H∞ filtering for stochastic time- delay systems

H

filtering for stochastic time- delay systems

Yumei Li

, Xinping Guan

, Dan Peng

, Jianhua Zhang

1)

Institute of Electrical Engineering Yanshan University, Qinhuangdao, 066004, China

2)

Department of Mathematics and System Science, Xinjiang University,Urumuqi, 830046, China xpguan@ysu.edu.cn

Abstract— This paper presents the robustH∞filter design for stochastic systems with time-varying delay.

The aim is to design a stable linear filter assuring exponential stability in mean-square and a prescribed H∞ performance level for the filtering error system.

Based on the application of the descriptor model transformation and free weighting matrices, delay- dependent sufficient conditions are proposed in terms of linear matrix inequalities(LMIs). Numerical ex- ample demonstrates the proposed approaches are effective and are an improvement over existing ones.

Index Terms—exponential stability in mean-square, robustH∞filter, time-varying delay, linear stochastic system, linear matrix inequalities(LMIs).

I. INTRODUCTION

In recent years, stochasticH_∞ filtering and con- trol problems with system models expressed by It˜o-type stochastic differential equations have be- come a interesting research topic and has gained extensive attention; (see [1]-[2] ), there have been a lot of studies on H_∞ control or state estima- tion in deterministic systems, (see [3]-[4]). Robust H_∞estimation problems for linear and nonlinear stochastic systems were discussed by [1] and [5], respectively.

This paper discusses robust H_∞ filtering for stochastic systems with time-varying delay. Here, we focus on the design of a linear state estimator such that the dynamics of the estimation error is stochastically exponentially stable in mean square, whose L₂-induced gain with respect to uncertain disturbance signal is less than a prescribed levelγ.

First, by extending the descriptor system approach introduced in [6] for deterministic systems with delay to the stochastic systems case. We represent the stochastic systems in the equivalent descriptor stochastic systems. Based on the equivalent de- scriptor form representation, we introduced a new type of Lyapunov-Krasovskii functional. Second, we extended the idea introduced in [7] to stochastic

0This work was supported in part by NSFC under Grant 60704009 and NSF of Hebei Province P.R. China under Grant F2005000390, F2006000270.

systems case. By using some zero equations, we in- troduce some free weighting matrices. To guarantee the existence of desired robustH_∞ filters, Delay- dependent sufficient conditions are proposed based on LMI algorithm and the minimumγobtained are less conservative than corresponding results in the literature. Numerical simulation example shows the results are effective and are an improvement over existing methods.

For convenience, we adopt the following nota- tions:

T r(A)(A^T) Trace (transpose) of the matrixA.

A≥0(A >0) Positive semi-definite (positive definite) matrixA.

L²([0,∞);Rⁿ) space of nonanticipative stochastic processes φ(t) with respect to filtration t satisfying

φ(t)²_L2

=E

_∞

0 φ(t)²dt <∞ L²₀([−τ ,0];Rⁿ) the family ofRⁿ−valued stochastic processes η(s), −τ ≤ s ≤ 0 such thatη(s)is0−measurable for every second and ₀

−τEη(s)²ds <∞

E{·} mathematical expectation operator with respect to the given probability measureP.

II. MAINRESULTS

Consider the following stochastic linear time- delay systems

dx(t) = [A₀x(t) +A_0dx(t−h(t) +B₀v(t)]dt(1) +[C₀x(t) +C_0dx(t−h(t))]dβ(t) x(t) = ϕ(t), t∈[−τ ,0]

dy(t) = [A₁x(t) +A_1dx(t−h(t) +B₁v(t)]dt(2) +[C₁x(t) +C_1dx(t−h(t))]dβ(t)

z(t) = Lx(t) (3)

where x(t) ∈ Rⁿ is the state vector, the time delay h(t) is a time-varying continuous function

Proceedings of the 7th

World Congress on Intelligent Control and Automation June 25 - 27, 2008, Chongqing, China

that satisfies

0≤h(t)≤τ (4)

and h˙(t)≤μ≤1 (5)

y(t) ∈ R^m is the measurement output.ϕ(t)is a continuous vector-valued initial function andϕ:=

{ϕ(s) :−τ ≤s≤0} ∈L²₀([−τ ,0], Rⁿ).z(t)∈ R^ris the state combination to be estimated,v(t)∈ L²([0,∞) ;Rⁿ) stands for the exogenous distur- bance signal.A_i,A_id,B_i,C_i,C_id,i= 1,2,andL are known constant matrices with appropriate di- mensions. Where the variablesβ(t)is 1-D Brown- ian motion satisfyingE{dβ(t)}= 0, E{dβ(t)²}= dt.

We take the following linear filter for the esti- mation ofz(t)

dx_f(t) = A_fx_f(t)dt+B_fdy(t)

x_f(0) = 0 (6)

z_f(t) = C_fx_f(t)

where x_f(t) ∈ Rⁿ,z(t)ˆ ∈ R^r,the constant matrices A_f, B_f, C_f are filter parameters to be designed. denoting ξ^T(t) = [x^T(t), x_f^T(t)] and

˜

z(t) =z(t)−z_f(t), then, we obtain the following augmented systems:

dξ(t) = Aξ(t) + ˜˜ A_dξ(t−h(t)) + ˜Bv(t)]dt +[ ˜Cξ(t) + ˜C_dξ(t−h(t))]dβ(t) (7) z˜ = ˜Lξ(t)

where A˜ =

A₀ 0 B_fA₁ A_f

,A˜_d=

A_0d 0 B_fA_1d 0

B˜ =

B₀ B_fB₁

,L˜=

L −C_f C˜ =

C₀ 0 B_fC₁ 0

,C˜_d=

C_0d 0 B_fC_1d 0

(8) Definition 1: System (7) with v(t) ≡ 0 is said to be exponentially stable in mean square if there exists a positive constantαsuch that

t→∞lim sup1

tlogEx(t)²≤ −α

The objective of this paper is to seek the filter parametersA_f,B_fandC_fsuch that the augmented system(7) with v(t) ≡ 0 is exponentially stable in mean square, More specifically, for a prescribed disturbance attenuation level γ >0, such that the performance index

Jˆ= _∞

0

(˜z^Tz˜−γ²v^Tv)dt (9)

is negative∀0=v(t)∈L²((0,∞), Rⁿ).

A. Delay-dependent H_∞ filtering

Before presenting the main results, we first give the following lemmas.

Lemma 1: Given scalars τ > 0 and μ <1, the system (7) with v(t) = 0 is exponentially stable in mean square, if there exist symmetric positive definite matrix P > 0, Q ≥ 0, Z > 0, R > 0, and appropriately dimensioned matricesN_i, T_j and Y_i,(i = 1,2,3,4;j = 1,2,3,4,5) such that the following LMI holds.

⎡

⎢⎢

⎢⎢

⎢⎢

⎣

Γ₁₁ Γ₁₂ Γ₁₃ Γ₁₄ τ Y₁ Y₁

∗ Γ₂₂ Γ₂₃ Γ₂₄ τ Y₂ Y₂

∗ ∗ Γ₃₃ Γ₃₄ τ Y₃ Y₃

∗ ∗ ∗ Γ₄₄ τ Y₄ Y₄

∗ ∗ ∗ ∗ −τ R 0

∗ ∗ ∗ ∗ ∗ −Z

⎤

⎥⎥

⎥⎥

⎥⎥

⎦

<0

(10) where an ellipsis∗denotes a block induced easily by symmetry, and

Γ₁₁ = N₁A˜+ ˜A^TN₁^T+T₁C˜+ ˜C^TT₁^T +Q+Y₁+Y₁^T Γ₁₂ = N₁A˜_d+ ˜A^TN₂^T+T₁C˜_d+ ˜C^TT₂^T−Y₁+Y₂^T Γ₁₃ = P−N₁+ ˜A^TN₃^T + ˜C^TT₃^T +Y₃^T

Γ₁₄ = A˜^TN₄^T−T₁+ ˜C^TT₄^T +Y₄^T

Γ₂₂ = N₂A˜_d+ ˜A^T_dN₂^T+T₂C˜_d+ ˜C_d^TT₂^T−(1−μ)Q

−Y₂−Y₂^T

Γ₂₃ = −N₂+ ˜A^T_dN₃^T + ˜C_d^TT₃^T −Y₃^T Γ₂₄ = A˜^T_dN₄^T−T₂+ ˜C_d^TT₄^T −Y₄^T

Γ₃₃ = −N₃−N₃^T+τ R, Γ₃₄=−N₄^T −T₃ Γ₄₄ = P−T₄^T −T₄+τ Z

Proof: For convenience, set

q(t) = Aξ(t) + ˜˜ A_dξ(t−h(t)) + ˜Bv(t)(11) g(t) = Cξ(t) + ˜˜ C_dξ(t−h(t)) (12) then system (7) becomes the following descriptor stochastic systems

dξ(t) = q(t)dt+g(t)dβ(t) (13)

˜

z = ˜Lξ(t)

Choose a Lyapunov-Krasovskii functional for sys- tem (13) to be

V(t) =⁴

i=1V_i(t) in which

V₁(t) =ξ(t)^TP ξ(t) V₂(t) =

_t

t−h(t)

ξ^T(s)Qξ(s)ds

V₃(t) = ₀

−τ

_t

t+θq^T(s)Rq(s)dsdθ V₄(t) =

₀

−τ

_t

t+θ

trace[g^T(s)Z(g(s)]dsdθ where P, Q, Z, R are symmetric positive definite matrices with appropriate dimensions. L be the weak infinitesimal operator of (13), the Newton- Leibniz formula provides

ξ(t)−ξ(t−h(t)) = _t

t−h(t)

q(s)ds+

_t

t−h(t)

g(s)dβ(s) (14) For appropriately dimensioned matricesN_i, Y_i and T_i (i= 1,2,3,4), equations in (11) - (12) ensure that

2

ξ^T(t)N₁+ξ^T(t−h(t))N₂+q^T(t)N₃+g^T(t)N₄

∗

Aξ(t) + ˜˜ A_dξ(t−h(t)) + ˜Bv(t)−q(t)

≡0 (15) 2

ξ^T(t)T₁+ξ^T(t−h(t))T₂+q^T(t)T₃+g^T(t)T₄

∗

Cξ(t) + ˜˜ C_dξ(t−h(t))−g(t)

≡0 (16) 2

ξ^T(t)Y₁+ξ^T(t−h(t))Y₂+q^T(t)Y₃+g^T(t)Y₄

∗ [ξ(t)−ξ(t−h(t))−κ−ς]≡0 (17)

τΛ− _t

t−h(t)

Λds≥0 (18)

where κ^T = _t

t−h(t)q^T(s)ds, ς^T = _t

t−h(t)g^T(s)dβ(s), Λ = η^T(t) ˜Y R⁻¹Y˜^Tη(t).

Adding the terms on the left of (15)-(18) to L_v=0V , Moreover, by Lemma[8], for any matrix Z >0

−2η^T(t) ˜Y ς ≤η^T(t) ˜Y Z⁻¹Y˜^Tη(t) +ς^TZς thenL_v=0V can be expressed as

L_v=0V ≤η^T(t)Ξη(t)− _t

t−h(t)

ΘR⁻¹Θ^Tds

− _t

t−h(t)trace

g^T(t)Zg(t)

ds+ς^TZς where

η^T(t) =

ξ^T(t), ξ^T(t−h(t)), q^T(t), g^T(t) Ξ = Γ +τY R˜ ⁻¹Y˜^T + ˜Y Z⁻¹Y˜^T Θ = η^T(t) ˜Y +q^T(s)R

Y˜^T =

Y₁^T Y₁^T Y₁^T Y₁^T

(19) Γ =

⎡

⎢⎢

⎣

Γ₁₁ Γ₁₂ Γ₁₃ Γ₁₄

∗ Γ₂₂ Γ₂₃ Γ₂₄

∗ ∗ Γ₃₃ Γ₃₄

∗ ∗ ∗ Γ₄₄

⎤

⎥⎥

⎦ (20)

Since E

ς^TZς

=E _t

t−h(t)trace

g^T(t)Zg(t) ds

It follows that

EL_v=0V(t)≤Eη^T(t)Ξη(t) (21) By Schur’s complement, Ξ < 0 is equivalent to LMI (10).

Setλ₀=λ_min(−Ξ), λ₁=λ_min(P),by (21) EL_v=0V(t)≤ −λ₀Eη^T(t)η(t)≤ −λ₀Ex^T(t)x(t)

(22) From the definitions of V(t), q(t) and g(t), there exist positive scalarsα_1,α₂ such that

λ₁x(t)²≤V(t)≤α₁x(t)²+α_{2 t}

t−2τx(s)²ds (23) Choose a scalarα₀>0 such that

α₀(α₁+ 2α₂τ e^2α0τ)≤λ₀ (24) then, by Itˆo differential formula, for t₀ ≥ 2τ , it has

Ee^α0tV(t)−Ee^α0t0V(t₀) =E _t

t0

L_v=0(e^α0sV(s))ds

≤ E _t

t0

e^α0s[α₀(α₁x(s)² +α₂

_s

s−2τx(u)²du)−λ₀x(s)²]ds By (23) and (24), getting

t→∞lim sup1

t logEx(t)²≤ −α₀

which implies system (13) is exponentially stable in mean square. The proof of Lemma 1 is completed.

Theorem 1: Consider the system (1)-(3) with (4) and (5). Given scalars τ > 0 and μ < 1, for a prescribed γ > 0, the H_∞ performance J <ˆ 0 holds for all nonzero v(t) ∈ L²((0,∞), Rⁿ), if there exist symmetric positive definite matrixP >

0, Q≥0, Z >0, R >0,scalarsh_i>0, e_i>0and appropriately dimensioned matricesS_{i ,}W_i and Y_i (i= 1,2,3,4) satisfying the following LMI

⎡

⎢⎢

⎢⎢

⎣

Γ τY˜ Y˜ K˜ F˜

∗ −τ R 0 0 0

∗ ∗ −Z 0 0

∗ ∗ ∗ −I 0

∗ ∗ ∗ ∗ −γ²I

⎤

⎥⎥

⎥⎥

⎦<0 (25)

If a solution of the LMI exists then the filter that guarantees the estimation error level of γ is given by (6) with

A_f =S₄⁻¹X₄, B_f =S₄⁻¹X₅ and C_f . where Γ and Y˜ are defined in (19) and (20), respectively ,

F˜=

⎡

⎢⎢

⎣ N₁B˜ N₂B˜ N₃B˜ N₄B˜

⎤

⎥⎥

⎦,K˜ =

⎡

⎢⎢

⎣ L˜ 0 0 0

⎤

⎥⎥

⎦ (26)

and Γ_ii =

Γ¹_ii Γ²_ii

∗ Γ⁴_ii

, Γ_ij =

Γ¹_ij Γ²_ij Γ³_ij Γ⁴_ij

(27)

Γ¹₁₁ = h₁S₁A₀+h₁A^T₀S₁^T+h₁X₂A₁+h₁A^T₁X₂^T +e₁W₁C₀+e₁C₀^TW₁^T +e₁X₃C₁

+e₁C₁^TX₃^T +Q₁+Y₁₁+Y₁₁^T

Γ²₁₁ = h₁X₁+h₁A^T₀S₃^T +h₁A^T₁X₅^T +e₁C₀^TW₃^T +e₁C₁^TX₆^T +Q₂+Y₁₂+Y₁₃^T

Γ₁₁⁴ = h₁X₄+h₁X₄^T+Q₃+Y₁₄+Y₁₄^T

Γ¹₁₂ = h₁S₁A_0d+h₁X₂A_1d+h₂A^T₀S₁^T+h₂A^T₁X₂^T +e₁W₁C_0d+e₁X₃C_1d+e₂C₀^TW₁^T

+e₂C₁^TX₃^T −Y₁₁+Y₂₁^T

Γ²₁₂ = h₂A^T₀S₃^T +h₂A^T₁X₅^T +e₂C₀^TW₃^T +e₂C₁^TX₆^T −Y₁₂+Y₂₃^T

Γ³₁₂ = h₁S₃A_0d+h₁X₅A_1d+h₂X₁^T +e₁W₃C_0d +e₁X₆C_1d−Y₁₃+Y₂₂^T

Γ⁴₁₂ = h₂X₄^T −Y₁₄+Y₂₄^T

Γ¹₁₃ = P₁−h₁S₁+h₃A^T₀S₁^T +h₃A^T₁X₂^T +e₃C₀^TW₁^T +e₃C₁^TX₃^T +Y₃₁^T Γ²₁₃ = P₂−h₁S₂+h₃A^T₀S₃^T +h₃A^T₁X₅^T

+e₃C₀^TW₃^T +e₃C₁^TX₆^T +Y₃₃^T

Γ³₁₃ = P₂^T −h₁S₃+h₃X₁^T +Y₃₂^T Γ⁴₁₃ = P₃−h₁S₄+h₃X₄^T +Y₃₄^T Γ¹₁₄ = h₄A^T₀S₁^T +h₄A^T₁X₂^T −e₁W₁

+e₄C₀^TW₁^T +e₄C₁^TX₃^T +Y₄₁^T Γ²₁₄ = h₄A^T₀S₃^T +h₄A^T₁X₅^T −e₁W₂ +e₄C₀^TW₃^T +e₄C₁^TX₆^T +Y₄₃^T Γ³₁₄ = h₄X₁^T −e₁W₃+Y₄₂^T

Γ⁴₁₄ = h₄X₄^T −e₁W₄+Y₄₄^T Γ⁴₁₄ = h₄X₄^T −e₁W₄+Y₄₄^T

Γ¹₁₅ = τ Y₁₁, Γ²₁₅=τ Y₁₂,Γ³₁₅=τ Y₁₃, Γ⁴₁₅=τ Y₁₄

Γ¹₁₆ = h₁S₁B₀+h₁X₂B₁ Γ²₁₆ = h₁S₃B₀+h₁X₅B₁

Γ¹₂₂ = h₂S₁A_0d+h₂X₂A_1d+h₂A^T_0dS₁^T +h₂A^T_1dX₂^T +e₂W₁C_0d+e₂X₃C_1d+e₂C_0d^TW₁^T

+e₂C_1d^TX₃^T−(1−μ)Q₁−Y₂₁−Y₂₁^T Γ²₂₂ = h₂A^T_0dS₃^T +h₂A^T_1dX₅^T +e₂C_0d^TW₃^T

+e₂C_1d^TX₆^T−(1−μ)Q₂−Y₂₂−Y₂₃^T Γ⁴₂₂ = −(1−μ)Q₃−Y₂₄−Y₂₄^T

Γ¹₂₃ = −h₂S₁+h₃A_0d^T S₁^T +h₃A^T_1dX₂^T +e₃C_0d^TW₁^T +e₃C_1d^TX₃^T −Y₃₁^T Γ²₂₃ = −h₂S₂+h₃A^T_0dS₃^T +h₃A^T_1dX₅^T

+e₃C_0d^TW₃^T +e₃C_1d^TX₆^T −Y₃₃^T Γ³₂₃ = −h₂S₃−Y₃₂^T, Γ⁴₂₃=−h₂S₄−Y₃₄^T Γ¹₂₄ = h₄A^T_0dS₁^T +h₄A^T_1dX₂^T +e₄C_0d^TW₁^T

+e₄C_1d^TX₃^T−e₂W₁−Y₄₁^T

Γ²₂₄ = h₄A^T_0dS₃^T +h₄A^T_1dX₅^T +e₄C_0d^TW₃^T +e₄C_1d^TX₆^T−e₂W₂−Y₄₃^T

Γ³₂₄ = −e₂W₃−Y₄₂^T, Γ⁴₂₄=−e₂W₄−Y₄₄^T Γ¹₂₅ = τ Y₂₁,Γ²₂₅=τ Y₂₂,Γ³₂₅=τ Y₂₃, Γ⁴₂₅=τ Y₂₄ Γ¹₂₆ = h₂S₁B₀+h₂X₂B₁

Γ²₂₆ = h₂S₃B₀+h₂X₅B₁ Γ¹₃₃ = −h₃S₁−h₃S₁^T+τ R₁ Γ²₃₃ = −h₃S₂−h₃S₃^T+τ R₂ Γ⁴₃₃ = −h₃S₄−h₃S₄^T+τ R₃

Γ¹₃₄ = −h₄S₁^T −e₃W₁, Γ²₃₄=−h₄S₃^T −e₃W₂ Γ³₃₄ = −h₄S₂^T −e₃W₃, Γ⁴₃₄=−h₄S₄^T −e₃W₄

Γ¹₃₅ = τ Y₃₁, Γ²₃₅=τ Y₃₂, Γ³₃₅=τ Y₃₃, Γ⁴₃₅=τ Y₃₄ Γ¹₃₆ = h₃S₁B₀+h₃X₂B₁,Γ²₃₆=h₃S₃B₀+h₃X₅B₁ Γ₄₄¹ = P₁−e₄W₁^T−e₄W₁+τ Z₁

Γ₄₄² = P₂−e₄W₃^T−e₄W₂+τ Z₂

Γ₄₄⁴ = P₃−e₄W₄^T −e₄W₄+τ Z₃

Γ¹₄₅ = τ Y₄₁,Γ²₄₅=τ Y₄₂,Γ³₄₅=τ Y₄₃,Γ⁴₄₅=τ Y₄₄ Γ¹₄₆ = h₄S₁B₀+h₄X₂B₁, Γ²₄₆=h₄S₃B₀+h₄X₅B₁

Y_i =

Y_i1 Y_i2 Y_i3 Y_i4

, i= 1,2,3,4 L˜ =

L −C_f

(28) Proof: First by the proof of the Lemma 1, we can deduce that the augmented system (7) with v(t) = 0 to be robust exponential stable in mean square. Second, we prove J <ˆ 0.for all nonzero v(t) ∈ L²((0,+∞), Rⁿ) with ξ(t,0) = 0,Note

that for anyT >0 J(Tˆ ) = E

_T

0

(z(t)˜ ²−γ²v(t)²)dt

≤ E _T

0

[z(t)˜ ²−γ²v(t)²+L_vV(ξ(t), t)]dt

= E _T

0

η(t) v(t)

_T Ψ

η(t) v(t)

dt

−E _T

0

_t

t−h(t)

ΘRΘ^Tdsdt

WhereF˜ andK˜ are defined in (26), and Ψ =

Ξ˜ F˜

∗ −γ²I

Ξ = Γ +˜ τY R˜ ⁻¹Y˜^T + ˜Y Z⁻¹Y˜^T + ˜KK˜(29)^T Therefore, ifΨ<0,then

J(Tˆ ) ≤ −λ_min(−Ψ)E _T

0 (ξ(t)²+v(t)²)dt

≤ −λ_min(−Ψ)E _T

0 v(t)²)dt <0 for any nonzero v(t) ∈ L²((0,+∞), Rⁿ),which yield J(t)ˆ < −λ_min(−Ψ)E_t

0v(t)²)dt <0. If we take

N_i = h_i

S₁ S₂ S₃ S₄

, Y_i=

Y_i1 Y_i2 Y_i3 Y_i4

(30) T_i = e_i

W₁ W₂ W₃ W₄

, i= 1,2,3,4 (31)

Q =

Q₁ Q₂

∗ Q₃

, P =

P₁ P₂

∗ P₃

R =

R₁ R₂

∗ R₃

, Z=

Z₁ Z₂

∗ Z₃

(32) Substituting (8) and (32) into (29), and letting S₂A_f =X₁, S₂B_f =X₂, W₂B_f =X₃, S₄A_f = X₄, S₄B_f = X₅, W₄B_f = X₆ then Ψ < 0 is equivalent to (25). From our assumption, A_f = S₄⁻¹X₄, B_f = S₄⁻¹X₅,and an H_∞ filter is con- structed as in the form of (6), and the proof of Theorem 1 is completed.

III. NUMERICAL SIMULATION Consider the stochastic time-varying delay sys- tem[10] with parameters as follows:

A₀ =

0 3

−4 −5

, B₀=

−0.4545 0.9090

A_0d =

−0.1 0 0.2 −0.2

, A_1d=

0.5 0.3

A₁ =

1 2 , L=

3 4

, B₁= 1 C₀ = C_0d=

0.5 0 0 0.5

, C₁=C_1d=

1 2 For the simple choice ofh₁= 0.7, h₂= 0.41, h₃= 0.7, h₄ = 0.2, e₁ = 0.5, e₂ = 0.3, e₃ = 0.1, e₄ = 0.5, e₅= 0.01,

The minimum achievable noise attenuation level obtained in [10] isγ= 0.6074, However, applying the delay-dependent criterion of Theorem 1, we obtain that the minimum achievable bound on the noise attenuation level is γ_min=8×10⁻⁷ (for τ = 0.1, μ= 0.3), which is far smaller than the result obtained in [10].When we take γ = 0.6074, the corresponding filter matrices obtained by Theorem 1 and [10] respectively are

A_f =

−9.7029 1.1090 0.9684 −9.4364

,B_f =

−0.1195

−0.1562 C_f =

0.0399 0.0965

(33) and

A_f =

0.3065 2.5820

−4.4618 −5.0860

,B_f =

0.1797

−0.5045 C_f =

−2.8091 −3.4756

(34) Set the initial conditions asx(0) = [0.1,0]^T, x_f = [0.04,0]^T respectively. The exogenous disturbance inputv(t) is set as random noise andv(0) = 0.1.

Fig.1 give the time responses of the system states and filter states for the filter (33) with v(t) = 0.

It shows the filtering error system (7) is exponen- tial stable in mean square with v(t) = 0.Fig.2 show the time response of the function ω(t) =

_∞

0 z(s)˜ ²ds ₀^∞v(s)²ds with v(t) = 0,It is seen that the maximum value of this function is less than 0.25, which reveals that the H_∞ per- formance level √

0.25 = 0.5 is much less than the prescribed level0.6074. Therefore, we can see that the designed H_∞ filter meets the specified requirements. When v(t) = 0,the time responses of the system statesx₁,x₂ and filter statex_f1,x_f2 for the filter (33) and (34) are displayed in Fig.3 and Fig.4 respectively. The simulation results imply that the desired goal is well achieved. Moreover, our method is better than the method in [10].

IV. CONCLUSION

The problem of robustH_∞ filtering for stochas- tic systems with time-varying delay has been ad- dressed in this paper. LMI-based technique, ex- ponentially stable in mean square linear filter are designed which guaranteeL₂ gain to be less than a prescribed levelγ >0.Delay-dependent sufficient condition for stochastic system is presented, the

0 200 400 600 800 1000

−0.06

−0.04

−0.02 0 0.02 0.04 0.06 0.08 0.1 0.12

x1 xf1 x2 xf2

Fig. 1. The time responses of statexandxf

0 200 400 600 800 1000

0 0.05 0.1 0.15 0.2 0.25

Fig. 2. the time response of the functionω(t)

0 200 400 600 800 1000

−0.08

−0.06

−0.04

−0.02 0 0.02 0.04 0.06 0.08 0.1

x2 xf2 [10]xf2

Fig. 3. The time responses of statex1 andxf1

0 200 400 600 800 1000

−0.08

−0.06

−0.04

−0.02 0 0.02 0.04 0.06 0.08 0.1

x2 xf2 [10]xf2

Fig. 4. The time responses of statex2andxf2

minimum γ obtained are less conservative than corresponding results in the literature due to ap- plying descriptor model transformation of the sys- tem and introducing some free weighting matrices.

Numerical example has clearly indicated the less conservatism of our design.

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