Stability and Stabilization of Networked Control System with Forward and Backward Random Time Delays

Abstract

This paper deals with the problem of stabilization for a class of networked control systems (NCSs) with random time delay via the state feedback control. Both sensor-to-controller and controller-to-actuator delays are modeled as Markov processes, and the resulting closed-loop system is modeled as a Markovian jump linear system (MJLS). Based on Lyapunov stability theorem combined with Razumikhin-based technique, a new delay-dependent stochastic stability criterion in terms of bilinear matrix inequalities (BMIs) for the system is derived. A state feedback controller that makes the closed-loop system stochastically stable is designed, which can be solved by the proposed algorithm. Simulations are included to demonstrate the theoretical result.

1. Introduction

Feedback control systems in which the control loops are closed through a real-time network are called networked control systems (NCSs) [1]. Recently, much attention has been paid to the study of stability analysis and controller design of NCSs [2, 3] due to their low cost, reduced weight and power requirements, simple installation and maintenance, and high reliability. Consequently, NCSs have been applied to various areas such as mobile sensor networks [4], remote surgery [5], haptics collaboration over the Internet [6–8], and automated highway systems and unmanned aerial vehicles [9, 10]. However, the sampling data and controller signals are transmitted through a network, so network-induced delays in NCSs are always inevitable [11, 12].

One of the main issues in NCSs is network-induced delays, which are usually the major causes for the deterioration of system performance and potential system instability [13]. For different scheduling protocols, the network-induced delay may be constant, or time-varying, but in most cases, it is random [14]. Hence, systems with random time delay attract considerable attention [15–18]. Based on stochastic control theory and a separation property, the effect of random delay is treated as an LQG problem in [15]. However, the network-induced random delay has to be less than one sampling interval. The results in [15] have recently been extended to the case with longer delays in [16]. It is noted that the given controller depends only on sensor-to-controller delay. In [17], a control problem for Bernoulli binary random delay is considered, and a linear matrix inequalities (LMIs) problem for the analysis of stochastic exponential mean square stability is established. The model-based NCSs with random transmission delay is studied in [18]. Sufficient conditions for almost sure stability and stochastic exponential mean square stability are presented.

On the other hand, the study of stochastic systems has attracted a great deal of attention [19–38]. Some of these results are applied to networked control systems with random time delays [39–43]. In [39, 40], the network-induced random delays are modeled as Markov chains such that the closed-loop systems are jump linear systems with one mode. It is noticed that in [39], the state feedback gain is mode independent, and in [40], the state feedback gain only depends on the delay from sensor to controller. Recently, stabilization of networked control systems with the sensor-to-controller and controller-to-actuator delays are considered in [41]. In [42, 43], a class of Markovian jump linear systems with time delays both in the system state and in the mode signal is considered. Based on Lyapunov method, a time-delayed, mode-dependent, and state feedback controller such that the closed-loop system is stochastically stable is designed. It is noticed that the time delay in the mode signal is constant in [42, 43], and the time delay in the mode signal is random. It is worth pointing out that in all of the aforementioned papers, the plant is in the discrete-time domain. To the best of the authors' knowledge, the stability and stabilization problems for NCSs with the plant being in the continuous-time domain have not been fully investigated to date. Especially for the case where both sensor-to-controller and controller-to-actuator network-induced delays are random and longer than one sampling interval, very few results related to NCSs have been available in the literature so far, which motivates the present study.

The aim of this paper is to consider a class of networked control systems with sensors and actuators connected to a controller via two communication networks in the continuous-time domain. Two Markov processes are introduced to describe sensor-to-controller transmission delay and the controller-to-actuator transmission delay. Based on Lyapunov stability theorem, a method for designing a mode-dependent state feedback controller that stabilizes this class of networked control systems is proposed. The existence of such a controller is given in terms of BMIs, which can be solved by the proposed algorithm.

This paper is organized as follows. In Section 2, the problem is stated and some useful definitions and lemmas are given, and then the main results of this paper are given in Section 3. Simulation results are presented in Section 4. Finally, the conclusions are provided in Section 5.

Notation.  ℝ^𝑛 denotes the 𝑛-dimensional Euclidean space, and 𝐼 is identity matrix. 𝐴^𝑇 stands for the transpose of the corresponding matrix 𝐴. The notation 𝐴≥0(𝐴>0) means that the matrix 𝐴 is a positive semidefinite (positive definite) matrix. For an arbitrary matrix 𝑌 and two symmetric matrices 𝑋 and 𝑍, 𝑋𝑌∗𝑍 denotes a symmetric matrix, where ∗ denotes a block matrix entry implied by symmetry, and ‖⋅‖ refers to the Euclidean norm for vectors and induced 2-norm for matrices. 𝔼(⋅) stands for the mathematical expectation operator, and ℙ(⋅) for probability operator.

2. Problem Formulation

Consider linear systems described by the differential equation𝑥̇=𝐴𝑥(𝑡)+𝐵𝑢(𝑡), (2.1) where 𝑥(𝑡)∈ℝ^𝑛 is the state vector, and 𝑢(𝑡)∈ℝ^𝑚 is the control input. Matrices 𝐴 and 𝐵 are known matrices of appropriate dimensions.

The plant is interconnected by a controller over a communication network, see Figure 1. The sensor and controller are periodically sampled with the sampling interval 𝑇. We describe the sensor-to-controller transmission delay as 𝜏_sc(𝑟_𝑡) and the controller-to-actuator transmission delay as 𝜏_ca(𝜂_𝑡). The mode switching of 𝜏_sc(𝑟_𝑡) is governed by the continuous-time discrete-state Markov process 𝑟_𝑡 taking the values in the finite set 𝜍_𝑟∶={1,…,𝑁_𝑟} with generator Λ=(𝜆_𝑖𝑗),𝑖,𝑗∈𝜍_𝑟 given by ℙ𝑟_𝑡+ℎ=𝑗∣𝑟_𝑡=𝜆=𝑖^𝑖𝑗ℎ+𝑜(ℎ),𝑖≠𝑗,1+𝜆_𝑖𝑖ℎ+𝑜(ℎ),𝑖=𝑗, (2.2) where 𝜆_𝑖𝑗 is the transition rate from mode 𝑖 to 𝑗 with 𝜆_𝑖𝑗≥0 when 𝑖≠𝑗 and 𝜆_𝑖𝑖=−∑^𝑁𝑟_{𝑗=1,𝑗≠𝑖}𝜆_𝑖𝑗, and 𝑜(ℎ) is such that lim_ℎ→0𝑜(ℎ)/ℎ=0. The mode switching of 𝜏_ca(𝜂_𝑡) is governed by the continuous-time discrete-state Markov process 𝜂_𝑡 taking the values in the finite set 𝜍_𝜂∶={1,…,𝑁_𝜂} with generator Π=(𝜋_𝑘𝑙),𝑘,𝑙∈𝜍_𝜂 given by ℙ𝜂_𝑡+ℎ=𝑙∣𝜂_𝑡=𝜋=𝑘^𝑘𝑙ℎ+𝑜(ℎ),𝑘≠𝑙,1+𝜋_𝑘𝑘ℎ+𝑜(ℎ),𝑘=𝑙, (2.3) with 𝜋_𝑘𝑙≥0 and 𝜋_𝑘𝑘=−∑^𝑁𝜂_{𝑙=1,𝑙≠𝑘}𝜋_𝑘𝑙.

Figure 1 

Illustration of NCSs over communication network.

Throughout the paper, the following assumption is needed for the considered networked control systems.

Assumption 2.1. The switching difference of consecutive delays is less than one sampling interval, that is, ℙ𝜏_sc𝑟_𝑡𝑘+1−𝜏_sc𝑟_𝑡𝑘ℙ𝜏≥𝑇=0,_ca𝜂_𝑡𝑘+1−𝜏_ca𝜂_𝑡𝑘≥𝑇=0, (2.4) where 𝑡_𝑘=𝑘𝑇 is the 𝑘th sampling instant.

Remark 2.2. Although Assumption 2.1 restricts that the switching difference of consecutive delays is less than one sampling interval 𝑇, this does not imply that the network delay 𝜏_sc(𝑟_𝑡𝑘) and 𝜏_ca(𝜂_𝑡𝑘) are less than 𝑇.

According to Figure 1, for 𝑡_𝑘≤𝑡<𝑡_𝑘+1, the control law has the form:𝑢(𝑡)=𝐾𝑟_𝑡,𝜂_𝑡𝑥𝑡_𝑘−𝜏_sc𝑟_𝑡−𝜏_ca𝜂_𝑡. (2.5)

Define the time delay 𝜏(𝑟_𝑡,𝜂_𝑡) as follows:𝜏𝑟_𝑡,𝜂_𝑡=𝑡−𝑡_𝑘+𝜏_sc𝑟_𝑡+𝜏_ca𝜂_𝑡, (2.6) which can be illustrated by Figure 2.

Figure 2 

Illustration of the time delay.

Then, we have𝑢(𝑡)=𝐾𝑟_𝑡,𝜂_𝑡𝑥𝑡−𝜏𝑟_𝑡,𝜂_𝑡. (2.7) The associated upper bounds of 𝜏(𝑟_𝑡,𝜂_𝑡) are defined as𝜏=𝑇+max_{𝑖∈𝜍𝑟}𝜏_sc(𝑖)+max_{𝑘∈𝜍𝜂}𝜏_ca(𝑘). (2.8)

Applying controller (2.7) to the open-loop system (2.1) results in the closed-loop networked control system𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐵𝐾𝑟_𝑡,𝜂_𝑡𝑥𝑡−𝜏𝑟_𝑡,𝜂_𝑡,𝑥(𝜃)=𝜙(𝜃),𝜃∈[−𝜏,0], (2.9) where 𝜙(𝜃), 𝜃∈[−𝜏,0] is the initial function.

We have the following stochastic stability concept for system (2.9).

Definition 2.3. The system (2.9) is said to be stochastically stable if there exists a constant 𝕋(𝑟₀,𝜂₀,𝜙(⋅)) such that 𝔼^∞₀‖𝑥(𝑠)‖²𝑑𝑠∣𝑟₀,𝜂₀,𝜙(⋅)≤𝕋𝑟₀,𝜂₀,𝜙(⋅), (2.10) for any initial condition 𝑥(𝑟₀,𝜂₀,𝜙(⋅)).

The following lemmas will be essential for the proofs in Section 3.

Lemma 2.4 (see [44]). Given any real matrices Σ₁,Σ₂,Σ₃ of appropriate dimensions and a scalar 𝜖>0 such that Σ₃=Σ^𝑇₃>0, Then the following inequality holds: ^𝑇₁₂+^𝑇₂₁≤𝜖^𝑇₁₃₁+𝜖^𝑇−1₂⁻¹₃₂. (2.11)

For the delay functional differential equation,𝑥̇(𝑡)=𝑓𝑡,𝑥_𝑡, (2.12) where 𝑓∶[0,+∞)×𝐶[−𝜏,0],ℝ^𝑛⟶ℝ^𝑛, (2.13) is completely continuous, 𝑓(𝑡,0)=0, and 𝑥_𝑡(𝜃) is defined as 𝑥_𝑡(𝜃)=𝑥(𝑡+𝜃),𝜃∈[−𝜏,0]. (2.14) Then we have the following Razumikhin lemma.

Lemma 2.5 (see [45]). Suppose that 𝑢,𝑣,𝑤∶ℝ⁺→ℝ⁺ are continuous, strictly monotonous increasing functions, then 𝑢(𝑠),𝑣(𝑠), and 𝑤(𝑠) are positive for 𝑠>0, and 𝑢(0)=𝑣(0)=0. If there is a continuous function 𝑉∶[−𝜏,+∞)×ℝ^𝑛→ℝ⁺ such that 𝑢(‖𝑥‖)≤𝑉(𝑡,𝑥)≤𝑣(‖𝑥‖),𝑡∈[−𝜏,+∞),𝑥∈ℝ, (2.15) and there is a continuous nondecreasing function 𝑝(𝑠)>𝑠 for 𝑠>0, and for any 𝑡₀∈ℝ⁺, 𝑉̇(𝑡,𝑥)≤−𝑤(‖𝑥‖) (2.16) if 𝑉(𝑥(𝑡+𝜃,𝑡+𝜃))<𝑝(𝑉(𝑡,𝑥)),𝜃∈[−𝜏,0],𝑡≥𝑡₀, (2.17) then the zero solution of (2.12) is uniformly asymptotically stable.

3. Main Results

The following theorem provides sufficient conditions for existence of a mode-dependent state feedback controller for the system (2.9).

Theorem 3.1. Consider the closed-loop system (2.9) satisfying Assumption 2.1. If there exist symmetric matrix 𝑄(𝑖,𝑘)>0, matrix 𝑌(𝑖,𝑘), and positive scalar 𝜖₁, 𝜖₂ such that the following matrix inequalities hold for all 𝑖∈𝜍_𝑟 and 𝑘∈𝜍_𝜂, 𝐽(𝑖,𝑘)𝜑₁(𝑖,𝑘)𝜑₂(𝑖,𝑘)∗−𝜓₁0∗∗−𝜓₂<0,−𝜀₁𝑄()()()𝑖,𝑘𝐴𝑄𝑖,𝑘∗−𝑄𝑖,𝑘<0,−𝜀₂𝑄()()()𝑖,𝑘𝐵𝑌𝑖,𝑘∗−𝑄𝑖,𝑘<0, (3.1)(3.2)(3.3) where 𝐽(𝑖,𝑘)=𝑄(𝑖,𝑘)𝐴^𝑇+𝐴𝑄(𝑖,𝑘)+𝑌^𝑇(𝑖,𝑘)𝐵^𝑇+𝐵𝑌(𝑖,𝑘)+𝜏𝜖₁+3𝜖₂𝑄(𝑖,𝑘)+𝜆_𝑖𝑖𝑄(𝑖,𝑘)+𝜋_𝑘𝑘𝑄(),𝜑𝑖,𝑘₁(i,𝑘)=𝜆_𝑖,1𝑄(𝑖,𝑘),…,𝜆_{𝑖,𝑖−1}𝑄(𝑖,𝑘),𝜆_𝑖,𝑖+1𝑄(𝑖,𝑘),…,𝜆_{𝑖,𝑁𝑟}𝑄(,𝜑𝑖,𝑘)₂(𝑖,𝑘)=√𝜋_𝑘,1𝑄(𝑖,𝑘),…,√𝜋_{𝑘,𝑘−1}𝑄(𝑖,𝑘),√𝜋_𝑘,𝑘+1𝑄(𝑖,𝑘),…,𝜋_{𝑘,𝑁𝜂}𝑄(,𝜓𝑖,𝑘)₁=diag𝑄(1,𝑘),…,𝑄(𝑖−1,𝑘),𝑄(𝑖+1,𝑘),…,𝑄𝑁_𝑟,𝜓,𝑘₂=diag𝑄(𝑖,1),…,𝑄(𝑖,𝑘−1),𝑄(𝑖,𝑘+1),…,𝑄𝑖,𝑁_𝜂, (3.4) with 𝑃(𝑖,𝑘)=𝑄⁻¹(𝑖,𝑘), then the system is stochastically stable with the state feedback gain: 𝐾(𝑖,𝑘)=𝑌(𝑖,𝑘)𝑄⁻¹(𝑖,𝑘). (3.5)

Proof. Consider the following Lyapunov candidate: 𝑉𝑥(𝑡),𝑟_𝑡,𝜂_𝑡=𝑥^𝑇(𝑡)𝑃𝑟_𝑡,𝜂_𝑡𝑥(𝑡), (3.6) where 𝑃(𝑟_𝑡,𝜂_𝑡) is the positive symmetric matrix. From (3.6), it follows that 𝛽₁‖𝑥(𝑡)‖²≤𝑉𝑥(𝑡),𝑟_𝑡,𝜂_𝑡≤𝛽₂‖𝑥(𝑡)‖², (3.7) where 𝛽₁=min_{𝑟𝑡∈𝜍𝑟,𝜂𝑡∈𝜍𝜂}𝜆_min𝑃𝑟_𝑡,𝜂_𝑡,𝛽₂=max_{𝑟𝑡∈𝜍𝑟,𝜂𝑡∈𝜍𝜂}𝜆_max𝑃𝑟_𝑡,𝜂_𝑡. (3.8) Note that 𝑥𝑡−𝜏𝑟_𝑟,𝜂_𝑡=𝑥(𝑡)−⁰_{𝑟−𝜏𝑟,𝜂𝑡}()(𝑡)−𝑥̇𝑡+𝜃𝑑𝜃=𝑥⁰_{−𝜏(𝑟𝑟,𝜂𝑡)}𝐴𝑥(𝑡+𝜃),+𝐵𝐾𝑟_𝑡,𝜂_𝑡𝑥𝑡−𝜏𝑟_𝑟,𝜂_𝑡+𝜃𝑑𝜃. (3.9) Thus, the closed-loop system (2.9) can be rewritten as 𝑥̇(𝑡)=𝐴+𝐵𝐾𝑟_𝑡,𝜂_𝑡𝑥(𝑡)−𝐵𝐾𝑟_𝑡,𝜂_𝑡⁰_{𝑟−𝜏𝑟,𝜂𝑡}𝐴𝑥(𝑡+𝜃)+𝐵𝐾𝑟_𝑡,𝜂_𝑡𝑥𝑡−𝜏𝑟_𝑟,𝜂_𝑡+𝜃𝑑𝜃. (3.10) Let ℒ(⋅) be the weak infinitesimal generator of {𝑥(𝑡),𝑟_𝑡,𝜂_𝑡,𝑡≥0}, then for 𝑟_𝑡=𝑖∈𝜍_𝑟,𝜂_𝑡=𝑘∈𝜍_𝜂, we haveℒ𝑉(𝑥(𝑡),𝑖,𝑘)=𝑥̇^𝑇(𝑡)𝑃(𝑖,𝑘)𝑥(𝑡)+𝑥^𝑇(𝑡)𝑃(𝑖,𝑘)𝑥̇(𝑡)+^𝑁𝑟_𝑗=1𝜆_𝑖𝑗𝑥^𝑇(𝑡)𝑃𝑗,𝑘𝑥(𝑡)+^𝑁𝜂_𝑙=1𝜋_𝑘𝑙𝑥^𝑇(𝑡)𝑃(𝑖,𝑙)𝑥(𝑡)+𝑥^𝑇(𝑡)𝐴^𝑇𝑃(𝑖,𝑘)+𝑃(𝑖,𝑘)𝐴+𝐾^𝑇(𝑖,𝑘)𝐵^𝑇𝑃(𝑖,𝑘)+𝑃(𝑖,𝑘)𝐵𝐾(𝑖,𝑘)+^𝑁𝑟_𝑗=1𝜆_𝑖𝑗𝑃𝑗,𝑘+^𝑁𝜂_𝑙=1𝜋_𝑘𝑙𝑃(𝑖,𝑙)𝑥−2⁰_{−𝜏(𝑖,𝑘)}𝑥^𝑇(𝑡)𝑃(𝑖,𝑘)𝐵𝐾(𝑖,𝑘)×[𝐴𝑥(𝑡+𝜃)+𝐵𝐾(𝑖,𝑘)𝑥(𝑡−𝜏(𝑖,𝑘)+𝜃)]𝑑𝜃. (𝑡)(3.11)According to Lemma 2.4, we have −2⁰_{−𝜏(𝑖,𝑘)}𝑥^𝑇(𝑡)𝑃()()×[()()𝑥(()(𝜖𝑖,𝑘𝐵𝐾𝑖,𝑘𝐴𝑥𝑡+𝜃+𝐵𝐾𝑖,𝑘𝑡−𝜏𝑖,𝑘+𝜃)]𝑑𝜃≤𝜏𝑖,𝑘)[₁⁻¹𝑥^𝑇(𝑡)𝑃(𝑖,𝑘)𝐵𝐾(𝑖,𝑘)𝐴𝑃⁻¹(𝑖,𝑘)×𝐴^𝑇𝐾^𝑇(𝑖,𝑘)𝐵^𝑇𝑃(𝑖,𝑘)𝑥(𝑡)+𝜖₂⁻¹𝑥^𝑇(𝑡)×𝑃(𝑖,𝑘)𝐵𝐾(𝑖,𝑘)𝐵𝐾(𝑖,𝑘)𝑃⁻¹(𝑖,𝑘)𝐾^𝑇(𝑖,𝑘)×𝐵^𝑇𝐾^𝑇(𝑖,𝑘)𝐵^𝑇𝑃(𝑖,𝑘)𝑥(𝑡)]+𝜖₁⁰_{−𝜏(𝑖,𝑘)}𝑥^𝑇(𝑡+𝜃)𝑃(𝑖,𝑘)𝑥(𝑡+𝜃)𝑑𝜃+𝜖₂⁰_{−𝜏(𝑖,𝑘)}𝑥^𝑇(𝑡−𝜏(𝑖,𝑘)+𝜃)𝑃(𝑖,𝑘)𝑥(𝑡−𝜏(𝑖,𝑘)+𝜃)𝑑𝜃. (3.12) From (3.2), (3.3), and Lemma 2.5, we can obtain 𝐴𝑃⁻¹(𝑖,𝑘)𝐴^𝑇<𝜖₁𝑃⁻¹(),()𝑃𝑖,𝑘𝐵𝐾𝑖,𝑘⁻¹(𝑖,𝑘)𝐾^𝑇(𝑖,𝑘)𝐵^𝑇<𝜖₂𝑃⁻¹(𝑖,𝑘), (3.13) which yields −2⁰_{−𝜏(𝑖,𝑘)}𝑥^𝑇(𝑡)𝑃()()×[()()𝑥(()()𝜖𝑖,𝑘𝐵𝐾𝑖,𝑘𝐴𝑥𝑡+𝜃+𝐵𝐾𝑖,𝑘𝑡−𝜏𝑖,𝑘+𝜃)]𝑑𝜃≤2𝜏𝑖,𝑘₂𝑥^𝑇(𝑡)𝑃(𝑖,𝑘)𝑥(𝑡)+𝜖₁⁰_{−𝜏(𝑖,𝑘)}𝑥^𝑇(𝑡+𝜃)𝑃(𝑖,𝑘)𝑥(𝑡+𝜃)𝑑𝜃+𝜖₂⁰_{()−𝜏𝑖,𝑘}𝑥^𝑇(𝑡−𝜏(𝑖,𝑘)+𝜃)𝑃(𝑖,𝑘)×𝑥(𝑡−𝜏(𝑖,𝑘)+𝜃)𝑑𝜃. (3.14) Following Lemma 2.5, for −2𝜏≤𝜃≤0, we assume that for any 𝛿>1, the following inequality holds: 𝑉𝑥(𝑡+𝜃),𝑟_𝑡+𝜃,𝜂_𝑡+𝜃<𝛿𝑉𝑥(𝑡),𝑟_𝑡,𝜂_𝑡, (3.15) then we have ℒ𝑉(𝑥(𝑡),𝑖,𝑘)≤𝑥^𝑇(𝑡)ℋ(𝜏(𝑖,𝑘),𝛿)𝑥(𝑡), (3.16) where ℋ(𝜏(𝑖,𝑘),𝛿) is given by ℋ(𝜏(𝑖,𝑘),𝛿)=𝐴^𝑇𝑃(𝑖,𝑘)+𝑃(𝑖,𝑘)𝐴+𝐾^𝑇(𝑖,𝑘)𝐵^𝑇𝑃(𝑖,𝑘)+𝑃(𝑖,𝑘)𝐵𝐾(𝑖,𝑘)+^𝑁𝑟_𝑗=1𝜆_𝑖𝑗𝑃+𝑗,𝑘_{𝑁𝜂𝑙=1}𝜋_𝑘𝑙𝑃(𝑖,𝑙)+2𝜏(𝑖,𝑘)𝜖₂𝑃(𝑖,𝑘)+𝜏(𝑖,𝑘)𝜖₁𝛿𝑃(𝑖,𝑘)+𝜏(𝑖,𝑘)𝜖₂𝛿𝑃(𝑖,𝑘), (3.17) for some positive scalars 𝜖₁ and 𝜖₂. before and after multiplying ℋ(𝜏(𝑖,𝑘),𝛿) by 𝑄(𝑖,𝑘)=𝑃⁻¹(𝑖,𝑘) and its transpose, it gives ℋ(𝜏())()𝐴𝑖,𝑘,𝛿=𝑄𝑖,𝑘^𝑇+𝐴𝑄(𝑖,𝑘)+𝑄(𝑖,𝑘)𝐾^𝑇(𝑖,𝑘)𝐵^𝑇+𝐵𝐾(𝑖,𝑘)𝑄(𝑖,𝑘)+𝑄(𝑖,𝑘)^𝑁𝑟_𝑗=1𝜆_𝑖𝑗𝑃𝑄()()𝑗,𝑘𝑖,𝑘+𝑄𝑖,𝑘^𝑁𝜂_𝑙=1𝜋_𝑘𝑙𝑃(𝑖,𝑙)𝑄(𝑖,𝑘)+2𝜏(𝑖,𝑘)𝜖₂𝑄(𝑖,𝑘)+𝜏(𝑖,𝑘)𝜖₁𝛿𝑄(𝑖,𝑘)+𝜏(𝑖,𝑘)𝜖₂𝛿𝑄(𝑖,𝑘). (3.18) Since 0≤𝜏(𝑖,𝑘)≤𝜏, (3.19) we have from (3.16) that ℒ𝑉(𝑥(𝑡),𝑖,𝑘)≤𝑥^𝑇(𝑡)ℋ(𝜏,𝛿)𝑥(𝑡). (3.20) From (3.1) and Lemma 2.5, it follows that ℋ(𝜏,𝛿=1)<0, (3.21) which is equivalent to ℋ(𝜏,𝛿=1)<0. (3.22) Using the continuity properties of the eigenvalues of ℋ with respect to 𝛿, then there exists a 𝛿>1 sufficiently small such that (3.21) still holds. Thus, for such a 𝛿, we have ℋ(𝜏,𝛿)<0, (3.23) which yields ℒ𝑉𝑥(𝑡),𝑟_𝑡,𝜂_𝑡≤−𝛽‖𝑥(𝑡)‖², (3.24) where 𝛽=min_{𝑖∈𝜍𝑟,𝑘∈𝜍𝜂}𝜆_min(−ℋ(𝜏,𝛿))>0. (3.25) Applying Dynkin's formula, we have 𝔼[𝑉(𝑥(𝑡),𝑖,𝑘)]−𝔼𝑉𝑥₀,𝑟₀,𝜂₀=𝔼^𝑡₀ℒ𝑉𝑥(𝑠),𝑟_𝑠,𝜂_𝑠𝑑𝑠∣𝑥₀,𝑟₀,𝜂₀≤−𝛽𝔼^𝑡₀‖𝑥(𝑠)‖²𝑑𝑠∣𝑥₀,𝑟₀,𝜂₀. (3.26) Note that 𝔼[𝑉(𝑥(𝑡),𝑖,𝑘)]≥0, (3.27) Then we can obtain 𝛽𝔼^𝑡₀‖𝑥(𝑠)‖²𝑑𝑠∣𝑥₀,𝑟₀,𝜂₀≤𝔼[𝑉(𝑥(𝑡),𝑖,𝑘)]+𝛽𝔼^𝑡₀‖𝑥(𝑠)‖²𝑑𝑠∣𝑥₀,𝑟₀,𝜂₀≤𝔼𝑉𝑥₀,𝑟₀,𝜂₀. (3.28) This completes the proof.

Remark 3.2. In case of constant transmission delay, that is, 𝜏_sc(𝑟_𝑡)=𝜏_sc,𝜏_ca(𝜂_𝑡)=𝜏_ca,𝜆_𝑖𝑗=0, and 𝜋_𝑘𝑙=0, Theorem 3.1 can be directly applied to systems with constant delay.
It should be noted that the terms 𝜖₁𝑄(𝑖,𝑘) and 𝜖₂𝑄(𝑖,𝑘) in (3.1)–(3.3) are bilinear. Therefore, we propose the following algorithm to solve these bilinear matrix inequality problems.
Step 1. Set 𝑄₀(𝑖,𝑘)>0, and 𝑌₀(𝑖,𝑘) such that the following LMI holds: 𝐽̃(𝑖,𝑘)𝜑₁(𝑖,𝑘)𝜑₂(𝑖,𝑘)∗−𝜓₁0∗∗−𝜓₂<0, (3.29) where 𝐽̃(𝑖,𝑘)=𝑄(𝑖,𝑘)𝐴^𝑇+𝐴𝑄(𝑖,𝑘)+𝑌^𝑇(𝑖,𝑘)𝐵^𝑇+𝐵𝑌(𝑖,𝑘)+𝜆_𝑖𝑖𝑄(𝑖,𝑘)+𝜋_𝑘𝑘𝑄(𝑖,𝑘). (3.30)Step 2. For 𝑄(𝑖,𝑘)>0 given in the previous step, find 𝜖_1𝑠,𝜖_2𝑠, and 𝑌_𝑠(𝑖,𝑘) by solving the following convex optimization problem: max_{𝑌()𝑖,𝑘,𝜖1,𝜖2}𝜏𝑌(𝑖,𝑘),𝜖₁,𝜖₂,s.t.(3.1)-(3.3)holdfor𝑄(𝑖,𝑘)>0fixed. (3.31)Step 3. For 𝑌(𝑖,𝑘), 𝜖₂, and 𝜖₁ given in the previous step, find 𝑄_𝑠(𝑖,𝑘)>0 by solving the following quasiconvex optimization problem max_{𝑄()𝑖,𝑘>0}𝜏(𝑄(,()-()()𝑖,𝑘))s.t.3.13.3holdfor𝑌𝑖,𝑘,𝜖₂,and𝜖₁fixed. (3.32)Step 4. Return to step 2 until the convergence of 𝜏 is attained with a desired precision.

Remark 3.3. For a given 𝑄(𝑖,𝑘), the considered optimization problem consists of minimizing an eigenvalue problem which is a convex one. On the other hand, for given 𝑌(𝑖,𝑘), 𝜖₁ and 𝜖₂, the considered optimization problem consists of minimizing a generalized eigenvalue problem which is a quasiconvex optimization problem. Therefore, the proposed algorithm gives a suboptimal solution.

4. Simulations

In this section, simulations of the position control for robotic manipulator ViSHaRD3 [46] are included to illustrate the effectiveness of the proposed method. Combining computed torque feedback approach [47] with friction compensation, the system is decoupled into three systems. The first and second joints of the ViSHaRD3 are 𝑑𝑞=𝑞+01𝑢(𝑡)𝑑𝑡𝑞̇011−50𝑞̇, (4.1) and the third is 𝑑𝑞=𝑞+01𝑢(𝑡)𝑑𝑡𝑞̇011−40𝑞̇. (4.2)

For simplicity, we only discuss the third joint of ViSHaRD3. Suppose that the sampling interval is 𝑇=0.01 s, and the Markov process 𝑟_𝑡 that governs the mode switching of the SC delay takes values in 𝜍_𝑟={1,2} and has the generator Λ=−332−2, (4.3) and the Markov process 𝜂_𝑡 that governs the mode switching of the CA delay takes values in 𝜂_𝑟={1,2} and has the generator Π=−112−2. (4.4)

Associated with modes 1 and 2, let the system have time delay 𝜏_sc(1)=0.03 s, 𝜏_ca(1)=0.02 s and 𝜏_sc(2)=0.025 s, 𝜏_ca(2)=0.015 s, respectively. From (2.8), we have 𝜏=0.06 s, and the initial condition is 𝜙(𝜃)=[−1,0]^𝑇,𝜃∈[−0.06,0]. By the proposed algorithm and Theorem 3.1, we can obtain the controllers as follows: 𝐾()=,𝐾()=,𝐾()=,𝐾()=.1,1−645.0596−15.91091,2−623.3689−15.49992,1−575.1361−14.22962,2−616.8428−15.3049 (4.5)

The simulations of the state response and the control input for the closed-loop system are depicted in Figures 3 and 4, respectively, which shows that the system is stochastically stable.

Figure 3 

State response of closed-loop system.

Figure 4 

Control input of closed-loop system.

5. Conclusions

In this paper, a technique of designing a mode-dependent state feedback controller for networked control systems with random time delays has been proposed. The main contribution of this paper is that both the sensor-to-controller and controller-to-actuator delays have been taken into account. Two Markov processes have been used to model these two time delays. Based on Lyapunov stability theorem combined with Razumikhin-based technique, some new delay-dependent stability criteria in terms of BMIs for the system are derived. A state feedback controller that makes the closed-loop system stochastically stable is designed, which can be solved by the proposed algorithm. Simulations results are presented to illustrate the validity of the design methodology.

Acknowledgments

This work is partially supported by the Youth Fund of Anhui Province no. 2010SQRL162 and the program of science and technology of Huainan no. 2011A08005.