# RobustH∞ Filtering for Uncertain Neutral Stochastic Systems with Markovian Jumping Parameters and Time Delay

Authors: Yajun Li; Zhaowen Huang
Journal: Mathematical Problems in Engineering
 (2015)

---

## Abstract

This paper deals with the robustH∞ filter design problem for a class of uncertain neutral stochastic systems with Markovian jumping parameters and time delay. Based on the Lyapunov-Krasovskii theory and generalized Finsler Lemma, a delay-dependent stability condition is proposed  to ensure not only that the filter error system is robustly stochastically stable but also that a prescribed H∞ performance level is satisfied for all admissible uncertainties. All obtained results are expressed in terms of linear matrix inequalities which can be easily solved by MATLAB LMI toolbox. Numerical examples are given to show that the results obtained are both less conservative and less complicated in computation.


---

## Body

## 1. Introduction

Time delay exists extensively in chemical process systems, communication systems, economic systems, microwave oscillator, and networked control systems. Meanwhile, because of the modeling inaccuracies and changes in the environment of the model parameter, uncertainties are unavoidable in the process of modeling. The appearance of time delay and uncertainties in many systems can often bring instability, oscillation, and poor performance; considerable attention has been focused on the stability analysis of uncertain time delays systems; see [1–6] and the references therein. On the other hand, a real system is often affected by external disturbances such as stochastic perturbations. The stochastic effects can also lead to oscillation, divergence, or other poor performances. Therefore, the stability study of stochastic systems with time delay has been paid great attention and a lot of significant results have been reported in the literature; see [7–12] and the references therein.In the past few decades, filtering problem has been a hot issue in the fields of signal processing. Kalman filtering has been successfully applied in many fields such as manufacturing systems, economic systems, and maneuvered target tracking. However, the exact requirement of known dynamics system and precise noise statistics have restricted its practical application. In such a case, we can resort toH∞ filtering [13–22] and L2-L∞ filtering [23–25]; see the references therein.Markovian jump systems, originally raised by Krasovskii and Lidskii [26], are famous for the description of many dynamical practical systems whose structure and parameters are subject to random changes. Therefore, the stability analysis and filtering problem for Markovian jump systems have been studied [27–34]. For example, the stability analysis of impulsive stochastic neural networks with Markovian jump are studied in [27, 29, 31, 34]; H∞ control and mode-dependent H∞ filtering for discrete-time Markovian jump linear systems with partly unknown transition probabilities are, respectively, investigated in [28, 30, 32]. The design of reduced-order H∞ filter for Markovian jumping systems with time delay is studied in [15]. The L2-L∞ filter design for stochastic time-varying delay systems with Markovian jumping parameters is considered in [33].Many methods are proposed in the process of robust stochastic stability analysis and filtering design, which develops from the early solving Riccati equation to model transformation method and cross-terms bounding technique [5], free-weighting matrices method [11, 32], slack matrix variables [17–19, 24, 28], and delay-partitioning method [20]. However, model transformations may give rise to additional dynamics of the original systems [13], and cross-terms bounding techniques can bring conservatism. Moreover, as pointed out in [33], under certain circumstance, free matrix variables may not lessen the conservatism. In recent years, another popular method called Finsler Lemma is carried out so as to decrease the computational cost as well [5], and the Finsler Lemma in deterministic setting is extended to generalized Finsler Lemma in stochastic systems in [5, 33, 34].On the other hand, many dynamical systems can be modeled by neutral systems which are organized by neutral functional differential equations. Other than retarded time delay systems containing delays only in states, a neutral time delay system contains delays in both its state and its derivatives of state. Recently, for neutral stochastic time delay systems, the stability analysis and filter design problem are mainly addressed in [4, 9, 19]. It is necessary to point out that the delay-dependent robust H∞ filtering for uncertain neutral stochastic time delay system is studied in [19, 24]. Robust H∞ filter design for neutral stochastic uncertain systems with time-varying delay is studied in [13]. To the best of the authors’ knowledge, the H∞ filtering problem has not been reported about uncertain neutral stochastic systems with Markovian jumping parameters and time delay, which motivates the present study.Motivated by the works in [5, 13, 17, 18], the robust H∞ filtering for uncertain neutral stochastic systems with Markovian jumping parameters and time delay is considered in this paper. By generalized Finsler Lemma, the robust stochastic stability condition is obtained. The presented condition is simple and efficient. Finally, the effectiveness of the approach is verified by illustrative examples including a comparison with some recent results.Throughout this paper,Rn denotes the n-dimensional Euclidean space. Rn×m is the set of n×m real matrices. I is the identity matrix. · denotes Euclidean norm for vectors and · denotes the spectral norm of matrices. N denotes the set of all natural numbers; that is, N={0,1,2,…}. (Ω,F,{Ft}t≥0,P) is a complete probability space with filtration {Ft}t≥0 satisfying the usual conditions. MT stands for the transpose of the matrix M. For symmetric matrices X and Y, the notation X>Y (resp., X≥Y) means that the X-Y is positive definite (resp., positive semidefinite). ∗ denotes a block that is readily inferred by symmetry. E{·} stands for the mathematical expectation operator with respect to the given probability measure P.


## 2. Problem Description

Consider the following uncertain neutral stochastic systems with Markovian jumping parameters and time delay:(1)dxt-Grtxt-h=At,rttxt+A1t,rttxt-h+Brtvtdt+Dt,rttxt+D1t,rttxt-h+D2rtvtdwt,dyt=Crtxt+C1rtxt-h+C2rtvtdt+Erttxt+E1rtxt-h+E2rtvtdwt,zt=L1rtxt+L2rtxt-h+L3rtvt,xθ=ψτ,rt=r0∈S,∀τ∈-h,0,where x(t)∈Rn is the state vector, y(t)∈Rm is the measured output, v(t)∈Rq is the disturbance input in L2[0,∞), and z(t)∈Rp is the signal to be estimated. A(t,rt)(t), A1(t,rt)(t), B(rt), D(t,rt)(t), D1(t,rt)(t), D2(rt), C(rt), C1(rt), C2(rt), E(rt), E1(rt), E2(rt), L1(rt), L2(rt), and L3(rt) are the matrix functions of the random jumping process r(t), where r(t) is a finite-state Markovian jump process representing the system mode; that is, r(t) takes discrete values in given finite set S=1,2,…,N. Here ψ(·) is the initial condition and is assumed to be continuously differentiable on [-h,0]. Consider h>0 indicates the time delay. w(t) is a scalar Brownian motion (Wiener process) defined on the complete probability space (Ω,F,{Ft}t≥0,P) satisfying(2)Edwt=0,Edw2t=dt.The transition probability matrix of systems (1) is given by (3)Prrt+Δ=j∣rt=i=πijΔ+oΔ,j≠i1+πiiΔ+oΔ,j=i,where Δ>0, limΔ→0o(Δ)/Δ=0, πij≥0, ∀j≠i, is the transition rate from mode i at time t to mode j at time t+Δ, and (4)πii=-∑j=1,j≠ij=Nπij<0.For the purpose of simplicity, in this paper, for eachr(t)=i∈S, A(t,rt)(t), A1(t,rt)(t), and B(rt) are denoted by Ai(t), A1i(t), Bi, and so on. In systems (1), (5)Ait=Ai+ΔAit,A1it=A1i+ΔA1it,Dit=Di+ΔDit,D1it=D1i+ΔD1it,and Ai, A1i, Bi, Di, D1i, D2i, Ci, C1i, C2i, Ei, E1i, E2i, L1i, L2i, and L3i are known real constant matrices with appropriate dimensions. ΔAi(t), ΔA1i(t), ΔDi(t), and ΔD1i(t) are unknown matrices representing norm-bounded parameter uncertainties, which are assumed to satisfy(6)ΔAitΔA1itΔDitΔD1it=M1iM2iFitN1iN2i,where M1i, M2i, N1i, and N2i are known real constant matrices and Fi(·):R→Rk×l is an unknown time-varying matrix function satisfying (7)FiTtFit≤I,∀t.The parameter uncertainties ΔAi(t), ΔA1i(t), ΔDi(t), and ΔD1i(t) are said to be admissible if both (6) and (7) hold.In this paper, we make the following assumption on the matrixGi in systems (1).Assumption 1.
The matrixGi in systems (1) satisfies (8)ρGi<1,where the notation ρ(Gi) denotes the spectral radius of Gi.We now consider a full-order filter for systems (1) with the following form: (9)Σf:dx^t=Afix^tdt+Bfidyt,z^t=Cfix^t,where x^(t)∈Rn is the filter state, z^(t)∈Rp is the estimation of z(t), and Afi, Bfi, and Cfi(i∈S) are the filter parameters with compatible dimensions to be determined.Define(10)ξt=xTtx^TtT,et=zt-z^t.Then the filtering error systems can be obtained as follows: (11)dξt-G-iKξt-h=A-itξt+A-1itKξt-h+B-ivtdt+D-itξt+D-1itKξt-h+D-2ivtdwt,et=L-1iξt+L2iKξt-h+L3ivt,where (12)A-it=A-i+ΔA-it,A-1it=A-1i+ΔA-1it,D-it=D-i+ΔD-it,D-1it=D-1i+ΔD-1itwith (13)A-i=Ai0BfiCiAfi,ΔA-it=ΔAit000,A-1i=A1iBfiC1i,ΔA-1it=ΔA1it0,B-i=BiBfiC2i,D-i=Di0BfiEi0,ΔD-it=ΔDit000,D-1i=D1iBfiE1i,ΔD-1it=ΔD1it0,D-2i=D2iBfiE2iK=I0,L-1i=L1i-Cfi,G-i=Gi0.Then the problem of robust H∞ filtering to be addressed in this paper is formulated as follows: given the uncertain stochastic delay systems (1) and a prescribed attenuation level γ>0, design linear stochastic filter (Σf) as the form of (9) such that the filtering error systems (11) are robustly stochastically stable and under zero initial conditions, the following inequality holds: (14)et2<γvt2for all nonzero v(t)∈L2[0,∞) and all admissible uncertainties.Before concluding this section, we introduce the following Lemmas, which will be used in the derivation of our main results in the next section.Lemma 2.
For any vectorsx,y∈Rn and any scalar ϵ>0, matrices D, F, E are real matrices of appropriate dimensions with FTF≤I, then the following inequality hold: (15)2xTDFEy≤ϵ-1xTDDTx+ϵyTEETy.Proposition 3 ([5], generalized Finsler Lemma (GFL)).
Consider stochastic vectorθ∈Rn, symmetric and positive matrix Θ∈Rn×n, and matrix B∈Rm×n with rank (B)=r<n. Let B⊥ represent the right orthogonal complement of B, that is, BB⊥=0, then the following four statements are equivalent: (T1)
E{θTΘθ}<0, ∀θ≠0, t>t0, E{Bθ}=0;(T2)
B⊥TΘB⊥<0;(T3)
∃ϵ∈R:Θ-ϵBTB<0;(T4)
∃Λ∈Rn×m:Θ+ΛB+BTΛT<0.Remark 4.
Based on generalized Finsler Lemma, the stability of neutral stochastic systems with time delay has been studied in [5, 34]. In addition, it should be noted that the stochastic systems in [5, 34] are not Markovian jump systems. And the filtering problem for stochastic time delay systems with Markovian jumping parameters is considered in [20]. However, it should be pointed out that the systems in [20] do not include any analysis of neutral phenomena. So H∞ filtering for neutral stochastic systems with Markovian jumping parameters and time delay is considered in this paper.


## 3. Main Results

Theorem 5.
Consider the uncertain neutral stochastic Markovian jump systems (1). For given scalars γ>0, h>0, systems (1) are robustly stochastically stable for all admissible uncertainties satisfying (6) and (7), if there exist symmetric positive matrices Pi>0, R>0, Q=Q1Q2∗Q3>0, and scalar ϵi>0 satisfying (16)Ωi=Ω11iΩ12i-KTRGiPiB-ihA-iTKTRD-iTPiL-1iTPiM-1i∗Ω22iΩ23i-G-iTPiB-ihA-1iTKTRD-1iTPiL2iT-G-iTPiM-1i∗∗Ω33i00000∗∗∗-γ2IhB-iTKTRD-2iTPiL3iT0∗∗∗∗-R00hRKM-1i∗∗∗∗∗-Pi0PiM-2i∗∗∗∗∗∗-I0∗∗∗∗∗∗∗-ϵi<0,where (17)Ω11i=PiA-i+A-iTPi+∑j=1NπijPj+KTQ1K+ϵiN~1iTN~1i-KTRK,Ω12i=PiA-1i-A-iTPiG-i-∑j=1NπijPjG-i+KTQ2+ϵiN~1iTN2i+KTRGi+I,Ω22i=-G-iTPiA-1i-A-1iTPiG-i+G-iT∑j=1NπijPjG-i+Q3-Q1+ϵiN2iTN2i-Gi+ITRGi+I,Ω23i=-Q2+Gi+ITRGi,Ω33i=-Q3-GiTRGi,N~1i=N1iK=N1i0,M-1i=M1i0,M-2i=M2i0.Proof.
For the purpose of convenience, the following notations are adopted:(18)z-t=ξt-G-iKξt-h,fvt,ξt,i=A-itξt+A-1itKξt-h+B-ivt,gvt,ξt,i=D-itξt+D-1itKξt-h+D-2ivt,and then the filtering error systems (11) become (19)dz-t=fvt,ξt,idt+gvt,ξt,idwt.Choose the Lyapunov-Krasovskii functional candidate as follows: (20)Vt,ξt,i=z-TtPiz-t+∫t-htηTsKTQKηsds+h∫-h0∫t+βtfvTs,ξs,iKTRKfvs,ξs,idsdβ,where η(t)=ξT(t)ξT(t-h)T. According to Itô’s differential formula, the stochastic differential along systems (11) is (21)dVt,ξt,i=LVt,ξt,idt+2z-TtPigvt,ξt,idwt,where (22)LVt,ξt,i=2z-TtPifvt,ξt,i+gvTt,ξt,iPigvt,ξt,i+z-Tt∑j=1NπijPjz-t+ηTtKTQKηt-ηTt-hKTQKηt-h+h2fvTt,ξt,iKTRKfvt,ξt,i-∫t-htfvTs,ξs,iKThRKfvs,ξs,ids.
Firstly, we show that the filtering error systems (11) with v(t)=0 are robustly stochastically stable.
Taking mathematical expectation on both sides of system (19) and by virtue of E{dw(t)=0}, we obtain (23)Edz-t=Efvt,ξt,idt.Integrating both sides of (23) from t-h to t, we have (24)∫t-htE-Iξt+G-iK+Iξt-h-G-iKξt-2h+hfvs,ξs,ids=0.From the definition of fv(t,ξ(t),i) with v(t)=0 and (24), it is easy to obtain (25)∫t-htEBiςids=0,where (26)Bi=-IA-itA-1it000-KKG-i+I-KG-ihI,ςi=fvTt,ξt,iξTtξTt-hKTξTt-2hKTfvTs,ξs,iKTT.
The right orthogonal complements ofBi is (27)Bi⊥=A-iTtI00KThA-1iTt0I0-KG-i+IhT000IKG-ihTT.
Taking mathematical expectation on both sides of (21) and then substituting (22) into (21), we have (28)dEVt,ξt,idt=ELVt,ξt,i≤E1h∫t-htςiTΓiςids=1h∫t-htEςiTΓiςids,where (29)Γi=h2KTRKPi-PiG-i00∗Γ22iΓ23i00∗∗Γ33i-Q20∗∗∗-Q30∗∗∗∗-h2R,Γ22i=∑j=1NπijPj+D-iTtPiD-it+KTQ1K,Γ23i=-∑j=1NπijPjG-i+D-iTtPiD-1it+KTQ2,Γ33i=G-iT∑j=1NπijPjG-i+D-1iTtPiD-1it+Q3-Q1.In order to prove the robust stochastic stability of the filtering error systems (11) with v(t)=0, it suffices to show(30)EςiTΓiςi<0.
By virtue of Proposition3, (30) is equivalent to (31)EΓ^i=EBi⊥TΓiBi⊥<0,where (32)Γ^i=Γ^11iΓ^12i-KTRKG-i∗Γ^22iΓ^23i∗∗Γ^33i,Γ^11i=h2A-iTtKTRKA-it+PiA-it+A-iTtPi+∑j=1NπijPj+D-iTtPiD-it+kTQ1K-KTRK,Γ^12i=h2A-iTtKTRKA-1it+PiA-1it-A-iTtPiG-i-∑j=1NπijPjG-i+D-iTtPiD-1it+KTQ2+KTRKG-i+I,Γ^22i=h2A-1iTtKTRKA-1it-G-iTPiA-1it-A-1iTtPiG-i+G-iT∑j=1NπijPjG-i+D-1iTtPiD-1it+Q3-Q1-KG-i+ITRKG-i+I,Γ^23i=-Q2+KG-i+ITRKG-i,Γ^33i=-Q3-KG-iTRKG-i.It is obvious that Γ^i<0 are implied by (16) according to Schur complements. Therefore, if (16) is feasible, then filtering error systems (11) with v(t)=0 are robustly stochastically stable.
Next, we will establish theH∞ performance for the filtering error systems (11) under the zero initial condition.
From the definition offv(t,ξ(t),i) and together with (24), it implies (33)∫t-htEBviςvids=0,where (34)Bvi=-IA-itA-1it00B-i0-KKG-i+I-KG-ihI0,ςvi=fvTt,ξt,iξTtξTt-hKTξTt-2hKTfvTs,ξs,iKTvTtT.The right orthogonal complements of Bvi are (35)Bvi⊥=A-iTtI00KTh0A-1iTt0I0-KG-i+IhT0000IKG-ihT0B-iT0000IT.
Noticing (21)-(22) together with E{dw(t)}=0, we can obtain (36)dEVt,ξt,idt=ELVt,ξt,i≤E1h∫t-htςviTΓviςvids=1h∫t-htEςviTΓviςvids,where (37)Γvi=ΓiΓv∗Γ66i,Γv=0Γ26iTΓ36iT00T,Γ26i=D-iTtPiD-2i,Γ36i=D-1iTtPiD-2i,Γ66i=D-2iTPiD-2i,and set (38)J=E∫0∞eTtet-γ2vTtvtdt.Adding the right side of (38) to both sides of (36) and integrating both sides of (36) from 0 to ∞ and then taking the zero initial condition into account, we can acquire (39)J≤1h∫0∞∫t-htEςviTHviςvidsdt,where (40)Hvi=Γvi+ΠTΠ+diag0,0,0,0,0,-γ2I,and Π=0L-1iL2iK00L3i.
According to Proposition3, E{ςviTHviςvi}<0 is equivalent to (41)Θi=Bvi⊥THviBvi⊥=Θ11iΘ12i-KTRKG-iΘ14i∗Θ22iΘ23iΘ24i∗∗Θ33i0∗∗∗Θ44i<0,where (42)Θ11i=Γ^11i+L-1iTL-1i,Θ12i=Γ^12i+L-1iTL2i,Θ22i=Γ^22i+L2iTL2i,Θ23i=Γ^23i,Θ33i=Γ^33i,Θ14i=h2A-iTtKTRKB-i+PiB-i+D-iTtPiD-2i+L-1iTL3i,Θ24i=h2A-1iTtKTRKB-i-G-iTPiB-i+D-1iTtPiD-2i+L2iTL3i,Θ44i=h2B-iTKTRKB-i+D-2iTPiD-2i+L3iTL3i-γ2I.
According to Schur complement and (12), we can obtain (43)Θi=Π11iΠ12i-KTRGiPiB-ihA-iTKTRD-iTPiL-1iT∗Π22iΠ23i-GiTPiB-ihA-1iTKTRD-1iTPiL2iT∗∗Π33i0000∗∗∗-γ2IhB-iTKTRD-2iTPiL3iT∗∗∗∗-R00∗∗∗∗∗-Pi0∗∗∗∗∗∗-I+Θ11iΘ12i00hΔA-iTtKTRΔD-iTtPi0Θ21iΘ22i00hΔA-1iTtKTRΔD-1iTtPi000000000000000hRKΔA-ithRKΔA-1it00000PiΔD-iTtPiΔD-1it000000000000,where (44)Π11i=PiA-i+A-iTPi+∑j=1NπijPj+kTQ1K-KTRK,Π12i=PiA-1i-A-iTPiG-i-∑j=1NπijPjG-i+KTQ2+KTRKG-i+I,Π22i=-G-iTPiA-1i-A-1iTPiG-i+G-iT∑j=1NπijPjG-i+Q3-Q1-KG-i+ITRKG-i+I,Π23i=-Q2+KG-i+ITRKG-i,Π33i=-Q3-KG-iTRKG-i,Θ11i=PiΔA-it+ΔA-iTtPi,Θ12i=PiΔA-1it-ΔA-iTtPiG-i,Θ21i=ΔA-1iTtPi-G-iTPiΔA-it,Θ22i=-G-iTPiΔA-1it-ΔA-1iTtPiG-i.
Then by Lemma2, it can be seen that (45)Θ11iΘ12i00hΔA-iTtKTRΔD-iTtPi0Θ21iΘ22i00hΔA-1iTtKTRΔD-1iTtPi000000000000000hRKΔA-ithRKΔA-1it00000PiΔD-iTtPiΔD-1it000000000000=ΥFitΩ+ΩTFitTΥT≤1ϵiΥΥT+ϵiΩTΩ,where Υ=M-1iTPi-M-1iTPiG-i00hM-1iKTRM-2iTPi0T,   Ω=N~1iN2i00000.
By (41), (46), and Schur complements, Θi<0 holds if and only if Ωi<0. This completes the proof.Remark 6.
Theorem5 is established based on GFL. For the sake of reducing the computational complexity, similar to [6, 8, 10], the first two equivalent conditions of Proposition 3 are adopted in this paper.
Now we are in a position to present theH∞ filter design for uncertain neutral stochastic system with Markovian jumping parameters and time delay based on Theorem 5.Theorem 7.
Consider systems (1), for given scalars h>0, γ>0; then there exists a linear stochastic full-order filter with the form (9), such that filter error systems (11) are robustly stochastically stable and satisfy prescribed H∞ disturbance attenuation level γ for all admissible uncertainties (6) and (7) if there exist symmetric positive matrices Xi>0, Fi>0, Q=Q1Q2∗Q3>0, and R>0 and matrices AFi, BFi, CFi, scalars ϵi>0, such that the following LMI holds: (46)Υ11iΥ12iΥ13i-RGiΥ15ihAiTRΥ17iΥ18iL1iT-CFiTΥ110i∗Υ22iΥ23i0Υ25i000-CFiT-FiM1i∗∗Υ33iΥ34iΥ35ihA1iTRΥ37iΥ38iL2iT-GiTXiM1i∗∗∗Υ44i000000∗∗∗∗-γ2IhBiTRΥ57iΥ58iL3iT0∗∗∗∗∗-R000hRM1i∗∗∗∗∗∗-Xi+Fi00Υ710i∗∗∗∗∗∗∗-Fi0-FiM2i∗∗∗∗∗∗∗∗-I0∗∗∗∗∗∗∗∗∗-ϵiI<0,where (47)Υ11i=Xi-FiAi+AiTXi-Fi+∑j=1NπijXj-Fj+Q1-R+ϵiN1iTN1i,Υ12i=-AiTFi+CiTBFiT+AFiT,Υ22i=AFi+AFiT+∑j=1NπijFj,Υ13i=Xi-FiA1i-AiTXiGi+CiTBFiTGi+AFiTGi+∑j=1NπijFj-XjGi+Q2+RGi+I+ϵiN1iTN2i,Υ23i=-FiA1i+BFiC1i+AFiTGi+∑j=1NπijFjGi,Υ33i=-GiTXiA1i-A1iTXiTGi+GiTBFiC1i+C1iTBFiTGi+GiT∑j=1NπijXjGi+Q3-Q1-Gi+ITRGi+I+ϵiN2iTN2i,Υ34i=-Q2+Gi+ITRGi,Υ44i=-Q3-GiTRGi,Υ15i=Xi-FiBi,Υ25i=-FiBi+BFiC2i,Υ35i=-GiTXiBi+GiTBFiC2i,Υ17i=DiTXi-Fi,Υ37i=D1iTXi-Fi,Υ57i=D2iTXi-Fi,Υ18i=-DiTFi+EiTBFiT,Υ38i=-D1iTFi+E1iTBFiT,Υ58i=-D2iTFi+E2iTBFiT,Υ110i=Xi-FiM1i,Υ710i=Xi-FiM2i.
Meanwhile, the filter parameters are given by(48)Afi=Fi-1AFi,Bfi=Fi-1BFi,Cfi=CFi.Proof.
We note that, from (48), it is easy to see Xi-Fi-FiFi>0, and Xi>Fi>0. Define (49)Pi=Xi-Fi-FiFi;then applying Schur complement, Xi-FiFi-1Fi=Xi-Fi>0 guarantees Pi>0. Let (50)J=diagT,I,I,I,I,T,I,I,I,where T=I0II. Substituting Pi and (12) into (16), then pre- and postmultiplying (16) by JT and J, respectively, and using (51), the desired result (48) follows immediately. This completes the proof.Remark 8.
Theorem7 considers the H∞ filtering problem for uncertain neutral stochastic time delay systems with Markovian jumping parameters. It should be noted that the proposed conditions are formulated in terms of LMIs. Therefore, by MATLAB LMI toolbox, for given different h or γ, the lower bound of performance index γ and the upper bound of h can be efficiently obtained by solving a generalized eigenvalue problem.
Now we would like to proceed to present theH∞ filtering for uncertain neutral stochastic time delay systems without Markovian jumping parameters. Considering the system (Σ) without the Markovian jumping parameters, the following systems can be obtained: (51)ΣD:dxt-Gxt-h=Atxt+A1txt-h+Bvtdt+Dtxt+D1txt-h+D2vtdwt,(52)dyt=Cxt+C1xt-h+C2vtdt+Ext+E1xt-h+E2vtdwt,(53)zt=L1xt+L2xt-h+L3vt,(54)xθ=ψτ,rt=r0∈S,∀τ∈-h,0,where (55)At=A+ΔAt,A1t=A1+ΔA1t,Dt=D+ΔDt,D1t=D1+ΔD1t,and ΔA(t), ΔA1(t), ΔD(t), and ΔD1(t) are unknown matrices satisfying (56)ΔAtΔA1tΔDtΔD1t=M1M2FtN1N2.
We can obtain the following Corollary9 for system (ΣD).Corollary 9.
Consider the system(ΣD), for given scalars h>0, γ>0; then there exists a linear stochastic full-order filter with the form (9), if there exist symmetric positive matrices F>0, X>0, R>0, Q=Q1Q2∗Q3>0, matrices AF, BF, CF, and scalar ϵ>0, such that the following LMI holds: (57)Ξ=Ξ11Ξ12Ξ13-RGΞ15hATRΞ17Ξ18L1T-CFTX-FM1∗Ξ22Ξ230Ξ25000-CFT-FM1∗∗Ξ33Ξ34Ξ35hA1TRΞ37Ξ38L2T-GTXM1∗∗∗Ξ44000000∗∗∗∗-γ2IhBTRΞ57Ξ58L3T0∗∗∗∗∗-R000hRM1∗∗∗∗∗∗-X+F00X-FM2∗∗∗∗∗∗∗-F0-FM2∗∗∗∗∗∗∗∗-I0∗∗∗∗∗∗∗∗∗-ϵI<0,where (58)Ξ11=X-FA+ATX-F+Q1-R+ϵN1TN1,Ξ12=-ATF+CTBFT+AFT,Ξ13=X-FA1-ATXG+CTBFTG+AFTG+ϵN1TN2+Q2+RG+I,Ξ15=X-FB,Ξ17=DTX-F,Ξ18=-DTF+ETBFT,Ξ22=AF+AFT,Ξ23=-FA1+BFC1+AFTG,Ξ25=-FB+BFC2,Ξ33=-GTXA1-A1TXG+GTBFC1+C1TBFTG+Q3-Q1-G+ITRG+I+ϵN2TN2,Ξ34=-Q2+G+ITRG,Ξ35=-GTXB+GTBFC2,Ξ37=D1TX-F,Ξ38=-D1TF+E1TBFT,Ξ44=-Q3-GTRG,Ξ57=D2TX-F,Ξ58=-D2TF+E2TBFT;then the robust H∞ filtering problem is solvable. Furthermore, the parameters of the desired robust H∞ filter can be given as (59)Af=F-1AF,Bf=F-1BF,Cf=CF.Remark 10.
The proof of Corollary9 follows the same lines as that in the proof of Theorem 7, so the detailed procedure is omitted here. When D2=0, E=0, E1=0, and E2=0, systems (52)–(55) in this paper reduces to systems (1) in [18]. It is noticed that the filtering problem studied in [18] is a special case of this paper. For comparisons of our results with that in [18], see Example 1 in detail.


## 4. Numerical Examples

In this section, numerical examples and simulations are given to illustrate the validity and benefits of the proposed approach.Example 1.
Consider systems (52)–(55) without Markovian jumping parameters as follows: (60)A=-100-0.9,A1=-10-1-1,B=0.20-0.10.1,D=0.2000.2,D1=0.100.3-0.2,D2=0.2000.2,C=0.1000.1,C1=00.15-0.10.1,C2=0.20.100.1,E=0.50.51.00,E1=1.00.20.3-1,E2=0.20.50.30.4,L1=-0.10.10-0.1,L2=0.1-0.1500.15,L3=0.20-0.150.1,M1=0.2000.2,M2=0.10.100.2,G=0.100.10.1,N1=0.1000.1,N2=0.10.100.1.
For different given noise attenuation levelsγ, the upper bounds of delay for systems (52)–(55) in Corollary 9 of this paper are presented in Table 1. For different given time delays, the lower bounds of noise attenuation level γ for systems (52)–(55) in Corollary 9 of this paper are provided in Table 2.
In particular, whenD2=0, E=0, E1=0, and E2=0, systems (52)–(55) in this paper reduce to systems (1) in [18]. For different given noise attenuation levels γ, the comparison of the upper bounds of delay in [18] with our results is presented in Table 3, and, for different given time delays, the lower bounds of noise attenuation level γ for systems (1) in [18] and the same systems in this paper are provided in Table 4.
Besides, Theorem  2 in [13] fails to give a feasible solution. The number of decision variables of Theorem  3 in [18] is 13n2+5n+2/2, which is the same as that in Corollary 9 of this paper. From Tables 3 and 4, we can see that our proposed method is less conservative than that in [18].
Now in the case whenγ=2.4, hmax=2.167, we resort to the MATLAB LMI control toolbox to solve the LMI (59), and the feasible solution can be obtained as follows: (61)dx^t=-2.6806-0.1892-8.4386-2.0417x^tdt+0.02320.0476-0.15730.3199dyt,z^t=-0.31270.04740.1536-0.0621x^t.The initial conditions are also taken as x(0)=1-1T and x^(0)=-22T. The simulation results of the state response of the system are plotted in Figures 1–3. The filter state x1(t) and its estimation x^1(t) and state x2(t) and its estimation x^2(t) are given in Figures 1 and 2, respectively. Figure 3 depicts the estimation error e(t)=z(t)-z^(t). The simulation results demonstrate that the designed H∞ filters are feasible, effective and the stochastic stability of the error systems is ensured.Table 1
Upper bounds ofh for different γ (Example 1).
γ
0.3
0.8
1.2
1.8
2.4
3.0
h by Corollary 9
1.079
1.786
1.954
2.089
2.167
2.219Table 2
Lower bounds ofγ for different h (Example 1).
h
0.5
1.0
1.5
2.0
2.2
2.4
γ by Corollary 9
0.275
0.291
0.473
1.364
2.745
10.940Table 3
Upper bounds ofh for different γ (Example 1).
γ
0.3
0.8
1.2
1.8
2.4
3.0
h by Theorem 3 [18]
1.157
1.808
1.926
1.991
2.025
2.046
h by Corollary 9
1.409
2.138
2.263
2.335
2.371
2.394Table 4
Lower bounds ofγ for different h (Example 1).
h
0.5
1
1.5
2
2.1
2.2
γ by Theorem 3 [18]
0.264
0.270
0.630
1.919
8.235
infeasible
γ by Corollary 9
0.264
0.266
0.340
0.712
0.766
0.925Figure 1
Statex1(t) and its estimation x^1(t) in Example 1.Figure 2
Statex2(t) and its estimation x^2(t) in Example 1.Figure 3
The error responsese1(t) and e2(t) in Example 1.Example 2.
Consider systems (1) with Markovian jumping parameters as follows.
Mode  1. Consider(62)A1=-2.000-1.9,A11=-10-1-1,B1=0.150.40.20.3,D1=-0.100-0.1,D11=0.2000.2,G1=-0.500-0.5,C1=0.2000.2,C11=0.2000.2,C21=0.30.90.20.1,E1=1.00.20.3-1,E11=0.20.50.30.4,E21=0.50.51.00,L11=0.80.60.40.3,L21=0.40.60.50.8,L31=0.60.50.30.4,M11=0.2000.2,M21=0.2000.2,N11=0.1000.1,N21=0.3000.3.
Mode  2. Consider(63)A2=0.5000.3,A12=-1.00-1.0-0.5,B2=0.150.40.20.31,D2=0.2000.3,D12=0.3000.3,G2=-0.600-0.6,C2=0.1000.1,C12=0.1000.1,C22=0.30.90.20.15,E2=0.50.61.20,E12=1.00.30.4-1.0,E22=0.250.50.30.4,L12=0.80.50.60.8,L22=0.40.80.30.5,L32=0.51.00.60.4,M12=0.3000.3,M22=0.1000.1,N12=0.2000.2,N22=0.2000.2,π11=-1,π22=-2,π12=1,π21=2.In this section, the purpose of this example is to design a full-order filter in the form of (9) such that the filtering error system is robustly stochastically stable for all admissible uncertainties and satisfies the required H∞ performance level.
By Theorem7, for different given noise attenuation levels γ, the upper bounds of delay for systems (1) are presented in Table 5. For different given time delays, the lower bounds of noise attenuation level γ for systems (1) are provided in Table 6.
Now in the case whenγ=2, hmax=0.232, we resort to the MATLAB LMI control toolbox to solve the LMI (48), and we obtain the solution as follows: (64)Q1=9.6525-1.9471-1.94711.6603,Q2=-10.1587-0.22972.2216-1.6233,Q3=10.8069-0.0873-0.08732.3599,R=39.1583-0.3727-0.37278.8082,X1=2.83760.15530.15532.4043,X2=4.37240.47300.47305.8205,F1=0.29420.28960.28961.1395,F2=1.27830.90200.90202.8727,ϵ1=1.1107,ϵ2=8.1439,AF1=-0.9169-0.6283-1.6111-2.5633,BF1=0.12620.07300.23920.3601,CF1=0.56900.92570.40041.0310,AF2=-1.4295-1.3599-2.9406-2.5256,BF2=1.8563-0.73693.0463-1.0766,CF2=1.16021.10860.73640.9807.
Therefore, the full-order stochastic filter parameters are given as follows:(65)Af1=-2.30010.1251-0.8294-2.2813,Bf1=0.2963-0.08370.13460.3373,Cf1=0.56900.92570.40041.0310,Af2=-0.5087-0.5697-0.8639-0.7003,Bf2=0.9043-0.40080.7765-0.2489,Cf2=1.16021.10860.73640.9807.The initial conditions are also taken as x(0)=-11.5T and x^(0)=-0.51T. The simulation results of the state response of the system are plotted in Figures 4–7. The filter state x1(t) and its estimation x^1(t) and state x2(t) and its estimation x^2(t) are, respectively, given in Figures 4 and 5. Assuming the simulation step size Δ=0.1, one of the possible realizations of the Markovian jumping mode is plotted in Figure 6. Figure 7 depicts the estimation error e(t)=z(t)-z^(t). It is clearly observed from the simulation results that the designed H∞ filter satisfies the specified requirements and the expected objectives are well achieved.Table 5
Upper bounds ofh for different γ (Example 2).
γ
0.5
2
3
4
5
h by Theorem 7
0.104
0.232
0.311
0.343
0.361Table 6
Lower bounds ofγ for different h (Example 2).
h
0.1
0.2
0.3
0.4
γ by Theorem 7
1.493
1.806
2.774
15.047Figure 4
Statex1(t) and its estimation x^1(t) in Example 2.Figure 5
Statex2(t) and its estimation x^2(t) in Example 2.Figure 6
Markovian jumping mode in Example2.Figure 7
The error responsese1(t) and e2(t) in Example 2.Remark 3.
In order to reduce conservatism, many methods such as cross-terms bounding techniques, free-weighting matrices method, and cross-terms bounding techniques are often adopted in the stability analysis of stochastic systems. In this paper, the generalized Finsler Lemma is utilized in uncertain neutral stochastic systems, which can bring the low conservatism and less computational cost.


## 5. Conclusions

In this paper, the robust filtering problem for a class of uncertain neutral stochastic systems with Markovian jumping parameters and time delay has been considered. Based on the Lyapunov-Krasovskii functional theory and generalized Finsler Lemma, a delay-dependent sufficient condition is proposed for the existence ofH∞ filters which reduces the conservatism. The obtained result ensures the robust stability and a prescribed H∞ performance level of the filtering error system for all admissible uncertainties. Two numerical examples and simulations have been presented to demonstrate the usefulness and effectiveness of the proposed filter design method.



---

*Source: 808973-2015-10-18.xml*