Journal of Applied MathematicsVolume 2012 (2012), Article ID 615790, 16 pagesdoi:10.1155/2012/615790

Research Article

Finite-Time Filtering for Singular Stochastic Systems

Caixia Liu, Yingqi Zhang, and Huixia Sun

College of Science, Henan University of Technology, Zhengzhou 450001, China

Received 14 July 2011; Revised 5 November 2011; Accepted 5 November 2011

Academic Editor: F. Marcellán

Copyright © 2012 Caixia Liu 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 addresses the problem of finite-time filtering for one family of singular stochastic systems with parametric uncertainties and time-varying norm-bounded disturbance. Initially, the definitions of singular stochastic finite-time boundedness and singular stochastic finite-time boundedness are presented. Then, the filtering is designed for the class of singular stochastic systems with or without uncertain parameters to ensure singular stochastic finite-time boundedness of the filtering error system and satisfy a prescribed performance level in some given finite-time interval. Furthermore, sufficient criteria are presented for the solvability of the filtering problems by employing the linear matrix inequality technique. Finally, numerical examples are given to illustrate the validity of the proposed methodology.

1. Introduction

Singular systems also referred to as descriptor systems or generalized state-space systems represent one family of dynamical systems since it generalizes the linear system model and has extensive applications in economics systems, power systems, mechanics systems, chemical processes, and so on; see for more practical examples [1, 2] and the references therein. Many control results in state-space systems have been extended to singular systems, such as stability, stabilization, control, and the filtering problems, for instance, see [3–6] and the references therein. Meanwhile, Markovian jump systems are referred to as one special family of hybrid systems and stochastic systems, which are very appropriate to model plants whose structure is subject to random abrupt changes, see the reference [7]. Thus, many attracting results have been studied, such as stochastic stability and stabilization [8, 9], robust control [10–12], guaranteed cost control [13], and other issues. For more details, the readers may be refered to [7, 14] and the references therein. Recently, the problem of state estimation for singular Markovian jump systems has also attracted considerable attention. As far as we know, the traditional Kalman filtering requires the exact knowledge of statistics of the noise signals. To overcome the limitations regarding the system uncertainties and the statistical properties, the filtering problem has been proposed and tackled for both the continuous-time case and the discrete-time one including without or with time-delay and full-order or reduced-order [15–20]. For more details, we refer the readers to [7, 21] and the references therein.

On the other hand, in many practical processes, many concerned problems are the practical ones which described that system state does not exceed some bound during some time interval. Compared with classical Lyapunov asymptotical stability, in order to deal with the transient performance of control systems, finite-time stability or short-time stability was introduced in [22]. Employing linear matrix inequality (LMI) theory and Lyapunov function approach, some appealing results were obtained to ensure finite-time stability, finite-time boundedness, and finite-time stabilization of various systems including linear systems, nonlinear systems, and stochastic systems. For instance, Amato et al. [23] investigated the output feedback finite-time stabilization for continuous linear system. Zhang and An [24] considered finite-time control problems for linear stochastic system. For more details of the literature related to finite-time stability, the reader is referred to [25–34] and the references therein. However, to date and to the best of our knowledge, the filtering problem for singular stochastic systems has not investigated in finite-time interval. The problem is important and challenging in many practice applications, which motivates us for this study.

This paper deals with the problem of finite-time filtering for one family of singular stochastic systems with parametric uncertainties and time-varying norm-bounded disturbance. Our results are totally different from those previous results, although some studies on filtering and finite-time stability for singular stochastic systems have been addressed, see [19–21, 31, 32, 35]. The main aim of this paper is to design an filtering which guarantees the filtering error system singular stochastic finite-time boundedness and satisfies a prescribed performance level in the given finite-time interval. Sufficient criteria are presented for the solvability of the filtering problems by applying the LMI technique. Finally, simulation examples are presented to demonstrate the validity of the developed theoretical results.

Notations. Throughout the paper, and denote the sets of component real vectors and real matrices, respectively. The superscript stands for matrix transposition or vector. denotes the expectation operator with respective to some probability measure . In addition, the symbol denotes the term that is induced by symmetry and stands for a block-diagonal matrix. and denote the smallest and the largest eigenvalue of matrix , respectively. Notations . and . denote the supremum and infimum, respectively. Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for algebraic operations.

2. Problem Formulation

In this paper, let us consider the dynamics of continuous-time singular system with Markovian jumps: where is the state variable, is the measurement output of the system, is the signal to be estimated, and is a singular matrix with ; is continuous-time Markov stochastic process taking values in a finite space with transition matrix , and the transition probabilities are described as follows: where , satisfies , and for all ; and are uncertain matrices and satisfy where is an unknown, time-varying matrix function and satisfies for all ; moreover, the disturbance input satisfies and the matrices , and are coefficient matrices and of appropriate dimension for all .

In this paper, we construct the following full-order filter: where is the filter state, is the filter output, and , and are to design the filter matrices with appropriate dimensions.

Define and combining (2.1) and (2.5), one can obtain the following filtering error dynamics as follows: where For notational simplicity, in the sequel, for each possible , a matrix will be denoted by ; for instance, will be denoted by , by , and so on.

Throughout the paper, we need the following definitions and lemmas.

Definition 2.1 (regular and impulse free, see [21]). (i) The singular system with Markovian jumps (2.1) is said to be regular in time interval if the characteristic polynomial is not identically zero for all .(ii) The singular systems with Markovian jumps (2.1) is said to be impulse free in time interval , if for all .

Definition 2.2 (singular stochastic finite-time boundedness (SSFTB)). The singular system with Markovian jumps (2.6) which satisfies (2.4) is said to be SSFTB with respect to , with , , if the stochastic system (2.6) is regular and impulse free in time interval and satisfies

Remark 2.3. SSFTB implies that not only is dynamical mode of the filtering error system finite-time bounded but also whole mode of the one is finite-time bounded since the static mode is regular and impulse free.

Definition 2.4 (singular stochastic finite-time boundedness (SSFTB)). The singular system with Markovian jumps (2.6) is said to be SSFTB with respect to , if the singular system with Markovian jumps (2.6) is SSFTB with respect to and under the zero-initial condition, the output error satisfies the cost constrained function for any nonzero which satisfies (2.4), where is a prescribed positive scalar.

Definition 2.5 (see [9]). Let be the stochastic function, define its weak infinitesimal operator of stochastic process by

Lemma 2.6 (see [36]). For matrices , and of appropriate dimensions, where is a symmetric matrix, then holds for all matrix satisfying for all , if and only if there exists a positive constant , such that the following inequality: holds.

Lemma 2.7 (see [36]). The linear matrix inequality is equivalent to , where and .

Lemma 2.8. The following items are true. (i) Assume that , there exist two orthogonal matrices and such that has the decomposition as where with for all . Partition , , and with and . (ii) If satisfies then with and satisfying (2.13) if and only if with . In addition, when is nonsingular, one has and . Furthermore, satisfying (2.14) can be parameterized as where , , and is an arbitrary parameter matrix. (iii) If is a nonsingular matrix, and are two symmetric positive definite matrices, and satisfy (2.14), is a diagonal matrix from (2.16), and the following equality holds: Then the symmetric positive definite matrix is a solution of (2.17).

Proof. One only requires to prove that (ii) and (iii) hold. Let Then by (2.13) and (2.14), it follows that condition if and only if and . In addition, when is nonsingular, it follows that and . Noting that (2.13) and is an orthogonal matrix, thus we have where , with a parameter matrix . Thus (ii) is true.By (i) and (ii), noticing and , we have Thus, is a solution of (2.17). This completes the proof of the lemma.

In the paper, our main objective is to concentrate on designing the filter of system (2.1) which guarantees the resulting filtering error dynamic system (2.6) SSFTB.

3. Main Results

In this section, firstly we give SSFTB analysis results of the filtering problem for nominal system (2.1). Then these results will be extended to the uncertain systems. Linear matrix inequality conditions are established to show the nominal system or the uncertain system (2.6) is finite-time boundedness, and the output error and disturbance satisfy the constrain condition (2.9).

Lemma 3.1. The filtering error system (2.6) is SSFTB with respect to , if there exists a scalar , a set of nonsingular matrices with , two sets of symmetric positive definite matrices with , with , and for all such that the following inequalities hold:

Proof. Firstly, one proves the filtering error system (2.6) is regular and impulse free in time interval . By Lemma 2.7 and noting that condition (3.1b), one has Now, we choose two orthogonal matrices and such that has the decomposition as where with for all . Partition , and with and . Denote Noting that condition (3.1a) and is a nonsingular matrix, by Lemma 2.8, we have and . Pre and postmultiplying by and , it can easily obtain . Therefore is nonsingular, which implies that system (2.6) is regular and impulse free in time interval .Let us consider the quadratic Lyapunov function candidate for system (2.6). Computing , the derivative of along the solution of system (2.6), we obtain where . From (3.1b) and (3.5), we obtain Further, (3.6) can be rewritten as Integrating (3.7) from 0 to , with , we obtain Noting that and condition (3.1c), we have Taking into account that we obtain Therefore, it follows that condition (3.1d) implies for all . This completes the proof of the lemma.

Lemma 3.2. The filtering error system (2.6) is SSFTB with respect to , if there exists a scalar , a set of nonsingular matrices with , a set of symmetric positive definite matrices with , and for all such that (3.1a), (3.1c) and the following inequalities hold:

Proof. Noting that Thus, condition (3.12a) implies that Let for all , by Lemma 3.1, conditions (3.1a), (3.1c), (3.12b), and (3.14) guarantee that system (2.6) is SSFTB with respect to . Therefore, we only need to prove that (2.9) holds. Let and noting that (3.5) and (3.14), we obtain Then using the similar proof as Lemma 3.1, condition (2.9) can be easily obtained and thus is omitted. Therefore, the proof of the lemma is completed.

Denote , and . Using Lemmas 2.7 and 3.2, we obtain the following theorem.

Theorem 3.3. The nominal filtering error system (2.6) is SSFTB with respect to with , if there exists a scalar , a set of nonsingular matrices with , a set of positive definite matrices with , three sets of matrices with , with , with , for all such that hold, where , and . In addition, the desired filter parameters can be chosen by Noting that is nonsingular matrix, by Lemma 2.8, there exist two orthogonal matrices and , such that has the decomposition as where with for all . Partition , , and with and . Let , from (3.16a), is of the following form , and can be expressed as where and with a parameter matrix . If we choose being a symmetric positive definite matrix, then is a symmetric positive definite matrix. Furthermore, the symmetric positive definite matrix is a solution of (3.16c), and satisfies

From the above discussion, we have the following theorem.

Theorem 3.4. The nominal filtering error system (2.6) is SSFTB with respect to with , if there exists a scalar , a set of positive definite matrices with , four sets of matrices with , with , with , and with ,   for all such that (3.16d) and the following linear matrix inequality hold, where , , , and are from the form (3.19); Moreover, other matrical variables are the same as Theorem 3.3.

By Theorems 3.3 and 3.4 and applying Lemmas 2.6–2.8, one can obtain the results stated as follows.

Theorem 3.5. The uncertain filtering error system (2.6) is SSFTB with respect to with , if there exists a scalar , a set of positive definite matrices with , four sets of matrices with , with , with , with , and a set of positive scalars , for all such that (3.16d) and the following linear matrix inequality hold, where , , , and are from the form (3.19); Moreover, other matrical variables are the same as Theorem 3.3.

Remark 3.6. Theorems 3.4 and 3.5 extend the filtering problem of singular stochastic systems to the finite-time filtering problem of singular stochastic systems. In fact, if we fix without condition (3.16d), we can obtain sufficient conditions of the filtering of singular stochastic systems.Let , then one can check that condition (3.16d) can be guaranteed by imposing the conditions

Remark 3.7. The feasibility of conditions stated in Theorem 3.4 and Theorem 3.5 can be turned into the following LMIs-based feasibility problem with a fixed parameter , respectively:

4. Simulation Examples

In this section, numerical results are given to illustrate the effectiveness of the suggested method.

Example 4.1. Consider a two-mode singular stochastic system (2.1) with uncertain parameters as follows: Mode , Mode , and , , where satisfies for all and . In addition, the switching between the two modes is described by the transition rate matrix .Then, we choose , by Theorem 3.5, the optimal bound with minimum value of relies on the parameter . We can find feasible solution when . Figures 1 and 2 show the optimal values with different value of . Noting that when , it yields the optimal values and . Then, by using the program fminsearch in the optimization toolbox of Matlab starting at , the locally convergent solution can be derived as with , and the optimal values , .

Figure 1: The local optimal bound of .

Figure 2: The local optimal bound of .

Remark 4.2. From the above example and Remark 3.7, condition (3.22) in Theorem 3.5 is not strict in LMI form, however, one can find the parameter by an unconstrained nonlinear optimization approach, which a locally convergent solution can be obtained by using the program fminsearch in the optimization toolbox of Matlab.

Example 4.3. Consider a two-mode singular stochastic system (2.1) with uncertain parameters as follows: Moreover, other matrical variables and the transition rate matrix are defined similarly as Example 4.1.Let , then the feasible solution of the above filtering error system can be found when , Theorem 3.5 yields the optimal values , , and Thus, the above filtering error system is stochastically stable and the calculated minimum performance satisfies .

5. Conclusion

In this paper, we deal with the problem of finite-time filtering for a class of singular stochastic systems with parametric uncertainties and time-varying norm-bounded disturbance. Designed algorithms are provided to guarantee the filtering error system SSFTB and satisfy a prescribed performance level in a given finite-time interval, which can be reduced to feasibility problems involving restricted linear matrix equalities with a fixed parameter. Numerical examples are given to demonstrate the validity of the proposed methodology.

Acknowledgments

The authors would like to thank the reviewers and the editors for their very helpful comments and suggestions to improve the presentation of the paper. The paper was supported by the National Natural Science Foundation of P.R. China under Grant 60874006, by the Doctoral Foundation of Henan University of Technology under Grant 2009BS048, by the Natural Science Foundation of Henan Province of China under Grant 102300410118, by the Foundation of Henan Educational Committee under Grant 2011A120003 and 2011B110009, and by the Foundation of Henan University of Technology under Grant 09XJC011.

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