Mathematical Problems in Engineering

Volume 2013, Article ID 945705, 7 pages

http://dx.doi.org/10.1155/2013/945705

## Filter Design for Large-Scale Systems with Missing Measurements

^{1}College of Automation, Nanjing University of Posts and Telecommunications, Nanjing 210003, China^{2}School of Information and Control Engineering, Nanjing University of Information Science & Technology, Nanjing 210044, China

Received 29 May 2013; Accepted 22 July 2013

Academic Editor: Bochao Zheng

Copyright © 2013 Ying Zhou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper is concerned with 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 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 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 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 controller design for a class of large-scale systems with missing measurements is considered. In [9], a state feedback 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, 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 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 subsystems , ; the dynamics of the th subsystem is described by where is the state vector of the th subsystem at time , is the measurement output, is the controlled output, is the disturbance vector belonging to , , , , and are known real constant matrices with appropriate dimensions, and is the interconnection matrix of the subsystem of and .

The measurement with missing data can be characterized by where is the actual measured state, the stochastic variable is a Bernoulli distributed white noise sequence taking the values of 0 and 1 with certain probability and is a known positive constant.

In order to observe the states of the system (1), we consider the following filter of order described by where is the state estimate of system (1) and is the observer gain to be determined later.

Define the state estimation error by and the filter error output is denoted by Then it follows from (1), (2), and (4) that Since it contains stochastic quantities , 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 , there exist constants and , such that
where and .

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 ;(2)under zero-initial condition, the filter error satisfies
where 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 be a Lyapunov functional. If there exist constants , , , and such that
**
then the sequence satisfies
**
The main results are concluded into the following theorems.*

Theorem 3. *Given and , the filter error system (8) is exponentially mean-square asymptotically stable if there exist positive definite matrices , and gain matrix , satisfying
**
where
*

*Proof. * When , define the Lyapunov functional
where , are positive definite matrices. It follows from (8) that
Noting that , , we have
where , , and .

By Schur complement, inequality (13) implies . Then we have
where , and then from (18), we get
where .

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 if there exist positive definite matrices , and gain matrices , that satisfy linear matrix inequality
**
and equation
**
where , , , and , then the error system (8) is exponentially mean-square asymptotically stable.*

*Proof. * Through left- and right-multiplying (13) by , we have
For the definitions of the matrices , , , and , 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 , and , . Moreover, the matrices are given by , , and .

Theorem 5. *Given if there exist positive definite matrices , and gain matrix , that satisfy the following linear matrix inequality:
**
where , , , and , , , , , , and are the same as (13), then the filter error system (8) is exponentially mean-square asymptotically stable and achieves the prescribed performance.*

*Proof. *When , (23) is equivalent to (13), so the filter error system is exponentially mean-square asymptotically stable.

When , define the Lyapunov functional as
then
where . By the Schur complement, inequality (23) implies , and then we have
Since the system is exponentially mean-square asymptotically stable, it is straightforward to see that
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 if there exist positive definite matrices , and gain matrices , that satisfy the following linear matrix inequality
**
then the filter error system (8) is exponentially mean-square asymptotically stable and achieves the prescribed performance.*

*Proof. *Through left- and right-multiplying (23) by , we have
Similar to the proof of Theorem 4, we define , , , and . 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 performance. The proof is completed.

#### 4. Numerical Simulations

Consider a linear discrete-time large-scale system which consists of two interconnected subsystems: Choose the disturbance input . The initial state values of original system and its observer are , and , , respectively. Suppose that stochastic sequence obeys Bernoulli distribution with probability , given . For simplicity, let and . We can obtain the following parameters in Theorem 6 by using the MATLAB YALMIP Toolbox: The Lyapunov function solution matrices and observer parameter are given by The simulation results are shown in Figures 1, 2, 3, and 4.

It can be verified that and the filter error system satisfies the prescribed performance.

#### 5. Conclusion

In this paper, the 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 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).

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