Robust Estimation for Discrete Markov System with Time-Varying Delay and Missing Measurements

Abstract

This paper addresses the ℋ_∞ filtering problem for time-delayed Markov jump systems (MJSs) with intermittent measurements. Within network environment, missing measurements are taken into account, since the communication channel is supposed to be imperfect. A Bernoulli process is utilized to describe the phenomenon of the missing measurements. The original system is transformed into an input-output form consisting of two interconnected subsystems. Based on scaled small gain (SSG) theorem and proposed Lyapunov-Krasovskii functional (LKF), the scaled small gains of the subsystems are analyzed, respectively. New conditions for the existence of the ℋ_∞ filters are established, and the corresponding ℋ_∞ filter design scheme is proposed. Finally, a simulation example is provided to demonstrate the effectiveness of the proposed approach.

1. Introduction

In recent years, networked control systems (NCSs) have found a great deal of applications in a range of engineering areas [1–6]. Compared to traditional control systems, the NCSs hold many magnificent advantages, such as reduced power and weight requirements, low cost, improved flexibility and suppleness, high reliability, and facilitated system diagnosis and maintenance. Consequently, NCSs have drawn extensive investigation, and a great deal of significant results have been reported [7–12].

On the other hand, some unexpected phenomena appear in the networked environment, since the digital communication channels have been introduced, on which all measured signals are transmitted with limited capacity. Intermittent measurements and time-delay are widely emerged in many kinds of NCSs, which could always lead to poor performance even instability of the systems. Some research issues in networked environment, such as filtering problems have become much more complicated since most traditional approaches are proposed on the assumption that the communication channel between the physical plant and filter is perfect. In this sense, a great attention has been attracted to the filtering problems with limited capacity and led to many important results. To mention them, [13] addresses the filtering problem with communication delay, [14] investigates the 𝐻_∞ filtering problem for nonlinear stochastic systems, and robust filtering problems subject to stochastic missing measurements which described by different methods are introduced in [15–17].

Based on the direct Lyapunov method, a great deal of stability criteria have been reported [9, 18]. Meanwhile, an input-output approach has been introduced to analyze the stability of the systems by transforming the original system into a form of interconnection of two subsystems via employing a two-term approximation for the delayed variables [19]. Recently, Observing that the resulting performance largely depends on the error between the delayed variable and the approximation, an approach for estimating the delay by the upper and the lower delay bounds was introduced [20], of which the approximation error is much smaller than the error obtained by other one-term approaches [21, 22]. However, to the best of the authors’ knowledge, corresponding results for time-delay MJSs with limited capacity channel have been relatively few, which motivates the research in this paper.

In this paper, we investigate the ℋ_∞ filtering problem for time-delayed MJSs with intermittent measurements. The communication channel between the plant and the filter is supposed to be imperfect. And the phenomenon of missing measurements is modeled by employing a stochastic variable satisfying Bernoulli random binary distribution. By using a new two-term approximation to the delay variables, the system is transformed into an interconnected systems comprised of two subsystems. Based on the scaled small gain (SSG) theorem and proposed Lyapunov-Krasovskii functional (LKF), sufficient conditions for the existence of the ℋ_∞ filters and the corresponding ℋ_∞ filter design method are proposed. Finally, a numerical example is provided to show effectiveness of the approach.

The remainder of this paper is organized as follows. In Section 2, the problem of ℋ_∞ filtering is formulated for time-delayed MJSs with intermittent measurements; Section 3 presents the results for ℋ_∞ performance analysis, and ℋ_∞ filter design problem is solved in Section 4; a numerical example is provided to demonstrate the effectiveness of the design filtering technique in Section 5, and we conclude the paper with Section 6.

Notation. Throughout this paper, ℝ^𝑛 represents the 𝑛-dimensional Euclidean space, ℝ^𝑛×𝑚 is the set of all 𝑛×𝑚 real matrices, the superscripts “−1” and “𝑇,” respectively, stand for the matrix inverse and matrix transpose. Sym{𝐴} is the shorten notation for 𝐴+𝐴^𝑇, and the notation 𝑃>0 (resp., 𝑃≥0), for 𝑃∈ℝ^𝑛×𝑛 means that 𝑃 is real symmetric and positive definite (resp., semidefinite). The symmetric elements of the symmetric matrix is represented by an asterisk (∗), and the block-diagonal matrices are denoted by diag{⋯}. 𝐆₁∘𝐆₂ means the series connection of mapping 𝐆₁ and 𝐆₂. 𝐸{⋅} denotes the expectation operator with respect to probability measure, and for vector 𝑥(𝑘), ‖𝑥‖_𝐸2=𝐸{∑^∞_𝑛=0‖𝑥(𝑛)‖²}^1/2.

2. Problem Formulation

Consider the following Markov jump system with the time-varying delay: 𝑥_𝑘+1=𝐴𝑟_𝑘𝑥_𝑘+𝐴_𝑑𝑟_𝑘𝑥_{𝑘−𝜏𝑘}+𝐵𝑟_𝑘𝜔_𝑘,𝑦_𝑘=𝐶𝑟_𝑘𝑥_𝑘+𝐶_𝑑𝑟_𝑘𝑥_{𝑘−𝜏𝑘}+𝐷𝑟_𝑘𝜔_𝑘,𝑧_𝑘=𝐿𝑟_𝑘𝑥_𝑘,𝑥_𝑘=𝜓_𝑘,𝑘=−𝑑₂,…,−1,0, (1) where 𝑥_𝑘∈ℝ^𝑛 denotes system state; 𝑦_𝑘∈ℝ^𝑚 is the measured output; 𝜔_𝑘∈ℝ^𝑞 is the measurement noise which belongs 𝐿₂[0,∞); 𝑧_𝑘∈ℝ^𝑝 is the signal to be estimated; and 𝜓_𝑘 is a given real-valued initial condition. Let {𝑟_𝑘,𝑘∈ℤ} be a Markov chain that takes values in a given finite set 𝒮={1,2,…,𝑠} with transition probability 𝑝_𝑖𝑗=Pr{𝑟_𝑘+1=𝑗∣𝑟_𝑘=𝑖}, where 𝑝_𝑖𝑗≥0 for 𝑖,𝑗∈𝒮 and ∑^𝑠_𝑗=1𝑝_𝑖𝑗=1. And 𝜏_𝑘 is a positive integer, denoting the time-varying delay satisfying 𝑑₁≤𝜏(𝑘)≤𝑑₂, (2) where 𝑑₁≥0 and 𝑑₂>0 are the minimum and the maximum delay, respectively. The delay interval is defined as 𝑑_𝑡=𝑑₂−𝑑₁.

Then, we consider the following mode-independent filter with full order: 𝑥ℱ∶^𝑘+1=𝐴_𝐹𝑥_𝑘+𝐵_𝐹𝑦_𝐹(𝑘),𝑧_𝑘=𝐿_𝐹𝑥_𝑘, (3) where 𝑥_𝑘∈ℝ^𝑛 is the filter state vector and 𝑧_𝑘∈ℝ^𝑝 is the output of the filter; 𝐴_𝐹∈ℝ^𝑛×𝑛, 𝐵_𝐹∈ℝ^𝑛×𝑚, and 𝐶_𝐹∈ℝ^𝑝×𝑛 are the filter gains to be determined.

In the network the data transmitted in the digital channel may be lost at a certain probability as shown in Figure 1. As a result, the output of the physical plant 𝑦_𝑘 is no more equivalent to the input of the filter, which is denoted by 𝑦_𝐹(𝑘). In this paper, this phenomenon is modeled via a Bernoulli process: 𝑦_𝐹(𝑘)=𝛼_𝑘𝑦_𝑘, (4) where 𝛼_𝑘 is Bernoulli process, which models the intermittent transmission between the plant and the filter. When 𝛼(𝑘)=1, the data is transmitted to the filter successfully. While when 𝛼(𝑘)=0, the transmission fails and the data is lost. It can be described as: Prob𝛼_𝑘=1=𝐸𝛼_𝑘=𝛼,Prob𝛼_𝑘=0=1−𝛼. (5)

Figure 1 

Filtering problem with intermittent measurements.

Combining (3) and (5), we have 𝑥ℱ∶^𝑘+1=𝐴_𝐹𝑥_𝑘+𝐵_𝐹𝛼_𝑘𝑦_𝑘,𝑧_𝑘=𝐿_𝐹𝑥_𝑘. (6) Defining 𝛼=𝛼_𝑘−𝛼 yields 𝐸{𝛼}=0 and 𝐸{(𝛼)²}=𝛼(1−𝛼).

Then, denote 𝜉_𝑘=𝑥^𝑇_𝑘𝑥^𝑇_𝑘^𝑇,𝑒_𝑘=𝑧_𝑘−𝑧_𝑘. (7) By connecting (1) with (3), the filtering error dynamic can be described by 𝜉𝒮∶^𝑘+1=𝐴(𝑖)𝜉_𝑘+𝐴_𝑑(𝑖)𝐾𝜉_{𝑘−𝜏𝑘}+𝐵(𝑖)𝜔_𝑘,𝑒_𝑘=𝐿(𝑖)𝜉_𝑘, (8) where 𝐴(𝑖)=𝐴₁(𝑖)+𝛼𝐴₂(𝑖),𝐴_𝑑(𝑖)=𝐴_𝑑1(𝑖)+𝛼𝐴_𝑑2(𝑖),𝐵(𝑖)=𝐵₁(𝑖)+𝛼𝐵₂(𝑖),𝐾=𝐼0,𝐴₁(𝑖)=𝐴(𝑖)0𝛼𝐵_𝐹𝐶(𝑖)𝐴_𝐹,𝐴₂(𝑖)=𝐵00_𝐹𝐶(𝑖)0,𝐴_𝑑1(𝑖)=𝐴_𝑑(𝑖)𝛼𝐵_𝐹𝐶_𝑑(𝑖),𝐴_𝑑2(𝑖)=0𝐵_𝐹𝐶_𝑑(𝑖),𝐵₁(𝑖)=𝐵(𝑖)𝛼𝐵_𝐹𝐷(𝑖),𝐵₂(𝑖)=0𝐵_𝐹𝐷(𝑖),𝐿(𝑖)=𝐿(𝑖)−𝐿_𝐹. (9)

Moreover, we introduce the definition of stochastic stability which is vital and important in this paper.

Definition 1 (see [23]). For the system in (1) with 𝜔_𝑘=0, the system is stochastically stable if for any 𝜓_𝑘∈ℝ^𝑛, 𝐸^∞_𝑘=0𝑥^𝑇_𝑘𝑥_𝑘∣𝜓(0)<∞. (10)

Definition 2 (see [23]). A mapping 𝐆∶𝑢_𝑘→𝑦_𝑘 is input-output stable in mean-square if there exists 𝛾≥0 such that 𝑦_𝑘𝐸2=‖𝐆(𝑢)‖_𝐸2≤𝛾𝑢_𝑘𝐸2. (11)

Then, the ℋ_∞ filtering design for MJSs with intermittent measurements to be addressed in this paper can be formulated as follows.

Consider system (1) with intermittent measurements modeled by (5). Given positive integers 𝑑₂>𝑑₁ and a prescribed scalar 𝛾>0, the filter gains 𝐴_𝐹, 𝐵_𝐹, and 𝐿_𝐹 of filter (3) are designed such that for any time-delay 𝜏_𝑘 satisfying (2), the filter error system (8) is stochastically stable, and ‖𝑒‖_𝐸2≤𝛾‖𝜔‖_𝐸2 holds under zero initial conditions.

3. Main Results

In this section, by developing SSG Theorem to stochastic systems and a new model transformation to system (8), we will focus on the filter performance analysis. Sufficient conditions are presented, under which system ℰ is stochastically stable with a prescribed ℋ_∞ performance index.

3.1. Filtering Performance Analysis

Lemma 3 (see [24]). One considers the following system consisting of two subsystems as shown in Figure 2: 𝒴₁∶𝜁_𝑘=𝐆𝑢_𝑘−𝛿_𝑘,𝒴₂∶𝛿_𝑘=Δ𝜁_𝑘. (12)
The closed loop system is input-output stable in mean-square for all the subsystems 𝐆 and Δ, if ‖𝑇_𝜁∘𝐆∘𝑇_𝛿‖<1 holds for some nonsingular matrices 𝑇_𝜁 and 𝑇_𝛿 with ‖𝑇_𝛿∘Δ∘𝑇_𝜁‖≤1.

Figure 2 

The interconnection subsystems.

Proof. According to (12) and the condition in Lemma 3, by defining Δ𝑢_𝑘=𝑢_𝑘−𝛿_𝑘, and without loss of generality, it can be assumed that 𝐆(Δ𝑢_𝑘)_𝐸2≤𝑘₁Δ𝑢_𝑘𝐸2,Δ(𝜁_𝑘)_𝐸2≤𝑘₂𝜁_𝑘𝐸2,0<𝑘₁<1,0<𝑘₂≤1. (13)
Obviously, we have 𝑇𝑦_𝑘𝐸2≤𝑘₁𝑇Δ𝑢_𝑘𝐸2=𝑘₁𝑇(𝑢_𝑘−Δ(𝑦_𝑘))_𝐸2≤𝑘₁𝑇𝑢_𝑘𝐸2+𝑘₂𝑇𝑦_𝑘𝐸2,𝑇𝑦_𝑘𝐸2≤𝑘₁1−𝑘₁𝑘₂𝑇𝑢_𝑘𝐸2. (14)
Since 𝑘₁𝑘₂<1, system (12) is input-output stable in mean-square, which completes the proof.

Then, to separate the uncertainties 𝜏_𝑘 from the filtering error system, the time-delay system 𝒮 in (8) is transformed into the interconnection of two subsystems introduced in Lemma 3.

Notice that the time-varying delay can be written as 𝑥_{𝑘−𝜏𝑘}=12𝑥_𝑘−𝑑1+12𝑥_𝑘−𝑑2+𝑑₁₂2𝛿_𝑘, (15) where (1/2)(𝑥_𝑘−𝑑1+𝑥_𝑘−𝑑2) is the approximation of the time-varying delay 𝑥_{𝑘−𝜏𝑘}, and (𝑑₁₂/2)𝛿_𝑘 is the approximation error. Then, we can describe system 𝒮 as 𝜉𝒮∶^𝑘+1=𝐴(𝑖)𝜉_𝑘+12𝐴_𝑑(𝑖)𝐾𝜉_𝑘−𝑑1+12𝐴_𝑑(𝑖)𝐾𝜉_𝑘−𝑑2+𝑑₁₂2𝐴_𝑑(𝑖)𝛿_𝑘+𝐵(𝑖)𝜔_𝑘,𝑒_𝑘=𝐿(𝑖)𝜉_𝑘. (16)

Defining 𝜁_𝑘=𝑥_𝑘+1−𝑥_𝑘 and 𝛽(𝑖)≜1,when𝑖⩽𝑘−𝜏_𝑘−1,−1,when𝑖>𝑘−𝜏_𝑘−1, (17) we can obtain 𝛿_𝑘=2𝑑₁₂𝑥_{𝑘−𝜏𝑘}−12𝑥_𝑘−𝑑1−12𝑥_𝑘−𝑑2=1𝑑₁₂^{𝑘−𝜏𝑘−1}_{𝑖=𝑘−𝑑2}𝜁_𝑖−^{𝑘−𝑑1−1}_{𝑖=𝑘−𝜏𝑘}𝜁_𝑖=1𝑑₁₂^{𝑘−𝑑1−1}_{𝑖=𝑘−𝑑2}𝛽(𝑖)𝜁_𝑖. (18)

From (18), we know that mapping Δ∶𝜁_𝑘→𝛿_𝑘 includes the uncertainty in 𝜏_𝑘 leaving only two constant delays 𝑑₁ and 𝑑₂ in (16). Then the system in (16) can be described as 𝒮₁∶𝜉_𝑘+1𝜁_𝑘𝑒_𝑘=Φ₁𝑑₁₂2𝐴_𝑑(𝑖)Φ𝐵(𝑖)₂𝑑₁₂2𝐴_𝑑(𝑖)Φ𝐵(𝑖)₃00_𝐆𝜙_𝑘𝛿_𝑘𝜔_𝑘,𝒮₂∶𝛿_𝑘=Δ𝜁_𝑘, (19) where 𝜙_𝑘=𝜉^𝑇_𝑘𝜉^𝑇_𝑘−𝑑1𝐾^𝑇𝜉^𝑇_𝑘−𝑑2𝐾^𝑇𝑇,Φ₁=𝐴(𝑖)12𝐴_𝑑(𝑖)12𝐴_𝑑(𝑖),Φ₂=(𝐴(𝑖))𝐾1−𝐼2𝐴_𝑑(𝑖)12𝐴_𝑑(𝑖),Φ₃=𝐿00. (20)

Furthermore, the following lemma shows that the scaled small gain of the mapping Δ has an upper bound.

Lemma 4. 𝛾(Δ_𝑇) has an upper bound: 𝛾Δ_𝑇=sup𝑇𝛿_𝑘𝐸2𝑇𝜁_𝑘𝐸2≤1. (21)

Proof. By using Jensen's inequality, it can be derived under zero initial condition according to the fact that |𝛽(𝑖)|=1, 𝑇𝛿_𝑘²_𝐸2=𝐸^∞_𝑗=0𝛿^𝑇_𝑗𝑍𝛿_𝑗≤2𝑑²₁₂𝐸^∞_𝑖=0𝑑₁₂^{𝑖−𝑑1−1}_{𝑗=𝑖−𝑑2}𝛽²𝑗𝛿^𝑇_𝑗𝛿_𝑗=1𝑑₁₂𝐸^−𝑑1−1_{𝑗=−𝑑2}^∞_𝑖=0𝛿^𝑇_𝑖+𝑗𝛿_𝑖+𝑗≤1𝑑₁₂𝐸^−𝑑1−1_{𝑗=−𝑑2}^∞_𝑖=0𝛿^𝑇_𝑖𝛿_𝑖≤𝐸^∞_𝑗=0𝜁^𝑇_𝑗𝑍𝜁_𝑗=𝑇𝜁_𝑘²_𝐸2, (22) where 𝑍=𝑇^𝑇𝑇, which implies 𝛾(Δ_𝑇)≤1. The proof is completed.

Now, we are in the position to propose the sufficient conditions under which the filtering error system (8) is stochastically stable with a prescribed ℋ_∞ performance attention level 𝛾. First we assume the filter matrices 𝐴_𝐹, 𝐵_𝐹, and 𝐿_𝐹 are fixed and the corresponding conditions are established in the following theorem.

Theorem 5. Consider system (1), and assume the filter matrices 𝐴_𝐹, 𝐵_𝐹, and 𝐿_𝐹 are fixed. Given integers 𝑑₂>𝑑₁≥0 and a constant 𝛾>0, the filtering error system (16) is input-output stable in mean-square with an ℋ_∞ performance attention level 𝛾 for any time-varying delay satisfying (2) if there exist matrices 𝑃>0, 𝑄_𝑖>0(𝑖=1,2), 𝑅_𝑖>0(𝑖=1,2), 𝑍>0 satisfying ΞΨ≜¹𝑑₁Γ^𝑇₃𝑅₁𝑑₂Γ^𝑇₃𝑅₂Γ^𝑇₃𝑍Γ^𝑇₄Γ^𝑇₁𝑃𝑓Γ^𝑇₂𝑃∗diag−𝑅₁,−𝑅₂,−𝑍,−𝐼_𝑝,−𝑃,−𝑃<0, (23) where Ξ₁=Ξ₁₁𝐾^𝑇𝑅₁𝐾^𝑇𝑅₂Ξ00∗diag₂,Ξ₃,−𝑍,−𝛾²𝐼_𝑞,Γ₁=𝐴₁(𝑖)12𝐴_𝑑1(𝑖)12𝐴_𝑑1(𝑖)12𝑑₁₂𝐴_𝑑1(𝑖)𝐵₁(𝑖),Γ₂=𝐴₂(𝑖)12𝐴_𝑑2(𝑖)12𝐴_𝑑2(𝑖)12𝑑₁₂𝐴_𝑑2(𝑖)𝐵₂(𝑖),Γ₃=(𝐴(𝑖))𝐾1−𝐼2𝐴_𝑑(𝑖)12𝐴_𝑑(𝑖)12𝑑₁₂𝐴_𝑑(𝑖)𝐵(𝑖),Γ₄=𝐿(𝑖)0_{𝑝×(3𝑛+𝑞)},𝑓=𝜂1−𝜂,Ξ₁₁=−𝑃(𝑖)+𝐾^𝑇𝑄₁+𝑄₂−𝑅₁−𝑅₂Ξ𝐾,₂=−𝑄₁−𝑅₁,Ξ₃=−𝑄₂−𝑅₂. (24)

Proof. Choose the following Lyapunov-Krasovskii functional: 𝑉(𝑘)=𝑉₁(𝑘)+𝑉₂(𝑘)+𝑉₃(𝑘), (25) with 𝑉₁(𝑘)=𝜉^𝑇_𝑘𝑃(𝑖)𝜉_𝑘,𝑉₂(𝑘)=^𝑘−1_{𝑖=𝑘−𝑑1}𝜉^𝑇_𝑖𝐾^𝑇𝑄₁𝐾𝜉_𝑖+^𝑘−1_{𝑖=𝑘−𝑑2}𝜉^𝑇_𝑖𝐾^𝑇𝑄₂𝐾𝜉_𝑖,𝑉₃(𝑘)=𝑑₁⁻¹_{𝑖=−𝑑1}^𝑘−1_{𝑗=𝑘+𝑖}𝜁^𝑇_𝑗𝑅₁𝜁_𝑗+𝑑₂⁻¹_{𝑖=−𝑑2}^𝑘−1_{𝑗=𝑘+𝑖}𝜁^𝑇_𝑗𝑅₂𝜁_𝑗, (26) where 𝑃>0 and 𝑄_𝑖>0 are the matrices to be determined.
Defining Δ𝑉(𝑘)=𝐸{𝑉(𝑘+1)}−𝑉(𝑘), we have Δ𝑉₁(𝑘)=𝐸𝜉^𝑇_𝑘+1𝑃𝑖⁺𝜉_𝑘+1−𝜉_𝑘𝑃(𝑖)𝜉_𝑘=𝜑^𝑇_𝑘Γ^𝑇₁𝑃𝑖⁺Γ₁+𝑓²Γ^𝑇₂𝑃𝑖⁺Γ₂𝜑_𝑘−𝜉_𝑘𝑃(𝑖)𝜉_𝑘,Δ𝑉₂(𝑘)=𝜉^𝑇_𝑘𝐾^𝑇𝑄₁𝐾𝜉_𝑘−𝜉^𝑇_𝑘−𝑑1𝐾^𝑇𝑄₁𝐾𝜉_𝑘−𝑑1+𝜉^𝑇_𝑘𝐾^𝑇𝑄₂𝐾𝜉_𝑘−𝜉^𝑇_𝑘−𝑑2𝐾^𝑇𝑄₂𝐾𝜉_𝑘−𝑑2,Δ𝑉₃(𝑘)=𝑑²₁𝜁^𝑇_𝑘𝑅₁𝜁_𝑘+𝑑²₂𝜁^𝑇_𝑘𝑅₂𝜁_𝑘−𝑑₁^𝑘−1_{𝑗=𝑘−𝑑1}𝜁^𝑇_𝑗𝑅₁𝜁_𝑗−𝑑₂^𝑘−1_{𝑗=𝑘−𝑑2}𝜁^𝑇_𝑗𝑅₂𝜁_𝑗, (27) where 𝜑_𝑘=𝜉^𝑇_𝑘𝜉^𝑇_𝑘−𝑑1𝐾^𝑇𝜉^𝑇_𝑘−𝑑2𝐾^𝑇𝛿^𝑇_𝑘𝜔^𝑇_𝑘^𝑇.
Moreover, define 𝐽_𝑠=𝐸^∞_𝑘=0𝑦^𝑇_𝑘𝑍𝑦_𝑘−𝛿^𝑇_𝑘𝑍𝛿_𝑘+𝑒^𝑇_𝑘𝑒_𝑘−𝛾²𝜔^𝑇_𝑘𝜔_𝑘. (28)
Together with (25), and according to 𝑉(0)=0 and 𝑉(∞)≥0, we have 𝐽_𝑠≤𝐽_𝑠+𝑉(∞)−𝑉(0)=𝐸^∞_𝑘=0Δ𝑉(𝑘)+𝜁^𝑇_𝑘𝑍𝜁_𝑘−𝛿^𝑇_𝑘𝑍𝛿_𝑘+𝑒^𝑇_𝑘𝑒_𝑘−𝛾²𝜔^𝑇_𝑘𝜔_𝑘=𝐸^∞_𝑘=0𝜑^𝑇_𝑘Ξ₁+Γ^𝑇₃𝑑²₁𝑅₁+𝑑²₂𝑅₂+𝑍Γ₃+Γ^𝑇₄Γ₄+Γ^𝑇₁𝑃𝑖⁺Γ₁+𝑓Γ^𝑇₂𝑃𝑖⁺Γ₂𝜑_𝑘, (29)
It is obvious that Δ𝑉(𝑘)+𝜁^𝑇_𝑘𝑍𝜁_𝑘−𝛿^𝑇_𝑘𝑍𝛿_𝑘+𝑒^𝑇_𝑘𝑒_𝑘−𝛾²𝜔^𝑇_𝑘𝜔_𝑘<0, if Ξ₁+Γ^𝑇₃𝑑²₁𝑅₁+𝑑²₂𝑅₂+𝑍Γ₃+Γ^𝑇₄Γ₄+Γ^𝑇₁𝑃𝑖⁺Γ₁+𝑓Γ^𝑇₂𝑃𝑖⁺Γ₂<0, (30) which, by Schur complement, is equivalent to (23). Therefore, defining 𝑍=𝑇^𝑇𝑇, we have 𝐽_𝑠<0, if (23) holds, which means 𝑇𝜁_𝑘²_𝐸2+𝑒_𝑘²_𝐸2≤𝑇𝛿_𝑘²_𝐸2+𝛾²𝜔_𝑘²_𝐸2, (31) if Theorem 5 holds, which means the SSG of subsystem 𝒮₁ satisfying 𝛾(𝐆_𝑇)<1. By Lemmas 3 and 4, the filtering error system (16) is input-output stable in mean-square. Furthermore, we can obtain ‖𝑒_𝑘‖²_𝐸2≤𝛾²‖𝜔_𝑘‖²_𝐸2 from (31) together with (22). The proof is completed.

Theorem 5 guarantees the input-output stability in mean-square of system in (19). Through the following research, it can be proved that the system in (19) is stochastically stable by Lyapunov-Krasovskii functional method.

Theorem 6. For the forward system 𝒮₁ in (19) with 𝜔_𝑘=0, if there exists an LKF 𝑉(𝑘) and 𝑍>0 satisfying Δ𝑉(𝑘)+𝜁^𝑇_𝑘𝑍𝜁_𝑘−𝛿^𝑇_𝑘𝑍𝛿_𝑘<0, then the system in (19) is stochastically stable by the following LKF: 𝑉_𝑐(𝑘)(𝑘)+1=𝑉𝑑₁₂^−𝑑1−1_{𝑖=−𝑑2}^𝑘−1_{𝑗=𝑘+𝑖}𝜁^𝑇_𝑘𝑍𝜁_𝑘. (32)

Proof. Defining Δ𝑉_𝑐(𝑘)=𝑉_𝑐(𝑘+1)−𝑉_𝑐(𝑘); we have Δ𝑉_𝑐(𝑘)(𝑘)+1=Δ𝑉𝑑₁₂^−𝑑1−1_{𝑖=−𝑑2}𝜁^𝑇_𝑘𝑍𝜁_𝑘−𝜁^𝑇_𝑘+𝑖𝑍𝜁_𝑘+𝑖=Δ𝑉(𝑘)+𝜁^𝑇_𝑘𝑍𝜁_𝑘−1𝑑₁₂^{𝑘−𝑑1−1}_{𝑖=𝑘−𝑑2}𝛽(𝑖)𝜁^𝑇_𝑖𝑍𝛽(𝑖)𝜁_𝑖,≤Δ𝑉(𝑘)+𝜁^𝑇_𝑘𝑍𝜁_𝑘−1𝑑²₁₂^{𝑘−𝑑1−1}_{𝑖=𝑘−𝑑2}𝛽(𝑖)𝜁_𝑖^𝑇𝑍^{𝑘−𝑑1−1}_{𝑖=𝑘−𝑑2}𝛽(𝑖)𝜁_𝑖=Δ𝑉(𝑘)+𝜁^𝑇_𝑘𝑍𝜁_𝑘−𝛿^𝑇_𝑘𝑍𝛿_𝑘<0, (33) which is implied by (23). Let 𝜆>0, we have Δ𝑉_𝑐(𝑘)=𝐸𝑉_𝑐(𝑘+1)−𝑉_𝑐(𝑘)≤−𝜃𝑥^𝑇_𝑘𝑥_𝑘<0. (34)
By summing up the inequality on both sides from 𝑘=0,…,𝑅, it can be obtained that 𝐸𝑉_𝑐(𝑅)−𝑉_𝑐(0)≤−𝜃^𝑅_𝑘=0𝑥^𝑇_𝑘𝑥_𝑘<0. (35)
When 𝑅→∞, we have 𝜃𝑥^𝑇_𝑘𝑥_𝑘²_𝐸2≤𝑉_𝑐(0)−𝐸𝑉_𝑐(∞)≤𝑉_𝑐(0)<∞, (36) which implies that the systems in (19) are stochastically stable. The proof is completed.

4. ℋ_∞ Filter Design

Theorem 7. Consider the Markov jump system (1). For given integers 𝑑₂>𝑑₁≥0 and a constant 𝛾>0, if there exist matrices 𝑅_𝑖, 𝑆, 𝑄₁>0, 𝑄₂>0, 𝑅₁>0, 𝑅₂>0, 𝑍>0, 𝐴_𝐹, 𝐵_𝐹, and 𝐿_𝐹 such that ΞΠ≜¹𝑑²₁Γ^𝑇₃𝑅₁𝑑²₂Γ^𝑇₃𝑅₂Γ^𝑇₃𝑍Γ^𝑇₄Γ^𝑇₁𝑓Γ^𝑇₂∗diag−𝑅₁,−𝑅₂,−𝑍,−𝐼_𝑝,Θ,Θ<0,𝑆−𝑅_𝑖<0, (37) where Ξ₁=Ξ₁₁𝐾^𝑇𝑅₁𝐾^𝑇𝑅₂00∗diag−𝑄₁−𝑅₁,−𝑄₂−𝑅₂,−𝑍,−𝛾²𝐼_𝑞,Γ₁=Γ^𝑇_𝐴𝑅+𝜂Γ^𝑇_𝐹1+Γ^𝑇_𝐹2Γ^𝑇_𝐴𝑆+𝜂Γ^𝑇_𝐹1+Γ^𝑇_𝐹2^𝑇,Γ₃=𝐴(𝑖)012𝐴_𝑑(𝑖)12𝐴_𝑑(𝑖)12𝑑₁₂𝐴_𝑑(𝑖)𝐵(𝑖),Γ₂=Γ^𝑇_𝐹1Γ^𝑇_𝐹1^𝑇,Γ₄=𝐿(𝑖)−𝐿_𝐹0_{𝑝×(3𝑛+𝑞)},Γ_𝐴=𝐴(𝑖)012𝐴_𝑑(𝑖)12𝐴_𝑑(𝑖)𝑑₁₂2𝐴_𝑑(𝑖)𝐵(𝑖),Γ_𝐹1=𝐵_𝐹𝐶(𝑖)012𝐵_𝐹𝐶_𝑑(𝑖)12𝐵_𝐹𝐶_𝑑(𝑖)𝑑₁₂2𝐵_𝐹𝐶_𝑑(𝑖)𝐵_𝐹𝐷(𝑖),Γ_𝐹2=0_𝑛×𝑛𝐴_𝐹0_{𝑛×(3𝑛+𝑞)},Ξ₁₁=−𝑃(𝑖)+𝐾^𝑇𝑄₁+𝑄₂−𝑅₁−𝑅₂,𝐾,Θ=𝑅−𝑆−𝑆−𝑆𝑅=−^𝑠_𝑖=1𝑝_𝑖𝑗𝑅_𝑖, (38) then, there exists an ℋ_∞ filter of the type (3) such that the filtering error system in (8) is stochastically stable with a prescribed ℋ_∞ performance level 𝛾. Furthermore, the filter gains are given by 𝐴_𝐹𝐵_𝐹𝐿_𝐹0=𝑆⁻¹00𝐼𝐴_𝐹𝐵_𝐹𝐿_𝐹0. (39)

Proof. Suppose there exist matrices 𝑃(𝑖)>0, 𝑄_𝑖>0, 𝑅_𝑖>0, (𝑖=1,2), 𝑍>0, 𝐴_𝐹, 𝐵_𝐹, and 𝐿_𝐹 satisfying (23). Partition 𝑃 as 𝑃(𝑖)≜𝑃_1𝑖𝑃₂𝑃^𝑇₂𝑃₃, (40) and define matrices 𝐽 and 𝐽₁ as follows: 𝐽≜𝐼00𝑃₃⁻¹𝑃^𝑇₂,𝐽₁≜diag{𝐽,𝐼,𝐼,𝐼,𝐼,𝐼,𝐼,𝐽,𝐽}. (41)
According to the construction of 𝐽 and 𝐽₁ and the fact that 𝑃₂ and 𝑃₃ are nonsingular even disturbed by a small perturbation, it is obvious that matrix 𝐽 is invertible. Let 𝑅_𝑖≜𝑃_1𝑖,𝑆≜𝑃₂𝑃₃⁻¹𝑃^𝑇₂. (42)
Then, multiply (23) by 𝐽^𝑇₁ from the left side and its transpose from the right side, respectively, and define 𝐴_𝐹𝐵_𝐹𝐿_𝐹0≜𝑃₂0𝐴0𝐼^𝐹𝐵_𝐹𝐿_𝐹0𝑃₃⁻¹𝑃^𝑇₂00𝐼, (43) we obtain (37).
On the other hand, suppose that matrices 𝑅_𝑖>0, 𝑆>0, 𝑄₁>0, 𝑄₂>0, 𝑅₁>0, 𝑅₂>0, 𝑍>0, 𝐴_𝐹, 𝐵_𝐹, and 𝐿_𝐹 satisfy (37). Introduce 𝑃_1𝑖≜𝑅_𝑖 and nonsingular square matrices 𝑃₂ and 𝑃₃ satisfying 𝑆=𝑃₂𝑃₃⁻¹𝑃^𝑇₂. By Schur complement, 𝑃(𝑖)>0 is equivalent to 𝑅_𝑖>0 and 𝑅_𝑖−𝑆=𝑃_1𝑖−𝑃₂𝑃₃⁻¹𝑃^𝑇₂>0. Then, without loss of generality, setting 𝑃₂^−𝑇𝑃₃=𝐼 yields (39). The proof is completed.

5. Numerical Example

In this section, we provide an example to demonstrate the effectiveness of the filter design method proposed in the preceding sections.

Consider system (1) with the following parameters: 𝐴₁=,𝐴−0.100.01−0.59−0.10.020.1−0.06−0.1_𝑑1=,𝐵0.010.0200.0100.0100−0.01₁=−0.2020.3830.139^𝑇,𝐶₁=,𝐶−0.02−0.01−0.020.050.020.021_𝑑1=,𝐿−0.010−0.020.010.020₁=0.20.50.1,𝐷₁=0.10.47^𝑇,𝐴₂=,𝐴−0.10.10.04−0.51−0.210.010.1−0.04−0.1_𝑑2=,𝐵0.010.0200.0100.01000.01₂=−0.1340.2480.117^𝑇,𝐶₂=,𝐶0.01−0.040.0150.0310.0420.071_𝑑2=,𝐿0.010−0.010.020−0.01₂=0.10.30.2,𝐷₂=0.230.15^𝑇, (44) and the mode switching is governed by a Markov chain with transition probability: 𝑝₁₁=0.8,𝑝₁₂𝑝=0.2;₂₁=0.7,𝑝₂₂=0.3. (45)

Suppose 𝑑₁=0 and 𝑑₂=1 and the initial condition 𝑥(0)=000^𝑇, 𝑥_𝐹(0)=00^𝑇. Then, by Theorem 7, a filter in the form of (3) is designed such that the system (8) is stochastically stable with a guaranteed ℋ_∞ performance level 𝛾.

Suppose the packet loss parameter is 𝜂=0.8 as shown in Figure 3. By (37), it can be obtained that the minimum ℋ_∞ performance 𝛾_min=0.2801, and ,𝑅𝑆=0.64170.24090.49630.24090.4799−0.21050.4963−0.21050.9024₁=,𝑅3.54911.14361.42311.14361.1211−0.38081.4231−0.38082.9697₂=,𝑄4.48481.23664.01991.23661.18420.08954.01990.08956.1286₁₁=,𝑄0.28610.05670.27080.05670.01320.04260.27080.04260.3200₁₂=,𝑄0.65480.10100.83360.10100.01910.11310.83360.11311.1341₂₁=,𝑄0.31410.06770.28250.06770.01640.05130.28250.05130.3075₂₂=,𝑅0.61090.09340.77520.09340.01740.10470.77520.10471.0582₁₁=,𝑅0.36300.05160.21460.05160.01160.00780.21460.00780.2992₁₂=,𝑅0.60510.08880.81730.08880.01960.10330.81730.10331.1513₂₁=,𝑅0.13210.01830.07250.01830.00350.00490.07250.00490.0786₂₂=,,0.12880.02110.16790.02110.00580.02140.16790.02140.2356𝑍=0.16420.02330.22290.02330.00850.01930.22290.01930.3438 (46) and the filter matrices are as follows: 𝐴_𝐹=,−0.2262−0.0798−0.0365−0.06780.0340−0.0505−0.1747−0.13940.0068𝐵_𝐹=,0.0390−0.10990.2651−0.0906−0.1568−0.0028𝐿_𝐹=−0.1391−0.48500.3233. (47)

Figure 3 

Packet loss parameter 𝜂(𝑘).

In addition, by (39), we have 𝐴_𝐹=,𝐵−0.5488−0.05920.00270.20240.0525−0.11570.1555−0.1096−0.0210_𝐹=,−0.8717−0.44681.25240.15810.59790.2796𝐿_𝐹=−0.1391−0.48500.3233. (48)

To illustrate the performance of the filter, it is assumed that the external disturbance 𝜔_𝑘=2/(𝑘+10). By calculation, the switching of Markov chain is shown in Figure 4. Figure 5 shows the system state 𝑥_𝑘 and its estimation 𝑥_𝑘 and signal 𝑧_𝑘 and its estimation 𝑧_𝑘. Moreover, it can be obtained that the ‖𝑒(𝑘)‖_𝐸2=0.0166 and ‖𝜔(𝑘)‖_𝐸2=0.3754 as shown in Figure 5, that yields 𝛾=0.0626 which is below the minimum 𝛾_min=1.7051, demonstrating the effectiveness of the proposed filter design.

Figure 4 

Switching states of Markov chain.

(a) 𝑥1 and estimation 𝑥1

(b) 𝑥2 and estimation 𝑥2

(c) 𝑥3 and estimation 𝑥3

(d) signal 𝑧1−𝑧1

Figure 5 

Original signal and estimation signal.

6. Conclusion

This paper addresses the ℋ_∞ filtering problem for time-delay Markov jump systems subject to intermittent measurements. The communication link failure from the plant to the filter is modeled by a stochastic variable satisfying the Bernoulli random binary distribution. The system is transformed into an input-output form consisted of two interconnected subsystems. Based on the SSG Theorem developed to stochastic systems and the proposed Lyapunov-Krasovskii function, sufficient conditions under which the ℋ_∞ filter can be achieved with the prescribed ℋ_∞ performance index is established. Finally, a numerical example is presented to demonstrate the effectiveness of the proposed filter design scheme.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work is supported by National Natural Science Foundation of China (Grant no. 61304102) and Natural Science Foundation of Liaoning Province of China (Grant no. 2013020002).