# H

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

---

## Abstract

This paper is concerned with theH∞ control issue for a class of networked control systems (NCSs) with packet dropouts and time-varying delays. Firstly, the addressed NCS is modeled as a Markovian discrete-time switched system with two subsystems; by using the average dwell time method, a sufficient condition is obtained for the mean square exponential stability of the closed-loop NCS with a desired H∞ disturbance attenuation level. Then, the desired H∞ controller is obtained by solving a set of linear matrix inequalities (LMIs). Finally, a numerical example is given to illustrate the effectiveness of the proposed method.


---

## 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 network-based control scheme reduces system wiring and has low cost, high reliability, information sharing, and remote control [1, 2]. Nevertheless, the introduction of communication networks also brings some new problems and challenges, such as time-delay, packet dropout, quantization, and band-limited channel [3–8], which all might be potential sources of poor performance, even of instability.Random delay and packet dropout in NCS are two major causes for the deterioration of system stability; various approaches have been developed for the NCS with random communication delays and packet dropout in [9–17]. The time delay occurs in various physical, industrial, and engineering systems and is a source of poor performance and instability of systems. In [9, 10], the uncertainties of the delays are transformed into those of the system models with uncertain parameters. The delay is limited to take finite values during a sampling period, and the NCS is ultimately modeled as a discrete-time switched system with a finite number of subsystems [11, 12]. In [13–15], the delay is assumed to be random and follows some specific distribution laws, which may not be exactly known prior in practice. And in some literature the delay is separated into a nominal part and an uncertain part; in this way, the NCS is represented as an uncertain system with norm-bounded uncertainties or polytopic uncertainties. Another important issue in NCS control problem is packet dropout; most of the NCS models are presented by using the Bernoulli random binary distributed sequence methods or the Markov chain. For NCSs, let the binary-valued function denote the data transmission status from sensor to controller and controller to actuator, respectively, where 1 means successful packet communication and 0 is the case of packet dropout [16]. Reference [17] proposes an iterative method to model NCSs with bounded packet dropout as MJLSs with partly unknown transition probabilities.On the other hand, in view of abrupt variation in the structures, such as component failures, sudden environmental disturbance, and abrupt variations of the operating points of NCSs, it is more appropriate to model such class of systems as a special class of stochastic hybrid systems with finite operation modes. And packet dropout (time-delay) of the next sampling moment may have a close relation to the previous moment, so it is reasonable to model NCS as the Markov switched system. The mean square stabilization of a class of Markovian NCS is studied in [18], and the average dwell time (ADT) approach is applied to investigate the stability of the NCS in [19]. However, to the best of our knowledge, the problems of mean square exponential stability and control for the NCS have not been fully investigated to date. This motivates the present study.With the motivation of the above reasons, we consider the mean square exponentialH∞ performance for NCS with random delay and packet dropout. The main contribution can be summarized as follows: (i) an NCS model with random delay and packet dropout is proposed firstly; the packet dropout process is modeled as a finite state Markov chain and the resulting closed-loop system is a Markovian switching system; (ii) the parameter-dependent Lyapunov function is applied for stability analysis and control synthesis, and sufficient conditions for the robustly mean square exponential stability of the closed-loop system are given by using the ADT method [20], where the convergence of the Markov chain is utilized; and (iii) a state feedback controller is designed by using a cone complementary linearization approach to ensure that the closed-loop system is mean square exponentially stable and achieves the disturbance attenuation level.The paper is organized as follows. In Section2, the NCS with packet dropouts and time-varying delays is modeled as a class of the Markovian discrete-time switched system with two subsystems. The mean square exponential stability of the closed-loop NCS with a desired H∞ disturbance attenuation level is developed in Section 3 and the desired H∞ controller is formulated in a set of LMIs. A numerical example is provided in Section 4. Finally, Section 5 concludes this paper.Notation 1.
Throughout the paper, the superscript ‘‘-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 a 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 places. 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. L2[k0,∞) stands for the space of square integrable functions on [k0,∞).


## 2. Model of Networked Control System

Consider the following system:(1)x˙(t)=Apx(t)+Bpu(t)+f(x,t)+Hpw(t),z(t)=Cx(t),
where x(t)∈Rn, u(t)∈Rm, and z(t)∈RP are the state vector, control input vector, and controlled output vector, respectively, and 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, and 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; the sensor is time-driven with sampling periodT,(2)
the network-induced delayτk≜τksc+τkca satisfies 0<τmin≤τk≤τmax<T,(3)
the zero-order hold device does not update the output value until the new value arrives.The output value of the discrete-time state-feedback controller corresponding tox(k) is denoted by
(3)u(k)=Kx(k).Consider the plant input:(4)u(k)={u^(k)ifu^(k)andx(k)is  successfully  transmitted,u(k-1)ifu^(k)orx(k)is  lost  during  transmission.Discretizing system (1) in one period, we can obtain the discrete state equation of the NCS:
(5)x(k+1)=Ax(k)+B0(τk)u(k)+B1(τk)u(k-1)+f~(x,k)+Hw(k),
where A=eApT, B0(τk)=∫0T-τkeApsdsBp, B1(τk)=∫T-τkTeApsdsBp, H=∫0TeApsdsHp, and f~(x,k)=∫0TeApsdsf(x,k).By using the Jordan form of the matrixAp, B0(τk) is rewritten as [21]
(6)B0(τk)=F0+∑i=1vηi(τk)Fi
with v≤n, where F0 and Fi are constant matrices, ηi(τk)=ea(T-τk)cos(b(T-τk)) and the eigenvalue of A is λ=a+ib.Then,{B0(τk)∣k∈N} is a subset of co(F) with
(7)F={F0+∑i=1vη~iFi∣η~i=η¯i,η_i,i=1,2,…,v}={F~i∣i=1,2,…,2v},
where η-i=maxηi(τk), η_i=minηi(τk), F is the set of vertices, and co(·) denotes the convex hull. Thus we obtain
(8)B0(τk)=∑i=12vξi(k)F~i,
with ∑i=12vξi(k)=1, ξi(k)∈[0,1].Defining an augmented vectorx-(k)=[xT(k)uT(k-1)]T, during each sampling period, two cases may arise, which can be listed as follows.S
1: no packet dropout happens; (5) can be written as
(9)x-(k+1)=A-1(k)x-(k)+f-(x,k)+H-w(k),z(k)=C-x-(k),
where
(10)A-1(k)=[A+B0(τk)KB1(τk)K0]=[AB-B0(τk)00]+[B0(τk)I][K0],B=∫0TeApsdsBp,B1(τk)=B-B0(τk),f-(x,k)=[f~(x,k)0],H-=[H0],C-=[C0].
Substituting (8) into (9) gives rise to
(11)A-1(k)=∑i=12vξi(k)Ai,
where
(12)Ai=A~i+B~i[K0]=A~i+B~iK-,A~i=[AB-F~i00],B~i=[F~iI],K-=[K0].S
2: packet dropout happens; (5) can be written as
(13)x-(k+1)=A-2(k)x-(k)+f-(x,k)+H-w(k),z(k)=C-x-(k),
where A-2=[AB0I].From (2)–(9), the nonlinear uncertainty f-(x,k) satisfies
(14)f-T(x,k)f-(x,k)=f~T(x,k)f~(x,k)≤x-T(k)UTUx-(k),
where U is a known constant positive-definite matrix.By the above analysis and assumptions, we can see that networked control system can be described by the following switched system with two modes:(15)x-(k+1)=A-σ(k)(k)x-(k)+f-(x,k)+H-w(k),z(k)=C-x-(k),
where σ(k) is called a switching signal. σ(k)=1 represents no packet dropout, while σ(k)=2 implies packet dropout. The switching characteristics between the two modes are often assumed as the Markov chain, and πrl is transition probability from mode r to l, r,l=1,2; therefore, σ(k) of the Markov chain has ergodicity and satisfied the following condition:
(16)limn→∞πrl(n)=πl,r,l=1,2,
where πl is the limitation of state l. So {π1,π2} is the stationary distribution of the Markov chain.For an arbitrary switching sequenceσ(k) and any given integer k>0, let k0 imply the initial time, and k0<k1<k2<⋯kq<⋯<k, q≥1 represent the switching instants. Denote T1[k0,k) as the all sequence of the time period in which subsystem 1 is active during the time interval [k0,k). Similarly, T2[k0,k) represents the all period sequence that subsystem 2 is active during the time interval [k0,k).Lemma 1 (Schur complement [22]).
For a given matrixS=[S11S12S12TS22], where S11,S22 are square matrices, the following conditions are equivalent:(1)
S<0;(2)
S11<0,S22-S12TS11-1S12<0;(3)
S22<0,S11-S12S22-1S12T<0.Lemma 2 (see [23]).
The stochastic stability in discrete time implies the stochastic stability in continuous time.Definition 3 (see [24]).
The closed-loop system (15) is mean square exponentially stable with w(k)=0, if there exists δ>0, 0<β<1, such that
(17)E{∥x-(k)∥2}<δβk-k0E{∥x-(k0)∥2}
for all initial condition (x-(k0),σ(k0)).Definition 4 (see [25]).
For anyk>k0≥0, let Nσ[k0,k) denote the total number of the switching of σ(k) during the interval [k0,k). If
(18)Nσ[k0,k)≤N0+k-k0Ta
holds for a given N0≥0, Ta>0, then the constant Ta is called the average dwell time and N0 is the chatter bound. For simplicity, we choose N0=0 without loss of generality.Definition 5 (see [20]).
Given scalarsγ>0 and 0<λ<1, the closed-loop system (15) is robustly exponentially stable with an exponential H∞ performance γ if the following conditions are satisfied: (a)
the closed-loop system (15) with ω(k)≡0 is exponentially stable;(b)
under the zero-initial condition, it holds that(19)∑k=k0∞E{λkzT(k)z(k)}<γ2∑k=k0∞E{wT(k)w(k)},∀w(k)∈L2[k0,∞).


## 3. Main Results

The following theorems present a sufficient condition for the mean square stability of the considered system and theH∞ controller design method.


### 3.1. Stability Analysis

In this subsection, sufficient conditions for the existence of mean square exponential stability of system (15) with ω(k)≡0 are given in the following theorem.Theorem 6.
System (15) is mean square exponentially stable with a decay rate λρ, if there exist positive definite matrices Pi, Q, scalars μ≥1, λ1, and λ2, such that(20)[AiT(π11Pj+π12Q)Ai-λ1Pi+UTUAiT(π11Pj+π12Q)*(π11Pj+π12Q)-I]<0,(21)[A-2T(π21Pj+π22Q)A2--λ2Q+UTUA-2T(π21Pj+π22Q)*(π21Pj+π22Q)-I]<0i,j=1,2,3,…,2v,(22)1μQ≤Pi≤μQ,(23)0<λ<1,(24)max{π12,π21}<-lnλlnμ,
where λ=λ1π21/(π12+π21)λ2π12/(π12+π21),ρ=1+max{π12,π21}·(lnμ/lnλ).Proof.
For the system (15), define the following Lyapunov function:
(25)Vσ(k)(x-(k),ξ(k))=x-T(k)P~σ(k)x-(k),
where
(26)P~σ(k)=∑i=12vξi(k)Pi,forσ(k)=1,P~σ(k)=Q,forσ(k)=2.
For subsystem 1, it follows from (15) that
(27)ΔV1[x-(k+1)]=E[V1(x-(k+1),ξ(k+1))]-λ1V1(x-(k),ξ(k))=x-T(k+1)(π11∑j=12vξj(k+1)Pj+π12Q)x-(k+1)=-λ1x-T(k)(∑i=12vξi(k)Pi)x-(k)≤(∑i=12vξi(k)Aix-(k)+f-(x,k))T=×(π11∑j=12vξj(k+1)Pj+π12Q)=·(∑i=12vξi(k)Aix-(k)+f-(x,k))=-λ1x-T(k)(∑i=12vξi(k)Pi)x-(k)=+x-T(k)UTUx-(k)-f-T(x,k)f-(x,k)=∑i=12vξi(k)∑j=12vξj(k+1)[x-(k)f-(x,k)]TΘ[x-(k)f-(x,k)],
where
(28)Θ=[AiT(π11Pj+π12Q)Ai-λ1Pi+UTUAiT(π11Pj+π12Q)*(π11Pj+π12Q)-I].
From inequality (20), one obtains
(29)E[V1(x-(k+1),ξ(k+1))]<λ1V1(x-(k),ξ(k)).
In the same way, for subsystem 2, we obtain
(30)ΔV2[x-(k+1)]=E[V2(x-(k+1),ξ(k+1))]-λ2V2(x-(k),ξ(k))=x-T(k+1)(π21∑j=12vξj(k+1)Pj+π22Q)x-(k+1)=-λ2x-T(k)Qx-(k)≤(A2-x-(k)+f-(x,k))T=×(π21∑j=12vξj(k+1)Pj+π22Q)=·(A2-x-(k)+f-(x,k))-λ2x-T(k)Qx-(k)=+x-T(k)UTUx-(k)-f-T(x,k)f-(x,k);
then
(31)E[V2(x-(k+1),ξ(k+1))]<λ2V2(x-(k),ξ(k)).
Considering the condition (22), we get that
(32)V1(x-(k),ξ(k))≤μV2(x-(k-),ξ(k-)),V2(x-(k),ξ(k))≤μV1(x-(k-),ξ(k-)).
Then for kq<k<kq+1, we get
(33)E[Vσ(k)(x-(k),ξ(k))]<E[μλσ(kq)(k-kq)Vσ(kq-1)(x-(kq),ξ(kq))]<E[μλσ(kq)(k-kq)λσ(kq-1)(kq-kq-1)Vσ(kq-1)(x-(kq-1),ξ(kq-1))]⋮<E[μNσ[k0,k)λ1T1[k0,k)λ2T2[k0,k)Vσ(k0)(x-(k0),ξ(k0))].
Note that the Markov chain is stationary (16); then
(34)E[T1[k0,k)]=π1(k-k0),E[T2[k0,k)]=π2(k-k0).
Therefore, we can obtain that
(35)E[Vσ(k)(x-(k),ξ(k))]<E[μNσ[k0,k)λ1π1(k-k0)λ2π2(k-k0)Vσ(k0)(x-(k0),ξ(k0))]=E[μNσ[k0,k)λ1(π21/(π12+π21))(k-k0)=E×λ2(π12/(π12+π21))(k-k0)Vσ(k0)(x-(k0),ξ(k0))]=E[μNσ[k0,k)λ(k-k0)Vσ(k0)(x-(k0),ξ(k0))]=E[λNσ[k0,k)(lnμ/lnλ)λ(k-k0)Vσ(k0)(x-(k0),ξ(k0))]=E[(λ1+(Nσ[k0,k)/(k-k0))·(lnμ/lnλ))(k-k0)]=×E[Vσ(k0)(x-(k0),ξ(k0))].
From Definition 4 we have that
(36)k-k0Nσ[k0,k)≥Ta.
And from [24], we can get 1/E(Ta)≤max{π12,π21}; then combining (23) and (24), we can know that
(37)0<λ1+(Nσ[k0,k)/(k-k0))·(lnμ/lnλ)<1,
which ensure the convergence of E[Vσ(k)(x-(k),ξ(k))].
In this case,(38)E[Vσ(k)(x-(k),ξ(k))]<E[(λ1+(1/Ta)·(lnμ/lnλ))(k-k0)]=×E[Vσ(k0)(x-(k0),ξ(k0))]=λ(k-k0)[1+(1/E(Ta))·(lnμ/lnλ)]E[Vσ(k0)(x-(k0),ξ(k0))]≤(λρ)(k-k0)E[Vσ(k0)(x-(k0),ξ(k0))].
Furthermore
(39)E[a∥x-(k)∥2]≤E[Vσ(k)(x-(k),ξ(k))]<(λρ)(k-k0)E[b∥x-(k0)∥2].
Then
(40)E[∥x-(k)∥2]≤ba(λρ)(k-k0)E[∥x-(k0)∥2].
Therefore, by Definition 3, system (15) is mean square exponentially stable.Remark 7.
From Lemma2, we know that system (1) is also mean square exponentially stable.


### 3.2.H∞ Performance Analysis and Controller Design

In this subsection, we are in the position to prove the main result. TheH∞ controller design method is given in the following theorem.Theorem 8.
For given scalarsλ1, λ2, γ, and μ≥1, if there exist positive definite matrices Pi, Xi, S, Q, and matrix K-=[K0] of appropriate dimensions, i,j=1,2,3,…,2v, such that (22)–(24) and the following inequalities:
(41)[-λ1Pi00Γ14Γ15C-TUT*-I0π11Iπ12I00**-γ2Iπ11H-Tπ12H-T00***-Xj000****-S00*****-I0******-I]<0,
where Γ14=π11A~iT+π11K-TB~iT,  Γ15=π12A~iT+π12K-TB~iT,
(42)[-λ2Q00π21A-2Tπ22A-2TC-TUT*-I0π21Iπ22I00**-γ2Iπ21H-Tπ22H-T00***-Xj000****-S00*****-I0******-I]<0,XiPi=I,SQ=I,i,j=1,2,3,…,2v
hold, then system (15) with the controller gain matrix K- has robustly mean square exponential stability with H∞ disturbance attenuation level γ.Proof.
It is easy to obtain that (20) and (21) can be deduced from (41) and (42), respectively. Then from Theorem 6, it can be verified that closed-loop system (15) is mean square exponentially stable with w(k)=0.
For the nonzerow(k), using the same Lyapunov function candidates as in Theorem 6, the following relations can be obtained:
(43)ΔV1[x-(k+1)]=E[V1(x-(k+1),ξ(k+1))]-λ1V1(x-(k),ξ(k))≤(∑i=12vξi(k)Aix-(k)+f-(x,k)+H-w(k))T≤×(π11∑j=12vξj(k+1)Pj+π12Q)≤·(∑i=12vξi(k)Aix-(k)+f-(x,k)+H-w(k))≤-λ1x-T(k)(∑i=12vξi(k)Pi)x-(k)≤+x-T(k)UTUx-(k)-f-T(x,k)f-(x,k),ΔV2[x-(k+1)]=E[V2(x-(k+1),ξ(k+1))]-λ2V2(x-(k),ξ(k))≤(A2-x-(k)+f-(x,k)+H-w(k))T≤×(π21∑j=12vξj(k+1)Pj+π22Q)≤·(A2-x-(k)+f-(x,k)+H-w(k))≤-λ2x-T(k)Qx-(k)+x-T(k)UTUx-(k)≤-f-T(x,k)f-(x,k).
From inequalities (43), we have
(44)ΔV1[x-(k+1)]+zT(k)z(k)-γ2wT(k)w(k)≤∑i=12vξi(k)∑j=12vξj(k+1)ηT(k)Ξ1η(k),(45)ΔV2[x-(k+1)]+zT(k)z(k)-γ2wT(k)w(k)≤∑j=12vξj(k+1)ηT(k)Ξ2η(k),
where
(46)η(k)=[x-T(k)f-T(x,k)wT(k)]T,Ξ1=[ψ1AiT(π11Pj+π12Q)AiT(π11Pj+π12Q)H-*π11Pj+π12Q-I(π11Pj+π12Q)H-**φ1],Ξ2=[ψ2A-2T(π21Pj+π22Q)A-2T(π21Pj+π22Q)H-*π21Pj+π22Q-I(π21Pj+π22Q)H-**φ2],ψ1=AiT(π11Pj+π12Q)Ai-λ1Pi+C-TC-+UTU,ψ2=A-2T(π21Pj+π22Q)A2--λ2Q+C-TC-+UTU,φ1=H-T(π11Pj+π12Q)H--γ2I,φ2=H-T(π21Pj+π22Q)H--γ2I.
In terms of the Schur complement, we obtain
(47)[-λ1Pi+C-TC-+U-TU-00π11AiTπ12AiT*-I0π11Iπ12I**-γ2Iπ11H-Tπ12H-T***-Pj-10****-Q-1]<0,
where Ai=A~i+B~iK-,
(48)[-λ2Q+C-TC-+U-TU-00π21A-2Tπ22A-2T*-I0π21Iπ22I**-γ2Iπ21H-Tπ22H-T***-Pj-10****-Q-1]<0.
In light of Lemma 1, if equalities (47) and (48) hold, then combining (44) and (45), we have that
(49)E[V1(x-(k+1),ξ(k+1))]<λ1V1(x-(k),ξ(k))-J(k),E[V2(x-(k+1),ξ(k+1))]<λ2V2(x-(k),ξ(k))-J(k),
where J(k)=zT(k)z(k)-γ2wT(k)w(k).
Combining (22) and (49), it can be seen that
(50)E[Vσ(k)(x-(k),ξ(k))]<E[μλσ(kq)k-kqVσ(kq-1)(kq-1)-∑s=kqk-1λσ(kq)k-s-1J(s)]<E{μλσ(kq)k-kq[λkq-1kq-kq-1Vσ(kq-1)(kq-1)-∑s=kq-1kq-1λσ(kq-1)kq-s-1J(s)]-∑s=kqk-1λσ(kq)k-s-1J(s)}⋮<E[∑s=kqk-1λσ(kq)k-s-1J(s)μNσ[k0,k)λ1T1[k0,k)λ2T2[k0,k)Vσ(k0)(k0)<-∑s=k0k-1μNσ[s,k)λ1T1[s,k-1)λ2T2[s,k-1)J(s)].
Since Vσ(k)>0 and the zero-initial state assumption, it can be seen that
(51)E[∑s=k0k-1μNσ[s,k)λ1T1[s,k-1)λ2T2[s,k-1)J(s)]<0.
From (34), (51) can be written as
(52)E[∑s=k0k-1μNσ[s,k)λk-1-sJ(s)]<0.
Multiplying both sides of inequality (52) by -Nσ[0,k), we can obtain
(53)E[μ-Nσ[0,k)∑s=k0k-1μNσ[s,k)λ(k-1-s)zT(s)z(s)]<E[μ-Nσ[0,k)∑s=k0k-1μNσ[s,k)λ(k-1-s)γ2wT(s)w(s)],
which is equivalent to
(54)E[∑s=k0k-1μ-Nσ[0,s)λ(k-1-s)zT(s)z(s)]<E[∑s=k0k-1μ-Nσ[0,s)λ(k-1-s)γ2wT(s)w(s)].
Then, from Definition 4 and (24)
(55)Nσ[0,s)≤sTa<s·max{π12,π21}<s·(-lnλlnμ),
we have
(56)E[∑s=k0k-1μ-Nσ[0,s)λ(k-1-s)zT(s)z(s)]>E[∑s=k0k-1μs·lnλ/lnμλ(k-1-s)zT(s)z(s)]=E[∑s=k0k-1λ(k-1)zT(s)z(s)],E[∑s=k0k-1μ-Nσ[0,s)λ(k-1-s)γ2wT(s)w(s)]<E[∑s=k0k-1λ(k-1-s)γ2wT(s)w(s)].
Therefore
(57)E[∑s=k0k-1λ(k-1)zT(s)z(s)]<E[∑s=k0k-1λ(k-1-s)γ2wT(s)w(s)],
which implies that
(58)E[∑s=k0∞zT(s)z(s)∑k=s+1∞λ(k-1)]<E[∑s=k0∞γ2wT(s)w(s)∑k=s+1∞λ(k-1-s)].
Then
(59)E[∑s=k0∞λszT(s)z(s)]<E[∑s=k0∞γ2wT(s)w(s)].
By Definition 5, system (15) has an exponential H∞ performance γ. This completes the proof.Remark 9.
It should be pointed out that the conditions proposed in Theorem8 are not standard LMIs. In this paper, it is suggested to use the cone complementarity linearization (CCL) algorithm to solve this problem [26]; a nonlinear constraint can be converted to a linear optimization problem with a rank constraint.Remark 10.
In this paper, the mean square exponentialH∞ performance of the system (15) can be guaranteed, which means the noise attenuation performance is different when the decay degree of the system is different, and the decay degree has a close relation with the elements of the transition probabilities. Note that the scalar λ in the sequel symbolizes the decreasing rate of the Lyapunov function to be constructed for each subsystem from Theorem 6. Then, if λ→1, the evaluated performance index will approach the normal H∞ performance for the whole time domain.


## 3.1. Stability Analysis

In this subsection, sufficient conditions for the existence of mean square exponential stability of system (15) with ω(k)≡0 are given in the following theorem.Theorem 6.
System (15) is mean square exponentially stable with a decay rate λρ, if there exist positive definite matrices Pi, Q, scalars μ≥1, λ1, and λ2, such that(20)[AiT(π11Pj+π12Q)Ai-λ1Pi+UTUAiT(π11Pj+π12Q)*(π11Pj+π12Q)-I]<0,(21)[A-2T(π21Pj+π22Q)A2--λ2Q+UTUA-2T(π21Pj+π22Q)*(π21Pj+π22Q)-I]<0i,j=1,2,3,…,2v,(22)1μQ≤Pi≤μQ,(23)0<λ<1,(24)max{π12,π21}<-lnλlnμ,
where λ=λ1π21/(π12+π21)λ2π12/(π12+π21),ρ=1+max{π12,π21}·(lnμ/lnλ).Proof.
For the system (15), define the following Lyapunov function:
(25)Vσ(k)(x-(k),ξ(k))=x-T(k)P~σ(k)x-(k),
where
(26)P~σ(k)=∑i=12vξi(k)Pi,forσ(k)=1,P~σ(k)=Q,forσ(k)=2.
For subsystem 1, it follows from (15) that
(27)ΔV1[x-(k+1)]=E[V1(x-(k+1),ξ(k+1))]-λ1V1(x-(k),ξ(k))=x-T(k+1)(π11∑j=12vξj(k+1)Pj+π12Q)x-(k+1)=-λ1x-T(k)(∑i=12vξi(k)Pi)x-(k)≤(∑i=12vξi(k)Aix-(k)+f-(x,k))T=×(π11∑j=12vξj(k+1)Pj+π12Q)=·(∑i=12vξi(k)Aix-(k)+f-(x,k))=-λ1x-T(k)(∑i=12vξi(k)Pi)x-(k)=+x-T(k)UTUx-(k)-f-T(x,k)f-(x,k)=∑i=12vξi(k)∑j=12vξj(k+1)[x-(k)f-(x,k)]TΘ[x-(k)f-(x,k)],
where
(28)Θ=[AiT(π11Pj+π12Q)Ai-λ1Pi+UTUAiT(π11Pj+π12Q)*(π11Pj+π12Q)-I].
From inequality (20), one obtains
(29)E[V1(x-(k+1),ξ(k+1))]<λ1V1(x-(k),ξ(k)).
In the same way, for subsystem 2, we obtain
(30)ΔV2[x-(k+1)]=E[V2(x-(k+1),ξ(k+1))]-λ2V2(x-(k),ξ(k))=x-T(k+1)(π21∑j=12vξj(k+1)Pj+π22Q)x-(k+1)=-λ2x-T(k)Qx-(k)≤(A2-x-(k)+f-(x,k))T=×(π21∑j=12vξj(k+1)Pj+π22Q)=·(A2-x-(k)+f-(x,k))-λ2x-T(k)Qx-(k)=+x-T(k)UTUx-(k)-f-T(x,k)f-(x,k);
then
(31)E[V2(x-(k+1),ξ(k+1))]<λ2V2(x-(k),ξ(k)).
Considering the condition (22), we get that
(32)V1(x-(k),ξ(k))≤μV2(x-(k-),ξ(k-)),V2(x-(k),ξ(k))≤μV1(x-(k-),ξ(k-)).
Then for kq<k<kq+1, we get
(33)E[Vσ(k)(x-(k),ξ(k))]<E[μλσ(kq)(k-kq)Vσ(kq-1)(x-(kq),ξ(kq))]<E[μλσ(kq)(k-kq)λσ(kq-1)(kq-kq-1)Vσ(kq-1)(x-(kq-1),ξ(kq-1))]⋮<E[μNσ[k0,k)λ1T1[k0,k)λ2T2[k0,k)Vσ(k0)(x-(k0),ξ(k0))].
Note that the Markov chain is stationary (16); then
(34)E[T1[k0,k)]=π1(k-k0),E[T2[k0,k)]=π2(k-k0).
Therefore, we can obtain that
(35)E[Vσ(k)(x-(k),ξ(k))]<E[μNσ[k0,k)λ1π1(k-k0)λ2π2(k-k0)Vσ(k0)(x-(k0),ξ(k0))]=E[μNσ[k0,k)λ1(π21/(π12+π21))(k-k0)=E×λ2(π12/(π12+π21))(k-k0)Vσ(k0)(x-(k0),ξ(k0))]=E[μNσ[k0,k)λ(k-k0)Vσ(k0)(x-(k0),ξ(k0))]=E[λNσ[k0,k)(lnμ/lnλ)λ(k-k0)Vσ(k0)(x-(k0),ξ(k0))]=E[(λ1+(Nσ[k0,k)/(k-k0))·(lnμ/lnλ))(k-k0)]=×E[Vσ(k0)(x-(k0),ξ(k0))].
From Definition 4 we have that
(36)k-k0Nσ[k0,k)≥Ta.
And from [24], we can get 1/E(Ta)≤max{π12,π21}; then combining (23) and (24), we can know that
(37)0<λ1+(Nσ[k0,k)/(k-k0))·(lnμ/lnλ)<1,
which ensure the convergence of E[Vσ(k)(x-(k),ξ(k))].
In this case,(38)E[Vσ(k)(x-(k),ξ(k))]<E[(λ1+(1/Ta)·(lnμ/lnλ))(k-k0)]=×E[Vσ(k0)(x-(k0),ξ(k0))]=λ(k-k0)[1+(1/E(Ta))·(lnμ/lnλ)]E[Vσ(k0)(x-(k0),ξ(k0))]≤(λρ)(k-k0)E[Vσ(k0)(x-(k0),ξ(k0))].
Furthermore
(39)E[a∥x-(k)∥2]≤E[Vσ(k)(x-(k),ξ(k))]<(λρ)(k-k0)E[b∥x-(k0)∥2].
Then
(40)E[∥x-(k)∥2]≤ba(λρ)(k-k0)E[∥x-(k0)∥2].
Therefore, by Definition 3, system (15) is mean square exponentially stable.Remark 7.
From Lemma2, we know that system (1) is also mean square exponentially stable.


## 3.2.H∞ Performance Analysis and Controller Design

In this subsection, we are in the position to prove the main result. TheH∞ controller design method is given in the following theorem.Theorem 8.
For given scalarsλ1, λ2, γ, and μ≥1, if there exist positive definite matrices Pi, Xi, S, Q, and matrix K-=[K0] of appropriate dimensions, i,j=1,2,3,…,2v, such that (22)–(24) and the following inequalities:
(41)[-λ1Pi00Γ14Γ15C-TUT*-I0π11Iπ12I00**-γ2Iπ11H-Tπ12H-T00***-Xj000****-S00*****-I0******-I]<0,
where Γ14=π11A~iT+π11K-TB~iT,  Γ15=π12A~iT+π12K-TB~iT,
(42)[-λ2Q00π21A-2Tπ22A-2TC-TUT*-I0π21Iπ22I00**-γ2Iπ21H-Tπ22H-T00***-Xj000****-S00*****-I0******-I]<0,XiPi=I,SQ=I,i,j=1,2,3,…,2v
hold, then system (15) with the controller gain matrix K- has robustly mean square exponential stability with H∞ disturbance attenuation level γ.Proof.
It is easy to obtain that (20) and (21) can be deduced from (41) and (42), respectively. Then from Theorem 6, it can be verified that closed-loop system (15) is mean square exponentially stable with w(k)=0.
For the nonzerow(k), using the same Lyapunov function candidates as in Theorem 6, the following relations can be obtained:
(43)ΔV1[x-(k+1)]=E[V1(x-(k+1),ξ(k+1))]-λ1V1(x-(k),ξ(k))≤(∑i=12vξi(k)Aix-(k)+f-(x,k)+H-w(k))T≤×(π11∑j=12vξj(k+1)Pj+π12Q)≤·(∑i=12vξi(k)Aix-(k)+f-(x,k)+H-w(k))≤-λ1x-T(k)(∑i=12vξi(k)Pi)x-(k)≤+x-T(k)UTUx-(k)-f-T(x,k)f-(x,k),ΔV2[x-(k+1)]=E[V2(x-(k+1),ξ(k+1))]-λ2V2(x-(k),ξ(k))≤(A2-x-(k)+f-(x,k)+H-w(k))T≤×(π21∑j=12vξj(k+1)Pj+π22Q)≤·(A2-x-(k)+f-(x,k)+H-w(k))≤-λ2x-T(k)Qx-(k)+x-T(k)UTUx-(k)≤-f-T(x,k)f-(x,k).
From inequalities (43), we have
(44)ΔV1[x-(k+1)]+zT(k)z(k)-γ2wT(k)w(k)≤∑i=12vξi(k)∑j=12vξj(k+1)ηT(k)Ξ1η(k),(45)ΔV2[x-(k+1)]+zT(k)z(k)-γ2wT(k)w(k)≤∑j=12vξj(k+1)ηT(k)Ξ2η(k),
where
(46)η(k)=[x-T(k)f-T(x,k)wT(k)]T,Ξ1=[ψ1AiT(π11Pj+π12Q)AiT(π11Pj+π12Q)H-*π11Pj+π12Q-I(π11Pj+π12Q)H-**φ1],Ξ2=[ψ2A-2T(π21Pj+π22Q)A-2T(π21Pj+π22Q)H-*π21Pj+π22Q-I(π21Pj+π22Q)H-**φ2],ψ1=AiT(π11Pj+π12Q)Ai-λ1Pi+C-TC-+UTU,ψ2=A-2T(π21Pj+π22Q)A2--λ2Q+C-TC-+UTU,φ1=H-T(π11Pj+π12Q)H--γ2I,φ2=H-T(π21Pj+π22Q)H--γ2I.
In terms of the Schur complement, we obtain
(47)[-λ1Pi+C-TC-+U-TU-00π11AiTπ12AiT*-I0π11Iπ12I**-γ2Iπ11H-Tπ12H-T***-Pj-10****-Q-1]<0,
where Ai=A~i+B~iK-,
(48)[-λ2Q+C-TC-+U-TU-00π21A-2Tπ22A-2T*-I0π21Iπ22I**-γ2Iπ21H-Tπ22H-T***-Pj-10****-Q-1]<0.
In light of Lemma 1, if equalities (47) and (48) hold, then combining (44) and (45), we have that
(49)E[V1(x-(k+1),ξ(k+1))]<λ1V1(x-(k),ξ(k))-J(k),E[V2(x-(k+1),ξ(k+1))]<λ2V2(x-(k),ξ(k))-J(k),
where J(k)=zT(k)z(k)-γ2wT(k)w(k).
Combining (22) and (49), it can be seen that
(50)E[Vσ(k)(x-(k),ξ(k))]<E[μλσ(kq)k-kqVσ(kq-1)(kq-1)-∑s=kqk-1λσ(kq)k-s-1J(s)]<E{μλσ(kq)k-kq[λkq-1kq-kq-1Vσ(kq-1)(kq-1)-∑s=kq-1kq-1λσ(kq-1)kq-s-1J(s)]-∑s=kqk-1λσ(kq)k-s-1J(s)}⋮<E[∑s=kqk-1λσ(kq)k-s-1J(s)μNσ[k0,k)λ1T1[k0,k)λ2T2[k0,k)Vσ(k0)(k0)<-∑s=k0k-1μNσ[s,k)λ1T1[s,k-1)λ2T2[s,k-1)J(s)].
Since Vσ(k)>0 and the zero-initial state assumption, it can be seen that
(51)E[∑s=k0k-1μNσ[s,k)λ1T1[s,k-1)λ2T2[s,k-1)J(s)]<0.
From (34), (51) can be written as
(52)E[∑s=k0k-1μNσ[s,k)λk-1-sJ(s)]<0.
Multiplying both sides of inequality (52) by -Nσ[0,k), we can obtain
(53)E[μ-Nσ[0,k)∑s=k0k-1μNσ[s,k)λ(k-1-s)zT(s)z(s)]<E[μ-Nσ[0,k)∑s=k0k-1μNσ[s,k)λ(k-1-s)γ2wT(s)w(s)],
which is equivalent to
(54)E[∑s=k0k-1μ-Nσ[0,s)λ(k-1-s)zT(s)z(s)]<E[∑s=k0k-1μ-Nσ[0,s)λ(k-1-s)γ2wT(s)w(s)].
Then, from Definition 4 and (24)
(55)Nσ[0,s)≤sTa<s·max{π12,π21}<s·(-lnλlnμ),
we have
(56)E[∑s=k0k-1μ-Nσ[0,s)λ(k-1-s)zT(s)z(s)]>E[∑s=k0k-1μs·lnλ/lnμλ(k-1-s)zT(s)z(s)]=E[∑s=k0k-1λ(k-1)zT(s)z(s)],E[∑s=k0k-1μ-Nσ[0,s)λ(k-1-s)γ2wT(s)w(s)]<E[∑s=k0k-1λ(k-1-s)γ2wT(s)w(s)].
Therefore
(57)E[∑s=k0k-1λ(k-1)zT(s)z(s)]<E[∑s=k0k-1λ(k-1-s)γ2wT(s)w(s)],
which implies that
(58)E[∑s=k0∞zT(s)z(s)∑k=s+1∞λ(k-1)]<E[∑s=k0∞γ2wT(s)w(s)∑k=s+1∞λ(k-1-s)].
Then
(59)E[∑s=k0∞λszT(s)z(s)]<E[∑s=k0∞γ2wT(s)w(s)].
By Definition 5, system (15) has an exponential H∞ performance γ. This completes the proof.Remark 9.
It should be pointed out that the conditions proposed in Theorem8 are not standard LMIs. In this paper, it is suggested to use the cone complementarity linearization (CCL) algorithm to solve this problem [26]; a nonlinear constraint can be converted to a linear optimization problem with a rank constraint.Remark 10.
In this paper, the mean square exponentialH∞ performance of the system (15) can be guaranteed, which means the noise attenuation performance is different when the decay degree of the system is different, and the decay degree has a close relation with the elements of the transition probabilities. Note that the scalar λ in the sequel symbolizes the decreasing rate of the Lyapunov function to be constructed for each subsystem from Theorem 6. Then, if λ→1, the evaluated performance index will approach the normal H∞ performance for the whole time domain.


## 4. Numerical Example

In this section, we present an example to illustrate the effectiveness of the proposed approach. Consider the following system:(60)x˙(t)=[-110-0.1]x(t)+[00.1]u(t)+[0.06x1sinx10.01x2cosx2]+[0.050.01]w(t),z(t)=[0.10.5]x(t).Let the sampling period beT=0.3s, and 0≤τk≤0.1s. Assume that the transition probability matrix of stochastic switching signals is given as P=[0.80.20.60.4]; the corresponding matrices are given by
(61)A~1=[0.54880.2219000.97040000],A~2=[0.54880.22190.100.97040000],A~3=[0.54880.2219-0.098000.97040.0098000],A~4=[0.54880.22190.002000.97040.0098000],A-2=[0.54880.22190.003700.97040.0296001],B~1=[0.00370.02961],B~2=[-0.09630.02961],B~3=[0.10170.01981],B~4=[0.10170.01981],H-=[0.01160.00300],U=diag{0.2,0.2,0},C-=[0.10.50].
For subsystem 2 without state feedback, A-2 is an unstable matrix, and λ2>1. By Theorem 6, we can get 0<λ1<1. Take λ1=0.45, λ2=2; then λ=0.6533<1, which satisfies the condition (16). It is assumed that γ=2; solving LMIs (41) and (42) in Theorem 8, we get the following solutions:
(62)X1=[2.3889-0.18090.9960-0.18090.0703-0.87370.9960-0.873740.8886],X2=[6.4614-0.1005-31.2945-0.10050.0663-1.2282-31.2945-1.2282275.1873],X3=[7.8714-0.370343.1660-0.37030.0723-1.670943.1660-1.6709343.6616],X4=[2.4000-0.12274.0524-0.12270.2325-11.72204.0524-11.7220145.1491],S=[9.7289-2.48599.0787-2.48590.7924-4.35089.0787-4.350885.3273].
Then the controller gain can be obtained:
(63)K=[-0.8405-14.2332].
The state trajectories of the NCS and the corresponding switching signal are shown in Figures 1 and 2, respectively, where the initial condition x0=[-11]T and w(k)=0.05exp(-0.01k).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 robustly mean square exponentially stable and theH∞ disturbance attenuation level γ=2.


## 5. Conclusions

In this paper, a discrete-time switched system with two subsystems has been presented to model the NCS with time delay and packet dropout. A new approach by using the average dwell time method is proposed to study the robust stabilization andH∞ control of the addressed NCS. Finally, a numerical example has been given to demonstrate the effectiveness of the proposed method.



---

*Source: 635941-2013-04-30.xml*