𝐻_∞ Control for Networked Control Systems with Time Delays and Packet Dropouts

Abstract

This paper is concerned with the 𝐻_∞ 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.

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 exponential 𝐻_∞ 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 Section 2, 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; 𝑅^𝑛 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: 𝑥̇(𝑡)=𝐴_𝑝𝑥(𝑡)+𝐵_𝑝𝑢(𝑡)+𝑓(𝑥,𝑡)+𝐻_𝑝𝑤(𝑡),𝑧(𝑡)=𝐶𝑥(𝑡), (1) where 𝑥(𝑡)∈𝑅^𝑛, 𝑢(𝑡)∈𝑅^𝑚, and 𝑧(𝑡)∈𝑅^𝑃 are the state vector, control input vector, and controlled output vector, respectively, and 𝑤(𝑡)∈𝑅^𝑑 is the exogenous disturbance signal belonging to 𝐿₂[0,∞). 𝐴_𝑝, 𝐵_𝑝, 𝐻_𝑝, and 𝐶 are known real matrices with appropriate dimensions. 𝑓∶Ω×[𝑡₀,∞)→𝑅^𝑛(Ω⊂𝑅^𝑛) is the nonlinear function vector, and 𝑓(0,𝑡₀)=0. 𝑓 satisfies the local Lipschitz condition, that is, 𝑓𝑥₁,𝑡−𝑓𝑥₂,𝑡₂≤𝛼𝑥₁−𝑥₂₂,∀𝑥₁,𝑥₂∈Ω⊂𝑅^𝑛,∀𝑡∈𝑡₀,∞, (2) 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 by 𝜏^sc and 𝜏^ca, 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 period 𝑇,(2)the network-induced delay 𝜏_𝑘≜𝜏^sc_𝑘+𝜏^ca_𝑘 satisfies 0<𝜏_min≤𝜏_𝑘≤𝜏_max<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 to 𝑥(𝑘) is denoted by 𝑢(𝑘)=𝐾𝑥(𝑘). (3)

Consider the plant input: 𝑢(𝑘)=𝑢(𝑘)if𝑢(𝑘)and𝑥(𝑘)issuccessfullytransmitted,𝑢(𝑘−1)if𝑢(𝑘)or𝑥(𝑘)islostduringtransmission. (4)

Discretizing system (1) in one period, we can obtain the discrete state equation of the NCS: 𝑥(𝑘+1)=𝐴𝑥(𝑘)+𝐵₀𝜏_𝑘𝑢(𝑘)+𝐵₁𝜏_𝑘𝑢(𝑘−1)+𝑓(𝑥,𝑘)+𝐻𝑤(𝑘), (5) where 𝐴=𝑒^𝐴𝑝𝑇, 𝐵₀(𝜏_𝑘)=∫^{𝑇−𝜏𝑘}₀𝑒^𝐴𝑝𝑠𝑑𝑠𝐵_𝑝, 𝐵₁(𝜏_𝑘)=∫^𝑇_{𝑇−𝜏𝑘}𝑒^𝐴𝑝𝑠𝑑𝑠𝐵_𝑝, 𝐻=∫^𝑇₀𝑒^𝐴𝑝𝑠𝑑𝑠𝐻_𝑝, and 𝑓(𝑥,𝑘)=∫^𝑇₀𝑒^𝐴𝑝𝑠𝑑𝑠𝑓(𝑥,𝑘).

By using the Jordan form of the matrix 𝐴_𝑝, 𝐵₀(𝜏_𝑘) is rewritten as [21] 𝐵₀𝜏_𝑘=𝐹₀+^𝑣_𝑖=1𝜂_𝑖𝜏_𝑘𝐹_𝑖 (6) with 𝑣≤𝑛, where 𝐹₀ and 𝐹_𝑖 are constant matrices, 𝜂_𝑖(𝜏_𝑘)=𝑒^{𝑎(𝑇−𝜏𝑘)}cos(𝑏(𝑇−𝜏_𝑘)) and the eigenvalue of 𝐴 is 𝜆=𝑎+𝑖𝑏.

Then, {𝐵₀(𝜏_𝑘)∣𝑘∈𝑁} is a subset of co(𝐹) with 𝐹=𝐹₀+^𝑣_𝑖=1𝜂_𝑖𝐹_𝑖∣𝜂_𝑖=𝜂_𝑖,𝜂_𝑖=𝐹,𝑖=1,2,…,𝑣_𝑖∣𝑖=1,2,…,2^𝑣, (7) where 𝜂_𝑖=max𝜂_𝑖(𝜏_𝑘), 𝜂_𝑖=min𝜂_𝑖(𝜏_𝑘), 𝐹 is the set of vertices, and co(⋅) denotes the convex hull. Thus we obtain 𝐵₀𝜏_𝑘=^2𝑣_𝑖=1𝜉_𝑖(𝑘)𝐹_𝑖, (8) with ∑^2𝑣_𝑖=1𝜉_𝑖(𝑘)=1, 𝜉_𝑖(𝑘)∈[0,1].

Defining an augmented vector 𝑥(𝑘)=[𝑥^𝑇(𝑘)𝑢^𝑇(𝑘−1)]^𝑇, during each sampling period, two cases may arise, which can be listed as follows.

𝑆₁: no packet dropout happens; (5) can be written as 𝑥(𝑘+1)=𝐴₁(𝑘)𝑥(𝑘)+𝑓(𝑥,𝑘)+(𝑘),𝑧(𝑘)=𝐻𝑤𝐶𝑥(𝑘), (9) where 𝐴₁(𝑘)=𝐴+𝐵₀𝜏_𝑘𝐾𝐵₁𝜏_𝑘=𝐾0𝐴𝐵−𝐵₀𝜏_𝑘+𝐵00⁰𝜏_𝑘𝐼𝐾0,𝐵=^𝑇₀𝑒^𝐴𝑝𝑠𝑑𝑠𝐵_𝑝,𝐵₁𝜏_𝑘=𝐵−𝐵₀𝜏_𝑘,𝑓()=𝑓()0𝑥,𝑘,𝑥,𝑘𝐻0𝐻=,𝐶=𝐶0. (10) Substituting (8) into (9) gives rise to 𝐴₁(𝑘)=^2𝑣_𝑖=1𝜉_𝑖(𝑘)𝐴_𝑖, (11) where 𝐴_𝑖=𝐴_𝑖+𝐵_𝑖𝐾0=𝐴_𝑖+𝐵_𝑖𝐴𝐾,_𝑖=𝐴𝐵−𝐹_𝑖,𝐵00^𝑖=𝐹_𝑖𝐼,𝐾=𝐾0. (12)

𝑆₂: packet dropout happens; (5) can be written as 𝑥(𝑘+1)=𝐴₂(𝑘)𝑥(𝑘)+𝑓(𝑥,𝑘)+(𝑘),𝑧(𝑘)=𝐻𝑤𝐶𝑥(𝑘), (13) where 𝐴₂=^𝐴𝐵_0𝐼.

From (2)–(9), the nonlinear uncertainty 𝑓(𝑥,𝑘) satisfies 𝑓^𝑇(𝑥,𝑘)𝑓(𝑥,𝑘)=𝑓^𝑇(𝑥,𝑘)𝑓(𝑥,𝑘)≤𝑥^𝑇(𝑘)𝑈^𝑇𝑈𝑥(𝑘), (14) 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: 𝑥(𝑘+1)=𝐴_𝜎(𝑘)(𝑘)𝑥(𝑘)+𝑓(𝑥,𝑘)+(𝑘),𝑧(𝑘)=𝐻𝑤𝐶𝑥(𝑘), (15) 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 𝜋_𝑟𝑙 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: lim_𝑛→∞𝜋^(𝑛)_𝑟𝑙=𝜋_𝑙,𝑟,𝑙=1,2, (16) where 𝜋_𝑙 is the limitation of state 𝑙. So {𝜋₁,𝜋₂} is the stationary distribution of the Markov chain.

For an arbitrary switching sequence 𝜎(𝑘) and any given integer 𝑘>0, let 𝑘₀ imply the initial time, and 𝑘₀<𝑘₁<𝑘₂<⋯𝑘_𝑞<⋯<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 matrix 𝑆=^𝑆11𝑆12_{𝑆𝑇12𝑆22}, where 𝑆₁₁,𝑆₂₂ are square matrices, the following conditions are equivalent:(1)𝑆<0;(2)𝑆₁₁<0,𝑆₂₂−𝑆^𝑇₁₂𝑆⁻¹₁₁𝑆₁₂<0;(3)𝑆₂₂<0,𝑆₁₁−𝑆₁₂𝑆⁻¹₂₂𝑆^𝑇₁₂<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 𝐸‖𝑥(𝑘)‖²<𝛿𝛽^𝑘−𝑘0𝐸𝑥𝑘₀² (17) for all initial condition (𝑥(𝑘₀),𝜎(𝑘₀)).

Definition 4 (see [25]). For any 𝑘>𝑘₀≥0, let 𝑁_𝜎[𝑘₀,𝑘) denote the total number of the switching of 𝜎(𝑘) during the interval [𝑘₀,𝑘). If 𝑁_𝜎𝑘₀,𝑘≤𝑁₀+𝑘−𝑘₀𝑇_𝑎 (18) holds for a given 𝑁₀≥0, 𝑇_𝑎>0, then the constant 𝑇_𝑎 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 scalars 𝛾>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 ^∞_𝑘=𝑘0𝐸𝜆^𝑘𝑧^𝑇(𝑘)𝑧(𝑘)<𝛾^2∞_𝑘=𝑘0𝐸𝑤^𝑇(𝑘)𝑤(𝑘),∀𝑤(𝑘)∈𝐿₂𝑘₀,∞. (19)

3. Main Results

The following theorems present a sufficient condition for the mean square stability of the considered system and the 𝐻_∞ 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 𝜆^𝜌, if there exist positive definite matrices 𝑃_𝑖, 𝑄, scalars 𝜇≥1, 𝜆₁, and 𝜆₂, such that𝐴^𝑇_𝑖𝜋₁₁𝑃_𝑗+𝜋₁₂𝑄𝐴_𝑖−𝜆₁𝑃_𝑖+𝑈^𝑇𝑈𝐴^𝑇_𝑖𝜋₁₁𝑃_𝑗+𝜋₁₂𝑄∗𝜋₁₁𝑃_𝑗+𝜋₁₂𝑄−𝐼<0,𝐴^𝑇₂𝜋₂₁𝑃_𝑗+𝜋₂₂𝑄𝐴₂−𝜆₂𝑄+𝑈^𝑇𝑈𝐴^𝑇₂𝜋₂₁𝑃_𝑗+𝜋₂₂𝑄∗𝜋₂₁𝑃_𝑗+𝜋₂₂𝑄−𝐼<0𝑖,𝑗=1,2,3,…,2^𝑣,1𝜇𝑄≤𝑃_𝑖≤𝜇𝑄,0<𝜆<1,max𝜋₁₂,𝜋₂₁<−ln𝜆,ln𝜇 (20)(21)(22)(23)(24) where 𝜆=𝜆^{𝜋21/(𝜋12+𝜋21)}₁𝜆^{𝜋12/(𝜋12+𝜋21)}₂,𝜌=1+max{𝜋₁₂,𝜋₂₁}⋅(ln𝜇/ln𝜆).

Proof. For the system (15), define the following Lyapunov function: 𝑉_𝜎(𝑘)(𝑥(𝑘),𝜉(𝑘))=𝑥^𝑇(𝑘)𝑃_𝜎(𝑘)𝑥(𝑘), (25) where 𝑃_𝜎(𝑘)=^2𝑣_𝑖=1𝜉_𝑖(𝑘)𝑃_𝑖,for𝜎(𝑘)𝑃=1,_𝜎(𝑘)=𝑄,for𝜎(𝑘)=2. (26) For subsystem 1, it follows from (15) that Δ𝑉₁[𝑥(𝑉𝑘+1)]=𝐸₁(𝑥(𝑘+1),𝜉(𝑘+1))−𝜆₁𝑉₁(𝑥(𝑘)(𝑘=,𝜉))𝑥^𝑇(𝑘+1)𝜋²₁₁^𝑣_𝑗=1𝜉_𝑗(𝑘+1)𝑃_𝑗+𝜋₁₂𝑄𝑥(𝑘+1)−𝜆₁𝑥^𝑇(𝑘)^2𝑣_𝑖=1𝜉_𝑖(𝑘)𝑃_𝑖𝑥(𝑘)≤^2𝑣_𝑖=1𝜉_𝑖(𝑘)𝐴_𝑖𝑥(𝑘)+𝑓(𝑥,𝑘)^𝑇×𝜋²₁₁^𝑣_𝑗=1𝜉_𝑗(𝑘+1)𝑃_𝑗+𝜋₁₂𝑄⋅^2𝑣_𝑖=1𝜉_𝑖(𝑘)𝐴_𝑖𝑥(𝑘)+𝑓(𝑥,𝑘)−𝜆₁𝑥^𝑇(𝑘)^2𝑣_𝑖=1𝜉_𝑖(𝑘)𝑃_𝑖𝑥(𝑘)+𝑥^𝑇(𝑘)𝑈^𝑇𝑈𝑥(𝑘)−𝑓^𝑇(𝑥,𝑘)𝑓()=𝑥,𝑘_2𝑣𝑖=1𝜉_𝑖(𝑘)^2𝑣_𝑗=1𝜉_𝑗(𝑘+1)𝑥(𝑘)𝑓(𝑥,𝑘)^𝑇Θ𝑥(𝑘)𝑓(),𝑥,𝑘 (27) where 𝐴Θ=^𝑇𝑖𝜋₁₁𝑃_𝑗+𝜋₁₂𝑄𝐴_𝑖−𝜆₁𝑃_𝑖+𝑈^𝑇𝑈𝐴^𝑇_𝑖𝜋₁₁𝑃_𝑗+𝜋₁₂𝑄∗𝜋₁₁𝑃_𝑗+𝜋₁₂𝑄.−𝐼 (28) From inequality (20), one obtains 𝐸𝑉₁(𝑥(𝑘+1),𝜉(𝑘+1))<𝜆₁𝑉₁(𝑥(𝑘),𝜉(𝑘)). (29) In the same way, for subsystem 2, we obtain Δ𝑉₂[𝑥(𝑉𝑘+1)]=𝐸₂(𝑥(𝑘+1),𝜉(𝑘+1))−𝜆₂𝑉₂(𝑥(𝑘)(𝑘=,𝜉))𝑥^𝑇(𝑘+1)𝜋²₂₁^𝑣_𝑗=1𝜉_𝑗(𝑘+1)𝑃_𝑗+𝜋₂₂𝑄𝑥(𝑘+1)−𝜆₂𝑥^𝑇(𝑘)𝑄𝑥(𝑘)≤𝐴₂𝑥(𝑘)+𝑓(𝑥,𝑘)^𝑇×𝜋²₂₁^𝑣_𝑗=1𝜉_𝑗(𝑘+1)𝑃_𝑗+𝜋₂₂𝑄⋅𝐴₂𝑥(𝑘)+𝑓(𝑥,𝑘)−𝜆₂𝑥^𝑇(𝑘)𝑄𝑥(𝑘)+𝑥^𝑇(𝑘)𝑈^𝑇𝑈𝑥(𝑘)−𝑓^𝑇(𝑥,𝑘)𝑓(𝑥,𝑘); (30) then 𝐸𝑉₂(𝑥(𝑘+1),𝜉(𝑘+1))<𝜆₂𝑉₂(𝑥(𝑘),𝜉(𝑘)). (31) Considering the condition (22), we get that 𝑉₁(𝑥(𝑘),𝜉(𝑘))≤𝜇𝑉₂(𝑥(𝑘⁻),𝜉(𝑘⁻,𝑉))₂(𝑥(𝑘),𝜉(𝑘))≤𝜇𝑉₁(𝑥(𝑘⁻),𝜉(𝑘⁻)). (32) Then for 𝑘_𝑞<𝑘<𝑘_𝑞+1, we get 𝐸𝑉_𝜎(𝑘)(𝑥(𝑘),𝜉(𝑘))<𝐸𝜇𝜆^{(𝑘−𝑘𝑞)}_𝜎𝑘𝑞𝑉_{𝜎(𝑘𝑞−1)}𝑥𝑘_𝑞,𝜉𝑘_𝑞<𝐸𝜇𝜆^{(𝑘−𝑘𝑞)}_𝜎𝑘𝑞𝜆^{(𝑘𝑞−𝑘𝑞−1)}_{𝜎𝑘𝑞−1}𝑉_{𝜎(𝑘𝑞−1)}𝑥𝑘_𝑞−1,𝜉𝑘_𝑞−1⋮<𝐸𝜇^{𝑁𝜎[𝑘0,𝑘)}𝜆^𝑇1[𝑘0₁^,𝑘)𝜆^𝑇2[𝑘0₂^,𝑘)𝑉_𝜎(𝑘0)𝑥𝑘₀,𝜉𝑘₀. (33) Note that the Markov chain is stationary (16); then 𝐸𝑇¹𝑘₀,𝑘=𝜋₁𝑘−𝑘₀,𝐸𝑇²𝑘₀,𝑘=𝜋₂𝑘−𝑘₀. (34) Therefore, we can obtain that 𝐸𝑉_𝜎(𝑘)(𝑥(𝑘)(𝑘𝜇,𝜉))<𝐸^{𝑁𝜎[𝑘0,𝑘)}𝜆^{𝜋1(𝑘−𝑘0)}₁𝜆^{𝜋2(𝑘−𝑘0)}₂𝑉_𝜎(𝑘0)𝑥𝑘₀,𝜉𝑘₀=𝐸𝜇^{𝑁𝜎[𝑘0,𝑘)}𝜆^{(𝜋21/(𝜋12+𝜋21))(𝑘−𝑘0)}₁×𝜆^{(𝜋12/(𝜋12+𝜋21))(𝑘−𝑘0)}₂𝑉_𝜎(𝑘0)𝑥𝑘₀,𝜉𝑘₀=𝐸𝜇^{𝑁𝜎[𝑘0,𝑘)}𝜆^{(𝑘−𝑘0)}𝑉_𝜎(𝑘0)𝑥𝑘₀,𝜉𝑘₀=𝐸𝜆^{𝑁𝜎[𝑘0,𝑘)(ln𝜇/ln𝜆)}𝜆^{(𝑘−𝑘0)}𝑉_𝜎(𝑘0)𝑥𝑘₀,𝜉𝑘₀=𝐸𝜆^{1+(𝑁𝜎[𝑘0,𝑘)/(𝑘−𝑘0))⋅(ln𝜇/ln𝜆)(𝑘−𝑘0)}×𝐸𝑉_𝜎(𝑘0)𝑥𝑘₀,𝜉𝑘₀. (35) From Definition 4 we have that 𝑘−𝑘₀𝑁_𝜎𝑘₀,𝑘≥𝑇_𝑎. (36) And from [24], we can get 1/𝐸(𝑇_𝑎)≤max{𝜋₁₂,𝜋₂₁}; then combining (23) and (24), we can know that 0<𝜆^{1+(𝑁𝜎[𝑘0,𝑘)/(𝑘−𝑘0))⋅(ln𝜇/ln𝜆)}<1, (37) which ensure the convergence of 𝐸[𝑉_𝜎(𝑘)(𝑥(𝑘),𝜉(𝑘))].
In this case, 𝐸𝑉_𝜎(𝑘)(𝑥(𝑘)(𝑘𝜆,𝜉))<𝐸^{1+(1/𝑇𝑎)⋅(ln𝜇/ln𝜆)(𝑘−𝑘0)}×𝐸𝑉_𝜎(𝑘0)𝑥𝑘₀,𝜉𝑘₀=𝜆^{(𝑘−𝑘0)[1+(1/𝐸(𝑇𝑎))⋅(ln𝜇/ln𝜆)]}𝐸𝑉_𝜎(𝑘0)𝑥𝑘₀,𝜉𝑘₀≤𝜆^{𝜌(𝑘−𝑘0)}𝐸𝑉_𝜎(𝑘0)𝑥𝑘₀,𝜉𝑘₀. (38) Furthermore 𝐸𝑎‖𝑥(𝑘)‖²≤𝐸𝑉_𝜎(𝑘)(𝑥(𝑘)(𝑘<𝜆,𝜉))_{𝜌(𝑘−𝑘0)}𝐸𝑏𝑥𝑘₀². (39) Then 𝐸‖𝑥(𝑘)‖²≤𝑏𝑎𝜆^{𝜌(𝑘−𝑘0)}𝐸𝑥𝑘₀². (40) Therefore, by Definition 3, system (15) is mean square exponentially stable.

Remark 7. From Lemma 2, 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. The 𝐻_∞ controller design method is given in the following theorem.

Theorem 8. For given scalars 𝜆₁, 𝜆₂, 𝛾, and 𝜇≥1, if there exist positive definite matrices 𝑃_𝑖, 𝑋_𝑖, 𝑆, 𝑄, and matrix 𝐾=[𝐾0] of appropriate dimensions, 𝑖,𝑗=1,2,3,…,2^𝑣, such that (22)–(24) and the following inequalities: −𝜆₁𝑃_𝑖00Γ₁₄Γ₁₅𝐶^𝑇𝑈^𝑇∗−𝐼0√𝜋₁₁𝐼√𝜋₁₂𝐼00∗∗−𝛾²𝐼√𝜋₁₁𝐻^𝑇√𝜋₁₂𝐻^𝑇00∗∗∗−𝑋_𝑗000∗∗∗∗−𝑆00∗∗∗∗∗−𝐼0∗∗∗∗∗∗−𝐼<0, (41) where Γ₁₄=√𝜋₁₁𝐴^𝑇_𝑖+√𝜋₁₁𝐾^𝑇𝐵^𝑇_𝑖, Γ₁₅=√𝜋₁₂𝐴^𝑇_𝑖+√𝜋₁₂𝐾^𝑇𝐵^𝑇_𝑖, −𝜆₂𝑄00√𝜋₂₁𝐴^𝑇₂√𝜋₂₂𝐴^𝑇₂𝐶^𝑇𝑈^𝑇∗−𝐼0√𝜋₂₁𝐼√𝜋₂₂𝐼00∗∗−𝛾²𝐼√𝜋₂₁𝐻^𝑇√𝜋₂₂𝐻^𝑇00∗∗∗−𝑋_𝑗𝑋000∗∗∗∗−𝑆00∗∗∗∗∗−𝐼0∗∗∗∗∗∗−𝐼<0,_𝑖𝑃_𝑖=𝐼,𝑆𝑄=𝐼,𝑖,𝑗=1,2,3,…,2^𝑣 (42) 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 nonzero 𝑤(𝑘), using the same Lyapunov function candidates as in Theorem 6, the following relations can be obtained: Δ𝑉₁[𝑥(𝑉𝑘+1)]=𝐸₁(𝑥(𝑘+1),𝜉(𝑘+1))−𝜆₁𝑉₁(𝑥(𝑘)(𝑘≤,𝜉))_2𝑣𝑖=1𝜉_𝑖(𝑘)𝐴_𝑖𝑥(𝑘)+𝑓(𝑥,𝑘)+𝐻𝑤(𝑘)^𝑇×𝜋²₁₁^𝑣_𝑗=1𝜉_𝑗(𝑘+1)𝑃_𝑗+𝜋₁₂𝑄⋅^2𝑣_𝑖=1𝜉_𝑖(𝑘)𝐴_𝑖𝑥(𝑘)+𝑓(𝑥,𝑘)+𝐻𝑤(𝑘)−𝜆₁𝑥^𝑇(𝑘)^2𝑣_𝑖=1𝜉_𝑖(𝑘)𝑃_𝑖𝑥(𝑘)+𝑥^𝑇(𝑘)𝑈^𝑇𝑈𝑥(𝑘)−𝑓^𝑇(𝑥,𝑘)𝑓(𝑥,𝑘),Δ𝑉₂[𝑥(𝑉𝑘+1)]=𝐸₂(𝑥(𝑘+1),𝜉(𝑘+1))−𝜆₂𝑉₂(𝑥(𝑘)(𝑘≤,𝜉))𝐴₂𝑥(𝑘)+𝑓(𝑥,𝑘)+𝐻𝑤(𝑘)^𝑇×𝜋²₂₁^𝑣_𝑗=1𝜉_𝑗(𝑘+1)𝑃_𝑗+𝜋₂₂𝑄⋅𝐴₂𝑥(𝑘)+𝑓(𝑥,𝑘)+𝐻𝑤(𝑘)−𝜆₂𝑥^𝑇(𝑘)𝑄𝑥(𝑘)+𝑥^𝑇(𝑘)𝑈^𝑇𝑈𝑥(𝑘)−𝑓^𝑇(𝑥,𝑘)𝑓(𝑥,𝑘). (43) From inequalities (43), we have Δ𝑉₁[𝑥(𝑘+1)]+𝑧^𝑇(𝑘)𝑧(𝑘)−𝛾²𝑤^𝑇(𝑘)𝑤(𝑘)≤^2𝑣_𝑖=1𝜉_𝑖(𝑘)^2𝑣_𝑗=1𝜉_𝑗(𝑘+1)𝜂^𝑇(𝑘)Ξ₁𝜂(𝑘),Δ𝑉₂[𝑥(𝑘+1)]+𝑧^𝑇(𝑘)𝑧(𝑘)−𝛾²𝑤^𝑇(𝑘)𝑤(𝑘)≤^2𝑣_𝑗=1𝜉_𝑗(𝑘+1)𝜂^𝑇(𝑘)Ξ₂𝜂(𝑘), (44)(45) where 𝜂(𝑘)=𝑥^𝑇(𝑘)𝑓^𝑇(𝑥,𝑘)𝑤^𝑇(𝑘)^𝑇,Ξ₁=𝜓₁𝐴^𝑇_𝑖𝜋₁₁𝑃_𝑗+𝜋₁₂𝑄𝐴^𝑇_𝑖𝜋₁₁𝑃_𝑗+𝜋₁₂𝑄𝐻∗𝜋₁₁𝑃_𝑗+𝜋₁₂𝑄−𝐼𝜋₁₁𝑃_𝑗+𝜋₁₂𝑄𝐻∗∗𝜑₁,Ξ₂=𝜓₂𝐴^𝑇₂𝜋₂₁𝑃_𝑗+𝜋₂₂𝑄𝐴^𝑇₂𝜋₂₁𝑃_𝑗+𝜋₂₂𝑄𝐻∗𝜋₂₁𝑃_𝑗+𝜋₂₂𝑄−𝐼𝜋₂₁𝑃_𝑗+𝜋₂₂𝑄𝐻∗∗𝜑₂,𝜓₁=𝐴^𝑇_𝑖𝜋₁₁𝑃_𝑗+𝜋₁₂𝑄𝐴_𝑖−𝜆₁𝑃_𝑖+𝐶^𝑇𝐶+𝑈^𝑇𝜓𝑈,₂=𝐴^𝑇₂𝜋₂₁𝑃_𝑗+𝜋₂₂𝑄𝐴₂−𝜆₂𝑄+𝐶^𝑇𝐶+𝑈^𝑇𝜑𝑈,₁=𝐻^𝑇𝜋₁₁𝑃_𝑗+𝜋₁₂𝑄𝐻−𝛾²𝜑𝐼,₂=𝐻^𝑇𝜋₂₁𝑃_𝑗+𝜋₂₂𝑄𝐻−𝛾²𝐼. (46) In terms of the Schur complement, we obtain −𝜆₁𝑃_𝑖+𝐶^𝑇𝐶+𝑈^𝑇𝑈00√𝜋₁₁𝐴^𝑇_𝑖√𝜋₁₂𝐴^𝑇_𝑖∗−𝐼0√𝜋₁₁𝐼√𝜋₁₂𝐼∗∗−𝛾²𝐼√𝜋₁₁𝐻^𝑇√𝜋₁₂𝐻^𝑇∗∗∗−𝑃_𝑗⁻¹0∗∗∗∗−𝑄⁻¹<0, (47) where 𝐴_𝑖=𝐴_𝑖+𝐵_𝑖𝐾, −𝜆₂𝑄+𝐶^𝑇𝐶+𝑈^𝑇𝑈00√𝜋₂₁𝐴^𝑇₂√𝜋₂₂𝐴^𝑇₂∗−𝐼0√𝜋₂₁𝐼√𝜋₂₂𝐼∗∗−𝛾²𝐼√𝜋₂₁𝐻^𝑇√𝜋₂₂𝐻^𝑇∗∗∗−𝑃_𝑗⁻¹0∗∗∗∗−𝑄⁻¹<0. (48) In light of Lemma 1, if equalities (47) and (48) hold, then combining (44) and (45), we have that 𝐸𝑉₁(𝑥(𝑘+1),𝜉(𝑘+1))<𝜆₁𝑉₁(𝑥(𝑘)(𝑘(𝑘),𝐸𝑉,𝜉))−𝐽₂(𝑥(𝑘+1),𝜉(𝑘+1))<𝜆₂𝑉₂(𝑥(𝑘),𝜉(𝑘))−𝐽(𝑘), (49) where 𝐽(𝑘)=𝑧^𝑇(𝑘)𝑧(𝑘)−𝛾²𝑤^𝑇(𝑘)𝑤(𝑘).
Combining (22) and (49), it can be seen that 𝐸𝑉_𝜎(𝑘)(𝑥(𝑘),𝜉(𝑘))<𝐸𝜇𝜆^{𝑘−𝑘𝑞}_𝜎𝑘𝑞𝑉_{𝜎(𝑘𝑞−1)}𝑘_𝑞−1−^𝑘−1_{𝑠=𝑘𝑞}𝜆_𝜎𝑘^{𝑘−𝑠−1}_𝑞𝐽(𝑠)<𝐸𝜇𝜆^{𝑘−𝑘𝑞}_𝜎𝑘𝑞𝜆^{𝑘𝑞−𝑘𝑞−1}_𝑘𝑞−1𝑉_{𝜎(𝑘𝑞−1)}𝑘_𝑞−1−^𝑘𝑞−1_{𝑠=𝑘𝑞−1}𝜆^𝑘𝑞_𝜎𝑘^−𝑠−1_𝑞−1𝐽(𝑠)−^𝑘−1_{𝑠=𝑘𝑞}𝜆_𝜎𝑘^{𝑘−𝑠−1}_𝑞𝐽(𝑠)⋮<𝐸𝜇^{𝑁𝜎[𝑘0,𝑘)}𝜆^𝑇1[𝑘0₁^,𝑘)𝜆^𝑇2[𝑘0₂^,𝑘)𝑉_𝜎(𝑘0)𝑘₀−^𝑘−1_𝑠=𝑘0𝜇^{𝑁𝜎[𝑠,𝑘)}𝜆^𝑇1₁^{[𝑠,𝑘−1)}𝜆^𝑇2₂^{[𝑠,𝑘−1)}𝐽(𝑠). (50) Since 𝑉_𝜎(𝑘)>0 and the zero-initial state assumption, it can be seen that 𝐸^𝑘−1_𝑠=𝑘0𝜇^{𝑁𝜎[𝑠,𝑘)}𝜆^𝑇1₁^{[𝑠,𝑘−1)}𝜆^𝑇2₂^{[𝑠,𝑘−1)}𝐽(𝑠)<0. (51) From (34), (51) can be written as 𝐸^𝑘−1_𝑠=𝑘0𝜇^{𝑁𝜎[𝑠,𝑘)}𝜆^{𝑘−1−𝑠}𝐽(𝑠)<0. (52) Multiplying both sides of inequality (52) by −𝑁_𝜎[0,𝑘), we can obtain 𝐸𝜇^{−𝑁𝜎[0,𝑘)𝑘−1}_𝑠=𝑘0𝜇^{𝑁𝜎[𝑠,𝑘)}𝜆^{(𝑘−1−𝑠)}𝑧^𝑇(𝑠)𝑧(𝑠)<𝐸𝜇^{−𝑁𝜎[0,𝑘)𝑘−1}_𝑠=𝑘0𝜇^{𝑁𝜎[𝑠,𝑘)}𝜆^{(𝑘−1−𝑠)}𝛾²𝑤^𝑇(𝑠)𝑤(𝑠), (53) which is equivalent to 𝐸^𝑘−1_𝑠=𝑘0𝜇^{−𝑁𝜎[0,𝑠)}𝜆^{(𝑘−1−𝑠)}𝑧^𝑇(𝑠)𝑧(𝑠)<𝐸^𝑘−1_𝑠=𝑘0𝜇^{−𝑁𝜎[0,𝑠)}𝜆^{(𝑘−1−𝑠)}𝛾²𝑤^𝑇(𝑠)𝑤(𝑠). (54) Then, from Definition 4 and (24) 𝑁_𝜎[)≤𝑠0,𝑠𝑇_𝑎<𝑠⋅max𝜋₁₂,𝜋₂₁<𝑠⋅−ln𝜆,ln𝜇 (55) we have 𝐸^𝑘−1_𝑠=𝑘0𝜇^{−𝑁𝜎[0,𝑠)}𝜆^{(𝑘−1−𝑠)}𝑧^𝑇(𝑠)𝑧(𝑠)>𝐸^𝑘−1_𝑠=𝑘0𝜇^{𝑠⋅ln𝜆/ln𝜇}𝜆^{(𝑘−1−𝑠)}𝑧^𝑇(𝑠)𝑧(𝑠)=𝐸^𝑘−1_𝑠=𝑘0𝜆^(𝑘−1)𝑧^𝑇(𝑠)𝑧(𝑠),𝐸^𝑘−1_𝑠=𝑘0𝜇^{−𝑁𝜎[0,𝑠)}𝜆^{(𝑘−1−𝑠)}𝛾²𝑤^𝑇(𝑠)𝑤(𝑠)<𝐸^𝑘−1_𝑠=𝑘0𝜆^{(𝑘−1−𝑠)}𝛾²𝑤^𝑇(𝑠)𝑤(𝑠). (56) Therefore 𝐸^𝑘−1_𝑠=𝑘0𝜆^(𝑘−1)𝑧^𝑇(𝑠)𝑧(𝑠)<𝐸^𝑘−1_𝑠=𝑘0𝜆^{(𝑘−1−𝑠)}𝛾²𝑤^𝑇(𝑠)𝑤(𝑠), (57) which implies that 𝐸^∞_𝑠=𝑘0𝑧^𝑇(𝑠)𝑧(𝑠)^∞_𝑘=𝑠+1𝜆^(𝑘−1)<𝐸^∞_𝑠=𝑘0𝛾²𝑤^𝑇(𝑠)𝑤(𝑠)^∞_𝑘=𝑠+1𝜆^{(𝑘−1−𝑠)}. (58) Then 𝐸^∞_𝑠=𝑘0𝜆^𝑠𝑧^𝑇(𝑠)𝑧(𝑠)<𝐸^∞_𝑠=𝑘0𝛾²𝑤^𝑇(𝑠)𝑤(𝑠). (59) 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 Theorem 8 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 exponential 𝐻_∞ 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: 𝑥̇(𝑡)=𝑥(𝑡)+0𝑢(𝑡)+−110−0.10.10.06𝑥₁sin𝑥₁0.01𝑥₂cos𝑥₂+𝑤(𝑡),𝑧(𝑡)=𝑥(𝑡).0.050.010.10.5 (60)

Let the sampling period be 𝑇=0.3s, and 0≤𝜏_𝑘≤0.1s. Assume that the transition probability matrix of stochastic switching signals is given as 𝑃=^0.80.2_0.60.4; the corresponding matrices are given by 𝐴₁=,𝐴0.54880.2219000.97040000₂=,𝐴0.54880.22190.100.97040000₃=,𝐴0.54880.2219−0.098000.97040.0098000₄=,0.54880.22190.002000.97040.0098000𝐴₂=,𝐵0.54880.22190.003700.97040.0296001¹=1,𝐵0.00370.0296₂=1,𝐵−0.09630.0296₃=1,𝐵0.10170.0198₄=1,0.10170.01980,{},𝐻=0.01160.0030𝑈=diag0.2,0.2,0𝐶=0.10.50. (61) For subsystem 2 without state feedback, 𝐴₂ 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: 𝑋₁=,𝑋2.3889−0.18090.9960−0.18090.0703−0.87370.9960−0.873740.8886₂=,𝑋6.4614−0.1005−31.2945−0.10050.0663−1.2282−31.2945−1.2282275.1873₃=,𝑋7.8714−0.370343.1660−0.37030.0723−1.670943.1660−1.6709343.6616₄=,.2.4000−0.12274.0524−0.12270.2325−11.72204.0524−11.7220145.1491𝑆=9.7289−2.48599.0787−2.48590.7924−4.35089.0787−4.350885.3273 (62) Then the controller gain can be obtained: 𝐾=−0.8405−14.2332. (63) 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 the 𝐻_∞ 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 and 𝐻_∞ control of the addressed NCS. Finally, a numerical example has been given to demonstrate the effectiveness of the proposed method.

Acknowledgment

This work was supported by the National Natural Science Foundation of China under Grants nos. 60974027 and 61273120.