𝐻_∞ Filter Design for Large-Scale Systems with Missing Measurements

Abstract

This paper is concerned with 𝐻_∞ filter design problem for large-scale systems with missing measurements. The occurrence of missing measurements is assumed to be a Bernoulli distributed sequence with known probability. The new full-dimensional filter is designed to make the filter error system exponentially mean-square stable and achieve a prescribed 𝐻_∞ performance. Sufficient conditions are derived in terms of linear matrix inequality (LMI) for the existence of the filter, and the parameters of filter are obtained by solving the LMI. Finally, the numerical simulation results illustrate the effectiveness of the proposed scheme.

1. Introduction

In many practical applications, due to the limitations imposed by the network, missing measurements often occur due to the network link transmission errors, network congestion, and so forth. Currently, the research of filter and controller design for systems with missing measurements has attracted more attention [1–7]. In [1], the robust control problem with missing measurements was investigated, where the missing measurements were described by a binary switch sequence satisfied conditional probability distribution. The similar model was employed in [2–4], where the filtering problem was investigated in [2, 3], and the distributed state estimation problem was studied in [4]. In [5], the quantized 𝐻_∞ control problem is investigated for a class of nonlinear stochastic time-delay network-based systems with probabilistic data missing. In [6], the filtering problem with packet loss was considered using Markov chains to describe probabilistic losses. The problem of robust 𝐻_∞ filtering for discrete-time switched systems with missing measurements under asynchronous switching is considered in [7].

Most of the existing research is focused on general linear or nonlinear discrete system. However, many actual systems are large-scale systems which are composed of interconnected subsystems. Although ideas of decentralized control of large-scale systems have attracted much attention in the literature during the past two decades, the research about large-scale systems with missing measurements is seldom. In [8], a decentralized 𝐻_∞ controller design for a class of large-scale systems with missing measurements is considered. In [9], a state feedback 𝐻_∞ controller is designed for a class of linear discrete-time large-scale system with both measurement data and control data missing simultaneously.

In this paper, 𝐻_∞ filter is considered for a class of large-scale systems with missing measurements. We apply Bernoulli distributed sequence to describe the occurrence of missing measurements, and the linear discrete-time large-scale system is modeled as interconnection of 𝑁 subsystems with missing measurements. Then, we design a new decentralized filter. Sufficient conditions are derived in terms of linear matrix inequality (LMI) which is easy to be solved by using MATLAB LMI Toolbox for the decentralized stabilization of this class of large-scale system.

2. Problem Formulation

Consider the linear discrete-time large-scale system comprising 𝑁 subsystems Ξ_𝑖, 𝑖=1,2,…,𝑁; the dynamics of the 𝑖th subsystem is described by Ξ_𝑖∶𝑥_𝑖(𝑘+1)=𝐴_𝑖𝑥_𝑖(𝑘)+𝐵_𝑖𝑤_𝑖(𝑘)+∑^𝑁_{𝑗=1𝑗≠𝑖}𝐺_𝑖𝑗𝑥_𝑗(𝑘)𝑧_𝑖(𝑘)=𝐶_𝑖𝑥_𝑖(𝑘)+𝐷_𝑖𝑤_𝑖(𝑘)𝑦_𝑖(𝑘)=𝑥_𝑖(𝑘), (1) where 𝑥_𝑖(𝑘)∈𝑅^𝑛𝑖 is the state vector of the 𝑖th subsystem at time 𝑘, 𝑦_𝑖(𝑘)∈𝑅^𝑞𝑖 is the measurement output, 𝑧_𝑖(𝑘)∈𝑅^𝑝𝑖 is the controlled output, 𝑤_𝑖(𝑘)∈𝑅^𝑟𝑖 is the disturbance vector belonging to 𝑙₂[0,∞), 𝐴_𝑖, 𝐵_𝑖, 𝐶_𝑖, and 𝐷_𝑖 are known real constant matrices with appropriate dimensions, and 𝐺_𝑖𝑗∈𝑅^{𝑛𝑖×𝑛𝑗} is the interconnection matrix of the subsystem of 𝑗 and 𝑖.

The measurement with missing data can be characterized by 𝑥_𝑖(𝑘)=𝛼_𝑖(𝑘)𝑥_𝑖(𝑘), (2) where 𝑥_𝑖(𝑘)∈𝑅^𝑚𝑖 is the actual measured state, the stochastic variable 𝛼_𝑖(𝑘)∈𝑅 is a Bernoulli distributed white noise sequence taking the values of 0 and 1 with certain probability prob𝛼_𝑖(𝑘)=1=𝐸𝛼_𝑖(𝑘)∶=𝛼_𝑖,prob𝛼_𝑖(𝑘)=0=1−𝐸𝛼_𝑖(𝑘)∶=1−𝛼_𝑖, (3) and 0<𝛼_𝑖<1 is a known positive constant.

In order to observe the states of the system (1), we consider the following filter of order 𝑛 described by 𝑥_𝑖(𝑘+1)=𝐴_𝑖𝑥_𝑖(𝑘)+^𝑁_{𝑗=1𝑗≠𝑖}𝐺_𝑖𝑗𝑥_𝑗(𝑘)+𝐾_𝑖𝛼_𝑖𝑥_𝑖(𝑘)−𝑥_𝑖(𝑘),𝑧_𝑖(𝑘)=𝐶_𝑖𝑥_𝑖(𝑘),𝑦_𝑖(𝑘)=𝑥_𝑖(𝑘), (4) where 𝑥_𝑖(𝑘)∈𝑅^𝑛𝑖 is the state estimate of system (1) and 𝐾_𝑖 is the observer gain to be determined later.

Define the state estimation error by 𝜀_𝑖(𝑘)=𝑥_𝑖(𝑘)−𝑥_𝑖(𝑘), (5) and the filter error output is denoted by 𝑒_𝑖(𝑘)=𝑧_𝑖(𝑘)−𝑧_𝑖(𝑘). (6) Then it follows from (1), (2), and (4) that 𝜀_𝑖(𝑘+1)=𝐴_𝑖+𝛼_𝑖𝐾_𝑖𝜀_𝑖(𝑘)+^𝑁_{𝑗=1𝑗≠𝑖}𝐺_𝑖𝑗𝜀_𝑗(𝑘)+𝛼_𝑖(𝑘)−𝛼_𝑖𝐾_𝑖𝑥_𝑖(𝑘)+𝐵_𝑖𝑤_𝑖(𝑘),𝑒_𝑖(𝑘)=𝑧_𝑖(𝑘)−𝑧_𝑖(𝑘)=𝐶_𝑖𝜀_𝑖(𝑘)+𝐷_𝑖𝑤_𝑖(𝑘). (7)(8) Since it contains stochastic quantities 𝛼_𝑖(𝑘), the filter error system (8) is actually a stochastic parameter system. Then we use the following definition.

Definition 1 (see [10]). The filter error system (8) is said to be exponentially mean-square asymptotically stable if with 𝑤(𝑘)=0, there exist constants 𝜅>0 and 0<𝜏<1, such that 𝐸‖𝑒(𝑘)‖²<𝜅𝜏^𝑘𝐸‖𝑒(0)‖²,∀𝑒(𝑘)≠0, (9) where 𝑒(𝑘)=𝑒^𝑇₁(𝑘)⋯𝑒^𝑇_𝑁(𝑘)^𝑇and 𝑤(𝑘)=𝑤^𝑇₁(𝑘)⋯𝑤^𝑇_𝑁(𝑘)^𝑇.
With this definition, our objective is to design the full-order filter of form (4), such that(1)the filter error system (8) is exponentially mean-square asymptotically stable with 𝑤(𝑘)=0;(2)under zero-initial condition, the filter error 𝑒(𝑘) satisfies ^∞_𝑘=0𝐸‖𝑒(𝑘)‖²<𝛾^2∞_𝑘=0𝐸‖𝑤(𝑘)‖², (10)where 𝛾 is a given positive constant.

3. Main Results

For investigating the stability conditions of the filter error system (8), the following lemma is needed.

Lemma 2 (see [10]). Let 𝑉(𝜂(𝑘)) be a Lyapunov functional. If there exist constants 𝜆≥0, 𝜇>0, 𝑣>0, and 0<𝜓<1 such that 𝜇𝜂(𝑘)²≤𝑉𝜂(𝑘)≤𝑣𝜂(𝑘)²,𝐸𝑉𝜂(𝑘+1)∣𝜂(𝑘)−𝑉𝜂(𝑘)≤𝜆−𝜓𝑉𝜂(𝑘), (11) then the sequence 𝜂(𝑘) satisfies 𝐸𝜂(𝑘)²≤𝑣𝜇𝜂(0)²1−𝜓^𝑘+𝜆.𝜇𝜓 (12) The main results are concluded into the following theorems.

Theorem 3. Given 0<𝛼_𝑖<1 and 𝑤(𝑘)=0, the filter error system (8) is exponentially mean-square asymptotically stable if there exist positive definite matrices 𝑃_1𝑖=𝑃^𝑇_1𝑖, 𝑃_2𝑖=𝑃^𝑇_2𝑖 and gain matrix 𝐾_𝑖,𝑖=1,2,…,𝑁, satisfying −𝑃₁0𝑄^𝑇₁+𝛼𝐾000−𝑃₂0𝑄^𝑇₁𝐾𝑄₁+𝛼𝐾0−𝑃₁⁻¹000𝑄₁0−𝑃₂⁻¹00𝐾00−𝛽²𝑃₁⁻¹<0, (13) where 𝑄₁=𝐴₁𝐺₁₂⋯𝐺_1𝑁𝐺₂₁𝐴₂⋯𝐺_2𝑁𝐺⋮⋮⋱⋮_𝑁1𝐺_𝑁2⋯𝐴_𝑁,𝑃₁=diag𝑃₁₁,𝑃₁₂,…,𝑃_1𝑁,𝑃₂=diag𝑃₂₁,𝑃₂₂,…,𝑃_2𝑁,𝐾=diag𝐾₁,𝐾₂,…,𝐾_𝑁,𝛽=diag𝛽₁,𝛽₂,…,𝛽_𝑁,𝛽_𝑖=1−𝛼_𝑖𝛼_𝑖^1/2,𝑖=1,2,…,𝑁,𝛼=diag𝛼₁,𝛼₂,…,𝛼_N. (14)

Proof. When 𝑤(𝑘)=0, define the Lyapunov functional 𝑉(𝑘)=^𝑁_𝑖=1𝜀^𝑇_𝑖(𝑘)𝑃_1𝑖𝜀_𝑖(𝑘)+^𝑁_𝑖=1𝑥^𝑇_𝑖(𝑘)𝑃_2𝑖𝑥_𝑖(𝑘), (15) where 𝑃_1𝑖=𝑃^𝑇_1𝑖, 𝑃_2𝑖=𝑃^𝑇_2𝑖 are positive definite matrices. It follows from (8) that 𝑉()(𝑘)=𝑘+1−𝑉_𝑁𝑖=1𝐴_𝑖+𝛼_𝑖𝐾_𝑖𝜀_𝑖(𝑘)+^𝑁_{𝑗=1𝑗≠𝑖}𝐺_𝑖𝑗𝜀_𝑗(𝑘)+𝛼_𝑖(𝑘)−𝛼_𝑖𝐾_𝑖𝑥_𝑖(𝑘)^𝑇𝑃_1𝑖×𝐴_𝑖+𝛼_𝑖𝐾_𝑖𝜀_𝑖(𝑘)+^𝑁_{𝑗=1𝑗≠𝑖}𝐺_𝑖𝑗𝜀_𝑗(𝑘)+𝛼_𝑖(𝑘)−𝛼_𝑖𝐾_𝑖𝑥_𝑖(𝑘)+^𝑁_𝑖=1𝐴_𝑖𝑥_𝑖(𝑘)+^𝑁_{𝑗=1𝑗≠𝑖}𝐺_𝑖𝑗𝑥_𝑗(𝑘)^𝑇𝑃_2𝑖×𝐴_𝑖𝑥_𝑖(𝑘)+^𝑁_{𝑗=1𝑗≠𝑖}𝐺_𝑖𝑗𝑥_𝑗(𝑘)−^𝑁_𝑖=1𝜀^𝑇_𝑖(𝑘)𝑃_1𝑖𝜀_𝑖(𝑘)−^𝑁_𝑖=1𝑥^𝑇_𝑖(𝑘)𝑃_2𝑖𝑥_𝑖(𝑘). (16) Noting that 𝐸{𝛼_𝑖(𝑘)−𝛼_𝑖}=0, 𝐸{(𝛼_𝑖(𝑘)−𝛼_𝑖)²}=(1−𝛼_𝑖)𝛼_𝑖≜𝛽²_𝑖, we have 𝐸{𝑉((𝑘)=𝑘+1)}−𝑉_𝑁𝑖=1𝐴_𝑖+𝛼_𝑖𝐾_𝑖𝜀_𝑖(𝑘)+^𝑁_{𝑗=1𝑗≠𝑖}𝐺_𝑖𝑗𝜀_𝑗(𝑘)^𝑇𝑃_1𝑖×𝐴_𝑖+𝛼_𝑖𝐾_𝑖𝜀_𝑖(𝑘)+^𝑁_{𝑗=1𝑗≠𝑖}𝐺_𝑖𝑗𝜀_𝑗(𝑘)+^𝑁_𝑖=1𝐴_𝑖𝑥_𝑖(𝑘)+^𝑁_{𝑗=1𝑗≠𝑖}𝐺_𝑖𝑗𝑥_𝑗(𝑘)^𝑇𝑃_2𝑖×𝐴_𝑖𝑥_𝑖(𝑘)+^𝑁_{𝑗=1𝑗≠𝑖}𝐺_𝑖𝑗𝑥_𝑗(𝑘)+^𝑁_𝑖=1𝛽²_𝑖𝐾_𝑖𝑥_𝑖(𝑘)^𝑇𝑃_1𝑖𝐾_𝑖𝑥_𝑖(𝑘)−^𝑁_𝑖=1𝜀^𝑇_𝑖(𝑘)𝑃_1𝑖𝜀_𝑖(𝑘)−^𝑁_𝑖=1𝑥^𝑇_𝑖(𝑘)𝑃_2𝑖𝑥_𝑖(𝑘)=𝜂^𝑇(𝑘)𝜃₁𝜂(𝑘), (17) where 𝑥(𝑘)=𝑥^𝑇₁(𝑘)⋯𝑥^𝑇_𝑁(𝑘)^𝑇, 𝜀(𝑘)=𝜀^𝑇₁(𝑘)⋯𝜀^𝑇_𝑁(𝑘)^𝑇, and 𝜂(𝑘)=[𝜀^𝑇(𝑘),𝑥^𝑇(𝑘)]^𝑇.
By Schur complement, inequality (13) implies 𝜃₁<0. Then we have 𝐸{𝑉(𝑘+1)}−𝑉(𝑘)=𝜂^𝑇(𝑘)𝜃₁𝜂(𝑘)≤−𝜆_min−𝜃₁𝜂^𝑇(𝑘)𝜂(𝑘)<−𝛾𝜂^𝑇(𝑘)𝜂(𝑘), (18) where 0<𝛾<min{𝜆_min(−𝜃₁),𝜎}, and then from (18), we get 𝐸{𝑉(𝑘+1)}−𝑉(𝑘)<−𝛾𝜂^𝑇(𝑘)𝜂(𝑘)𝛾<−𝜎𝑉𝜂(𝑘)=−𝜓𝑉𝜂(𝑘), (19) where 𝜓=𝛾/𝜎∈(0,1).
From Definition 1 and Lemma 2, we can conclude that the filter error system (8) is exponentially mean-square asymptotically stable. This completes the proof.

In the sequel, we further provide method for solving matrix inequality (13) that is not a linear matrix inequality (LMI).

Theorem 4. Given 0<𝛼_𝑖<1 if there exist positive definite matrices 𝑀₁=𝑀^𝑇₁, 𝑀₂=𝑀^𝑇₂ and gain matrices 𝑁₁, 𝑁₂ that satisfy linear matrix inequality ^{−𝑀10𝑀1𝑄𝑇1+𝛼𝑁1000−𝑀20𝑀2𝑄𝑇1𝛽𝑁2}_{𝑄1𝑀1+𝛼𝑁10−𝑀1000𝑄1𝑀20−𝑀200𝛽𝑁200−𝑀1}<0 (20) and equation 𝑁₁𝑃₁=𝑁₂𝑃₂=𝐾, (21) where 𝑀₁=𝑃₁⁻¹, 𝑀₂=𝑃₂⁻¹, 𝑁₁=𝐾𝑃₁⁻¹, and N₂=KP₂⁻¹, then the error system (8) is exponentially mean-square asymptotically stable.

Proof. Through left- and right-multiplying (13) by diag{𝑃₁⁻¹,𝑃₂⁻¹,𝐼,𝐼,𝛽𝐼}, we have ^{−𝑃1−10𝑃1−1𝑄1+𝛼𝐾𝑇000−𝑃2−10𝑃2−1𝑄𝑇1𝛽𝑃2−1𝐾}_{𝑄1+𝑃𝛼𝐾1}⁻¹_0−𝑃1⁻¹_{000𝑄1𝑃2−10−𝑃2−100𝛽𝐾𝑃2−100−𝑃1−1}<0. (22) For the definitions of the matrices 𝑀₁, 𝑀₂, 𝑁₁, and 𝑁₂, the matrix inequality (13) is equivalent to (20) and (21). This completes the proof.

By solving the linear matrix inequality (20) and (21), we have 𝑀₁, 𝑀₂ and 𝑁₁, 𝑁₂. Moreover, the matrices are given by 𝑃₁=𝑀₁⁻¹, 𝑃₂=𝑀₂⁻¹, and 𝐾=𝑁₁𝑃₁=𝑁₂𝑃₂.

Theorem 5. Given 0<𝛼_𝑖<1 if there exist positive definite matrices 𝑃_1𝑖=𝑃^𝑇_1𝑖,𝑃_2𝑖=𝑃^𝑇_2𝑖 and gain matrix 𝐾_𝑖,𝑖=1,2,…,𝑁, that satisfy the following linear matrix inequality: −𝑃₁∗∗∗∗∗∗0−𝑃₂∗∗∗∗∗00−𝛾²𝑄𝐼∗∗∗∗₁+𝛼𝐾0𝐵−𝑃₁⁻¹∗∗∗0𝑄₁𝐵0−𝑃₂⁻¹∗∗0𝐾000−𝛽²𝑃₁⁻¹∗𝐶0𝐷000−𝐼<0, (23) where 𝐶=diag{𝐶₁,𝐶₂,…,𝐶_𝑁}, 𝐷=diag{𝐷₁,𝐷₂,…,𝐷_𝑁}, 𝐵=diag{𝐵₁,𝐵₂,…,𝐵_𝑁}, and Q₁, P₁, P₂, K, 𝛼, 𝛽_𝑖, and 𝛽 are the same as (13), then the filter error system (8) is exponentially mean-square asymptotically stable and achieves the prescribed 𝐻_∞ performance.

Proof. When 𝑤(𝑘)=0, (23) is equivalent to (13), so the filter error system is exponentially mean-square asymptotically stable.
When 𝑤(𝑘)≠0, define the Lyapunov functional as 𝑉(𝑘)=^𝑁_𝑖=1𝜀^𝑇_𝑖(𝑘)𝑃_1𝑖𝜀_𝑖(𝑘)+^𝑁_𝑖=1𝑥^𝑇_𝑖(𝑘)𝑃_2𝑖𝑥_𝑖(𝑘), (24) then 𝐸{𝑉((𝑘)𝑒𝑘+1)}−𝑉+𝐸^𝑇(𝑘)𝑒(𝑘)−𝛾²𝐸𝑤^𝑇(𝑘)𝑤(𝑘=_𝑁𝑖=1)𝐴_𝑖+𝛼_𝑖𝐾_𝑖𝜀_𝑖(𝑘)+^𝑁_{𝑗=1𝑗≠𝑖}𝐺_𝑖𝑗𝜀_𝑗(𝑘)+𝐵_𝑖𝑤_𝑖(𝑘)^𝑇×𝑃_1𝑖×𝐴_𝑖+𝛼_𝑖𝐾_𝑖𝜀_𝑖(𝑘)+^𝑁_{𝑗=1𝑗≠𝑖}𝐺_𝑖𝑗𝜀_𝑗(𝑘)+𝐵_𝑖𝑤_𝑖(𝑘)+^𝑁_𝑖=1𝐴_𝑖𝑥_𝑖(𝑘)+^𝑁_{𝑗=1𝑗≠𝑖}𝐺_𝑖𝑗𝑥_𝑗(𝑘)+𝐵_𝑖𝑤_𝑖(𝑘)^𝑇𝑃_2𝑖×𝐴_𝑖𝑥_𝑖(𝑘)+^𝑁_{𝑗=1𝑗≠𝑖}𝐺_𝑖𝑗𝑥_𝑗(𝑘)+𝐵_𝑖𝑤_𝑖(𝑘)+𝛽²_𝑖^𝑁_𝑖=1𝐾_𝑖𝑥_𝑖(𝑘)^𝑇𝑃_1𝑖𝐾_𝑖𝑥_𝑖(𝑘)+𝐸^𝑁_𝑖=1𝐶_𝑖𝜀_𝑖(𝑘)+𝐷_𝑖𝑤_𝑖(𝑘)^𝑇×𝐶_𝑖𝜀_𝑖(𝑘)+𝐷_𝑖𝑤_𝑖(𝑘)−^𝑁_𝑖=1𝜀^𝑇_𝑖(𝑘)𝑃_1𝑖𝜀_𝑖(𝑘)−^𝑁_𝑖=1𝑥^𝑇_𝑖(𝑘)𝑃_2𝑖𝑥_𝑖(𝑘)−𝛾²𝐸𝑤^𝑇(𝑘)𝑤(𝑘=)𝜂^𝑇(𝑘)𝜃₂𝜂(𝑘), (25) where 𝜂(𝑘)=[𝜀^𝑇(𝑘),𝑥^𝑇(𝑘),𝑤^𝑇(𝑘)]^𝑇. By the Schur complement, inequality (23) implies 𝜃₂<0, and then we have 𝐸{𝑉(𝑘+1)}−𝑉(𝑘)+𝐸𝑒^𝑇(𝑘)𝑒(𝑘)−𝛾²𝐸𝑤^𝑇(𝑘)𝑤(𝑘)<0,_∞𝑘=0𝐸𝑒^𝑇(𝑘)𝑒(𝑘)<𝛾^2∞_𝑘=0𝐸𝑤^𝑇(𝑘)𝑤(𝑘{𝑉(0{𝑉(∞.)+𝐸)}−𝐸)} (26) Since the system is exponentially mean-square asymptotically stable, it is straightforward to see that ^∞_𝑘=0𝐸‖𝑒(𝑘)‖²<𝛾^2∞_𝑘=0𝐸‖𝑤(𝑘)‖², (27) under the zero-initial condition. This completes the proof.

In the sequel, we further present how to convert the matrix inequality (23) into an LMI with matrix equality constraint.

Theorem 6. Given 0<𝛼_𝑖<1 if there exist positive definite matrices 𝑀^𝑇₁=𝑀₁, 𝑀^𝑇₂=𝑀₂ and gain matrices 𝑁₁, 𝑁₂ that satisfy the following linear matrix inequality −𝑀₁∗∗∗∗∗∗0−𝑀₂∗∗∗∗∗00−𝛾²𝑄𝐼∗∗∗∗₁𝑀₁+𝛼𝑁₁0𝐵−𝑀₁∗∗∗0𝑄₁𝑀₂𝐵0−𝑀₂∗∗0𝛽𝑁₂000−𝑀₁∗𝐶𝑀₁0𝐷000−𝐼<0,𝐾=𝑁₁𝑃₁=𝑁₂𝑃₂, (28) then the filter error system (8) is exponentially mean-square asymptotically stable and achieves the prescribed H_∞ performance.

Proof. Through left- and right-multiplying (23) by diag{𝑃₁⁻¹,𝑃₂⁻¹,𝐼,𝐼,𝐼,𝛽𝐼,𝐼}>0, we have −𝑃₁⁻¹∗∗∗∗∗∗0−𝑃₂⁻¹∗∗∗∗∗00−𝛾²𝑄𝐼∗∗∗∗₁+𝛼𝐾𝑃₁⁻¹0𝐵−𝑃₁⁻¹∗∗∗0𝑄₁𝑃₂⁻¹𝐵0−𝑃₂⁻¹∗∗0𝛽𝐾𝑃₂⁻¹000−𝑃₁⁻¹∗𝐶𝑃₁⁻¹0𝐷000−𝐼<0. (29) Similar to the proof of Theorem 4, we define 𝑀₁=𝑃₁⁻¹, 𝑀₂=𝑃₂⁻¹, 𝑁₁=𝐾𝑃₁⁻¹, and 𝑁₂=𝐾𝑃₂⁻¹. Then the matrix inequality (29) is equivalent to (23). From Theorem 5, we can conclude that the filter error system (8) is exponentially mean-square asymptotically stable and achieves the prescribed 𝐻_∞ performance. The proof is completed.

4. Numerical Simulations

Consider a linear discrete-time large-scale system which consists of two interconnected subsystems: 𝑥₁(𝑘+1)=𝑥−130−0.1¹(𝑘)+𝑥0.20.30.10.2²(𝑘)+𝑤1001¹(𝑘)=𝐴₁𝑥₁(𝑘)+𝐺₁₂𝑥₂(𝑘)+𝐸₁𝑤₁(𝑘),𝑧₁=𝑥10−10.8¹(𝑘)+𝑤1110¹(𝑘)=𝐶₁𝑥₁(𝑘)+𝐷₁(𝑘)𝑤₁(𝑘),𝑦₁(𝑘)=𝑥₁(𝑘),𝑥₂(𝑘+1)=𝑥−110−0.2²(𝑘)+𝑥0.30.10.20.2¹(𝑘)+𝑤1001²(𝑘)=𝐴₂𝑥₂(𝑘)+𝐺₂₁𝑥₁(𝑘)+𝐸₂𝑤₂(𝑘),𝑧₂=𝑥101−0.8²(𝑘)+𝑤1101²(𝑘)=𝐶₂𝑥₂(𝑘)+𝐷₂(𝑘)𝑤₂(𝑘),𝑦₂(𝑘)=𝑥₂(𝑘). (30) Choose the disturbance input 𝑤₁(𝑘)=𝑤₂(𝑘)=0.01^sin(100𝑘)_sin(100𝑘). The initial state values of original system and its observer are 𝑥₁(0)=²₁, 𝑥₂(0)=²₂ and 𝑥₁(0)=⁰₀, 𝑥₂(0)=₀⁻¹, respectively. Suppose that stochastic sequence obeys Bernoulli distribution with probability 𝐸{𝛼₁(𝑘)∣𝛼₁(𝑘)=1}=𝐸{𝛼₂(𝑘)∣𝛼₂(𝑘)=1}=0.996, given 𝛾=1. For simplicity, let 𝑀=𝑃₁⁻¹=𝑃₂⁻¹ and 𝑁=𝐾𝑃₁⁻¹=𝐾𝑃₂⁻¹. We can obtain the following parameters in Theorem 6 by using the MATLAB YALMIP Toolbox: 𝑀₁=1.35490.40140.40140.3819,𝑀₂=,𝑁1.10580.78810.78811.6449₁=0.1755−0.3232−0.32320.0041,𝑁₂=.0.2464−0.3847−0.38470.3580 (31) The Lyapunov function solution matrices and observer parameter are given by 𝑃₁₁=𝑃₂₁=𝑀₁⁻¹=,𝑃1.0718−1.1264−1.12643.8021₁₂=𝑃₂₂=𝑀₂⁻¹=,𝐾1.3732−0.6579−0.65790.9231₁=𝑁₁𝑃₁=,𝐾0.5521−1.4264−0.35100.3789₂=𝑁₂𝑃₂=.0.5919−0.5172−0.76370.5835 (32) The simulation results are shown in Figures 1, 2, 3, and 4.

Figure 1 

Filter error 𝑒11(𝑘).

Figure 2 

Filter error 𝑒12(𝑘).

Figure 3 

Filter error 𝑒21(𝑘).

Figure 4 

Filter error 𝑒22(𝑘).

It can be verified that ∑^∞_𝑘=0𝐸{‖𝑒(𝑘)‖²}<𝛾²∑^∞_𝑘=0{‖𝑤(𝑘)‖²} and the filter error system satisfies the prescribed 𝐻_∞ performance.

5. Conclusion

In this paper, the 𝐻_∞ filter for a class of linear discrete-time large-scale system has been designed, where the measurements are probably missing. The missing probability is assumed to obey Bernoulli distribution. By employing the Lyapunov stability theory combined with stochastic analysis method, a filter is designed to reconstruct the states of original system such that the filter error system is exponentially stable in the sense of mean square and achieves the prescribed 𝐻_∞ performance.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (61104103, 60904025, 61203028); Natural Science Foundation of Jiangsu Province of China (BK2011826); Jiangsu Government Scholarship for Overseas Studies; Natural Science Foundation for Colleges and Universities in Jiangsu Province (10KJB120001); Climbing Program of Nanjing University of Telecommunications and Posts (NY210013, NY210014).