About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 791296, 7 pages
http://dx.doi.org/10.1155/2013/791296
Research Article

Resilient - Filtering of Uncertain Markovian Jumping Systems within the Finite-Time Interval

College of Electrical Engineering and Automation, Anhui University, Hefei 230601, China

Received 16 November 2012; Revised 10 March 2013; Accepted 17 March 2013

Academic Editor: Gani Stamov

Copyright © 2013 Shuping He. 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 studies the resilient - filtering problem for a class of uncertain Markovian jumping systems within the finite-time interval. The objective is to design such a resilient filter that the finite-time - gain from the unknown input to an estimation error is minimized or guaranteed to be less than or equal to a prescribed value. Based on the selected Lyapunov-Krasovskii functional, sufficient conditions are obtained for the existence of the desired resilient - filter which also guarantees the stochastic finite-time boundedness of the filtering error dynamic systems. In terms of linear matrix inequalities (LMIs) techniques, the sufficient condition on the existence of finite-time resilient - filter is presented and proved. The filter matrices can be solved directly by using the existing LMIs optimization techniques. A numerical example is given at last to illustrate the effectiveness of the proposed approach.

1. Introduction

More recently, the finite-time stability and control problems have received great attention in the literature; see [16]. Compared with the Lyapunov stable dynamical systems, a finite-time stable dynamical system does not require the steady-state behavior of control dynamics over an infinite-time interval and the asymptotic pattern of system trajectories. The main attention may be related to the transient characteristics of the dynamical systems over a fixed finite-time interval, for instance, keeping the acceptable values in a prescribed bound in the presence of saturations. However, more details are related to the stability and control problems of various dynamic systems, and very few reports in the literature consider the filtering problems.

Since the Kalman filtering theory [7] has been introduced in the early 1960s, the filtering problem has been extensively investigated. In the filtering scheme, its objective is to estimate the unavailable state variables (or a linear combination of the states) of a given system. During the past decades, many filtering schemes have been developed, such as Kalman filtering [8], filtering [9], reduced-order filtering [10], and - filtering [11]. Then, extension of this effort to the problem of resilient Kalman filtering with respect to estimator gain perturbations was considered in [12]. And the resilient filtering [13] was also raised. Among the filtering schemes, the resilient - filtering was not considered. In practical engineering applications, the peak values of filtering error should always be considered. Compared with the filtering scheme, the external disturbances are both assumed to be energy bounded; but - filtering setting requires the mapping from the external disturbances to the filtering error is minimized or no larger than some prescribed level in terms of the - performance norm.

In this paper, we have studied the resilient finite-time - filtering problem for uncertain Markouvian Jumping Systems (MJSs). Firstly, the augmented filtering error dynamic system is constructed based on the state estimated filter with resilient filtering parameters. Secondly, a sufficient condition is established on the existence of the robust finite-time filter such that the filtering error dynamic MJSs are finite-time bounded and satisfy a prescribed level of - disturbance attenuation with the finite-time interval. And the design criterion is presented by means of LMIs techniques. Subsequently, the robust finite-time - filter matrices can be solved directly by using the existing LMIs optimization algorithms. In order to illustrate the proposed result, a numerical example is given at last.

Let us introduce some notations. The symbols and stand for an -dimensional Euclidean space and the set of all real matrices, respectively, and denote the matrix transpose and matrix inverse, diag represents the block-diagonal matrix of and , and , respectively, denote the maximal and minimal eigenvalues of a real matrix , denotes the Euclidean norm of vectors, denotes the mathematics statistical expectation of the stochastic process or vector, stands for a positive-definite matrix, is the unit matrix with appropriate dimensions, 0 is the zero matrix with appropriate dimensions, and means the symmetric terms in a symmetric matrix.

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 us consider a class of linear uncertain MJSs defined in the probability space and described by the following differential equations: where is the state, is the measured output, is the unknown input, is the controlled output, and and are, respectively, the initial states and mode. , , , , and are known mode-dependent constant matrices with appropriate dimensions. The jump parameter represents a continuous-time discrete-state Markov stochastic process taking values on a finite set with transition rate matrix , , and has the following transition probability from mode at time to mode at time as where and .

In this relation, is the transition probability rates from mode at time to mode at time , and

For presentation convenience, we denote , , , , , , and as , , , , , , and , respectively.

And the matrices with the symbol are considered as the uncertain matrices satisfying the following conditions: where , , and are known mode-dependent matrices with appropriate dimensions, and is the time-varying unknown matrix function with Lebesgue norm measurable elements satisfying .

Remark 1. It is always impossible to obtain the exact mathematical model of practical dynamics due to the complexity process, the environmental noises, and the difficulties of measuring various and uncertain parameters, and so forth; thus, the model of practical dynamics to be controlled almost contains some types of uncertainties. In general, the uncertainties in (1) satisfying the restraining conditions (4) and are said to be admissible. The unknown mode-dependent matrix can also be allowed to be state dependent; that is, , as long as is satisfied.

We now consider the following resilient filter: where is the filter state, is the filer output, is the initial estimation states, and the mode-dependent matrices , , and are unknown filter parameters to be designed for each value . and are uncertain filter parameter matrices satisfying the following conditions: where , , , and are defined similarly as (4).

The objective of this paper is to design the resilient - filter of uncertain MJSs in (1) and obtain an estimate of the signal such that the defined guaranteed performance criteria are minimized in an - estimation error sense. Define and , such that the filtering error dynamic MJSs are given by

Let , and we have where

The external disturbance is varying and satisfies the constraint condition with respect to the finite-time interval as follows: where is a positive scalar.

Definition 2. Given a time-constant , the filtering error dynamic MJSs (8) (setting ) are said to be stochastically finite-time stable (FTS) with respect to , if where , , , and is the weight coefficient matrix.

Definition 3. Given a time-constant , the filtering error dynamic MJSs (8) are stochastically finite-time bounded (FTB) with respect to , wherein , , , if condition (10) holds.

Definition 4. For the filtering error dynamic MJSs (8), if there exist filter parameters , , and , as well as a positive scalar , such that the filtering error dynamic MJSs (8) are stochastically 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 (10): where = , = .
Then, the resilient filter (5) is called the stochastic finite-time - filter of the uncertain dynamic MJSs (1) with -disturbance attenuation.

3. Main Results

In this section, we will study the robust stochastic finite-time resilient filtering problem for the filtering error dynamic MJSs (8) in an - estimation error sense. Before proceeding with the study, the following lemma is needed.

Lemma 5 (see [14]). 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

Theorem 6. For given , , , and , the filtering error dynamic MJSs (8) are stochastically FTB with respect to and have a - performance level , if there exist positive constants, , , and mode-dependent symmetric positive-definite matrices , such that where , , and .

Proof. Let the mode at time be ; that is, . Take the stochastic Lyapunov-Krasovskii functional as
Then, we introduce a weak infinitesimal generator (see [15, 16]), acting on , for all , which is defined as
The time derivative of along the trajectories of the filtering error dynamic MJSs (8) is given by
Considering the - filtering performance for the dynamic filtering error system (8), we introduce the following cost function by Definition 4 with :
It follows from relation (16) that ; that is,
Then, multiplying the previous inequality by , we have
Integrating the above inequality from to , we have
Considering , as well as the zero initial condition; that is, , for , then it follows that
Then, it can be verified from the defined Lyapunov-Krasovskii functional that
By (15) and within the finite-time interval , we can also get
Since the previous inequality is always true for any , the following relation:
It is easy to get the following result:
Therefore, the cost function inequality (10) can be guaranteed, which implies .
Denote that , , and . From equality (24), we have
On the other hand, it results from the stochastic Lyapunov-Krasovskii functional that
Then, we can get
It implies that for all , we have by condition (17). This completes the proof.

Theorem 7. For given , , , and , the filtering error dynamic MJSs (8) are stochastically FTB with respect to with and have a prescribed - performance level , if there exist a set of mode-dependent symmetric positive-definite matrices , a set of mode-dependent matrices , , and a positive scalar and mode-dependent sequences , satisfying the following matrix inequalities for all : where  +  + ,  − ,  +  −  + .
Moreover, the suitable filter parameters can be given as

Proof. For convenience, we set . Then, we can get the following relations according to matrix inequalities (15) and (16): where
Referring to Lemma 5, and can be presented as the following form: where , ,  ,  ,   = , and  .
Then, inequalities (15) and (16) are equivalent to LMIs (38) and (39) by letting and .
On the other hand, we set , and LMI (35) implies that
Then, recalling condition (17), we can get LMI (36). This completes the proof.

To obtain an optimal finite-time - filtering performance against unknown inputs, uncertainties, and model errors, the attenuation lever can be reduced to the minimum possible value such that LMIs (33)–(36) are satisfied. The optimization problem can be described as follows:

Remark 8. In Theorems 6 and 7, if is a variable to be solved, then (17) and (36) can be always satisfied as long as is sufficiently large. For these, we can also fix and look for the optimal admissible or guaranteeing the stochastic finite-time boundedness of desired filtering error dynamic properties.

4. Numeral Examples

Example 9. Consider a class of MJSs with two operation modes described as follows.
Mode 1:
Mode 2: The mode switching is governed by a Markov chain that has the following transition rate matrix:

In this paper, we choose the initial values for , , , and . Then, we fix and look for the optimal admissible with different which can guarantee the stochastic finite-time boundedness of desired filtering error dynamic properties. Figure 1 gives the optimal minimal admissible with different initial upper bound .

791296.fig.001
Figure 1: The optimal minimal upper bound with different initial .

For , we solve LMIs (33)–(36) by Theorem 7 and the optimization algorithm (43) and get the following optimal resilient finite-time - filter: And then, we can also get the attenuation lever as and the relevant upper bound .

To show the effectiveness of the designed optimal resilient finite-time - filter, we assume that the unknown inputs are unknown white noise with noise power 0.05 over a finite-time interval . For the selected initial conditions and , the simulation results of the jumping modes and the response of system states are shown in Figures 2, 3, and 4. It is clear from the simulation figures that the estimated states can track the real states smoothly.

791296.fig.002
Figure 2: System response with state .
791296.fig.003
Figure 3: System response with state .
791296.fig.004
Figure 4: System response with output error .

5. Conclusion

The resilient finite-time - filtering problems for a class of stochastic MJSs with uncertain parameters have been studied. By using the Lyapunov-Krasovskii functional approach and LMIs optimization techniques, a sufficient condition is derived such that the filtering dynamic error MJSs are finite-time bounded and satisfy a prescribed level of - disturbance attenuation in a finite time-interval. The robust resilient filter gains can be solved directly by using the existing LMIs. Simulation results illustrate the effectiveness of the proposed a pproach.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China (Grant no. 61203051), the Joint Specialized Research Fund for the Doctoral Program of Higher Education (Grant no. 20123401120010), and the Key Program of Natural Science Foundation of Education Department of Anhui Province (Grant no. KJ2012A014).

References

  1. P. Dorato, “Short time stability in linear time-varying systems,” in Proceedings of the IRE international Convention Record, pp. 83–87, 1961.
  2. F. Amato, R. Ambrosino, C. Cosentino, and G. De Tommasi, “Input-output finite time stabilization of linear systems,” Automatica, vol. 46, no. 9, pp. 1558–1562, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. Y. Zhang, C. Liu, and X. Mu, “Robust finite-time stabilization of uncertain singular Markovian jump systems,” Applied Mathematical Modelling, vol. 36, no. 10, pp. 5109–5121, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. J. Zhou, S. Xu, and H. Shen, “Finite-time robust stochastic stability of uncertain stochastic delayed reaction—diffusion genetic regulatory networks,” Neurocomputing, vol. 74, no. 17, pp. 2790–2796, 2011.
  5. S. He and F. Liu, “Finite-time H control of nonlinear jump systems with time-delays via dynamic observer-based state feedback,” IEEE Transactions on Fuzzy Systems, vol. 20, no. 4, pp. 605–614, 2012.
  6. H. Liu, Y. Shen, and X. Zhao, “Delay-dependent observer-based H finite-time control for switched systems with time-varying delay,” Nonlinear Analysis: Hybrid Systems, vol. 6, no. 3, pp. 885–898, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. R. E. Kalman, “A new approach to linear filtering and prediction problems,” Journal of Basic Engineering D, vol. 82, pp. 35–45, 1960.
  8. P. Shi, E.-K. Boukas, and R. K. Agarwal, “Kalman filtering for continuous-time uncertain systems with Markovian jumping parameters,” Institute of Electrical and Electronics Engineers, vol. 44, no. 8, pp. 1592–1597, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. Y. He, G. P. Liu, D. Rees, and M. Wu, “H filtering for discrete-time systems with time-varying delay,” Signal Processing, vol. 89, no. 3, pp. 275–282, 2009. View at Publisher · View at Google Scholar · View at Scopus
  10. S. Xu and J. Lam, “Reduced-order H filtering for singular systems,” Systems & Control Letters, vol. 56, no. 1, pp. 48–57, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. H. Zhang, A. S. Mehr, and Y. Shi, “Improved robust energy-to-peak filtering for uncertain linear systems,” Signal Processing, vol. 90, no. 9, pp. 2667–2675, 2010. View at Publisher · View at Google Scholar · View at Scopus
  12. G.-H. Yang and J. L. Wang, “Robust nonfragile Kalman filtering for uncertain linear systems with estimator gain uncertainty,” Institute of Electrical and Electronics Engineers, vol. 46, no. 2, pp. 343–348, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. M. S. Mahmoud, “Resilient linear filtering of uncertain systems,” Automatica, vol. 40, no. 10, pp. 1797–1802, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. Y. Y. Wang, L. Xie, and C. E. de Souza, “Robust control of a class of uncertain nonlinear systems,” Systems & Control Letters, vol. 19, no. 2, pp. 139–149, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. X. Feng, K. A. Loparo, Y. Ji, and H. J. Chizeck, “Stochastic stability properties of jump linear systems,” Institute of Electrical and Electronics Engineers, vol. 37, no. 1, pp. 38–53, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. X. Mao, “Stability of stochastic differential equations with Markovian switching,” Stochastic Processes and their Applications, vol. 79, no. 1, pp. 45–67, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet