Elsevier

Automatica

Volume 109, November 2019, 108528
Automatica

Brief paper
H filtering for singular Markovian jump systems with partly unknown transition rates

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Abstract

This paper suggests H filtering for singular Markovian jump systems (SMJSs) with partly unknown transition rates. While the existing researches in the literature have derived only the sufficient conditions, this paper successfully demonstrates the necessary and sufficient condition for the existence of H filtering for SMJSs with partly unknown transition rates. First, a new bounded real lemma for SMJS is derived in terms. Next, the matrix inequalities which come from the bounded real lemma for the filtering error system with partly unknown transition rates are reformulated into the strict LMIs. Two numerical examples are presented to show the effectiveness of the proposed result.

Introduction

Filtering methods which estimate states or unmeasurable variables have been important research topics in control and signal processing areas for the last few decades (NoRgaard, Poulsen, & Ravn, 2000). Among them, H filtering has been widely used to estimate the output of systems with external disturbance (Xu, Zhang, & Zhao, 2018).

On the other hand, Markovian jump systems (MJSs), which are a special class of hybrid and stochastic systems, can be modeled as a set of subsystems with Markov chain taking values in a finite set (Cao, Niu, & Zhao, 2018). Especially, singular Markovian jump systems (SMJSs), which describe the singular systems with suddenly changing parameters, were also extensively studied (Kwon, Park, Park, Park et al., 2017, Xu and Lam, 2006). These researches were performed for SMJSs with exactly known transition rates.

In practical systems, however, it is difficult to determine the exact values of the transition rates because measuring the transition rate is sometimes impossible or requires a high cost. Therefore, there have been many studies on SMJSs with imperfect transition rates such as partly unknown transition rates (Li & Zhang, 2016) or uncertain transition rates (Kao, Xie, & Wang, 2014). In particular, with the assumption that the lower bounds of the unknown self-transition rates are known, the paper (Zhang & Lam, 2010) derived the necessary and sufficient condition of the stochastic stability criterion of MJS with partly unknown transition rates.

The goal of H filtering for SMJS is to find a full-order filter which allows the filtering error system to be stochastically admissible with γ-disturbance attenuation. Therefore, the recent studies have been focused on deriving the solvable conditions for filter parameters from the stochastic admissibility with γ-disturbance attenuation criterion of the filtering error system. The papers (Boukas, 2008, Jianwei, 2007, Wu et al., 2010) suggested H filtering for SMJS with time-delay or model uncertainty in terms of non-strict linear matrix inequalities (LMIs) (Boukas, 2008, Jianwei, 2007) and strict LMIs (Wu et al., 2010). Besides, the authors Wang, Zhang, and Zhang (2014) considered H filtering for SMJS with various transition rates: completely known, partly unknown and uncertain transition rates. These studies were based on only the sufficient condition for stochastic admissibility with γ-disturbance attenuation criterion of the filtering error system. To the best of our knowledge, however, there have been no studies which derive the necessary and sufficient condition for H filtering for SMJSs, which is the motivation behind this study.

This paper proposes H filtering for continuous-time SMJSs with partly unknown transition rates. First, this paper derives a new bounded real lemma which shows the necessary and sufficient condition of stochastic admissibility with γ-disturbance attenuation for SMJS with completely known transition rates. Since the bounded real lemma for the filtering error system is expressed in terms of non-convex matrix inequality, this paper converts it to strict LMIs with the help of the variable elimination lemma. The numerical examples show the effectiveness of proposed H filtering.

The notations used in this paper are fairly standard. For xRn, xT means the transpose of x. For symmetric matrices X and Y, the expression X>()Y denotes that the matrix XY is positive (semi) definite. For any matrices Ai and a set S={s1,s2,,sk}, [Ai]iS=[As1As2Ask] and diag[Ai]iS=diag(As1,As2,,Ask). Also, AA=diag(A,A) and rank(A) represents the dimension of the vector space spanned by its columns. For a square matrix X, He(X)=X+XT. I is an identity matrix with appropriate dimension. In symmetric block matrices, the notation () is used as an ellipsis for terms induced by symmetry.

Section snippets

Problem statement

Consider a continuous-time singular Markov jump system (SMJS) with external disturbance Eẋ(t)=A(rt)x(t)+B(rt)w(t),y(t)=F(rt)x(rt),z(t)=C(rt)x(t), where x(t)Rnx, w(t)Rnw, z(t)Rnz and y(t)Rny denote the state, the disturbance input, the performance output and the measured output, respectively. To describe the singular property of SMJS, the matrix ERnx×nx is supposed to be singular with rank(E)=r<nx. Here, to deal with the singular property of SMJS, we define two full column rank matrices ER

Main result

Before considering H filtering, a bounded real lemma for SMJS is established.

Numerical examples

Example 1

Consider the SMJS (1)–(3) with the following system parameters: A1=5122.5,A2=5141,A3=10.515.5,B1=1.31T,B2=0.50.2T,B3=0.70.5T,F1=0.91.5,F2=1.10.2,F3=0.21,C1=11,C2=1.53,C3=10.5,E=1000,EL=ER=10. and two types of the transition rate matrix are given as

Πk=10.580.420.52.21.71.20.51.7,Πu.k=1???2.21.71.2??. To construct H filter with minimal H performance γ, we solve an optimization problem which minimizes γ2 subject to Theorem 2, and obtain the filtering result (Fig. 1). Also,

Conclusion

This paper considered H filtering for SMJS with partly unknown transition rates. First, a bounded real lemma for SMJS was proposed in terms of strict LMIs. Since the criterion for H filtering for SMJS was expressed as non-convex conditions, elimination lemma was used to reformulate it into in terms of strict LMIs. Finally, the result was extended to SMJS with partly unknown transition rates. Two numerical examples showed the validity of the proposed results.

Chan-eun Park received her B.S. degree from School of Electrical Engineering and Computer Science at Ulsan National Institute of Science and Technology (UNIST), Korea, in 2014, and the M.S. degree from the Division of Electrical and Computer Engineering at Pohang University of Science and Technology (POSTECH), Korea, in 2016. She is currently a Ph.D. candidate in Electric and Electrical Engineering at POSTECH. Her research interest includes stochastic system, singular system and robust control

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Chan-eun Park received her B.S. degree from School of Electrical Engineering and Computer Science at Ulsan National Institute of Science and Technology (UNIST), Korea, in 2014, and the M.S. degree from the Division of Electrical and Computer Engineering at Pohang University of Science and Technology (POSTECH), Korea, in 2016. She is currently a Ph.D. candidate in Electric and Electrical Engineering at POSTECH. Her research interest includes stochastic system, singular system and robust control theory.

Nam Kyu Kwon received his B.S. degree and Ph.D. degree from the Division of Electrical and Computer Engineering at Pohang University of Science and Technology (POSTECH), Korea, in 2010 and 2017, respectively. He is currently an assistant professor in Yeungnam University. His research interest includes artificial intelligent and robust control theory.

In Seok Park received his B.S. degree from School of Electrical Engineering and Computer Science at Kyungpook National University (KNU), Korea, in 2014, and the M.S. degree from the Division of Electrical and Computer Engineering at Pohang University of Science and Technology (POSTECH), Korea, in 2016. He is currently a Ph.D. candidate in Electric and Electrical Engineering at POSTECH. His research interest includes fuzzy system, stochastic system and robust control theory.

PooGyeon Park received his B.S. degree and M.S. degree in Control and Instrumentation Engineering from Seoul National University, Korea, in 1988 and 1990, respectively, and the Ph.D. degree in Electrical Engineering from Stanford University, U.S.A., in 1995. Since 1996, he has been affiliated with the Division of Electrical and Computer Engineering at Pohang University of Science and Technology (POSTECH), where he is currently a professor. His current research interests include robust, LPV, and network-related control theories, delayed systems, fuzzy systems, and signal processing.

This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT, and Future Planning (2017R1D1A1A09000787). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Antonio Vicino under the direction of Editor Torsten Söderström

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