Network-Based H ∞ Filter Design for Linear System with Random Delays

This paper investigates the filtering problem for a class of network-based systems with random network-induced delays. The considered randomdelay between the sensor and the filter is assumed to be satisfying a certain stochastic characteristic. Considering the probability of the delay taking value in different intervals, a stochastic variable satisfying Bernoulli randombinary distribution is introduced and a new systemmodel is established by employing the information of the probability distribution. By using a properly constructed Lyapunov function, sufficient conditions for the existence of the H ∞ filters are presented in terms of linear matrix inequalities, which are dependent on the occurrence probability of both the random communication delays. The filter parameter is then characterized by the solution to a set of LMIs. A simulation example is employed to show the effectiveness of the proposed method.


Introduction
With the wide application of networks in the complex dynamical processes such as advanced aircraft and manufacturing processes, the networked control systems (NCSs) have attracted great attention during the past few years, see [1][2][3][4][5].However, they also present some challenging problems arising from the fact that data typically travel through the communication networks from sensors to the controller and from controller to the actuators.In particular, due to the finite bandwidth for data transmission over networks, random delay and packets loss are inevitable in networked systems where a common medium is used for data transfers.In many situations, they can even destabilize the system.Hence, the uncertainties in the measurement transmission have attracted increasing interest.
In recent years, lots of outstanding results have been published on sensor delays, packet dropouts, and missing measurements, see [6][7][8].In [6], based on a Markov chain, the authors considered mixed uncertainties in the measurement transmission and developed a unified/combined model to accommodate random delay, packet dropouts and missing measurements.The authors in [7] invested a class of linear uncertain discrete-time stochastic systems with randomly varying sensor delay.The authors in [8] studied a delaydependent  ∞ filtering for Markovian jump systems with time-varying delays, and criterion was derived for the  ∞ performance analysis of the filtering-error systems, which could lead to much less conservative analysis results.The filtering problem for network-based systems with random delay and packets loss has attracted increasing attention during the past years, see [9][10][11][12][13][14].For example, in [9], the filtering problem of network-based system with long timevarying delay was considered.The authors in [12] studied the robust filtering problem for a class of uncertain discrete time-delay systems with missing measurements.In [13], the authors designed  ∞ filter for nonlinear systems with timedelay via a T-S fuzzy model approach.
However, to the best of our knowledge, the filtering problems for NCSs with random communication delays and missing measurements has not been fully investigated yet, which still remains as a challenging research issue in the literature.Moreover, the filtering problem for network-based systems with random delay and event triggered scheme has not been investigated in the existing literature, which motivates our present work.
In this paper, the filtering problem is addressed for communication delays and missing measurements.The delay is modeled as a Bernoulli process.By using the Lyapunov method and the LMI technique, a sufficient condition to guarantee the filtering error system exponentially meansquare stable is derived.A convex optimization problem is also formulated to design the optimal  ∞ filter.A simulation example is used to demonstrate the effectiveness of the proposed design procedures.
Notation.R  and R × denote the -dimensional Eculidean space, and the set of  ×  real matrices; the superscript "" stands for matrix transposition;  is the identity matrix of appropriate dimension; ‖ ⋅ ‖ stands for the Euclidean vector norm or the induced matrix 2-norm as appropriate; the notation  > 0 (resp.,  ≥ 0), for  ∈ R × means that the matrix  is real symmetric positive definite (resp., positive semidefinite).When  is a stochastic variable, E{} stands for the expectation of .For a matrix  and two symmetric matrices  and , [  * ] denotes a symmetric matrix, where * denotes the entries implied by symmetry.

System Description
In this paper, we consider the following system: where () ∈ R  , () ∈ R  , and () ∈ R  denote the state vector, measurement vector, and the signal vector to be estimated, respectively; , , , and  are parameter matrices with appropriate dimensions.We consider the following full order linear dynamic filter: where   () ∈ R  is the filter state, ŷ() ∈ R  is the filter input, and   () ∈ R  is the estimated signal, and   ,   ,   , and   are filter parameters to be determined.We use () to denote the time-delay in the network between the sensor and the filter, and it is assumed that () Assumption 1. () changes randomly and for a constant  0 ∈ [ 0 ,  2 ], the probability of () ∈ [ 1 ,  0 ] and () ∈ [ 0 ,  2 ] can be known.
Remark 3. The introduction of () is motivated by [7,12,15], where the Bernoulli distributed sequence () is used to model the missing message of the systems.Different from [7,12,15], () is used in this paper to describe the time-varying delay taking values in different intervals.
In the following, based on Theorem 10, we will design the filter parameters   ,   ,   ,   , in (4).

Simulation Examples
Consider the system ( 1 To illustrate the performance of the designed filter, choose the disturbance function as follows: otherwise. (45) Figure 1 shows the error-estimation signal of (), and Figure 2 shows the state of filtering-error system with the initial values   (0) = [0.5 − 0.5].

Conclusion
This paper has investigated the problem of filtering design for linear time-delay system.The probability distribution of the delay taking values in some intervals is assumed to be known a priori.Corresponding to the probability of the delay taking value in different intervals, a stochastic variable satisfying Bernoulli random binary distribution has been introduced and a new modeling method is presented to describe the overall filtering error system.Sufficient conditions are derived to guarantee the stochastic stability and a  ∞ performance level for the filtering error system.An optimal filter design problem is also provided by optimizing the filtering performances.Simulations are conducted to evaluate and demonstrate the performance of the proposed approach.

Figure 2 :
Figure 2: The state curves of filtering error system.