Modeling and Output Feedback Control of Networked Control Systems with Both Time Delays; and Packet Dropouts

Abstract

This paper is concerned with the problem of modeling and output feedback controller design for a class of discrete-time networked control systems (NCSs) with time delays and packet dropouts. A Markovian jumping method is proposed to deal with random time delays and packet dropouts. Different from the previous studies on the issue, the characteristics of networked communication delays and packet dropouts can be truly reflected by the unified model; namely, both sensor-to-controller (S-C) and controller-to-actuator (C-A) time delays, and packet dropouts are modeled and their history behavior is described by multiple Markov chains. The resulting closed-loop system is described by a new Markovian jump linear system (MJLS) with Markov delays model. Based on Lyapunov stability theory and linear matrix inequality (LMI) method, sufficient conditions of the stochastic stability and output feedback controller design method for NCSs with random time delays and packet dropouts are presented. A numerical example is given to illustrate the effectiveness of the proposed method.

1. Introduction

Networked control systems (NCSs) are systems in which control loops are closed over a real-time communication network. The fact that controllers, sensors, and actuators are not connected through point-to-point connections but through a multipurpose network offers advantages, such as low cost, decreased wiring, ease of installation and maintenance, and high efficiency, reliability, and flexibility [1–5]. Consequently, NCSs are applied in a broad range such as manufacturing plants, vehicles, aircrafts, spacecrafts, and remote surgery. However, incorporation of a network in control loops results in various constraints such as time delays and packet dropouts, due to limited bandwidth, quantization errors caused by hybrid nature of NCSs, variable sampling or transmission intervals due to multiple nodes, clock asynchronization among local and remote nodes, network security and safety, and network security due to shared communication networks [6–8]. It is generally known that any of these networked-induced communication imperfections and constraints can degrade closed-loop performance or, even worse, can harm closed-loop stability of NCSs. Therefore, it is important to know how these effects influence the stability properties.

Recently, systematic approaches to analyze stability of NCSs subject to only one of these networked-induced imperfections are well developed. For instance, the effects of packet dropouts are studied in [9–14], and of time delays in [15–24], of quantization in [7, 25–28], of time-varying transmission intervals and communication constraints in [29–31] and [32–35], respectively. However, in NCSs all the aforementioned limitations and constraints are present simultaneously. Unfortunately, most of the results are available that study single problem of these imperfections. In this paper, we will focus on the modeling and stability of NCSs with random time delays and packet dropouts.

In fact, packet dropout and time delay are both important issues that could severely destabilize NCSs. To study these issues, much research has been done on the effect of packet dropout and time delay on NCSs. As to the packet dropout, many approaches have been developed. Stabilization problem of NCSs with both arbitrary and Markovian packet losses is considered in [9]. The 𝐻_∞ control problem for NCSs with packet dropouts is studied in [10, 13]. A new NCSs model is considered for NCSs with single- and multiple-packet transmissions in [11]. The control problem for NCSs with Bernoulli-distributed stochastic packet losses is revisited in [14]. As to the time delay, great efforts have been made on stability analysis and controller design for NCSs with interval time-varying delay, uncertain time delay, large time delay, or random time delay, as presented in [15–23] (for more details, please refer to the literature therein).

So far, most of the results considered the network-induced delay and data packet loss separately, while in practice, the time delay and packet dropout are present simultaneously in NCSs. Fewer results are available that study combinations of time delay and packet dropout in the literature [36–42]. Furthermore, in [36–38], the switched system method is studied for NCSs with both delay and packet dropout. The two-mode-dependent state feedback controller design for NCSs with time delays and packet dropouts is addressed by augmenting the state variable approach in [39]. In [40] the robust 𝐻_∞ filtering problem for a class of uncertain nonlinear networked systems with both stochastic time-varying communication delays and packet dropouts is investigated. In [41] the problem of 𝐻_∞ controller design for NCSs with time delay and packet dropout by applying the linear estimation-based time delay and packet dropout compensation method is considered. The receding horizon 𝐻_∞ control problem for a class of NCSs with random delay and packet disordering is investigated by using the receding optimization principle in [42]. To the best of the authors’ knowledge, up to now, little attention has been paid to the study of unified time delays and packet dropouts model and output feedback controller design for NCSs based on Markovian jump linear system (MJLS) with Markovian time delays model, which motivates the study of this paper.

In this paper, we address the unified model, stability analysis, and output feedback controller design of NCSs with the sensor-to-controller (S-C) and controller-to-actuator (C-A) random time delays and packet dropouts under an MJLS [43] framework. The present paper involves three contributions compared to the previous relevant works. The first is in the unified new model of NCSs with random network time delays and packet dropouts. The second one is that the Lyapunov functional method is used for stochastic stability analysis for NCSs. Compared with the extended matrix method, the Lyapunov functional avoids augmenting the dimension of the closed-loop system state, thus avoiding the complicated computation and conservatism caused by extended matrix method. The third one is the output feedback controller which is used to stabilize the closed-loop system, and the results are applied to a classical angular positioning system to illustrate the effectiveness of the approach.

Notations. In the sequel, matrices are assumed to have appropriate dimensions. ℝ^𝑛 and ℝ^𝑛×𝑚 denote, respectively, the 𝑛 dimensional Euclidean space and the set of all 𝑛×𝑚 real matrices. The notations 𝐴>0 and 𝐴<0 are used to denote a positive and negative definite matrices, respectively. diag(𝐴₁,…,𝐴_𝑛) refers to an 𝑛×𝑛 diagonal matrix with 𝐴_𝑖 as its 𝑖th diagonal entry. 𝐼 and 0 denote identity matrix and zero matrix with appropriate dimensions, respectively. The superscript 𝑇 denotes the transpose for vectors or matrices. 𝔼[⋅] denotes the mathematical expectation operator. The symbol ∗ within a matrix represents the symmetric entries.

2. Problem Description

In this paper, we consider the following linear discrete-time system: 𝑥_𝑝((𝑘+1)𝑇)=𝐴𝑥_𝑝()(),𝑦()𝑘𝑇+𝐵𝑢𝑘𝑇𝑘𝑇=𝐶𝑥_𝑝(𝑘𝑇), (1) where 𝑥_𝑝(𝑘𝑇)∈ℝ^𝑛 is the system state, 𝑢(𝑘𝑇)∈ℝ^𝑚 is the control input, and 𝑦(𝑘𝑇)∈ℝ^𝑝 is the measurable output. 𝑘𝑇 and 𝑇 are the sampling instant and the sampling period of the sensor. 𝐴, 𝐵, and 𝐶 are known real constant matrices with appropriate dimensions. In this paper, for simplicity of presentation, 𝑘𝑇 is denoted by 𝑘.

Considering the same assumption in [16], the sensor, the controller, and the actuator are time driven and are connected over a network medium. The framework of the system over a network medium is depicted in Figure 1. The holder is chosen as zero-order hold (ZOH) holder. Under these assumptions, it is known that the controller and actuator update at the instant 𝑘 only will always use the most recent data; otherwise, it will maintain the old data. In the NCS as in Figure 1, network-induced time delays and packet dropouts exist in the communication links between the S-C and C-A.

Figure 1 

The structure of the NCS with random delays and/or packet dropouts.

Random time delays exist in the S-C and C-A, as shown in Figure 1. Random but bounded scalar 𝑑(𝑘)≥0 and 𝜏(𝑘)≥0 are the random S-C and C-A time delays, respectively. It is assumed that both 𝑑(𝑘) and 𝜏(𝑘) are bounded, which are 0≤𝑑≤𝑑(𝑘)≤𝑑, 0≤𝜏≤𝜏(𝑘)≤𝜏, where 𝑑=min{𝑑(𝑘)}, 𝑑=max{𝑑(𝑘)}, 𝜏=min{𝜏(𝑘)}, and 𝜏=max{𝜏(𝑘)}. One way to model the delays 𝑑(𝑘) and 𝜏(𝑘) is using the finite state Markov chains as shown in [19, 44–46]. The main advantages of the Markov model is as follows: the dependency between delays is taken into account since in real networks the current time delays are usually related with the previous delays [45]. In this paper 𝑑(𝑘) and 𝜏(𝑘) are modeled as two homogeneous Markov chains that take value in 𝑆₂={0,1,…,𝑠₂} and 𝑆₃={0,1,…,𝑠₃}. 𝜆={𝜆_𝑚𝑛} and 𝛾={𝛾_𝑞𝑝} denote transition probability matrices of 𝑑(𝑘) and 𝜏(𝑘), respectively, with probabilities 𝜆_𝑚𝑛 and 𝜋_𝑖𝑗, which are defined in (7).

In Figure 1, 𝑆₁ and 𝑆₂ denote the network switches between the S-C and C-A, respectively. 𝛼 and 𝛽(𝛼=0,1;𝛽=0,1) denote the states of 𝑆₁ and 𝑆₂. When 𝑆₁(𝑆₂) is in state 𝛼=0(𝛽=0), the packet is received successfully and the 𝑦(𝑘)=𝑦(𝑘−𝑑(𝑘))(𝑢(𝑘)=𝑢(𝑘−𝜏(𝑘))). Whereas when 𝑆₁(𝑆₂) is in state 𝛼=1(𝛽=1), the packet is lost and the switch output (control input) is held at the previous value 𝑦(𝑘)=𝑦(𝑘−1)(𝑢(𝑘)=𝑢(𝑘−1)). The behavior of the S-C and C-A time delays and packet dropouts can be modeled as 𝑦(𝑘)=(1−𝛼)𝑦(𝑘−𝑑(𝑘))+𝛼𝑦(),𝑢(𝑘)=𝑘−11−𝛽𝑢(𝑘−𝜏(𝑘))+𝛽𝑢(𝑘−1), (2) where 𝛼=0,if𝑆₁isclosedandthepacketisreceived1,otherwise,𝛽=0,if𝑆₂isclosedthepacketisreceived1,otherwise. (3) From the above analysis, the random data packet dropouts in S-C and C-A can be modeled a discrete-time homogeneous Markov chain 𝜇(𝑘) with four modes. Four modes of Markov chain 𝜇(𝑘)=1, 𝜇(𝑘)=2, 𝜇(𝑘)=3, and 𝜇(𝑘)=4 are corresponding to four states (𝛼=0,𝛽=0), (𝛼=0,𝛽=1), (𝛼=1,𝛽=0), and (𝛼=1,𝛽=1) of network switches 𝑆₁ and 𝑆₂.

This paper studies the stability of system (1) under an output feedback controller; that is, 𝑢(𝑘)=𝐾𝑦(𝑘), (4) where 𝐾 is the output feedback controller gain.

Remark 1. There are two ways to design the controller (4). One is adopting a mode-independent controller, and another one is using a mode-dependent controller (the controller gain dependent on the information of time delays and packet losses). However, the mode-independent controller (4) can avoid the controller gain switching in high frequency following the multimodes of Markov chains 𝜇(𝑘), 𝜏(𝑘), and 𝑑(𝑘) in system (5). Hence, in this paper, we choose the mode-independent controller.

Let 𝑥(𝑘)=^𝑥𝑇_𝑝^{(𝑘)𝑦𝑇(𝑘−1)𝑢𝑇(𝑘−1)𝑇} be the augmented state vector. Applying the controller (4) to (1) and (2), we can obtain the closed-loop system for the NCS with random time delays and packet dropouts in Figure 1 as 𝑥(𝑘+1)=𝐴𝜇(𝑘)𝑥(𝑘)+³_𝑠=1𝐵_𝑠𝜇(𝑘)𝐻_𝑠𝑥𝑘−𝜂_𝑠(𝑘),𝑥(𝑘)=𝜑(𝑘),𝑘=−𝑑−𝜏,−𝑑−𝜏+1,…,0, (5) where 𝐴𝜇(𝑘)=,𝐴0𝛽𝐵0𝛼𝐼000𝛽𝐼𝐵₁𝜇(𝑘)=0()𝐶01−𝛼,𝐵₂𝜇(𝑘)=𝐵^∼₂𝜇(𝑘)𝐾,𝐵₃𝜇(𝑘)=𝐵^∼₃𝜇(𝑘)𝐾𝐶,_∼𝐵₂𝜇(𝑘)=𝛼𝐵0𝛼𝐼,1−𝛽1−𝛽_∼𝐵₃𝜇(𝑘=()𝐵0()𝐼,𝐻)1−𝛼1−𝛽1−𝛼1−𝛽₁=𝐼00,𝐻₂=,𝐻0𝐼0₃=,𝜂𝐼00₁(𝑘)=𝑑(𝑘),𝜂₂(𝑘)(𝑘),𝜂=𝜏₃(𝑘)=𝑑(𝑘−𝜏(𝑘))+𝜏(𝑘)≈𝑑(𝑘)+𝜏(𝑘),𝑑=max{𝑑(𝑘)},𝑑=min{𝑑(𝑘)},𝜏=max{𝜏(𝑘)},𝜏=min{𝜏(𝑘)}, (6)𝜑(𝑘), 𝑘=−𝑑−𝜏, −𝑑−𝜏+1,…,0 is the system initial value.

Remark 2. In (5), we assume that delay steps 𝜂₃(𝑘)=𝑑(𝑘−𝜏(𝑘))+𝜏(𝑘)≈𝑑(𝑘)+𝜏(𝑘). There are three reasons to support the ideal hypothesis: the first one is that 𝑑(𝑘) and 𝑑(𝑘−𝜏(𝑘)) take values in the same finite set 𝑆₂={0,1,…,𝑑_max}; the second one is that 𝜂₃(𝑘)=𝑑(𝑘−𝜏(𝑘))+𝜏(𝑘) and 𝑑(𝑘)+𝜏(𝑘) take values in the same finite set {0,1,2,…,𝑑_max+𝜏_max}; and the third one is that for system (5), 𝜂₃(𝑘) denotes the delay steps of system state 𝑥(𝑘); using (𝑑(𝑘)+𝜏(𝑘)) approximation equal to (𝑑(𝑘−𝜏(𝑘))+𝜏(𝑘)) will not affect the performance of system (5). Therefore, 𝜂₃(𝑘)=𝑑(𝑘−𝜏(𝑘))+𝜏(𝑘) will be replaced by 𝜂₃(𝑘)=𝑑(𝑘)+𝜏(𝑘) in the following.

In system (5), {𝜇(𝑘),𝑘∈ℤ}, {𝑑(𝑘),𝑘∈ℤ} and {𝜏(𝑘),𝑘∈ℤ}, are three finite state discrete-time homogeneous Markov chains. The three Markov chains take values in the finite sets 𝑆₁={1,2,…,𝑠₁}, 𝑆₂={0,1,…,𝑠₂}, and 𝑆₃={0,1,…,𝑠₃} with transition probabilities Pr𝜇(𝑘+1)=𝑗∣𝜇(𝑘)=𝑖=𝜋_𝑖𝑗,𝜋_𝑖=Pr𝜋₀,{𝑑()(𝑘)}=𝑖Pr𝑘+1=𝑛∣𝑑=𝑚=𝜆_𝑚𝑛,𝜆_𝑚=Pr𝜆₀,𝜏()(𝑘)=𝑚Pr𝑘+1=𝑞∣𝜏=𝑝=𝛾_𝑝𝑞,𝛾_𝑝=Pr𝛾₀=𝑝, (7) where 𝜋_𝑖𝑗≥0, 𝜆_𝑚𝑛≥0, and 𝛾_𝑝𝑞≥0 for all 𝑖,𝑗∈𝑆₁,  𝑚,𝑛∈𝑆₂, 𝑝,𝑞∈𝑆₃ and ^𝑠1_𝑗=1𝜋_𝑖𝑗=1,^𝑠2_𝑛=0𝜆_𝑚𝑛=1,_𝑠3𝑞=0𝛾_𝑝𝑞=1. (8) For 𝜇(𝑘)=𝑖, 𝑖∈𝑆₁, when 𝑖=1, 𝑖=2, 𝑖=3, and 𝑖=4, the 𝛼 and 𝛽 in (5) take values (𝛼=0,𝛽=0), (𝛼=0,𝛽=1), (𝛼=1,𝛽=0), and (𝛼=1,𝛽=1), respectively. 𝐴(𝜇(𝑘)), 𝐵₁(𝜇(𝑘)), 𝐵^∼₂(𝜇(𝑘)), and 𝐵^∼₃(𝜇(𝑘)) are known constant matrices of appropriate dimensions.

Remark 3. The closed-loop system (5) is a Markovian jump linear system with multiple Markov chains, which describe the behavior of the S-C and C-A time delays and packet dropouts, respectively. This enables us to analyze and synthesize such NCSs by applying Markovian jump linear system theory. Note that the problem of unified modeling and output feedback control for NCSs with both S-C and C-A time delays and packet dropouts modeled by multiple Markov chains has not been done in the literature.

Definition 4 (see [11, 47]). System (5) is stochastically stable if there exists a constant 𝐶 such that ^∞_𝑘=0𝔼‖𝑥(𝑘)‖²𝑥₀,𝜇₀,𝑑₀,𝜏₀≤𝐶Υ𝑥₀,𝜑_𝑖,−𝑑_max−𝜏_max≤𝑖≤0,𝜇₀,𝑑₀,𝜏₀, (9) where Υ(𝑥₀,𝜑_𝑖,−𝑑−𝜏≤𝑖≤0,𝜇₀,𝑑₀,𝜏₀) is a nonnegative function and the system initial values satisfy Υ(0,0,…,0)=0.

3. Main Results

By applying a new Lyapunov functional, sufficient conditions for the stochastic stability, the synthesis of controller design for system (5) will be established in this section.

Theorem 5. For system (5), given random but bounded scalar 𝑑(𝑘)∈[^𝑑𝑑] and 𝜏(𝑘)∈[^𝜏𝜏], if for each mode 𝑖∈𝑆₁, 𝑚∈𝑆₂, and 𝑝∈𝑆₃, there exist matrices 𝑃_{𝑖,𝑚,𝑝}>0, 𝑄_1𝑠>0, 𝑄_2𝑠>0, 𝑄_3𝑠>0, 𝑅_1𝑠>0, and 𝑅_2𝑠>0, 𝑠=1,2,3, such that the following matrix inequalities: Ξ₁Ξ₂Ξ₃Ξ₄∗−𝑃_{𝑖,𝑚,𝑝}00∗∗−𝑅₁0∗∗∗−𝑅₂<0, (10) where Ξ₁=^{Π𝑖,𝑚,𝑝Ω1200∗−𝑄3−Ω22−2ΨΨΩ24+Ψ}_{∗∗−𝑄2−Ψ0∗∗∗−𝑄1−Ω44−Ψ},Ξ₂=𝑃_{𝑖,𝑚,𝑝}𝐴_𝑖𝑃_{𝑖,𝑚,𝑝}Λ00^𝑇,Ξ₃=Ξ₃₁Ξ₃₂Ξ₃₃,Ξ₄=Ξ₄₁Ξ₄₂Ξ₄₃,Ξ₃₁=𝑡₁𝑅₁₁𝐻₁(𝐴_𝑖−𝐼)𝑡₁𝑅₁₁Λ00^𝑇,Ξ₃₂=𝑡₂𝑅₁₂𝐻₂(𝐴_𝑖−𝐼)𝑡₂𝑅₁₂Λ00^𝑇,Ξ₃₃=𝑡₃𝑅₁₃𝐻₃(𝐴_𝑖−𝐼)𝑡₃𝑅₁₃Λ00^𝑇,Ξ₄₁=(𝑡₁−𝑡₁)𝑅₂₁𝐻₁(𝐴_𝑖−𝐼)(𝑡₁−𝑡₁)𝑅₂₁Λ00^𝑇,Ξ₄₂=(𝑡₂−𝑡₂)𝑅₂₂𝐻₂(𝐴_𝑖−𝐼)(𝑡₂−𝑡₂)𝑅₂₂Λ00^𝑇,Ξ₄₃=(𝑡₃−𝑡₃)𝑅₂₃𝐻₃(𝐴_𝑖−𝐼)(𝑡₃−𝑡₃)𝑅₂₃Λ00^𝑇,𝑃_{𝑖,𝑚,𝑝}=^𝑠1𝑠_𝑗=1^2𝑠_𝑛=0³_𝑞=0𝜋_𝑖𝑗𝜆_𝑚𝑛𝛾_𝑝𝑞𝑃_{𝑗,𝑛,𝑞},𝑅₁=diag𝑅₁₁,𝑅₁₂,𝑅₁₃,𝑅₂=diag𝑅₂₁,𝑅₂₂,𝑅₂₃,Π_{𝑖,𝑚,𝑝}=−𝑃_{𝑖,𝑚,𝑝}+³_𝑠=1𝐻^𝑇_𝑠𝑄_1𝑠+𝑄_2𝑠+𝑡_𝑠−𝑡_𝑠+1𝑄_3𝑠𝐻_𝑠−Ω₁₁,Ω₁₁=³_𝑠=1𝐻^𝑇_𝑠𝑅_1𝑠𝐻_𝑠,Ω₁₂=𝐻^𝑇₁𝑅₁₁𝐻^𝑇₂𝑅₁₂𝐻^𝑇₃𝑅₁₃,Ω₂₂=diag2𝑅₁₁,2𝑅₁₂,2𝑅₁₃,Ω₂₄=Ω₄₄=diag𝑅₁₁,𝑅₁₂,𝑅₁₃,Ψ=diag𝑅₂₁,𝑅₂₂,𝑅₂₃,𝑄₁=diag𝑄₁₁,𝑄₁₂,𝑄₁₃,𝑄₂=diag𝑄₂₁,𝑄₂₂,𝑄₂₃,𝑄₃=diag𝑄₃₁,𝑄₃₂,𝑄₃₃,Λ=𝐵_𝑖1𝐵_𝑖2𝐵_𝑖3,Λ=𝐻₁𝐵_𝑖1𝐻₂𝐵_𝑖2𝐻₃𝐵_𝑖3,𝐴_𝑖=𝐴𝜇(𝑘),𝐵_𝑖1=𝐵₁𝜇(𝑘),𝐵_𝑖2=𝐵₂𝜇(𝑘),𝐵_𝑖3=𝐵₃𝜇(𝑘),𝑡₁=𝑑,𝑡₁=𝑑,𝑡₂=𝜏,𝑡₂=𝜏,𝑡₃=𝑑+𝜏,𝑡₃=𝑑+𝜏, (11)𝐴(𝜇(𝑘)), 𝐵₁(𝜇(𝑘)), 𝐵₂(𝜇(𝑘)), and 𝐵₃(𝜇(𝑘)) defined in (5) hold for all 𝑖,𝑗∈𝑆₁, 𝑚,𝑛∈𝑆₂, and 𝑝,𝑞∈𝑆₃, then system (5) is stochastically stable.

Proof. It is given in the appendix.

Theorem 5 gives a sufficient condition for the stochastic stability of system (5). However, it should be noted that the conditions (10) are no more LMI conditions. To handle this, the equivalent LMI conditions are given in Theorem 6 by cone complementarity linearization (CCL) algorithm.

Theorem 6. Consider system (5) with random but bounded scalar 𝑑(𝑘)∈[^𝑑𝑑] and 𝜏(𝑘)∈[^𝜏𝜏]. There exists an output feedback controller (4) such that the resulting closed-loop system is stochastically stable if for each mode 𝑖∈𝑆₁, 𝑚∈𝑆₂, and 𝑝∈𝑆₃, there exist matrices 𝑃_{𝑖,𝑚,𝑝}>0, 𝑋_{𝑖,𝑚,𝑝}>0, 𝑄_1𝑠>0, 𝑄_2𝑠>0, 𝑄_3𝑠>0, 𝑅_1𝑠>0, 𝑅_2𝑠>0, 𝑈_1𝑠>0, and 𝑈_2𝑠>0, 𝑠=1,2,3 and 𝐾 such that Ξ₁Ξ₂Ξ₃Ξ₄∗−𝑋_{𝑗,𝑛,𝑞}00∗∗−𝑈_1𝑠0∗∗∗−𝑈_2𝑠𝑃<0,_{𝑖,𝑚,𝑝}𝑋_{𝑖,𝑚,𝑝}=𝐼,𝑅_1𝑠𝑈_1𝑠=𝐼,𝑅_2𝑠𝑈_2𝑠=𝐼,𝑠=1,2,3, (12)(13) where Ξ₂=ℒ_{𝑖,𝑚,𝑝}𝐴_𝑖ℒ_{𝑖,𝑚,𝑝}Λ00^𝑇,Ξ₃=Ξ₃₁Ξ₃₂Ξ₃₃,Ξ₄=Ξ₄₁Ξ₄₂Ξ₄₃,Ξ₃₁=𝑡₁𝐻₁(𝐴_𝑖−𝐼)𝑡₁Λ00^𝑇,Ξ₃₂=𝑡₂𝐻₂(𝐴_𝑖−𝐼)𝑡₂Λ00^𝑇,Ξ₃₃=𝑡₃𝐻₃(𝐴_𝑖−𝐼)𝑡₃Λ00^𝑇,Ξ₄₁=(𝑡₁−𝑡₁)𝐻₁(𝐴_𝑖−𝐼)(𝑡₁−𝑡₁)Λ00^𝑇,Ξ₄₂=(𝑡₂−𝑡₂)𝐻₂(𝐴_𝑖−𝐼)(𝑡₂−𝑡₂)Λ00^𝑇,Ξ₄₃=(𝑡₃−𝑡₃)𝐻₃(𝐴_𝑖−𝐼)(𝑡₃−𝑡₃)Λ00^𝑇,𝑋_{𝑗,𝑛,𝑞}=diag{𝑋_1,0,0,𝑋_1,0,1,…,𝑋_{𝑗,𝑛,𝑞},,…,𝑋_{𝑠1,𝑠2,𝑠3}}_{𝑠11+𝑠21+𝑠3items},ℒ_{𝑖,𝑚,𝑝}𝐴_𝑖^𝑇=𝜋_𝑖1𝜆_𝑚0𝛾_𝑝0𝐴_𝑖^𝑇𝜋_𝑖1𝜆_𝑚0𝛾_𝑝1𝐴_𝑖^𝑇⋯𝜋_𝑖𝑗𝜆_𝑚𝑛𝛾_𝑝𝑞𝐴_𝑖^𝑇⋯𝜋_𝑖𝑠1𝜆_𝑚𝑠2𝛾_𝑝𝑠3𝐴_𝑖^𝑇_{(𝑠1(1+𝑠2)(1+𝑠3))items},ℒ_{𝑖,𝑚,𝑝}Λ^𝑇=𝜋_𝑖1𝜆_𝑚0𝛾_𝑝0Λ^𝑇𝜋_𝑖1𝜆_𝑚0𝛾_𝑝1Λ^𝑇⋯𝜋_𝑖𝑗𝜆_𝑚𝑛𝛾_𝑝𝑞Λ^𝑇⋯𝜋_𝑖𝑠1𝜆_𝑚𝑠2𝛾_𝑝𝑠3Λ^𝑇_{(𝑠1(1+𝑠2)(1+𝑠3))items}, (14)Ξ₁, 𝐴_𝑖, Λ, and Λ are defined in Theorem 5. Moreover, if (12) and (13) have solutions, the controller gain is given by 𝐾.

Proof. By Schur complement, (12) is equivalent to Ξ₁Ξ₂Ξ₃Ξ₄∗−𝑃⁻¹_{𝑗,𝑛,𝑞}00∗∗−𝑅⁻¹_1𝑠0∗∗∗−𝑅⁻¹_2𝑠<0,𝑠=1,2,3. (15) Let 𝑃⁻¹_{𝑗,𝑛,𝑞}=𝑋_{𝑗,𝑛,𝑞}, 𝑅⁻¹_1𝑠=𝑈_1𝑠, and 𝑅⁻¹_2𝑠=𝑈_2𝑠; then we can obtain (12) and (13). This completes the proof.

The conditions in Theorem 6 are a set of LMIs with some matrix inverse constraints. Although they are nonconvex, which prevents us from solving them using the existing convex optimization tool, we can use the CCL algorithm to transform this problem into the nonlinear minimization problem with LMI constraints as follows: minTrace³_𝑠=1𝑅_1𝑠𝑈_1𝑠+𝑅_2𝑠𝑈_2𝑠+^𝑠1𝑠_𝑗=1^2𝑠_𝑛=0³_𝑞=0𝑃_{𝑖,𝑚,𝑝}𝑋_{𝑖,𝑚,𝑝},(i)()𝑅s.t.)(12ii^1𝑠𝐼∗𝑈_1𝑠{},()𝑅>0,𝑠=1,2,3iii^2𝑠𝐼∗𝑈_2𝑠{},()𝑃>0,𝑠=1,2,3iv^{𝑖,𝑚,𝑝}𝐼∗𝑋_{𝑖,𝑚,𝑝}>0,𝑖∈𝑆₁,𝑚∈𝑆₂,𝑝∈𝑆₃. (16)(17) The above nonlinear minimization problem can be solved by the CCL algorithm presented in the following.

Algorithm 7. Consider the following steps.
Step  1. Find a feasible solution satisfying LMIs (i), (ii), (iii), and (iv) in (17), set as (𝑅⁰_1𝑠,𝑈⁰_1𝑠,𝑅⁰_2𝑠,𝑈⁰_2𝑠,𝑃⁰_{𝑖,𝑚,𝑝},𝑋⁰_{𝑖,𝑚,𝑝}), 𝑠={1,2,3} and 𝑘=0.
Step  2. Solve the following LMI optimization problem for variables (𝑅_1𝑠,𝑈_1𝑠,𝑅_2𝑠,𝑈_2𝑠,𝑠={1,2,3},𝑃_{𝑖,𝑚,𝑝},𝑋_{𝑖,𝑚,𝑝},𝐾⁰): Minimize trace {∑³_𝑠=1((𝑅^𝑘_1𝑠𝑈_1𝑠+𝑅_1𝑠𝑈^𝑘_1𝑠)+(𝑅^𝑘_2𝑠𝑈_2𝑠+𝑅_2𝑠𝑈^𝑘_2𝑠))+∑^𝑠1_𝑗=1∑^𝑠2_𝑛=0∑^𝑠3_𝑞=0(𝑃^𝑘_{𝑖,𝑚,𝑝}𝑋_{𝑖,𝑚,𝑝}+𝑃_𝑖,𝑚𝑋^𝑘_{𝑖,𝑚,𝑝})} subject to LMIs (17). Set 𝑅^𝑘+1_1𝑠=𝑅_1𝑠, 𝑈^𝑘+1_1𝑠=𝑈_1𝑠, 𝑅^𝑘+1_2𝑠=𝑅_2𝑠, 𝑈^𝑘+1_2𝑠=𝑈_2𝑠, 𝑠={1,2,3}, 𝑃^𝑘+1_{𝑖,𝑚,𝑝}=𝑃_{𝑖,𝑚,𝑝}, 𝑋^𝑘+1_{𝑖,𝑚,𝑝}=𝑋_{𝑖,𝑚,𝑝}, and 𝐾^𝑘+1=𝐾.
Step  3. If LMI (i) in (17) is satisfied, then exit the iteration. If LMI (i) in (17) is not satisfied, let 𝑘=𝑘+1 and then return to Step 2.

4. Numerical Example

In this section, a simulation example is given to illustrate the usefulness of the developed method. Consider a classical angular positioning system [42, 48, 49] in Figure 2, which consists of a rotating antenna at the origin of the plane, driven by an electric motor. Assume that the angular position of the antenna 𝜃 (rad), the angular position of the moving object 𝜃_𝑟 (rad), and the angular velocity of the antenna 𝜃̇ (rad s⁻¹) are measurable. The state variables are chosen as [_𝜃^̇_𝜃]^𝑇 and the output is 𝑦=𝜃^𝑇. The control problem is to use the input voltage to the motor to rotate the antenna so that it always point in the direction of a moving object in the plant. The output feedback controller is designed for the following values of the matrices 𝐴, 𝐵, and 𝐶: ,.𝐴=10.099500.99,𝐵=0.00390.0783𝐶=1.40.8−0.20.4 (18)

Figure 2 

The angular positioning system.

The stochastic jumping parameter 𝜇(𝑘)∈{1,2,3,4} and the random delays involved in system (5) which are 𝑑(𝑘)=𝑚, 𝑚∈{0,1,2} and 𝜏(𝑘)=𝑝, 𝑝∈{0,1}, and their transition matrices are given by ,.𝜋=0.40.30.20.10.20.50.20.10.30.20.40.10.10.20.20.5,𝜆=0.50.20.30.40.50.10.30.20.5𝛾=0.50.50.80.2 (19)

Figures 3, 4, and 5 show part of the simulation of the stochastic jumping parameter 𝜇(𝑘), S-C delay 𝑑(𝑘), and C-A delay 𝜏(𝑘) governed by their corresponding transition probability matrices, respectively.

Figure 3 

Values of 𝜇(𝑘).

Figure 4 

S-C random delays 𝑑(𝑘).

Figure 5 

C-A random delays 𝜏(𝑘).

The initial value 𝑥(0)=[^0.010]^𝑇. By Theorem 6, we can obtain the gain matrices 𝐾 of controller (4) are constructed as 𝐾=[^{−0.0088−0.0117}].

The state trajectories are shown in Figure 6, where two curves represent state trajectories under the controller gains 𝐾 and Figure 6 indicates that system (5) is stochastically stable.

Figure 6 

State trajectories under 𝐾.

5. Conclusion

The output feedback stabilization problem for NCSs with both S-C and C-A random time delays and packet dropouts is investigated in this paper. The S-C and C-A random time delays and packet dropouts are modeled by independent multiple Markov chains. Then the resulting closed-loop system is modeled as a Markovian jump linear system with Markov delays. Sufficient conditions on stochastic stability and stabilization are obtained by Lyapunov stability theory and linear matrix inequalities method. The CCL algorithm is employed to obtain the output feedback controller. Finally, an example is presented to illustrate the effectiveness of the approach.

Appendix

A. Proof of Theorem 5

Proof. For the closed-loop system (5), construct a Lyapunov-Krasovskii functional as 𝑉(𝑘)=𝑉₁(𝑘)+𝑉₂(𝑘)+𝑉₃(𝑘)+𝑉₄(𝑘), (A.1) where 𝑉₁(𝑘)=𝑥(𝑘)^𝑇𝑃𝜇(𝑘)(𝑘)(𝑘)𝑥(𝑘),𝑉,𝑑,𝜏₂(𝑘)=³_𝑠=1^𝑘−1_{𝑙=𝑘−𝑡𝑠}𝑥(𝑙)^𝑇𝐻^𝑇_𝑠𝑄_1𝑠𝐻_𝑠𝑥(𝑙)+³_𝑠=1^𝑘−1_{𝑙=𝑘−𝑡𝑠}𝑥(𝑙)^𝑇𝐻^𝑇_𝑠𝑄_2𝑠𝐻_𝑠𝑥(𝑙),𝑉₃(𝑘)=³_𝑠=1^{−𝜂𝑠+1}_{𝜃=−𝜂𝑠+2}^𝑘−1_{𝑙=𝑘+𝜃−1}𝑥(𝑙)^𝑇𝐻^𝑇_𝑠𝑄_3𝑠𝐻_𝑠𝑥(𝑙)+³_𝑠=1^𝑘−1_{𝑙=𝑘−𝜂𝑠(𝑘)}𝑥(𝑙)^𝑇𝐻^𝑇_𝑠𝑄_3𝑠𝐻_𝑠𝑥(𝑙),𝑉₄(𝑘)=³⁰_{𝑠=1𝜃=−𝜂𝑠+1}^𝑘−1_{𝑙=𝑘+𝜃−1}𝜂_𝑠𝛿(𝑙)^𝑇𝐻^𝑇_𝑠𝑅_1𝑠𝐻_𝑠𝛿(𝑙)+³_𝑠=1^−𝜂𝑠_{𝜃=−𝜂𝑠+1}^𝑘−1_{𝑙=𝑘+𝜃−1}𝜂_𝑠−𝜂_𝑠𝛿(𝑙)^𝑇𝐻^𝑇_𝑠𝑅_2𝑠𝐻_𝑠𝛿(𝑙), (A.2)𝛿(𝑙)=𝑥(𝑙+1)−𝑥(𝑙), 𝑃(𝜇(𝑘), 𝑑(𝑘), 𝜏(𝑘)), 𝑄_1𝑠, 𝑄_2𝑠, 𝑄_3𝑠, 𝑅_1𝑠, 𝑅_2𝑠>0, 𝑠={1,2,3}, 𝑡₁, 𝑡₁,𝑡₂, 𝑡₂, 𝑡₃, and 𝑡₃ are defined in Theorem 5. In the following, when 𝜇(𝑘)=𝑖, 𝑑(𝑘)=𝑚, and 𝜏(𝑘)=𝑝, we will denote 𝑃(𝜇(𝑘),𝑑(𝑘),𝜏(𝑘)), 𝐴(𝜇(𝑘)) and 𝐵_𝑠(𝜇(𝑘)) by 𝑃_{𝑖,𝑚,𝑝}, 𝐴_𝑖 and 𝐵_𝑖𝑠, respectively.
Let 𝜉(𝑘)=^{𝑥(𝑘)𝑇𝜂(𝑘)𝑇𝜂(𝑘)𝑇𝜂(𝑘)𝑇𝑇}, 𝜂(𝑘)=𝐻₁𝑥(𝑘−𝑚)^𝑇𝐻₂𝑥(𝑘−𝑝)^𝑇𝐻₃𝑥(𝑘−𝑚−𝑝)^𝑇𝑇,𝜂(𝑘)=𝐻₁𝑥(𝑘−𝑑)^𝑇𝐻₂𝑥(𝑘−𝜏)^𝑇𝐻₃𝑥(𝑘−𝑑−𝜏)^𝑇𝑇,𝜂(𝑘)=𝐻₁𝑥𝑘−𝑑^𝑇𝐻₂𝑥(𝑘−𝜏)^𝑇𝐻₃𝑥𝑘−𝑑−𝜏^𝑇𝑇. (A.3)
Then, along the solution of system (5) we have 𝔼Δ𝑉₁(𝑘)=𝑥(𝑘+1)^{𝑇𝑠1𝑠}_𝑗=1^2𝑠_𝑛=0³_𝑞=0𝜋_𝑖𝑗𝜆_𝑚𝑛𝛾_𝑝𝑞𝑃_{𝑗,𝑛,𝑞}×𝑥(𝑘+1)−𝑥^𝑇(𝑘)𝑃_{𝑖,𝑚,𝑝}𝑥(𝑘)=𝜉^𝑇(𝑘)𝐴^𝑇_𝑖Λ^𝑇00𝑃_{𝑖,𝑚,𝑝}𝐴_𝑖(𝑘)Λ00×𝜉−𝑥^𝑇(𝑘)𝑃_{𝑖,𝑚,𝑝}𝑥(𝑘), (A.4) where 𝑃_{𝑖,𝑚,𝑝} and Λ are defined in Theorem 5.
Consider that 𝔼Δ𝑉₂(𝑘)=³_𝑠=1𝐻_𝑠𝑥(𝑘)^𝑇𝑄_1𝑠+𝑄_2𝑠𝐻_𝑠𝑥(𝑘)−𝐻_𝑠𝑥𝑘−𝑡_𝑠^𝑇𝑄_1𝑠𝐻_𝑠𝑥𝑘−𝑡_𝑠−𝐻_𝑠𝑥𝑘−𝑡_𝑠^𝑇𝑄_2𝑠𝐻_𝑠𝑥𝑘−𝑡_𝑠,𝔼Δ𝑉₃(𝑘)=³_𝑠=1𝑡_𝑠−𝑡_𝑠+1𝑥(𝑘)^𝑇𝐻^𝑇_𝑠𝑄_3𝑠𝐻_𝑠𝑥(𝑘)−^{𝑘−𝑡𝑠}_{𝜃=𝑘−𝑡𝑠+1}𝑥(𝑙)^𝑇𝐻^𝑇_𝑠𝑄_3𝑠𝐻_𝑠𝑥(𝑙)+^𝑘−1_{𝑙=𝑘−𝜂𝑠()𝑘+1+1}−^𝑘−1_{𝑙=𝑘−𝜂𝑠(𝑘)+1}×𝑥(𝑙)^𝑇𝐻^𝑇_𝑠𝑄_3𝑠𝐻_𝑠𝑥(𝑙)−𝑥𝑘−𝜂_𝑠(𝑘)^𝑇×𝐻^𝑇_𝑠𝑄_3𝑠𝐻_𝑠𝑥𝑘−𝜂_𝑠(𝑘). (A.5)(A.6) Note that ³_𝑠=1^𝑘−1_{𝑙=𝑘−𝜂𝑠()𝑘+1+1}𝑥^𝑇(𝑙)𝐻^𝑇_𝑠𝑄_3𝑠𝐻_𝑠𝑥(𝑙)=³_𝑠=1^𝑘−1_{𝑙=𝑘−𝑡𝑠+1}+^{𝑘−𝑡𝑠}_{𝑙=𝑘−𝜂𝑠()𝑘+1+1}𝑥^𝑇(𝑙)𝐻^𝑇_𝑠𝑄_3𝑠𝐻_𝑠𝑥(𝑙)≤³_𝑠=1^𝑘−1_{𝑙=𝑘−𝜂𝑠(𝑘)+1}+^{𝑘−𝑡𝑠}_{𝑙=𝑘−𝑡𝑠+1}𝑥^𝑇(𝑙)𝐻^𝑇_𝑠𝑄_3𝑠𝐻_𝑠𝑥(𝑙). (A.7) By combining (A.6) and (A.7), we have 𝔼Δ𝑉₃=³_𝑠=1𝑡_𝑠−𝑡_𝑠+1𝑥(𝑘)^𝑇𝐻^𝑇_𝑠𝑄_3𝑠𝐻_𝑠𝑥(𝑘)−𝑥𝑘−𝜂_𝑠(𝑘)^𝑇𝐻^𝑇_𝑠𝑄_3𝑠𝐻_𝑠𝑥𝑘−𝜂_𝑠(𝑘),𝔼Δ𝑉₄=³_𝑠=1𝑡²_𝑠𝛿^𝑇(𝑘)𝐻^𝑇_𝑠𝑅_1𝑠𝐻_𝑠𝛿(𝑘)−^𝑘−1_{𝑙=𝑘−𝑡𝑠}𝑡_𝑠𝛿^𝑇(𝑙)𝐻^𝑇_𝑠𝑅_1𝑠𝐻_𝑠𝛿(𝑙)+(𝑡_𝑠−𝑡_𝑠)²𝛿^𝑇(𝑘)𝐻^𝑇_𝑠𝑅_2𝑠𝐻_𝑠𝛿(𝑘)−^{𝑘−𝑡𝑠−1}_{𝑙=𝑘−𝑡𝑠}𝑡_𝑠−𝑡_𝑠𝛿^𝑇(𝑙)𝐻^𝑇_𝑠𝑅_2𝑠𝐻_𝑠𝛿(𝑙). (A.8)(A.9) By Jensen’s inequality, we can get ³_𝑠=1^𝑘−1_{𝑙=𝑘−𝑡𝑠}𝑡_𝑠𝛿^𝑇(𝑙)𝐻^𝑇_𝑠𝑅_1𝑠𝐻_𝑠𝛿(𝑙)=³_𝑠=1^{𝑘−𝜂𝑘−1}_{𝑙=𝑘−𝑡𝑠}+^𝑘−1_{𝑙=𝑘−𝜂𝑘}×𝑡_𝑠−𝜂_𝑘+𝜂_𝑘𝛿^𝑇(𝑙)𝐻^𝑇_𝑠𝑅_1𝑠𝐻_𝑠𝛿(𝑙)≥³_𝑠=1𝑥𝑘−𝑡_𝑠−𝑥𝑘−𝜂_𝑘^𝑇𝐻^𝑇_𝑠𝑅_1𝑠𝐻_𝑠×𝑥𝑘−𝑡_𝑠−𝑥𝑘−𝜂_𝑘+𝑥𝑘−𝜂_𝑘−𝑥(𝑘)^𝑇×𝐻^𝑇_𝑠𝑅_1𝑠𝐻_𝑠𝑥𝑘−𝜂_𝑘−𝑥(𝑘)≥𝜉(𝑘)^𝑇Ω₁₁−Ω₁₂00∗Ω₂₂0−Ω₂₄∗∗00∗∗∗Ω₄₄𝜉(𝑘), (A.10) where Ω₁₁, Ω₁₂, Ω₂₂, Ω₂₄, and Ω₄₄ are defined in Theorem 5.
Similarly, we have ³_𝑠=1^{𝑘−𝑡𝑠−1}_{𝑙=𝑘−𝑡𝑠}𝑡_𝑠−𝑡_𝑠𝛿^𝑇(𝑙)𝐻^𝑇_𝑠𝑅_2𝑠𝐻_𝑠𝛿(𝑙)=³_𝑠=1^{𝑘−𝜂𝑘−1}_{𝑙=𝑘−𝑡𝑠}+^{𝑘−𝑡𝑠−1}_{𝑙=𝑘−𝜂𝑘}×𝑡_𝑠−𝜂_𝑘+𝜂_𝑘−𝑡_𝑠𝛿^𝑇(𝑙)𝐻^𝑇_𝑠𝑅_2𝑠𝐻_𝑠𝛿(𝑙)≥³_𝑠=1𝑡_𝑠−𝜂_𝑘^{𝑘−𝜂𝑘−1}_{𝑙=𝑘−𝑡𝑠}𝛿^𝑇(𝑙)𝐻^𝑇_𝑠𝑅_2𝑠𝐻_𝑠𝛿(𝑙)+𝜂_𝑘−𝑡_𝑠^{𝑘−𝑡𝑠−1}_{𝑙=𝑘−𝜂𝑘}𝛿^𝑇(𝑙)𝐻^𝑇_𝑠𝑅_2𝑠𝐻_𝑠𝛿(𝑙≥_{3𝑠=1𝑘−𝜂𝑘−1𝑙=𝑘−𝑡𝑠})𝛿(𝑙)^𝑇𝐻^𝑇_𝑠𝑅_2𝑠𝐻_𝑠^{𝑘−𝜂𝑘−1}_{𝑙=𝑘−𝑡𝑠}𝛿(𝑙)+^{𝑘−𝑡𝑠−1}_{𝑙=𝑘−𝜂𝑘}𝛿(𝑙)^𝑇𝐻^𝑇_𝑠𝑅_2𝑠𝐻_𝑠^{𝑘−𝑡𝑠−1}_{𝑙=𝑘−𝜂𝑘}𝛿(𝑙)≥𝜉(𝑘)^𝑇𝜉(𝑘),0000∗2Ψ−Ψ−Ψ∗∗Ψ0∗∗∗Ψ (A.11) where Ψ is defined in Theorem 5.
By combining (A.9), (A.10), and (A.11), we have 𝔼Δ𝑉₄≤𝜉(𝑘)^𝑇𝑡₁𝐻₁(𝐴_𝑖−𝐼)^𝑇𝑡₁Λ^𝑇00×𝑅₁₁𝑡₁𝐻₁𝐴_𝑖−𝐼𝑡₁+Λ00𝑡₂𝐻₂𝐴_𝑖−𝐼^𝑇𝑡₂Λ^𝑇00×𝑅₁₂𝑡₂𝐻₂𝐴_𝑖−𝐼𝑡₁+Λ00𝑡₃𝐻₃𝐴_𝑖−𝐼^𝑇𝑡₃Λ^𝑇00×𝑅₁₃𝑡₃𝐻₃𝐴_𝑖−𝐼𝑡₁+Λ00𝑡₁−𝑡₁𝐻₁(𝐴_𝑖−𝐼)^𝑇𝑡₁−𝑡₁Λ^𝑇00×𝑅₂₁𝑡₁−𝑡₁𝐻₁𝐴_𝑖−𝐼𝑡₁−𝑡₁+Λ00𝑡₂−𝑡₂𝐻₂(𝐴_𝑖−𝐼)^𝑇𝑡₂−𝑡₂Λ^𝑇00×𝑅₂₂𝑡₂−𝑡₂𝐻₂𝐴_𝑖−𝐼𝑡₂−𝑡₂+Λ00𝑡₃−𝑡₃𝐻₃(𝐴_𝑖−𝐼)^𝑇𝑡₃−𝑡₃Λ^𝑇00×𝑅₂₃𝑡₃−𝑡₃𝐻₃𝐴_𝑖−𝐼𝑡₃−𝑡₃+Λ00−Ω₁₁Ω₁₂00∗−Ω₂₂0Ω₂₄∗∗00∗∗∗−Ω₄₄+𝜉(𝑘),0000∗−2ΨΨΨ∗∗−Ψ0∗∗∗−Ψ (A.12) where Λ is defined in Theorem 5.
By combining (A.4), (A.5), (A.8), and (A.12), we have 𝔼[Δ𝑉]≤𝜉^𝑇(𝑘)×^{Π𝑖,𝑚,𝑝Ω1200∗−𝑄3−Ω22−2ΨΨΩ24+Ψ}_{∗∗−𝑄2−Ψ0∗∗∗−𝑄1−Ω44−Ψ}+𝐴^𝑇_𝑖Λ^𝑇00𝑃_{𝑖,𝑚,𝑝}𝐴_𝑖+Λ00𝑡₁𝐻₁𝐴_𝑖−𝐼^𝑇𝑡₁Λ^𝑇00×𝑅₁₁𝑡₁𝐻₁𝐴_𝑖−𝐼𝑡₁+Λ00𝑡₂𝐻₂𝐴_𝑖−𝐼^𝑇𝑡₂Λ^𝑇00×𝑅₁₂𝑡₂𝐻₂𝐴_𝑖−𝐼𝑡₂+Λ00𝑡₃𝐻₃𝐴_𝑖−𝐼^𝑇𝑡₃Λ^𝑇00×𝑅₁₃𝑡₃𝐻₃𝐴_𝑖−𝐼𝑡₃+Λ00𝑡₁−𝑡₁𝐻₁(𝐴_𝑖−𝐼)^𝑇𝑡₁−𝑡₁Λ^𝑇00×𝑅₂₁𝑡₁−𝑡₁𝐻₁𝐴_𝑖−𝐼𝑡₁−𝑡₁+Λ00𝑡₂−𝑡₂𝐻₂(𝐴_𝑖−𝐼)^𝑇𝑡₂−𝑡₂Λ^𝑇00×𝑅₂₂𝑡₂−𝑡₂𝐻₂𝐴_𝑖−𝐼𝑡₂−𝑡₂+Λ00𝑡₃−𝑡₃𝐻₃(𝐴_𝑖−𝐼)^𝑇𝑡₃−𝑡₃Λ^𝑇00×𝑅₂₃𝑡₃−𝑡₃𝐻₃𝐴_𝑖−𝐼𝑡₃−𝑡₃(𝑘)Λ00×𝜉=𝜉^𝑇(𝑘)Θ_{𝑖,𝑚,𝑝}𝜉(𝑘), (A.13) where 𝑄₁, 𝑄₂, 𝑄₃, and Π_{𝑖,𝑚,𝑝} are defined in Theorem 5.
By Schur complement and from (10), we have Θ_{𝑖,𝑚,𝑝}<0. Therefore, 𝔼[Δ𝑉]≤−𝝀_min−Θ_{𝑖,𝑚,𝑝}𝜉(𝑘)^𝑇𝜉(𝑘)≤−𝜂𝑥(𝑘)^𝑇𝑥(𝑘), (A.14) where 𝜆_min(−Θ_{𝑖,𝑚,𝑝}) denotes the minimal eigenvalue of −Θ_{𝑖,𝑚,𝑝} and 𝜂=inf{𝜆_min(−Θ_{𝑖,𝑚,𝑝})}. From (A.14), it is seen that for any 𝑡>0𝔼[𝑉(𝑘+1)]−𝔼[𝑉(0)]≤−𝜂^𝑡_𝑘=0𝐸𝑥(𝑘)^𝑇𝑥(𝑘). (A.15) Furthermore ^𝑡_𝑘=0𝔼𝑥(𝑘)^𝑇𝑥(𝑘≤1)𝜂𝔼[𝑉(0)]. (A.16) By taking limit as 𝑡→∞, we have ^∞_𝑘=0𝔼𝑥(𝑘)^𝑇𝑥(𝑘≤1)𝜂𝔼[𝑉(0)]<∞. (A.17) According to Definition 4, the closed-loop system (5) is stochastically stable. This completes the proof.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (Grant nos. 61174070, 61104107), the Foundation of Key Laboratory of Autonomous Systems and Networked Control, Ministry of Education, P.R. China, the Natural Science Foundation of Guangdong Province (Grant no. S2011040004702), China Postdoctoral Science Foundation (Grant no. 20110491334), the Project-Sponsored by SRF for ROCS, SEM (Grant no. 20100172120028), and the Foundation of Shenzhen University (Grant nos. JG2012035,00035653).