H_∞ Filtering for a Class of Piecewise Homogeneous Markovian Jump Nonlinear Systems

Abstract

𝐻_∞ filtering problem for a class of piecewise homogeneous Markovian jump nonlinear systems is investigated. The aim of this paper is to design a mode-dependent filter such that the filtering error system is stochastically stable and satisfies a prescribed 𝐻_∞ disturbance attenuation level. By using a new mode-dependent Lyapunov-Krasovskii functional, mixed mode-dependent sufficient conditions on stochastic stability are formulated in terms of linear matrix inequalities (LMIs). Based on this, the mode-dependent filter is obtained. A numerical example is given to illustrate the effectiveness of the proposed main results.

1. Introduction

The filtering problem has received significant attention in the past decade. Current efforts on this topic can be divided into two classes: the Kalman filtering approach and the 𝐻_∞ filtering approach. As we all know, Kalman filtering approach is based on the assumption that the system is exactly known, and its disturbances are stationary Gaussian noises with known statistics. These assumptions limit the application scope of the Kalman filtering technique when there are uncertainties in either the exogenous input signals or the system model [1]. To overcome the restriction described above, 𝐻_∞ filtering has been introduced as an alternative filtering technique [2–6].

On the other hand, Markovian jump systems are an active area of research. It switches from one mode to another in a random way, and the switching between the modes is governed by a Markovian process with discrete and finite state space. These models serve as convenient tools for analyzing plants that are subjected to random abrupt parameter changes due to, for instance, component and/or interconnection failures, sudden environmental changes, or change of the operating point of a linearized model of a nonlinear plant. A wide class of industrial system applications experience time delays due to various reasons including inherent physical properties (mass transport flow, recycling), data transmission delays or finite capabilities of information exchange [7]. When considering the continuous systems with time-varying delay, the systems can be clarified into two types, one is slow time-varying delay systems, that is, the derivative of the time delay is less than one, for example, [2, 8, 9], and the other is fast time-varying delay systems, that is, there are no constraints on the derivative of the time delay. Both Lyapunov-Krasovskii and Lyapunov-Razumikhin approaches are fundamental for time-delay systems, and some existing work usually do not require the derivative of the time delay to be less than one, see, for example, [10, 11]. Due to their extensive practical applications, considerable attention has been devoted to Markovian jump systems with time delays. The issues of stability and control have been well investigated; see, for example, [9, 11–29] and references therein. In [30–32], the sliding mode control of Markovian jump singular systems was studied, and, new integral-type sliding surface functions were designed. Moreover, strict LMI conditions of the stochastic stability were proposed in [30, 31], which are easy to be checked by Matlab LMI toolbox. In [32], a suitable switching surface function and a sliding mode control law were designed to ensure the attraction of the sliding surface when the system changes from one mode to another under Markovian switching, and the slack matrix approach was used to derive less conservative LMI conditions assuring stochastic admissibility. The filtering problem for Markovian jump time-delay systems was reported in [8, 33–40]. Many nonlinear physical systems can be represented as a connection of a linear dynamical system and a nonlinear element. Filtering for Markovian jump nonlinear system is an important research area that has attracted considerable interest [41–43]. It should be pointed out that the above-mentioned references assume that the Markovian processes are homogeneous, that is, the considered transition probabilities (TPs) in Markovian process are assumed to be time invariant. However, the assumption cannot always be satisfied in real applications, and the ideal assumption on TPs inevitably limits the applications of the established results to some extent [44]. Therefore, it is important and necessary to pay attention to the study of Markovian jump systems with time-varying TPs. Recently, the problem of 𝐻_∞ estimation for discrete-time Markovian jump linear system with time-varying TPs has been investigated in [44]. The 𝐻_∞ control problem has been conducted for a class of discrete-time Markovian jump systems with time-varying TPs in [45], where the average dwell-time switching is used to describe the variation among the TPs. The stochastic stability analysis of piecewise homogeneous Markovian jump neural networks with mixed time delays has been studied in [46]. But, the time-varying delays in [46] are independent of jump mode. To the best of our knowledge, no results have been given for piecewise homogeneous Markovian jump nonlinear systems with mode-dependent time-varying delays. With the appearance of time-varying TPs and mode-dependent time-varying discrete and distributed delays, the main difficulties are as follows: (1) the new Lyapunov functional should be constructed to deal with above problem; (2) since the system involves joint jump processes and mode-dependent time-varying delays, the calculation of derivative of the Lyapunov functional and the using of inequality techniques become more complicated. Moreover, the Lyapunov matrix 𝑃_𝑖 is assumed to be diagonal matrix in some existing literature, which leads to some conservativeness. Therefore, the key problems in this research are: (1) how to choose a Lyapunov function to derive a sufficient stochastic stability condition for the considered systems; (2) how to use the inequality techniques and calculate the parameters of the filter such that the resulting sufficient conditions are less conservative? Which has motivated this paper.

In this study, we are concerned to develop an efficient approach for 𝐻_∞ filtering problem of piecewise homogeneous Markovian jump system. The system under study involves mode-dependent time-varying discrete and distributed delays and inherent sector-like nonlinearities. By using a novel Lyapunov-Krasovskii functional, mixed mode-dependent sufficient condition on stochastic stability with an 𝐻_∞ performance is derived in terms of LMIs. Based on this, the existence condition of the desired filter which guarantees stochastic stability and an 𝐻_∞ performance of the corresponding filtering error system is presented. A numerical example is provided to show the effectiveness of the proposed results.

Notation. Throughout this paper, ℝ^𝑛 denotes the 𝑛-dimensional Euclidean space. (Ω,ℱ,𝒫) is a probability space, Ω is the sample space, ℱ is the 𝜎-algebra of subsets of the sample space, and 𝒫 is the probability measure on ℱ. ℰ{⋅} refers to the expectation operator with respect to some probability measure 𝒫. We use diag{⋅,⋅,⋅} as a block diagonal matrix. 𝐴>0  (<0) means 𝐴 is a symmetric positive (negative) definite matrix. 𝐴^𝑇 denotes the transpose of matrix 𝐴, 𝐼 is the identity matrix with compatible dimension.

2. System Description and Definitions

Fix a probability space (Ω,ℱ,𝒫) and consider the following stochastic Markovian jump system with mode-dependent time-varying delays: 𝑥̇(𝑡)=𝐴𝑟_𝑡𝑥(𝑡)+𝐴₁𝑟_𝑡𝑥𝑡−𝜏𝑡,𝑟_𝑡+𝐴₂𝑟_𝑡^𝑡_{𝑡−𝜏𝑡,𝑟𝑡}𝑥(𝑠)d𝑠+𝐵𝑟_𝑡𝑓(𝑥(𝑡))+𝐷₁𝑟_𝑡𝜔(𝑡),𝑦(𝑡)=𝐶𝑟_𝑡𝑥(𝑡)+𝐶₁𝑟_𝑡𝑥𝑡−𝜏𝑡,𝑟_𝑡+𝐶₂𝑟_𝑡^𝑡_{𝑡−𝜏𝑡,𝑟𝑡}𝑥(𝑠)d𝑠+𝐸𝑟_𝑡𝑔(𝑥(𝑡))+𝐷₂𝑟_𝑡𝜔(𝑡),𝑧(𝑡)=𝐻𝑟_𝑡𝑥(𝑡),𝑥(𝑡)=𝜙(𝑡),𝑡∈[−𝜏,0], (2.1) where 𝑥(𝑡)∈ℝ^𝑛 is the state vector; 𝜔(𝑡) is the exogenous disturbance input which belongs to 𝐿₂[0∞); 𝑦(𝑡)∈ℝ^𝑝 is the measured output; 𝑧(𝑡)∈ℝ^𝑞 is the signal to be estimated; 𝜙(𝑡) is a compatible vector-valued initial function defined on [−𝜏,0]; 𝐴(𝑟_𝑡), 𝐴₁(𝑟_𝑡), 𝐴₂(𝑟_𝑡), 𝐵(𝑟_𝑡), 𝐷₁(𝑟_𝑡),  𝐶(𝑟_𝑡), 𝐶₁(𝑟_𝑡), 𝐶₂(𝑟_𝑡), 𝐸(𝑟_𝑡), 𝐷₂(𝑟_𝑡) and 𝐻(𝑟_𝑡) are real constant matrices with appropriate dimensions. 𝜏(𝑡,𝑟_𝑡) is the mode-dependent time-varying delay. The process {𝑟_𝑡,𝑡⩾0} is described by a Markov chain with finite state space 𝑆₁={1,2,…,𝑁}, and its transition probability matrix, Π^{(𝜎𝑡+Δ)}=[𝜋^{(𝜎𝑡+Δ)}_𝑖𝑙]_𝑁×𝑁(𝑖,𝑙∈𝑆₁), is governed by Pr𝑟_𝑡+Δ=𝑙∣𝑟_𝑡=𝜋=𝑖^{(𝜎𝑡+Δ)𝑖𝑙}Δ+𝑜(Δ),𝑙≠𝑖,1+𝜋^{(𝜎𝑡+Δ)}_𝑖𝑖Δ+𝑜(Δ),𝑙=𝑖, (2.2) where Δ>0 and lim_Δ→0(𝑜(Δ)/Δ)=0; 𝜋^{(𝜎𝑡+Δ)}_𝑖𝑙⩾0 for 𝑙≠𝑖 is the transition rate from mode 𝑖 at time 𝑡 to mode 𝑙 at time 𝑡+Δ and 𝜋^{(𝜎𝑡+Δ)}_𝑖𝑖=−∑^𝑁_{𝑙=1,𝑙≠𝑖}𝜋^{(𝜎𝑡+Δ)}_𝑖𝑙. In this study, we assume that 𝜎_𝑡 vary in another finite set 𝑆₂={1,2,…,𝑀}, and the variations are considered as the stochastic variation. The variation of 𝜎_𝑡 is governed by a higher-level transition probability (HTP) matrix Λ=[𝜆_𝑗𝑘]_𝑀×𝑀(𝑗,𝑘∈𝑆₂) and the TPs of Markov chain satisfy Pr𝜎_𝑡+Δ=𝑘∣𝜎_𝑡=𝜆=𝑗^𝑗𝑘Δ+𝑜(Δ),𝑘≠𝑗,1+𝜆_𝑗𝑗Δ+𝑜(Δ),𝑘=𝑗, (2.3) where 𝜆_𝑗𝑘⩾0 for 𝑘≠𝑗 is the transition rate from mode 𝑗 at time 𝑡 to mode 𝑘 at time 𝑡+Δ and 𝜆_𝑗𝑗=−∑^𝑀_{𝑘=1,𝑘≠𝑗}𝜆_𝑗𝑘. The stochastic processes 𝑟_𝑡 and 𝜎_𝑡 are assumed to be independent throughout this paper. For vector-valued functions 𝑓(𝑥(𝑡)) and 𝑔(𝑥(𝑡)), we assume: 𝑓(𝑥)−𝑓𝑦−𝑀₁𝑥−𝑦^𝑇𝑓(𝑥)−𝑓𝑦−𝑀₂𝑔(𝑥)𝑦𝑥−𝑦⩽0,−𝑔−𝐿₁𝑥−𝑦^𝑇𝑔(𝑥)−𝑔𝑦−𝐿₂𝑥−𝑦⩽0, (2.4)

where for all 𝑥,𝑦∈ℝ^𝑛 and 𝑀₁,𝑀₂,𝐿₁,𝐿₂∈ℝ^𝑛×𝑛 are the known constant matrices. In what follows, for implicity of presentations and without loss of generality, we always assume that 𝑓(0)=0 and 𝑔(0)=0.

For simplicity, a matrix 𝑅(𝑟_𝑡) will be denoted by 𝑅_𝑖. For example, 𝐴(𝑟_𝑡) is denoted by 𝐴_𝑖, 𝐴₁(𝑟_𝑡) is denoted by 𝐴_1𝑖 and 𝜏(𝑟_𝑡,𝑡) is denoted by 𝜏_𝑖(𝑡), (𝑖∈𝑆₁). When the mode is in 𝑟_𝑡=𝑖, the mode-dependent time-varying delay satisfies 0⩽𝜏_𝑖(𝑡)⩽𝜏_𝑖⩽𝜏,𝜏̇_𝑖(𝑡)⩽𝜇_𝑖, (2.5) where 𝜏=max{𝜏_𝑖}.

In this study, the following full-order linear filter is proposed to estimate the signal 𝑧(𝑡): 𝑥̇(𝑡)=𝐴_𝐹𝑖𝑗𝑥(𝑡)d𝑡+𝐵_𝐹𝑖𝑗𝑦(𝑡),𝑧(𝑡)=𝐶_𝐹𝑖𝑗𝑥(𝑡),𝑥(0)=0, (2.6) where 𝑥(𝑡) is the filter state vector, and (𝐴_𝐹𝑖𝑗𝐵_𝐹𝑖𝑗𝐶_𝐹𝑖𝑗) are appropriately dimensioned filter matrices to be determined.

Define the estimation error by 𝑒(𝑡)=𝑧(𝑡)−𝑧(𝑡), we obtain the following filtering error system: 𝜉̇(𝑡)=𝐴_𝑖𝑗𝜉(𝑡)+𝐴_1𝑖𝑗𝐾𝜉𝑡−𝜏_𝑖(𝑡)+𝐴_2𝑖𝑗𝐾^𝑡_{𝑡−𝜏𝑖(𝑡)}𝜉(𝑠)d𝑠+𝐵_𝑖𝑗𝑓(𝐾𝜉(𝑡))+𝐸_𝑖𝑗𝑔(𝐾𝜉(𝑡))+𝐷_𝑖𝑗𝜔(𝑡),𝑒(𝑡)=𝐻_𝑖𝑗𝜉(𝑡),𝑥(𝑡)=𝜙(𝑡),∀𝑡∈[−𝜏,0], (2.7) where 𝜉(𝑡)=[𝑥^𝑇(𝑡)𝑥^𝑇(𝑡)]^𝑇,  𝜙(𝑡)=[𝜙^𝑇(𝑡)0^𝑇]^𝑇, and 𝐴_𝑖𝑗=𝐴_𝑖0𝐵_𝐹𝑖𝑗𝐶_𝑖𝐴_𝐹𝑖𝑗,𝐴_1𝑖𝑗=𝐴_1𝑖𝐵_𝐹𝑖𝑗𝐶_1𝑖,𝐴_2𝑖𝑗=𝐴_2𝑖𝐵_𝐹𝑖𝑗𝐶_2𝑖,𝐵_𝑖𝑗=𝐵_𝑖0,𝐸_𝑖𝑗=0𝐵_𝐹𝑖𝑗𝐸_𝑖,𝐷_𝑖𝑗=𝐷_1𝑖𝐵_𝐹𝑖𝑗𝐷_2𝑖,𝐻_𝑖𝑗=𝐻_𝑖−𝐶_𝐹𝑖𝑗,𝐾=𝐼0. (2.8)

Remark 2.1. According to the definitions of homogeneous Markovian chain and nonhomogeneous Markovian chain in [44, 47], one can see that the Markovian chain 𝜎_𝑡 in this paper is homogeneous, while the Markovian chain 𝑟_𝑡 is neither homogeneous nor nonhomogeneous, but a state between them, which can be called the finite piecewise homogeneous Markovian chain.

Remark 2.2. In this paper, the nonlinear functions 𝑓(𝑥(𝑡)) and 𝑔(𝑥(𝑡)) are said to belong to sectors, which means that the nonlinearities are bounded by sectors. The nonlinear descriptions in (2.4) are quite general that include the usual Lipschitz conditions as a special case [2].

The following lemma and definitions are introduced, which will be used in the proof of the main results.

Lemma 2.3 (see [48]). For any matrix 𝑀>0, scalar 𝛾>0, vector function 𝜔∶[0,𝛾]→𝑅^𝑛 such that the integrations concerned are well defined, the following inequality holds: ^𝛾₀𝜔^𝑇(𝑠)d𝑠𝑀^𝛾₀𝜔(𝑠)d𝑠⩽𝛾^𝛾₀𝜔^𝑇(𝑠)𝑀𝜔(𝑠)d𝑠. (2.9)

Definition 2.4. The filtering error system (2.7) with 𝜔(𝑡)=0 is said to be stochastically stable, if for any initial 𝜑(𝑡)∈ℝ^𝑛 defined on [−𝜏,0] and modes 𝑟_𝑡 and 𝜎_𝑡, the following relation holds: [7] lim_𝑇→∞ℰ^𝑇₀𝜉𝑠,𝑟₀,𝜎₀,𝜑²d𝑠<∞. (2.10)

Definition 2.5. Given a scalar 𝛾>0, the filtering error system (2.7) is said to be stochastically stable with an 𝐻_∞ performance 𝛾, if for every system mode 𝑟_𝑡, the filtering error system (2.7) with 𝜔(𝑡)=0 is stochastically stable and, under zero initial condition, it satisfies ‖𝑒‖₂⩽𝛾‖𝜔‖₂ for any nonzero 𝜔(𝑡)∈𝐿₂[0,∞].

3. Main Results

In this section, we first propose a delay-dependent sufficient condition for stochastic stability with the 𝐻_∞ performance of filtering error system (2.7). Now, consider the following Lyapunov-Krasovskii functional for systems (2.7): 𝑉𝜉_𝑡,𝑡,𝑖,𝑗=⁶_𝑛=1𝑉_𝑛𝜉_𝑡,𝑡,𝑖,𝑗, (3.1) where 𝑉₁𝜉_𝑡,𝑡,𝑖,𝑗=𝜉^𝑇(𝑡)𝑃𝑟_𝑡,𝜎_𝑡𝜉(𝑡),𝑉₂𝜉_𝑡,𝑡,𝑖,𝑗=^𝑡_{𝑡−𝜏(𝑡,𝑟𝑡)}𝜉^𝑇(𝑠)𝐾^𝑇𝑄₁𝑟_𝑡,𝜎_𝑡𝐾𝜉(𝑠)d𝑠+^𝑡_{𝑡−𝜏(𝑟𝑡)}𝜉^𝑇(𝑠)𝐾^𝑇𝑄₂𝑟_𝑡,𝜎_𝑡(𝑠)𝑉𝐾𝜉d𝑠,₃𝜉_𝑡,𝑡,𝑖,𝑗=^𝑡_{𝑡−𝜏(𝑡,𝑟𝑡)}^𝑡_𝜃𝜉^𝑇(𝑠)𝐾^𝑇d𝑠𝑅₁𝑟_𝑡,𝜎_𝑡^𝑡_𝜃(𝑠)𝑉𝐾𝜉d𝑠d𝜃,₄𝜉_𝑡,𝑡,𝑖,𝑗=𝜏⁰_−𝜏^𝑡_𝑡+𝜃𝜉̇^𝑇(𝑠)𝐾^𝑇𝜉̇(𝑠)𝑉𝑍𝐾d𝑠d𝜃,₅𝜉_𝑡,𝑡,𝑖,𝑗=⁰_−𝜏^𝑡_𝑡+𝜃𝜉^𝑇(𝑠)𝐾^𝑇𝑅₂𝐾𝜉(𝑠)d𝑠d𝜃+⁰_{𝑟−𝜏𝑡}^𝑡_𝑡+𝜃𝜉^𝑇(𝑠)𝐾^𝑇𝐺₁𝑟_𝑡(𝑠)𝑉𝐾𝜉d𝑠d𝜃,₆𝜉_𝑡,𝑡,𝑖,𝑗=^𝜏₀⁰_−𝜃^𝑡_𝑡+𝑠𝜉^𝑇(𝛼)𝐾^𝑇𝑅₃𝐾𝜉(𝛼)d𝛼d𝑠d𝜃+^𝜏₀^𝑡_𝑡−𝜃(𝑠−𝑡+𝜃)𝜉^𝑇(𝑠)𝐾^𝑇𝐺₂𝐾𝜉(𝑠)d𝑠d𝜃. (3.2)

Let ℒ be the weak infinitesimal generator of the random process {𝜉_𝑡,𝑟_𝑡,𝜎_𝑡}. Then, for each 𝑖∈𝑆₁, 𝑗∈𝑆₂, the stochastic differential of 𝑉₁(𝜉_𝑡,𝑡,𝑖,𝑗) along the trajectory of system (2.7) is given by ℒ𝑉₁𝜉_𝑡,𝑡,𝑖,𝑗=2𝜉^𝑇(𝑡)𝑃_𝑖𝑗×𝐴_𝑖𝑗𝜉(𝑡)+𝐴_1𝑖𝑗𝐾𝜉𝑡−𝜏_𝑖(𝑡)+𝐴_2𝑖𝑗𝐾^𝑡_{𝑡−𝜏𝑖(𝑡)}𝜉(𝑠)d𝑠+𝐵_𝑖𝑗𝑓((𝑡+𝐾𝜉))𝐸_𝑖𝑗𝑔(𝐾𝜉(𝑡))+𝐷_𝑖𝑗𝜔(𝑡)+𝜉^𝑇(𝑡)_𝑙∈𝑆1𝜋^(𝑗)_𝑖𝑙𝑃_𝑙𝑗+_𝑘∈𝑆2𝜆_𝑗𝑘𝑃_𝑖𝑘𝜉(𝑡),ℒ𝑉₂𝜉_𝑡,𝑡,𝑖,𝑗=ℒ^𝑡_{𝑡−𝜏(𝑡,𝑟𝑡)}𝜉^𝑇(𝑠)𝐾^𝑇𝑄₁𝑟_𝑡,𝜎_𝑡𝐾𝜉(𝑠)d𝑠+ℒ^𝑡_{𝑡−𝜏(𝑟𝑡)}𝜉^𝑇(𝑠)𝐾^𝑇𝑄₂𝑟_𝑡,𝜎_𝑡𝐾𝜉(𝑠)d𝑠=lim_Δ→0+1Δ𝐸_𝑟^𝑡+Δ_{𝑡+Δ−𝜏𝑡+Δ,𝑡+Δ}𝜉^𝑇(𝑠)𝐾^𝑇𝑄₁𝑟_𝑡+Δ,𝜎_𝑡+Δ𝐾𝜉(𝑠)d𝑠−^𝑡_{𝑡−𝜏𝑖(𝑡)}𝜉^𝑇(𝑠)𝐾^𝑇𝑄_1𝑖𝑗𝐾𝜉(𝑠)d𝑠+lim_Δ→0+1Δ𝐸_𝑟^𝑡+Δ_{𝑡+Δ−𝜏𝑡+Δ}𝜉^𝑇(𝑠)𝐾^𝑇𝑄₂𝑟_𝑡+Δ,𝜎_𝑡+Δ(𝑠)−𝐾𝜉d𝑠_{𝑡𝑡−𝜏𝑖(𝑡)}𝜉^𝑇(𝑠)𝐾^𝑇𝑄_2𝑖𝑗𝐾𝜉(𝑠)d𝑠=𝜉^𝑇(𝑡)𝐾^𝑇𝑄_1𝑖𝑗𝐾𝜉(𝑡)−1−𝜏̇_𝑖(𝑡)𝜉^𝑇𝑡−𝜏_𝑖(𝑡)𝐾^𝑇𝑄_1𝑖𝑗𝐾𝜉𝑡−𝜏_𝑖(𝑡)+𝜉^𝑇(𝑡)𝐾^𝑇𝑄_2𝑖𝑗𝐾𝜉(𝑡)−𝜉^𝑇𝑡−𝜏_𝑖𝐾^𝑇𝑄_2𝑖𝑗𝐾𝜉𝑡−𝜏_𝑖+_𝑙∈𝑆1𝜋^(𝑗)_𝑖𝑙^𝑡_{𝑡−𝜏𝑙(𝑡)}𝜉^𝑇(𝑠)𝐾^𝑇𝑄_1𝑙𝑗𝐾𝜉(𝑠)d𝑠+_𝑘∈𝑆2𝜆_𝑗𝑘^𝑡_{𝑡−𝜏𝑖(𝑡)}𝜉^𝑇(𝑠)𝐾^𝑇𝑄_1𝑖𝑘𝐾𝜉(𝑠)d𝑠+_𝑙∈𝑆1𝜋^(𝑗)_𝑖𝑙^𝑡_{𝑡−𝜏𝑙}𝜉^𝑇(𝑠)𝐾^𝑇𝑄_2𝑙𝑗𝐾𝜉(𝑠)d𝑠+_𝑘∈𝑆2𝜆_𝑗𝑘^𝑡_{𝑡−𝜏𝑖}𝜉^𝑇(𝑠)𝐾^𝑇𝑄_2𝑖𝑘𝐾𝜉(𝑠)d𝑠⩽𝜉^𝑇(𝑡)𝐾^𝑇𝑄_1𝑖𝑗𝐾𝜉(𝑡)−1−𝜇_𝑖𝜉^𝑇𝑡−𝜏_𝑖(𝑡)𝐾^𝑇𝑄_1𝑖𝑗𝐾𝜉𝑡−𝜏_𝑖(𝑡)+𝜉^𝑇(𝑡)𝐾^𝑇𝑄_2𝑖𝑗𝐾𝜉(𝑡)−𝜉^𝑇𝑡−𝜏_𝑖𝐾^𝑇𝑄_2𝑖𝑗𝐾𝜉𝑡−𝜏_𝑖+𝜋^(𝑗)_𝑖𝑖^𝑡_{𝑡−𝜏𝑖(𝑡)}𝜉^𝑇(𝑠)𝐾^𝑇𝑄_1𝑖𝑗(𝑠)+𝐾𝜉d𝑠_{𝑡𝑡−𝜏}𝜉^𝑇(𝑠)𝐾^𝑇_{𝑙∈𝑆1,𝑙≠𝑖}𝜋^(𝑗)_𝑖𝑙𝑄_1𝑙𝑗(𝑠)+𝐾𝜉d𝑠_{𝑡𝑡−𝜏𝑖(𝑡)}𝜉^𝑇(𝑠)𝐾^𝑇_𝑘∈𝑆2𝜆_𝑗𝑘𝑄_1𝑖𝑘𝐾𝜉(𝑠)d𝑠+𝜋^(𝑗)_𝑖𝑖^𝑡_{𝑡−𝜏𝑖}𝜉^𝑇(𝑠)𝐾^𝑇𝑄_2𝑖𝑗𝐾𝜉(𝑠)d𝑠+^𝑡_𝑡−𝜏𝜉^𝑇(𝑠)𝐾^𝑇_{𝑙∈𝑆1,𝑙≠𝑖}𝜋^(𝑗)_𝑖𝑙𝑄_2𝑙𝑗(𝑠)+𝐾𝜉d𝑠_{𝑡𝑡−𝜏𝑖}𝜉^𝑇(𝑠)𝐾^𝑇_𝑘∈𝑆2𝜆_𝑗𝑘𝑄_2𝑖𝑘𝐾𝜉(𝑠)d𝑠.ℒ𝑉₃𝜉_𝑡,𝑡,𝑖,𝑗=−1−𝜏̇_𝑖(𝑡)^𝑡_{𝑡−𝜏𝑖(𝑡)}𝜉^𝑇(𝑠)𝐾^𝑇d𝑠𝑅_1𝑖𝑗^𝑡_{𝑡−𝜏𝑖(𝑡)}𝐾𝜉(𝑠)d𝑠+2^𝑡_{𝑡−𝜏𝑖(𝑡)}𝜉^𝑇(𝑡)𝐾^𝑇𝑅_1𝑖𝑗^𝑡_𝜃𝐾𝜉(𝑠)d𝑠d𝜃+_𝑙∈𝑆1𝜋^(𝑗)_𝑖𝑙^𝑡_{𝑡−𝜏𝑙(𝑡)}^𝑡_𝜃𝜉^𝑇(𝑠)𝐾^𝑇d𝑠𝑅_1𝑙𝑗^𝑡_𝜃𝐾𝜉(𝑠)d𝑠d𝜃+_𝑘∈𝑆2𝜆_𝑗𝑘^𝑡_{𝑡−𝜏𝑖(𝑡)}^𝑡_𝜃𝜉^𝑇(𝑠)𝐾^𝑇𝑅_1𝑖𝑘𝐾𝜉(𝑠)d𝑠d𝜃. (3.3) Using Lemma 2.3 and considering (2.5), it can be deduced that ℒ𝑉₃𝜉_𝑡,𝑡,𝑖,𝑗⩽−1−𝜇_𝑖^𝑡_{𝑡−𝜏𝑖(𝑡)}𝜉^𝑇(𝑠)𝐾^𝑇d𝑠𝑅_1𝑖𝑗^𝑡_{𝑡−𝜏𝑖(𝑡)}𝐾𝜉(𝑠)d𝑠+𝜉^𝑇(𝑡)𝐾^𝑇12𝜏²_𝑖𝑅_1𝑖𝑗𝐾𝜉(𝑡)+^𝑡_{𝑡−𝜏𝑖(𝑡)}^𝑡_𝜃𝜉^𝑇(𝑠)𝐾^𝑇𝑅_1𝑖𝑗𝐾𝜉(𝑠)d𝑠d𝜃+_𝑙≠𝑖𝜋^(𝑗)_𝑖𝑙^𝑡_{𝑡−𝜏𝑙(𝑡)}(𝑡−𝜃)^𝑡_𝜃𝜉^𝑇(𝑠)𝐾^𝑇𝑅_1𝑙𝑗𝐾𝜉(𝑠)d𝑠d𝜃+_𝑘≠𝑗𝜆_𝑗𝑘^𝑡_{𝑡−𝜏𝑖(𝑡)}(𝑡−𝜃)^𝑡_𝜃𝜉^𝑇(𝑠)𝐾^𝑇𝑅_1𝑖𝑘𝐾𝜉(𝑠)d𝑠d𝜃⩽−1−𝜇_𝑖^𝑡_{𝑡−𝜏𝑖(𝑡)}𝜉^𝑇(𝑠)𝐾^𝑇d𝑠𝑅_1𝑖𝑗^𝑡_{𝑡−𝜏𝑖(𝑡)}𝐾𝜉(𝑠)d𝑠+𝜉^𝑇(𝑡)𝐾^𝑇12𝜏²_𝑖𝑅_1𝑖𝑗𝐾𝜉(𝑡)+^𝑡_{𝑡−𝜏𝑖(𝑡)}^𝑡_𝜃𝜉^𝑇(𝑠)𝐾^𝑇𝑅_1𝑖𝑗𝐾𝜉(𝑠)d𝑠d𝜃+_𝑙≠𝑖𝜋^(𝑗)_𝑖𝑙^𝑡_{𝑡−𝜏𝑙(𝑡)}𝜉^𝑇(𝑠)𝐾^𝑇𝑅_1𝑙𝑗𝐾𝜉(𝑠)^𝑡_{𝑡−𝜏𝑙(𝑡)}(𝑡−𝜃)d𝜃d𝑠+_𝑘≠𝑗𝜆_𝑗𝑘^𝑡_{𝑡−𝜏𝑖(𝑡)}𝜉^𝑇(𝑠)𝐾^𝑇𝑅_1𝑖𝑘𝐾𝜉(𝑠)^𝑡_{𝑡−𝜏𝑖(𝑡)}(𝑡−𝜃)d𝜃d𝑠⩽−1−𝜇_𝑖^𝑡_{𝑡−𝜏𝑖(𝑡)}𝜉^𝑇(𝑠)𝐾^𝑇d𝑠𝑅_1𝑖𝑗^𝑡_{𝑡−𝜏𝑖(𝑡)}𝐾𝜉(𝑠)d𝑠+𝜉^𝑇(𝑡)𝐾^𝑇12𝜏²_𝑖𝑅_1𝑖𝑗𝐾𝜉(𝑡)+^𝑡_𝑡−𝜏^𝑡_𝜃𝜉^𝑇(𝑠)𝐾^𝑇𝑅_1𝑖𝑗𝐾𝜉(𝑠)d𝑠d𝜃+_𝑙≠𝑖12𝜏²_𝑙𝜋^(𝑗)_𝑖𝑙^𝑡_𝑡−𝜏𝜉^𝑇(𝑠)𝐾^𝑇𝑅_1𝑙𝑗(𝑠)+1𝐾𝜉d𝑠2_𝑘≠𝑗𝜏²_𝑖𝜆_𝑗𝑘^𝑡_{𝑡−𝜏𝑖(𝑡)}𝜉^𝑇(𝑠)𝐾^𝑇𝑅_1𝑖𝑘𝐾𝜉(𝑠)d𝑠. (3.4) In addition, it is not difficult to get ℒ𝑉₄𝜉_𝑡,𝑡,𝑖,𝑗=𝜏²𝜉̇^𝑇(𝑡)𝐾^𝑇𝑍𝐾𝜉̇(𝑡)−𝜏^𝑡_𝑡−𝜏𝜉̇^𝑇(𝑠)𝐾^𝑇𝑍𝐾𝜉̇(𝑠)d𝑠,ℒ𝑉₅𝜉_𝑡,𝑡,𝑖,𝑗⩽𝜏𝜉^𝑇(𝑡)𝐾^𝑇𝑅₂𝐾𝜉(𝑡)−^𝑡_𝑡−𝜏𝜉^𝑇(𝑠)𝐾^𝑇𝑅₂𝐾𝜉(𝑠)d𝑠+𝜏_𝑖𝜉^𝑇(𝑡)𝐾^𝑇𝐺_1𝑖(𝑡)−1𝐾𝜉𝜏^𝑡_{𝑡−𝜏𝑖(𝑡)}𝜉^𝑇(𝑠)𝐾^𝑇d𝑠𝐺_1𝑖^𝑡_{𝑡−𝜏𝑖(𝑡)}(𝑠)+𝐾𝜉d𝑠_{0−𝜏𝑡𝑡+𝜃}𝜉^𝑇(𝑠)𝐾^𝑇_{𝑙∈𝑆1,𝑙≠𝑖}𝜋^(𝑗)_𝑖𝑙𝐺_1𝑖𝐾𝜉(𝑠)d𝑠d𝜃,ℒ𝑉₆𝜉_𝑡=1,𝑡,𝑖,𝑗2𝜏²𝜉^𝑇(𝑡)𝐾^𝑇𝑅₃𝐾𝜉(𝑡)−^𝑡_𝑡−𝜏^𝑡_𝜃𝜉^𝑇(𝑠)𝐾^𝑇𝑅₃(𝑠)+1𝐾𝜉d𝑠d𝜃2𝜏²𝜉^𝑇(𝑡)𝐾^𝑇𝐺₂𝐾𝜉(𝑡)−⁰_−𝜏^𝑡_𝑡+𝜃𝜉^𝑇(𝑠)𝐾^𝑇𝐺₂𝐾𝜉(𝑠)d𝑠d𝜃. (3.5)(3.6)(3.7)

Next, following a similar method of [46], to (3.5), denote 𝛿₁(𝑡)=^𝑡_{𝑡−𝜏𝑖(𝑡)}𝐾𝜉̇(𝑠)d𝑠,𝛿₂(𝑡)=^{𝑡−𝜏𝑖(𝑡)}_𝑡−𝜏𝐾𝜉̇(𝑠)d𝑠. (3.8) When 0<𝜏_𝑖(𝑡)<𝜏, according to Jensen’s inequality, we have that 𝜏^𝑡_𝑡−𝜏𝜉̇^𝑇(𝑠)𝐾^𝑇𝑍𝐾𝜉̇(𝑠)d𝑠=𝜏^𝑡_{𝑡−𝜏𝑖(𝑡)}𝜉̇^𝑇(𝑠)𝐾^𝑇𝑍𝐾𝜉̇(𝑠)d𝑠+𝜏^{𝑡−𝜏𝑖(𝑡)}_𝑡−𝜏𝜉̇^𝑇(𝑠)𝐾^𝑇𝜉̇(𝑠)⩾𝜏𝑍𝐾d𝑠𝜏_𝑖(𝑡)𝛿₁(𝑡)^𝑇𝑍𝛿₁(𝑡)+𝜏𝜏−𝜏_𝑖(𝑡)𝛿₂(𝑡)^𝑇𝑍𝛿₂(𝑡)=𝛿₁(𝑡)^𝑇𝑍𝛿₁(𝑡)+𝜏−𝜏_𝑖(𝑡)𝜏_𝑖(𝑡)𝛿₁(𝑡)^𝑇𝑍𝛿₁(𝑡)+𝛿₂(𝑡)^𝑇𝑍𝛿₂(𝑡)+𝜏_𝑖(𝑡)𝜏−𝜏_𝑖(𝑡)𝛿₂(𝑡)^𝑇𝑍𝛿₂(𝑡). (3.9) It is clear that [49] 𝜏−𝜏_𝑖(𝑡)𝜏_𝑖𝛿(𝑡)¹−(𝑡)𝜏_𝑖(𝑡)𝜏−𝜏_𝑖𝛿(𝑡)²(𝑡)^𝑇𝑍𝑆∗𝑍𝜏−𝜏_𝑖(𝑡)𝜏_𝑖(𝑡)𝛿₁(𝑡)−𝜏_𝑖(𝑡)𝜏−𝜏_𝑖(𝑡)𝛿₂(𝑡)⩾0, (3.10) which implies 𝜏−𝜏_𝑖(𝑡)𝜏_𝑖(𝑡)𝛿₁(𝑡)^𝑇𝑍𝛿₁(𝑡)+𝜏_𝑖(𝑡)𝜏−𝜏_𝑖(𝑡)𝛿₂(𝑡)^𝑇𝑍𝛿₂(𝑡)⩾𝛿₁(𝑡)^𝑇𝑆𝛿₂(𝑡)+𝛿₂(𝑡)^𝑇𝑆^𝑇𝛿₁(𝑡)^𝑇. (3.11) Then, we can get from (3.9) and (3.11) that 𝜏^𝑡_𝑡−𝜏𝜉̇^𝑇(𝑠)𝐾^𝑇𝑍𝐾𝜉̇(𝑠)d𝑠⩾𝛿₁(𝑡)^𝑇𝑍𝛿₁(𝑡)+𝛿₂(𝑡)^𝑇𝑍𝛿₂(𝑡)+𝛿₁(𝑡)^𝑇𝑆𝛿₂(𝑡)+𝛿₂(𝑡)^𝑇𝑆^𝑇𝛿₁(𝑡)=𝛿₁(𝑡)𝛿₂(𝑡)^𝑇𝑍𝑆𝛿∗𝑍¹(𝑡)𝛿₂(𝑡). (3.12) Note that when 𝜏_𝑖(𝑡)=0 or 𝜏_𝑖(𝑡)=𝜏, we have 𝛿₁(𝑡)=0 or 𝛿₂(𝑡)=0, respectively. So relation (3.12) still holds. It is clear that (3.12) implies −𝜏^𝑡_𝑡−𝜏𝜉̇^𝑇(𝑠)𝐾^𝑇𝑍𝐾𝜉̇(𝑠)d𝑠⩽𝜒^𝑇(𝑡)Ω𝜒(𝑡), (3.13) where 𝜒(𝑡)=[𝜉^𝑇(𝑡)𝜉^𝑇(𝑡−𝜏_𝑖(𝑡))𝐾^𝑇𝜉^𝑇(𝑡−𝜏)𝐾^𝑇]^𝑇Ω=−𝐾^𝑇𝑍𝐾𝐾^𝑇(𝑍−𝑆)𝐾^𝑇𝑆∗−2𝑍+𝑆+𝑆^𝑇𝑍−𝑆∗∗−𝑍. (3.14)

The following equation is true for any matrix 𝑁 with appropriate dimensions: 0=2𝜉̇^𝑇(𝑡)𝐾^𝑇𝑁×−𝐾𝜉̇(𝑡)+𝐾𝐴_𝑖𝑗𝜉(𝑡)+𝐾𝐴_1𝑖𝑗𝐾𝜉𝑡−𝜏_𝑖(𝑡)+𝐾𝐴_2𝑖𝑗𝐾^𝑡_{𝑡−𝜏𝑖(𝑡)}𝜉(𝑠)d𝑠+𝐾𝐵_𝑖𝑗𝑓(𝐾𝜉(𝑡))+𝐾𝐸_𝑖𝑗𝑔(𝐾𝜉(𝑡))+𝐾𝐷_𝑖𝑗𝑤(𝑡). (3.15) From (2.4), it is clear that [2] 𝜉(𝑡)𝑓(𝐾𝜉(𝑡))^𝑇𝐾^𝑇𝑀₁𝐾𝐾^𝑇𝑀₂𝜉(𝑡)∗𝐼𝑓(𝐾𝜉(𝑡))⩽0𝜉(𝑡)𝑔(𝐾𝜉(𝑡))^𝑇𝐾^𝑇𝐿₁𝐾𝐾^𝑇𝐿₂𝜉(𝑡)∗𝐼𝑔(𝐾𝜉(𝑡))⩽0, (3.16)(3.17) where 𝑀₁=(1/2)(𝑀^𝑇₁𝑀₂+𝑀^𝑇₂𝑀₁), 𝑀₂=(−1/2)(𝑀^𝑇₁+𝑀^𝑇₂), 𝐿₁=(1/2)(𝐿^𝑇₁𝐿₂+𝐿^𝑇₂𝐿₁), 𝐿₂=−(1/2)(𝐿^𝑇₁+𝐿^𝑇₂). It implies from (3.17) and (3.18) that there exist 𝜀₁>0 and 𝜀₂>0 such that −𝜀₁𝜉(𝑡)𝑓(𝐾𝜉(𝑡))^𝑇𝐾^𝑇𝑀₁𝐾𝐾^𝑇𝑀₂𝜉(𝑡)∗𝐼𝑓(𝐾𝜉(𝑡))⩾0,−𝜀₂𝜉(𝑡)𝑔(𝐾𝜉(𝑡))^𝑇𝐾^𝑇𝐿₁𝐾𝐾^𝑇𝐿₂𝜉(𝑡)∗𝐼𝑔(𝐾𝜉(𝑡))⩾0. (3.18)(3.19) We define 𝜂_𝑖(𝑡)=𝜉^𝑇(𝑡)𝜉^𝑇𝑡−𝜏_𝑖(𝑡)𝐾^𝑇𝜉^𝑇𝑡−𝜏_𝑖𝐾^𝑇𝜉̇^𝑇(𝑡)𝐾^𝑇𝜉^𝑇(𝑡−𝜏^𝑇_{𝑡𝑡−𝜏𝑖(𝑡)})𝐾𝜉^𝑇(𝑠)𝐾^𝑇d𝑠𝑓^𝑇(𝐾𝜉(𝑡))𝑔^𝑇(𝐾𝜉(𝑡))^𝑇. (3.20) From the above discussion, we have ℒ𝑉𝜉_𝑡⩽𝜂,𝑖,𝑗^𝑖(𝑡)𝑤(𝑡)^𝑇Σ_𝑖𝑗Φ_1𝑖𝑗𝜂∗0^𝑖(𝑡)𝑤(𝑡)+^𝑡_{𝑡−𝜏𝑖(𝑡)}𝜉^𝑇(𝑠)𝐾^𝑇_𝑘∈𝑆2𝜆_𝑗𝑘𝑄_1𝑖𝑘+𝜏²_𝑖2_𝑘≠𝑗𝜆_𝑗𝑘𝑅_1𝑖𝑘+𝜋^(𝑗)_𝑖𝑖𝑄_1𝑖𝑗(𝑠)+𝐾𝜉d𝑠_{𝑡𝑡−𝜏}𝜉^𝑇(𝑠)𝐾^𝑇_{𝑙∈𝑆1,𝑙≠𝑖}𝜋^(𝑗)_𝑖𝑙𝑄_1𝑙𝑗+𝑄_2𝑙𝑗+𝜏²_𝑙2𝑅_1𝑙𝑗−𝑅₂(𝑠)+𝐾𝜉d𝑠_{𝑡𝑡−𝜏𝑖}𝜉^𝑇(𝑠)𝐾^𝑇_𝑘∈𝑆2𝜆_𝑗𝑘𝑄_2𝑖𝑘+𝜋^(𝑗)_𝑖𝑖𝑄_2𝑖𝑗(𝑠)+𝐾𝜉d𝑠_{𝑡𝑡−𝜏𝑡𝜃}𝜉^𝑇(𝑠)𝐾^𝑇𝑅_1𝑖𝑗−𝑅₃𝐾𝜉(𝑠)d𝑠d𝜃+⁰_−𝜏^𝑡_𝑡+𝜃𝜉^𝑇(𝑠)𝐾^𝑇_{𝑙∈𝑆1,𝑙≠𝑖}𝜋^(𝑗)_𝑖𝑙𝐺_1𝑖−𝐺₂𝐾𝜉(𝑠)d𝑠d𝜃, (3.21) where Σ_𝑖𝑗=Σ₁₁Σ₁₂0𝑁𝐾𝐴_𝑖𝑗^𝑇𝐾^𝑇𝑆𝑃_𝑖𝑗𝐴_2𝑖𝑗Σ₁₇Σ₁₈∗Σ₂₂0𝑁𝐾𝐴_1𝑖𝑗^𝑇𝑍−𝑆00∗∗−𝑄_2𝑖𝑗00000∗∗∗Σ₄₄0𝑁𝐾𝐴_2𝑖𝑗𝑁𝐾𝐵_𝑖𝑗𝑁𝐾𝐸_𝑖𝑗∗∗∗∗−𝑍000∗∗∗∗∗Σ₆₆00∗∗∗∗∗∗−𝜀₁𝐼0∗∗∗∗∗∗∗−𝜀₂𝐼,Σ₁₁=𝑃_𝑖𝑗𝐴_𝑖+𝐴^𝑇_𝑖𝑃_𝑖𝑗+_𝑙∈𝑆1𝜋^(𝑗)_𝑖𝑙𝑃_𝑙𝑗+_𝑘∈𝑆2𝜆_𝑗𝑘𝑃_𝑖𝑘+𝐾^𝑇𝑄_1𝑖𝑗+𝑄_2𝑖𝑗+𝜏𝑅₂1−𝑍+2𝜏²_𝑖𝑅_1𝑖𝑗+12𝜏²𝑅₃+𝜏_𝑖𝐺_1𝑖+12𝜏²𝐺₂×𝐾−𝜀₁𝐾^𝑇𝑀₁𝐾−𝜀₂𝐾^𝑇𝐿₁Σ𝐾,₁₂=𝑃_𝑖𝑗𝐴_1𝑖𝑗+𝐾^𝑇(𝑍−𝑆),Σ₁₇=𝑃_𝑖𝑗𝐵_𝑖𝑗−𝜀₁𝐾^𝑇𝑀₂,Σ₁₈=𝑃_𝑖𝑗𝐸_𝑖𝑗−𝜀₂𝐾^𝑇𝐿₂,Σ₂₂=−1−𝜇_𝑖𝑄_1𝑖𝑗−2𝑍+𝑆+𝑆^𝑇,Σ₄₄=𝜏²𝑍−𝑁−𝑁^𝑇,Σ₆₆=−1−𝜇_𝑖𝑅_1𝑖𝑗−1𝜏𝐺_1𝑖,Φ_1𝑖𝑗=𝐷^𝑇_𝑖𝑗𝑃_𝑖𝑗00𝐷^𝑇_𝑖𝑗𝐾^𝑇𝑁^𝑇0000^𝑇. (3.22) Therefore, we have the following result for the 𝐻_∞ performance analysis.

Theorem 3.1. Given scalars 𝜏, 𝜏_𝑖, and 𝜇_𝑖, the filtering error system (2.7) is stochastically stable with an 𝐻_∞ performance 𝛾 for any time delay 𝜏_𝑖(𝑡) satisfying (2.5), if there exist matrices 𝑃_𝑖𝑗>0, 𝑄_1𝑖𝑗>0, 𝑄_2𝑖𝑗>0, 𝑅_1𝑖𝑗>0, 𝑅₂>0, 𝑅₃>0, 𝑍>0, 𝐺_1𝑖>0, 𝐺₂>0, and matrices 𝑆, 𝑁 such that for each 𝑖∈𝑆₁, 𝑗∈𝑆₂Σ_𝑖𝑗Φ_1𝑖𝑗Φ_2𝑖𝑗∗−𝛾²𝐼0∗∗−𝐼<0,_𝑘∈𝑆2𝜆_𝑗𝑘𝑄_1𝑖𝑘+𝜏²_𝑖2_𝑘≠𝑗𝜆_𝑗𝑘𝑅_1𝑖𝑘+𝜋^(𝑗)_𝑖𝑖𝑄_1𝑖𝑗<0,_{𝑙∈𝑆1,𝑙≠𝑖}𝜋^(𝑗)_𝑖𝑙𝑄_1𝑙𝑗+𝑄_2𝑙𝑗+𝜏²_𝑙2𝑅_1𝑙𝑗−𝑅₂<0,_𝑘∈𝑆2𝜆_𝑗𝑘𝑄_2𝑖𝑘+𝜋^(𝑗)_𝑖𝑖𝑄_2𝑖𝑗𝑅<0,_1𝑖𝑗<𝑅₃,_{𝑙∈𝑆1,𝑙≠𝑖}𝜋^(𝑗)_𝑖𝑙𝐺_1𝑖<𝐺₂, (3.23)(3.24)(3.25)(3.26)(3.27)(3.28) where Φ_2𝑖𝑗=𝐻_𝑖0000000^𝑇. (3.29)

Proof. Using the Schur complement formula to (3.23), it can be seen that inequality (3.23) is equivalent to Σ_𝑖𝑗+Φ_2𝑖𝑗Φ^𝑇_2𝑖𝑗Φ_1𝑖𝑗∗−𝛾²𝐼<0, (3.30) which implies Σ_𝑖𝑗<0. Now, we show that the filtering error system (2.7) with 𝜔(𝑡)=0 is stochastically stable. If 𝜔(𝑡)=0, from (3.1), (3.21), (3.24)–(3.28), and Σ_𝑖𝑗<0, there exists a scalar 𝜆>0 such that ℒ𝑉𝜉_𝑡,𝑡,𝑖,𝑗⩽−𝜆‖𝑥(𝑡)‖². (3.31) Therefore, for any 𝑇>0, by Dynkin’s formula, we have ℰ𝑉𝜉_𝑡,𝑡,𝑖,𝑗−ℰ𝑉𝜉₀,0,0,0⩽−𝜆ℰ^𝑇₀‖𝑥(𝑠)‖²d𝑠, (3.32) which yields ℰ^𝑇₀‖𝑥(𝑠)‖²1d𝑠⩽𝜆ℰ𝑉𝜉₀,0,0,0<∞. (3.33) Thus, the filtering error system (2.7) with 𝜔(𝑡)=0 is stochastically stable by Definition 2.4.
In the sequel, we will deal with the 𝐻_∞ performance of the filtering error system (2.7). Using (3.30) and 𝐻_∞ performance, we have ℰℒ𝑉𝜉_𝑡,𝑡,𝑖,𝑗+𝑒^𝑇(𝑡)𝑒(𝑡)−𝛾²𝜔^𝑇(𝑡)𝜔(𝑡⩽𝜂_𝑖)(𝑡)𝜔(𝑡)^𝑇Σ_𝑖𝑗+Φ_2𝑖𝑗Φ^𝑇_2𝑖𝑗Φ_1𝑖𝑗∗−𝛾²𝐼𝜂_𝑖(𝑡)𝜔(𝑡)<0. (3.34) Noting that the zero initial condition, then it follows from (3.34) that 𝐽=ℰ^∞₀𝑒^𝑇(𝑡)𝑒(𝑡)−𝛾²𝜔^𝑇(𝑡)𝜔(𝑡)d𝑡⩽ℰ^∞₀𝑒^𝑇(𝑡)𝑒(𝑡)−𝛾²𝜔^𝑇(𝑡)𝜔(𝑡)+ℒ𝑉𝜉_𝑡,𝑡,𝑖,𝑗d𝑡<0. (3.35) Hence, if (3.23)–(3.28) hold, 𝐽<0 can be guaranteed. That is, ‖𝑒‖₂⩽𝛾‖𝜔‖₂ for all nonzero 𝜔(𝑡). Therefore, the filtering error system (2.7) is stochastically stable with the 𝐻_∞ performance 𝛾 by Definition 2.5. This completes the proof.

Remark 3.2. A new stochastic stability criterion is obtained in Theorem 3.1 by constructing a novel mode-dependent Lyapunov functional. The Lyapunov functional in this paper uses all information about 𝑟_𝑡, 𝜎_𝑡 and 𝜏(𝑡,𝑟_𝑡). The Lyapunov matrices 𝑃(𝑟_𝑡,𝜎_𝑡), 𝑄₁(𝑟_𝑡,𝜎_𝑡), 𝑄₂(𝑟_𝑡,𝜎_𝑡), and 𝑅₁(𝑟_𝑡,𝜎_𝑡) depend on both the system mode 𝑟_𝑡 and the higher-level Markovian chain 𝜎_𝑡. Compared with the mode-independent Lyapunov matrices [40, 42], the mode-dependent Lyapunov matrices can reduce the conservativeness since they provide additional degrees of freedom which are very important for deriving LMIs solutions in general. Hence, the Lyapunov functional in this paper is more general and the condition on stability is more applicable.

Remark 3.3. It should be pointed out that the aim of the introduction of 𝑉₃(𝜉_𝑡,𝑡,𝑖,𝑗) is to propose a stability condition which depends not only on the delay upper bound 𝜏, but also on the subsystems’ delay upper bounds 𝜏_𝑖, in other words, if 𝑉₃(𝜉_𝑡,𝑡,𝑖,𝑗) is not considered, the obtained stability condition only depends on the delay upper bound 𝜏. Hence, the introduction of 𝑉₃(𝜉_𝑡,𝑡,𝑖,𝑗) may reduce some conservativeness.

Based on Theorem 3.1, the 𝐻_∞ filter synthesis problem can be developed in terms of LMIs for the system (2.1) with higher-level Markovian chain.

Theorem 3.4. Consider the systems (2.1). Given scalars 𝜏, 𝜏_𝑖 and 𝜇_𝑖, the filtering error system (2.7) is stochastically stable with an 𝐻_∞ performance 𝛾 for any time delay 𝜏_𝑖(𝑡) satisfying (2.5), if there exist matrices 𝑃_1𝑖𝑗>0, 𝑄_1𝑖𝑗>0, 𝑄_2𝑖𝑗>0, 𝑅_1𝑖𝑗>0, 𝑅₂>0, 𝑅₃>0, 𝑍>0, 𝐺_1𝑖>0, 𝐺₂>0, 𝑆_𝑖𝑗>0, 𝑈_𝑙𝑖𝑗>0, 𝑉_𝑘𝑖𝑗>0, 𝑆_𝑙𝑖𝑗, 𝑇_𝑘𝑖𝑗, 𝐴_𝐹𝑖𝑗, 𝐵_𝐹𝑖𝑗, 𝐶_𝐹𝑖𝑗, and matrices 𝑆, 𝑁 such that for each 𝑖∈𝑆₁, 𝑗∈𝑆₂Σ₁₁Σ₁₂Σ₁₃0𝑁𝐴_𝑖^𝑇𝑆Σ₁₄Σ₁₅Σ₁₆Σ₁₇𝐻^𝑇_𝑖∗Σ₂₂Σ₂₃000Σ₂₄𝑆_𝑖𝑗𝐵_𝑖𝐵_𝐹𝑖𝑗𝐸_𝑖Σ₂₅−𝐶^𝑇_𝐹𝑖𝑗∗∗Σ₃₃0𝑁𝐴_1𝑖^𝑇𝑍−𝑆00000∗∗∗−𝑄_2𝑖𝑗0000000∗∗∗∗Σ₅₅0𝑁𝐴_2𝑖𝑁𝐵_𝑖0𝑁𝐷_1𝑖0∗∗∗∗∗−𝑍00000∗∗∗∗∗∗Σ₇₇0000∗∗∗∗∗∗∗−𝜀₁𝐼000∗∗∗∗∗∗∗∗−𝜀₂𝐼00∗∗∗∗∗∗∗∗∗−𝛾²𝐼0∗∗∗∗∗∗∗∗∗∗−𝐼<0,_𝑘∈𝑆2𝜆_𝑗𝑘𝑄_1𝑖𝑘+𝜏²_𝑖2_𝑘≠𝑗𝜆_𝑗𝑘𝑅_1𝑖𝑘+𝜋^(𝑗)_𝑖𝑖𝑄_1𝑖𝑗<0,_{𝑙∈𝑆1,𝑙≠𝑖}𝜋^(𝑗)_𝑖𝑙𝑄_1𝑙𝑗+𝑄_2𝑙𝑗+𝜏²_𝑙2𝑅_1𝑙𝑗−𝑅₂<0,_𝑘∈𝑆2𝜆_𝑗𝑘𝑄_2𝑖𝑘+𝜋^(𝑗)_𝑖𝑖𝑄_2𝑖𝑗𝑅<0,_1𝑖𝑗<𝑅₃,_{𝑙∈𝑆1,𝑙≠𝑖}𝜋^(𝑗)_𝑖𝑙𝐺_1𝑖<𝐺₂, (3.36)(3.37)(3.38)(3.39)(3.40)(3.41) where Σ₁₁=𝑃_1𝑖𝑗𝐴_𝑖+𝐴^𝑇_𝑖𝑃_1𝑖𝑗+𝐵_𝐹𝑖𝑗𝐶_𝑖+𝐶^𝑇_𝑖𝐵^𝑇_𝐹𝑖𝑗+_𝑙∈𝑆1𝜋^(𝑗)_𝑖𝑙𝑃_1𝑙𝑗+_𝑘∈𝑆2𝜆_𝑗𝑘𝑃_1𝑖𝑘+𝑄_1𝑖𝑗+𝑄_2𝑖𝑗+12𝜏²_𝑖𝑅_1𝑖𝑗+𝜏𝑅₂+12𝜏²𝑅₃−𝑍+𝜏_𝑖𝐺_1𝑖+12𝜏²𝐺₂−𝜀₁𝑀₁−𝜀₂𝐿₁,Σ₁₂=𝐴_𝐹𝑖𝑗+𝐴^𝑇_𝑖𝑆^𝑇_𝑖𝑗+𝐶^𝑇_𝑖𝐵^𝑇_𝐹𝑖𝑗+𝜋^(𝑗)_𝑖𝑖+𝜆_𝑗𝑗𝑆_𝑖𝑗+_𝑙≠𝑖𝜋^(𝑗)_𝑖𝑙𝑆_𝑙𝑖𝑗+_𝑘≠𝑗𝜆_𝑗𝑘𝑇_𝑘𝑖𝑗,Σ₁₃=𝑃_1𝑖𝑗𝐴_1𝑖+𝐵_𝐹𝑖𝑗𝐶_1𝑖+𝑍−𝑆,Σ₁₄=𝑃_1𝑖𝑗𝐴_2𝑖+𝐵_𝐹𝑖𝑗𝐶_2𝑖,Σ₁₅=𝑃_1𝑖𝑗𝐵_𝑖−𝜀₁𝑀₂,Σ₁₆=𝐵_𝐹𝑖𝑗𝐸_𝑖−𝜀₂𝐿₂,Σ₁₇=𝑃_1𝑖𝑗𝐷_1𝑖+𝐵_𝐹𝑖𝑗𝐷_2𝑖,Σ₂₂=𝐴_𝐹𝑖𝑗+𝐴^𝑇_𝐹𝑖𝑗+𝜋^(𝑗)_𝑖𝑖+𝜆_𝑗𝑗𝑆_𝑖𝑗+_{𝑙∈𝑆1,𝑙≠𝑖}𝜋^(𝑗)_𝑖𝑙𝑈_𝑙𝑖𝑗+_{𝑘∈𝑆2,𝑘≠𝑗}𝜆_𝑗𝑘𝑉_𝑘𝑖𝑗,Σ₂₃=𝑆_𝑖𝑗𝐴_1𝑖+𝐵_𝐹𝑖𝑗𝐶_1𝑖,Σ₂₄=𝑆_𝑖𝑗𝐴_2𝑖+𝐵_𝐹𝑖𝑗𝐶_2𝑖,Σ₂₅=𝑆_𝑖𝑗𝐷_1𝑖+𝐵_𝐹𝑖𝑗𝐷_2𝑖,Σ₃₃=−1−𝜇_𝑖𝑄_1𝑖𝑗−2𝑍+𝑆+𝑆^𝑇,Σ₅₅=𝜏²𝑍−𝑁−𝑁^𝑇,Σ₇₇=−1−𝜇_𝑖𝑅_1𝑖𝑗−1𝜏𝐺_1𝑖. (3.42) In this case, the parameters of the desired filter can be chosen by 𝐴_𝐹𝑖𝑗=𝑆⁻¹_𝑖𝑗𝐴_𝐹𝑖𝑗,𝐵_𝐹𝑖𝑗=𝑆⁻¹_𝑖𝑗𝐵_𝐹𝑖𝑗,𝐶_𝐹𝑖𝑗=𝐶_𝐹𝑖𝑗. (3.43)

Proof. For each 𝑟_𝑡=𝑖∈𝑆₁, 𝜎_𝑡=𝑗∈𝑆₂, we define a matrix 𝑃_𝑖𝑗>0 by 𝑃_𝑖𝑗=^{𝑃1𝑖𝑗𝑃2𝑖𝑗}_{∗𝑃3𝑖𝑗}. By invoking a small perturbation, if necessary, we can assume that 𝑃_2𝑖𝑗 and 𝑃_3𝑖𝑗 are nonsingular. Thus, we can introduce the following invertible matrix 𝐽=^𝐼0_0𝑃⁻¹_{3𝑖𝑗𝑃}^𝑇_2𝑖𝑗. Pre- and postmultiplying (3.23) by diag [𝐽^𝑇,𝐼,𝐼,𝐼,𝐼,𝐼,𝐼,𝐼,𝐼,𝐼,𝐼] and its transpose, respectively. Then, we define 𝑆_𝑖𝑗=𝑃_2𝑖𝑗𝑃⁻¹_3𝑖𝑗𝑃^𝑇_2𝑖𝑗,𝐴_𝐹𝑖𝑗=𝑃_2𝑖𝑗𝐴_𝐹𝑖𝑗𝑃⁻¹_3𝑖𝑗𝑃^𝑇_2𝑖𝑗,𝐵_𝐹𝑖𝑗=𝑃_2𝑖𝑗𝐵_𝐹𝑖𝑗,𝐶_𝐹𝑖𝑗=𝐶_𝐹𝑖𝑗𝑃⁻¹_3𝑖𝑗𝑃^𝑇_2𝑖𝑗,𝑆_𝑙𝑖𝑗=𝑃_2𝑙𝑗𝑃⁻¹_3𝑖𝑗𝑃^𝑇_2𝑖𝑗(𝑙≠𝑖),𝑇_𝑘𝑖𝑗=𝑃_2𝑖𝑘𝑃⁻¹_3𝑖𝑗𝑃^𝑇_2𝑖𝑗𝑘≠𝑗,𝑈_𝑙𝑖𝑗=𝑃_2𝑖𝑗𝑃^−𝑇_3𝑖𝑗𝑃_3𝑙𝑗𝑃⁻¹_3𝑖𝑗𝑃^𝑇_2𝑖𝑗(),𝑉𝑙≠𝑖_𝑘𝑖𝑗=𝑃_2𝑖𝑗𝑃^−𝑇_3𝑖𝑗𝑃_3𝑖𝑘𝑃⁻¹_3𝑖𝑗𝑃^𝑇_2𝑖𝑗𝑘≠𝑗. (3.44) It is easy to obtain (3.36).
On the other hand, according to (3.44), we have 𝐴_𝐹𝑖𝑗=𝑃⁻¹_2𝑖𝑗𝐴_𝐹𝑖𝑗𝑃^−𝑇_2𝑖𝑗𝑃_3𝑖𝑗,𝐵_𝐹𝑖𝑗=𝑃⁻¹_2𝑖𝑗𝐵_𝐹𝑖𝑗,𝐶_𝐹𝑖𝑗=𝐶_𝐹𝑖𝑗𝑃^−𝑇_2𝑖𝑗𝑃_3𝑖𝑗. (3.45) From (2.6), the transfer function from measured output 𝑦(𝑡) to estimated signal 𝑧(𝑡) can be described by 𝑇𝑧_𝑦=𝐶_𝐹𝑖𝑗𝑠𝐼−𝐴_𝐹𝑖𝑗⁻¹𝐵_𝐹𝑖𝑗=𝐶_𝐹𝑖𝑗𝑃^−𝑇_2𝑖𝑗𝑃_3𝑖𝑗𝑠𝐼−𝑃⁻¹_2𝑖𝑗𝐴_𝐹𝑖𝑗𝑃^−𝑇_2𝑖𝑗𝑃_3𝑖𝑗⁻¹𝑃⁻¹_2𝑖𝑗𝐵_𝐹𝑖𝑗=𝐶_𝐹𝑖𝑗𝑠𝐼−𝑆⁻¹_𝑖𝑗𝐴_𝐹𝑖𝑗⁻¹𝑆⁻¹_𝑖𝑗𝐵_𝐹𝑖𝑗. (3.46) Therefore, we can conclude from (3.46) that the parameters of the filter in (2.6) can be constructed by (3.43). This completes the proof.

Remark 3.5. It should be pointed out that some existing work in control and filter design of Markovian jump systems, the Lyapunov matrix 𝑃_𝑖 is assumed to be diagonal matrix, for example, see [42]. It is well known that such assumption can lead to much more conservative result. Although the first diagonal element Σ₁₁ in (3.23) includes 𝑃_𝑖𝑗, 𝑃_𝑙𝑗, and 𝑃_𝑖𝑘 in this paper, 𝑃_𝑖𝑗 is not assumed to be diagonal matrices.

Remark 3.6. In [30–32, 38], the authors have achieved some excellent work of Markovian jump singular systems. Due to the presence of the singular matrix 𝐸, the issues of stability and control of such systems are more difficult and complicated. However, there is no results on piecewise homogeneous Markovian jump singular systems in the existing work, and the problem of control for such system is an interesting issue.

Remark 3.7. Theorem 3.4 solves the filtering problem of a class of piecewise homogeneous Markovian jump nonlinear systems. The obtained conditions are formulated in terms of LMIs, which could be easily checked by using the LMI toolbox in Matlab. The feasible solutions to the conditions presented in Theorem 3.4 will depend on both the mode 𝑟_𝑡 and the higher-level Markovian chain 𝜎_𝑡, which ensure that the error system is stochastically stable. A numerical example verifies the validity of the designed filter in Section 4.

4. A Numerical Example

In this section, a numerical example will be presented to show the validity of the main results derived above.

Example 4.1. Let us consider the stochastic system (2.1) with the following system of matrices: 𝐴(1)=−801−12,𝐴₁(1)=10.311,𝐴₂(1)=,𝐵(1)=0.400−0.40.70.80.10.6,𝐷₁(1)=0.70.1,𝑀₁=𝐿₁=,𝑀0.010.010.010.02₂=𝐿₂=(1)=−0.010.01−0.03−0.02,𝐶1.9−2.1,𝐶₁(1)=,𝐶1.50₂(1)=11,𝐸(1)=0.80.6,𝐷₂(1)(1)=,𝐴(2)==0.8,𝐻0.40.2−900−9,𝐴₁(2)=0.80.610.8,𝐴₂(2)=,𝐵(2)=−0.800−0.60.912.30.6,𝐷₁(2)=0.20.2,𝑀₁=𝐿₁=,𝑀0.010.010.010.02₂=𝐿₂=(2)=−0.010.01−0.03−0.02,𝐶1.8−2.2,𝐶₁(2)=,𝐶1.20.1₂(2)=1−0.8,𝐸(2)=0.80.6,𝐷₂(2)=1,𝐻(2)=0.90.1. (4.1) The piecewise homogeneous TP matrices are given as Π¹=−0.50.50.6−0.6,Π²=−0.70.70.4−0.4,Π³=−0.80.80.6−0.6. (4.2) The HTP matrix for the Markovian chain is considered as follows: Λ=−0.80.40.40.4−0.90.50.70.8−1.5. (4.3) In this example, we assume 𝜏₁=𝜏₂=1, 𝜇₁=0.8, 𝜇₂=0.5. For 𝛾=1.2. By solving LMIs (3.36)–(3.41), the filter matrices are obtained as 𝐴_𝐹11=−7.6750−2.3822−3.7133−4.8336,𝐵_𝐹11=0.04960.3182,𝐶_𝐹11=,𝐴0.00371.3061_𝐹12=−10.9509−3.4284−2.9982−4.2265,𝐵_𝐹12=0.73220.1760,𝐶_𝐹12=,𝐴−0.0053−0.0017_𝐹13=−11.2973−4.8499−5.0689−5.7131,𝐵_𝐹13=1.02090.7196,𝐶_𝐹13=,𝐴−0.0041−0.0029_𝐹21=−5.5850−6.2702−4.3917−15.3643,𝐵_𝐹21=0.41440.9799,𝐶_𝐹21=,𝐴0.00021.2670_𝐹22=−6.2896−4.2309−4.7375−10.6039,𝐵_𝐹22=−0.0779−0.3519,𝐶_𝐹22=𝐴−0.0142−0.0107_𝐹23=−7.9621−6.4413−6.1046−16.3128,𝐵_𝐹23=−0.0841−0.3711,𝐶_𝐹23=−0.0150−0.0117. (4.4) For simulation purposes, we assume the initial condition 𝑥(0)=[0.10.1]^𝑇 and 𝜔(𝑡)=1/(0.5+1.2𝑡). The time delays are 𝜏₁(𝑡)=0.2+0.8sin(𝑡), 𝜏₂(𝑡)=0.5+0.5cos(𝑡). The nonlinear functions 𝑓(𝑥(𝑡)) and 𝑔(𝑥(𝑡)) are selected as 𝑓(𝑥(𝑡))=𝑔(𝑥(𝑡))=0.02𝑥₁(𝑡)sin²𝑥₁(𝑡)−0.01𝑥₁(𝑡)−𝑥₂(𝑡)−0.01𝑥₁(𝑡), (4.5)

Figures 1–5 illustrate the simulation results. A case for stochastic variation with HTP matrix is shown in Figure 1, and possible realizations of the Markov jumping mode of system and delay are plotted in Figure 2, where the initial modes are assumed to be 𝑟₀=1 and 𝜎₀=1. Figure 3 shows the state responses of real states 𝑥₁(𝑡) and its estimate 𝑥₁(𝑡). Figure 4 shows the state responses of real states 𝑥₂(𝑡) and its estimate 𝑥₂(𝑡). Figure 5 is the simulation result of the estimation error response of 𝑒(𝑡). The simulation results demonstrate that the designed 𝐻_∞ filters are feasible and effective.

Figure 1 

Variation of TP matrices subject to HTP.

Figure 2 

System jumping mode.

Figure 3 

The state responses of 𝑥1(𝑡) and 𝑥1(𝑡).

Figure 4 

The state responses of 𝑥2(𝑡) and 𝑥2(𝑡).

Figure 5 

The estimation error response of 𝑒(𝑡).

5. Conclusion

The problem of 𝐻_∞ filtering for a class of Markovian jump nonlinear systems is investigated in this paper. The piecewise homogeneous Markovian chain and mode-dependent time-varying delays are considered in the model. By using the Lyapunov-Krasovskii functional, mixed mode-dependent sufficient conditions are developed to design stable filters. A numerical example demonstrates the effectiveness of the given method.

Acknowledgments

This work was supported by the fund of Sichuan Provincial Key Laboratory of signal and INFORMATION PROCESSING, Xihua University (SZJJ2009-002), the Fundamental Research Funds for the Central Universities (103.1.2E022050205), and the National Basic Research Program of China (2010CB732501).