Mathematical Problems in Engineering

Volume 2013, Article ID 574571, 8 pages

http://dx.doi.org/10.1155/2013/574571

## Robust Filtering for a Class of Uncertain Markovian Jump Systems with Time Delays

^{1}Reliability and Systems Engineering School, Beihang University, Beijing 100191, China^{2}China Astronaut Research and Training Center, Beijing 100094, China^{3}School of Electrical and Automation Engineering, Nanjing Normal University, 78 Bancang Street, Nanjing 210042, China

Received 5 May 2013; Accepted 1 July 2013

Academic Editor: Jun Hu

Copyright © 2013 Yi Yang and Junwei Lu. 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 problem of robust filtering for a class of uncertain time-delay systems with Markovian jumping parameters. The system under consideration is subject to norm-bounded time-varying parameter uncertainties. The problem to be addressed is the design of a Markovian jump filter such that the filter error dynamics are stochastically stable and a prescribed bound on the -induced gain from the noise signals to the filter error is guaranteed for all admissible uncertainties. A sufficient condition for the existence of the desired robust filter is given in terms of two sets of coupled algebraic Riccati inequalities. When these algebraic Riccati inequalities are feasible, the expression of a desired filter is also presented. Finally, an illustrative numerical example is provided.

#### 1. Introduction

In the past decades, much attention has been focused on the celebrated Kalman filtering which seems to be one of the most popular estimation approaches; see, for example, [1]. Such a kind of filtering approach assumes that the system is subjected to stationary Gaussian noises with known statistics. However, in practical applications, the statistics of the noise sources may not be exactly known. To deal with this problem, an alternative approach named filtering was proposed, and a great number of results on this topic have been reported in the literature; see, for example, [2–5]. Note that in the filtering setting, the exogenous input signals are assumed to belong to; furthermore, no exact statistics are required to be known. When parameter uncertainties appear not only in the exogenous input but also in the system model, the problem of robust filtering has been investigated and algebraic Riccati equation approach and linear matrix inequality (LMI) approach have been adopted to solve this problem; see [6–9] and the references therein.

Systems with Markovian jumping parameters have been extensively studied due to their both practical and theoretical importance; many issues, such as filtering, stability analysis, and synthesis, on this kind of systems have been studied [10–13]. The problem offiltering for Markovian jumping systems has received much attention. Sufficient conditions for the solvability of this problem were presented in [14], and a design methodology was also proposed based on LMI approach. These results were further extended to uncertain Markovian jump linear systems in [15–18].

In this paper, we consider the problem of robustfiltering for a class of continuous-time uncertain systems with Markovian jump parameters in all system matrices and time delays in the state variables. We consider uncertain systems with norm-bounded time-varying parameter uncertainties. The objective of this paper is to design a Markovian jump linear filter which guarantees both the stochastic stability and a prescribedperformance of the filtering error dynamics for all admissible uncertainties. An algebraic Riccati inequalities approach is developed to solve the previous problem and the desired filter can be constructed by solving two sets of coupled algebraic Riccati inequalities.

*Notation*. Throughout this paper, for symmetric matrices and , the notation (resp., ) means that the matrix is positive semidefinite (resp., positive definite);is the identity matrix with appropriate dimension. The notation represents the transpose of the matrixdenotes the expectation operator with respect to some probability measure ; is the space of square-integrable vector functions over; refers to the Euclidean vector norm;stands for the usualnorm, whiledenotes the norm in;, is a probability space;is used to denote the minimum eigenvalue of the matrix Matrices, if not explicitly stated, are assumed to have compatible dimensions.

#### 2. Problem Formulation

Fix a complete probability space and consider the following class of uncertain stochastic linear systems with Markovian jumping parameters and time delay: where is the system state; is the measurement;is the noise signal which belongs to;is a linear combination of state variables to be estimated;is the time delay of the system; andis the continuous initial-value function. The parameter represents a continuous discrete-state Markov process taking values in a finite setwith transition probability matrix given by where,, and is the transition rate from modeat time to modeat time and In system , , and are appropriately dimensioned real-valued matrix functions of . For simplicity of notations, in the sequel, for each possible , , we will denote the matrices associated with the mode by where , , , , , and , are known constant matrices representing the nominal system for each and , , and are unknown matrices representing time-varying parameter uncertainties and are assumed to be of the form where , , , , , and , for each , are known constant matrices and , , and , for each , are the uncertain time-varying matrices satisfying It is assumed that all the elements of , , and are Lebesgue measurable. , , and , for each , are said to be admissible if both (6) and (7) hold.

Throughout the paper we will use the following concept of stochastic stability.

*Definition 1. *Consider the following stochastic jump system:
The system (8) is said to be stochastically stable, if, for finite defined on and , there exists a scalar such that
where denotes the solution of system (8) at time under the initial conditions and .

Now, for each , consider a Markovian filter of the form where is the estimator state and the matrices , , and are to be chosen. The filtering error is defined by

Then the robust filtering problem to be dealt with in this paper can be formulated as determining a filter of the form (10) such that, for all admissible uncertainties , , and , , the following requirements are satisfied:(R1) the augmented system from system and the filter is stochastically stable;(R2) with zero initial conditions, the following holds: for all nonzero , where is a prescribed scalar.

Before concluding this section, we introduce the following lemma which will be used in the proof of our main results in the next section.

Lemma 2 (see [19, 20]). *Let , , , , and be real matrices of appropriate dimensions with and satisfying . Then one has the following:*(1)* for any scalar ,
*(2)* for any scalar such that ,
*(3)*. *

#### 3. Main Results

In this section, a Riccati-like inequality approach is proposed to design a robust filter for uncertain stochastic linear systems with Markovian jumping parameters and time delay. Our main result is presented in the next theorem.

Theorem 3. *Consider the uncertain jump linear system. If there exist scalars , , , and and matrices , , , , and , such that , , and the following coupled Riccati matrix inequalities hold:
**
where
**
for , then the robust filtering problem is solvable. In this case, a suitable robust filter is given in form (10) with parameters as follows:
*

*Proof. *Let the mode at time be , that is; , , and define
then from system and filter , it is easy to show that
Therefore, by defining , we obtain the augmented system from and as
where
For , , we define a matrix by
Next, we will show that under the conditions of the theorem, the following matrix inequality
holds, when , , where
To this end, we use Lemma 2 to obtain the following matrix inequalities:
where (7) and
are used and
Hence
Substituting (16) and (17) into the right-hand side of the previous inequality and using algebraic manipulations we have that (23) holds.

Now, we will show the stochastic stability of the augmented system when . In the following we will simply use to stand for the solution of the system at time with the initial conditions and .

Introduce the following stochastic Lyapunov functional candidate for system :
It can then be shown that the weak infinitesimal generator of the random process is given by
for any , . Then, from Lemma 2, it follows that
Using (23), we have
where . From (15), it is easy to see that .

Now using Dynkin’s formula, we have, for each , , ,
On the other hand, for each , , we can show that
where and . The previous inequality and (33) imply that
where and . Then, it can be verified that
Therefore,
Taking limit as , we have
Note that there always exists a scalar such that
Considering (38), we have that the augmented system is stochastically stable. Next we will show that
for all nonzero . To this end, we introduce
For any , , , we have
Thus, under zero initial condition, for any , , , ,
Hence
Finally, from (23), (40) follows immediately. This completes the proof.

*Remark 4. *Theorem 3 provides a sufficient condition for the solvability of robust filtering problem for uncertain stochastic time-delay systems with Markovian jumping parameters, and the desired filter can be constructed by solving two sets of coupled algebraic Riccati inequalities.

In the case when , that is, there is only one mode operation, we have , , and system reduces to the following uncertain time-delay system with no jumping parameters: where , , and are unknown matrices and are assumed to be of the form where , , , , , and are known constant matrices and , , and are the uncertain time-varying matrices satisfying Then, from Theorem 3, we have the following robust filtering result for the previous system.

Corollary 5. *Consider the uncertain time-delay system . If there exist scalars , , and matrices , , , and , such that , and the following coupled Riccati matrix inequalities hold:
**
where
**
then the robust filtering problem is solvable. In this case, a suitable robust filter is given by
**
where
*

*Remark 6. *The solvability for robust filtering problem for uncertain continuous delay-free systems can be easily derived from Corollary 5, in this case. It can be shown that the result coincides with that proposed in [8].

#### 4. Numerical Example

In this section, we will give a numerical example to demonstrate the applicability of the proposed approach.

Consider the uncertain time-delay stochastic linear systems with Markovian jumping parameters in form (1) with two modes. For mode the dynamics of the system are described as follows: For mode , the dynamics of the system are described as follows: Let the transition probability matrix be given by In this example, we set . By solving (15), we obtain Therefore, using Theorem 3, a suitable Markovian robust filter can be constructed with parameters given as follows:

#### 5. Conclusions

In this paper, we have studied the problem of robust filtering for a class of uncertain Markovian jump systems with time-delay and norm-bounded time-varying parameter uncertainties. A Markovian jump filter is designed which guarantees the stochastic stability of the filter error dynamics and a prescribed bound on the -induced gain from the noise signals to the filter error irrespective of the parameter uncertainties. It has been shown that the desired robust filter can be constructed by solving two sets of coupled algebraic Riccati inequalities.

#### Acknowledgments

This work was supported in part by the National Natural Science Foundation of China (61104132) and Director of the foundation (SJ201002).

#### References

- B. D. O. Anderson and J. B. Moore,
*Optimal Filtering*, Prentice-Hall, Englewood Cliffs, NJ, USA, 1979. - D. S. Bernstein and W. M. Haddad, “Steady-state Kalman filtering with an ${H}_{\infty}$ error bound,”
*Systems & Control Letters*, vol. 12, no. 1, pp. 9–16, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. J. Grimble and A. Elsayed, “Solution of the ${H}_{\infty}$ optimal linear filtering problem for discrete-time systems,”
*Institute of Electrical and Electronics Engineers*, vol. 38, no. 7, pp. 1092–1104, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. M. Nagpal and P. P. Khargonekar, “Filtering and smoothing in an ${H}_{\infty}$ setting,”
*Institute of Electrical and Electronics Engineers*, vol. 36, no. 2, pp. 152–166, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. O. S. Teixeira, J. Chandrasekar, H. J. Palanthandalam-Madapusi, L. A. B. Tôrres, L. A. Aguirre, and D. S. Bernstein, “Gain-constrained Kalman filtering for linear and nonlinear systems,”
*IEEE Transactions on Signal Processing*, vol. 56, no. 9, pp. 4113–4123, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - M. Fu, C. E. de Souza, and L. Xie, “${H}_{\infty}$ estimation for uncertain systems,”
*International Journal of Robust and Nonlinear Control*, vol. 2, pp. 87–105, 1992. View at Google Scholar - H. Li and M. Fu, “An LMI approach to robust ${H}_{\infty}$ filtering for linear systems,” in
*Proceedings of the 34th IEEE Conference Decision and Control*, pp. 3608–3613, New Orleans, La, USA, December 1995. - L. Xie and C. E. de Souza, “On robust filtering for linear systems with parameter uncertainty,” in
*Proceedings of the 34th IEEE Conference Decision and Control*, pp. 2087–2092, New Orleans, La, USA, December 1995. - L. Xie, C. E. de Souza, and M. Fu, “${H}_{\infty}$ estimation for discrete-time linear uncertain systems,”
*International Journal of Robust and Nonlinear Control*, vol. 1, pp. 111–123, 1991. View at Google Scholar - H. J. Chizeck, A. S. Willsky, and D. Castañon, “Discrete-time Markovian-jump linear quadratic optimal control,”
*International Journal of Control*, vol. 43, no. 1, pp. 213–231, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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 - Y. Ji and H. J. Chizeck, “Controllability, stabilizability, and continuous-time Markovian jump linear quadratic control,”
*Institute of Electrical and Electronics Engineers*, vol. 35, no. 7, pp. 777–788, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. Zhang and Y. Li, “Exponential ${L}_{2}$
-${H}_{\infty}$ filtering for distributed delay systems with Markovian jumping
parameters,”
*Signal Processing*, vol. 93, pp. 206–216, 2013. View at Google Scholar - C. E. de Souza and M. D. Fragoso, “${H}_{\infty}$ filtering for Markovian jump linear systems,” in
*Proceedings of the 35th IEEE Conference on Decision and Control*, pp. 4814–4818, Kobe, Japan, December 1996. - C. E. de Souza and M. D. Fragoso, “Robust ${H}_{\infty}$ filtering for Markovian jump linear systems,” in
*Proceedings of the 35th IEEE Conference Decision and Control*, pp. 4808–4813, Kobe, Japan, December 1996. - C. E. de Souza and M. D. Fragoso, “Robust ${H}_{\infty}$ filtering for uncertain Markovian jump linear systems,”
*International Journal of Robust and Nonlinear Control*, vol. 12, no. 5, pp. 435–446, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Wen, Z. Zeng, and T. Huang, “Reliable ${H}_{\infty}$ filter design for a class of mixed-delay Markovian jump
systems with stochastic nonlinearities and multiplicative noises via delay-partitioning method,”
*International Journal of Control, Automation and Systems*, vol. 10, pp. 711–720, 2012. View at Google Scholar - R. Zhang, Y. Zhang, C. Hu, M. Q.-H. Meng, and Q. He, “Delay-range-dependent ${H}_{\infty}$ filtering for two-dimensional Markovian jump systems with interval delays,”
*IET Control Theory & Applications*, vol. 5, no. 18, pp. 2191–2199, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - X. Li and C. E. de Souza, “Criteria for robust stability and stabilization of uncertain linear systems with state delay,”
*Automatica*, vol. 33, no. 9, pp. 1657–1662, 1997. View at Publisher · View at Google Scholar · View at MathSciNet - X. Li and C. E. de Souza, “Delay-dependent robust stability and stabilization of uncertain linear delay systems: a linear matrix inequality approach,”
*Institute of Electrical and Electronics Engineers*, vol. 42, no. 8, pp. 1144–1148, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet