# Delay-DependentH∞ Control for Networked Control Systems with Large Delays

Authors: Yilin Wang; Hamid Reza Karimi; Zhengrong Xiang
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

We consider the problems of robust stability andH∞ 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 where 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 where the subsystem occurrence probabilities are uncertain. Finally, a simulation example is presented to show the efficacy of the proposed method.


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## Body

## 1. Introduction

Networked control systems (NCSs) 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 [1, 2]. Nevertheless, the introduction of communication network also brings some new problems and challenges, such as time-delay, packet dropout, quantization, and band-limited channel [3–11], which all might be potential sources to poor performance, even instability.Network-induced delay is one of the main problems in NCS [12, 13] 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, [14] 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 [15, 16]. Reference [17] 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 [18, 19]. 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 NCS with long time-varying or random delays, and one aspect of the difficulties lies in providing an appropriate modeling method for such NCSs. 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 NCS varies from sampling period to sampling period. Reference [21] deals with the problem of stability and stabilization controller design for NCS with long time-varying delays. The NCS 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].In most models, NCSs 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 [20, 22–24]. 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 of the NCS and the feature of stochastic long time-delay NCSs. With the motivation of the above reasons, it is natural to consider the stability and controller design problems for NCS 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 andH∞ performance analysis for a class of NCSs with a continuous-time nonlinear plant, and the delay larger than the sampling period may be an arbitrary value in a finite interval; a discrete-time stochastic switched NCS model is proposed. A mode-dependent 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 Section2, we model the NCS 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 NCS in Section 3. Simulation results are given in Section 4 to verify the proposed scheme. Finally, Section 5 concludes the paper.Notation.Throughout the paper, the superscripts “−1” and “T” stand for the inverse and transpose of a matrix, respectively; Rn denotes the  n-dimensional Euclidean space and the notation P>0 means that  P is real symmetric positive definite matrix. E{x} is the expectation of the stochastic variable x. I and 0 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. λmin(Q) and λmax(Q) are defined as the minimum and maximum eigenvalue of Q, respectively. The set of all positive integers is represented by Z+.


## 2. Model for Networked Control System

The plant is a continuous-time system described by(1)x˙(t)=Apx(t)+Bpu(t)+f(x,t)+Hpw(t),z(t)=Cx(t),
where x(t)∈Rn, u(t)∈Rm, z(t)∈Rq are the state vector, control input vector, and controlled output vector, respectively, w(t)∈Rd is the exogenous disturbance signal belonging to L2[0,∞). Ap, Bp, Hp, and C are known real matrices with appropriate dimensions. f:Ω×[t0,∞)→Rn(Ω⊂Rn)  is the nonlinear function vector, f(0,t0)=0. f  satisfies the local Lipschitz condition, that is,
(2)∥f(x1,t)-f(x2,t)∥2≤α∥x1-x2∥2,∀x1,x2∈Ω⊂Rn,∀t∈[t0,∞),
where α>0 is a known constant.In the considered NCS, 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 byτsc and τca, respectively. The assumptions in the above NCS are as follows.(1)
The discrete-time state-feedback controller and the actuator are event-driven, and the sensor is time-driven with sampling periodT.(2)
The network-induced delayτk≜τksc+τkca  satisfies T≤τk<NT during the kth sampling period, where N is a given positive number and N≥2.(3)
In order to improve the real-time ability of NCS, the data packet will be discarded and be held at previous value once the time delay of this data packet is longer thanNT.Define the indicator function(3)ξ(k)=[ξ2(k),…,ξl(k),…,ξN(k)]T
with l=2,…,N and
(4)ξl(k)={1,thelthsubsystemisactive,0,thelthsubsystemisnotactive.Remark 1.
At any time instantk, if the long delay τk∈[(i-1)T,iT), then we can get ξi=1, ξj=0, where j≠i, i,j=2,…,N, and ∑i=2Nξi=1; that is, we can instantaneously identify which subsystem is active at any time instant k, so we can define the switching rule as follows σ(k):ℤ+={0,1,2,…}→ℓ={2,3,…,N}.
Then, the system (1) can be written into discrete time model during the interval [kT,(k+1)T), where k is a nonnegative integer, we get
(5)x(k+1)=Ax(k)+∑l=2Nξl[Bl0(τk)u(k-l+1)+Bl1(τk)u(k-l)]+f-(x,k)+Hw(k),z(k)=Cx(k),
where A=eApT, Bl0(τk)=∫0lT-τkeApsdsBp, Bl1(τk)=∫lT-τkTeApsdsBp, H=∫0TeApsdsHp, and
(6)f-(x,k)=∫0TeApsdsf(x,k).
According to Assumption 2,τk∈[T,NT) is stochastic variable; therefore, Bl0(τk) and  Bl0(τk) are also stochastic matrices. Define a scalar  τk′, and the parameters in (6) could be transformed as follows:
(7)Bl0(τk)=∫0lT-τkeApsdsBp=∫0lT-τk′eApsdsBp+eAp(lT-τk′)∫0τk′-τkeApsdsBp,Bl1(τk)=∫lT-τkTeApsdsBp=∫lT-τk′TeApsdsBp-eAp(lT-τk′)∫0τk′-τkeApsdsBp.
For convenience of the following proof, we can defineΓl0=∫0lT-τk′eApsdsBp,  Γl1=∫lT-τk′TeApsdsBp,  Gl=eAp(lT-τk′),  F(τk′)=∫0τk′-τkeApsds, and E=Bp, where FT(τk′)F(τk′)≤I,  l∈ℓ. Then, we get
(8)Bl0(τk)=Γl0+GlF(τk′)E,Bl1(τk)=Γl1-GlF(τk′)E.
The discrete-time state feedback controller is(9)u(k)=K(k)x(k),
where x(k)∈Rn and u(k)∈Rm 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 instant kT to be considered.
Consider the plant input:(10)u(k)={u^(k),ifu^(k)andx(k)issuccessfullytransmitted,u(k-1),ifu^(k)orx(k)islostduringtransmission.
Then the resulting closed-loop NCS is shown as follows:(11)x(k+1)=Ax(k)+∑l=2Nξl[Bl0(τk)Klx(k-l+1)+Bl1(τk)Klx(k-l)]+f-(x,k)+Hw(k).
Let  x~(k)=[x(k)x(k-1)], (11) leads to
(12)x~(k+1)=A~x~(k)+∑l=2NξlB~lK~lx~(k-l+1)+f~(x,k)+H~w(k),z(k)=C~x~(k),
where
(13)A~=[A0I0],K~l=[Kl00Kl],f~(x,k)=[f-(x,k)0],H~=[H0],C~=[C0],B~l=[Bl0Bl100]=[Γl0Γl100]+[Gl0]F(τk′)[E-E]=Γ~l+G~lF(τk′)E~.
From (2)–(6), the nonlinear f~(x,k)  satisfies
(14)f~T(x,k)f~(x,k)=f-T(x,k)f-(x,k)≤x~T(k)UTUx~(k)
with U=[αλmax(Π)I000] is a known constant positive-definite matrix, where Π=(∫0TeApsds)T∫0TeApsds.
Now, for each subsystem, we define the subsystem occurrence probabilityβl. Subsequently, we have a priori information about ξl, for all l∈ℓ. As discussed previously, the probability of  ξl=1 is represented by βl, that is,
(15)Prob{ξl=1}=βl,Prob{ξl=0}=1-βl.
We consider two cases:(1) the value of βl is precisely known; (2)  βl is subject to an uncertainty. A general form for ξl is depicted as follows
(16)Prob{ξl=1}=βl∈[β1l,β2l],
where β1l and β2l are two constants satisfying 0≤β1l≤β2l≤1.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 matrixS=[S11S12S12TS22], where S11, S22 are square matrixes, then the following conditions are equivalent: (1)
S<0;(2)
S11<0,  S22-S12TS11-1S12<0;(3)
S22<0,  S11-S12S22-1S12T<0.Lemma 4 (see [28]).
LetD,  E, and F be matrices with appropriate dimensions. If  FTF≤I, then for any scalar  ε>0, one has
(17)DFE+ETFTDT≤εDDT+ε-1ETE.Lemma 5 (see [29]).
Assume thati∈ℓ={2,3,…,N}, and N is a given positive number and N≥2, then for any positive-definite matrix P∈Rn×n, one has
(18)(N-1)∑s=k-N+1k-1xT(s)Px(s)≥(∑s=k-i+1k-1xT(s))P(∑s=k-i+1k-1x(s)),k∈Z+.Definition 6 (see [30]).
The system (12) with w(k)≡0 is said to be stochastically stable if for every finite x~0=x~(0), and the following inequality holds:
(19)E{∑k=0∞∥x~(k)∥2∣x~0}<∞.Definition 7 (see [31]).
The closed-loop system (12) 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 (12) with ω(k)≡0 is stochastically stable.(b)
Under the zero-initial condition, it holds that(20)∑k=0∞E{zT(k)z(k)}<γ2∑k=0∞E{wT(k)w(k)},∀w(k)∈L2[0,∞).


## 3. Main Results

### 3.1. Stochastic Stability Analysis

With Lemma2, the stability of system (1) can be converted into the stability of system (12). Then a sufficient condition for stochastic stability of system (12) with w(k)=0 is given in the following theorems.


#### 3.1.1. Stability with the Known Occurrence Probability

Theorem 8.
Suppose that the occurrence probability for each subsystem is known. The networked control system (12) is stochastically stable if there exist positive definite matrices Pi∈R2n×2n, Q∈R2n×2n, R∈R2n×2n and Si∈R2n×2n, i∈ℓ with appropriate dimensions satisfying the following LMI:
(21)[Φ11Φ12Φ13*Φ220**Φ33]<0,i,j∈ℓ,
where
(22)Φ11=[Ωi0⋯-SiT⋯00-SiT*-Q⋯0⋯000**⋱⋮⋱⋮⋮⋮***-Q⋯000****⋱⋮⋮⋮*****-Q00******-I0*******-R],Φ12=[β2A~TPj⋯βiA~TPj⋯βNA~TPjβ2(B~2K~2)TPj⋯0⋯0⋮⋱⋮⋱⋮0⋯βi(B~iK~i)TPj⋯0⋮⋱⋮⋱⋮0⋯0⋯βN(B~NK~N)TPjβ2Pj⋯βiPj⋯βNPj0⋯0⋯0],Φ13=[β2(A~-I)TR⋯βi(A~-I)TR⋯βN(A~-I)TRβ2(B~2K~2)TR⋯0⋯0⋮⋱⋮⋱⋮0⋯βi(B~iK~i)TR⋯0⋮⋱⋮⋱⋮0⋯0⋯βN(B~NK~N)TRβ2R⋯βiR⋯βNR0⋯0⋯0],Φ22=-diag{Pj,…,Pj,…,Pj}︸N-1,Φ33=-1(N-1)2diag{R,…,R,…,R}︸N-1,Ωi=-Pi+Si+SiT+UTU+(N-1)Q.Proof.
Construct Lyapunov function candidates for closed-loop system (12) as follows:
(23)V(x~(k))=V1(x~(k))+V2(x~(k))+(N-1)V3(x~(k)),
where
(24)V1(x~(k))=x~T(k)Pσ(k)x~(k),V2(x~(k))=∑θ=-N-2∑s=k+θ+1k-1x~T(s)Qx~(s),V3(x~(k))=∑θ=-N-2∑s=k+θ+1k-1eT(s)Re(s).Pσ(k)>0,  Q>0, and R>0 are matrices to be determined. The new variable e(k) satisfies the following equation:
(25)e(k)=x~(k+1)-x~(k).
Then, we can further write (25) in the equivalent descriptor form
(26)e(k)=(A~-I)x~(k)+∑l=2NξlB~lK~lx~(k-l+1)+f~(x,k).
Assume that theith and jth modes are active at times k and k+1, respectively. That is,  τk∈[(i-1)T,iT) and τk+1∈[(j-1)T,jT) for any i,j∈ℓ. Along the solution of the system (12) with w(k)=0 and using Lemma 5, we have
(27)ΔV1(x~(k))=E[V1(x~(k+1))]-V1(x~(k))=E[x~T(k+1)Pjx~(k+1)]-x~T(k)Pix~(k)=E[A~x~(k)+∑l=2NξlB~lK~lx~(k-l+1)+f~(x,k)]T×Pj[A~x~(k)+∑l=2NξlB~lK~lx~(k-l+1)+f~(x,k)]-x~T(k)Pix~(k)+2x~T(k)SiT[x~(k)-x~(k-i+1)-∑s=k-i+1k-1e(s)],ΔV2(x~(k))=E(V2(x~(k+1)))-V2(x~(k))=∑θ=-N-2x~T(k)Qx~(k)-∑s=k-N+1k-1x~T(s)Qx~(s)≤(N-1)x~T(k)Qx~(k)-∑s=k-N+1k-1x~T(s)Qx~(s),ΔV3(x~(k))=E(V3(x~(k+1)))-V3(x~(k))=∑θ=-N-2eT(k)Re(k)-∑s=k-N+1k-1eT(s)Re(s)≤(N-1)eT(k)Re(k)-1N-1(∑s=k-i+1k-1eT(s))R(∑s=k-i+1k-1e(s)).
Denote v(k)=∑s=k-i+1k-1e(s), for all i∈ℓ  and combine (27) with (14), then we can obtain
(28)ΔV(x~(k))=ΔV1(x~(k))+ΔV2(x~(k))+(N-1)ΔV3(x~(k))≤ΔV1(x~(k))+ΔV2(x~(k))+(N-1)ΔV3(x~(k))+x~T(k)UTUx~(k)-f~T(x,k)f~(x,k)=ηT(k)Φ(i,j)η(k),
where ηT(k)=[x~T(k)x~T(k-1)⋯x~T(k-i+1)⋯x~T(k-N+1)f~T(x,k)vT(k)], and
(29)Φ(i,j)=Φ11+∑l=2NβlΦ~1lTPjΦ~1l+(N-1)2∑l=2NβlΦ~2lTRΦ~2l,
with
(30)Φ~1l=[A~0⋯0︸l-2B~lK~l0⋯0︸N-lI0],Φ~2l=[(A~-I)0⋯0︸l-2B~lK~l0⋯0︸N-lI0],∀i,j,l∈ℓ.
By Schur complement lemma, the inequality (21) guarantees Φ(i,j)<0. Thus, with the above relations, the inequality (28) can be rewritten as follows:
(31)ΔV(x~(k))=E{V(x~(k+1)∣σ(k+1)=j)}-V(x~(k)∣σ(k)=i)≤-λmin(-Φ(i,j))∥x~(k)∥2=-μ∥x~(k)∥2.
From the previous inequalities, we can obtain
(32)E{V(x~(k+1))∣σ(k+1)=j}-V(x~(0)∣σ(0))≤-μE{∑k=0k∥x~(k)∥2}
which implies that
(33)E{∑k=0∞∥x~(k)∥2}≤1μE{V(x~(0))}≤∞.
Therefore, by Definition6, it can be obtained that the closed-loop system (12) is stochastically stable.   The proof is completed.


#### 3.1.2. Stability with Uncertain Active Probabilities

In Section3.1.1, 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. Next, the stochastic stability criterion of the closed-loop system (12) with uncertain subsystem occurrence probabilities is given in the following theorem.Theorem 9.
Suppose that the range of occurrence probability for each subsystem is known. The networked control system (12) is stochastically stable if there exist positive definite matrices Pi∈R2n×2n, Q∈R2n×2n,  R∈R2n×2n and Si∈R2n×2n, i∈ℓ with appropriate dimensions satisfying the following LMI:
(34)[Φ11Φ12′Φ13′*Φ220**Φ33]<0,i,j∈ℓ,
where
(35)Φ12′=[β22A~TPj⋯β2iA~TPj⋯β2NA~TPjβ22(B~2K~2)TPj⋯0⋯0⋮⋱⋮⋱⋮0⋯β2i(B~iK~i)TPj⋯0⋮⋱⋮⋱⋮0⋯0⋯β2N(B~NK~N)TPjβ22Pj⋯β2iPj⋯β2NPj0⋯0⋯0],Φ13′=[β22(A~-I)TR⋯β2i(A~-I)TR⋯β2N(A~-I)TRβ22(B~2K~2)TR⋯0⋯0⋮⋱⋮⋱⋮0⋯β2i(B~iK~i)TR⋯0⋮⋱⋮⋱⋮0⋯0⋯β2N(B~NK~N)TRβ22R⋯β2iR⋯β2NR0⋯0⋯0].Proof.
By Schur complement lemma, (34) implies
(36)Φ11+∑l=2Nβ2lΦ~1lTPjΦ~1l+(N-1)2∑l=2Nβ2lΦ~2lTRΦ~2l<0.
Note that, for positive definite matricesPj and R, and 0≤β1l≤βl≤β2l≤1,
(37)∑l=2NβlΦ~1lTPjΦ~1l≤∑l=2Nβ2lΦ~1lTPjΦ~1l,∑l=2NβlΦ~2lTRΦ~2l≤∑l=2Nβ2lΦ~2lTRΦ~2l,∀j=ℓ.
Thus, if the LMI (34) holds, then we obtain
(38)Φ11+∑l=2NβlΦ~1lTPjΦ~1l+(N-1)2∑l=2NβlΦ~2lTRΦ~2l<0.
According to Theorem8, the discrete-time stochastic switched system (12) with uncertain subsystem occurrence probabilities is stochastically stable. This completes the proof.


### 3.2.H∞ Controller Synthesis with Uncertain Active Probabilities

This section is devoted to synthesizing a controller given in the formu(k)=Kix(k), for all i∈ℓ that guarantees the closed-loop system is robustly stable with the noise attenuation level γ.Theorem 10.
Suppose that the range of occurrence probability for each subsystem is known. The closed-loop system (12) is stochastically stable and achieves the given disturbance attenuation performance if there exist constants εi>0, and positive definite matrices Pi∈R2n×2n, Q∈R2n×2n, R∈R2n×2n, Xi∈R2n×2n, Y∈R2n×2n and Si∈R2n×2n, i∈ℓ with appropriate dimensions, and feedback gain matrix Ki satisfying matrix inequalities:
(39)[Θ11Θ12Θ13Θ14*Θ2200**Θ330***Θ44]<0,PjXj=I,RY=I,∀i,j∈ℓ,
where
(40)Θ11=[℧i0⋯-SiT⋯00-SiT0*-Q⋯0⋯0000**⋱⋮⋱⋮⋮⋮⋮***-Q⋯0000****⋱⋮⋮⋮⋮*****-Q000******-I0⋮*******-R0********-γ2I],Θ12=[β22A~T⋯β2iA~T⋯β2NA~Tβ22(Γ~2K~2)T⋯0⋯0⋮⋱⋮⋱⋮0⋯β2i(Γ~iK~i)T⋯0⋮⋱⋮⋱⋮0⋯0⋯β2N(Γ~NK~N)Tβ22I⋯β2iI⋯β2NI0⋯0⋯0β22H~T⋯β2iH~T⋯β2NH~T],Θ13=[β22(A~-I)T⋯β2i(A~-I)T⋯β2N(A~-I)Tβ22(Γ~2K~2)T⋯0⋯0⋮⋱⋮⋱⋮0⋯β2i(Γ~iK~i)T⋯0⋮⋱⋮⋱⋮0⋯0⋯β2N(Γ~NK~N)Tβ22I⋯β2iI⋯β2NI0⋯0⋯0β22H~T⋯β2iH~T⋯β2NH~T],Θ14=[0⋯0⋯0β22(E~K~2)T⋯0⋯0⋮⋱⋮⋱⋮0⋯β2i(E~K~i)T⋯0⋮⋱⋮⋱⋮0⋯0⋯β2N(E~K~N)T0⋯0⋯00⋯0⋯00⋯0⋯0],Θ22=diag{δ2,…,δi,…,δN}︸N-1,δi=-Xj+εiG~iG~iT,Θ33=diag{θ2,…,θi,…,θN}︸N-1,θi=-1(N-1)2Y+εiG~iG~iT,Θ44=diag{-ε2I,…,-εiI,…,-εNI}︸N-1,℧i=-Pi+Si+SiT+(N-1)Q+UTU+C~TC~.Proof.
From Theorem8, the closed-loop system (12) with w(k)=0 is stochastically stable.
Assume that theith and jth modes are activated at instants k and k+1, respectively. For nonzero  w(k), using the same Lyapunov function candidates as in Theorem 8, we have
(41)ΔV1(x~(k))=E[V1(x~(k+1))]-V1(x~(k))=E[x~T(k+1)Pjx~(k+1)]-x~T(k)Pix~(k)=E[(A~x~(k)+∑l=2NξlB~lK~lx~(k-l+1)+f~(x,k)+H~w(k)∑l=2NξlB~lK~lx~(k-l+1))T×Pj(A~x~(k)+∑l=2NξlB~lK~lx~(k-l+1)+f~(x,k)+H~w(k)∑l=2NξlB~lK~lx~(k-l+1))]-x~T(k)Pix~(k)+2x~T(k)SiT[x~(k)-x~(k-i+1)-∑s=k-i+1k-1e(s)],ΔV2(x~(k))=E(V2(x~(k+1)))-V2(x~(k))=∑θ=-N-2x~T(k)Qx~(k)-∑s=k-N+1k-1x~T(s)Qx~(s)≤(N-1)x~T(k)Qx~(k)-∑s=k-N+1k-1x~T(s)Qx~(s),ΔV3(x~(k))=E(V3(x~(k+1)))-V3(x~(k))=∑θ=-N-2eT(k)Re(k)-∑s=k-N+1k-1eT(s)Re(s)≤(N-1)eT(k)Re(k)-1N-1(∑s=k-i+1k-1eT(s))R(∑s=k-i+1k-1e(s)).
From inequations (41), and combining the nonlinear condition (14), we have
(42)ΔV(x~(k))=ΔV1(x~(k))+ΔV(x~(k))2+(N-1)ΔV3(x~(k))≤ΔV1(x~(k))+ΔV2(x~(k))2+(N-1)ΔV3(x~(k))+x~T(k)UTUx~(k)-f~T(x,k)f~(x,k)+E{zT(k)z(k)}-γ2E{wT(k)w(k)}-E{zT(k)z(k)}+γ2E{wT(k)w(k)}=ζT(k)Θ(i,j)ζ(k)-E{zT(k)z(k)}+γ2E{wT(k)w(k)},
where ζT(k)=[x~T(k)x~T(k-1)⋯x~T(k-i+1)⋯x~T(k-N+1)f~T(x,k)vT(k)w(k)], and
(43)Θ(i,j)=Θ11+∑l=2NβlΘ~1lTPjΘ~1l+(N-1)2∑l=2NβlΘ~2lTRΘ~2l,∀l∈ℓ.
with
(44)Θ~1l=[A~0⋯0︸l-2B~lK~l0⋯0︸N-lI0H~],Θ~2l=[(A~-I)0⋯0︸l-2B~lK~l0⋯0︸N-lI0H~].
Using the condition (13) and by Lemmas 3 and 4, we can obtain
(45)∑l=2NϕlTFT(τk′)φl+∑l=2NφlTF(τk′)ϕl≤∑l=2Nεl-1ϕlTϕl+∑l=2NεlφlTφl,
where
(46)ϕl=[0⋯0︸l-1βlE~K~l0⋯0︸3N-l+1],φl=[0⋯0︸N+l+1G~lT0⋯0︸N-2G~lT0⋯0︸N-1],∀l∈ℓ.
Using the similar analysis methods as in Theorem9, the inequality (39) guarantees Θ(i,j)<0.
From Theorem8, the following inequality can be obtained:
(47)ΔV(x~(k))=E{V(x~(k+1))}-V(x~(k))≤-E{zT(k)z(k)}+γ2E{wT(k)w(k)}+min{λmin(-Θ(i,j))}∥x~(k)∥2≤-E{zT(k)z(k)}+γ2E{wT(k)w(k)}.
Taking expectation and summing up fromk=0 to ∞ on both sides of inequality (47), it can be obtained that the above inequality (47) is equivalent to
(48)E[V(x~(∞))]-V(x~(0))≤-∑k=0∞E{zT(k)z(k)}+γ2∑k=0∞E{wT(k)w(k)}
which implies that
(49)∑k=0∞E{zT(k)z(k)}≤γ2∑k=0∞E{wT(k)w(k)}.
Therefore, the closed-loop system (12) is stochastically stable with disturbance attenuation level γ. This completes the proof.Remark 11.
It should be pointed out that the sufficient conditions proposed in Theorem10 are not standard LMI condition anymore. The subsystem occurrence probabilities are not coupled with the Lyapunov weighting matrix Pj 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 12.
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 13.
In Theorems9 and 10, the upper bound of occurrence 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.


## 3.1. Stochastic Stability Analysis

With Lemma2, the stability of system (1) can be converted into the stability of system (12). Then a sufficient condition for stochastic stability of system (12) with w(k)=0 is given in the following theorems.


### 3.1.1. Stability with the Known Occurrence Probability

Theorem 8.
Suppose that the occurrence probability for each subsystem is known. The networked control system (12) is stochastically stable if there exist positive definite matrices Pi∈R2n×2n, Q∈R2n×2n, R∈R2n×2n and Si∈R2n×2n, i∈ℓ with appropriate dimensions satisfying the following LMI:
(21)[Φ11Φ12Φ13*Φ220**Φ33]<0,i,j∈ℓ,
where
(22)Φ11=[Ωi0⋯-SiT⋯00-SiT*-Q⋯0⋯000**⋱⋮⋱⋮⋮⋮***-Q⋯000****⋱⋮⋮⋮*****-Q00******-I0*******-R],Φ12=[β2A~TPj⋯βiA~TPj⋯βNA~TPjβ2(B~2K~2)TPj⋯0⋯0⋮⋱⋮⋱⋮0⋯βi(B~iK~i)TPj⋯0⋮⋱⋮⋱⋮0⋯0⋯βN(B~NK~N)TPjβ2Pj⋯βiPj⋯βNPj0⋯0⋯0],Φ13=[β2(A~-I)TR⋯βi(A~-I)TR⋯βN(A~-I)TRβ2(B~2K~2)TR⋯0⋯0⋮⋱⋮⋱⋮0⋯βi(B~iK~i)TR⋯0⋮⋱⋮⋱⋮0⋯0⋯βN(B~NK~N)TRβ2R⋯βiR⋯βNR0⋯0⋯0],Φ22=-diag{Pj,…,Pj,…,Pj}︸N-1,Φ33=-1(N-1)2diag{R,…,R,…,R}︸N-1,Ωi=-Pi+Si+SiT+UTU+(N-1)Q.Proof.
Construct Lyapunov function candidates for closed-loop system (12) as follows:
(23)V(x~(k))=V1(x~(k))+V2(x~(k))+(N-1)V3(x~(k)),
where
(24)V1(x~(k))=x~T(k)Pσ(k)x~(k),V2(x~(k))=∑θ=-N-2∑s=k+θ+1k-1x~T(s)Qx~(s),V3(x~(k))=∑θ=-N-2∑s=k+θ+1k-1eT(s)Re(s).Pσ(k)>0,  Q>0, and R>0 are matrices to be determined. The new variable e(k) satisfies the following equation:
(25)e(k)=x~(k+1)-x~(k).
Then, we can further write (25) in the equivalent descriptor form
(26)e(k)=(A~-I)x~(k)+∑l=2NξlB~lK~lx~(k-l+1)+f~(x,k).
Assume that theith and jth modes are active at times k and k+1, respectively. That is,  τk∈[(i-1)T,iT) and τk+1∈[(j-1)T,jT) for any i,j∈ℓ. Along the solution of the system (12) with w(k)=0 and using Lemma 5, we have
(27)ΔV1(x~(k))=E[V1(x~(k+1))]-V1(x~(k))=E[x~T(k+1)Pjx~(k+1)]-x~T(k)Pix~(k)=E[A~x~(k)+∑l=2NξlB~lK~lx~(k-l+1)+f~(x,k)]T×Pj[A~x~(k)+∑l=2NξlB~lK~lx~(k-l+1)+f~(x,k)]-x~T(k)Pix~(k)+2x~T(k)SiT[x~(k)-x~(k-i+1)-∑s=k-i+1k-1e(s)],ΔV2(x~(k))=E(V2(x~(k+1)))-V2(x~(k))=∑θ=-N-2x~T(k)Qx~(k)-∑s=k-N+1k-1x~T(s)Qx~(s)≤(N-1)x~T(k)Qx~(k)-∑s=k-N+1k-1x~T(s)Qx~(s),ΔV3(x~(k))=E(V3(x~(k+1)))-V3(x~(k))=∑θ=-N-2eT(k)Re(k)-∑s=k-N+1k-1eT(s)Re(s)≤(N-1)eT(k)Re(k)-1N-1(∑s=k-i+1k-1eT(s))R(∑s=k-i+1k-1e(s)).
Denote v(k)=∑s=k-i+1k-1e(s), for all i∈ℓ  and combine (27) with (14), then we can obtain
(28)ΔV(x~(k))=ΔV1(x~(k))+ΔV2(x~(k))+(N-1)ΔV3(x~(k))≤ΔV1(x~(k))+ΔV2(x~(k))+(N-1)ΔV3(x~(k))+x~T(k)UTUx~(k)-f~T(x,k)f~(x,k)=ηT(k)Φ(i,j)η(k),
where ηT(k)=[x~T(k)x~T(k-1)⋯x~T(k-i+1)⋯x~T(k-N+1)f~T(x,k)vT(k)], and
(29)Φ(i,j)=Φ11+∑l=2NβlΦ~1lTPjΦ~1l+(N-1)2∑l=2NβlΦ~2lTRΦ~2l,
with
(30)Φ~1l=[A~0⋯0︸l-2B~lK~l0⋯0︸N-lI0],Φ~2l=[(A~-I)0⋯0︸l-2B~lK~l0⋯0︸N-lI0],∀i,j,l∈ℓ.
By Schur complement lemma, the inequality (21) guarantees Φ(i,j)<0. Thus, with the above relations, the inequality (28) can be rewritten as follows:
(31)ΔV(x~(k))=E{V(x~(k+1)∣σ(k+1)=j)}-V(x~(k)∣σ(k)=i)≤-λmin(-Φ(i,j))∥x~(k)∥2=-μ∥x~(k)∥2.
From the previous inequalities, we can obtain
(32)E{V(x~(k+1))∣σ(k+1)=j}-V(x~(0)∣σ(0))≤-μE{∑k=0k∥x~(k)∥2}
which implies that
(33)E{∑k=0∞∥x~(k)∥2}≤1μE{V(x~(0))}≤∞.
Therefore, by Definition6, it can be obtained that the closed-loop system (12) is stochastically stable.   The proof is completed.


### 3.1.2. Stability with Uncertain Active Probabilities

In Section3.1.1, 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. Next, the stochastic stability criterion of the closed-loop system (12) with uncertain subsystem occurrence probabilities is given in the following theorem.Theorem 9.
Suppose that the range of occurrence probability for each subsystem is known. The networked control system (12) is stochastically stable if there exist positive definite matrices Pi∈R2n×2n, Q∈R2n×2n,  R∈R2n×2n and Si∈R2n×2n, i∈ℓ with appropriate dimensions satisfying the following LMI:
(34)[Φ11Φ12′Φ13′*Φ220**Φ33]<0,i,j∈ℓ,
where
(35)Φ12′=[β22A~TPj⋯β2iA~TPj⋯β2NA~TPjβ22(B~2K~2)TPj⋯0⋯0⋮⋱⋮⋱⋮0⋯β2i(B~iK~i)TPj⋯0⋮⋱⋮⋱⋮0⋯0⋯β2N(B~NK~N)TPjβ22Pj⋯β2iPj⋯β2NPj0⋯0⋯0],Φ13′=[β22(A~-I)TR⋯β2i(A~-I)TR⋯β2N(A~-I)TRβ22(B~2K~2)TR⋯0⋯0⋮⋱⋮⋱⋮0⋯β2i(B~iK~i)TR⋯0⋮⋱⋮⋱⋮0⋯0⋯β2N(B~NK~N)TRβ22R⋯β2iR⋯β2NR0⋯0⋯0].Proof.
By Schur complement lemma, (34) implies
(36)Φ11+∑l=2Nβ2lΦ~1lTPjΦ~1l+(N-1)2∑l=2Nβ2lΦ~2lTRΦ~2l<0.
Note that, for positive definite matricesPj and R, and 0≤β1l≤βl≤β2l≤1,
(37)∑l=2NβlΦ~1lTPjΦ~1l≤∑l=2Nβ2lΦ~1lTPjΦ~1l,∑l=2NβlΦ~2lTRΦ~2l≤∑l=2Nβ2lΦ~2lTRΦ~2l,∀j=ℓ.
Thus, if the LMI (34) holds, then we obtain
(38)Φ11+∑l=2NβlΦ~1lTPjΦ~1l+(N-1)2∑l=2NβlΦ~2lTRΦ~2l<0.
According to Theorem8, the discrete-time stochastic switched system (12) with uncertain subsystem occurrence probabilities is stochastically stable. This completes the proof.


## 3.1.1. Stability with the Known Occurrence Probability

Theorem 8.
Suppose that the occurrence probability for each subsystem is known. The networked control system (12) is stochastically stable if there exist positive definite matrices Pi∈R2n×2n, Q∈R2n×2n, R∈R2n×2n and Si∈R2n×2n, i∈ℓ with appropriate dimensions satisfying the following LMI:
(21)[Φ11Φ12Φ13*Φ220**Φ33]<0,i,j∈ℓ,
where
(22)Φ11=[Ωi0⋯-SiT⋯00-SiT*-Q⋯0⋯000**⋱⋮⋱⋮⋮⋮***-Q⋯000****⋱⋮⋮⋮*****-Q00******-I0*******-R],Φ12=[β2A~TPj⋯βiA~TPj⋯βNA~TPjβ2(B~2K~2)TPj⋯0⋯0⋮⋱⋮⋱⋮0⋯βi(B~iK~i)TPj⋯0⋮⋱⋮⋱⋮0⋯0⋯βN(B~NK~N)TPjβ2Pj⋯βiPj⋯βNPj0⋯0⋯0],Φ13=[β2(A~-I)TR⋯βi(A~-I)TR⋯βN(A~-I)TRβ2(B~2K~2)TR⋯0⋯0⋮⋱⋮⋱⋮0⋯βi(B~iK~i)TR⋯0⋮⋱⋮⋱⋮0⋯0⋯βN(B~NK~N)TRβ2R⋯βiR⋯βNR0⋯0⋯0],Φ22=-diag{Pj,…,Pj,…,Pj}︸N-1,Φ33=-1(N-1)2diag{R,…,R,…,R}︸N-1,Ωi=-Pi+Si+SiT+UTU+(N-1)Q.Proof.
Construct Lyapunov function candidates for closed-loop system (12) as follows:
(23)V(x~(k))=V1(x~(k))+V2(x~(k))+(N-1)V3(x~(k)),
where
(24)V1(x~(k))=x~T(k)Pσ(k)x~(k),V2(x~(k))=∑θ=-N-2∑s=k+θ+1k-1x~T(s)Qx~(s),V3(x~(k))=∑θ=-N-2∑s=k+θ+1k-1eT(s)Re(s).Pσ(k)>0,  Q>0, and R>0 are matrices to be determined. The new variable e(k) satisfies the following equation:
(25)e(k)=x~(k+1)-x~(k).
Then, we can further write (25) in the equivalent descriptor form
(26)e(k)=(A~-I)x~(k)+∑l=2NξlB~lK~lx~(k-l+1)+f~(x,k).
Assume that theith and jth modes are active at times k and k+1, respectively. That is,  τk∈[(i-1)T,iT) and τk+1∈[(j-1)T,jT) for any i,j∈ℓ. Along the solution of the system (12) with w(k)=0 and using Lemma 5, we have
(27)ΔV1(x~(k))=E[V1(x~(k+1))]-V1(x~(k))=E[x~T(k+1)Pjx~(k+1)]-x~T(k)Pix~(k)=E[A~x~(k)+∑l=2NξlB~lK~lx~(k-l+1)+f~(x,k)]T×Pj[A~x~(k)+∑l=2NξlB~lK~lx~(k-l+1)+f~(x,k)]-x~T(k)Pix~(k)+2x~T(k)SiT[x~(k)-x~(k-i+1)-∑s=k-i+1k-1e(s)],ΔV2(x~(k))=E(V2(x~(k+1)))-V2(x~(k))=∑θ=-N-2x~T(k)Qx~(k)-∑s=k-N+1k-1x~T(s)Qx~(s)≤(N-1)x~T(k)Qx~(k)-∑s=k-N+1k-1x~T(s)Qx~(s),ΔV3(x~(k))=E(V3(x~(k+1)))-V3(x~(k))=∑θ=-N-2eT(k)Re(k)-∑s=k-N+1k-1eT(s)Re(s)≤(N-1)eT(k)Re(k)-1N-1(∑s=k-i+1k-1eT(s))R(∑s=k-i+1k-1e(s)).
Denote v(k)=∑s=k-i+1k-1e(s), for all i∈ℓ  and combine (27) with (14), then we can obtain
(28)ΔV(x~(k))=ΔV1(x~(k))+ΔV2(x~(k))+(N-1)ΔV3(x~(k))≤ΔV1(x~(k))+ΔV2(x~(k))+(N-1)ΔV3(x~(k))+x~T(k)UTUx~(k)-f~T(x,k)f~(x,k)=ηT(k)Φ(i,j)η(k),
where ηT(k)=[x~T(k)x~T(k-1)⋯x~T(k-i+1)⋯x~T(k-N+1)f~T(x,k)vT(k)], and
(29)Φ(i,j)=Φ11+∑l=2NβlΦ~1lTPjΦ~1l+(N-1)2∑l=2NβlΦ~2lTRΦ~2l,
with
(30)Φ~1l=[A~0⋯0︸l-2B~lK~l0⋯0︸N-lI0],Φ~2l=[(A~-I)0⋯0︸l-2B~lK~l0⋯0︸N-lI0],∀i,j,l∈ℓ.
By Schur complement lemma, the inequality (21) guarantees Φ(i,j)<0. Thus, with the above relations, the inequality (28) can be rewritten as follows:
(31)ΔV(x~(k))=E{V(x~(k+1)∣σ(k+1)=j)}-V(x~(k)∣σ(k)=i)≤-λmin(-Φ(i,j))∥x~(k)∥2=-μ∥x~(k)∥2.
From the previous inequalities, we can obtain
(32)E{V(x~(k+1))∣σ(k+1)=j}-V(x~(0)∣σ(0))≤-μE{∑k=0k∥x~(k)∥2}
which implies that
(33)E{∑k=0∞∥x~(k)∥2}≤1μE{V(x~(0))}≤∞.
Therefore, by Definition6, it can be obtained that the closed-loop system (12) is stochastically stable.   The proof is completed.


## 3.1.2. Stability with Uncertain Active Probabilities

In Section3.1.1, 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. Next, the stochastic stability criterion of the closed-loop system (12) with uncertain subsystem occurrence probabilities is given in the following theorem.Theorem 9.
Suppose that the range of occurrence probability for each subsystem is known. The networked control system (12) is stochastically stable if there exist positive definite matrices Pi∈R2n×2n, Q∈R2n×2n,  R∈R2n×2n and Si∈R2n×2n, i∈ℓ with appropriate dimensions satisfying the following LMI:
(34)[Φ11Φ12′Φ13′*Φ220**Φ33]<0,i,j∈ℓ,
where
(35)Φ12′=[β22A~TPj⋯β2iA~TPj⋯β2NA~TPjβ22(B~2K~2)TPj⋯0⋯0⋮⋱⋮⋱⋮0⋯β2i(B~iK~i)TPj⋯0⋮⋱⋮⋱⋮0⋯0⋯β2N(B~NK~N)TPjβ22Pj⋯β2iPj⋯β2NPj0⋯0⋯0],Φ13′=[β22(A~-I)TR⋯β2i(A~-I)TR⋯β2N(A~-I)TRβ22(B~2K~2)TR⋯0⋯0⋮⋱⋮⋱⋮0⋯β2i(B~iK~i)TR⋯0⋮⋱⋮⋱⋮0⋯0⋯β2N(B~NK~N)TRβ22R⋯β2iR⋯β2NR0⋯0⋯0].Proof.
By Schur complement lemma, (34) implies
(36)Φ11+∑l=2Nβ2lΦ~1lTPjΦ~1l+(N-1)2∑l=2Nβ2lΦ~2lTRΦ~2l<0.
Note that, for positive definite matricesPj and R, and 0≤β1l≤βl≤β2l≤1,
(37)∑l=2NβlΦ~1lTPjΦ~1l≤∑l=2Nβ2lΦ~1lTPjΦ~1l,∑l=2NβlΦ~2lTRΦ~2l≤∑l=2Nβ2lΦ~2lTRΦ~2l,∀j=ℓ.
Thus, if the LMI (34) holds, then we obtain
(38)Φ11+∑l=2NβlΦ~1lTPjΦ~1l+(N-1)2∑l=2NβlΦ~2lTRΦ~2l<0.
According to Theorem8, the discrete-time stochastic switched system (12) with uncertain subsystem occurrence probabilities is stochastically stable. This completes the proof.


## 3.2.H∞ Controller Synthesis with Uncertain Active Probabilities

This section is devoted to synthesizing a controller given in the formu(k)=Kix(k), for all i∈ℓ that guarantees the closed-loop system is robustly stable with the noise attenuation level γ.Theorem 10.
Suppose that the range of occurrence probability for each subsystem is known. The closed-loop system (12) is stochastically stable and achieves the given disturbance attenuation performance if there exist constants εi>0, and positive definite matrices Pi∈R2n×2n, Q∈R2n×2n, R∈R2n×2n, Xi∈R2n×2n, Y∈R2n×2n and Si∈R2n×2n, i∈ℓ with appropriate dimensions, and feedback gain matrix Ki satisfying matrix inequalities:
(39)[Θ11Θ12Θ13Θ14*Θ2200**Θ330***Θ44]<0,PjXj=I,RY=I,∀i,j∈ℓ,
where
(40)Θ11=[℧i0⋯-SiT⋯00-SiT0*-Q⋯0⋯0000**⋱⋮⋱⋮⋮⋮⋮***-Q⋯0000****⋱⋮⋮⋮⋮*****-Q000******-I0⋮*******-R0********-γ2I],Θ12=[β22A~T⋯β2iA~T⋯β2NA~Tβ22(Γ~2K~2)T⋯0⋯0⋮⋱⋮⋱⋮0⋯β2i(Γ~iK~i)T⋯0⋮⋱⋮⋱⋮0⋯0⋯β2N(Γ~NK~N)Tβ22I⋯β2iI⋯β2NI0⋯0⋯0β22H~T⋯β2iH~T⋯β2NH~T],Θ13=[β22(A~-I)T⋯β2i(A~-I)T⋯β2N(A~-I)Tβ22(Γ~2K~2)T⋯0⋯0⋮⋱⋮⋱⋮0⋯β2i(Γ~iK~i)T⋯0⋮⋱⋮⋱⋮0⋯0⋯β2N(Γ~NK~N)Tβ22I⋯β2iI⋯β2NI0⋯0⋯0β22H~T⋯β2iH~T⋯β2NH~T],Θ14=[0⋯0⋯0β22(E~K~2)T⋯0⋯0⋮⋱⋮⋱⋮0⋯β2i(E~K~i)T⋯0⋮⋱⋮⋱⋮0⋯0⋯β2N(E~K~N)T0⋯0⋯00⋯0⋯00⋯0⋯0],Θ22=diag{δ2,…,δi,…,δN}︸N-1,δi=-Xj+εiG~iG~iT,Θ33=diag{θ2,…,θi,…,θN}︸N-1,θi=-1(N-1)2Y+εiG~iG~iT,Θ44=diag{-ε2I,…,-εiI,…,-εNI}︸N-1,℧i=-Pi+Si+SiT+(N-1)Q+UTU+C~TC~.Proof.
From Theorem8, the closed-loop system (12) with w(k)=0 is stochastically stable.
Assume that theith and jth modes are activated at instants k and k+1, respectively. For nonzero  w(k), using the same Lyapunov function candidates as in Theorem 8, we have
(41)ΔV1(x~(k))=E[V1(x~(k+1))]-V1(x~(k))=E[x~T(k+1)Pjx~(k+1)]-x~T(k)Pix~(k)=E[(A~x~(k)+∑l=2NξlB~lK~lx~(k-l+1)+f~(x,k)+H~w(k)∑l=2NξlB~lK~lx~(k-l+1))T×Pj(A~x~(k)+∑l=2NξlB~lK~lx~(k-l+1)+f~(x,k)+H~w(k)∑l=2NξlB~lK~lx~(k-l+1))]-x~T(k)Pix~(k)+2x~T(k)SiT[x~(k)-x~(k-i+1)-∑s=k-i+1k-1e(s)],ΔV2(x~(k))=E(V2(x~(k+1)))-V2(x~(k))=∑θ=-N-2x~T(k)Qx~(k)-∑s=k-N+1k-1x~T(s)Qx~(s)≤(N-1)x~T(k)Qx~(k)-∑s=k-N+1k-1x~T(s)Qx~(s),ΔV3(x~(k))=E(V3(x~(k+1)))-V3(x~(k))=∑θ=-N-2eT(k)Re(k)-∑s=k-N+1k-1eT(s)Re(s)≤(N-1)eT(k)Re(k)-1N-1(∑s=k-i+1k-1eT(s))R(∑s=k-i+1k-1e(s)).
From inequations (41), and combining the nonlinear condition (14), we have
(42)ΔV(x~(k))=ΔV1(x~(k))+ΔV(x~(k))2+(N-1)ΔV3(x~(k))≤ΔV1(x~(k))+ΔV2(x~(k))2+(N-1)ΔV3(x~(k))+x~T(k)UTUx~(k)-f~T(x,k)f~(x,k)+E{zT(k)z(k)}-γ2E{wT(k)w(k)}-E{zT(k)z(k)}+γ2E{wT(k)w(k)}=ζT(k)Θ(i,j)ζ(k)-E{zT(k)z(k)}+γ2E{wT(k)w(k)},
where ζT(k)=[x~T(k)x~T(k-1)⋯x~T(k-i+1)⋯x~T(k-N+1)f~T(x,k)vT(k)w(k)], and
(43)Θ(i,j)=Θ11+∑l=2NβlΘ~1lTPjΘ~1l+(N-1)2∑l=2NβlΘ~2lTRΘ~2l,∀l∈ℓ.
with
(44)Θ~1l=[A~0⋯0︸l-2B~lK~l0⋯0︸N-lI0H~],Θ~2l=[(A~-I)0⋯0︸l-2B~lK~l0⋯0︸N-lI0H~].
Using the condition (13) and by Lemmas 3 and 4, we can obtain
(45)∑l=2NϕlTFT(τk′)φl+∑l=2NφlTF(τk′)ϕl≤∑l=2Nεl-1ϕlTϕl+∑l=2NεlφlTφl,
where
(46)ϕl=[0⋯0︸l-1βlE~K~l0⋯0︸3N-l+1],φl=[0⋯0︸N+l+1G~lT0⋯0︸N-2G~lT0⋯0︸N-1],∀l∈ℓ.
Using the similar analysis methods as in Theorem9, the inequality (39) guarantees Θ(i,j)<0.
From Theorem8, the following inequality can be obtained:
(47)ΔV(x~(k))=E{V(x~(k+1))}-V(x~(k))≤-E{zT(k)z(k)}+γ2E{wT(k)w(k)}+min{λmin(-Θ(i,j))}∥x~(k)∥2≤-E{zT(k)z(k)}+γ2E{wT(k)w(k)}.
Taking expectation and summing up fromk=0 to ∞ on both sides of inequality (47), it can be obtained that the above inequality (47) is equivalent to
(48)E[V(x~(∞))]-V(x~(0))≤-∑k=0∞E{zT(k)z(k)}+γ2∑k=0∞E{wT(k)w(k)}
which implies that
(49)∑k=0∞E{zT(k)z(k)}≤γ2∑k=0∞E{wT(k)w(k)}.
Therefore, the closed-loop system (12) is stochastically stable with disturbance attenuation level γ. This completes the proof.Remark 11.
It should be pointed out that the sufficient conditions proposed in Theorem10 are not standard LMI condition anymore. The subsystem occurrence probabilities are not coupled with the Lyapunov weighting matrix Pj 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 12.
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 13.
In Theorems9 and 10, the upper bound of occurrence 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.


## 4. Numerical Example

In this section, we present an example to illustrate the effectiveness of the proposed approach. Consider system (1) with parameters as follows:
(50)x˙(t)=[0-23-5]x(t)+[0.41]u(t)+[0.4x1sinx10.4x2cosx2]+[0.20.1]w(t),z(t)=[01]x(t).Here, the nonlinear functionf(x,t) satisfies sector condition (2), and we can obtain that  α=0.4. Choose the sampling period T=0.1s and suppose 0.1≤τk<0.3s, so we can know that the upper bound of the long time delays is N=3, and the NCS can be modeled with two subsystems, where the corresponding system matrices are obtained as follows:
(51)A=[0.9746-0.15580.23370.5850],Γ20=[0.04730.1342],Γ21=[-0.0162-0.0512],Γ30=[0.05640.1726],Γ31=[-0.0252-0.0896],G2=[0.9205-0.23670.35500.3289],G3=[0.8418-0.28190.42280.1371],E=[0.41],H=[0.01900.0103],U=diag{0.2,0.2,0,0}.By using iterative algorithm-CCL and solving the constraint conditions in Theorem10, we can obtain the upper bounds of the occurrence probability for the first subsystem and the second subsystem that are β22=0.4366 and β23=0.7035, respectively.Then, the controller can be obtained as follows:(52)K2=[-0.1610-0.5509],K3=[-2.3499-3.0658].Then the range of the subsystem occurrence probability isβ2∈[0.2965,0.4366] and β3∈[0.5634,0.7035]. Assume that the actual subsystem occurrence probabilities for the subsystem are β2=0.4 and β3=0.6, the initial state of the system is  x0=[-11]T  and w(k)=0.05exp(-0.01k), and the state trajectories of the NCS and the corresponding switching signal are shown in Figures 1 and 2, respectively.Figure 1
State trajectories of the closed-loop system.Figure 2
The stochastic switching signal.From simulation results, it can be seen that the NCS is stochastically stable and hasH∞  disturbance attenuation level γ=1.2.It should be pointed that the methods proposed in the literature [22–25] cannot be used to deal with the H∞ control of the given NCS because of the uncertain environment and the feature of stochastic long-time delays.


## 5. Conclusion

In this paper, the problems ofH∞ control have been studied for a class of nonlinear NCS with long-time delays. We first model NCS 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 where 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 NCS with communication constraints.



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*Source: 643174-2013-04-16.xml*