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

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 prescribedH ∞ 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.


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][2][3][4][5][6][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][3][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 discretetime 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 largescale 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.

Problem Formulation
Consider the linear discrete-time large-scale system comprising  subsystems Ξ  ,  = 1, 2, . . ., ; 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  2 [0, ∞),   ,   ,   , 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 0 <   < 1 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 x () ∈    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 () =   () − ẑ () =     () +     () .
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 () = 0, there exist constants  > 0 and 0 <  < 1, such that where .With this definition, our objective is to design the fullorder filter of form ( 4), such that (1) the filter error system ( 8) is exponentially meansquare asymptotically stable with () = 0; (2) under zero-initial condition, the filter error () satisfies where  is a given positive constant.

Main Results
For investigating the stability conditions of the filter error system (8), the following lemma is needed.
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 <   < 1 if there exist positive definite matrices  1 =   1 ,  2 =   2 and gain matrices  1 ,  2 that satisfy linear matrix inequality and equation where , and N 2 = KP −1 2 , then the error system (8) is exponentially mean-square asymptotically stable.
In the sequel, we further present how to convert the matrix inequality (23) into an LMI with matrix equality constraint.Theorem 6.Given 0 <   < 1 if there exist positive definite matrices   1 =  1 ,   2 =  2 and gain matrices  1 ,  2 that satisfy the following linear matrix inequality then the filter error system (8) is exponentially mean-square asymptotically stable and achieves the prescribed H ∞ performance.

Numerical Simulations
Consider a linear discrete-time large-scale system which consists of two interconnected subsystems: Choose the disturbance input  1 () =  2 () = 0.01 [ sin(100) sin(100) ].The initial state values of original system and its observer are We can obtain the following parameters in Theorem 6 by using the MATLAB YALMIP Toolbox:

Conclusion
In this paper, the  ∞ filter for a class of linear discretetime 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.