# Finite-TimeH∞ Filtering for Singular Stochastic Systems

Authors: Caixia Liu; Yingqi Zhang; Huixia Sun
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

This paper addresses the problem of finite-timeH∞ 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.


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## Body

## 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-timeH∞ 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 ℝn×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. λmin(P) and λmax(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:(2.1)Eẋ(t)=[A(rt)+ΔA(rt)]x(t)+[B(rt)+ΔB(rt)]w(t),y(t)=C(rt)x(t)+D(rt)w(t),z(t)=L(rt)x(t)+G(rt)w(t),
where x(t)∈ℝn is the state variable, y(t)∈ℝq1 is the measurement output of the system, z(t)∈ℝq2 is the signal to be estimated, and E is a singular matrix with rank(E)=r<n; {rt,t≥0} is continuous-time Markov stochastic process taking values in a finite space 𝕄:={1,2,…,N} with transition matrix Γ=(πij)N×N, and the transition probabilities are described as follows:(2.2)Pij=Pr(rt+Δ=j∣rt=i)={πijΔ+o(Δ),ifi≠j,1+πijΔ+o(Δ),ifi=j,
where limΔ→0o(Δ)/Δ=0, πij satisfies πij≥0(i≠j), and πii=-∑j=1,j≠iNπij for all i,j∈𝕄; ΔA(rt) and ΔB(rt) are uncertain matrices and satisfy(2.3)[ΔA(rt),ΔB(rt)]=F(rt)Δ(rt)[E1(rt),E2(rt)],
where Δ(rt) is an unknown, time-varying matrix function and satisfies ΔT(rt)Δ(rt)≤I for all rt∈𝕄; moreover, the disturbance input w(t)∈ℝp satisfies(2.4)∫0TwT(t)w(t)dt≤d2,d≥0,
and the matrices A(rt),B(rt),C(rt),D(rt),L(rt), and G(rt) are coefficient matrices and of appropriate dimension for all rt∈𝕄.In this paper, we construct the following full-order filter:(2.5)Efx̃̇(t)=Af(rt)x̃(t)+Bf(rt)y(t),z̃(t)=Cf(rt)x̃(t),
where x̃(t)∈ℝn is the filter state, z̃(t)∈ℝq2 is the filter output, and Ef,Af(rt),Bf(rt), and Cf(rt) are to design the filter matrices with appropriate dimensions.Definex¯(t)=[xT(t)xT(t)-x̃T(t)]T,e(t)=z(t)-z̃(t) and combining (2.1) and (2.5), one can obtain the following filtering error dynamics as follows:(2.6)E¯x̅̇(t)=A¯(rt)x¯(t)+B¯(rt)w(t),e(t)=L¯(rt)x¯(t)+G(rt)w(t),
where(2.7)E¯=[E0E-EfEf],A¯(rt)=[A(rt)+ΔA(rt)0A(rt)+ΔA(rt)-Af(rt)-Bf(rt)C(rt)Af(rt)],B¯(rt)=[B(rt)+ΔB(rt)B(rt)+ΔB(rt)-Bf(rt)D(rt)],L¯(rt)=[L(rt)-Cf(rt)Cf(rt)].
For notational simplicity, in the sequel, for each possible rt=i,i∈𝕄, a matrix K(rt) will be denoted by Ki; for instance, A(rt) will be denoted by Ai, B(rt) by Bi, 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-Ai-ΔAi) 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-Ai-ΔAi))=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
(2.8)E{x¯T(0)E¯TR¯iE¯x¯(0)}≤c12⟹E{x¯T(t)E¯TR¯iE¯x¯(t)}<c22,∀t∈[0,T].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 stochasticH∞ 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
(2.9)E{∫0TeT(t)e(t)dt}<γ2∫0TwT(t)w(t)dt,
for any nonzero w(t) which satisfies (2.4), where γ is a prescribed positive scalar.Definition 2.5 (see [9]).
LetV(x(t),rt=i,t>0) be the stochastic function, define its weak infinitesimal operator L of stochastic process {(x(t),rt=i),t≥0} by
(2.10)LV(x(t),rt=i,t)=Vt(x(t),i,t)+Vx(x(t),i,t)ẋ(t,i)+∑j=1NπijV(x(t),j,t).Lemma 2.6 (see [36]).
For matricesY,M, and N of appropriate dimensions, where Y is a symmetric matrix, then
(2.11)Y+MF(t)N+NTFT(t)MT<0
holds for all matrix F(t) satisfying FT(t)F(t)≤I for all t∈ℝ, if and only if there exists a positive constant ϵ, such that the following inequality:
(2.12)Y+ϵ-1MMT+ϵNTN<0
holds.Lemma 2.7 (see [36]).
The linear matrix inequalityS=[S11S12*S22]<0 is equivalent to S22<0,S11-S12S22-1S12T<0, where S11=S11T and S22=S22T.Lemma 2.8.
The following items are true.
(i) Assume thatrank(E)=r, there exist two orthogonal matrices U and V such that E has the decomposition as
(2.13)E=U[Σr0*0]VT=U[Ir0*0]VT,
where Σr=diag{δ1,δ1,…,δr} with δk>0 for all k=1,2,…,r. Partition U=[U1U2], V=[V1V2], and 𝒱=[V1ΣrV2] with EV2=0 and U2TE=0.
(ii) IfP satisfies
(2.14)ETP=PTE≥0,
then P̃=UTP𝒱-T with U and 𝒱 satisfying (2.13) if and only if
(2.15)P̃=[P110P21P22],
with P11≥0∈ℝr×r. In addition, when P is nonsingular, one has P11>0 and det(P22)≠0. Furthermore, P satisfying (2.14) can be parameterized as
(2.16)P=UXUTE+U2YVT,
where X=diag{P11,Λ}, Y=[P21P22], and Λ∈ℝ(n-r)×(n-r) is an arbitrary parameter matrix.
(iii) IfP 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:
(2.17)ETP=ETR1/2QR1/2E.
Then the symmetric positive definite matrix Q=R-1/2UXUTR-1/2 is a solution of (2.17).Proof.
One only requires to prove that (ii) and (iii) hold. Let(2.18)P̃=[P11P12P21P22].
Then by (2.13) and (2.14), it follows that condition P̃=UTP𝒱-T if and only if P12=0 and P11≥0∈ℝr×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
(2.19)P=U[P110P21P22]VT=(U[P110*Λ]UT)(U[Ir0*0]VT)+U[00P21P22]VT=U[P110*Λ]UTE+[U1U2][00P21P22]VT=UXUTE+U2YVT,
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), noticingU2TE=0 and P=UXUTE+U2Y𝒱T, we have
(2.20)ETP=ET(UXUTE+U2YVT)=ETUXUTE.
Thus, Q=R-1/2UXUTR-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∈ℝ2n×2n, two sets of symmetric positive definite matrices {Q¯1i,i∈𝕄} with Q¯1i∈ℝ2n×2n, {Q¯2i,i∈𝕄} with Q¯2i∈ℝp×p, and for all i∈𝕄 such that the following inequalities hold:(3.1a)E̅TP̅i=P̅iTE̅≥0,(3.1b)[A̅iTP̅+P̅iTA̅i+∑j=1NπijE̅TP̅j-αE̅TP̅iP̅iTB̅i*-Q̅2i]<0,(3.1c)E̅TP̅i=E̅TR̅i1/2Q̅1iR̅i1/2E̅,(3.1d)c12sup⁡i∈M⁡{λmax⁡⁡(Q̅1i)}+d2sup⁡i∈M⁡{λmax⁡⁡(Q̅2i)}<c22e-αTinf⁡i∈M⁡{λmin⁡(Q̅1i)}.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
(3.2)A¯iTP¯i+P¯iTA¯i+∑j=1NπijE¯TP¯j-αE¯TP¯i<0.
Now, we choose two orthogonal matrices U¯ and V¯ such that E¯ has the decomposition as
(3.3)E¯=U¯[Σr¯0*0]V¯T=U¯[Ir¯0*0]V¯T,
where Σr¯=diag{δ1,δ2,…,δr¯} with δk>0 for all k=1,2,…,r. Partition U¯=[U¯1U¯2], V¯=[V¯1V¯2] and 𝒱¯=[V¯1Σr¯V¯2] with E¯V¯2=0 and U¯2TE¯=0. Denote
(3.4)U¯TA¯iV¯-T=[A¯11iA¯12iA¯21iA¯22i],U¯TP¯iV¯-T=[P¯11iP¯12iP¯21iP¯22i].
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¯22iTP¯22i+P¯22iTA¯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 candidateV(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
(3.5)LV(x¯(t),i)=ξT(t)[A¯iTP¯i+P¯iTA¯i+∑j=1NπijE¯TP¯jP¯iTB¯i*0]ξ(t),
where ξ(t)=[x¯T(t),wT(t)]T. From (3.1b) and (3.5), we obtain
(3.6)E{LV(x¯(t),i)}<αE{V(x¯(t),i)}+wT(t)Q¯2iw(t).
Further, (3.6) can be rewritten as
(3.7)E{L[e-αtV(x¯(t),i)]}<e-αtwT(t)Q¯2iw(t).
Integrating (3.7) from 0 to t, with t∈[0,T], we obtain
(3.8)e-αtE{V(x¯(t),i)}<E{V(x¯(0),i=r0)}+∫0te-ατwT(τ)Q¯2iw(τ)dτ.
Noting that α≥0,t∈[0,T] and condition (3.1c), we have
(3.9)E{x¯T(t)E¯TP¯ix¯(t)}=E{V(x¯(t),i)}<eαtE{V(x¯(0),i=r0)}+eαt∫0te-ατwT(τ)Q¯2iw(τ)dτ≤eαt[sup⁡i∈M{λmax⁡(Q¯1i)}c12+sup⁡i∈M{λmax⁡(Q2i)}d2].
Taking into account that
(3.10)E{x¯T(t)E¯TP¯ix¯(t)}=E{x¯T(t)E¯TR¯i1/2Q¯1iR¯i1/2E¯x¯(t)}≥inf⁡i∈M{λmin⁡(Q¯1i)}E{xT(t)E¯TR¯iE¯x(t)},
we obtain
(3.11)E{x¯T(t)E¯TR¯iE¯x¯(t)}≤sup⁡i∈M{λmax⁡(Q¯1i-1)}E{x¯T(t)E¯TP¯ix¯(t)}<eαTsup⁡i∈M{λmax⁡(Q¯1i)}c12+sup⁡i∈M{λmax⁡(Q¯2i)}d2inf⁡i∈M{λmin⁡(Q¯1i)}.
Therefore, it follows that condition (3.1d) implies 𝔼{xT(t)E¯TR¯iE¯x(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∈ℝ2n×2n, a set of symmetric positive definite matrices {Q¯1i,i∈𝕄} with Q¯1i∈ℝ2n×2n, and for all i∈𝕄 such that (3.1a), (3.1c) and the following inequalities hold:(3.12a)[A̅iTP̅i+P̅iTA̅i+∑j=1NπijE̅TP̅j+L̅iTL̅i-αE̅TP̅iL̅iTGi+P̅iTB̅i*GiTGi-γ2e-αTI]<0,(3.12b)d2γ2<c22inf⁡i∈M⁡{λmin⁡(Q̅1i)}.Proof.
Noting that(3.13)[L¯iTL¯iL¯iTGi*GiTGi]=[L¯iTGiT][L¯iGi]≥0.
Thus, condition (3.12a) implies that
(3.14)[A¯iTP¯i+P¯iTA¯i+∑j=1NπijE¯TP¯j-αE¯TP¯iP¯iTB¯i*-γ2e-αTI]<0.
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
(3.15)E{LV(x¯(t),i)}<αE{V(x¯(t),i)}+γ2e-αTwT(t)w(t)-E{eT(t)e(t)}.
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.DenoteP¯i=diag{Pi,Pi},Q¯1i=diag{Q1i,Q1i},Mi=PiTAfi,Ni=PiTBfi, 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,R¯i,γ,d) with R¯i=diag{Ri,Ri}, if there exists a scalar α≥0, a set of nonsingular matrices {Pi,i∈𝕄} with Pi∈ℝn×n, a set of positive definite matrices {Q1i,i∈𝕄} with Q1i∈ℝn×n, three sets of matrices {Mi,i∈𝕄} with Mi∈ℝn×n, {Ni,i∈𝕄} with Ni∈ℝn×q1, {Cfi,i∈𝕄} with Cfi∈ℝq2×n, for all i∈𝕄 such that(3.16a)ETPi=PiTE≥0,(3.16b)[Θ1i***PiTAi-Mi-NiCiΘ2iPiTBi-NiDi*BiTPi*-γ2e-αTI*Li-CfiCfiGi-I]<0,(3.16c)ETPi=ETRi1/2Q1iRi1/2E,(3.16d)d2γ2<c22inf⁡i∈M⁡{λmin⁡(Q1i)}hold, where Θ1i=PiTAi+AiTPi+∑j=1NπijETPj-αETPi, and Θ2i=Mi+MiT+∑j=1NπijETPj-αETPi.
In addition, the desired filter parameters can be chosen by(3.17)Afi=Pi-TMi,Bfi=Pi-TNi,Cfi=Cfi,Ef=E.
Noting thatPi is nonsingular matrix, by Lemma 2.8, there exist two orthogonal matrices U and V, such that E has the decomposition as
(3.18)E=U[Σr0*0]VT=U[Ir0*0]VT,
where Σr=diag{δ1,δ2,…,δr} with δk>0 for all k=1,2,…,r. Partition U=[U1U2], V=[V1V2], and 𝒱=[V1ΣrV2] with EV2=0 and U2TE=0. Let P̃i=UTPi𝒱-T,  from (3.16a), P̃i is of the following form [P11i0P21iP22i], and Pi can be expressed as
(3.19)Pi=UXiUTE+U2YiVT,
where Xi=diag{P11i,Λi} and Yi=[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=Ri-1/2UXiUTRi-1/2 is a solution of (3.16c), and Pi satisfies
(3.20)ETPi=PiTE=ETUXiUTE.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{Ri,Ri}, if there exists a scalar α≥0, a set of positive definite matrices {Xi,i∈𝕄} with Xi∈ℝn×n, four sets of matrices {Yi,i∈𝕄} with Yi∈ℝ(n-r)×n, {Mi,i∈𝕄} with Mi∈ℝn×n, {Ni,i∈𝕄} with Ni∈ℝn×q1, and {Cfi,i∈𝕄} with Cfi∈ℝq2×n,   for all i∈𝕄 such that (3.16d) and the following linear matrix inequality
(3.21)[Ξ1i***PiTAi-Mi-NiCiΞ2iPiTBi-NiDi*BiTPi*-γ2e-αTI*Li-CfiCfiGi-I]<0
hold, where Ξ1i=PiTAi+AiTPi+∑j=1NπijETPj-αETPi, Ξ2i=Mi+MiT+∑j=1NπijETPj-αETPi, Pi=UXiUTE+U2Yi𝒱T, Xi and Yi are from the form (3.19); Moreover, other matrical variables are the same as Theorem 3.3.By Theorems3.3 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={Ri,Ri}, if there exists a scalar α≥0, a set of positive definite matrices {Xi,i∈𝕄} with Xi∈ℝn×n, four sets of matrices {Yi,i∈𝕄} with Yi∈ℝ(n-r)×n, {Mi,i∈𝕄} with Mi∈ℝn×n, {Ni,i∈𝕄} with Ni∈ℝn×q1, {Cfi,i∈𝕄} with Cfi∈ℝq2×n, and a set of positive scalars {ϵi,i∈𝕄}, for all i∈𝕄 such that (3.16d) and the following linear matrix inequality
(3.22)[Υ1i****PiTAi-Mi-NiCiΥ2iPiTBi-NiDi**BiTPi+ϵiE2iTE1i*ϵiE2iTE2i-γ2e-αTI**FiTPiFiTPi0-ϵiI*Li-CfiCfiGi0-I]<0
hold, where Υ1i=PiTAi+AiTPi+∑j=1NπijETPj+ϵiE1iTE1i-αETPi, Υ2i=Mi+MiT+∑j=1NπijETPj-αETPi, Pi=UXiUTE+U2Yi𝒱T, Xi and Yi are from the form (3.19); Moreover, other matrical variables are the same as Theorem 3.3.Remark 3.6.
Theorems3.4 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.
LetI<Q1i<ηI, then one can check that condition (3.16d) can be guaranteed by imposing the conditions
(3.23)I<Ri-1/2UXiUTRi-1/2<ηI,d2γ2-c22<0.Remark 3.7.
The feasibility of conditions stated in Theorem3.4 and Theorem 3.5 can be turned into the following LMIs-based feasibility problem with a fixed parameter α, respectively:
(3.24)min⁡⁡(γ2+c22)Xi,Yi,Mi,Ni,Cfi,ηs.t.(3.21)and(3.23),min⁡⁡(γ2+c22)Xi,Yi,Mi,Ni,Cfi,ϵi,ηs.t.(3.22)and(3.23).


## 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,
(4.1)A1=[0.21132.510.132],B1=[111],L1=[10.11]T,C1=[111]T,F1=[0.050000.04000.010],E11=[0.200.030.030.0200.0500.01],E21=[0.030.020.05],
(ii) Mode #2,
(4.2)A2=[-4111-31121],B2=[101],L2=[0.712]T,C2=[0.841]T,F2=[0.050000.04000.010.02],E12=[0.0200.030.030.0300.0200.1],E22=[0.030.020.02],
and E=diag{1,1,0},D1=0.2,G1=0.3,D2=0.1,G2=0.5,d=0.6, Δi=diag{r1(i),r2(i),r3(i)}, where rj(i) satisfies |rj(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 chooseR1=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
(4.3)Af1=[-0.89061.50440.395864.153747.069251.020714.904815.643114.4887],Bf1=[2.2274-63.2281-13.4543],Cf1=[0.9315-0.35460.6824],(4.4)Af2=[25.0076170.668440.7148-149.7287-851.1311-192.867520.5802107.732925.9671],Bf2=[-44.9909223.8116-27.9168],Cf2=[0.39160.37221.6748],
with α=1.3209, and the optimal values γ=8.1261, c2=4.8758.Figure 1
The local optimal bound ofγ.Figure 2
The local optimal bound ofc2.Remark 4.2.
From the above example and Remark3.7, 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:
(4.5)A1=[-320-3-2.50101],A2=[130-1-2.50-10-4.8].
Moreover, other matrical variables and the transition rate matrix are defined similarly as Example 4.1.
LetR1=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
(4.6)Af1=[35.992347.863242.1297-125.9936-141.5997-122.030731.997834.460431.8623],Bf1=[-47.6632137.2043-33.8527],Cf1=[0.82940.11360.8294],Af2=[16.876573.669625.7772-137.4627-607.3156-225.4097-17.9446-72.5666-36.9766],Bf2=[-15.8299141.986217.5990],Cf2=[1.12901.59753.6412].
Thus, the above filtering error system is stochastically stable and the calculated minimum H∞ performance γ satisfies ∥Twz∥<3.7770.


## 5. Conclusion

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



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*Source: 615790-2012-02-19.xml*