Robust 𝐻_∞ Filtering for a Class of Uncertain Markovian Jump Systems with Time Delays

Abstract

This paper studies the problem of robust 𝐻_∞ filtering for a class of uncertain time-delay systems with Markovian jumping parameters. The system under consideration is subject to norm-bounded time-varying parameter uncertainties. The problem to be addressed is the design of a Markovian jump filter such that the filter error dynamics are stochastically stable and a prescribed bound on the ℒ₂-induced gain from the noise signals to the filter error is guaranteed for all admissible uncertainties. A sufficient condition for the existence of the desired robust 𝐻_∞ filter is given in terms of two sets of coupled algebraic Riccati inequalities. When these algebraic Riccati inequalities are feasible, the expression of a desired 𝐻_∞ filter is also presented. Finally, an illustrative numerical example is provided.

1. Introduction

In the past decades, much attention has been focused on the celebrated Kalman filtering which seems to be one of the most popular estimation approaches; see, for example, [1]. Such a kind of filtering approach assumes that the system is subjected to stationary Gaussian noises with known statistics. However, in practical applications, the statistics of the noise sources may not be exactly known. To deal with this problem, an alternative approach named 𝐻_∞ filtering was proposed, and a great number of results on this topic have been reported in the literature; see, for example, [2–5]. Note that in the 𝐻_∞ filtering setting, the exogenous input signals are assumed to belong toℒ₂[0,∞); furthermore, no exact statistics are required to be known. When parameter uncertainties appear not only in the exogenous input but also in the system model, the problem of robust 𝐻_∞ filtering has been investigated and algebraic Riccati equation approach and linear matrix inequality (LMI) approach have been adopted to solve this problem; see [6–9] and the references therein.

Systems with Markovian jumping parameters have been extensively studied due to their both practical and theoretical importance; many issues, such as filtering, stability analysis, and synthesis, on this kind of systems have been studied [10–13]. The problem of𝐻_∞filtering for Markovian jumping systems has received much attention. Sufficient conditions for the solvability of this problem were presented in [14], and a design methodology was also proposed based on LMI approach. These results were further extended to uncertain Markovian jump linear systems in [15–18].

In this paper, we consider the problem of robust𝐻_∞filtering for a class of continuous-time uncertain systems with Markovian jump parameters in all system matrices and time delays in the state variables. We consider uncertain systems with norm-bounded time-varying parameter uncertainties. The objective of this paper is to design a Markovian jump linear filter which guarantees both the stochastic stability and a prescribed𝐻_∞performance of the filtering error dynamics for all admissible uncertainties. An algebraic Riccati inequalities approach is developed to solve the previous problem and the desired 𝐻_∞filter can be constructed by solving two sets of coupled algebraic Riccati inequalities.

Notation. Throughout this paper, for symmetric matrices 𝑋 and 𝑌, the notation 𝑋≥𝑌 (resp., 𝑋>𝑌) means that the matrix 𝑋−𝑌 is positive semidefinite (resp., positive definite);𝐼is the identity matrix with appropriate dimension. The notation 𝑀^𝑇 represents the transpose of the matrix𝑀;ℰ{⋅}denotes the expectation operator with respect to some probability measure 𝒫; 𝐿₂[0,∞)is the space of square-integrable vector functions over[0,∞); |⋅|refers to the Euclidean vector norm;‖⋅‖₂stands for the usual𝐿₂[0,∞)norm, while‖⋅‖_𝐸2denotes the norm in𝐿₂((Ω,ℱ,𝒫),[0,∞));(Ω,ℱ,𝒫) is a probability space;𝜆_min(𝑀)is used to denote the minimum eigenvalue of the matrix 𝑀.Matrices, if not explicitly stated, are assumed to have compatible dimensions.

2. Problem Formulation

Fix a complete probability space(Ω,ℱ,𝒫) and consider the following class of uncertain stochastic linear systems with Markovian jumping parameters and time delay: 𝑥̇(𝑡)=[𝐴(𝑟(𝑡((𝑡𝑥(𝑡)+𝐴Σ∶))+Δ𝐴𝑡,𝑟))]_𝑑(𝑟(𝑡))+Δ𝐴_𝑑((𝑡()(𝑟(𝑡𝜔(𝑡),𝑦(𝑡)=[𝐶(𝑟(𝑡((𝑡(𝑡)(𝑟(𝑡𝜔(𝑡),𝑧(𝑡)(𝑟(𝑡𝑥(𝑡),𝑥(𝑡)(𝑡)[](0)𝑡,𝑟))×𝑥𝑡−𝑑+𝐵))))+Δ𝐶𝑡,𝑟))]×𝑥+𝐷))=𝐿))=𝜙,∀𝑡∈−𝑑,0,𝑑>0,𝑟=𝑟₀, (1) where 𝑥(𝑡)∈ℝ^𝑛is the system state; 𝑦(𝑡)∈ℝ^𝑟is the measurement;𝜔(𝑡)∈ℝ^𝑚is the noise signal which belongs to𝐿₂[0,∞);𝑧(𝑡)∈ℝ^𝑙is a linear combination of state variables to be estimated;𝑑>0is the time delay of the system; and𝜙(𝑡)∈ℝ^𝑛is the continuous initial-value function. The parameter 𝑟(𝑡)represents a continuous discrete-state Markov process taking values in a finite set𝒮={1,2,…,𝑠}with transition probability matrix Π≜{𝜋_𝑖𝑗}given by Pr𝑟()(𝑡)=𝜋𝑡+ℎ=𝑗∣𝑟=𝑖^𝑖𝑗ℎ+𝑜(ℎ),𝑖≠𝑗,1+𝜋_𝑖𝑖ℎ+𝑜(ℎ),𝑖=𝑗, (2) whereℎ>0,lim_ℎ→∞(𝑜(ℎ)/ℎ)=0, and 𝜋_𝑖𝑗≥0is the transition rate from mode𝑖at time 𝑡 to mode𝑗at time𝑡+ℎ and 𝜋_𝑖𝑖=−^𝑠_{𝑗=1,𝑗≠𝑖}𝜋_𝑖𝑗. (3) In  system Σ, 𝐴(𝑟(𝑡)),Δ𝐴(𝑡,𝑟(𝑡)),𝐴_𝑑(𝑟(𝑡)),Δ𝐴_𝑑(𝑡,𝑟(𝑡)),𝐵(𝑟(𝑡)),𝐶(𝑟(𝑡)),Δ𝐶(𝑡,𝑟(𝑡),𝐷(𝑟(𝑡)), and 𝐿(𝑟(𝑡)) are appropriately dimensioned real-valued matrix functions of 𝑟(𝑡). For simplicity of notations, in the sequel, for each possible 𝑟(𝑡)=𝑖, 𝑖∈𝒮, we will denote the matrices associated with the𝑖th mode by 𝐴_𝑖≜𝐴(𝑟(𝑡)),Δ𝐴_𝑖(𝑡)((𝑡,𝐴≜Δ𝐴𝑡,𝑟))_𝑑𝑖≜𝐴_𝑑(𝑟(𝑡)),Δ𝐴_𝑑𝑖(𝑡)≜Δ𝐴_𝑑((𝑡,𝐵𝑡,𝑟))_𝑖≜𝐵(𝑟(𝑡)),𝐶_𝑖≜𝐶(𝑟(𝑡)),Δ𝐶_𝑖(𝑡)≜Δ𝐶(𝑡,𝑟(𝑡)),𝐷_𝑖≜𝐷(𝑟(𝑡)),𝐿_𝑖≜𝐿(𝑟(𝑡)), (4)(5) where 𝐴_𝑖, 𝐴_𝑑𝑖, 𝐵_𝑖, 𝐶_𝑖, 𝐷_𝑖, and 𝐿_𝑖, are known constant matrices representing the nominal system for each 𝑖∈𝒮 and Δ𝐴_𝑖(𝑡), Δ𝐴_𝑑𝑖(𝑡), and Δ𝐶_𝑖(𝑡) are unknown matrices representing time-varying parameter uncertainties and are assumed to be of the form Δ𝐴_𝑖(𝑡)=𝑀_𝐴𝑖𝐹_𝐴𝑖(𝑡)𝑁_𝐴𝑖,Δ𝐴_𝑑𝑖(𝑡)=𝑀_𝑑𝑖𝐹_𝑑𝑖(𝑡)𝑁_𝑑𝑖,Δ𝐶_𝑖(𝑡)=𝑀_𝐶𝑖𝐹_𝐶𝑖(𝑡)𝑁_𝐶𝑖, (6) where 𝑀_𝐴𝑖, 𝑁_𝐴𝑖, 𝑀_𝑑𝑖, 𝑁_𝑑𝑖, 𝑀_𝐶𝑖, and 𝑁_𝐶𝑖, for each 𝑖∈𝒮, are known constant matrices and 𝐹_𝐴𝑖(𝑡), 𝐹_𝑑𝑖(𝑡), and 𝐹_𝐶𝑖(𝑡), for each 𝑖∈𝒮, are the uncertain time-varying matrices satisfying 𝐹_𝐴𝑖(𝑡)^𝑇𝐹_𝐴𝑖(𝑡)≤𝐼,𝐹_𝑑𝑖(𝑡)^𝑇𝐹_𝑑𝑖(𝑡)𝐹≤𝐼,_𝐶𝑖(𝑡)^𝑇𝐹_𝐶𝑖(𝑡)≤𝐼,∀𝑖∈𝒮. (7) It is assumed that all the elements of 𝐹_𝐴𝑖(𝑡), 𝐹_𝑑𝑖(𝑡), and 𝐹_𝐶𝑖(𝑡) are Lebesgue measurable. Δ𝐴_𝑖(𝑡), Δ𝐴_𝑑𝑖(𝑡), and Δ𝐶_𝑖(𝑡), for each 𝑖∈𝒮, are said to be admissible if both (6) and (7) hold.

Throughout the paper we will use the following concept of stochastic stability.

Definition 1. Consider the following stochastic jump system: 𝑥̇(𝑡)=𝐴(𝑟(𝑡))𝑥(𝑡)+𝐴_𝑑(𝑟(𝑡𝑥(),𝑥(𝑡)(𝑡)[](0)))𝑡−𝑑=𝜙,∀𝑡∈−𝑑,0,𝑑>0,𝑟=𝑟₀. (8) The system (8) is said to be stochastically stable, if, for finite 𝜙(𝑡)∈ℝ^𝑛 defined on [−𝑑,0] and 𝑟₀∈𝒮, there exists a scalar 𝑐>0such that lim_𝑡→∞ℰ^𝑡₀𝑥^𝑇𝜏,𝜙,𝑟₀𝑥𝜏,𝜙,𝑟₀𝑑𝜏∣𝜙,𝑟₀≤𝑐sup_{−𝑑≤𝜏≤0}𝜙(𝜏)², (9) where 𝑥(𝑡,𝜙,𝑟₀) denotes the solution of system (8) at time 𝑡 under the initial conditions 𝜙(𝑡) and 𝑟₀.

Now, for each 𝑖∈𝒮, consider a Markovian filter of the form Σ_𝐹∶𝑥̇(𝑡)=𝐴_𝑖𝑥(𝑡)+𝐾_𝑖𝑦(𝑡),𝑧(𝑡)=𝐿_𝑖𝑥(𝑡), (10) where 𝑥(𝑡)∈ℝ^𝑛 is the estimator state and the matrices 𝐴_𝑖, 𝐾_𝑖, and 𝐿_𝑖 are to be chosen. The filtering error is defined by 𝑒(𝑡)≜𝑧(𝑡)−𝑧(𝑡). (11)

Then the robust 𝐻_∞ filtering problem to be dealt with in this paper can be formulated as determining a filter of the form (10) such that, for all admissible uncertainties Δ𝐴_𝑖(𝑡), Δ𝐴_𝑑𝑖(𝑡), and Δ𝐶_𝑖(𝑡), 𝑖∈𝒮, the following requirements are satisfied:(R1) the augmented system from system Σ and the filter Σ_𝐹 is stochastically stable;(R2) with zero initial conditions, the following holds: ‖𝑒(𝑡)‖_𝐸2<𝛾‖𝜔(𝑡)‖₂, (12) for all nonzero 𝜔(𝑡)∈𝐿₂[0,∞), where 𝛾>0 is a prescribed scalar.

Before concluding this section, we introduce the following lemma which will be used in the proof of our main results in the next section.

Lemma 2 (see [19, 20]). Let 𝐴, 𝐷, 𝐸, 𝐹, and 𝑃 be real matrices of appropriate dimensions with 𝑃>0 and 𝐹 satisfying 𝐹^𝑇𝐹≤𝐼. Then one has the following:(1) for any scalar 𝜖>0, 𝐷𝐹𝐸+(𝐷𝐹𝐸)^𝑇≤𝜖⁻¹𝐷𝐷^𝑇+𝜖𝐸^𝑇𝐸, (13)(2) for any scalar 𝜖>0such that 𝑃−𝜖𝐸^𝑇𝐸>0, (𝐴+𝐷𝐹𝐸)𝑃⁻¹(𝐴+𝐷𝐹𝐸)^𝑇≤𝐴𝑃−𝜖𝐸^𝑇𝐸⁻¹𝐴^𝑇+𝜖⁻¹𝐷𝐷^𝑇, (14)(3)𝐴𝐷+(𝐴𝐷)^𝑇≤𝐴𝑃𝐴^𝑇+𝐷^𝑇𝑃⁻¹𝐷.

3. Main Results

In this section, a Riccati-like inequality approach is proposed to design a robust 𝐻_∞ filter for uncertain stochastic linear systems with Markovian jumping parameters and time delay. Our main result is presented in the next theorem.

Theorem 3. Consider the uncertain jump linear systemΣ. If there exist scalars 𝜀_1𝑖>0, 𝜀_2𝑖>0, 𝜀_3𝑖>0, and 𝜀_4𝑖>0 and matrices 𝑃_1𝑖>0, 𝑃_2𝑖>0, 𝑖=1,…,𝑠, 𝑄₁>0, and 𝑄₂>0, such that 𝑄₁−𝜀_3𝑖𝑁^𝑇_𝑑𝑖𝑁_𝑑𝑖>0, 𝑖=1,…,𝑠, and the following coupled Riccati matrix inequalities hold: 𝐴^𝑇_𝑖𝑃_1𝑖+𝑃_1𝑖𝐴_𝑖+^𝑠_𝑗=1𝜋_𝑖𝑗𝑃_1𝑗+𝑃_1𝑖𝑊_𝑖𝑃_1𝑖+𝜀_1𝑖𝑁^𝑇_𝐴𝑖𝑁_𝐴𝑖+𝜀_2𝑖𝑁^𝑇_𝐶𝑖𝑁_𝐶𝑖+𝑄₁𝐴<0,_𝑇𝑖𝑃_2𝑖+𝑃_2𝑖𝐴_𝑖+^𝑠_𝑗=1𝜋_𝑖𝑗𝑃_2𝑗+𝑃_2𝑖𝑈_𝑖𝑃_2𝑖+𝐿^𝑇_𝑖𝐿_𝑖+𝑄₂<0, (15) where 𝑊_𝑖=𝜀⁻¹_1𝑖𝑀_𝐴𝑖𝑀^𝑇_𝐴𝑖+𝜀⁻¹_3𝑖𝑀_𝑑𝑖𝑀^𝑇_𝑑𝑖+𝐴_𝑑𝑖𝑄₁−𝜀_3𝑖𝑁^𝑇_𝑑𝑖𝑁_𝑑𝑖⁻¹×𝐴^𝑇_𝑑𝑖+𝛾⁻²𝐵_𝑖𝐵^𝑇_𝑖,𝑈_𝑖=𝜀⁻¹_1𝑖𝑀_𝐴𝑖𝑀^𝑇_𝐴𝑖+𝜀⁻¹_3𝑖𝑀_𝑑𝑖𝑀^𝑇_𝑑𝑖+𝐴_𝑑𝑖𝑄₁−𝜀_3𝑖𝑁^𝑇_𝑑𝑖𝑁_𝑑𝑖⁻¹𝐴^𝑇_𝑑𝑖+𝛾⁻²𝐵_𝑖𝐼−𝐷^𝑇_𝑖𝐻_𝑖⁻¹𝐷_𝑖𝐵^𝑇_𝑖,𝐴_𝑖=𝐴_𝑖+𝜀⁻¹_1𝑖𝑀_𝐴𝑖𝑀^𝑇_𝐴𝑖+𝜀⁻¹_3𝑖𝑀_𝑑𝑖𝑀^𝑇_𝑑𝑖+𝐴_𝑑𝑖𝑄₁−𝜀_3𝑖𝑁^𝑇_𝑑𝑖𝑁_𝑑𝑖⁻¹𝐴^𝑇_𝑑𝑖𝑃_1𝑖+𝛾⁻²𝐵_𝑖𝐼−𝐷^𝑇_𝑖𝐻_𝑖⁻¹𝐷_𝑖𝐵^𝑇_𝑖𝑃_1𝑖−𝐵_𝑖𝐷^𝑇_𝑖𝐻_𝑖⁻¹𝐶_𝑖,𝐻_𝑖=𝐷_𝑖𝐷^𝑇_𝑖+𝛾²𝜀⁻¹_2𝑖𝑀_𝐶𝑖𝑀^𝑇_𝐶𝑖+𝜀_4𝑖𝐼, (16) for 𝑖=1,…,𝑠, then the robust 𝐻_∞  filtering problem is solvable. In this case, a suitable robust 𝐻_∞  filter Σ_𝐹 is given in form (10) with parameters as follows: 𝐾_𝑖=𝐵_𝑖𝐷^𝑇_𝑖𝐻_𝑖⁻¹,𝐿_𝑖=𝐿_𝑖,𝑖∈𝒮. (17)

Proof. Let the mode at time 𝑡 be 𝑖, that is; 𝑟(𝑡)=𝑖, 𝑖∈𝒮, and define 𝑥(𝑡)=𝑥(𝑡)−𝑥(𝑡), (18) then from system Σ and filter Σ_𝐹, it is easy to show that 𝑥̇‌(𝑡)=𝐴_𝑖𝑥(𝑡)+𝐴_𝑖+Δ𝐴_𝑖(𝑡)−𝐴_𝑖−𝐾_𝑖𝐶_𝑖+Δ𝐶_𝑖(𝑡)𝑥(𝑡)+𝐴_𝑑𝑖+Δ𝐴_𝑑𝑖(𝑡)𝑥(𝑡−𝑑)+𝐵_𝑖−𝐾_𝑖𝐷_𝑖𝜔(𝑡). (19) Therefore, by defining 𝑥(𝑡)^𝑇≜[𝑥(𝑡)^𝑇,𝑥(𝑡)^𝑇]^𝑇, we obtain the augmented system from Σ and Σ_𝐹 as Σ_aug∶𝑥̇(𝑡)=𝐴_𝑖(𝑡)𝑥(𝑡)+𝐴_𝑑𝑖(𝑡)𝑥(𝑡−𝑑)+𝐵_𝑖𝜔(𝑡),𝑒(𝑡)=𝐶_𝑖𝑥(𝑡),𝑥(𝑡)=𝜑(𝑡),∀𝑡∈[−𝑑,0],𝑑>0,𝑟(0)=𝑟₀, (20) where 𝐴_𝑖(𝑡)=𝐴_𝑖+Δ𝐴_𝑖(𝑡),𝐴_𝑑𝑖(𝑡)=𝐴_𝑑𝑖+Δ𝐴_𝑑𝑖(𝑡),𝐴_𝑖=𝐴_𝑖0𝐴_𝑖−𝐴_𝑖−𝐾_𝑖𝐶_𝑖𝐴_𝑖,Δ𝐴_𝑖(𝑡)=Δ𝐴_𝑖(𝑡)0Δ𝐴_𝑖(𝑡)−𝐾_𝑖Δ𝐶_𝑖(𝑡)0,𝐴_𝑑𝑖=𝐴_𝑑𝑖0𝐴_𝑑𝑖0,Δ𝐴_𝑑𝑖(𝑡)=Δ𝐴_𝑑𝑖(𝑡)0Δ𝐴_𝑑𝑖(𝑡)0,𝐵_𝑖=𝐵_𝑖𝐵_𝑖−𝐾_𝑖𝐷_𝑖,𝐶_𝑖=0𝐿_𝑖. (21) For 𝑟(𝑡)=𝑖, 𝑖∈𝒮, we define a matrix 𝑃_𝑖>0 by 𝑃_𝑖=𝑃_1𝑖00𝑃_2𝑖. (22) Next, we will show that under the conditions of the theorem, the following matrix inequality 𝑍_𝑖(𝑡)∶=𝐴_𝑖(𝑡)^𝑇𝑃_𝑖+𝑃_𝑖𝐴_𝑖(𝑡)+^𝑠_𝑗=1𝜋_𝑖𝑗𝑃_𝑗+𝑃_𝑖𝐴_𝑑𝑖(𝑡)𝑄⁻¹𝐴_𝑑𝑖(𝑡)^𝑇𝑃_𝑖+𝑄+𝐶^𝑇_𝑖𝐶_𝑖+𝛾⁻²𝑃_𝑖𝐵_𝑖𝐵^𝑇_𝑖𝑃_𝑖≤𝑉_1𝑖00𝑉_2𝑖 (23) holds, when 𝑟(𝑡)=𝑖, 𝑖∈𝒮, where 𝑄𝑄=¹00𝑄₂𝑉>0,_1𝑖=𝐴^𝑇_𝑖𝑃_1𝑖+𝑃_1𝑖𝐴_𝑖+^𝑠_𝑗=1𝜋_𝑖𝑗𝑃_1𝑗+𝑃_1𝑖𝑊_𝑖𝑃_1𝑖+𝜀_1𝑖𝑁^𝑇_𝐴𝑖𝑁_𝐴𝑖+𝜀_2𝑖𝑁^𝑇_𝐶𝑖𝑁_𝐶𝑖+𝑄₁,𝑉_2𝑖=𝐴^𝑇_𝑖𝑃_2𝑖+𝑃_2𝑖𝐴_𝑖+^𝑠_𝑗=1𝜋_𝑖𝑗𝑃_2𝑗+𝑃_2𝑖𝑈_𝑖𝑃_2𝑖+𝐿^𝑇_𝑖𝐿_𝑖+𝑄₂. (24) To this end, we use Lemma 2 to obtain the following matrix inequalities: 𝑃_𝑖Δ𝐴_𝑖(𝑡)+Δ𝐴_𝑖(𝑡)^𝑇𝑃_𝑖≤𝜀⁻¹_1𝑖𝑃_𝑖𝑀_𝐴𝑖𝑀_𝐴𝑖𝑀^𝑇_𝐴𝑖𝑀^𝑇_𝐴𝑖𝑃_𝑖+𝜀_1𝑖𝑁^𝑇_𝐴𝑖0𝑁_𝐴𝑖0+𝜀⁻¹_2𝑖𝑃_𝑖0−𝐾_𝑖𝑀_𝐶𝑖0−𝑀^𝑇_𝐶𝑖𝐾^𝑇_𝑖𝑃_𝑖+𝜀_21𝑖𝑁^𝑇_𝐶𝑖0𝑁_𝐶𝑖0=𝐻_1𝑖𝜀⁻¹_1𝑖𝑃_1𝑖𝑀_𝐴𝑖𝑀^𝑇_𝐴𝑖𝑃_2𝑖𝜀⁻¹_1𝑖𝑃_2𝑖𝑀_𝐴𝑖𝑀^𝑇_𝐴𝑖𝑃_1𝑖𝐻_2𝑖,𝑃_𝑖𝐴_𝑑𝑖(𝑡)𝑄⁻¹𝐴_𝑑𝑖(𝑡)^𝑇𝑃_𝑖≤𝑃_𝑖𝐴_𝑑𝑖𝑄−𝜀_3𝑖𝑁^𝑇_𝑑𝑖0𝑁_𝑑𝑖0⁻¹×𝐴^𝑇_𝑑𝑖𝑃_𝑖+𝜀⁻¹_3𝑖𝑃_𝑖𝑀_𝑑𝑖𝑀_𝑑𝑖𝑀^𝑇_𝑑𝑖𝑀^𝑇_𝑑𝑖𝑃_𝑖=𝐺_1𝑖𝐺^𝑇_2𝑖𝐺_2𝑖𝐺_3𝑖, (25) where (7) and 𝑄₁−𝜀_3𝑖𝑁^𝑇_𝑑𝑖𝑁_𝑑𝑖>0,𝑖=1,…,𝑠, (26) are used and 𝐻_1𝑖=𝜀⁻¹_1𝑖𝑃_1𝑖𝑀_𝐴𝑖𝑀^𝑇_𝐴𝑖𝑃_1𝑖+𝜀_1𝑖𝑁^𝑇_𝐴𝑖𝑁_𝐴𝑖+𝜀_2𝑖𝑁^𝑇_𝐶𝑖𝑁_𝐶𝑖,𝐻_2𝑖=𝑃_2𝑖𝜀⁻¹_1𝑖𝑀_𝐴𝑖𝑀^𝑇_𝐴𝑖+𝜀⁻¹_2𝑖𝐾_𝑖𝑀_𝐶𝑖𝑀^𝑇_𝐶𝑖𝐾^𝑇_𝑖𝑃_2𝑖,𝐺_1𝑖=𝑃_1𝑖𝜀⁻¹_3𝑖𝑀_𝑑𝑖𝑀^𝑇_𝑑𝑖+𝐴_𝑑𝑖𝑄₁−𝜀_3𝑖𝑁^𝑇_𝑑𝑖𝑁_𝑑𝑖⁻¹𝐴^𝑇_𝑑𝑖𝑃_1𝑖,𝐺_2𝑖=𝑃_2𝑖𝐴_𝑑𝑖𝑄₁−𝜀_3𝑖𝑁^𝑇_𝑑𝑖𝑁_𝑑𝑖⁻¹𝐴^𝑇_𝑑𝑖+𝜀⁻¹_3𝑖𝑀_𝑑𝑖𝑀^𝑇_𝑑𝑖𝑃_1𝑖,𝐺_3𝑖=𝑃_2𝑖𝐴_𝑑𝑖𝑄₁−𝜀_3𝑖𝑁^𝑇_𝑑𝑖𝑁_𝑑𝑖⁻¹𝐴^𝑇_𝑑𝑖+𝜀⁻¹_3𝑖𝑀_𝑑𝑖𝑀^𝑇_𝑑𝑖𝑃_2𝑖. (27) Hence 𝑍_𝑖(𝑡)≤𝐴^𝑇_𝑖𝑃_𝑖+𝑃_𝑖𝐴_𝑖+^𝑠_𝑗=1𝜋_𝑖𝑗𝑃_𝑗+𝑄+𝐶^𝑇_𝑖𝐶_𝑖+𝛾⁻²𝑃_𝑖𝐵_𝑖𝐵^𝑇_𝑖𝑃_𝑖+𝐻_1𝑖+𝐺_1𝑖𝜀⁻¹_1𝑖𝑃_1𝑖𝑀_𝐴𝑖𝑀^𝑇_𝐴𝑖𝑃_2𝑖+𝐺^𝑇_2𝑖𝜀⁻¹_1𝑖𝑃_2𝑖𝑀_𝐴𝑖𝑀^𝑇_𝐴𝑖𝑃_1𝑖+𝐺_2𝑖𝐻_2𝑖+𝐺_3𝑖. (28) Substituting (16) and (17) into the right-hand side of the previous inequality and using algebraic manipulations we have that (23) holds.
Now, we will show the stochastic stability of the augmented system Σ_aug when 𝜔(𝑡)=0. In the following we will simply use 𝑥(𝑡) to stand for the solution of the system Σ_aug at time 𝑡 with the initial conditions 𝜑(𝑡) and 𝑟₀.
Introduce the following stochastic Lyapunov functional candidate for system Σ_aug: 𝑉(𝑥(𝑡),𝑟(𝑡))=𝑥(𝑡)^𝑇𝑃(𝑟(𝑡))𝑥(𝑡)+^𝑡_𝑡−𝑑𝑥(𝜏)^𝑇𝑄𝑥(𝜏)𝑑𝜏. (29) It can then be shown that the weak infinitesimal generator 𝒜 of the random process {𝑥(𝑡),𝑟(𝑡)} is given by 𝒜𝑉(𝑥(𝑡),𝑖)=lim_ℎ⟶∞1ℎ[ℰ{𝑉(𝑥()(∣𝑥(𝑡),𝑟(𝑡)}(𝑥(𝑡)(𝑡)=𝑥(𝑡)𝑡+ℎ,𝑟𝑡+ℎ))=𝑖−𝑉,𝑟=𝑖)]_𝑇𝐴_𝑖(𝑡)^𝑇𝑃_𝑖+𝑃_𝑖𝐴_𝑖(𝑡)+^𝑠_𝑗=1𝜋_𝑖𝑗𝑃_𝑗×𝑥(𝑡)+𝑥(𝑡)+𝑄_𝑇𝑃_𝑖𝐴_𝑑𝑖(𝑡)𝑥()+𝑥()𝑡−𝑑𝑡−𝑑^𝑇𝐴_𝑑𝑖(𝑡)^𝑇𝑃_𝑖𝑥(𝑡)−𝑥(𝑡−𝑑)𝑄𝑥(𝑡−𝑑), (30) for any 𝑟(𝑡)=𝑖, 𝑖∈𝒮. Then, from Lemma 2, it follows that (𝑥(𝑡))≤𝑥(𝑡)𝒜𝑉,𝑖_𝑇𝐴_𝑖(𝑡)^𝑇𝑃_𝑖+𝑃_𝑖𝐴_𝑖(𝑡)+^𝑠_𝑗=1𝜋_𝑖𝑗𝑃_𝑗+𝑃_𝑖𝐴_𝑑𝑖(𝑡)𝑄⁻¹𝐴_𝑑𝑖(𝑡)^𝑇𝑃_𝑖+𝑄𝑥(𝑡). (31) Using (23), we have 𝒜𝑉(𝑥(𝑡),𝑖)≤𝑥(𝑡)^𝑇𝑉_1𝑖00𝑉_2𝑖𝑥(𝑡)≤−𝛽₁𝑥(𝑡)^𝑇𝑥(𝑡), (32) where 𝛽₁=min_𝑖∈𝒮{𝜆_min(−𝑉_1𝑖),𝜆_min(−𝑉_2𝑖)}. From (15), it is easy to see that 𝛽₁>0.
Now using Dynkin’s formula, we have, for each 𝑟(𝑡)=𝑖, 𝑖∈𝒮, 𝑡>0, ℰ{𝑉(𝑥(𝑡),𝑖)}−𝑉𝑥₀,𝑟₀=ℰ^𝑡₀𝒜𝑉(𝑥(𝜏),𝑟(𝜏))𝑑𝜏≤−𝛽₁^𝑡₀ℰ𝑥(𝜏)^𝑇𝑥(𝜏)𝑑𝜏. (33) On the other hand, for each 𝑟(𝑡)=𝑖, 𝑖∈𝒮, we can show that ℰ{𝑉(𝑥(𝑡),𝑖)}=ℰ𝑥(𝑡)^𝑇𝑃_𝑖𝑥(𝑡)+ℰ^𝑡_𝑡−𝑑𝑥(𝜏)^𝑇𝑄𝑥(𝜏)𝑑𝜏≥𝛽₂ℰ𝑥(𝑡)^𝑇𝑥(𝑡), (34) where 𝛽₂=min_𝑖∈𝒮{𝜆_min(𝑃_𝑖)} and 𝛽₂>0. The previous inequality and (33) imply that ℰ𝑥(𝑡)^𝑇𝑥(𝑡)≤−𝜆₁^𝑡₀ℰ𝑥(𝜏)^𝑇𝑥(𝜏)𝑑𝜏+𝜆₂𝑉𝑥₀,𝑟₀, (35) where 𝜆₁=𝛽₁𝛽₂⁻¹>0 and 𝜆₂=𝛽₂⁻¹>0. Then, it can be verified that ℰ𝑥(𝑡)^𝑇𝑥(𝑡)≤𝜆₂exp−𝜆₁𝑡𝑉𝑥₀,𝑟₀. (36) Therefore, ℰ^𝑡₀𝑥(𝜏)^𝑇𝑥(𝜏)𝑑𝜏∣𝜑,𝑟₀≤𝜆₁⁻¹𝜆₂1−exp−𝜆₁𝑡𝑉𝑥₀,𝑟₀. (37) Taking limit as 𝑡→∞, we have lim_𝑡→∞ℰ^𝑡₀𝑥(𝜏)^𝑇𝑥(𝜏)𝑑𝜏∣𝜑,𝑟₀≤𝜆₁⁻¹𝜆₂𝑉𝑥₀,𝑟₀. (38) Note that there always exists a scalar 𝑐>0 such that 𝜆₁⁻¹𝜆₂𝑉𝑥₀,𝑟₀≤𝑐sup_{−𝑑≤𝜏≤0}𝜑(𝜏)². (39) Considering (38), we have that the augmented system Σ_aug is stochastically stable. Next we will show that ‖𝑒(𝑡)‖_𝐸2<𝛾‖𝜔(𝑡)‖₂, (40) for all nonzero 𝜔(𝑡)∈𝐿₂[0,∞). To this end, we introduce 𝐽=ℰ^∞₀𝑒(𝜏)^𝑇𝑒(𝜏)−𝛾²𝜔(𝜏)^𝑇𝜔(𝜏)𝑑𝜏. (41) For any 𝑟(𝑡)=𝑖, 𝑖∈𝒮, 𝜔(𝑡)≠0, we have 𝒜𝑉(𝑥(𝑡),𝑖)=𝑥(𝑡)^𝑇𝐴_𝑖(𝑡)^𝑇𝑃_𝑖+𝑃_𝑖𝐴_𝑖(𝑡)+^𝑠_𝑗=1𝜋_𝑖𝑗𝑃_𝑗×𝑥(𝑡)+𝑥(𝑡)+𝑄_𝑇𝑃_𝑖𝐴_𝑑𝑖(𝑡)𝑥()+𝑥()𝑡−𝑑𝑡−𝑑^𝑇𝐴_𝑑𝑖(𝑡)^𝑇𝑃_𝑖𝑥(𝑡)−𝑥(𝑡−𝑑)^𝑇𝑄𝑥()+𝑥(𝑡)𝑡−𝑑_𝑇𝑃_𝑖𝐵_𝑖𝜔(𝑡)+𝜔(𝑡)^𝑇𝐵^𝑇_𝑖𝑃_𝑖𝑥(𝑡). (42) Thus, under zero initial condition, for any 𝑡>0, 𝑟(𝑡)=𝑖, 𝑖∈𝒮, 𝜔(𝑡)≠0, 𝐽₁=ℰ^𝑡₀𝑒(𝜏)^𝑇𝑒(𝜏)−𝛾²𝜔(𝜏)^𝑇𝜔(𝜏)𝑑𝜏=ℰ^𝑡₀𝑒(𝜏)^𝑇𝑒(𝜏)−𝛾²𝜔(𝜏)^𝑇𝜔(𝜏)+𝒜𝑉(𝑥(𝜏),𝑖)𝑑𝜏−ℰ^𝑡₀𝒜𝑉(𝑥(𝜏),𝑖)𝑑𝜏=ℰ^𝑡₀𝑥(𝜏)^𝑇𝑍_𝑖(𝜏)𝑥(𝜏)−𝑥(𝑡−𝑑)^𝑇−𝑥(𝑡)^𝑇𝑃_𝑖𝐴_𝑑𝑖(𝑡)𝑄⁻¹×𝑄𝑥(𝑡−𝑑)−𝑄⁻¹𝐴_𝑑𝑖(𝑡)^𝑇𝑃_𝑖𝑥(𝑡)−𝛾²𝜔(𝜏)^𝑇−𝛾⁻²𝑥(𝑡)^𝑇𝑃_𝑖𝐵_𝑖×𝜔(𝜏)−𝛾⁻²𝐵^𝑇_𝑖𝑃_𝑖𝑥(𝑡){𝑉(𝑥(𝑡).𝑑𝜏−ℰ,𝑖)} (43) Hence 𝐽≤ℰ^∞₀𝑥(𝜏)^𝑇𝑍_𝑖(𝜏)𝑥(𝜏)𝑑𝜏. (44) Finally, from (23), (40) follows immediately. This completes the proof.

Remark 4. Theorem 3 provides a sufficient condition for the solvability of robust 𝐻_∞ filtering problem for uncertain stochastic time-delay systems with Markovian jumping parameters, and the desired filter can be constructed by solving two sets of coupled algebraic Riccati inequalities.

In the case when 𝒮={1}, that is, there is only one mode operation, we have 𝜋_𝑖𝑖=0, 𝑖∈𝒮={1}, and system Σ reduces to the following uncertain time-delay system with no jumping parameters: Σ₁∶𝑥̇(𝑡)=[(𝑡𝑥(𝑡)+𝐴𝐴+Δ𝐴)]_𝑑+Δ𝐴_𝑑(𝑡𝑥()(𝑡),𝑦(𝑡)=[(𝑡𝑥(𝑡)(𝑡),𝑧(𝑡)(𝑡),𝑥(𝑡)(𝑡)[](0))𝑡−𝑑+𝐵𝜔𝐶+Δ𝐶)]+𝐷𝜔=𝐿𝑥=𝜙,∀𝑡∈−𝑑,0,𝑑>0,𝑟=𝑟₀, (45) where Δ𝐴(𝑡), Δ𝐴_𝑑(𝑡), and Δ𝐶(𝑡) are unknown matrices and are assumed to be of the form Δ𝐴(𝑡)=𝑀_𝐴𝐹_𝐴(𝑡)𝑁_𝐴,Δ𝐴_𝑑(𝑡)=𝑀_𝑑𝐹_𝑑(𝑡)𝑁_𝑑,Δ𝐶(𝑡)=𝑀_𝐶𝐹_𝐶(𝑡)𝑁_𝐶, (46) where 𝑀_𝐴, 𝑁_𝐴𝑖, 𝑀_𝑑, 𝑁_𝑑, 𝑀_𝐶, and 𝑁_𝐶 are known constant matrices and 𝐹_𝐴(𝑡), 𝐹_𝑑(𝑡), and 𝐹_𝐶(𝑡) are the uncertain time-varying matrices satisfying 𝐹_𝐴(𝑡)^𝑇𝐹_𝐴(𝑡)≤𝐼,𝐹_𝑑(𝑡)^𝑇𝐹_𝑑(𝑡)𝐹≤𝐼,_𝐶(𝑡)^𝑇𝐹_𝐶(𝑡)≤𝐼. (47) Then, from Theorem 3, we have the following robust 𝐻_∞ filtering result for the previous system.

Corollary 5. Consider the uncertain time-delay system (Σ₁). If there exist scalars 𝜀_𝑖>0, 𝑖=1,…,4, and matrices 𝑃₁>0, 𝑃₂>0, 𝑄₁>0, and 𝑄₂>0, such that 𝑄₁−𝜀₃𝑁^𝑇_𝑑𝑁_𝑑>0, and the following coupled Riccati matrix inequalities hold: 𝐴^𝑇𝑃₁+𝑃₁𝐴+𝑃₁𝑊𝑃₁+𝜀₁𝑁^𝑇_𝐴𝑁_𝐴+𝜀₂𝑁^𝑇_𝐶𝑁_𝐶+𝑄₁𝐴<0_𝑇𝑃₂+𝑃₂𝐴+𝑃₂𝑈𝑃₂+𝐿^𝑇𝐿+𝑄₂<0, (48) where 𝑊=𝜀₁⁻¹𝑀_𝐴𝑀^𝑇_𝐴+𝜀₃⁻¹𝑀_𝑑𝑀^𝑇_𝑑+𝐴_𝑑𝑄₁−𝜀₃𝑁^𝑇_𝑑𝑁_𝑑⁻¹𝐴^𝑇_𝑑+𝛾⁻²𝐵𝐵^𝑇,𝑈=𝜀₁⁻¹𝑀_𝐴𝑀^𝑇_𝐴+𝜀₃⁻¹𝑀_𝑑𝑀^𝑇_𝑑+𝐴_𝑑𝑄₁−𝜀₃𝑁^𝑇_𝑑𝑁_𝑑⁻¹𝐴^𝑇_𝑑+𝛾⁻²𝐵𝐼−𝐷^𝑇𝐻⁻¹𝐷𝐵^𝑇,𝐴=𝐴+𝜀₁⁻¹𝑀_𝐴𝑀^𝑇_𝐴+𝜀₃⁻¹𝑀_𝑑𝑀^𝑇_𝑑+𝐴_𝑑𝑄₁−𝜀₃𝑁^𝑇_𝑑𝑁_𝑑⁻¹𝐴^𝑇_𝑑𝑃₁+𝛾⁻²𝐵𝐼−𝐷^𝑇𝐻⁻¹𝐷𝐵^𝑇𝑃₁−𝐵𝐷^𝑇𝐻⁻¹𝐶,𝐻=𝐷𝐷^𝑇+𝛾²𝜀₂⁻¹𝑀_𝐶𝑀^𝑇_𝐶+𝜀₄𝐼, (49) then the robust 𝐻_∞  filtering problem is solvable. In this case, a suitable robust 𝐻_∞  filter is given by 𝑥̇(𝑡)=𝐴𝑥(𝑡)+(𝑡),𝑧(𝑡)=𝐿𝑥(𝑡),𝐾𝑦 (50) where 𝐾=𝐵𝐷^𝑇𝐻⁻¹,𝐿=𝐿. (51)

Remark 6. The solvability for robust 𝐻_∞ filtering problem for uncertain continuous delay-free systems can be easily derived from Corollary 5, in this case. It can be shown that the result coincides with that proposed in [8].

4. Numerical Example

In this section, we will give a numerical example to demonstrate the applicability of the proposed approach.

Consider the uncertain time-delay stochastic linear systems with Markovian jumping parameters in form (1) with two modes. For mode 1, the dynamics of the system are described as follows: 𝐴₁=−5200−4100−5,𝐴_𝑑1=,𝐵−100.200.510.500₁=1−2.3010.50,𝐶₁=,𝐷−102011₁=1001,𝑀_𝐶1=,𝑁0.5100.5_𝐶1=00.510−0.51,𝑀_𝐴1=,𝑀10010.51_𝑑1=00.50.510.50,𝑁^𝑇_𝐴1=0,𝑁−0.20.1_𝑇𝑑1=0−0.10.2,𝐿₁=.0.20000.3−0.1 (52) For mode 2, the dynamics of the system are described as follows: 𝐴₂=−610.5−1−7000.5−6,𝐴_𝑑2=,𝐵0100.5−1−0.1001₂=−10.5−3101,𝐶₂=,𝐷0.512−102₂=−10.50.21,𝑀_𝐶2=,𝑁0.1−0.30.50_𝐶2=−10.20.30.50.50,𝑀_𝐴2=,𝑀0.50.300.50.21_𝑑2=−0.60.5010.51,𝑁^𝑇_𝐴2=00,𝑁0.1_𝑇𝑑2=000.2,𝐿₂=.−0.20.100.300.5 (53) Let the transition probability matrix be given by Π=.−0.50.51−1 (54) In this example, we set 𝛾=2.8. By solving (15), we obtain 𝑃₁₁=,𝑃2.15920.3579−0.05890.35793.20180.1337−0.05890.13374.0867₁₂=,𝑄2.9083−0.0024−0.2047−0.00241.68550.0031−0.20470.00312.0020₁=,𝑃11.1343−0.4670−0.3949−0.467010.5734−0.6361−0.3949−0.636111.8166₂₁=,𝑃0.21570.10940.02210.10940.33240.05240.02210.05240.1848₂₂=,𝑄0.1572−0.01610.0122−0.01610.0989−0.01760.0122−0.01760.1414₂=,𝜀0.7630−0.0624−0.0257−0.06240.7896−0.0205−0.0257−0.02050.8370₁₁=12.1023,𝜀₂₁𝜀=9.2498,₃₁=12.1293,𝜀₄₁𝜀=0.0032,₁₂=12.1346,𝜀₂₂𝜀=10.1634,₃₂=12.1219,𝜀₄₂=0.0012. (55) Therefore, using Theorem 3, a suitable Markovian robust 𝐻_∞ filter can be constructed with parameters given as follows: 𝐴₁=,𝐾−3.55044.44960.33040.0239−3.83660.99410.29270.4860−4.7540₁=,𝐿0.9411−2.2211−0.18210.88650.2611−0.0911₁=,𝐴0.20000.3−0.1₂=,𝐾−5.96950.0301−1.2747−2.8680−9.2528−4.38900.78400.5184−7.3116₁=,𝐿0.9452−0.01622.7441−0.42870.18260.7602₂=.−0.20.100.300.5 (56)

5. Conclusions

In this paper, we have studied the problem of robust 𝐻_∞ filtering for a class of uncertain Markovian jump systems with time-delay and norm-bounded time-varying parameter uncertainties. A Markovian jump filter is designed which guarantees the stochastic stability of the filter error dynamics and a prescribed bound on the ℒ₂-induced gain from the noise signals to the filter error irrespective of the parameter uncertainties. It has been shown that the desired robust 𝐻_∞ filter can be constructed by solving two sets of coupled algebraic Riccati inequalities.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China (61104132) and Director of the foundation (SJ201002).