystem occurrence probabilities for the subsystem are β 2 =.4 and β 3 =.6 theinitialstateof the system is x = [ T and w(k) =.5 exp(.k) and the state trajectories of the CS and the corresponding switching signal are shown in Figures and 2respectively. From simulation results it can be seen that the CS is stochastically stable and has H disturbance attenuation level γ =.2. It should be pointed that the methods proposed in the literature [22 25cannotbeusedtodealwiththeH control of the given CS because of the uncertain environment and the feature of stochastic long-time delays. 5. Conclusion In this paper the problems of H control have been studied foraclassofnonlinearcswithlong-timedelays.wefirst model CS as a switched system with a priori information on the subsystem occurrence probabilities. Sufficient condition for the existence of the stochastic stability is established for the case subsystem occurrence probabilities are known. Then the obtained result is generalized to a more practical scenario when the subsystem occurrence probabilities are subject to uncertainties. The controller design method can be used to design a mode-dependent controller such that the closed-loop system is stochastically stable and achieves H disturbance attenuation level. Finally a numerical example is provided to show the correctness and effectiveness of the proposed method. Our further work will focus on extending the proposed method to solve the tracking control problem of CS with communication constraints. Acknowledgment This work was supported by the ational atural Science Foundation of China under Grant nos and

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