Finite-Time 𝐻_∞ Filtering for Singular Stochastic Systems

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, ℝ^𝑛 and ℝ^𝑛×𝑚 denote the sets of 𝑛 component real vectors and 𝑛×𝑚 real matrices, respectively. The superscript 𝑇 stands for matrix transposition or vector. 𝔼{⋅} denotes the expectation operator with respective to some probability measure ℙ. In addition, the symbol ∗ denotes the term that is induced by symmetry and diag{⋯} stands for a block-diagonal matrix. 𝜆_min(𝑃) and 𝜆_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:𝐸𝑥̇(𝑡)=𝐴𝑟𝑦_𝑡(𝑡)+=Δ𝐶𝐴𝑟𝑟_𝑡𝑡𝑥(𝑡)+𝐵𝑟_𝑡+Δ𝐵𝑟_𝑡𝑤(𝑡),𝑧(𝑡)=𝐿𝑟_𝑡𝑥(𝑡)+𝐷𝑟_𝑡𝑤(𝑡),𝑥(𝑡)+𝐺𝑟_𝑡𝑤(𝑡),(2.1) where 𝑥(𝑡)∈ℝ^𝑛 is the state variable, 𝑦(𝑡)∈ℝ^𝑞1 is the measurement output of the system, 𝑧(𝑡)∈ℝ^𝑞2 is the signal to be estimated, and 𝐸 is a singular matrix with rank(𝐸)=𝑟<𝑛; {𝑟_𝑡,𝑡≥0} is continuous-time Markov stochastic process taking values in a finite space 𝕄∶={1,2,…,𝑁} with transition matrix Γ=(𝜋_𝑖𝑗)_𝑁×𝑁, and the transition probabilities are described as follows:𝑃_𝑖𝑗=𝑃𝑟𝑟_𝑡+Δ=𝑗∣𝑟_𝑡=𝑖=𝜋^𝑖𝑗1+Δ𝜋+_𝑖𝑗𝑜(Δ),if𝑖≠𝑗,Δ+𝑜(Δ),if𝑖=𝑗,(2.2) where lim_Δ→0𝑜(Δ)/Δ=0, 𝜋_𝑖𝑗 satisfies 𝜋_𝑖𝑗≥0(𝑖≠𝑗), and 𝜋_𝑖𝑖=−∑^{𝑁𝑗=1,𝑗≠𝑖}𝜋_𝑖𝑗 for all 𝑖,𝑗∈𝕄; Δ𝐴(𝑟_𝑡) and Δ𝐵(𝑟_𝑡) are uncertain matrices and satisfyΔ𝐴𝑟_𝑡,Δ𝐵𝑟_𝑡=𝐹𝑟_𝑡Δ𝑟_𝑡𝐸₁𝑟_𝑡,𝐸₂𝑟_𝑡,(2.3) where Δ(𝑟_𝑡) is an unknown, time-varying matrix function and satisfies Δ^𝑇(𝑟_𝑡)Δ(𝑟_𝑡)≤𝐼 for all 𝑟_𝑡∈𝕄; moreover, the disturbance input 𝑤(𝑡)∈ℝ^𝑝 satisfies^𝑇0𝑤^𝑇(𝑡)𝑤(𝑡)𝑑𝑡≤𝑑²,𝑑≥0,(2.4) and the matrices 𝐴(𝑟_𝑡),𝐵(𝑟_𝑡),𝐶(𝑟_𝑡),𝐷(𝑟_𝑡),𝐿(𝑟_𝑡), and 𝐺(𝑟_𝑡) are coefficient matrices and of appropriate dimension for all 𝑟_𝑡∈𝕄.

In this paper, we construct the following full-order filter:𝐸_𝑓𝑥𝑧̇̃̃((𝑡𝑡))==𝐴𝐶_𝑓𝑓𝑟_𝑡𝑥̃(𝑡)+𝐵_𝑓𝑟_𝑡𝑦(𝑡),𝑟_𝑡𝑥̃(𝑡),(2.5) where 𝑥̃(𝑡)∈ℝ^𝑛 is the filter state, 𝑧(̃𝑡)∈ℝ^𝑞2 is the filter output, and 𝐸_𝑓,𝐴_𝑓(𝑟_𝑡),𝐵_𝑓(𝑟_𝑡), and 𝐶_𝑓(𝑟_𝑡) are to design the filter matrices with appropriate dimensions.

Define 𝑥(𝑡)=[𝑥^𝑇(𝑡)𝑥^𝑇(𝑡)−𝑥̃^𝑇(𝑡)]^𝑇,𝑒(𝑡)=𝑧(𝑡)−𝑧(̃𝑡) and combining (2.1) and (2.5), one can obtain the following filtering error dynamics as follows:𝐸𝑒𝑥(̇𝑡()𝑡)==𝐴𝑟_𝑡𝑥(𝑡)+𝐵𝑟_𝑡𝑤(𝑡),𝐿𝑟_𝑡𝑥(𝑡)+𝐺𝑟_𝑡𝑤(𝑡),(2.6) where𝐸=⎡⎢⎢⎣𝐸−𝐸𝐸_𝑓0𝐸_𝑓⎤⎥⎥⎦,𝐴𝑟_𝑡=⎡⎢⎢⎣𝐴𝑟^𝑡𝐴𝑟_𝑡+Δ𝐴𝑟_𝑡0+Δ𝐴𝑟_𝑡−𝐴_𝑓𝑟_𝑡−𝐵_𝑓𝑟_𝑡𝐶𝑟_𝑡𝐴_𝑓𝑟_𝑡⎤⎥⎥⎦,𝐵𝑟_𝑡=⎡⎢⎢⎣𝐵𝑟^𝑡𝐵𝑟_𝑡+Δ𝐵𝑟_𝑡+Δ𝐵𝑟_𝑡−𝐵_𝑓𝑟_𝑡𝐷𝑟_𝑡⎤⎥⎥⎦,𝐿𝑟_𝑡=𝐿𝑟_𝑡−𝐶_𝑓𝑟_𝑡𝐶_𝑓𝑟_𝑡.(2.7) For notational simplicity, in the sequel, for each possible 𝑟_𝑡=𝑖,𝑖∈𝕄, a matrix 𝐾(𝑟_𝑡) will be denoted by 𝐾_𝑖; for instance, 𝐴(𝑟_𝑡) will be denoted by 𝐴_𝑖, 𝐵(𝑟_𝑡) by 𝐵_𝑖, 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(𝑠𝐸−𝐴_𝑖−Δ𝐴_𝑖) is not identically zero for all 𝑡∈[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(𝑠𝐸−𝐴_𝑖−Δ𝐴_𝑖))=rank(𝐸) for all 𝑡∈[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,𝑅_𝑖,𝑑), with 𝑐₁<𝑐₂, 𝑅_𝑖>0, if the stochastic system (2.6) is regular and impulse free in time interval [0,𝑇] and satisfies 𝔼𝑥^𝑇(0)𝐸^𝑇𝑅_𝑖𝐸𝑥(0)≤𝑐²¹⟹𝔼𝑥^𝑇(𝑡)𝐸^𝑇𝑅_𝑖𝐸𝑥(𝑡)<𝑐²²,∀𝑡∈[0,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,𝑅_𝑖,𝛾,𝑑), if the singular system with Markovian jumps (2.6) is SSFTB with respect to (𝑐₁,𝑐₂,𝑇,𝑅_𝑖,𝑑) and under the zero-initial condition, the output error 𝑒(𝑡) satisfies the cost constrained function 𝔼^𝑇0𝑒^𝑇(𝑡)𝑒(𝑡)𝑑𝑡<𝛾²^𝑇0𝑤^𝑇(𝑡)𝑤(𝑡)𝑑𝑡,(2.9) for any nonzero 𝑤(t) which satisfies (2.4), where 𝛾 is a prescribed positive scalar.

Definition 2.5 (see [9]). Let 𝑉(𝑥(𝑡),𝑟_𝑡=𝑖,𝑡>0) be the stochastic function, define its weak infinitesimal operator 𝐿 of stochastic process {(𝑥(𝑡),𝑟_𝑡=𝑖),𝑡≥0} by 𝐿𝑉𝑥(𝑡),𝑟_𝑡=𝑖,𝑡=𝑉_𝑡(𝑥(𝑡),𝑖,𝑡)+𝑉_𝑥(𝑥(𝑡),𝑖,𝑡)𝑥̇(𝑡,𝑖)+^𝑁_𝑗=1𝜋_𝑖𝑗𝑉(𝑥(𝑡),𝑗,𝑡).(2.10)

Lemma 2.6 (see [36]). For matrices 𝑌,𝑀, and 𝑁 of appropriate dimensions, where 𝑌 is a symmetric matrix, then 𝑌+𝑀𝐹(𝑡)𝑁+𝑁^𝑇𝐹^𝑇(𝑡)𝑀^𝑇<0(2.11) holds for all matrix 𝐹(𝑡) satisfying 𝐹^𝑇(𝑡)𝐹(𝑡)≤𝐼 for all 𝑡∈ℝ, if and only if there exists a positive constant 𝜖, such that the following inequality: 𝑌+𝜖⁻¹𝑀𝑀^𝑇+𝜖𝑁^𝑇𝑁<0(2.12) holds.

Lemma 2.7 (see [36]). The linear matrix inequality 𝑆=^𝑆11𝑆12_∗𝑆22<0 is equivalent to 𝑆₂₂<0,𝑆₁₁−𝑆₁₂𝑆⁻¹²²𝑆^𝑇12<0, where 𝑆₁₁=𝑆^𝑇11 and 𝑆₂₂=𝑆^𝑇22.

Lemma 2.8. The following items are true.
(i) Assume that rank(𝐸)=𝑟, there exist two orthogonal matrices 𝑈 and 𝑉 such that 𝐸 has the decomposition as 𝐸=𝑈⎡⎢⎢⎣Σ^𝑟∗00⎤⎥⎥⎦𝑉_𝑇=𝑈⎡⎢⎢⎣𝐼^𝑟∗00⎤⎥⎥⎦𝒱_𝑇,(2.13) where Σ_𝑟=diag{𝛿₁,𝛿₁,…,𝛿_𝑟} with 𝛿_𝑘>0 for all 𝑘=1,2,…,𝑟. Partition 𝑈=[𝑈₁𝑈₂], 𝑉=[𝑉₁𝑉₂], and 𝒱=[𝑉₁Σ_𝑟𝑉₂] with 𝐸𝑉₂=0 and 𝑈^𝑇2𝐸=0.
(ii) If 𝑃 satisfies 𝐸^𝑇𝑃=𝑃^𝑇𝐸≥0,(2.14) then 𝑃=𝑈^𝑇𝑃𝒱^−𝑇 with 𝑈 and 𝒱 satisfying (2.13) if and only if 𝑃=⎡⎢⎢⎣𝑃¹¹𝑃₂₁0𝑃₂₂⎤⎥⎥⎦,(2.15) with 𝑃₁₁≥0∈ℝ^𝑟×𝑟. In addition, when 𝑃 is nonsingular, one has 𝑃₁₁>0 and det(𝑃₂₂)≠0. Furthermore, 𝑃 satisfying (2.14) can be parameterized as 𝑃=𝑈𝑋𝑈^𝑇𝐸+𝑈₂𝑌𝒱^𝑇,(2.16) where 𝑋=diag{𝑃₁₁,Λ}, 𝑌=[𝑃₂₁𝑃₂₂], and Λ∈ℝ^{(𝑛−𝑟)×(𝑛−𝑟)} is an arbitrary parameter matrix.
(iii) If 𝑃 is a nonsingular matrix, 𝑅 and Λ 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: 𝐸^𝑇𝑃=𝐸^𝑇𝑅^1/2𝑄𝑅^1/2𝐸.(2.17) Then the symmetric positive definite matrix 𝑄=𝑅^−1/2𝑈𝑋𝑈^𝑇𝑅^−1/2 is a solution of (2.17).

Proof. One only requires to prove that (ii) and (iii) hold. Let 𝑃=⎡⎢⎢⎣𝑃¹¹𝑃₁₂𝑃₂₁𝑃₂₂⎤⎥⎥⎦.(2.18) Then by (2.13) and (2.14), it follows that condition 𝑃=𝑈^𝑇𝑃𝒱^−𝑇 if and only if 𝑃₁₂=0 and 𝑃₁₁≥0∈ℝ^𝑟×𝑟. In addition, when 𝑃 is nonsingular, it follows that 𝑃₁₁>0 and det(𝑃₂₂)≠0. Noting that (2.13) and 𝑈 is an orthogonal matrix, thus we have 𝑃=𝑈⎡⎢⎢⎣𝑃¹¹𝑃₂₁0𝑃₂₂⎤⎥⎥⎦𝒱_𝑇=⎛⎜⎜⎝𝑈⎡⎢⎢⎣𝑃¹¹∗Λ0⎤⎥⎥⎦𝑈_𝑇⎞⎟⎟⎠⎛⎜⎜⎝𝑈⎡⎢⎢⎣𝐼∗_𝑟00⎤⎥⎥⎦𝒱_𝑇⎞⎟⎟⎠+𝑈⎡⎢⎢⎣𝑃00₂₁𝑃₂₂⎤⎥⎥⎦𝒱_𝑇=𝑈⎡⎢⎢⎣𝑃¹¹∗Λ0⎤⎥⎥⎦𝑈_𝑇𝐸+𝑈₁𝑈₂⎡⎢⎢⎣𝑃00₂₁𝑃₂₂⎤⎥⎥⎦𝒱_𝑇=𝑈𝑋𝑈^𝑇𝐸+𝑈₂𝑌𝒱^𝑇,(2.19) where 𝑋=diag{𝑃₁₁,Λ}, 𝑌=[𝑃₂₁𝑃₂₂] with a parameter matrix Λ∈ℝ^{(𝑛−𝑟)×(𝑛−𝑟)}. Thus (ii) is true.
By (i) and (ii), noticing 𝑈^𝑇2𝐸=0 and 𝑃=𝑈𝑋𝑈^𝑇𝐸+𝑈₂𝑌𝒱^𝑇, we have 𝐸^𝑇𝑃=𝐸^𝑇𝑈𝑋𝑈^𝑇𝐸+𝑈₂𝑌𝒱^𝑇=𝐸^𝑇𝑈𝑋𝑈^𝑇𝐸.(2.20) Thus, 𝑄=𝑅^−1/2𝑈𝑋𝑈^𝑇𝑅^−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(𝑡) and disturbance 𝑤(𝑡) satisfy the constrain condition (2.9).

Lemma 3.1. The filtering error system (2.6) is SSFTB with respect to (𝑐₁,𝑐₂,𝑇,𝑅_𝑖,𝑑), if there exists a scalar 𝛼≥0, a set of nonsingular matrices {𝑃_𝑖,𝑖∈𝕄} with 𝑃_𝑖∈ℝ^2𝑛×2𝑛, two sets of symmetric positive definite matrices {𝑄_1𝑖,𝑖∈𝕄} with 𝑄_1𝑖∈ℝ^2𝑛×2𝑛, {𝑄_2𝑖,𝑖∈𝕄} with 𝑄_2𝑖∈ℝ^𝑝×𝑝, and for all 𝑖∈𝕄 such that the following inequalities hold:𝐸^𝑇𝑃_𝑖=𝑃^𝑇𝑖𝐸≥0,(3.1a)⎡⎢⎢⎢⎢⎣𝐴^𝑇𝑖𝑃+𝑃^𝑇𝑖𝐴_𝑖+^𝑁_𝑗=1𝜋_𝑖𝑗𝐸^𝑇𝑃_𝑗−𝛼𝐸^𝑇𝑃_𝑖𝑃^𝑇𝑖𝐵_𝑖∗−𝑄_2𝑖⎤⎥⎥⎥⎥⎦<0,(3.1b)𝐸^𝑇𝑃_𝑖=𝐸^𝑇𝑅^𝑖1/2𝑄_1𝑖𝑅^𝑖1/2𝐸,(3.1c)𝑐²¹sup_𝑖∈𝕄𝜆^max𝑄_1𝑖+𝑑²sup_𝑖∈𝕄𝜆^max𝑄_2𝑖<𝑐²²𝑒^−𝛼𝑇inf_𝑖∈𝕄𝜆^min𝑄_1𝑖.(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 𝐴^𝑇𝑖𝑃_𝑖+𝑃^𝑇𝑖𝐴_𝑖+^𝑁_𝑗=1𝜋_𝑖𝑗𝐸^𝑇𝑃_𝑗−𝛼𝐸^𝑇𝑃_𝑖<0.(3.2) Now, we choose two orthogonal matrices 𝑈 and 𝑉 such that 𝐸 has the decomposition as 𝐸=𝑈⎡⎢⎢⎣Σ^𝑟∗00⎤⎥⎥⎦𝑉^𝑇=𝑈⎡⎢⎢⎣𝐼^𝑟∗00⎤⎥⎥⎦𝒱^𝑇,(3.3) where Σ_𝑟=diag{𝛿₁,𝛿₂,…,𝛿_𝑟} with 𝛿_𝑘>0 for all 𝑘=1,2,…,𝑟. Partition 𝑈=[𝑈₁𝑈₂], 𝑉=[𝑉₁𝑉₂] and 𝒱=[𝑉₁Σ_𝑟𝑉₂] with 𝐸𝑉₂=0 and 𝑈^𝑇2𝐸=0. Denote 𝑈^𝑇𝐴_𝑖𝒱^−𝑇=⎡⎢⎢⎣𝐴_11𝑖𝐴_12𝑖𝐴_21𝑖𝐴_22𝑖⎤⎥⎥⎦,𝑈^𝑇𝑃_𝑖𝒱^−𝑇=⎡⎢⎢⎣𝑃_11𝑖𝑃_12𝑖𝑃_21𝑖𝑃_22𝑖⎤⎥⎥⎦.(3.4) Noting that condition (3.1a) and 𝑃_𝑖 is a nonsingular matrix, by Lemma 2.8, we have 𝑃_12𝑖=0 and det(𝑃_22𝑖)≠0. Pre and postmultiplying by 𝒱⁻¹ and 𝒱^−𝑇, it can easily obtain 𝐴^𝑇22𝑖𝑃_22𝑖+𝑃^𝑇22𝑖𝐴_22𝑖<0. Therefore 𝐴_22𝑖 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(𝑥(𝑡),𝑖)=𝑥^𝑇(𝑡)𝐸^𝑇𝑃_𝑖𝑥(𝑡) for system (2.6). Computing 𝐿𝑉, the derivative of 𝑉(𝑥(𝑡),𝑖) along the solution of system (2.6), we obtain 𝐿𝑉𝑥(𝑡),𝑖=𝜉^𝑇(𝑡)⎡⎢⎢⎢⎢⎣𝐴^𝑇𝑖𝑃_𝑖+𝑃^𝑇𝑖𝐴_𝑖+^𝑁_𝑗=1𝜋_𝑖𝑗𝐸^𝑇𝑃_𝑗𝑃^𝑇𝑖𝐵_𝑖∗0⎤⎥⎥⎥⎥⎦𝜉(𝑡),(3.5) where 𝜉(𝑡)=[𝑥^𝑇(𝑡),𝑤^𝑇(𝑡)]^𝑇. From (3.1b) and (3.5), we obtain 𝔼𝐿𝑉𝑥(𝑡),𝑖<𝛼𝔼𝑉𝑥(𝑡),𝑖+𝑤^𝑇(𝑡)𝑄_2𝑖𝑤(𝑡).(3.6) Further, (3.6) can be rewritten as 𝔼𝐿𝑒^−𝛼𝑡𝑉𝑥(𝑡),𝑖<𝑒^−𝛼𝑡𝑤^𝑇(𝑡)𝑄_2𝑖𝑤(𝑡).(3.7) Integrating (3.7) from 0 to 𝑡, with 𝑡∈[0,𝑇], we obtain 𝑒^−𝛼𝑡𝔼𝑉𝑥(𝑡),𝑖<𝐸𝑉𝑥(0),𝑖=𝑟₀+^𝑡0𝑒^−𝛼𝜏𝑤^𝑇(𝜏)𝑄_2𝑖𝑤(𝜏)𝑑𝜏.(3.8) Noting that 𝛼≥0,𝑡∈[0,𝑇] and condition (3.1c), we have 𝔼𝑥^𝑇(𝑡)𝐸^𝑇𝑃_𝑖𝑥(𝑡)=<𝑒𝔼_𝛼𝑡𝑉𝑥(𝑡),𝑖𝔼𝑉𝑥(0),𝑖=𝑟₀+𝑒^𝛼𝑡^𝑡0≤𝑒_𝛼𝑡𝑒^−𝛼𝜏𝑤^𝑇(𝜏)𝑄_2𝑖𝑤(𝜏)𝑑𝜏sup_𝑖∈𝕄𝜆^max𝑄_1𝑖𝑐²¹+sup_𝑖∈𝕄𝜆_max𝑄_2𝑖𝑑².(3.9) Taking into account that 𝔼𝑥^𝑇(𝑡)𝐸^𝑇𝑃_𝑖𝑥(𝑡)=≥𝔼inf_𝑖∈𝕄𝑥^𝑇(𝑡)𝐸^𝑇𝑅^𝑖1/2𝑄_1𝑖𝑅^𝑖1/2𝐸𝑥(𝑡)𝜆^min𝑄_1𝑖𝔼𝑥^𝑇(𝑡)𝐸^𝑇𝑅_𝑖𝐸𝑥(𝑡),(3.10) we obtain 𝔼𝑥^𝑇(𝑡)𝐸^𝑇𝑅_𝑖𝐸𝑥(𝑡)≤<s𝑒u_𝛼𝑇p_𝑖∈𝕄𝜆^max𝑄^−11𝑖𝔼𝑥^𝑇(𝑡)𝐸^𝑇𝑃_𝑖𝑥(𝑡)sup_𝑖∈𝕄𝜆^max𝑄_1𝑖𝑐²¹+sup_𝑖∈𝕄𝜆^max𝑄_2𝑖𝑑²inf_𝑖∈𝕄𝜆^min𝑄_1𝑖.(3.11) Therefore, it follows that condition (3.1d) implies 𝔼{𝑥^𝑇(𝑡)𝐸^𝑇𝑅_𝑖𝐸𝑥(𝑡)}<𝑐²² for all 𝑡∈[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,𝑅_𝑖,𝛾,𝑑), if there exists a scalar 𝛼≥0, a set of nonsingular matrices {𝑃_𝑖,𝑖∈𝕄} with 𝑃_𝑖∈ℝ^2𝑛×2𝑛, a set of symmetric positive definite matrices {𝑄_1𝑖,𝑖∈𝕄} with 𝑄_1𝑖∈ℝ^2𝑛×2𝑛, and for all 𝑖∈𝕄 such that (3.1a), (3.1c) and the following inequalities hold:⎡⎢⎢⎢⎢⎣𝐴^𝑇𝑖𝑃_𝑖+𝑃^𝑇𝑖𝐴_𝑖+^𝑁_𝑗=1𝜋_𝑖𝑗𝐸^𝑇𝑃_𝑗+𝐿^𝑇𝑖𝐿_𝑖−𝛼𝐸^𝑇𝑃_𝑖𝐿^𝑇𝑖𝐺_𝑖+𝑃^𝑇𝑖𝐵_𝑖∗𝐺^𝑇𝑖𝐺_𝑖−𝛾²𝑒^−𝛼𝑇𝐼⎤⎥⎥⎥⎥⎦<0,(3.12a)𝑑²𝛾²<𝑐²²inf_𝑖∈𝕄𝜆^min𝑄_1𝑖.(3.12b)

Proof. Noting that ⎡⎢⎢⎣𝐿^𝑇𝑖𝐿_𝑖𝐿^𝑇𝑖𝐺_𝑖∗𝐺^𝑇𝑖𝐺_𝑖⎤⎥⎥⎦=⎡⎢⎢⎣𝐿^𝑇𝑖𝐺^𝑇𝑖⎤⎥⎥⎦𝐿_𝑖𝐺_𝑖≥0.(3.13) Thus, condition (3.12a) implies that ⎡⎢⎢⎢⎢⎣𝐴^𝑇𝑖𝑃_𝑖+𝑃^𝑇𝑖𝐴_𝑖+^𝑁_𝑗=1𝜋_𝑖𝑗𝐸^𝑇𝑃_𝑗−𝛼𝐸^𝑇𝑃_𝑖𝑃^𝑇𝑖𝐵_𝑖∗−𝛾²𝑒^−𝛼𝑇𝐼⎤⎥⎥⎥⎥⎦<0.(3.14) Let 𝑄_2𝑖=−𝛾²𝑒^−𝛼𝑇𝐼 for all 𝑖∈𝕄, 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,𝑅_𝑖,𝑑). Therefore, we only need to prove that (2.9) holds. Let 𝑉(𝑥(𝑡),𝑖)=𝑥^𝑇(𝑡)𝐸^𝑇𝑃_𝑖𝑥(𝑡) and noting that (3.5) and (3.14), we obtain 𝔼𝐿𝑉𝑥(𝑡),𝑖<𝛼𝔼𝑉𝑥(𝑡),𝑖+𝛾²𝑒^−𝛼𝑇𝑤^𝑇(𝑡)𝑤(𝑡)−𝔼𝑒^𝑇(𝑡)𝑒(𝑡).(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 𝑃_𝑖=diag{𝑃_𝑖,𝑃_𝑖},𝑄_1𝑖=diag{𝑄_1𝑖,𝑄_1𝑖},𝑀_𝑖=𝑃^𝑇𝑖𝐴_𝑓𝑖,𝑁_𝑖=𝑃^𝑇𝑖𝐵_𝑓𝑖, and 𝐸_𝑓=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,𝑅_𝑖,𝛾,𝑑) with 𝑅_𝑖=diag{𝑅_𝑖,𝑅_𝑖}, if there exists a scalar 𝛼≥0, a set of nonsingular matrices {𝑃_𝑖,𝑖∈𝕄} with 𝑃_𝑖∈ℝ^𝑛×𝑛, a set of positive definite matrices {𝑄_1𝑖,𝑖∈𝕄} with 𝑄_1𝑖∈ℝ^𝑛×𝑛, three sets of matrices {𝑀_𝑖,𝑖∈𝕄} with 𝑀_𝑖∈ℝ^𝑛×𝑛, {𝑁_𝑖,𝑖∈𝕄} with 𝑁_𝑖∈ℝ^𝑛×𝑞1, {𝐶_𝑓𝑖,𝑖∈𝕄} with 𝐶_𝑓𝑖∈ℝ^𝑞2×𝑛, for all 𝑖∈𝕄 such that𝐸^𝑇𝑃_𝑖=𝑃^𝑇𝑖𝐸≥0,(3.16a)⎡⎢⎢⎢⎢⎢⎢⎣𝑃Θ_1𝑖∗∗∗_𝑇𝑖𝐴_𝑖−𝐵_𝑇𝑖𝑀_𝑖−𝑁_𝑖𝐶_𝑖Θ_2𝑖𝑃^𝑇𝑖𝐵_𝑖−𝑁_𝑖𝐷_𝑖∗𝐿𝑃_𝑖∗−𝛾²𝑒^−𝛼𝑇𝐼∗_𝑖−𝐶_𝑓𝑖𝐶_𝑓𝑖𝐺_𝑖−𝐼⎤⎥⎥⎥⎥⎥⎥⎦<0,(3.16b)𝐸^𝑇𝑃_𝑖=𝐸^𝑇𝑅^𝑖1/2𝑄_1𝑖𝑅^𝑖1/2𝐸,(3.16c)𝑑²𝛾²<𝑐²²inf_𝑖∈𝕄𝜆_min𝑄_1𝑖(3.16d)hold, where Θ_1𝑖=𝑃^𝑇𝑖𝐴_𝑖+𝐴^𝑇𝑖𝑃_𝑖+∑^𝑁𝑗=1𝜋_𝑖𝑗𝐸^𝑇𝑃_𝑗−𝛼𝐸^𝑇𝑃_𝑖, and Θ_2𝑖=𝑀_𝑖+𝑀^𝑇𝑖+∑^𝑁𝑗=1𝜋_𝑖𝑗𝐸^𝑇𝑃_𝑗−𝛼𝐸^𝑇𝑃_𝑖.
In addition, the desired filter parameters can be chosen by 𝐴_𝑓𝑖=𝑃^𝑖−𝑇𝑀_𝑖,𝐵_𝑓𝑖=𝑃^𝑖−𝑇𝑁_𝑖,𝐶_𝑓𝑖=𝐶_𝑓𝑖,𝐸_𝑓=𝐸.(3.17)
Noting that 𝑃_𝑖 is nonsingular matrix, by Lemma 2.8, there exist two orthogonal matrices 𝑈 and 𝑉, such that 𝐸 has the decomposition as 𝐸=𝑈⎡⎢⎢⎣Σ^𝑟∗00⎤⎥⎥⎦𝑉_𝑇=𝑈⎡⎢⎢⎣𝐼^𝑟∗00⎤⎥⎥⎦𝒱_𝑇,(3.18) where Σ_𝑟=diag{𝛿₁,𝛿₂,…,𝛿_𝑟} with 𝛿_𝑘>0 for all 𝑘=1,2,…,𝑟. Partition 𝑈=[𝑈₁𝑈₂], 𝑉=[𝑉₁𝑉₂], and 𝒱=[𝑉₁Σ_𝑟𝑉₂] with 𝐸𝑉₂=0 and 𝑈^𝑇2𝐸=0. Let 𝑃_𝑖=𝑈^𝑇𝑃_𝑖𝒱^−𝑇, from (3.16a), 𝑃_𝑖 is of the following form ^{𝑃11𝑖0𝑃}_21𝑖^𝑃_22𝑖, and 𝑃_𝑖 can be expressed as 𝑃_𝑖=𝑈𝑋_𝑖𝑈^𝑇𝐸+𝑈₂𝑌_𝑖𝒱^𝑇,(3.19) where 𝑋_𝑖=diag{𝑃_11𝑖,Λ_𝑖} and 𝑌_𝑖=[𝑃_21𝑖𝑃_22𝑖] with a parameter matrix Λ_𝑖. If we choose Λ_𝑖 being a symmetric positive definite matrix, then 𝑋_𝑖 is a symmetric positive definite matrix. Furthermore, the symmetric positive definite matrix 𝑄_1𝑖=𝑅^𝑖−1/2𝑈𝑋_𝑖𝑈^𝑇𝑅^𝑖−1/2 is a solution of (3.16c), and 𝑃_𝑖 satisfies 𝐸^𝑇𝑃_𝑖=𝑃^𝑇𝑖𝐸=𝐸^𝑇𝑈𝑋_𝑖𝑈^𝑇𝐸.(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,𝑅_𝑖,𝛾,𝑑) with 𝑅_𝑖=diag{𝑅_𝑖,𝑅_𝑖}, if there exists a scalar 𝛼≥0, a set of positive definite matrices {𝑋_𝑖,𝑖∈𝕄} with 𝑋_𝑖∈ℝ^𝑛×𝑛, four sets of matrices {𝑌_𝑖,𝑖∈𝕄} with 𝑌_𝑖∈ℝ^{(𝑛−𝑟)×𝑛}, {𝑀_𝑖,𝑖∈𝕄} with 𝑀_𝑖∈ℝ^𝑛×𝑛, {𝑁_𝑖,𝑖∈𝕄} with 𝑁_𝑖∈ℝ^𝑛×𝑞1, and {𝐶_𝑓𝑖,i∈𝕄} with 𝐶_𝑓𝑖∈ℝ^𝑞2×𝑛,   for all 𝑖∈𝕄 such that (3.16d) and the following linear matrix inequality ⎡⎢⎢⎢⎢⎢⎢⎣𝑃Ξ_1𝑖∗∗∗_𝑇𝑖𝐴_𝑖−𝐵𝑀_𝑖𝑇𝑖−𝑁_𝑖𝐶_𝑖Ξ_2𝑖𝑃^𝑇𝑖𝐵_𝑖−𝑁_𝑖𝐷_𝑖∗𝐿𝑃_𝑖∗−𝛾²𝑒^−𝛼𝑇𝐼∗_𝑖−𝐶_𝑓𝑖𝐶_𝑓𝑖𝐺_𝑖−𝐼⎤⎥⎥⎥⎥⎥⎥⎦<0(3.21) hold, where Ξ_1𝑖=𝑃^𝑇𝑖𝐴_𝑖+𝐴^𝑇𝑖𝑃_𝑖+∑^𝑁𝑗=1𝜋_𝑖𝑗𝐸^𝑇𝑃_𝑗−𝛼𝐸^𝑇𝑃_𝑖, Ξ_2𝑖=𝑀_𝑖+𝑀^𝑇𝑖+∑^𝑁𝑗=1𝜋_𝑖𝑗𝐸^𝑇𝑃_𝑗−𝛼𝐸^𝑇𝑃_𝑖, 𝑃_𝑖=𝑈𝑋_𝑖𝑈^𝑇𝐸+𝑈₂𝑌_𝑖𝒱^𝑇, 𝑋_𝑖 and 𝑌_𝑖 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,𝑅_𝑖,𝛾,𝑑) with 𝑅_𝑖={𝑅_𝑖,𝑅_𝑖}, if there exists a scalar 𝛼≥0, a set of positive definite matrices {𝑋_𝑖,𝑖∈𝕄} with 𝑋_𝑖∈ℝ^𝑛×𝑛, four sets of matrices {𝑌_𝑖,𝑖∈𝕄} with 𝑌_𝑖∈ℝ^{(𝑛−𝑟)×𝑛}, {𝑀_𝑖,𝑖∈𝕄} with 𝑀_𝑖∈ℝ^𝑛×𝑛, {𝑁_𝑖,𝑖∈𝕄} with 𝑁_𝑖∈ℝ^𝑛×𝑞1, {𝐶_𝑓𝑖,𝑖∈𝕄} with 𝐶_𝑓𝑖∈ℝ^𝑞2×𝑛, and a set of positive scalars {𝜖_𝑖,𝑖∈𝕄}, for all 𝑖∈𝕄 such that (3.16d) and the following linear matrix inequality ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣𝑃Υ_1𝑖∗∗∗∗_𝑇𝑖𝐵𝐴_𝑖−𝑀_𝑖−𝑁_𝑖𝐶_𝑖Υ_2𝑖𝑃^𝑇𝑖𝐵_𝑖−𝑁_𝑖𝐷_𝑖∗∗_𝑇𝑖𝑃_𝑖𝐹+𝜖_𝑖𝐸^𝑇2𝑖𝐸_1𝑖∗𝜖_𝑖𝐸^𝑇2𝑖𝐸_2𝑖−𝛾²𝑒^−𝛼𝑇𝐼∗∗_𝑇𝑖𝐿𝑃_𝑖𝐹^𝑇𝑖𝑃_𝑖0−𝜖_𝑖𝐼∗_𝑖−𝐶_𝑓𝑖𝐶_𝑓𝑖𝐺_𝑖0−𝐼⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦<0(3.22) hold, where Υ_1𝑖=𝑃^𝑇𝑖𝐴_𝑖+𝐴^𝑇𝑖𝑃_𝑖+∑^𝑁𝑗=1𝜋_𝑖𝑗𝐸^𝑇𝑃_𝑗+𝜖_𝑖𝐸^𝑇1𝑖𝐸_1𝑖−𝛼𝐸^𝑇𝑃_𝑖, Υ_2𝑖=𝑀_𝑖+𝑀^𝑇𝑖+∑^𝑁𝑗=1𝜋_𝑖𝑗𝐸^𝑇𝑃_𝑗−𝛼𝐸^𝑇𝑃_𝑖, 𝑃_𝑖=𝑈𝑋_𝑖𝑈^𝑇𝐸+𝑈₂𝑌_𝑖𝒱^𝑇, 𝑋_𝑖 and 𝑌_𝑖 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<𝑄_1𝑖<𝜂𝐼, then one can check that condition (3.16d) can be guaranteed by imposing the conditions 𝐼<𝑅^𝑖−1/2𝑈𝑋_𝑖𝑈^𝑇𝑅^𝑖−1/2<𝜂𝐼,𝑑²𝛾²−𝑐²²<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: 𝑋_𝑖min𝛾²+𝑐²²,𝑌_𝑖,𝑀_𝑖,𝑁_𝑖,𝐶_𝑓𝑖,𝜂s.t.m(i3n.21)𝛾a₂nd(3.23),𝑋_𝑖+𝑐²²,𝑌_𝑖,𝑀_𝑖,𝑁_𝑖,𝐶_𝑓𝑖,𝜖_𝑖,𝜂s.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, 𝐹𝐴₁₁=⎡⎢⎢⎢⎢⎣003..21213.5121⎤⎥⎥⎥⎥⎦,𝐵₁=⎡⎢⎢⎢⎢⎣111⎤⎥⎥⎥⎥⎦,𝐿₁=⎡⎢⎢⎢⎢⎣011.1⎤⎥⎥⎥⎥⎦^𝑇,𝐶₁=⎡⎢⎢⎢⎢⎣111⎤⎥⎥⎥⎥⎦^𝑇,=⎡⎢⎢⎢⎢⎣0.000500..00041000⎤⎥⎥⎥⎥⎦,𝐸₁₁=⎡⎢⎢⎢⎢⎣000..00.2350.000200..00013⎤⎥⎥⎥⎥⎦,𝐸₂₁=⎡⎢⎢⎢⎢⎣000...000325⎤⎥⎥⎥⎥⎦,(4.1)
(ii) Mode #2, 𝐹₂𝐴₂=⎡⎢⎢⎢⎢⎣−114−123111⎤⎥⎥⎥⎥⎦,𝐵₂=⎡⎢⎢⎢⎢⎣101⎤⎥⎥⎥⎥⎦,𝐿₂=⎡⎢⎢⎢⎢⎣012.7⎤⎥⎥⎥⎥⎦^𝑇,𝐶₂=⎡⎢⎢⎢⎢⎣041.8⎤⎥⎥⎥⎥⎦^𝑇,=⎡⎢⎢⎢⎢⎣0.000500..000410.0002⎤⎥⎥⎥⎥⎦,𝐸₁₂=⎡⎢⎢⎢⎢⎣000...0002320.000300.00.13⎤⎥⎥⎥⎥⎦,𝐸₂₂=⎡⎢⎢⎢⎢⎣000...000322⎤⎥⎥⎥⎥⎦,(4.2) and 𝐸=diag{1,1,0},𝐷₁=0.2,𝐺₁=0.3,𝐷₂=0.1,𝐺₂=0.5,𝑑=0.6, Δ_𝑖=diag{𝑟₁(𝑖),𝑟₂(𝑖),𝑟₃(𝑖)}, where 𝑟_𝑗(𝑖) satisfies |𝑟_𝑗(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 Γ=⁻¹¹²⁻².
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 𝐴_𝑓1=⎡⎢⎢⎢⎢⎣−61440...198509𝐶34078641175...50606449342151014...304928508877⎤⎥⎥⎥⎥⎦,𝐵_𝑓1=⎡⎢⎢⎢⎢⎣−−261.332..22472548413_𝑓1⎤⎥⎥⎥⎥⎦,=0.9315−0.35460.6824,(4.3)𝐴_𝑓2=⎡⎢⎢⎢⎢⎣−221504..905.0877026287𝐶−118705071..67.1633821491−421059..279.1684768175⎤⎥⎥⎥⎥⎦,𝐵_𝑓2=⎡⎢⎢⎢⎢⎣−−2242347...899191106698_𝑓2⎤⎥⎥⎥⎥⎦,=0.39160.37221.6748,(4.4) with 𝛼=1.3209, and the optimal values 𝛾=8.1261, 𝑐₂=4.8758.

Figure 1 

The local optimal bound of 𝛾.

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=⎡⎢⎢⎢⎢⎣−−133−220.5001⎤⎥⎥⎥⎥⎦,𝐴₂=⎡⎢⎢⎢⎢⎣−−111−230.5−400.8⎤⎥⎥⎥⎥⎦.(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 𝐴_𝑓1=⎡⎢⎢⎢⎢⎣−331512..599.9992793836𝐶−431744..184.6653092497−431212..218.2609237307⎤⎥⎥⎥⎥⎦,𝐵_𝑓1=⎡⎢⎢⎢⎢⎣−−1343773...268065432327_𝑓1⎤⎥⎥⎥⎥⎦,𝐴=0.82940.11360.8294_𝑓2,=⎡⎢⎢⎢⎢⎣−−111637.78..97446645267𝐶−−767302.76..56369166566−−223526.57..97477062967⎤⎥⎥⎥⎥⎦,𝐵_𝑓2=⎡⎢⎢⎢⎢⎣−1141715.5..989829690_𝑓229⎤⎥⎥⎥⎥⎦,=1.12901.59753.6412.(4.6) Thus, the above filtering error system is stochastically stable and the calculated minimum 𝐻_∞ performance 𝛾 satisfies ‖𝑇_𝑤𝑧‖<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.