Table of Contents Author Guidelines Submit a Manuscript
Journal of Applied Mathematics
VolumeΒ 2012Β (2012), Article IDΒ 615790, 16 pages
http://dx.doi.org/10.1155/2012/615790
Research Article

Finite-Time 𝐻∞ Filtering for Singular Stochastic Systems

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.Β MarcellΓ‘n

Copyright Β© 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 𝐻∞ 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 𝐻∞ finite-time boundedness are presented. Then, the 𝐻∞ 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 𝐻∞ 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, 𝐻∞ 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 𝐻∞ 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 𝐻∞ 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 𝐻∞ 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 𝐻∞ 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 𝐻∞ filtering which guarantees the filtering error system singular stochastic finite-time boundedness and satisfies a prescribed 𝐻∞ 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, ℝ𝑛 and β„π‘›Γ—π‘š denote the sets of 𝑛 component real vectors and π‘›Γ—π‘š real matrices, respectively. The superscript 𝑇 stands for matrix transposition or vector. 𝔼{β‹…} denotes the expectation operator with respective to some probability measure β„™. In addition, the symbol βˆ— denotes the term that is induced by symmetry and diag{β‹―} stands for a block-diagonal matrix. πœ†min(𝑃) and πœ†max(𝑃) denote the smallest and the largest eigenvalue of matrix 𝑃, respectively. Notations sup. and inf. 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:ξ€Ίπ΄ξ€·π‘ŸπΈΜ‡π‘₯(𝑑)=π‘‘ξ€Έξ€·π‘Ÿ+Δ𝐴𝑑π‘₯ξ€Ίπ΅ξ€·π‘Ÿξ€Έξ€»(𝑑)+π‘‘ξ€Έξ€·π‘Ÿ+Ξ”π΅π‘‘π‘€ξ€·π‘Ÿξ€Έξ€»(𝑑),𝑦(𝑑)=πΆπ‘‘ξ€Έξ€·π‘Ÿπ‘₯(𝑑)+π·π‘‘ξ€Έπ‘§ξ€·π‘Ÿπ‘€(𝑑),(𝑑)=𝐿𝑑π‘₯ξ€·π‘Ÿ(𝑑)+𝐺𝑑𝑀(𝑑),(2.1) where π‘₯(𝑑)βˆˆβ„π‘› is the state variable, 𝑦(𝑑)βˆˆβ„π‘ž1 is the measurement output of the system, 𝑧(𝑑)βˆˆβ„π‘ž2 is the signal to be estimated, and 𝐸 is a singular matrix with rank(𝐸)=π‘Ÿ<𝑛; {π‘Ÿπ‘‘,𝑑β‰₯0} is continuous-time Markov stochastic process taking values in a finite space π•„βˆΆ={1,2,…,𝑁} with transition matrix Ξ“=(πœ‹π‘–π‘—)𝑁×𝑁, and the transition probabilities are described as follows:π‘ƒπ‘–π‘—ξ€·π‘Ÿ=π‘ƒπ‘Ÿπ‘‘+Ξ”=π‘—βˆ£π‘Ÿπ‘‘ξ€Έ=ξ‚»πœ‹=𝑖𝑖𝑗Δ+π‘œ(Ξ”),if𝑖≠𝑗,1+πœ‹π‘–π‘—Ξ”+π‘œ(Ξ”),if𝑖=𝑗,(2.2) where limΞ”β†’0π‘œ(Ξ”)/Ξ”=0, πœ‹π‘–π‘— satisfies πœ‹π‘–π‘—β‰₯0(𝑖≠𝑗), and πœ‹π‘–π‘–βˆ‘=βˆ’π‘π‘—=1,π‘—β‰ π‘–πœ‹π‘–π‘— for all 𝑖,π‘—βˆˆπ•„; Δ𝐴(π‘Ÿπ‘‘) and Δ𝐡(π‘Ÿπ‘‘) are uncertain matrices and satisfyξ€Ίξ€·π‘ŸΞ”π΄π‘‘ξ€Έξ€·π‘Ÿ,Ξ”π΅π‘‘ξ€·π‘Ÿξ€Έξ€»=πΉπ‘‘ξ€ΈΞ”ξ€·π‘Ÿπ‘‘πΈξ€Έξ€Ί1ξ€·π‘Ÿπ‘‘ξ€Έ,𝐸2ξ€·π‘Ÿπ‘‘,ξ€Έξ€»(2.3) where Ξ”(π‘Ÿπ‘‘) is an unknown, time-varying matrix function and satisfies Δ𝑇(π‘Ÿπ‘‘)Ξ”(π‘Ÿπ‘‘)≀𝐼 for all π‘Ÿπ‘‘βˆˆπ•„; moreover, the disturbance input 𝑀(𝑑)βˆˆβ„π‘ satisfiesξ€œπ‘‡0𝑀𝑇(𝑑)𝑀(𝑑)𝑑𝑑≀𝑑2,𝑑β‰₯0,(2.4) and the matrices 𝐴(π‘Ÿπ‘‘),𝐡(π‘Ÿπ‘‘),𝐢(π‘Ÿπ‘‘),𝐷(π‘Ÿπ‘‘),𝐿(π‘Ÿπ‘‘), and 𝐺(π‘Ÿπ‘‘) are coefficient matrices and of appropriate dimension for all π‘Ÿπ‘‘βˆˆπ•„.

In this paper, we construct the following full-order filter:𝐸𝑓̇̃π‘₯(𝑑)=π΄π‘“ξ€·π‘Ÿπ‘‘ξ€ΈΜƒπ‘₯(𝑑)+π΅π‘“ξ€·π‘Ÿπ‘‘ξ€Έπ‘¦(𝑑),̃𝑧(𝑑)=πΆπ‘“ξ€·π‘Ÿπ‘‘ξ€ΈΜƒπ‘₯(𝑑),(2.5) where Μƒπ‘₯(𝑑)βˆˆβ„π‘› is the filter state, ̃𝑧(𝑑)βˆˆβ„π‘ž2 is the filter output, and 𝐸𝑓,𝐴𝑓(π‘Ÿπ‘‘),𝐡𝑓(π‘Ÿπ‘‘), and 𝐢𝑓(π‘Ÿπ‘‘) are to design the filter matrices with appropriate dimensions.

Define π‘₯(𝑑)=[π‘₯𝑇(𝑑)π‘₯𝑇(𝑑)βˆ’Μƒπ‘₯𝑇(𝑑)]𝑇,𝑒(𝑑)=𝑧(𝑑)βˆ’Μƒπ‘§(𝑑) and combining (2.1) and (2.5), one can obtain the following filtering error dynamics as follows:𝐸̇π‘₯(𝑑)=π΄ξ€·π‘Ÿπ‘‘ξ€Έπ‘₯(𝑑)+π΅ξ€·π‘Ÿπ‘‘ξ€Έπ‘€π‘’(𝑑),(𝑑)=πΏξ€·π‘Ÿπ‘‘ξ€Έπ‘₯ξ€·π‘Ÿ(𝑑)+𝐺𝑑𝑀(𝑑),(2.6) where⎑⎒⎒⎣𝐸=𝐸0πΈβˆ’πΈπ‘“πΈπ‘“βŽ€βŽ₯βŽ₯⎦,π΄ξ€·π‘Ÿπ‘‘ξ€Έ=βŽ‘βŽ’βŽ’βŽ£π΄ξ€·π‘Ÿπ‘‘ξ€Έξ€·π‘Ÿ+Δ𝐴𝑑0π΄ξ€·π‘Ÿπ‘‘ξ€Έξ€·π‘Ÿ+Ξ”π΄π‘‘ξ€Έβˆ’π΄π‘“ξ€·π‘Ÿπ‘‘ξ€Έβˆ’π΅π‘“ξ€·π‘Ÿπ‘‘ξ€ΈπΆξ€·π‘Ÿπ‘‘ξ€Έπ΄π‘“ξ€·π‘Ÿπ‘‘ξ€ΈβŽ€βŽ₯βŽ₯⎦,π΅ξ€·π‘Ÿπ‘‘ξ€Έ=βŽ‘βŽ’βŽ’βŽ£π΅ξ€·π‘Ÿπ‘‘ξ€Έξ€·π‘Ÿ+Ξ”π΅π‘‘ξ€Έπ΅ξ€·π‘Ÿπ‘‘ξ€Έξ€·π‘Ÿ+Ξ”π΅π‘‘ξ€Έβˆ’π΅π‘“ξ€·π‘Ÿπ‘‘ξ€Έπ·ξ€·π‘Ÿπ‘‘ξ€ΈβŽ€βŽ₯βŽ₯⎦,πΏξ€·π‘Ÿπ‘‘ξ€Έ=ξ‚ƒπΏξ€·π‘Ÿπ‘‘ξ€Έβˆ’πΆπ‘“ξ€·π‘Ÿπ‘‘ξ€ΈπΆπ‘“ξ€·π‘Ÿπ‘‘ξ€Έξ‚„.(2.7) For notational simplicity, in the sequel, for each possible π‘Ÿπ‘‘=𝑖,π‘–βˆˆπ•„, a matrix 𝐾(π‘Ÿπ‘‘) will be denoted by 𝐾𝑖; for instance, 𝐴(π‘Ÿπ‘‘) will be denoted by 𝐴𝑖, 𝐡(π‘Ÿπ‘‘) by 𝐡𝑖, 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 [0,𝑇] if the characteristic polynomial det(π‘ πΈβˆ’π΄π‘–βˆ’Ξ”π΄π‘–) is not identically zero for all π‘‘βˆˆ[0,𝑇].
(ii) The singular systems with Markovian jumps (2.1) is said to be impulse free in time interval [0,𝑇], if deg(det(π‘ πΈβˆ’π΄π‘–βˆ’Ξ”π΄π‘–))=rank(𝐸) for all π‘‘βˆˆ[0,𝑇].

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 (𝑐1,𝑐2,𝑇,𝑅𝑖,𝑑), with 𝑐1<𝑐2, 𝑅𝑖>0, if the stochastic system (2.6) is regular and impulse free in time interval [0,𝑇] and satisfies 𝔼π‘₯𝑇(0)𝐸𝑇𝑅𝑖𝐸π‘₯(0)≀𝑐21ξ‚†βŸΉπ”Όπ‘₯𝑇(𝑑)𝐸𝑇𝑅𝑖𝐸π‘₯(𝑑)<𝑐22[].,βˆ€π‘‘βˆˆ0,𝑇(2.8)

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 𝐻∞ finite-time boundedness (SS𝐻∞FTB)). The singular system with Markovian jumps (2.6) is said to be SS𝐻∞FTB with respect to (0,𝑐2,𝑇,𝑅𝑖,𝛾,𝑑), if the singular system with Markovian jumps (2.6) is SSFTB with respect to (𝑐1,𝑐2,𝑇,𝑅𝑖,𝑑) and under the zero-initial condition, the output error 𝑒(𝑑) satisfies the cost constrained function π”Όξ‚»ξ€œπ‘‡0𝑒𝑇(𝑑)𝑒(𝑑)𝑑𝑑<𝛾2ξ€œπ‘‡0𝑀𝑇,(𝑑)𝑀(𝑑)𝑑𝑑(2.9) for any nonzero 𝑀(t) which satisfies (2.4), where 𝛾 is a prescribed positive scalar.

Definition 2.5 (see [9]). Let 𝑉(π‘₯(𝑑),π‘Ÿπ‘‘=𝑖,𝑑>0) be the stochastic function, define its weak infinitesimal operator 𝐿 of stochastic process {(π‘₯(𝑑),π‘Ÿπ‘‘=𝑖),𝑑β‰₯0} by 𝐿𝑉π‘₯(𝑑),π‘Ÿπ‘‘ξ€Έ=𝑖,𝑑=𝑉𝑑(π‘₯(𝑑),𝑖,𝑑)+𝑉π‘₯(π‘₯(𝑑),𝑖,𝑑)Μ‡π‘₯(𝑑,𝑖)+𝑁𝑗=1πœ‹π‘–π‘—π‘‰(π‘₯(𝑑),𝑗,𝑑).(2.10)

Lemma 2.6 (see [36]). For matrices π‘Œ,𝑀, and 𝑁 of appropriate dimensions, where π‘Œ is a symmetric matrix, then π‘Œ+𝑀𝐹(𝑑)𝑁+𝑁𝑇𝐹𝑇(𝑑)𝑀𝑇<0(2.11) holds for all matrix 𝐹(𝑑) satisfying 𝐹𝑇(𝑑)𝐹(𝑑)≀𝐼 for all π‘‘βˆˆβ„, if and only if there exists a positive constant πœ–, such that the following inequality: π‘Œ+πœ–βˆ’1𝑀𝑀𝑇+πœ–π‘π‘‡π‘<0(2.12) holds.

Lemma 2.7 (see [36]). The linear matrix inequality 𝑆=𝑆11𝑆12βˆ—π‘†22ξ‚„<0 is equivalent to 𝑆22<0,𝑆11βˆ’π‘†12π‘†βˆ’122𝑆𝑇12<0, where 𝑆11=𝑆𝑇11 and 𝑆22=𝑆𝑇22.

Lemma 2.8. The following items are true.
(i) Assume that rank(𝐸)=π‘Ÿ, there exist two orthogonal matrices π‘ˆ and 𝑉 such that 𝐸 has the decomposition as ⎑⎒⎒⎣Σ𝐸=π‘ˆπ‘Ÿ0⎀βŽ₯βŽ₯βŽ¦π‘‰βˆ—0π‘‡βŽ‘βŽ’βŽ’βŽ£πΌ=π‘ˆπ‘Ÿ0⎀βŽ₯βŽ₯βŽ¦π’±βˆ—0𝑇,(2.13) where Ξ£π‘Ÿ=diag{𝛿1,𝛿1,…,π›Ώπ‘Ÿ} with π›Ώπ‘˜>0 for all π‘˜=1,2,…,π‘Ÿ. Partition π‘ˆ=[π‘ˆ1π‘ˆ2], 𝑉=[𝑉1𝑉2], and 𝒱=[𝑉1Ξ£π‘Ÿπ‘‰2] with 𝐸𝑉2=0 and π‘ˆπ‘‡2𝐸=0.
(ii) If 𝑃 satisfies 𝐸𝑇𝑃=𝑃𝑇𝐸β‰₯0,(2.14) then 𝑃=π‘ˆπ‘‡π‘ƒπ’±βˆ’π‘‡ with π‘ˆ and 𝒱 satisfying (2.13) if and only if ξ‚βŽ‘βŽ’βŽ’βŽ£π‘ƒπ‘ƒ=110𝑃21𝑃22⎀βŽ₯βŽ₯⎦,(2.15) with 𝑃11β‰₯0βˆˆβ„π‘ŸΓ—π‘Ÿ. In addition, when 𝑃 is nonsingular, one has 𝑃11>0 and det(𝑃22)β‰ 0. Furthermore, 𝑃 satisfying (2.14) can be parameterized as 𝑃=π‘ˆπ‘‹π‘ˆπ‘‡πΈ+π‘ˆ2π‘Œπ’±π‘‡,(2.16) where 𝑋=diag{𝑃11,Ξ›}, π‘Œ=[𝑃21𝑃22], and Ξ›βˆˆβ„(π‘›βˆ’π‘Ÿ)Γ—(π‘›βˆ’π‘Ÿ) is an arbitrary parameter matrix.
(iii) If 𝑃 is a nonsingular matrix, 𝑅 and Ξ› are two symmetric positive definite matrices, 𝑃 and 𝐸 satisfy (2.14), 𝑋 is a diagonal matrix from (2.16), and the following equality holds: 𝐸𝑇𝑃=𝐸𝑇𝑅1/2𝑄𝑅1/2𝐸.(2.17) Then the symmetric positive definite matrix 𝑄=π‘…βˆ’1/2π‘ˆπ‘‹π‘ˆπ‘‡π‘…βˆ’1/2 is a solution of (2.17).

Proof. One only requires to prove that (ii) and (iii) hold. Let ξ‚βŽ‘βŽ’βŽ’βŽ£π‘ƒπ‘ƒ=11𝑃12𝑃21𝑃22⎀βŽ₯βŽ₯⎦.(2.18) Then by (2.13) and (2.14), it follows that condition 𝑃=π‘ˆπ‘‡π‘ƒπ’±βˆ’π‘‡ if and only if 𝑃12=0 and 𝑃11β‰₯0βˆˆβ„π‘ŸΓ—π‘Ÿ. In addition, when 𝑃 is nonsingular, it follows that 𝑃11>0 and det(𝑃22)β‰ 0. Noting that (2.13) and π‘ˆ is an orthogonal matrix, thus we have βŽ‘βŽ’βŽ’βŽ£π‘ƒπ‘ƒ=π‘ˆ110𝑃21𝑃22⎀βŽ₯βŽ₯βŽ¦π’±π‘‡=βŽ›βŽœβŽœβŽπ‘ˆβŽ‘βŽ’βŽ’βŽ£π‘ƒ110⎀βŽ₯βŽ₯βŽ¦π‘ˆβˆ—Ξ›π‘‡βŽžβŽŸβŽŸβŽ βŽ›βŽœβŽœβŽπ‘ˆβŽ‘βŽ’βŽ’βŽ£πΌπ‘Ÿ0⎀βŽ₯βŽ₯βŽ¦π’±βˆ—0π‘‡βŽžβŽŸβŽŸβŽ βŽ‘βŽ’βŽ’βŽ£π‘ƒ+π‘ˆ0021𝑃22⎀βŽ₯βŽ₯βŽ¦π’±π‘‡βŽ‘βŽ’βŽ’βŽ£π‘ƒ=π‘ˆ110⎀βŽ₯βŽ₯βŽ¦π‘ˆβˆ—Ξ›π‘‡ξ‚ƒπ‘ˆπΈ+1π‘ˆ2ξ‚„βŽ‘βŽ’βŽ’βŽ£π‘ƒ0021𝑃22⎀βŽ₯βŽ₯βŽ¦π’±π‘‡=π‘ˆπ‘‹π‘ˆπ‘‡πΈ+π‘ˆ2π‘Œπ’±π‘‡,(2.19) where 𝑋=diag{𝑃11,Ξ›}, π‘Œ=[𝑃21𝑃22] with a parameter matrix Ξ›βˆˆβ„(π‘›βˆ’π‘Ÿ)Γ—(π‘›βˆ’π‘Ÿ). Thus (ii) is true.
By (i) and (ii), noticing π‘ˆπ‘‡2𝐸=0 and 𝑃=π‘ˆπ‘‹π‘ˆπ‘‡πΈ+π‘ˆ2π‘Œπ’±π‘‡, we have 𝐸𝑇𝑃=πΈπ‘‡ξ€·π‘ˆπ‘‹π‘ˆπ‘‡πΈ+π‘ˆ2π‘Œπ’±π‘‡ξ€Έ=πΈπ‘‡π‘ˆπ‘‹π‘ˆπ‘‡πΈ.(2.20) Thus, 𝑄=π‘…βˆ’1/2π‘ˆπ‘‹π‘ˆπ‘‡π‘…βˆ’1/2 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) SS𝐻∞FTB.

3. Main Results

In this section, firstly we give SS𝐻∞FTB 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 𝑒(𝑑) and disturbance 𝑀(𝑑) satisfy the constrain condition (2.9).

Lemma 3.1. The filtering error system (2.6) is SSFTB with respect to (𝑐1,𝑐2,𝑇,𝑅𝑖,𝑑), if there exists a scalar 𝛼β‰₯0, a set of nonsingular matrices {𝑃𝑖,π‘–βˆˆπ•„} with π‘ƒπ‘–βˆˆβ„2𝑛×2𝑛, two sets of symmetric positive definite matrices {𝑄1𝑖,π‘–βˆˆπ•„} with 𝑄1π‘–βˆˆβ„2𝑛×2𝑛, {𝑄2𝑖,π‘–βˆˆπ•„} with 𝑄2π‘–βˆˆβ„π‘Γ—π‘, and for all π‘–βˆˆπ•„ such that the following inequalities hold:𝐸𝑇𝑃𝑖=𝑃𝑇𝑖𝐸β‰₯0,(3.1a)βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£π΄π‘‡π‘–π‘ƒ+𝑃𝑇𝑖𝐴𝑖+𝑁𝑗=1πœ‹π‘–π‘—πΈπ‘‡π‘ƒπ‘—βˆ’π›ΌπΈπ‘‡π‘ƒπ‘–π‘ƒπ‘‡π‘–π΅π‘–βˆ—βˆ’π‘„2π‘–βŽ€βŽ₯βŽ₯βŽ₯βŽ₯⎦<0,(3.1b)𝐸𝑇𝑃𝑖=𝐸𝑇𝑅𝑖1/2𝑄1𝑖𝑅𝑖1/2𝐸,(3.1c)𝑐21supπ‘–βˆˆπ•„ξ‚†πœ†max𝑄1𝑖+𝑑2supπ‘–βˆˆπ•„ξ‚†πœ†max𝑄2𝑖<𝑐22π‘’βˆ’π›Όπ‘‡infπ‘–βˆˆπ•„ξ‚†πœ†min𝑄1𝑖.(3.1d)

Proof. Firstly, one proves the filtering error system (2.6) is regular and impulse free in time interval [0,𝑇]. By Lemma 2.7 and noting that condition (3.1b), one has 𝐴𝑇𝑖𝑃𝑖+𝑃𝑇𝑖𝐴𝑖+𝑁𝑗=1πœ‹π‘–π‘—πΈπ‘‡π‘ƒπ‘—βˆ’π›ΌπΈπ‘‡π‘ƒπ‘–<0.(3.2) Now, we choose two orthogonal matrices π‘ˆ and 𝑉 such that 𝐸 has the decomposition as 𝐸=π‘ˆβŽ‘βŽ’βŽ’βŽ£Ξ£π‘Ÿ0⎀βŽ₯βŽ₯βŽ¦βˆ—0𝑉𝑇=π‘ˆβŽ‘βŽ’βŽ’βŽ£πΌπ‘Ÿ0⎀βŽ₯βŽ₯βŽ¦βˆ—0𝒱𝑇,(3.3) where Ξ£π‘Ÿ=diag{𝛿1,𝛿2,…,π›Ώπ‘Ÿ} with π›Ώπ‘˜>0 for all π‘˜=1,2,…,π‘Ÿ. Partition π‘ˆ=[π‘ˆ1π‘ˆ2], 𝑉=[𝑉1𝑉2] and 𝒱=[𝑉1Ξ£π‘Ÿπ‘‰2] with 𝐸𝑉2=0 and π‘ˆπ‘‡2𝐸=0. Denote π‘ˆπ‘‡π΄π‘–π’±βˆ’π‘‡=⎑⎒⎒⎣𝐴11𝑖𝐴12𝑖𝐴21𝑖𝐴22π‘–βŽ€βŽ₯βŽ₯⎦,π‘ˆπ‘‡π‘ƒπ‘–π’±βˆ’π‘‡=βŽ‘βŽ’βŽ’βŽ£π‘ƒ11𝑖𝑃12𝑖𝑃21𝑖𝑃22π‘–βŽ€βŽ₯βŽ₯⎦.(3.4) Noting that condition (3.1a) and 𝑃𝑖 is a nonsingular matrix, by Lemma 2.8, we have 𝑃12𝑖=0 and det(𝑃22𝑖)β‰ 0. Pre and postmultiplying by π’±βˆ’1 and π’±βˆ’π‘‡, it can easily obtain 𝐴𝑇22𝑖𝑃22𝑖+𝑃𝑇22𝑖𝐴22𝑖<0. Therefore 𝐴22𝑖 is nonsingular, which implies that system (2.6) is regular and impulse free in time interval [0,𝑇].
Let us consider the quadratic Lyapunov function candidate 𝑉(π‘₯(𝑑),𝑖)=π‘₯𝑇(𝑑)𝐸𝑇𝑃𝑖π‘₯(𝑑) for system (2.6). Computing 𝐿𝑉, the derivative of 𝑉(π‘₯(𝑑),𝑖) along the solution of system (2.6), we obtain 𝐿𝑉π‘₯(𝑑),𝑖=πœ‰π‘‡βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£(𝑑)𝐴𝑇𝑖𝑃𝑖+𝑃𝑇𝑖𝐴𝑖+𝑁𝑗=1πœ‹π‘–π‘—πΈπ‘‡π‘ƒπ‘—π‘ƒπ‘‡π‘–π΅π‘–βŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦βˆ—0πœ‰(𝑑),(3.5) where πœ‰(𝑑)=[π‘₯𝑇(𝑑),𝑀𝑇(𝑑)]𝑇. From (3.1b) and (3.5), we obtain 𝔼𝐿𝑉𝑉π‘₯(𝑑),𝑖<𝛼𝔼π‘₯(𝑑),𝑖+𝑀𝑇(𝑑)𝑄2𝑖𝑀(𝑑).(3.6) Further, (3.6) can be rewritten as π”Όξ€½πΏξ€Ίπ‘’βˆ’π›Όπ‘‘π‘‰ξ€·π‘₯(𝑑),𝑖<π‘’βˆ’π›Όπ‘‘π‘€π‘‡(𝑑)𝑄2𝑖𝑀(𝑑).(3.7) Integrating (3.7) from 0 to 𝑑, with π‘‘βˆˆ[0,𝑇], we obtain π‘’βˆ’π›Όπ‘‘π”Όξ€½π‘‰ξ€·ξ€½π‘‰ξ€·π‘₯(𝑑),𝑖<𝐸π‘₯(0),𝑖=π‘Ÿ0+ξ€œξ€Έξ€Ύπ‘‘0π‘’βˆ’π›Όπœπ‘€π‘‡(𝜏)𝑄2𝑖𝑀(𝜏)π‘‘πœ.(3.8) Noting that 𝛼β‰₯0,π‘‘βˆˆ[0,𝑇] and condition (3.1c), we have 𝔼π‘₯𝑇(𝑑)𝐸𝑇𝑃𝑖𝑉π‘₯(𝑑)=𝔼π‘₯(𝑑),𝑖<𝑒𝛼𝑑𝔼𝑉π‘₯(0),𝑖=π‘Ÿ0ξ€Έξ€Ύ+π‘’π›Όπ‘‘ξ€œπ‘‘0π‘’βˆ’π›Όπœπ‘€π‘‡(𝜏)𝑄2𝑖𝑀(𝜏)π‘‘πœβ‰€π‘’π›Όπ‘‘ξ‚Έsupπ‘–βˆˆπ•„ξ‚†πœ†max𝑄1𝑖𝑐21+supπ‘–βˆˆπ•„ξ€½πœ†max𝑄2𝑖𝑑2ξ‚Ή.(3.9) Taking into account that 𝔼π‘₯𝑇(𝑑)𝐸𝑇𝑃𝑖π‘₯(𝑑)=𝔼π‘₯𝑇(𝑑)𝐸𝑇𝑅𝑖1/2𝑄1𝑖𝑅𝑖1/2𝐸π‘₯(𝑑)β‰₯infπ‘–βˆˆπ•„ξ‚†πœ†min𝑄1𝑖𝔼π‘₯𝑇(𝑑)𝐸𝑇𝑅𝑖,𝐸π‘₯(𝑑)(3.10) we obtain 𝔼π‘₯𝑇(𝑑)𝐸𝑇𝑅𝑖𝐸π‘₯(𝑑)≀supπ‘–βˆˆπ•„ξ‚†πœ†maxξ‚€π‘„βˆ’11𝑖𝔼π‘₯𝑇(𝑑)𝐸𝑇𝑃𝑖π‘₯(𝑑)<𝑒𝛼𝑇supπ‘–βˆˆπ•„ξ‚†πœ†max𝑄1𝑖𝑐21+supπ‘–βˆˆπ•„ξ‚†πœ†max𝑄2𝑖𝑑2infπ‘–βˆˆπ•„ξ‚†πœ†min𝑄1𝑖.(3.11) Therefore, it follows that condition (3.1d) implies 𝔼{π‘₯𝑇(𝑑)𝐸𝑇𝑅𝑖𝐸π‘₯(𝑑)}<𝑐22 for all π‘‘βˆˆ[0,𝑇]. This completes the proof of the lemma.

Lemma 3.2. The filtering error system (2.6) is SS𝐻∞FTB with respect to (0,𝑐2,𝑇,𝑅𝑖,𝛾,𝑑), if there exists a scalar 𝛼β‰₯0, a set of nonsingular matrices {𝑃𝑖,π‘–βˆˆπ•„} with π‘ƒπ‘–βˆˆβ„2𝑛×2𝑛, a set of symmetric positive definite matrices {𝑄1𝑖,π‘–βˆˆπ•„} with 𝑄1π‘–βˆˆβ„2𝑛×2𝑛, and for all π‘–βˆˆπ•„ such that (3.1a), (3.1c) and the following inequalities hold:βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£π΄π‘‡π‘–π‘ƒπ‘–+𝑃𝑇𝑖𝐴𝑖+𝑁𝑗=1πœ‹π‘–π‘—πΈπ‘‡π‘ƒπ‘—+πΏπ‘‡π‘–πΏπ‘–βˆ’π›ΌπΈπ‘‡π‘ƒπ‘–πΏπ‘‡π‘–πΊπ‘–+π‘ƒπ‘‡π‘–π΅π‘–βˆ—πΊπ‘‡π‘–πΊπ‘–βˆ’π›Ύ2π‘’βˆ’π›Όπ‘‡πΌβŽ€βŽ₯βŽ₯βŽ₯βŽ₯⎦<0,(3.12a)𝑑2𝛾2<𝑐22infπ‘–βˆˆπ•„ξ‚†πœ†min𝑄1𝑖.(3.12b)

Proof. Noting that βŽ‘βŽ’βŽ’βŽ£πΏπ‘‡π‘–πΏπ‘–πΏπ‘‡π‘–πΊπ‘–βˆ—πΊπ‘‡π‘–πΊπ‘–βŽ€βŽ₯βŽ₯⎦=βŽ‘βŽ’βŽ’βŽ£πΏπ‘‡π‘–πΊπ‘‡π‘–βŽ€βŽ₯βŽ₯βŽ¦ξ‚ƒπΏπ‘–πΊπ‘–ξ‚„β‰₯0.(3.13) Thus, condition (3.12a) implies that βŽ‘βŽ’βŽ’βŽ’βŽ’βŽ£π΄π‘‡π‘–π‘ƒπ‘–+𝑃𝑇𝑖𝐴𝑖+𝑁𝑗=1πœ‹π‘–π‘—πΈπ‘‡π‘ƒπ‘—βˆ’π›ΌπΈπ‘‡π‘ƒπ‘–π‘ƒπ‘‡π‘–π΅π‘–βˆ—βˆ’π›Ύ2π‘’βˆ’π›Όπ‘‡πΌβŽ€βŽ₯βŽ₯βŽ₯βŽ₯⎦<0.(3.14) Let 𝑄2𝑖=βˆ’π›Ύ2π‘’βˆ’π›Όπ‘‡πΌ for all π‘–βˆˆπ•„, by Lemma 3.1, conditions (3.1a), (3.1c), (3.12b), and (3.14) guarantee that system (2.6) is SSFTB with respect to (0,𝑐2,𝑇,𝑅𝑖,𝑑). Therefore, we only need to prove that (2.9) holds. Let 𝑉(π‘₯(𝑑),𝑖)=π‘₯𝑇(𝑑)𝐸𝑇𝑃𝑖π‘₯(𝑑) and noting that (3.5) and (3.14), we obtain 𝔼𝐿𝑉𝑉π‘₯(𝑑),𝑖<𝛼𝔼π‘₯(𝑑),𝑖+𝛾2π‘’βˆ’π›Όπ‘‡π‘€π‘‡ξ€½π‘’(𝑑)𝑀(𝑑)βˆ’π”Όπ‘‡ξ€Ύ.(𝑑)𝑒(𝑑)(3.15) 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 𝑃𝑖=diag{𝑃𝑖,𝑃𝑖},𝑄1𝑖=diag{𝑄1𝑖,𝑄1𝑖},𝑀𝑖=𝑃𝑇𝑖𝐴𝑓𝑖,𝑁𝑖=𝑃𝑇𝑖𝐡𝑓𝑖, and 𝐸𝑓=𝐸. Using Lemmas 2.7 and 3.2, we obtain the following theorem.

Theorem 3.3. The nominal filtering error system (2.6) is SS𝐻∞FTB with respect to (0,𝑐2,𝑇,𝑅𝑖,𝛾,𝑑) with 𝑅𝑖=diag{𝑅𝑖,𝑅𝑖}, if there exists a scalar 𝛼β‰₯0, a set of nonsingular matrices {𝑃𝑖,π‘–βˆˆπ•„} with π‘ƒπ‘–βˆˆβ„π‘›Γ—π‘›, a set of positive definite matrices {𝑄1𝑖,π‘–βˆˆπ•„} with 𝑄1π‘–βˆˆβ„π‘›Γ—π‘›, three sets of matrices {𝑀𝑖,π‘–βˆˆπ•„} with π‘€π‘–βˆˆβ„π‘›Γ—π‘›, {𝑁𝑖,π‘–βˆˆπ•„} with π‘π‘–βˆˆβ„π‘›Γ—π‘ž1, {𝐢𝑓𝑖,π‘–βˆˆπ•„} with πΆπ‘“π‘–βˆˆβ„π‘ž2×𝑛, for all π‘–βˆˆπ•„ such that𝐸𝑇𝑃𝑖=𝑃𝑇𝑖𝐸β‰₯0,(3.16a)⎑⎒⎒⎒⎒⎒⎒⎣Θ1π‘–π‘ƒβˆ—βˆ—βˆ—π‘‡π‘–π΄π‘–βˆ’π‘€π‘–βˆ’π‘π‘–πΆπ‘–Ξ˜2π‘–π‘ƒπ‘‡π‘–π΅π‘–βˆ’π‘π‘–π·π‘–βˆ—π΅π‘‡π‘–π‘ƒπ‘–βˆ—βˆ’π›Ύ2π‘’βˆ’π›Όπ‘‡πΏπΌβˆ—π‘–βˆ’πΆπ‘“π‘–πΆπ‘“π‘–πΊπ‘–βŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦βˆ’πΌ<0,(3.16b)𝐸𝑇𝑃𝑖=𝐸𝑇𝑅𝑖1/2𝑄1𝑖𝑅𝑖1/2𝐸,(3.16c)𝑑2𝛾2<𝑐22infπ‘–βˆˆπ•„ξ€½πœ†min𝑄1𝑖(3.16d)hold, where Θ1𝑖=𝑃𝑇𝑖𝐴𝑖+𝐴𝑇𝑖𝑃𝑖+βˆ‘π‘π‘—=1πœ‹π‘–π‘—πΈπ‘‡π‘ƒπ‘—βˆ’π›ΌπΈπ‘‡π‘ƒπ‘–, and Θ2𝑖=𝑀𝑖+𝑀𝑇𝑖+βˆ‘π‘π‘—=1πœ‹π‘–π‘—πΈπ‘‡π‘ƒπ‘—βˆ’π›ΌπΈπ‘‡π‘ƒπ‘–.
In addition, the desired filter parameters can be chosen by 𝐴𝑓𝑖=π‘ƒπ‘–βˆ’π‘‡π‘€π‘–,𝐡𝑓𝑖=π‘ƒπ‘–βˆ’π‘‡π‘π‘–,𝐢𝑓𝑖=𝐢𝑓𝑖,𝐸𝑓=𝐸.(3.17)
Noting that 𝑃𝑖 is nonsingular matrix, by Lemma 2.8, there exist two orthogonal matrices π‘ˆ and 𝑉, such that 𝐸 has the decomposition as ⎑⎒⎒⎣Σ𝐸=π‘ˆπ‘Ÿ0⎀βŽ₯βŽ₯βŽ¦π‘‰βˆ—0π‘‡βŽ‘βŽ’βŽ’βŽ£πΌ=π‘ˆπ‘Ÿ0⎀βŽ₯βŽ₯βŽ¦π’±βˆ—0𝑇,(3.18) where Ξ£π‘Ÿ=diag{𝛿1,𝛿2,…,π›Ώπ‘Ÿ} with π›Ώπ‘˜>0 for all π‘˜=1,2,…,π‘Ÿ. Partition π‘ˆ=[π‘ˆ1π‘ˆ2], 𝑉=[𝑉1𝑉2], and 𝒱=[𝑉1Ξ£π‘Ÿπ‘‰2] with 𝐸𝑉2=0 and π‘ˆπ‘‡2𝐸=0. Let 𝑃𝑖=π‘ˆπ‘‡π‘ƒπ‘–π’±βˆ’π‘‡, from (3.16a), 𝑃𝑖 is of the following form 𝑃11𝑖0𝑃21𝑖𝑃22𝑖, and 𝑃𝑖 can be expressed as 𝑃𝑖=π‘ˆπ‘‹π‘–π‘ˆπ‘‡πΈ+π‘ˆ2π‘Œπ‘–π’±π‘‡,(3.19) where 𝑋𝑖=diag{𝑃11𝑖,Λ𝑖} and π‘Œπ‘–=[𝑃21𝑖𝑃22𝑖] with a parameter matrix Λ𝑖. If we choose Λ𝑖 being a symmetric positive definite matrix, then 𝑋𝑖 is a symmetric positive definite matrix. Furthermore, the symmetric positive definite matrix 𝑄1𝑖=π‘…π‘–βˆ’1/2π‘ˆπ‘‹π‘–π‘ˆπ‘‡π‘…π‘–βˆ’1/2 is a solution of (3.16c), and 𝑃𝑖 satisfies 𝐸𝑇𝑃𝑖=𝑃𝑇𝑖𝐸=πΈπ‘‡π‘ˆπ‘‹π‘–π‘ˆπ‘‡πΈ.(3.20)

From the above discussion, we have the following theorem.

Theorem 3.4. The nominal filtering error system (2.6) is SS𝐻∞FTB with respect to (0,𝑐2,𝑇,𝑅𝑖,𝛾,𝑑) with 𝑅𝑖=diag{𝑅𝑖,𝑅𝑖}, if there exists a scalar 𝛼β‰₯0, a set of positive definite matrices {𝑋𝑖,π‘–βˆˆπ•„} with π‘‹π‘–βˆˆβ„π‘›Γ—π‘›, four sets of matrices {π‘Œπ‘–,π‘–βˆˆπ•„} with π‘Œπ‘–βˆˆβ„(π‘›βˆ’π‘Ÿ)×𝑛, {𝑀𝑖,π‘–βˆˆπ•„} with π‘€π‘–βˆˆβ„π‘›Γ—π‘›, {𝑁𝑖,π‘–βˆˆπ•„} with π‘π‘–βˆˆβ„π‘›Γ—π‘ž1, and {𝐢𝑓𝑖,iβˆˆπ•„} with πΆπ‘“π‘–βˆˆβ„π‘ž2×𝑛,   for all π‘–βˆˆπ•„ such that (3.16d) and the following linear matrix inequality ⎑⎒⎒⎒⎒⎒⎒⎣Ξ1π‘–π‘ƒβˆ—βˆ—βˆ—π‘‡π‘–π΄π‘–βˆ’π‘€π‘–βˆ’π‘π‘–πΆπ‘–Ξž2π‘–π‘ƒπ‘‡π‘–π΅π‘–βˆ’π‘π‘–π·π‘–βˆ—π΅π‘‡π‘–π‘ƒπ‘–βˆ—βˆ’π›Ύ2π‘’βˆ’π›Όπ‘‡πΏπΌβˆ—π‘–βˆ’πΆπ‘“π‘–πΆπ‘“π‘–πΊπ‘–βŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦βˆ’πΌ<0(3.21) hold, where Ξ1𝑖=𝑃𝑇𝑖𝐴𝑖+𝐴𝑇𝑖𝑃𝑖+βˆ‘π‘π‘—=1πœ‹π‘–π‘—πΈπ‘‡π‘ƒπ‘—βˆ’π›ΌπΈπ‘‡π‘ƒπ‘–, Ξ2𝑖=𝑀𝑖+𝑀𝑇𝑖+βˆ‘π‘π‘—=1πœ‹π‘–π‘—πΈπ‘‡π‘ƒπ‘—βˆ’π›ΌπΈπ‘‡π‘ƒπ‘–, 𝑃𝑖=π‘ˆπ‘‹π‘–π‘ˆπ‘‡πΈ+π‘ˆ2π‘Œπ‘–π’±π‘‡, 𝑋𝑖 and π‘Œπ‘– 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 SS𝐻∞FTB with respect to (0,𝑐2,𝑇,𝑅𝑖,𝛾,𝑑) with 𝑅𝑖={𝑅𝑖,𝑅𝑖}, if there exists a scalar 𝛼β‰₯0, a set of positive definite matrices {𝑋𝑖,π‘–βˆˆπ•„} with π‘‹π‘–βˆˆβ„π‘›Γ—π‘›, four sets of matrices {π‘Œπ‘–,π‘–βˆˆπ•„} with π‘Œπ‘–βˆˆβ„(π‘›βˆ’π‘Ÿ)×𝑛, {𝑀𝑖,π‘–βˆˆπ•„} with π‘€π‘–βˆˆβ„π‘›Γ—π‘›, {𝑁𝑖,π‘–βˆˆπ•„} with π‘π‘–βˆˆβ„π‘›Γ—π‘ž1, {𝐢𝑓𝑖,π‘–βˆˆπ•„} with πΆπ‘“π‘–βˆˆβ„π‘ž2×𝑛, and a set of positive scalars {πœ–π‘–,π‘–βˆˆπ•„}, for all π‘–βˆˆπ•„ such that (3.16d) and the following linear matrix inequality ⎑⎒⎒⎒⎒⎒⎒⎒⎒⎒⎣Ξ₯1π‘–π‘ƒβˆ—βˆ—βˆ—βˆ—π‘‡π‘–π΄π‘–βˆ’π‘€π‘–βˆ’π‘π‘–πΆπ‘–Ξ₯2π‘–π‘ƒπ‘‡π‘–π΅π‘–βˆ’π‘π‘–π·π‘–π΅βˆ—βˆ—π‘‡π‘–π‘ƒπ‘–+πœ–π‘–πΈπ‘‡2𝑖𝐸1π‘–βˆ—πœ–π‘–πΈπ‘‡2𝑖𝐸2π‘–βˆ’π›Ύ2π‘’βˆ’π›Όπ‘‡πΉπΌβˆ—βˆ—π‘‡π‘–π‘ƒπ‘–πΉπ‘‡π‘–π‘ƒπ‘–0βˆ’πœ–π‘–πΏπΌβˆ—π‘–βˆ’πΆπ‘“π‘–πΆπ‘“π‘–πΊπ‘–βŽ€βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯βŽ₯⎦0βˆ’πΌ<0(3.22) hold, where Ξ₯1𝑖=𝑃𝑇𝑖𝐴𝑖+𝐴𝑇𝑖𝑃𝑖+βˆ‘π‘π‘—=1πœ‹π‘–π‘—πΈπ‘‡π‘ƒπ‘—+πœ–π‘–πΈπ‘‡1𝑖𝐸1π‘–βˆ’π›ΌπΈπ‘‡π‘ƒπ‘–, Ξ₯2𝑖=𝑀𝑖+𝑀𝑇𝑖+βˆ‘π‘π‘—=1πœ‹π‘–π‘—πΈπ‘‡π‘ƒπ‘—βˆ’π›ΌπΈπ‘‡π‘ƒπ‘–, 𝑃𝑖=π‘ˆπ‘‹π‘–π‘ˆπ‘‡πΈ+π‘ˆ2π‘Œπ‘–π’±π‘‡, 𝑋𝑖 and π‘Œπ‘– 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 𝐻∞ filtering problem of singular stochastic systems to the finite-time 𝐻∞ filtering problem of singular stochastic systems. In fact, if we fix 𝛼=0 without condition (3.16d), we can obtain sufficient conditions of the 𝐻∞ filtering of singular stochastic systems.
Let 𝐼<𝑄1𝑖<πœ‚πΌ, then one can check that condition (3.16d) can be guaranteed by imposing the conditions 𝐼<π‘…π‘–βˆ’1/2π‘ˆπ‘‹π‘–π‘ˆπ‘‡π‘…π‘–βˆ’1/2<πœ‚πΌ,𝑑2𝛾2βˆ’π‘22<0.(3.23)

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 𝛼, respectively: 𝛾min2+𝑐22𝑋𝑖,π‘Œπ‘–,𝑀𝑖,𝑁𝑖,𝐢𝑓𝑖,πœ‚s.t.(3.21)and(𝛾3.23),min2+𝑐22𝑋𝑖,π‘Œπ‘–,𝑀𝑖,𝑁𝑖,𝐢𝑓𝑖,πœ–π‘–,πœ‚s.t.(3.22)and(3.23).(3.24)

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:
(i) Mode #1, 𝐴1=⎑⎒⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦0.21132.510.132,𝐡1=⎑⎒⎒⎒⎒⎣111⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,𝐿1=⎑⎒⎒⎒⎒⎣11⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦0.1𝑇,𝐢1=⎑⎒⎒⎒⎒⎣111⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦π‘‡,𝐹1=⎑⎒⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦0.050000.04000.010,𝐸11=⎑⎒⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦0.200.030.030.0200.0500.01,𝐸21=⎑⎒⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,0.030.020.05(4.1)
(ii) Mode #2, 𝐴2=⎑⎒⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦βˆ’4111βˆ’31121,𝐡2=⎑⎒⎒⎒⎒⎣101⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,𝐿2=⎑⎒⎒⎒⎒⎣12⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦0.7𝑇,𝐢2=⎑⎒⎒⎒⎒⎣41⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦0.8𝑇,𝐹2=⎑⎒⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦0.050000.04000.010.02,𝐸12=⎑⎒⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦0.0200.030.030.0300.0200.1,𝐸22=⎑⎒⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,0.030.020.02(4.2) and 𝐸=diag{1,1,0},𝐷1=0.2,𝐺1=0.3,𝐷2=0.1,𝐺2=0.5,𝑑=0.6, Δ𝑖=diag{π‘Ÿ1(𝑖),π‘Ÿ2(𝑖),π‘Ÿ3(𝑖)}, where π‘Ÿπ‘—(𝑖) satisfies |π‘Ÿπ‘—(𝑖)|≀1 for all 𝑖=1,2 and 𝑗=1,2,3. In addition, the switching between the two modes is described by the transition rate matrix ξ€ΊΞ“=βˆ’112βˆ’2ξ€».
Then, we choose 𝑅1=𝑅2=𝐼3,𝑇=2, by Theorem 3.5, the optimal bound with minimum value of 𝛾2+𝑐22 relies on the parameter 𝛼. We can find feasible solution when 0.34≀𝛼≀11.02. Figures 1 and 2 show the optimal values with different value of 𝛼. Noting that when 𝛼=2, it yields the optimal values 𝛾=9.4643 and 𝑐2=5.6786. Then, by using the program fminsearch in the optimization toolbox of Matlab starting at 𝛼=2, the locally convergent solution can be derived as 𝐴𝑓1=⎑⎒⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦βˆ’0.89061.50440.395864.153747.069251.020714.904815.643114.4887,𝐡𝑓1=⎑⎒⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,𝐢2.2274βˆ’63.2281βˆ’13.4543𝑓1=,0.9315βˆ’0.35460.6824(4.3)𝐴𝑓2=⎑⎒⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦25.0076170.668440.7148βˆ’149.7287βˆ’851.1311βˆ’192.867520.5802107.732925.9671,𝐡𝑓2=⎑⎒⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,πΆβˆ’44.9909223.8116βˆ’27.9168𝑓2=,0.39160.37221.6748(4.4) with 𝛼=1.3209, and the optimal values 𝛾=8.1261, 𝑐2=4.8758.

615790.fig.001
Figure 1: The local optimal bound of 𝛾.
615790.fig.002
Figure 2: The local optimal bound of 𝑐2.

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 𝛼 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: 𝐴1=⎑⎒⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ₯βŽ¦βˆ’320βˆ’3βˆ’2.50101,𝐴2=⎑⎒⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦.130βˆ’1βˆ’2.50βˆ’10βˆ’4.8(4.5) Moreover, other matrical variables and the transition rate matrix are defined similarly as Example 4.1.
Let 𝑅1=𝑅2=𝐼3, then the feasible solution of the above filtering error system can be found when 𝛼=0, Theorem 3.5 yields the optimal values 𝛾=3.7770, 𝑐2=2.2663, and 𝐴𝑓1=⎑⎒⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦35.992347.863242.1297βˆ’125.9936βˆ’141.5997βˆ’122.030731.997834.460431.8623,𝐡𝑓1=⎑⎒⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,πΆβˆ’47.6632137.2043βˆ’33.8527𝑓1=,𝐴0.82940.11360.8294𝑓2=⎑⎒⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦16.876573.669625.7772βˆ’137.4627βˆ’607.3156βˆ’225.4097βˆ’17.9446βˆ’72.5666βˆ’36.9766,𝐡𝑓2=⎑⎒⎒⎒⎒⎣⎀βŽ₯βŽ₯βŽ₯βŽ₯⎦,πΆβˆ’15.8299141.986217.5990𝑓2=.1.12901.59753.6412(4.6) Thus, the above filtering error system is stochastically stable and the calculated minimum 𝐻∞ performance 𝛾 satisfies ‖𝑇𝑀𝑧‖<3.7770.

5. Conclusion

In this paper, we deal with the problem of finite-time 𝐻∞ 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 𝐻∞ 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.

References

  1. L. Dai, Singular Control Systems, vol. 118 of Lecture Notes in Control and Information Sciences, Springer, Berlin, Germany, 1989. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  2. F. L. Lewis, β€œA survey of linear singular systems,” Circuits, Systems, and Signal Processing, vol. 5, no. 1, pp. 3–36, 1986. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  3. J. Y. Ishihara and M. H. Terra, β€œOn the Lyapunov theorem for singular systems,” IEEE Transactions on Automatic Control, vol. 47, no. 11, pp. 1926–1930, 2002. View at Publisher Β· View at Google Scholar
  4. I. Masubuchi, Y. Kamitane, A. Ohara, and N. Suda, β€œH∞ control for descriptor systems: a matrix inequalities approach,” Automatica, vol. 33, no. 4, pp. 669–673, 1997. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  5. Y. Xia, P. Shi, G. Liu, and D. Rees, β€œRobust mixed H∞ state-feedback control for continuous-time descriptor systems with parameter uncertainties,” Circuits, Systems, and Signal Processing, vol. 24, no. 4, pp. 431–443, 2005. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  6. L. Zhang, B. Huang, and J. Lam, β€œLMI synthesis of H2 and mixed H2/H∞ controllers for singular systems,” IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing, vol. 50, no. 9, pp. 615–626, 2003. View at Publisher Β· View at Google Scholar
  7. E.-K. Boukas, Control of Singular Systems with Random Abrupt Changes, Communications and Control Engineering Series, Springer, Berlin, Germany, 2008.
  8. E. K. Boukas, β€œStabilization of stochastic singular nonlinear hybrid systems,” Nonlinear Analysis, vol. 64, no. 2, pp. 217–228, 2006. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  9. X. Mao, β€œStability of stochastic differential equations with Markovian switching,” Stochastic Processes and their Applications, vol. 79, no. 1, pp. 45–67, 1999. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  10. P. Shi, E.-K. Boukas, and R. K. Agarwal, β€œControl of Markovian jump discrete-time systems with norm bounded uncertainty and unknown delay,” IEEE Transactions on Automatic Control, vol. 44, no. 11, pp. 2139–2144, 1999. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  11. E. K. Boukas, β€œOn robust stability of singular systems with random abrupt changes,” Nonlinear Analysis, vol. 63, no. 3, pp. 301–310, 2005. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  12. C. E. de Souza, β€œRobust stability and stabilization of uncertain discrete-time Markovian jump linear systems,” IEEE Transactions on Automatic Control, vol. 51, no. 5, pp. 836–841, 2006. View at Publisher Β· View at Google Scholar
  13. E.-K. Boukas and P. Shi, β€œStochastic stability and guaranteed cost control of discrete-time uncertain systems with Markovian jumping parameters,” International Journal of Robust and Nonlinear Control, vol. 8, no. 13, pp. 1155–1167, 1998. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  14. N. M. Krasovskii and E. A. Lidskii, β€œAnalytical design of controllers in systems with random attributes,” Automation and Remote Control Remote Control, vol. 22, pp. 1201–1294, 1961. View at Google Scholar
  15. A. Elsayed and M. J. Grimble, β€œA new approach to the H∞ design of optimal digital linear filters,” IMA Journal of Mathematical Control and Information, vol. 6, no. 2, pp. 233–251, 1989. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  16. M. S. Mahmoud, P. Shi, and A. Ismail, β€œRobust Kalman filtering for discrete-time Markovian jump systems with parameter uncertainty,” Journal of Computational and Applied Mathematics, vol. 169, no. 1, pp. 53–69, 2004. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  17. C. E. de Souza, A. Trofino, and K. A. Barbosa, β€œMode-independent H∞ filters for Markovian jump linear systems,” IEEE Transactions Automatic Control, vol. 51, no. 11, pp. 1837–1841, 2006. View at Publisher Β· View at Google Scholar
  18. S. Xu, T. Chen, and J. Lam, β€œRobust H∞ filtering for uncertain Markovian jump systems with mode-dependent time delays,” IEEE Transactions on Automatic Control, vol. 48, no. 5, pp. 900–907, 2003. View at Publisher Β· View at Google Scholar
  19. S. Xu, J. Lam, and Y. Zou, β€œH∞ filtering for singular systems,” IEEE Transactions on Automatic Control, vol. 48, no. 12, pp. 2217–2222, 2003. View at Publisher Β· View at Google Scholar
  20. S. Xu and J. Lam, β€œReduced-order H∞ filtering for singular systems,” Systems & Control Letters, vol. 56, no. 1, pp. 48–57, 2007. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  21. S. Xu and J. Lam, Robust Control and Filtering of Singular Systems, vol. 332 of Lecture Notes in Control and Information Sciences, Springer, Berlin, Germany, 2006.
  22. F. Amato, M. Ariola, and P. Dorato, β€œFinite-time control of linear systems subject to parametric uncertainties and disturbances,” Automatica, vol. 37, no. 9, pp. 1459–1463, 2001. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  23. F. Amato, M. Ariola, and C. Cosentino, β€œFinite-time stabilization via dynamic output feedback,” Automatica, vol. 42, no. 2, pp. 337–342, 2006. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  24. W. Zhang and X. An, β€œFinite-time control of linear stochastic systems,” International Journal of Innovative Computing, Information and Control, vol. 4, no. 3, pp. 689–696, 2008. View at Google Scholar
  25. Y. Yang, J. Li, and G. Chen, β€œFinite-time stability and stabilization of nonlinear stochastic hybrid systems,” Journal of Mathematical Analysis and Applications, vol. 356, no. 1, pp. 338–345, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  26. Q. Meng and Y. Shen, β€œFinite-time H∞ control for linear continuous system with norm-bounded disturbance,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 4, pp. 1043–1049, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  27. G. Garcia, S. Tarbouriech, and J. Bernussou, β€œFinite-time stabilization of linear time-varying continuous systems,” IEEE Transactions on Automatic Control, vol. 54, no. 2, pp. 364–369, 2009. View at Publisher Β· View at Google Scholar
  28. R. Ambrosino, F. Calabrese, C. Cosentino, and G. De Tommasi, β€œSufficient conditions for finite-time stability of impulsive dynamical systems,” IEEE Transactions on Automatic Control, vol. 54, no. 4, pp. 861–865, 2009. View at Publisher Β· View at Google Scholar
  29. F. Amato, R. Ambrosino, M. Ariola, and C. Cosentino, β€œFinite-time stability of linear time-varying systems with jumps,” Automatica, vol. 45, no. 5, pp. 1354–1358, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  30. F. Amato, M. Ariola, and C. Cosentino, β€œFinite-time control of discrete-time linear systems: analysis and design conditions,” Automatica, vol. 46, no. 5, pp. 919–924, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  31. Y. Zhang, C. Liu, and X. Mu, β€œRobust finite-time H∞ control of singular stochastic systems via static output feedback,” Applied Mathematics and Computation, vol. 218, no. 9, pp. 5629–5640, 2012. View at Publisher Β· View at Google Scholar
  32. Y. Zhang, C. Liu, and X. Mu, β€œStochastic finite-time guaranteed cost control of Markovian jumping singular systems,” Mathematical Problems in Engineering, vol. 2011, Article ID 431751, 20 pages, 2011. View at Publisher Β· View at Google Scholar
  33. S. He and F. Liu, β€œStochastic finite-time boundedness of Markovian jumping neural network with uncertain transition probabilities,” Applied Mathematical Modelling, vol. 35, no. 6, pp. 2631–2638, 2011. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  34. Y. Zhang, C. Liu, and X. Mu, β€œOn stochastic finite-time control of discrete-time fuzzy systems with packet dropout,” Discrete Dynamics in Nature and Society, vol. 2012, Article ID 752950, 18 pages, 2012. View at Publisher Β· View at Google Scholar
  35. S. Ma and E. K. Boukas, β€œRobust H∞ filtering for uncertain discrete Markov jump singular systems with mode-dependent time delay,” IET Control Theory & Applications, vol. 3, no. 3, pp. 351–361, 2009. View at Publisher Β· View at Google Scholar
  36. S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequality in Systems and Control Theory, SIAM, Philadelphia, Pa, USA, 1994.