Mathematical Problems in Engineering

Volumeย 2011, Article IDย 839648, 17 pages

http://dx.doi.org/10.1155/2011/839648

## Robust Filtering of Time-Delay Jump Systems with Respect to the Finite-Time Interval

^{1}Key Laboratory of Advanced Process Control for Light Industry, Jiangnan University, Ministry of Education, Wuxi 214122, China^{2}Institute of Automation, Jiangnan University, Wuxi 214122, China^{3}Control Systems Centre, School of Electrical and Electronic Engineering, University of Manchester, Manchester M60 1QD, UK

Received 10 September 2010; Accepted 22 September 2010

Academic Editor: Mingย Li

Copyright ยฉ 2011 Shuping He and Fei Liu. 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 studied the problem of stochastic finite-time boundedness and disturbance attenuation for a class of linear time-delayed systems with Markov jumping parameters. Sufficient conditions are provided to solve this problem. The filters are, respectively, designed for time-delayed Markov jump linear systems with/without uncertain parameters such that the resulting filtering error dynamic system is stochastically finite-time bounded and has the finite-time interval disturbance attenuation for all admissible uncertainties, time delays, and unknown disturbances. By using stochastic Lyapunov-Krasovskii functional approach, it is shown that the filter designing problem is in terms of the solutions of a set of coupled linear matrix inequalities. Simulation examples are included to demonstrate the potential of the proposed results.

#### 1. Introduction

Since the introduction of the framework of the class of Markov jump linear systems (MJLSs) by Krasovskii and Lidskii [1], we have seen increasing interest for this class of stochastic systems. It was used to model a variety of physical systems, which may experience abrupt changes in structures and parameters due to, for instance, sudden environment changes, subsystem switching, system noises, and failures occurring in components or interconnections and executor faults. For more information regarding the use of this class of systems, we refer the reader to Sworder and Rogers [2], Athans [3], Arrifano and Oliveira [4], and the references therein. It has been recognized that the time-delays and parameter uncertainties, which are inherent features of many physical processes, are very often the cause for poor performance of systems. In the past few years, considerable attention has been given to the robust control and filtering for linear uncertain time-delayed systems. A great amount of progress has been made on this general topic; see, for example, [5โ8], and the references therein. As for MJLSs with time-delays and uncertain parameters, the results of stochastic stability, robust controllability, observability, filtering, and fault detection have been well investigated, and recent results can be found in [9โ15].

It is now worth pointing out that the control performances mentioned above concern the desired behavior of control dynamics over an infinite-time interval and it always deals with the asymptotic property of system trajectories. But in some practical processes, a Lyapunov asymptotically stable system over an infinite-time interval does not mean that it has good transient characteristics, for instance, biochemistry reaction system, robot control system, and communication network system. Moreover, the main attention in these dynamics may be related to the behavior over a fixed finite-time interval. Therefore, we need to check the unacceptable values to see whether the system states remain within the prescribed bound in a fixed finite-time interval or not. To discuss this transient performance of control systems, Dorato [16] gave the concept of finite-time stability (or short-time stability [17, 18]) in the early 1960s. Then, some attempts on finite-time stability can be found in [19โ24]. More recently, the concept of finite-time stability has been revisited in the light of recent results coming from linear matrix inequalities (LMIs) techniques, which relate the computationally appealing conditions guaranteeing finite-time stabilization [25, 26] of dynamic systems. Towards each case above, more details are related to linear control dynamics [27โ29]. However, very few results in the literature consider the related control and filtering problems [30] of stochastic MJLSs in the finite-time interval. These motivate us to research this topic.

As we all know, since the Kalman filtering theory has been introduced in the early 1960s, the filtering problem has been extensively investigated, whose objective is to estimate the unavailable state variables (or a linear combination of the states) of a given system. During the past decades, the filtering technique regains increasing interest, and many filtering schemes have been developed. Among these filtering approaches, the filtering problem [31, 32] has received less attention. But in practical engineering applications, the peak values of filtering error should be considered. And compared with filtering, the stochastic noise disturbances are both assumed energy bounded in these two filtering techniques, but filtering setting requires the performance prescribed bounded from unknown noise disturbances to filtering error.

This paper is concerned with the robust filtering problem for a class of continuous time-delay uncertain systems with Markov jumping parameters. We aim at designing a robust filter such that, for all admissible uncertainties, time delays, and unknown disturbances, the filtering error dynamic system is stochastically finite-time bounded (FTB) and satisfies the given finite-time interval induced norm of the operator from the unknown disturbance to the output error. By using stochastic Lyapunov-Krasovskii functional approach, we show that the filter designing problem can be dealt with by solving a set of coupled LMIs. In order to illustrate the proposed results, two simulation examples are given at last.

In the sequel, unless otherwise specified, matrices are assumed to have compatible dimensions. The notations used throughout this paper are quite standard. and denote, respectively, the dimensional Euclidean space and the set of all real matrices. and denote the matrix transpose and matrix inverse. represents the block-diagonal matrix of and . If is a symmetric matrix, denote by and its smallest and largest eigenvalues, respectively. denotes, for example, for , . stands for the mathematics statistical expectation of the stochastic process or vector and is the Euclidean vector norm. is the space of dimensional square integrable function vector over . stands for a positive-definite matrix. is the unit matrix with appropriate dimensions. is the zero matrix with appropriate dimensions. In symmetric block matrices, we use โโ as an ellipsis for the terms that are introduced by symmetry.

The paper is organized as follows. In Section 2, we derive the new definitions about stochastic finite-time filtering of MJLSs. In Section 3, we give the main results of filtering problem of MJLSs and extend this to uncertain dynamic MJLSs in Section 4. In Section 5, we demonstrate two simulation examples to show the validity of the developed methods.

#### 2. Problem Formulation

Given a probability space where is the sample space, is the algebra of events and is the probability measure defined on . Let the random form process be the continuous-time discrete-state Markov stochastic process taking values in a finite set with transition probability matrix given by where and . is the transition probability rates from mode at time to mode at time , and .

Consider the following time-delay dynamic MJLSs over the probability space : where is the state, is the measured output, is the unknown input, is the controlled output, is the constant time-delay, is a vector-valued initial continuous function defined on the interval , and is the initial mode. , , , , , and are known mode-dependent constant matrices with appropriate dimensions, and represents a continuous-time discrete state Markov stochastic process with values in the finite set .

*Remark 2.1. *For convenience, when , we denote , , , , , and as , , , , , and . Notice that the time-delays in (2.2) are constant and only dependent of the system structure, and they are not dependent on the defined stochastic process. To simplify the study, we take the initial time and let the initial values and be fixed. At each mode, we assume that the time-delay MJLSs have the same dimension.

We now construct the following full-order linear filter for MJLSs (2.2) as where is the filter state, is the filter output, is a continuous vector-valued initial function, and the mode-dependent matrices , , and are unknown filter parameters to be designed for each value .

Define and , then we can get the following filtering error system: Let , the filtering error system (2.4) can be rewritten as where

The objective of this paper consists of designing the finite-time filter of time-delay MJLSs in (2.1) and obtaining an estimate of the signal such that the defined guaranteed performance criteria are minimized. For some given initial conditions [24โ27], the general idea of finite-time filtering can be formalized through the following definitions over a finite-time interval.

*Assumption. *The external disturbance is time-varying and satisfies

*Definition 2.2. *For a given time-constant , the filtering error MJLSs system (2.5) with is stochastically finite-time stable (FTS) if there exist positive matrix and scalars and , such that

*Definition 2.3 (FTB). * For a given time-constant , the filtering error (2.5) is stochastically finite-time bounded (FTB) with respect to if condition (2.8) holds.

*Remark 2.4. *Notice that FTB and FTS are open-loop concepts. FTS can be recovered as a particular case of FTB with and FTS leads to the concept of FTB in the presence of external inputs. FTB implies finite-time stability, but the converse is not true. It is necessary to point out that Lyapunov stability and FTS are independent concepts. Different with the concept of Lyapunov stability [33โ35] which is largely known to the control community, a stochastic MJLSs is FTS if, once we fix a finite time-interval [36, 37], its state remain within prescribed bounds during this time-interval. Moreover, an MJLS which is FTS may not be Lyapunov stochastic stability; conversely, a Lyapunov stochastically stable MJLS could be not FTS if its states exceed the prescribed bounds during the transients.

*Definition 2.5 (Feng et al. [34], Mao [35]). *Let be the stochastic positive functional; define its weak infinitesimal operator as

*Definition 2.6. *For the filtering error MJLSs (2.5), if there exist filter parameters , , and , and a positive scalar , such that (2.5) is FTB and under the zero-valued initial condition, the system output error satisfies the following cost function inequality for with attenuation and for all admissible with the constraint condition (2.7),
where , .

Then, the filter (2.3) is called the stochastic finite-time filter of time-delay dynamic MJLSs (2.2) with -disturbance attenuation.

*Remark 2.7. *In stochastic finite-time filtering process, the unknown noises are assumed to be arbitrary deterministic signals of bounded energy and the problem of this paper is to design a filter that guarantees a prescribed bounded for the finite-time interval induced norm of the operator from the unknown noise inputs to the output error , that is, the designed stochastic finite-time filter is supposed to satisfy inequality (2.10) with attenuation .

#### 3. Finite-Time Filtering for MJLSs

In this section, we will study the stochastic finite-time filtering problem for time-delay dynamic MJLSs (2.2).

Theorem 3.1. *For a given time-constant , the filtering error MJLSs (2.5) are stochastically FTB with respect to and has a prescribed performance level if there exist a set of mode-dependent symmetric positive-definite matrices and symmetric positive-definite matrix , satisfying the following matrix inequalities for all ,
**
where , , , , , and .*

*Proof. *For the given symmetric positive-definite matrices and , we define the following stochastic Lyapunov-Krasovskii functional as

Then referring to Definition 2.5 and along the trajectories of the resulting closed-loop MJLSs (2.7), we can derive the corresponding time derivative of as

Considering the filtering performance for the dynamic filtering error system (2.5), we introduce the following cost function by Definition 2.6 with ,

According to relation (3.1), it follows that , that is,

Then, multiplying the above inequality by , we have

In the following, we assume zero initial condition, that is, , for , and integrate the above inequality from to ; then
Recalling to the defined Lyapunov-Krasovskii functional, it can be verified that,
By (3.2) and within the finite-time interval , we can also get
Therefore, the cost function inequality (2.10) can be guaranteed by setting , which implies .

On the other hand, by integrating the above inequality (3.8) between 0 to , it yields
Denote , , , , and . Note that , ; then
From the selected stochastic Lyapunov-Krasovskii function, we can obtain

Then we can get
which implies for . This completes the proof.

Theorem 3.2. *For a given time-constant , the filtering error dynamic MJLSs (2.5) are FTB with respect to with and has a prescribed performance level if there exist a set of mode-dependent symmetric positive-definite matrices , symmetric positive-definite matrix , a set of mode-dependent matrices , , and and positive scalars , satisfying the following matrix inequalities for all ,
**
where , , and . **Moreover, the suitable filter parameters can be given as
*

*Proof. *For convenience, we set , . Then inequalities (3.1) and (3.2) are equivalent to LMIs (3.16) and (3.17) by letting , .On the other hand, by setting , LMIs (3.18) and (3.19) imply that
Then recalling condition (3.3), we can get LMI (3.20). This completes the proof.

*Remark 3.3. *It can be seen that if we choose the infinite time-interval, that is, , the main results in Theorems 3.1 and 3.2 can reduce to conclusions of regular filtering. And other filtering schemes, such as Kalman, , and filtering of stochastic jump systems can be also handled, referring to [9โ13, 29, 32]. When the delays in MJLSs (2.2) satisfy , it reduces to a delay-free system. We can immediately get the corresponding results implied in Theorems 3.1 and 3.2 by choosing the stochastic Lyapunov-Krasovskii functional as and following the similar proofs.

#### 4. Extension to Uncertain MJLSs

It has been recognized that the unknown disturbances and parameter uncertainties are inherent features of many physical process and often encountered in engineering systems, their presences must be considered in realistic filter design. For these, we consider the following stochastic time-delay MJLSs with uncertain parameters,

*Assumption. *The time-varying but norm-bounded uncertainties and satisfy
where , , and are known real constant matrices of appropriate dimensions, is unknown, time-varying matrix function satisfying , and the elements of are Lebesgue measurable for any .

*Remark 4.1. *In condition (4.2), is chosen as a full row rank matrix. And the parameter uncertainty structure is an extension of the admissible condition. In fact, it is always impossible to obtain the exact mathematical model of the practical dynamics due to environmental noises, complexity process, time-varying parameters, and many measuring difficulties, and so forth. These motivate us to consider system (4.1) containing uncertainties and . Moreover, the uncertainties and within (4.2) reflect the inexactness in mathematical modeling of jump dynamical systems. To simplify the study, and can be abbreviated as and . It is necessary to point out that the unknown mode-dependent matrix in (4.2) can also be allowed to be state-dependent, that is, , as long as is satisfied.

For this case, we can get the following filtering error system by letting :
where

Lemma 4.2 (Wang et al. [38]). *Let , , and be real matrices with appropriate dimensions. Then for all time-varying unknown matrix function satisfying , the following relation
**
holds if and only if there exists a positive scalar , such that
**By following the similar lines and the main proofs of Theorems 3.1 and 3.2 and using the above Lemma 4.2, one can get the results stated as follows. *

Theorem 4.3. *For a given time-constant , the filtering error MJLSs (4.3) with uncertainties are stochastically FTB with respect to and has a prescribed performance level if there exist a set of mode-dependent symmetric positive-definite matrices , symmetric positive-definite matrix , a set of mode-dependent matrices, , and and positive scalars , and satisfying LMIs (3.17)โ(3.20), and the following matrix inequalities for all ,
**
And the suitable stochastic finite-time filter can be derived by (3.21).*

*Remark 4.4. *Theorems 3.2 and 4.3 have presented the sufficient condition of designing the stochastic finite-time filter of time-delay MJLSs. Notice that the coupled LMIs (3.16)โ(3.20) (or LMIs (4.7), (3.17)โ(3.20)) are with respect to , , , , , , , , , , , , , and . Therefore, for given , , , and , we can take as optimal variable, that is, to obtain an optimal stochastic finite-time filter, the attenuation lever can be reduced to the minimum possible value such that LMIs (3.16)โ(3.20) (or LMIs (4.7), (3.17)โ(3.20)) are satisfied. The optimization problem [39] can be described as follows:

*Remark 4.5. *As we did in previous Remark 4.4, we can also fix and look for the optimal admissible or guaranteeing the stochastically finite-time boundedness of desired filtering error dynamic properties.

#### 5. Numeral Examples

*Example 5.1. *Consider a class of constant time-delay MJLSs with parameters described as follows:*Mode 1**Mode 2*

Let the transition rate matrix be . With the initial value for , , , and , we fix and look for the optimal admissible of different guaranteeing the stochastically finite-time boundedness of desired filtering error dynamic properties. Table 1 and Figure 1, respectively, give the optimal minimal admissible with different initial upper bound .

For , we solve LMIs (3.16)โ(3.20) by Theorem 3.2 and optimization algorithm (4.8) and get the following optimal filters as
And then, we can also get the attenuation lever as .

*Example 5.2. *Consider a class of constant time-delay MJLSs with uncertain parameters described as follows:
The modes, transition rate matrix, the matrices parameters and initial conditions are defined similarly as Example 5.1.

By solving LMIs (4.7), (3.17)โ(3.20) by Theorem 4.3 and Remark 4.4, we can get the optimal value , and the mode-dependent optimized filtering performance can be easily obtained as follows:

In this work, the simulation time is selected as . Assume the initial conditions are , and . The unknown inputs are selected as

The simulation results of jump mode (the estimation of changing between modes during the simulation with the initial mode 1), the response of system states (real states and estimated states) and filtering output error are shown in Figures 2โ5, which show the effective of the proposed approaches.

It is clear from Figures 3โ5 that the estimated states can track the real states smoothly. Furthermore, the presented filter guarantees a prescribed bounded for the induced finite-time norm of the operator from the unknown disturbance to the filtering output error with attenuation , which illustrates the effectiveness of the proposed techniques.

#### 6. Conclusions

In the paper, we have studied the design of stochastic finite-time filter for uncertain time-delayed MJLSs. It ensures the finite-time stability and finite-time boundedness for the filtering error dynamic MJLSs. By selecting the appropriate Lyapunov-Krasovskii function and applying matrix transformation and variable substitution, the main results are provided in terms of LMIs form. Simulation examples demonstrate the effectiveness of the developed techniques.

#### Acknowledgments

This work was partially supported by National Natural Science Foundation of P. R. China (no. 60974001, 60873264, and 61070214), National Natural Science Foundation of Jiangsu Province (no. BK2009068), Six Projects Sponsoring Talent Summits of Jiangsu Province, and Program for Postgraduate Scientific Research and Innovation of Jiangsu Province.

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