# ExponentialL2-L∞ Filtering for a Class of Stochastic System with Mixed Delays and Nonlinear Perturbations

Authors: Zhaohui Chen; Qi Huang
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

The delay-dependent exponentialL2-L∞ performance analysis and filter design are investigated for stochastic systems with mixed delays and nonlinear perturbations. Based on the delay partitioning and integral partitioning technique, an improved delay-dependent sufficient condition for the existence of the L2-L∞ filter is established, by choosing an appropriate Lyapunov-Krasovskii functional and constructing a new integral inequality. The full-order filter design approaches are obtained in terms of linear matrix inequalities (LMIs). By solving the LMIs and using matrix decomposition, the desired filter gains can be obtained, which ensure that the filter error system is exponentially stable with a prescribed L2-L∞ performance γ. Numerical examples are provided to illustrate the effectiveness and significant improvement of the proposed method.


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## Body

## 1. Introduction

Time delays are quite often encountered in various practical engineering systems, and they are regarded as one of the main sources causing instability and degrading performance of control systems [1–3]. Over the past decades, numerous results and various approaches on delay systems have been reported in the literatures. Many researchers have focused on the stability analysis, stabilization, and filtering for time-delay systems; see [4–9] and the references therein. Time delays are usually classified into discrete delays and distributed delays. In the existing literatures, discrete time-delay system [10–12], distributed time-delay system [13, 14], and mixed (including both discrete and distributed time delays) system [15–17] are considered.Since certain unavoidable stochastic perturbations are widely existing in many engineering systems, stochastic systems have gained considerable research attention over the past few years [18–20]. Stochastic dynamic modeling has come to play an important role in many fields of science and engineering. In the past years, many researchers have focused on the problems of stability and stabilization of stochastic time-delay systems. For instance, robust stabilization for a class of large-scale stochastic systems was investigated in [21], delay-dependent stability results for stochastic systems were presented in [22–26], and H∞ state feedback control and H∞ dynamic output feedback control for uncertain stochastic time-delay systems were investigated in [27, 28], respectively.In the field of stochastic dynamic system with time delays, the filtering problem, which is to estimate the unavailable state of variables of a given control system, is also an important issue. Kalman filtering scheme is a well-known effective way to deal with the filtering problem. However, it has some limitations in practical applications due to the fact that it assumes that the system and its disturbances are exactly known, that is, stationary Gaussian noised with known statistics. Under this view, recently,H∞ filtering, mixed H2/H∞ filtering and L2-L∞ filtering for stochastic time-delay systems have been widely studied [8, 9, 29–38]. In H2/H∞ filtering, and L2-L∞ filtering problems, the external disturbances are assumed to be bounded. In H2/H∞ filtering problem, it requires that the filtering error systems satisfy not only a prescribed H∞ disturbance attenuation level but also the H2 performance (minimum of the H2 norm of transfer function of the filter error systems), while in L2-L∞ filtering problem, it requires that the filtering error systems satisfy a prescribed L2-L∞ disturbance attenuation level. H∞ filtering and mixed H2/H∞ filtering problems of nonlinear stochastic systems are investigated in [30, 31]. In [32], a delay-independent robust L2-L∞ filtering design approach for uncertain stochastic time-delay system is investigated. It is well known that the delay-independent results are generally more conservative than the delay-dependent ones. Authors in [33–35] developed delay-dependent filtering for stochastic time-delay systems. Authors in [36] proposed a delay-dependent L2-L∞ filter design approach for stochastic time-delay systems, based on a delay partitioning technique presented in [37]. As the results showed, delay-partitioning can reduce conservatism to some extent. Authors in [38] investigated the problem of robust L2-L∞ filtering for stochastic systems with both discrete and distributed delays. Although the filtering problems for stochastic systems with time delays have been well investigated in the aforementioned literatures, most of them are dealing with linear stochastic time-delay systems. To the authors’ knowledge, the L2-L∞ filtering problems of stochastic time-delay systems with nonlinear perturbation are still insufficient. This motivates the authors to deal with the L2-L∞ filtering problem of a class of nonlinear stochastic time-delay systems.This paper focuses on the problems of delay-dependentL2-L∞ filtering for stochastic systems with mixed delays and nonlinear perturbations. By Lyapunov-Krasovskii approach based on the delay partitioning and integral partitioning technique, we first develop a delay-dependent sufficient condition for L2-L∞ performance analysis. And then, an improved delay-dependent sufficient condition is obtained for the existence of desired filter in the form of linear matrix inequalities (LMIs). The L2-L∞ performance analysis and filter design of linear stochastic system with mixed delays are also investigated. Finally, numerical examples are provided to show that the proposed method is effective and less conservative than some existing literatures.Notations. Throughout this paper, X>0(X<0) means that the matrix X is positive definite (negative definite). Rn denotes the n-dimensional Euclidean space; Rm×n is the set of all m×n real matrices; L2[0,∞) is the space of square-integrable vector functions over [0,∞). The superscript “T” represents the transpose; “*” denotes the symmetric terms in a matrix; diag() denotes a block-diagonal matrix; λmax() and λmin() denote the maximum eigenvalue and minimum eigenvalue, respectively. sym(X)=X+XT; |·| denotes the Euclidean vector norm; ∥·∥2 stands for the usual L2[0,∞) norm. (Ω,F,P) is a probability space with Ω the sample space, F the σ-algebra of subsets of Ω, and P the probability measure on F. E{·} denotes the expectation operator with respect to some probability measure P. 0 and I represent zero matrix and identity matrix with appropriate dimensions, respectively, unless we say otherwise.


## 2. Problem Formulation

Consider the following stochastic systems with mixed delays and nonlinear perturbations:(1)dx(t)=[∫t-dtx(s)dsAx(t)+A1x(t-h)+A2∫t-dtx(s)ds+A3f(x(t),x(t-h),t)+Avv(t)]dt+g(x(t),x(t-h),t)dω(t),dy(t)=[Cx(t)+C1x(t-h)+C2∫t-dtx(s)ds+Cvv(t)]dt,z(t)=Lx(t),x(t)=φ(t),∀t∈[-τ,0],
where x(t)∈Rn is the state; y(t)∈Rm is the measured output; z(t)∈Rp is the signal to be estimated; v(t)∈Rq is the disturbance input which belongs to L2[0,∞), which is the space of square-integrable vector functions; ω(t) is a one-dimensional Brownian motion defined on a complete probability space (Ω,F,P) satisfying E{dω(t)}=0 and E{dω2(t)}=dt; φ(t) is an initial function that is continuous on [-τ,0] with τ=max{h,d}. h and d are discrete and distributed constant delays, respectively. f(·,·,·):Rn×Rn×R→Rn is a nonlinear function, which satisfies
(2)|f(x,y,t)|2≤|F1x|2+|F2y|2,f(0,0,0)=0,
where F1∈Rn×n and F2∈Rn×n are known constant matrices; g(·,·,·):Rn×Rn×R→Rn is a nonlinear perturbance input function, satisfying
(3)|g(x,y,t)|2≤|G1x|2+|G2y|2,g(0,0,0)=0,
where G1∈Rn×n and G2∈Rn×n are known constant matrices.For system (1), we are interested in constructing the following full-order linear filter:
(4)dxf(t)=Afxf(t)dt+Bfdy(t),zf(t)=Cfxf(t),
where xf(t)∈Rn is the filter state; Af, Bf, and Cf are filter matrices to be determined.Define(5)ξT(t)=[xT(t),xfT(t)]T,e(t)=z(t)-zf(t).
Then, the filtering error system can be written as
(6)dξ(t)=[A-ξ(t)+A-1Hξ(t-h)+A-2H∫t-dtξ(s)ds+A-3f-(ξ(t),ξ(t-h),t)+A-vv(t)∫t-dtξ(s)ds]dt+g-(ξ(t),ξ(t-h),t)dω(t)e(t)=L-ξ(t),
where
(7)A-=[A0BfCAf],A-1=[A1BfC1],A-2=[A2BfC2],A-3=[A3000],A-v=[AvBfCv],H=[In0n],f-(ξ(t),ξ(t-h),t)=[f(x(t),x(t-h),t)0],g-(ξ(t),ξ(t-h),t)=[g(x(t),x(t-h),t)0],L-=[L-Cf].The objective of this paper is to design full-orderL2-L∞ filter (4) for the stochastic time-delay system (1) such that the filtering error system (6) satisfies the following two requirements:(i)
the filtering error system (6) with v(t)=0 is exponentially stable [39];(ii)
under the zero initial condition, the filtering error system (6) is stochastically asymptotically stable and achieves a prescribed L2-L∞ attenuation level γ. The filtering error e(t) satisfies
(8)∥e(t)∥E∞<γ∥v(t)∥2,with ∥e(t)∥E∞=suptE{|e(t)|2}, ∥v(t)∥2=∫0∞vT(t)v(t)dt for any nonzero v(t)∈L2[0,∞].Before presenting the main results of this paper, we introduce the following lemmas, which will be essential to our derivation.Lemma 1 (see [40]).
For a given symmetrical matrixS=(S11S12S12TS22), where S11=S11T, and S22=S22T, the linear matrix inequality S<0 is equivalent to
(9)S11<0,S22-S12S11-1S12T<0,orS22<0,S11-S12S22-1S12T<0.Lemma 2 (see [1]).
For any positive symmetric matrixW∈Rn×n, scalars δ1 and δ2 satisfying δ1<δ2, a vector function x:[δ1,δ2]→Rn, one has
(10)∫δ1δ2xT(s)Wx(s)ds≥1(δ2-δ1)(∫δ1δ2x(s)ds)TW(∫δ1δ2x(s)ds).Lemma 3 (see [14]).
For any positive symmetric matrixW∈Rn×n, scalars a and b satisfying a<b≤0, a vector function x:[a,b]→Rn, one has
(11)∫ab∫t+λtxT(s)Wx(s)dsdλ≥2a2-b2(∫ab∫t+λtx(s)dsdλ)TW(∫ab∫t+λtx(s)dsdλ).


## 3. Filtering Performance Analysis

In this section, a new delay-dependent condition of theL2-L∞ filtering performance analysis for system (1) will be presented. A Lyapunov-Krasovskii functional is constructed; based on the idea of delay partitioning and integral partitioning, the conservatism will be reduced. For the convenience of expression, assume that the filter matrices (Af, Bf, and Cf) are known.Theorem 4.
Consider the stochastic time-delay system (1). For given scalars γ>0, h>0, d>0, ρ>0, and ε>0 and integers r1≥1 and r2≥1, there exists a linear filter (4) such that the filtering error system (6) is stochastically asymptotically stable with a guaranteed L2-L∞ performance γ, if there exist symmetrical positive definite matrices P∈R2n×2n, Qi∈Rn×n, Ri∈Rn×n(i=1,2,…,r1), Wj∈Rn×n, and Zj∈Rn×n(j=1,2,…,r2) and matrix M∈Rn×n satisfying
(12)Φ=[Φ11Φ12Φ13PA-3PA-vA-THTM*Φ22000Φ26**Φ3300Φ36***-εI0A-3THTM****-IA-vTHTM*****Φ66]<0,(13)P≤ρI,(14)Γ=[PL-T*γ2I]>0,
where
(15)Φ11=PA-+A-TP+HTQ1H-HTR1H-∑j=1r2HT(22j-1Zj)H+(dr2)2∑j=1r2HTWjH+ρHTG1TG1H+εHTF1TF1H,Φ12=HTR1K+PA-1Kr1,Φ13=PA-2Kr2+HTΖ-,Z-=2md[Z113Z2⋯12r2-1Zr2],Φ22=[Q2-Q1-R2-R1R2⋯00*Q3-Q2-R3-R2⋯00⋮⋮⋱⋮⋮**⋯Qr1-Qr1-1-Rr1-Rr1-1Rr1**⋯*Φ-22],Φ-22=-Qr1-Rr1+ρG2TG2+εF2TF2,Φ26=Kr1TA-1THTM,Φ33=diag(-W1-2m2d2Z1,-W2-2m23d2Z2,…,-Wr2-2m2(2r2-1)d2Zr2),Φ36=Kr2TA-2THTM,Φ66=(hr1)2∑i=1r1Ri+(dr2)2∑j=1r22j-12Zj-M-MT,K=[In0n×(r1-1)n],Kr1=[0n×(r1-1)nIn],Kr2=[II⋯I]n×r2n.Proof.
First, show the asymptotic stability of system (6) with v(t)=0. For simplicity of notations, rewrite the filtering error system (6) as
(16)dξ(t)=u(t)dt+π(t)dω(t),
where
(17)u(t)≔A-ξ(t)+A-1Hξ(t-h)+A-2H∫t-dtξ(s)ds+A-3f-(ξ(t),ξ(t-h),t)+A-vv(t),π(t):=g-(ξ(t),ξ(t-h),t).
Next, denoteη(t)dt=dξ(t), and choose the following Lyapunov-Krasovskii functional:
(18)V(ξt,t)=ξT(t)Pξ(t)+∑i=1r1∫t-(i/r1)ht-((i-1)/r1)hξT(s)HTQiHξ(s)ds+∑i=1r1hr1∫-(i/r1)h-((i-1)/r1)h∫t+θtηT(s)HTRiHη(s)dsdθ+∑j=1r2dr2∫-(j/r2)d-((j-1)/r2)d∫t+θtξT(s)HTWjHξ(s)dsdθ+∑j=1r2∫-(j/r2)d-((j-1)/r2)d∫θ0∫t+βtηT(s)HTZjHη(s)dsdβdθ.
Then, by Itô differential formula, the stochastic differential along the trajectories of system (6) is
(19)dV(ξt,t)=LV(ξt)dt+2ξ(t)Pπ(t)dω(t),
where
(20)LV(ξt,t)=2ξT(t)Pu(t)+trace(πT(t)Pπ(t))+∑i=1r1ξT(t-i-1r1h)HTQiHξ(t-i-1r1h)-∑i=1r1ξT(t-ir1h)HTQiHξ(t-ir1h)+(hr1)2∑i=1r1ηT(t)HTRiHη(t)-∑i=1r1hr1∫t-(i/r1)ht-((i-1)/r1)hηT(s)HTRiHη(s)ds+(dr2)2∑j=1r2ξT(t)HTWiHξ(t)-∑j=1r2dr2∫t-(j/r2)dt-((j-1)/r2)dξT(s)HTWjHξ(s)ds+(dr2)2∑j=1r22j-12ηT(t)HTZjHη(t)-∑j=1r2∫-(j/r2)d-((j-1)/r2)d∫t+θtηT(t)HTZjHη(t)dsdθ.
By Lemma 2, we have
(21)-∑i=1r1hr1∫t-(i/r1)ht-((i-1)/r1)hηT(s)HTRiHη(s)ds≤-∑i=1r1(∫t-(i/r1)ht-((i-1)/r1)hη(s)ds)T×HTRiH(∫t-(i/r1)ht-((i-1)/r1)hη(s)ds)=-∑i=1r1(ξ(t-i-1r1h)-ξ(t-ir1h))T×HTRiH(ξ(t-i-1r1h)-ξ(t-ir1h)),(22)-∑j=1r2dr2∫t-(j/r2)dt-((j-1)/r2)dξT(s)HTWjHξ(s)ds≤-∑j=1r2(∫t-(j/r2)dt-((j-1)/r2)dξ(s)ds)T×HTWjH(∫t-(j/r2)dt-((j-1)/r2)dξ(s)ds).
By Lemma 3, we have
(23)-∑j=1r2∫-(j/r2)d-((j-1)/r2)d∫t+θtηT(t)HTZjHη(t)dsdθ≤-∑j=1r222j-1(r2d)2(∫-(j/r2)d-((j-1)/r2)d∫t+θtη(s)dsdθ)T×HTZjH(∫-(j/r2)d-((j-1)/r2)d∫t+θtη(s)dsdθ)=-∑j=1r222j-1(r2d)2(dr2ξ(t)-∫t-(j/r2)dt-((j-1)/r2)dξ(s)ds)T×HTZjH(dr2ξ(t)-∫t-(j/r2)dt-((j-1)/r2)dξ(s)ds).
From (16), for any appropriately dimensioned matrix M, we have
(24)2ηT(t)HTMTH[u(t)dt+π(t)dω(t)-η(t)dt]=0.
On the other hand, (2) implies that there exists ε>0 such that
(25)ξT(t)εHTF1TF1Hξ(t)+ξT(t-h)εHTF2TF2Hξ(t-h)-εf-Tf-≥0,
where we take f- for f-(x(t),x(t-h),t), for simplicity of notation.
Notice the fact of (3), and from (13), we have
(26)trace(πT(t)Pπ(t))≤ξT(t)ρHTG1TG1Hξ(t)+ξT(t-h)ρHTG2TG2Hξ(t-h).
Combine (20)–(26); then
(27)LV(xt,t)≤ζT(t)Φ-ζ(t)+2ηT(t)HTMTHπ(t)dω(t),
where(28)ζT(t)=[ξp1T(t)ξp2T(t)f-(t)ηT(t)HT],ξp1T=[ξT(t)ξT(t-1r1h)HTξT(t-2r1h)HT⋯ξT(t-h)HT],ξp2T(t)=[∫t-(1/r2)dtξT(s)HTds∫t-(2/r2)dt-(1/r2)dξT(s)HTds⋯∫t-dt-((r2-1)/r2)dξT(s)HTds],Φ-=[Φ11Φ12Φ13PA-3A-THTM*Φ2200Φ26**Φ330Φ36***-εIA-3THTM****Φ66].
Thus,(29)E{LV(ξt,t)}≤E{ζT(t)Φ-ζ(t)}.
By Schur complement lemma, it is easy to show thatΦ<0 implies Φ-<0. Combined with (29), these imply that, for any ζ(t)≠0, we have E{LV(ξt,t)}<0.
By Dynkin’s formula, there existsβ>0, such that
(30)eβtEV(ξt,t)≤EV(ξ0,0).
Recalling the Lyapunov-Krasovskii functional in (18), notice the fact that there always exists κ>0 satisfying
(31)|η(t)|2≤κ|ξ(t)|2
for any -τ≤t≤0 such that
(32)EV(ξ0,0)≤∑i=15αisup-τ≤s≤0E|ξ(s)|2,
where α1=λmax(P), α2=hmaxi{∥Qi∥}, α3=(κh3/2r1)maxi{∥Ri∥}, α4=(d3/2r2)maxj{∥Wj∥}, and α5=(κd3/6)maxj{∥Zj∥}.
On the other hand, from (18)
(33)EV(ξt,t)≥λmin(P)E|ξ(t)|2.
From (32) and (33), it can be easily obtained that
(34)E|ξ(t;φ~)|2≤αe-βtsup-τ≤s≤0E|ξ(s)|2,
where α=∑i=15αi/λmin(P) and φ~ is the initial condition of filtering error system (6). Then by exponential stability definition of stochastic systems [39], the filtering error system (6) with v(t)=0 is exponentially stable in the sense of mean square.
Now, we will establish theL2-L∞ performance for the filtering error system (6). To this end, we assume the zero initial condition ζ(t)=0 for t∈[-τ,0]. Under the initial condition, it is easy to see that, for any t>0,
(35)E{V(ξt,t)}=E{∫0tLV(ξs,s)ds}.
Define(36)J(t)=E{V(ξt,t)}-∫0tvT(s)v(s)ds.
Then, for any nonzero v(t)∈L2[0,∞) and t>0, combined with (29), (35)-(36), we have
(37)J(t)=E{∫0t[LV(ξs,s)-vT(s)v(s)]ds}≤E{∫0tϑT(s)Φϑ(s)ds},
where ϑT(t)=[ξp1T(t)ξp2T(t)f-TvT(t)ηT(t)HT]. Φ<0 ensuring that J(t)≤0. Thus,
(38)E{ξT(t)Pξ(t)}≤E{V(ξt,t)}≤∫0tvT(s)v(s)ds.
Moreover, by Schur complement, (14) holds if and only if
(39)L-TL-<γ2P.
It follows from (38) and (39) that
(40)E{|e(t)|2}=E{ξT(t)L-TL-ξ(t)}<γ2E{ξT(t)Pξ(t)}≤γ2E{V(ξt,t)}≤γ2∫0tvT(s)v(s)ds.
Therefore, if (12)–(14) hold, the filtering error system (6) is mean-square exponentially stable with a prescribed L2-L∞ performance γ under zero initial condition. This completes the proof.In system (1), if A3=0 and g(x(t),x(t-h),t)=Bx(t)+B1x(t-h)+B2∫t-dtx(s)ds+Bvv(t), then the linear stochastic system with mixed delays can be written as
(41)dx(t)=[∫t-dtx(s)ds+Avv(t)Ax(t)+A1x(t-h)+A2∫t-dtx(s)ds+Avv(t)]dt+[∫t-dtx(s)ds+Bvv(t)Bx(t)+B1x(t-h)+B2∫t-dtx(s)ds+Bvv(t)]dω(t),dy(t)=[Cx(t)+C1x(t-h)+C2∫t-dtx(s)ds+Cvv(t)]dt,z(t)=Lx(t),x(t)=φ(t),∀t∈[-τ,0]
which is the same as the system in [38] with constant delays. Thus, following the similar lines in Theorem 4, a sufficient condition can be obtained guaranteeing that there exists a linear filter (4) such that the filtering error system is exponentially stable and achieves a prescribed L2-L∞ performance γ.Corollary 5.
Consider the stochastic time-delay system (41). For given scalars γ>0, h>0, and d>0 and integers r1≥1 and r2≥1, there exists a linear filter (4) such that the corresponding filtering error system is exponentially stable with a guaranteed L2-L∞ performance γ, if there exist symmetrical positive definite matrices  P∈R2n×2n, Qi∈Rn×n, RiT∈Rn×n(i=1,2,…,r1), Wj∈Rn×n, and Zj∈Rn×n(j=1,2,…,r2) and matrix M∈Rn×n satisfying (14) and
(42)Φ~=[Φ~11Φ12Φ13PA-vA-THTM*Φ~2200Φ26**Φ330Φ36***-IA-vTHTM****Φ66]+BξTPBξ<0,
where Φ~11=Φ11-ρHTG1TG1H-εHTF1TF1H, Φ~22=Φ22-Kr1T(ρG2TG2+εF2TF2)Kr1, and Bξ=[B-B-1Kr1B-2Kr2B-v].


## 4. Filter Design

In this section, we will focus on the design ofL2-L∞ filter for stochastic system (1). Based on Theorem 4, a delay-dependent sufficient condition will be obtained in the forms of LMI, which ensures that the filtering error system (6) is stochastically asymptotically stable and achieves a prescribed L2-L∞ performance γ.Theorem 6.
Consider the stochastic time-delay system (1). For given scalars γ>0, h>0, d>0, ρ>0, and ε>0 and integers  r1≥1 and r2≥1, there exists a linear filter (4) such that the filtering error system (6) is stochastically asymptotically stable with a prescribed L2-L∞ performance γ, if there exist symmetrical positive definite matrices X∈Rn×n, Y∈Rn×n, Qi∈Rn×n, Ri∈Rn×n(i=1,2,…,r1), Wj∈Rn×n, and Zj∈Rn×n(j=1,2,…,r2) and matrices M∈Rn×n, A^f∈Rn×m, B^f∈Rn×n, and C^f∈Rn×p satisfying
(43)Υ=[Υ11Υ12Υ13Υ14Υ15Υ16*Φ22000Υ26**Φ3300Υ36***-εI0A3TM****-IAvTM*****Φ66]<0,(44)[X-ρIYY(1-ρ)Y]<0,(45)Π2≤I,(46)Λ=[XYLT*Y-C^fT**γ2I]>0,
where
(47)Υ11=[Υ111A^f+ATY+CTB^fT*A^f+A^fT],Υ12=[R1K1+(XA1+B^fC1)Kr1(YA1+B^fC1)Kr1],Υ13=[(XA2+B^fC2)Kr2+Z-(YA2+B^fC2)Kr2],Υ111=sym(XA+B^fC)+Q1-R1+(dr2)2∑j=1r2Wj-∑j=1r222j-1Zj+ρG1TG1+εF1TF1,Υ14=[XA3YA3],Υ15=[XAv+B^fCvYAv+B^fCv],Υ16=[ATM0n,n],Υ26=Kr1TA1TM,Υ36=Kr2TA2TM.
In this case, the parameters of a desired filter in the form of (4) can be taken as
(48)Af=Π1-1A^fΠ1-TΠ2,Bf=Π1-1B^f,Cf=C^fΠ1-TΠ2,
where Π1 and Π2 are nonsingular matrices satisfying 0<Π2=Π2T≤I, 0<Y=Π1Π2-1Π1T, and X>Y>0.Proof.
From (46), it can be seen that [XYYY]>0, and X>Y>0. For any positive definite and symmetric matrix Y, one can always find nonsingular matrix Π1∈Rn×n and 0<Π2=Π2T∈Rn×n, such that Y=Π1Π2-1Π1T.
Set(49)P=[XΠ1Π1TΠ2].
Then X-Π1Π2-1Π1T=X-Y>0, and P>0.
Define(50)Π=[In0n,n0n,nΠ2-1Π1T].
Then
(51)Π-1=[In0n,n0n,nΠ1-TΠ2].
SubstituteA^f=Π1AfΠ2-1Π1T, B^f=Π1Bf, C^f=CfΠ2-1Π1T, and Y=Π1Π2-1Π1T into (43), (44), and (46) and then pre- and postmultiply (43) by diag{Π-T,Ir1n,Ir2n,In,In,In} and its transpose, respectively. Premultiply and postmultiply (46) by diag{Π-T,Ip} and its transpose, respectively. Noticing that P=[XΠ1Π1TΠ2], using Schur complement Lemma, one can obtain (12) and (14).
On the other hand, (44) implies
(52)[XYYY]<ρ[I00Y].
Pre- and postmultiply (52) by Π-T and Π-1, respectively. Notice that Y=ΠΛ-1ΠT, one can obtain
(53)[XΠ1Π1TΠ2]<ρ[I00Π2].
By (45), it is easy to see that
(54)[XΠ1Π1TΠ2]<ρ[I00I].
So, (13) is satisfied.
Therefore, by Theorem4, the suitable filter parameters can be constructed by (48), which ensures the filtering error system (6) to be stochastically asymptotically stable with L2-L∞ performance γ. This completes the proof.Remark 7.
When deriving the results in Theorem6 based on Theorem 4, considering dealing with the LMI (13), we give a method to avoid nonlinear terms emerging. Using Matlab LMI toolbox, one can solve linear matrix inequalities (43)-(44) and (46). Then, by matrix diagonalization approach, one can easily find that diagonally positive matrix Π2 and nonsingular matrix Π1 satisfy Π2=Π1TY-1Π1. If the obtained matrix Π2 does not satisfy (45), one can take Π2 for Π2/max{eig(Π2)} and Π1 for max{eig(Π2)}Π1. Thus, the desired filter parameters can be obtained by (48).
Following the similar method in Theorem6, one can obtain a result of filter design for linear stochastic time-delay system (41).Corollary 8.
Consider the stochastic time-delay system (41). For given scalars γ>0, h>0, and d>0 and integers r1≥1 and r2≥1, there exists a linear filter (4) such that the corresponding filtering error system is stochastically asymptotically stable with a prescribed L2-L∞ performance γ, if there exist symmetrical positive definite X∈Rn×n, Y∈Rn×n, Qi∈Rn×n, Ri∈Rn×n(i=1,2,…,r1), Wj∈Rn×n, and Zj∈Rn×n(j=1,2,…,r2) and matrices M∈Rn×n, A^f∈Rn×m, B^f∈Rn×n, and C^f∈Rn×p satisfying (46) and
(55)Υ~=[Υ~11Υ12Υ13Υ15Υ16Υ17*Φ~2200Υ26Υ27**Φ330Υ36Υ37***-IAvTMΥ47****Φ660*****Υ77]<0,
where
(56)Υ~11=[Υ~111A^f+ATY+CTB^fT*A^f+A^fT],Υ~111=Υ111-ρG1TG1-εF1TF1,Υ17=[BTXBTY0n,n0n,n],Υ27=Kr1T[B1TXB1TY],Υ37=Kr2T[B2TXB2TY],Υ47=[BvTXBvTY],Υ77=[-X-Y*-Y].Remark 9.
The results presented in Theorem6 and Corollary 8 can be easily extended to the systems with only discrete or distributed delays and also to the robust performance analysis for uncertain stochastic systems with mixed delays.


## 5. Numerical Examples

Example 1.
Consider the stochastic time-delay system (1) with parameters
(57)A=[-1.50.5-1-3],A1=[-0.80.20.2-0.5],A2=[0.2000.2],A3=[0.1000.1],Av=[0.200-0.2],C=[2-0.5-1.50.5],C1=[0.150.1-0.10.1],C2=[0.5-0.20.60],Cv=[0.10.20.10.03],L=[0.1-0.200.1],ε=1,ρ=7.
Moreover, for the nonlinear functions, we letG1=G2=0.1I and F1=F2=0.1I. Given h=1 and γ=0.2, from Theorem 6, one can obtain the upper bound of time delay d, which is listed in Table 1.
In the case ofr1=2 and r2=2, the desired filter parameters can be obtained:
(58)Af=[-6.14602.26440.6168-4.5679],Bf=[-3.87491.88680.7066-0.1266],Cf=[-0.05680.00520.02020.0057].Table 1
The upper bound ofdmax for h=1 and γ=0.2.
Methods
d
max
Theorem6 (r1=1, r2=1)
7.481
Theorem6 (r1=2, r2=2)
8.190
Theorem6 (r1=3, r2=3)
8.317
Theorem6 (r1=5, r2=5)
8.379Example 2.
Consider the stochastic time-delay system (36)–(39) with parameters
(59)A=[-1.50.5-1-3],A1=[-0.80.20.2-0.5],A2=[0.2000.2],Av=[0.200-0.2],B=[-0.80.20.5-0.5],B1=[0.50.50.20.3],B2=[0.2000.2],Bv=[-0.2000.5],C=[2-0.5-1.50.5],C1=[0.150.1-0.10.1],C2=[0.5-0.20.60],Cv=[0.10.20.10.03],L=[0.1-0.200.1].
Givend=1 and γ=0.2, from Corollary 8, one can obtain the upper bound of time delay h. Table 2 lists the results of Corollary 8 and [38] with constant delays. It is easy to see that the proposed filter design method in this paper is less conservative than [38].
From Corollary8, in the case of r1=2 and r2=2, the desired filter parameters can be obtained:
(60)Af=[-0.0000-4.1570-0.0000-2.5527],Bf=[10.1507-11.12366.0149-6.1484],Cf=[0.00000.0065-0.0000-0.0027].Table 2
The upper bound ofhmax for d=1 and γ=0.2.
Methods
h
max
[38]
1.725
Corollary8 (r1=1, r2=1)
3.755
Corollary8 (r1=2, r2=1)
5.054
Corollary8 (r1=2, r2=2)
5.111
Corollary8 (r1=3, r2=3)
5.688
Corollary8 (r1=5, r2=5)
5.920Remark 3.
It can be seen from the results that the conservatism can be reduced with the increase of partition integers. However, it is necessary to point out that the less conservatism is at the cost of a higher computational complexity.


## 6. Conclusions

In this paper, a new approach has been developed to investigate the problems of delay-dependentL2-L∞ filter design for stochastic system with mixed delays and nonlinear perturbations. Based on the idea of delay partitioning and integral partitioning, using Lyapunov-Krasovskii functional approach, a delay-dependent sufficient condition has been established that ensures the filtering error system is exponentially stable with  L2-L∞ performance γ. By solving the LMIs, one can get the desired filter gain matrices. The results also depend on the partition integers with the increase of partition integers, the conservatism can be decreased. Finally, numerical examples are presented to demonstrate the effectiveness of the proposed approach.



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*Source: 659251-2013-12-29.xml*