Extended ℋ_∞ Estimation for Two-Dimensional Markov Jump Systems under Asynchronous Switching

Abstract

This paper is concerned with the problem of designing ℋ_∞ filters for a class of two-dimensional (2D) Markov jump systems under asynchronous switching. The problem under consideration is primarily motivated by a realistic situation that the switching of candidate filters may have a lag to the switching of system modes. Different from conventional techniques, by a suitable augmentation, the jumping process of the error system is represented by a two-component Markov chain. Then, the extended transition probabilities are provided for the error system. A stochastic Lyapunov function approach is proposed for the design of desired filters that ensure a prescribed ℋ_∞ performance for admissible asynchronous switching. Finally, a numerical example is given to illustrate the effectiveness of the developed method.

1. Introduction

Networked systems have been an active topic that has attracted increasing attention of researchers [1–3]. Recently, 2D systems have gained substantial research interests due to their potential applications in many engineering fields, especially in the areas of linear iterative circuit networks, which have been extensively studied in encoding, decoding networks for linear codes, and image processing, and so forth, [4]. The challenges associated with 2D systems are their structural and dynamical complexities, and thus numerous methods have been conducted for the analysis and synthesis of such systems [5–7]. In the context of state estimation, various filtering problems for 2D systems have been widely investigated. According to the types of noise signals and performance criteria, many important filtering approaches have been proposed in the literature including the minimum mean-square state estimation [8, 9], ℋ_∞ filtering [10–17], and 𝑙₂−𝑙_∞ filtering [17].

As is well known, Markov jump linear systems, which were first introduced in [18], have the ability to capture the abrupt changes that appear in the system structure or its parameters. Markov jump systems can be regarded as a kind of multimodal systems in which the transitions among different modes are governed by a Markov chain taking values in a finite set. The past few decades have witnessed a significant progress on various aspects of Markov jump systems. The practical motivations as well as theoretical results on analysis and design for such systems can be found in several references, for example, in [19–22] for 1D case and [7, 13] for 2D case.

In particular, a lot of efforts have been focused on the filtering problem of Markov jump systems because of its importance in theory and practice [13, 22, 23]. The existing results can be broadly classified into two categories: mode-dependent and mode-independent filtering. With an adaptation sense, the mode-dependent filter design has gained popularity owing to its less conservatism. However, most of the mode-dependent results are based on the critical assumption that the switches of filters are strictly synchronized with those of the system modes. In practical case, it is inevitable that operations related to identifying the system modes and specifying the matched filter will take time, and thus mismatch of the modes between the filter and the system generally exists. Most recently, research results concerning asynchronous filtering problems have been derived [16, 24]. In spite of these developments, the problems of asynchronous filtering for 2D Markov jump systems are not fully resolved.

In this paper, we focus on 2D systems described by the Roesser model subject to Markovian jump parameters. We address the issue of designing filters such that an upper bound on ℋ_∞ norm of the estimation error system for admissible asynchronous switching is minimized. By a suitable augmentation, a novel approach is adopted to model the jumping process of the error system. The ℋ_∞ analysis result is derived, followed by a stochastic parameter-dependent approach. The asynchronous filter is then designed such that the error system is mean-square asymptotically stable and has a prescribed ℋ_∞ performance level. It is shown from the derived results that the effect of the asynchronous behavior in reality can be comprehensively understood for 2D Markov jump systems.

Notations. The notation used throughout the paper is fairly standard. The superscript 𝑇 stands for matrix transposition, ℝ^𝑛 denotes the 𝑛 dimensional Euclidean space, and 𝑃>0 means that 𝑃 is real symmetrical and positive definite. 𝔼{⋅} stands for the expectation operation. 𝑙₂{[0,∞),[0,∞)} is the space of square summable sequences on {[0,∞),[0,∞)}. In symmetrical block matrices expressions, we use an asterisk (∗) to represent a term that is induced by symmetry and stand diag{⋯} for a block diagonal matrix. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.

2. Problem Formulation

Consider the following class of 2D Markov jump systems: 𝑥^ℎ_𝑖+1,𝑗𝑥^𝑣_𝑖,𝑗+1=𝐴𝑟_𝑖,𝑗𝑥^ℎ_𝑖,𝑗𝑥^𝑣_𝑖,𝑗+𝐵𝑟_𝑖,𝑗𝑤_𝑖,𝑗,𝑦_𝑖,𝑗=𝐶𝑟_𝑖,𝑗𝑥^ℎ_𝑖,𝑗𝑥^𝑣_𝑖,𝑗+𝐷𝑟_𝑖,𝑗𝑤_𝑖,𝑗,𝑧_𝑖,𝑗=𝐻𝑟_𝑖,𝑗𝑥^ℎ_𝑖,𝑗𝑥^𝑣_𝑖,𝑗+𝐿𝑟_𝑖,𝑗𝑤_𝑖,𝑗, (1) where 𝑥^ℎ_𝑖,𝑗∈ℝ^𝑛1, 𝑥^𝑣_𝑖,𝑗∈ℝ^𝑛2 represent the horizontal and vertical states, respectively, 𝑦_𝑖,𝑗∈ℝ^𝑚 is the measured output, 𝑧_𝑖,𝑗∈ℝ^𝑝 is the objective signal to be estimated, and 𝑤_𝑖,𝑗∈ℝ^𝑞 is the noise signal which belongs to 𝑙₂{[0,∞),[0,∞)}. 𝐴(𝑟_𝑖,𝑗), 𝐵(𝑟_𝑖,𝑗), 𝐶(𝑟_𝑖,𝑗), 𝐷(𝑟_𝑖,𝑗), 𝐻(𝑟_𝑖,𝑗), and 𝐿(𝑟_𝑖,𝑗) are appropriately dimensioned real valued system matrices. These matrices are functions of 𝑟_𝑖,𝑗, which is a discrete-time, discrete-state Markov chain taking values in a finite set 𝒯={1,2,…,𝑔} with transition probabilities 𝑝_𝑚𝑛=Pr𝑟_𝑖+1,𝑗=𝑛∣𝑟_𝑖,𝑗𝑟=𝑚=Pr_𝑖,𝑗+1=𝑛∣𝑟_𝑖,𝑗=𝑚, (2) where 𝑝_𝑚𝑛≥0 and ∑^𝑔_𝑛=1𝑝_𝑚𝑛=1, for all 𝑚∈𝒯. To simplify the notation, the system matrices are denoted by 𝒮_𝑚=(𝐴_𝑚, 𝐵_𝑚, 𝐶_𝑚, 𝐷_𝑚, 𝐻_𝑚, 𝐿_𝑚), when 𝑟_𝑖,𝑗=𝑚∈𝒯.

For the asynchronous phenomenon under consideration, that is, the switches of filter gains may not coincide precisely with those of system modes, we are interested in estimating the objective signal 𝑧_𝑖,𝑗 by a filter with the structure as follows: 𝑥^ℎ_𝑖+1,𝑗𝑥^𝑣_𝑖,𝑗+1=𝜏_𝑘𝐴_𝑓𝑧+1−𝜏_𝑘𝐴_𝑓𝑚𝑥^ℎ_𝑖,𝑗𝑥^𝑣_𝑖,𝑗+𝜏_𝑘𝐵_𝑓𝑧+1−𝜏_𝑘𝐵_𝑓𝑚𝑦_𝑖,𝑗,𝑧_𝑖,𝑗=𝜏_𝑘𝐶_𝑓𝑧+1−𝜏_𝑘𝐶_𝑓𝑚𝑥^ℎ_𝑖,𝑗𝑥^𝑣_𝑖,𝑗+𝜏_𝑘𝐷_𝑓𝑧+1−𝜏_𝑘𝐷_𝑓𝑚𝑦_𝑖,𝑗, (3) where 𝑥^ℎ_𝑖,𝑗∈ℝ^𝑛1, 𝑥^𝑣_𝑖,𝑗∈ℝ^𝑛2 are the filter states, 𝑧_𝑖,𝑗∈ℝ^𝑝 is the estimation of 𝑧_𝑖,𝑗, ℱ_𝑚=(𝐴_𝑓𝑚, 𝐵_𝑓𝑚, 𝐶_𝑓𝑚, 𝐷_𝑓𝑚) and ℱ_𝑧=(𝐴_𝑓𝑧, 𝐵_𝑓𝑧, 𝐶_𝑓𝑧, 𝐷_𝑓𝑧) are the filter gains corresponding to the current and previous stage, respectively, for all 𝑚,𝑧∈𝒯, and 𝜏_𝑘 is a Bernoulli distributed white sequence specified by Pr𝜏_𝑘=1=𝔼𝜏_𝑘𝜏=𝜙,Pr_𝑘=0=1−𝔼𝜏_𝑘=1−𝜙. (4) In addition, 𝜏_𝑘 and 𝑟_𝑖,𝑗 are mutually independent.

In view of (1) and (3), the estimation error 𝑧_𝑖,𝑗=𝑧_𝑖,𝑗−𝑧_𝑖,𝑗 can be represented by the following model: 𝑥^ℎ_𝑖+1,𝑗𝑥^𝑣_𝑖,𝑗+1=𝐴_𝑚,𝑙𝑥^ℎ_𝑖,𝑗𝑥^𝑣_𝑖,𝑗+𝐵_𝑚,𝑙𝑤_𝑖,𝑗,𝑧_𝑖,𝑗=𝐶_𝑚,𝑙𝑥^ℎ_𝑖,𝑗𝑥^𝑣_𝑖,𝑗+𝐷_𝑚,𝑙𝑤_𝑖,𝑗, (5) where 𝑥^ℎ_𝑖,𝑗=[𝑥^ℎ𝑇_𝑖,𝑗𝑥^ℎ𝑇_𝑖,𝑗]^𝑇, 𝑥^𝑣_𝑖,𝑗=[𝑥^𝑣𝑇_𝑖,𝑗𝑥^𝑣𝑇_𝑖,𝑗]^𝑇 and 𝐴_𝑚,𝑙=Ω^𝑇𝐴_𝑚0𝐵_𝑓𝑙𝐶_𝑚𝐴_𝑓𝑙Ω,𝐵_𝑚,𝑙=Ω^𝑇𝐵_𝑚𝐵_𝑓𝑙𝐷_𝑚,𝐶_𝑚,𝑙=𝐻_𝑚−𝐷_𝑓𝑙𝐶_𝑚−𝐶_𝑓𝑙Ω,𝐷_𝑚,𝑙=𝐿_𝑚−𝐷_𝑓𝑙𝐷_𝑚,Ω=𝐼00000𝐼00𝐼00000𝐼,∀𝑚,𝑙∈𝒯. (6)

Accordingly, the jumping process {𝒮_𝑚,ℱ_𝑙} in the error system (5) forms a two-component Markov chain 𝑟_𝑖,𝑗 on 𝒯×𝒯 with the extended transition probabilities 𝑝_{(𝑚,𝑙)(𝑛,𝑢)}=Pr{𝒮_𝑛,ℱ_𝑢∣𝒮_𝑚,ℱ_𝑙} given by 𝑝_{(𝑚,𝑙)(𝑛,𝑢)}=𝑝_𝑚𝑛,𝑢=𝑛,𝑢=𝑙𝜙𝑝_𝑚𝑛𝑝,𝑢≠𝑛,𝑢=𝑙1−𝜙_𝑚𝑛,𝑢=𝑛,𝑢≠𝑙0,𝑢≠𝑛,𝑢≠𝑙. (7)

Remark 1. An important feature of the augmented system (5) lies in the fact that a two-component Markov chain with 𝑔² modes is considered for the jumping process, which constitutes the most significant distinction from previous results on asynchronous switching. The scalar 𝜙 in extended transition probabilities (7) indicates the mismatch degree of the modes between the filter and the system.

The following definitions are needed in formulating the considered problem. For more details refer to [13] and the references therein.

Definition 2. The system (5) is said to be mean-square asymptotically stable if for 𝑤_𝑖,𝑗=0 and bounded boundary conditions, the following holds: lim_{𝑖+𝑗→∞}𝔼𝑥_𝑖,𝑗²=0, (8) where 𝑥_𝑖,𝑗=[𝑥^ℎ𝑇_𝑖,𝑗𝑥^𝑣𝑇_𝑖,𝑗]^𝑇.

Definition 3. Given a scalar 𝛾>0, the system in (5) is said to be mean-square asymptotically stable with an ℋ_∞ disturbance attenuation level 𝛾, if it is mean-square asymptotically stable, and under zero initial conditions satisfies 𝑧_{𝑖,𝑗𝐸}<𝛾𝑤_𝑖,𝑗2, (9) for all nonzero 𝑤_𝑖,𝑗, where 𝑧_{𝑖,𝑗𝐸}=𝔼^∞∞_𝑖=0𝑗=0𝑧_𝑖,𝑗²,𝑤_𝑖,𝑗2=^∞∞_𝑖=0𝑗=0𝑤_𝑖,𝑗². (10)
Then, the estimation problem of interest is stated as follows: given 𝛾>0, design a filter of the form in (3) such that the error system in (5) with extended transition probabilities (7) is mean-square asymptotically stable and has a prescribed ℋ_∞ disturbance attenuation level 𝛾.

3. Main Results

In this section, we will show the procedure to design the asynchronous ℋ_∞ filter, which guarantees that the error system is mean-square asymptotically stable and has a prescribed ℋ_∞ disturbance attenuation level 𝛾. We first provide a new analysis result to check if the ℋ_∞ norm of the error system is bounded by the asynchronous filter. The corresponding analysis result is formulated in the following theorem.

Theorem 4. Consider the system (5) with extended transition probabilities (7), and let 𝛾>0 be a given constant. If there exist matrices 𝑃_𝑚,𝑙=diag{𝑃^ℎ_𝑚,𝑙,𝑃^𝑣_𝑚,𝑙}>0, for all (𝑚,𝑙)∈𝒯×𝒯, such that −𝒫_𝑛,𝑢0𝒫_𝑛,𝑢𝐴_𝑚,𝑙𝒫_𝑛,𝑢𝐵_𝑚,𝑙∗−𝐼𝐶_𝑚,𝑙𝐷_𝑚,𝑙∗∗−𝑃_𝑚,𝑙0∗∗∗−𝛾²𝐼<0, (11) where 𝒫_𝑛,𝑢=∑_{(𝑛,𝑢)∈𝒯×𝒯}𝑝_{(𝑚,𝑙)(𝑛,𝑢)}𝑃_𝑛,𝑢, then the system in (5) is mean-square asymptotically stable and has a prescribed ℋ_∞ performance index 𝛾.

Proof. First, we handle the stochastic stability of the system in (5) with 𝑤_𝑖,𝑗≡0. Construct the following index: ℒ_𝑖,𝑗=𝔼𝑥^ℎ𝑇_𝑖+1,𝑗𝑃^ℎ𝑟_𝑖+1,𝑗𝑥^ℎ_𝑖+1,𝑗+𝑥^𝑣𝑇_𝑖,𝑗+1𝑃^𝑣𝑟_𝑖,𝑗+1𝑥^𝑣_𝑖,𝑗+1−𝑥^𝑇_𝑖,𝑗𝑃𝑟_𝑖,𝑗𝑥_𝑖,𝑗∣𝑥_𝑖,𝑗,𝑟_𝑖,𝑗=(𝑚,𝑙), (12) where 𝑃(𝑟_𝑖,𝑗)=diag{𝑃^ℎ(𝑟_𝑖,𝑗),𝑃^𝑣(𝑟_𝑖,𝑗)}, which is represented as 𝑃_𝑚,𝑙 when 𝑟_𝑖,𝑗=(𝑚,𝑙). Then, along the evolution of the system (5) with 𝑤_𝑖,𝑗≡0, it follows that ℒ_𝑖,𝑗=𝑥^𝑇_𝑖,𝑗𝐴^𝑇_{𝑚,𝑙()𝑛,𝑢∈𝒯×𝒯}𝑝_{(𝑚,𝑙)(𝑛,𝑢)}𝑃_𝑛,𝑢𝐴_𝑚,𝑙−𝑃_𝑚,𝑙𝑥_𝑖,𝑗. (13) By Schur's complement, (11) guarantees ℒ_𝑖,𝑗<0, which implies that lim_{𝑖+𝑗→∞}𝔼𝑥_𝑖,𝑗²=0. (14) Thus, the system is mean-square asymptotically stable; see [13] for more details.
Next, to establish the ℋ_∞ performance for the system, we consider the following index: 𝒥_𝑖,𝑗=𝔼𝑥^ℎ𝑇_𝑖+1,𝑗𝑃^ℎ𝑟_𝑖+1,𝑗𝑥^ℎ_𝑖+1,𝑗+𝑥^𝑣𝑇_𝑖,𝑗+1𝑃^𝑣𝑟_𝑖,𝑗+1𝑥^𝑣_𝑖,𝑗+1−𝑥^𝑇_𝑖,𝑗𝑃𝑟_𝑖,𝑗𝑥_𝑖,𝑗+𝑧^𝑇_𝑖,𝑗𝑧_𝑖,𝑗−𝛾²𝑤^𝑇_𝑖,𝑗𝑤_𝑖,𝑗∣𝑥_𝑖,𝑗,𝑟_𝑖,𝑗=(𝑚,𝑙). (15) It is inferred from (15) that 𝒥_𝑖,𝑗=𝜁^𝑇_𝑖,𝑗Φ_𝑚,𝑙𝜁_𝑖,𝑗, (16) where 𝜁_𝑖,𝑗=[𝑥^𝑇_𝑖,𝑗𝑤^𝑇_𝑖,𝑗]^𝑇 and Φ_𝑚,𝑙=𝐴^𝑇_𝑚,𝑙𝐵^𝑇_𝑚,𝑙^{(𝑛,𝑢)∈𝒯×𝒯}𝑝_{(𝑚,𝑙)(𝑛,𝑢)}𝑃_𝑛,𝑢𝐴_𝑚,𝑙𝐵_𝑚,𝑙+𝐶^𝑇_𝑚,𝑙𝐷^𝑇_𝑚,𝑙𝐶_𝑚,𝑙𝐷_𝑚,𝑙−𝑃_𝑚,𝑙00𝛾²𝐼. (17) Applying Schur's complement again, (11) yields 𝒥_𝑖,𝑗<0, which means that ‖𝑧_𝑖,𝑗‖_𝐸<𝛾‖𝑤_𝑖,𝑗‖₂. Hence, the system is mean-square asymptotically stable and has a prescribed ℋ_∞ disturbance attenuation performance. The proof is completed.

Remark 5. It is worth stressing that the two-component Markov chain adopted in (5) makes the analysis criterion more concise and clear. It is shown that only condition (11) is necessary to test whether the system (5) has a prescribed disturbance attenuation level in the ℋ_∞ sense for a given asynchronous filter. The analysis result can be used to clarify the relationship between the asynchronous switching and the ℋ_∞ performance level.

The developments, in the above, lead to the asynchronous filter design result in the next theorem.

Theorem 6. The system in (5) with extended transition probabilities (7) is mean-square asymptotically stable and has a prescribed ℋ_∞ performance index 𝛾>0, if there exist matrices 𝑃_1𝑚,𝑙=diag{𝑃^ℎ_1𝑚,𝑙,𝑃^𝑣_1𝑚,𝑙}>0, 𝑃_3𝑚,𝑙=diag{𝑃^ℎ_3𝑚,𝑙,𝑃^𝑣_3𝑚,𝑙}>0, 𝑃_2𝑚,𝑙=diag{𝑃^ℎ_2𝑚,𝑙,𝑃^𝑣_2𝑚,𝑙}, 𝑋_𝑙, 𝑆_𝑙, 𝑈_𝑙, and 𝐴_𝑓𝑙,𝐵_𝑓𝑙,𝐶_𝑓𝑙,𝐷_𝑓𝑙, for all (𝑚,𝑙)∈𝒯×𝒯, such that Σ¹_𝑚,𝑙Σ²_𝑚,𝑙0Σ⁴_𝑚,𝑙𝑋^𝑇_𝑙𝐵_𝑚+𝐵_𝑓𝑙𝐷_𝑚∗Σ³_𝑚,𝑙0Σ⁵_𝑚,𝑙𝑆^𝑇_𝑙𝐵_𝑚+𝐵_𝑓𝑙𝐷_𝑚∗∗−𝐼Σ⁶_𝑚,𝑙𝐿_𝑚−𝐷_𝑓𝑙𝐷_𝑚∗∗∗Σ⁷_𝑚,𝑙0∗∗∗∗−𝛾²𝐼<0, (18) where Σ¹_𝑚,𝑙=_{()𝑛,𝑢∈𝒯×𝒯}𝑝_{(𝑚,𝑙)(𝑛,𝑢)}𝑃_1𝑛,𝑢−𝑋^𝑇_𝑙−𝑋_𝑙,Σ²_𝑚,𝑙=_{()𝑛,𝑢∈𝒯×𝒯}𝑝_{(𝑚,𝑙)(𝑛,𝑢)}𝑃_2𝑛,𝑢−𝑈^𝑇_𝑙−𝑆_𝑙,Σ³_𝑚,𝑙=_{(𝑛,𝑢)∈𝒯×𝒯}𝑝_{(𝑚,𝑙)(𝑛,𝑢)}𝑃_3𝑛,𝑢−𝑈^𝑇_𝑙−𝑈_𝑙,Σ⁴_𝑚,𝑙=𝑋^𝑇_𝑙𝐴_𝑚+𝐵_𝑓𝑙𝐶_𝑚𝐴_𝑓𝑙,Σ⁵_𝑚,𝑙=𝑆^𝑇_𝑙𝐴_𝑚+𝐵_𝑓𝑙𝐶_𝑚𝐴_𝑓𝑙,Σ⁶_𝑚,𝑙=𝐻_𝑚−𝐷_𝑓𝑙𝐶_𝑚−𝐶_𝑓𝑙,Σ⁷_𝑚,𝑙=−𝑃_1𝑚,𝑙−𝑃_2𝑚,𝑙∗−𝑃_3𝑚,𝑙. (19) In this case, the admissible filter gains are given by 𝐴_𝑓𝑙=𝑈_𝑙^−𝑇𝐴_𝑓𝑙,𝐵_𝑓𝑙=𝑈_𝑙^−𝑇𝐵_𝑓𝑙,𝐶_𝑓𝑙=𝐶_𝑓𝑙,𝐷_𝑓𝑙=𝐷_𝑓𝑙. (20)

Proof. First, for a matrix 𝐺_𝑙, for all 𝑙∈𝒯, it is verified from the fact (𝐺_𝑙−𝒫_𝑛,𝑢)^𝑇(𝒫_𝑛,𝑢)⁻¹(𝐺_𝑙−𝒫_𝑛,𝑢)≥0 that 𝒫_𝑛,𝑢−𝐺^𝑇_𝑙−𝐺_𝑙≥−𝐺^𝑇_𝑙(𝒫_𝑛,𝑢)⁻¹𝐺_𝑙. Then, we know that 𝒫_𝑛,𝑢−𝐺^𝑇_𝑙−𝐺_𝑙0𝐺^𝑇_𝑙𝐴_𝑚,𝑙𝐺^𝑇_𝑙𝐵_𝑚,𝑙∗−𝐼𝐶_𝑚,𝑙𝐷_𝑚,𝑙∗∗−𝑃_𝑚,𝑙0∗∗∗−𝛾²𝐼<0 (21) leads to (11). Thus, consider the system in (5), and assume that the matrices 𝑃^ℎ_𝑚,𝑙, 𝑃^𝑣_𝑚,𝑙, 𝐺_𝑙 in (21) have the following forms: 𝑃^ℎ_𝑚,𝑙=𝑃^ℎ_1𝑚,𝑙𝑃^ℎ_2𝑚,𝑙∗𝑃^ℎ_3𝑚,𝑙,𝑃^𝑣_𝑚,𝑙=𝑃^𝑣_1𝑚,𝑙𝑃^𝑣_2𝑚,𝑙∗𝑃^𝑣_3𝑚,𝑙,𝐺_𝑙=𝐺_1𝑙𝐺_2𝑙𝐺_4𝑙𝐺_3𝑙,𝐺_𝑖𝑙=𝑋_𝑖𝑙𝑆_𝑖𝑙𝑈_𝑖𝑙𝑈_𝑖𝑙,𝑖=1,…,4. (22) Notice that Ω𝑃_𝑚,𝑙Ω^𝑇=𝑃_1𝑚,𝑙𝑃_2𝑚,𝑙∗𝑃_3𝑚,𝑙,Ω𝐺_𝑙Ω^𝑇=𝑋_𝑙𝑆_𝑙𝑈_𝑙𝑈_𝑙,Ω𝐺^𝑇_𝑙𝐴_𝑚,𝑙Ω^𝑇=𝑋^𝑇_𝑙𝐴_𝑚+𝐵_𝑓𝑙𝐶_𝑚𝐴_𝑓𝑙𝑆^𝑇_𝑙𝐴_𝑚+𝐵_𝑓𝑙𝐶_𝑚𝐴_𝑓𝑙,Ω𝐺^𝑇_𝑙𝐵_𝑚,𝑙=𝑋^𝑇_𝑙𝐵_𝑚+𝐵_𝑓𝑙𝐷_𝑚𝑆^𝑇_𝑙𝐵_𝑚+𝐵_𝑓𝑙𝐷_𝑚,𝐶_𝑚,𝑙Ω^𝑇=𝐻_𝑚−𝐷_𝑓𝑙𝐶_𝑚−𝐶_𝑓𝑙,𝐷_𝑚,𝑙=𝐿_𝑚−𝐷_𝑓𝑙𝐷_𝑚. (23) Furthermore, define matrix variables as 𝐴_𝑓𝑙=𝑈^𝑇_𝑙𝐴_𝑓𝑙,𝐵_𝑓𝑙=𝑈^𝑇_𝑙𝐵_𝑓𝑙,𝐶_𝑓𝑙=𝐶_𝑓𝑙,𝐷_𝑓𝑙=𝐷_𝑓𝑙. (24) It is obvious from the inequality in (18) that (11) in Theorem 4 is satisfied with the filter matrices as in (24) and 𝐴_𝑚,𝑙, 𝐵_𝑚,𝑙, 𝐶_𝑚,𝑙, 𝐷_𝑚,𝑙 in (5). Hence, it follows from Theorem 4 that the system (5) with extended transition probabilities (7) is mean-square asymptotically stable and has a prescribed ℋ_∞ performance. Meanwhile, the filter gains in (20) follow immediately from (24). This completes the proof.

Remark 7. Theorem 6 provides a sufficient condition for designing an admissible ℋ_∞ filter of the form in (3) for the system (1) under asynchronous switching. Compared with the well-known ℋ_∞ filtering results of 1D Markov jump systems, such as [23], the structure of matrix variables in Theorem 6 is slightly involved due to the structural complexity of the 2D system (1).

Remark 8. By solving the convex problem formulated in Theorem 6, the ℋ_∞ performance 𝛾 can be optimized in terms of the feasibility of the corresponding condition. The result in Theorem 6 indicates that the different 𝜙 in (7) brings about the different optimal 𝛾 achieved for the system (5). Thus, the effect of asynchronous switching can be readily known by comparing the ℋ_∞ performance indexes, which will be shown in the next section.

4. A Numerical Example

In this section, an example is provided to demonstrate the merits of our main result. Consider a 2D Markov jump system with three operation modes as follows: 𝐴₁=0.2010.1,𝐴₂=,𝐴0.8010.6₃=0.5010.9,𝐶₁=,𝐶0.21₂=0.81,𝐶₃=,𝐵0.51₁=𝐵₂=𝐵₃=,𝐷1000₁=𝐷₂=𝐷₃=,𝐻01₁=𝐻₂=𝐻₃=01. (25) The transition probability matrix is given by 𝑝𝑝=¹¹𝑝₁₂𝑝₁₃𝑝₂₁𝑝₂₂𝑝₂₃𝑝₃₁𝑝₃₂𝑝₃₃=0.50.30.20.60.10.30.10.20.7. (26) Moreover, 𝜙 is set as 0.35.

For this system, the optimized ℋ_∞ performance level is 2.8202 via the method in [16]. By using the filter design method in Theorem 6, the minimum ℋ_∞ cost is obtained 𝛾^∗=2.1694 as well as the resulting filter gain matrices 𝐴_𝑓1=0.4496−0.20090.42450.0413,𝐵_𝑓1=,𝐶−0.2091−0.1126_𝑓1=,𝐴−0.1020−0.8002_𝑓2=0.36070.02810.8642−0.0670,𝐵_𝑓2=,𝐶−0.0999−0.6389_𝑓2=,𝐴−0.0442−0.9759_𝑓3=0.29140.00820.5848−0.0314,𝐵_𝑓3=,𝐶−0.1127−0.8919_𝑓3=−0.0234−1.0341. (27) In addition, let the system initial condition 𝑥^𝑣_𝑖,0=0.8, 0≤𝑖≤15, and select the following noise signals: CaseI:𝑤_𝑖,𝑗=0.1,1≤𝑖,𝑗≤5,0,otherwise,CaseII:𝑤_𝑖,𝑗=0.5𝑒^−0.25𝑗sin0.5𝜋𝑗,1≤𝑖≤10,𝑗≥1,0,otherwise. (28) By applying the obtained filter, the filtering errors are depicted in Figures 1 and 2, respectively, from which we can see that 𝑧_𝑖,𝑗 converge to zero. The effectiveness of the designed filter is apparent.

Figure 1 

Filtering error 𝑧𝑖,𝑗 in Case I.

Figure 2 

Filtering error 𝑧𝑖,𝑗 in Case II.

Figure 3 shows the minimum 𝛾 versus 𝜙 for the design method in Theorem 6. It is observed from the curve's trend of the asynchronous design that the lower the probability 𝜙 is, the better the performance 𝛾 is. In this sense, the proposed method reveals a compromise between the asynchronous switching and the performance benefit. Thus, by comparing the ℋ_∞ performance indexes achieved in Figure 3, the effect of the asynchronous switching can be comprehensively understood just as mentioned before.

Figure 3 

Optimized ℋ∞ performance.

5. Conclusion

This paper is dedicated to the problem of ℋ_∞ estimation for a class of 2D Markov jump systems under asynchronous switching. A novel method has been proposed to model the process of the error system. The existence condition of asynchronous ℋ_∞ filters has been derived to ensure the stochastic stability and ℋ_∞ disturbance attenuation level of the error system. It is also shown from the obtained conditions that the effect of the asynchronous phenomenon can be easily known for 2D Markov jump systems. A numerical example is given to demonstrate the validity and the merits of the proposed theoretical results.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China (61105038), by the Specialized Research Fund for the Doctoral Program of Higher Education (20122302120069), by the Basic Research Plan in Shenzhen City (JCYJ20120613135212389 and ZYC201105170376A), by the Fundamental Research Funds for the Central Universities (HIT.NSRIF.2011129), by the Guangdong Innovative Research Team Program (201001D0104648280), by the Guangzhou Nansha Research Funds (201201015), and by the Guangdong Foshan Shunde Research Funds (2012YSQ7).