L2–L∞ filter design for a class of neutral stochastic time delay systems

Author’s Accepted Manuscript L2−L∞ filter design for a class of neutral stochastic

time delay systems Lin Li, Heyang Wang, Shaodan Zhang

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PII: DOI: Reference:

S0016-0032(15)00450-0 http://dx.doi.org/10.1016/j.jfranklin.2015.11.015 FI2505

To appear in: Journal of the Franklin Institute Received date: 11 February 2015 Revised date: 27 August 2015 Accepted date: 9 November 2015 Cite this article as: Lin Li, Heyang Wang and Shaodan Zhang, L2−L∞ filter design for a class of neutral stochastic time delay systems, Journal of the Franklin Institute, http://dx.doi.org/10.1016/j.jfranklin.2015.11.015 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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L2-L∞ Filter Design for a class of Neutral Stochastic Time Delay Systems Lin Lia,b,∗ and Heyang Wang b and Shaodan Zhang b

Abstract This paper is devoted to the problem of L2 -L∞ filtering for a class of neutral stochastic systems with different neutral timedelay, discrete delay and distributed delays. By constructing a new Lyapunov-Krasovskii functional, some novel delay-dependent mean-square exponential stability criteria are obtained in terms of linear matrix inequalities. In the derivation process, neither model transformation method nor free-weighting matrix approach is used. Based on the obtained stability criterion, sufficient condition for the existence of the full-order L2 -L∞ filter is given by introducing two appropriate slack matrix variables. Desired L2 -L∞ filter is designed such that the resulting filtering error system is mean-square exponential stable and a prescribed L2 -L∞ disturbance attenuation level is satisfied. Finally, numerical examples are included to illustrate the effectiveness and the benefits of the proposed method.

Index Terms L2 -L∞ filtering, neutral stochastic systems, discrete time delays, distributed time delays, Linear matrix inequality

I. I NTRODUCTION It is generally known that time delay is frequently encountered in many practical systems, such as electronic systems, economic systems, robotics systems, network systems and space navigation systems. The existence of time delays can severely degrade the closed-loop systems performance, in some cases, drive the systems to instability [1]. There are two types of time delay systems: retarded and neutral. The retarded type contains delays only in its states, while the neutral type contains delays both in its states and in the derivatives of its states [2]. Neutral time delay systems can be found in such places as population ecology, distributed networks containing lossless transmission lines, heat exchangers, etc [3]. Over the past decade, the neutral time delay systems have been extensively investigated in aspects such as system analysis and control synthesis including robust H∞ control [4]–[7], stability and stabilization [2], [8]–[11], non-fragile control [12], [13] and so on. Due to the widely application in communications and radar, dynamic reliability, automation, biological engineering, social sciences and so on, much attention has been focused on the analysis and synthesis of stochastic time delay systems [14]–[24]. In recent years, many important results on the stability of stochastic time delay systems have been developed. For instance, ∗ Corresponding Author: [email protected] a Shanghai Key Lab of Modern Optical System, and Engineering Research Center of Optical Instrument and System, Ministry of Education, University of Shanghai for Science and Technology, Shanghai 200093, PR China. b Department of Control Science, University of Shanghai for Science and Technology, Shanghai 200093, PR China.

2

motivated by the Fridman’s work in [25], the model transformation method together with cross-terms technique were extended to solve the exponential stability in the mean square of stochastic time-delay systems [26]. Based on the slack matrix approach, a less conservatism stability criterion was derived for a class of stochastic time-delay systems with nonlinearities and Markovian jump parameters [27]. Furthermore, by using the free-weighting matrix approach, new delay-dependent stability criteria for neutral stochastic delay systems were proposed in [28], [29]. In addition, delay-partitioning method is another popular technique, an augmented delay partition Lyapunov-Krasovskii functional was constructed to resolve the H ∞ filtering problem of neutral stochastic time-delay systems [30]. However, model transformations may lead to additional dynamics of the original system [31], while the introduction of slack matrices or free-weighting ones may not be useful to the reduction of conservatism in some cases, instead it can increase the computational burden [32]. And when the number of delay-partitioning increases, the matrix formulation becomes more complex and the computational burden and time-consuming grow bigger [10]. In [31], [33], the mean-square exponential stability was discussed, and delay-dependent stability criteria for neutral stochastic delay systems were derived based on the generalized Finsler lemma (GFL). Unfortunately, from the preface of [34], there are some errors in this GFL. Thus, new results on delay-dependent stability analysis for neutral stochastic delay systems were derived in [34], where neither the model transformation method nor the free-weighting matrix method was used. Under this framework, further considering three types of delays with different characters (neutral, discrete and distributed delays), there is a need to establish new stability criterion with less conservatism and lower computational cost for neutral stochastic systems with mixed different time delays. It is one of the motivations of this paper. During the past decades, there has been an increasing research interest in the filtering problem [35]–[43]. Although existing literatures have provided wealthy of conclusions to estimate system states for different systems (from continuous-time to discrete-time systems, linear to non-linear systems). The earliest can be traced back to 1960’s [44], where Kalman filtering was designed to extract the useful signal mixed with noise. It has been proven to be very useful and has received ongoing interest since it was proposed. But Kalman filtering is under the assumption that the noise must obey the Gaussian distribution. H ∞ filtering was developed to overcome the deficiency that this filter was insensitive to the exact knowledge of the statistics of noise signals [36]–[40], [45], [46]. However, the H ∞ filtering constraints the deviation from the point of overall energy, and it is not sensitive to the deviation of every single moment. In some high-performance systems, where the indicators usually are given by the upper and lower bounds, the worst-case must be exactly constrained. Therefore, the L 2 -L∞ filtering is valuable, because the worst-case peak value of the estimation error of the bounded energy value of the noise is minimized. Especially in the digital systems, it is more meaningful to limit the maximum value of the transmitted signal. The goal of L 2 -L∞ filtering, also called energy-to-peak filtering, is to design a stabilizing filter minimizing the peak value of the estimation error for any bounded energy disturbances [41], [42], [47]–[50]. To our knowledge, there are few results on L 2 -L∞ filtering for stochastic systems [48]–[50], especially for the neutral stochastic delay system, which is almost blank. Besides, the conventional method on L2 -L∞ performance is no longer valid for neutral stochastic delay systems, which motivates this paper, on the other hand. In this paper, L 2 -L∞ filtering for a class of neutral stochastic systems with mixed time delays is investigated. The main contributions of this paper can be highlighted as follows: (1) By introducing a new Lyapunov-Krasovskii functional (LKF), some new simpler and less conservative mean-square exponential stability criteria for neutral stochastic systems with both

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discrete and distributed time delays are derived. In the derivation process of the LKF, neither model transformation method nor the free-weighting matrix approach is used, and not any redundant free matrix variables are introduced. (2) Sufficient delay-dependent conditions for the solvability of the L 2 -L∞ filtering problem are established. Different from conventional methods, the L2 -L∞ condition is obtained by introducing two slack matrices. And the corresponding filter can be easily designed by solving three strict linear matrix inequalities (LMIs). (3) It shows the improvement our approach provided over previous method through numerical examples. Notation: Throughout this paper, R n and Rn×m denote, respectively, the n-dimensional Euclidean space and the set of all n × m real matrices. The superscripts −1 and T indicate the inverse and the transpose of a matrix, respectively. ∗ denotes the symmetry part of a symmetry matrix. I is the identity matrix of appropriate dimension. The symmetry matrix X > 0 (or X ≥ 0) means that X is positive definite ( or semi-positive definite). E{·} denotes the mathematical expectation operator with respect to some probability measure P (Ω, F , P) is a probability space with Ω the sample space and F the σ-algebra of subsets of the sample space. II. P ROBLEM F ORMULATION Consider the following neutral stochastic delay system described by 

t

x(s)ds + Bv(t)]dt d[x(t) − Dx(t − τ )] = [Ax(t) + A1 x(t − d) + A2 t−d  t + [Gx(t) + G1 x(t − d) + G2 x(s)ds + B1 v(t)]dω(t), t−d  t dy(t) = [Cx(t) + C1 x(t − d) + C2 x(s)ds + B2 v(t)]dt t−d  t x(s)ds + B3 v(t)]dω(t), + [Hx(t) + H1 x(t − d) + H2

(1)

t−d

z(t) = Lx(t), x(t) = φ(t), t ∈ [−dmax , 0] where x(t) ∈ Rn is the state vector, y(t) ∈ Rl is the measured output, z(t) ∈ R q is the signal to be estimated, v(t) ∈ R p is the exogenous disturbance input which belongs to L 2 [0, ∞), and ω(t) is one-dimensional Brownian motion defined on the complete probability space (Ω, F , P) satisfying E{dw(t)} = 0, E{dw2 (t)} = dt. In system (1), D, A, B, G, C, H, Ai , Gi , Ci , Hi , Bj (i = 1, 2, j = 1, 2, 3) and L are known real matrices of appropriate dimensions, in which, the spectrum radius of the matrix D, ρ(D), satisfies ρ(D) < 1. φ(t) ∈ R n is the given initial condition and Eφ(·)2 is uniformly bounded, τ is the neutral delay, d is the discrete and distributed time delays. Denote d max = max(d, τ ). Remark 1: In system (1), τ is called neutral time delay which is inherent and dependent on the neutral system itself. Meanwhile, d addresses the discrete and distributed time delays, which is related to the system operating conditions and surroundings. In general, these two kinds of time delay are different. That is to say τ = d. However, in [28], [29], [33], [34],

4

the discrete delay is assumed to be equal to the neutral delay. It is too special. Therefore, in this paper, we study the generalized neutral stochastic time delay systems of the form (1) with mixed different time delays. Now, consider the following full order filter dxf (t) = Af xf (t)dt + Bf dy(t),

(2)

zf (t) = Cf xf (t) where xf (t) ∈ Rn is the filter state, Af , Bf and Cf are the filter gain matrices to be determined. Define the augmented state vector and estimation error as ξ T (t) = [xT (t) xTf (t)], e(t) = z(t) − zf (t). Then, the filtering error system can be obtained as follows ˜ ˜ d[ξ(t) − Dξ(t − τ )] =[Aξ(t) + A˜1 ξ(t − d) + A˜2



t

t−d

˜ ˜ 1 ξ(t − d) + G ˜2 + [Gξ(t) +G



˜ ξ(s)ds + Bv(t)]dt t

t−d

˜1 v(t)]dw(t), ξ(s)ds + B

(3)

˜ e(t) =Lξ(t) ⎡ ⎤ ⎡ D ⎥ ⎢ A ˜ =⎢ D ⎣ ⎦ E, A˜ = ⎣ 0 Bf C

where

⎡

⎤

⎤ ⎤ ⎤ ⎡ ⎡ A A 0⎥ ⎢ 1 ⎥ ⎢ 2 ⎥ ⎦ , A˜1 = ⎣ ⎦ E, A˜2 = ⎣ ⎦ E, Af Bf C1 Bf C2

⎤

⎡

⎤

⎡

⎤

⎡

⎡

⎤

B ⎥ G ⎥ ⎢ G1 ⎥ ⎢ G2 ⎥ ⎢ B1 ⎥ ˜=⎢ ˜=⎢ B ⎣ ⎦, G ⎦ E, G˜2 = ⎣ ⎦ E, B˜1 = ⎣ ⎦, ⎦ E, G˜1 = ⎣ ⎣ B f H1 B f H2 Bf B3 Bf B2 Bf H ˜ L = [L − Cf ], E = In×n

0n×n .

For convenience, we introduce the following two definitions. Definition 1: The nominal neutral stochastic system (1) with v(t) = 0 is said to be mean-square exponential stable if there exist positive scalars λ > 0, c > 0 such that E  x(t) 2  ce−λt

sup

θ∈[−dmax ,0]

E  φ(θ) 2 , t  0.

Definition 2: Given a scalar γ > 0. The filtering error system (3) is said to be mean-square exponential stable with an L 2 -L∞ performance γ be satisfied, if it is mean-square exponential stable, and under zero initial condition, the following inequality holds for any nonzero v(t) ∈ L 2 [0, ∞). E  e 2∞ < γ 2  v 22 where E  e 2∞ = sup E(eT (t)e(t)),  v 22 = t

 0

∞

v T (t)v(t)dt.

5

Remark 2: Performance analysis has grown, in the past few decades, as one of the most important problems in control theory in addition to stability analysis. Many control problems, to a certain extent, can be equal to designing proper controller such that the closed-loop system is asymptotical stable and its performance satisfies some requirements. The general adopted performance indexes include H 2 index, H∞ index, L1 index and L2 -L∞ index. Thereinto, L 2 -L∞ performance is to minimize the peak value of some important signal for any bounded energy disturbances. Especially in the digital systems, the L 2 -L∞ performance is more meaningful. In this paper, we are focused on the problem of L 2 -L∞ filtering for neutral stochastic systems with mixed delays. The objective of this paper is to design a full order linear filter of the form (2) for the system (1) such that the resulting filtering error system (3) is mean-square exponential stable while a prescribed L 2 -L∞ performance level γ is satisfied. Such a filter is regarded as L2 -L∞ filter (or Energy-to-Peak filter). Lemma 1: [51] For any constant matrix 0 < R = R T ∈ Rn×n , any scalar h > 0, and the differentiable vector function x(t) ∈ Rn , we have

 −h

t

 x˙ (s)Rx(s)ds ˙  −( T

t−h

t

t−h

 x(s)ds) ˙ R( T

t

t−h

x(s)ds). ˙

Lemma 2: [34] If x(t) is the solution of the following system with h is a known constant time delay d[x(t) − Dx(t − h)] = f (t)dt + g(t)dω(t). For any compatible dimension matrix N , we have  T

E(x (t − h)N

t

t−h

g(s)dω(s)) = 0, t  h.

Lemma 3: (Itoˆ Formula) [52] Let x(t) be an Itˆo process satisfying the following equation dx(t) = a(t)dt + b(t)dω(t).  (t, x). Then F (t, x) Suppose that F (t, x) is a real-valued function with continuous partial derivatives F t (t, x), Fx (t, x), and Fxx

is an Itˆo process such that

 dF (t, x) =

Ft (t, x)

+

Fx (t, x)a(t)

+

1  F (t, x)b2 (t) 2 xx

dt + Fx (t, x)b(t)dω(t)

III. M AIN R ESULTS In this section, we will focus on the problem of L 2 -L∞ filtering for neutral stochastic delay systems (1). By constructing a new Lyapunov-Krasovskii functional, we will firstly give sufficient condition for the mean-square exponential stability of the system (3), which is proposed as follows. Theorem 1: Give scalars τ > 0 and d > 0. The filtering error system (3) with v(t) = 0 is mean-square exponentially stable,

6

⎤

⎡

Z2 ⎥ ⎦ such that the following inequality holds: Z3

⎢ Z1 if there exist positive definite matrices P , Q, R, R 1 , S and Z = ⎣ Z2T ⎡ 0 ⎢(1, 1) ⎢ ⎢ ∗ (2, 2) ⎢ ⎢ ⎢ ⎢ ∗ ∗ ⎢ ⎢ ⎢ ∗ ∗ ⎢ ⎢ ⎢ ∗ ∗ ⎢ ⎢ ⎢ ⎢ ∗ ∗ ⎢ ⎢ ⎢ ∗ ∗ ⎢ ⎢ ⎢ ∗ ∗ ⎢ ⎣ ∗ ∗

P A˜2

˜ + Z2 −A˜T P D

0

0

˜ GP

0

0

˜ − Z2 −A˜T1 P D

dA˜T1 P

˜ 1P G

−S

˜ −A˜T2 P D

0

0

˜ 2P G

∗

−Q + Z3

0

0

0

∗

∗

−Z3

0

0

∗

∗

∗

−dR

0

∗

∗

∗

∗

−P

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

⎤ T T ˜ ˜ dA R τ A R1 ⎥ ⎥ dA˜T1 R τ A˜T1 R1 ⎥ ⎥ ⎥ ⎥ dA˜T2 R τ A˜T2 R1 ⎥ ⎥ ⎥ 0 0 ⎥ ⎥ ⎥ 0 0 ⎥ ⎥<0 ⎥ ⎥ 0 0 ⎥ ⎥ ⎥ 0 0 ⎥ ⎥ ⎥ −dR 0 ⎥ ⎥ ⎦ ∗ −τ R1

(4)

where (1, 1) = P A˜ + A˜T P + Q + Z1 + d2 S, (2, 2) = −Z1 + P A˜1 + A˜T1 P . Proof: For the sake of simplicity, the following notations are introduced ˜ η(t) =ξ(t) − Dξ(t − τ ), ˜ f (t) =Aξ(t) + A˜1 ξ(t − d) + A˜2 ˜ ˜ 1 ξ(t − d) + G ˜2 g(t) =Gξ(t) +G



t

t−d  t t−d

˜ ξ(s)ds + Bv(t), ˜1 v(t). ξ(s)ds + B

Since v(t) = 0, the system (3) can be rewritten as dη(t) = f (t)dt + g(t)dω(t). Choose the following Lyapunov-Krasovskii functional candidate for system (3) as  T

t



t

V (t, ξ(t)) =η (t)P η(t) + ξ (s)Qξ(s)ds + ζ T (s)Zζ(s)ds t−τ t−d  0  t  0  t ξ T (s)Sξ(s)dsdθ + f T (s)Rf (s)dsdθ +d  +

−d t+θ 0  t

−τ

t+θ

T

−d

(5)

t+θ

f T (s)R1 f (s)dsdθ

where ζ T (t) = [ξ T (t) ξ T (t − τ )]. The positive definite matrix Z has been defined before. From Lemma 3, the stochastic differential dV (t, ξ(t)) can be obtained dV (t, ξ(t)) = LV (t)dt + 2η T (t)P g(t)dω(t)

(6)

7

with the infinitesimal operator LV (t) =2η T (t)P f (t) + g T (t)P g(t) + ξ T (t)Qξ(t) − ξ(t − τ )Qξ(t − τ ) + ζ T (t)Zζ(t) − ζ T (t − d)Zζ(t − d) + df T (t)Rf (t) + τ f T (t)R1 f (t) + d2 ξ T (t)Sξ(t)  t  t  t − f T (s)Rf (s)ds − f T (s)R1 f (s)ds − d ξ T (s)Sξ(s)ds. t−d

t−τ

(7)

t−d

According to system (1), it should be interpreted as its integral form  η(t) − η(0) =

f (s)ds +

It is easy to see that

0

0

0

 η(t) − η(t − d) =

t

g(s)dω(s),  t−d t−d f (s)ds + g(s)dω(s).

0

 η(t − d) − η(0) =



t



t

t−d

f (s)ds +

t

t−d

g(s)dω(s).

Hence  η(t) =η(t − d) +



t

t−d

f (s)ds +

t

t−d

˜ =ξ(t − d) − Dξ(t − τ − d) +



g(s)dω(s), 

t

t−d

f (s)ds +

t

t−d

g(s)dω(s).

Then, we have ˜ + A˜2 2η (t)P f (t) =2η (t)P [Aξ(t) T

T



t

ξ(s)ds]  ˜ + 2[ξ(t − d) − Dξ(t − d − τ) + t−d



t

t−d

f (s)ds +

t

t−d

(8) g(s)dω(s)] P A˜1 ξ(t − d). T

Further, it follows from Lemma 1 that  −d

t

t−d

 ξ (s)Sξ(s)ds  −( T

t

t−d

 ξ(s)ds) S( T

t

t−d

ξ(s)ds).

(9)

Substituting (8) and (9) into (7) yields  t T ˜ ˜ ˜ ξ(s)ds] LV (t) 2(ξ(t) − Dξ(t − τ )) P [Aξ(t) + A2 t−d  t  ˜ + 2[ξ(t − d) − Dξ(t − d − τ ) + f (s)ds + t−d

t

t−d

g(s)dω(s)]T P A˜1 ξ(t − d) + d2 ξ T (t)Sξ(t)

+ ξ T (t)Qξ(t) − ξ T (t − τ )Qξ(t − τ ) + ζ T (t)Zζ(t) − ζ T (t − d)Zζ(t − d)  t  t  t  t T T −( ξ(s)ds) S( ξ(s)ds) − f (s)Rf (s)ds − f T (s)R1 f (s)ds t−d

T

t−d

t−d

(10)

t−τ

T

+ g (t)P g(t) + f (t)(dR + τ R1 )f (t). By Lemma 2, we have

 E[(

t t−d

g(s)dω(s))T P A˜1 ξ(t − d)] = 0.

(11)

8

It then follows from (10) and (11) that E(LV (t)  E[

1 d



t

t−d

T (t, s)[M + G T P G + AT (dR + τ R1 )A] (t, s)ds].

(12)

where T

(t, s) =

t



− d) t−d ⎡ ˜ + Z2 0 P A˜2 −A˜T P D ⎢(1, 1) ⎢ ⎢ ∗ (2, 2) 0 0 ⎢ ⎢ ⎢ ˜ ∗ −S −A˜T2 P D ⎢ ∗ M =⎢ ⎢ ⎢ ∗ ∗ ∗ −Q + Z3 ⎢ ⎢ ⎢ ∗ ∗ ∗ ∗ ⎢ ⎣ ∗ ∗ ∗ ∗

˜ G ˜1 G ˜2 0 0 0 , G= G

˜ ˜ ˜ A = A A1 A2 0 0 0 . ξ T (t)

ξ T (t

ξ T (s)ds

ξ T (t

− τ)

ξ T (t

− d − τ)

0 ˜ − Z2 −A˜T1 P D 0 0 −Z3 ∗

f T (s)

,

⎤

0

⎥ ⎥ T ⎥ ˜ dA1 P ⎥ ⎥ ⎥ 0 ⎥ ⎥, ⎥ 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎦ −dR

From Schur complement formula, the inequality (4) in Theorem 1 is equivalent to M + G T P G + AT (dR + τ R1 )A < 0, i.e. E(LV (t) < 0, which indicates that the mean-square stability of the filtering error system (3) can be guaranteed. Now, we analyse the mean-square exponential stability of system (3). From (6) and (12), it is easy to prove that there exists a scalar α > 0 such that E(

dV (t) ) = E(LV (t, ξ(t)))  −αEξ(t)2 . dt

(13)

On the other hand, observing (5), there surely exist finite scalars 0 < k 1 < k2 such that k1 ξ(t)2   V (t, ξ(t))  k2 ξ(t)2 .

(14)

α dV (t, ξ(t)) )  − E(V (t, ξ(t))). dt k2

(15)

It follows from (13) and (14) that E( Integrating both side of (15) from 0 to t yields α

E(V (t, ξ(t)))  E(V (0, ξ(0)))e− k2 t . 2 ˜ ˜ is the known initial condition of system Since there exists a scalar β such that E(V (0, ξ(0)))  β sup E( φ(t) ), where φ(t) 2 ˜ . Then, we get (3) with uniformly bounded norm E φ(·)

E(ξ(t)2 ) 

α β 2 − k2 t ˜ sup E(φ(t) )e . k1

Therefore, from Definition 1, system (3) is mean-square exponentially stable. Thus, this completes the proof.

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Remark 3: In the proof of Theorem 1, the Itˆo’s formula is used to calculate the differential of Lyapunov-Krasovskii functional V (t, ξ(t)). It follows from Lemma 3 that the stochastic differential dV (t, ξ(t)) can be written as  dV (t, ξ(t)) =



Vt (t, ξ)

+

Vξ (t, ξ)f (t)

+

1 T  2 g (t)Vξξ (t, ξ)g(t)

dt + Vξ (t, ξ)g(t)dω(t)

(16)

 where Vξ (t, ξ) = 2η T (t)P , Vξξ (t, ξ) = 2P and

Vt (t, ξ) =ξ T (t)Qξ(t) − ξ(t − τ )Qξ(t − τ ) + ζ T (t)Zζ(t) − ζ T (t − d)Zζ(t − d) + df T (t)Rf (t) + τ f T (t)R1 f (t) + d2 ξ T (t)Sξ(t)  t  t  − f T (s)Rf (s)ds − f T (s)R1 f (s)ds − d t−d

t−τ

(17) t

t−d

ξ T (s)Sξ(s)ds

 Substituting the aforementioned V ξ (t, ξ), Vξξ (t, ξ) and the equation (17) into (16) yields the differential of the Lyapunov-

Krasovskii functional, that is the equation (6) in the proof of Theorem 1. Remark 4: Many results on the stochastic stability have been developed, see, e.g. [14], [17]–[19], [23], [24], [27], and the reference therein. Compared with the stability for stochastic delay systems, it is much more difficult to delay with the stability of neutral stochastic delay systems due to the existence of the neutral item, which makes the problem be complicated. Recently, in [28], the stability condition for neutral stochastic delays systems is obtained, nevertheless, the method in [28] can increase the conservatism since the using of Jensen inequality and bounding techniques. In order to reduce this conservatism, a generalized Finsler lemma(GFL) was proposed and adopted in [31], [33], meanwhile, some slack matrices are introduced and Newton-Leibniz formula is used in [53]. However, it follows from [34] that there are some errors in this GFL(see Remark 4 in [34]), and the Newton-Leibniz formula is not valid in stochastic differential equations(see Equation(2) in [34]). In this case, new method for stochastic stability of neutral stochastic delay systems was derived in [34]. Motivated by this method, further considering different types of delays, Theorem 1 obtained a simpler and less conservative mean-square exponential stability criterion. It should be pointed out that in the derivation of Theorem 1, not any redundant free matrix variables has been introduced. Therefore the method in Theorem 1 will show less conservatism. Furthermore, the simplified stability conditions will be given in the following Corollary 1 and Corollary 2. If neglecting the distributed time delay, system (1) is reduced to d[x(t) − Dx(t − τ )] = [Ax(t) + A1 x(t − d)]dt + [Gx(t) + G1 x(t − d)]dw(t),

(18)

x(t) = φ(t), t ∈ [−dmax , 0]. Then, the following result is immediate. Corollary 1: Given scalars τ > 0, d > 0. The system (18) is mean-square exponentially stable, if there exist positive definite

10

TABLE I N UMBERS OF DECISION VARIABLES BY DIFFERENT METHODS .

Methods

[29]

Numbers of variables

37n2 +7n 2

⎢ Z1 matrices P , Q, R, R1 and Z = ⎣ Z2T ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

Corollary 2 7n2 +7n 2

⎤ Z2 ⎥ ⎦ such that Z3

⎡

⎡

[28] 17n2 +5n 2

(1,¯1) −AT P D + Z2

GT P

dAT R

τ AT R1

dAT1 P

GT1 P

dAT1 R

τ AT1 R1 ⎥ ⎥

∗

(2,¯2)

∗

∗

−Z3

0

0

0

0

∗

∗

∗

−dR

0

0

0

∗

∗

∗

∗

−P

0

0

∗

∗

∗

∗

∗

−dR

0

∗

∗

∗

∗

∗

∗

−τ R1

−AT1 P D

− Z2

⎤

0

0

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥<0 ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(19)

where (1,¯1) = P A + AT P + Q + Z1 , (2,¯2) = −Z1 + P A1 + AT1 P − Q + Z3 . More specifically, if the discrete delay is assumed to be equal the neutral delay in system (18), that is τ = d, we can obtain the following result. Corollary 2: Given scalar⎡τ = d > ⎤0. The system (18) is mean-square exponentially stable, if there exist positive definite ⎢ Z1 Z2 ⎥ matrices P , Q, R and Z = ⎣ ⎦ such that Z2T Z3 ⎡ T ¯ ⎢(1, 1) −A P D + Z2 ⎢ ⎢ ∗ (2,¯2) ⎢ ⎢ ⎢ ⎢ ∗ ∗ ⎢ ⎢ ⎢ ∗ ∗ ⎢ ⎢ ⎢ ∗ ∗ ⎢ ⎣ ∗ ∗

T

0

0

G P

−AT1 P D − Z2

dAT1 P

GT1 P

−Z3

0

0

∗

−dR

0

∗

∗

−P

∗

∗

∗

⎤ dA R⎥ ⎥ dAT1 R⎥ ⎥ ⎥ ⎥ 0 ⎥ ⎥<0 ⎥ 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎦ −dR T

(20)

where (1,¯1) and (2,¯2) are defined in Corollary 1. Remark 5: If setting Z 1 = Z3 , Z2 = 0, Corollary 2 will be reduced to Theorem 1 in [34]. Therefore, Theorem 1 in [34] is a special case of this paper. In Section IV, it shows, via a numerical example, that our obtained result is less conservative than that in [34]. Also, it should be noted that the numbers of decision variables in [28], [29] and Corollary 2 in this paper are , respectively, shown in Table I. Thus, our results is more computationally efficient than [28], [29]. Based on Theorem 1, we consider the L 2 -L∞ performance analysis, and the corresponding conclusion is stated as follows. Theorem 2: Given positive scalars τ , d and γ. The filtering error system (3) is mean-square exponentially stable with a

11

⎤

⎡ ⎢ Z1 prescribed L2 -L∞ performance γ, if there exist positive definite matrices P , Q, R, R 1 , S, T1 , T2 and Z = ⎣ Z2T that the following inequalities hold simultaneously ⎡ ⎤ T ˜ ⎢−T1 + T2 L ⎥ ⎣ ⎦ < 0, ∗ −γI ⎡ ⎢−P + T1 ⎣ ∗ ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

Z2 ⎥ ⎦, such Z3

(21)

⎤ ˜ PD ˜ − T2 −D P D ˜T

⎥ ⎦ < 0,

(22)

(1, 1)

0

P A˜2

˜ + Z2 −A˜T P D

0

0

˜ PB

˜ GP

∗

(2, 2)

0

0

˜ − Z2 −A˜T1 P D

dA˜T1 P

0

˜1P G

∗

∗

−S

˜ −A˜T2 P D

0

0

0

˜2P G

∗

∗

∗

−Q + Z3

0

0

˜TPB ˜ −D

0

∗

∗

∗

∗

−Z3

0

0

0

∗

∗

∗

∗

∗

−dR

0

0

∗

∗

∗

∗

∗

∗

−γI

˜1 P B

∗

∗

∗

∗

∗

∗

∗

−P

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

dA˜T R

τ A˜T R1

⎤

⎥ ⎥ τ A˜T1 R1 ⎥ ⎥ ⎥ T T dA˜2 R τ A˜2 R1 ⎥ ⎥ ⎥ ⎥ 0 0 ⎥ ⎥ ⎥ 0 0 ⎥ ⎥ ⎥ < 0. 0 0 ⎥ ⎥ ⎥ ˜T R τ B ˜ T R1 ⎥ ⎥ dB ⎥ ⎥ 0 0 ⎥ ⎥ ⎥ ⎥ −dR 0 ⎥ ⎦ ∗ −τ R1 dA˜T1 R

(23)

Proof: Similarly to the proof of Theorem 1, we get that (23) can guarantee the mean-square exponential stability of system (3). In the following, we establish the L 2 -L∞ performance. Under zero initial condition, we have E{V (0, ξ(0))} = 0. Define the following performance index:

 Jt = E{V (t, ξ(t))} − γ

0

t

v T (θ)v(θ))dθ

where V (t, ξ(t)) is defined as in (5). For any nonzero external disturbance v(t) ∈ L 2 [0, ∞), t > 0, we have  t Jt = E{V (t, ξ(t))} − E{V (0, ξ(0))} − γ v T (s)v(s)ds, 0  t  t = E{ dV (θ, ξ(θ))} − γ v T (s)v(s)ds, 0 0  t  t T LV (θ)dθ + 2η (θ)P g(θ)dω(θ)} − γ v T (θ)v(θ)dθ, = E{ 0 0  t = E{ [LV (θ) − γv T (θ)v(θ)]dθ},  0t ¯ + G¯T P G¯ + A¯T (dR + τ R1 )A]ς(θ, ¯ E ς T (θ, s)[M s)dθ, 0

(24)

12

where ⎡ ⎤ ˜ + Z2 ˜ 0 0 PB (1, 1) 0 P A˜2 −A˜T P D ⎢ ⎥ ⎢ ⎥ T T ⎢ ∗ ⎥ ˜ ˜ ˜ 0 (2, 2) 0 0 −A1 P D − Z2 dA1 P ⎢ ⎥ ⎢ ⎥ ⎢ ∗ ⎥ T ˜ ˜ 0 0 0 ∗ −S −A2 P D ⎢ ⎥ ⎢ ⎥ ⎢ ¯ =⎢ ∗ ˜TPB ˜⎥ M ⎥, ∗ ∗ −Q + Z 0 0 − D 3 ⎢ ⎥ ⎢ ⎥ ⎢ ∗ ⎥ 0 0 ∗ ∗ ∗ −Z3 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ∗ ∗ ∗ ∗ −dR 0 ⎢ ∗ ⎥ ⎣ ⎦ ∗ ∗ ∗ ∗ ∗ ∗ −γI



¯ ¯ ˜2 0 0 0 B ˜1 , A = A˜ A˜1 A˜2 0 0 0 B ˜ , ˜ G ˜1 G G= G

θ T T T T T T T ς (θ, s) = ξ (θ) ξ (θ − d) ξ (s)ds ξ (θ − τ ) ξ (θ − d − τ ) f (s) v(θ) . θ−d It is easy to get that the inequality (23) is equivalent to ¯ + G¯T P G¯ + A¯T (dR + τ R1 )A¯ < 0. M Thus, from (24), we easily obtain Jt < 0. That is to say

 E{V (t, ξ(t))} < γ

0

t

v T (θ)v(θ)dθ.

(25)

Next, from (21), it derives ˜ < γ(T1 − T2 ). ˜T L L ⎡

And from (22), we get

⎤

⎢ T1 ⎣ 0

⎡

0 ⎥ ⎢ P ⎦<⎣ −T2 −DT P

⎤ −P D ⎥ ⎦. DT P D

(26)

It follows form (25) and (26) that ˜ ˜ T Lξ(t)} < γE{ξ T (t)(T1 − T2 )ξ(t)}, E{eT (t)e(t)} = E{ξ T (t)L = γE{ξ T (t)T1 ξ(t) − ξ T (t − τ )T2 ξ(t − τ )}, ⎤⎡ ⎡ ⎤ ⎤T ⎡ −P D ⎥ ⎢ ξ(t) ⎥ ⎢ ξ(t) ⎥ ⎢ P < γE{⎣ ⎦⎣ ⎦}, ⎦ ⎣ −DT P DT P D ξ(t − τ ) ξ(t − τ )  γE{V (t, ξ(t))},  ∞ 2 v T (t)v(t)dt. <γ 0

(27)

13

It is obvious that E  e(t)  ∞ < γ  v(t) 2 . Then, from Definition 2, the L 2 -L∞ performance is satisfied. Thus, this completes the proof. Remark 6: Recently, in [48], [49], and [50], the L 2 -L∞ filtering for stochastic delay systems were studied. It is worth pointing out that the conventional method is adopted in them, see the inequality (16) in [50], the inequality (10) in [48], and the inequality (25) in [49]. However, for neutral stochastic system, this conventional method is no longer valid since the existence of neutral term. Currently, the authors in [30] said the problem of robust L 2 -L∞ filter design for uncertain neutral stochastic with mixed delays was investigated in this paper, however, where the H ∞ filtering was studied in fact. To our knowledge, it is almost blank on the study of L 2 -L∞ filtering for neutral stochastic delay systems. In Theorem 2, by introducing two slack matrices T 1 and T2 , the inequalities (21) and (22) are obtained that can ensure the L 2 -L∞ performance. The condition established in Theorem 2 is in terms of nonlinear matrix inequalities which is difficult to solve. Now we are in a position to present the solvability condition for the L 2 -L∞ filter design of the neutral stochastic time-delays system (1). Based on Theorem 2, the following theorem is derived. Theorem 3: Given positive scalars τ , d and γ. The filtering error system (3) is mean-square exponentially ⎤ stable with a ⎡ ¯ ¯ Z1 Z2 ⎥ ¯ R, ¯ R ¯ 1 , S, ¯ Z¯ = ⎢ prescribed L2 -L∞ performance γ, if there exist positive definite matrices X 1 , X, Q, ⎦, and matrices ⎣ Z¯2T Z¯3 Vi , (i = 1, 2, 3) such that the following LMIs hold: ⎡ ⎤ ˜11 + T˜21 −T˜12 + T˜22 LT − V3T − T ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ < 0, T (28) ∗ −T˜13 + T˜23 L ⎢ ⎥ ⎣ ⎦ ∗ ∗ −γI ⎡ X + T˜11 ⎢ 1 ⎢ ⎢ ∗ ⎢ ⎢ ⎢ ∗ ⎢ ⎣ ∗ ⎡ [1, 1] 0 ΦA 2 ⎢ ⎢ ⎢ ∗ [2, 2] 0 ⎢ ⎢ ⎢ ∗ ∗ −S¯ ⎢ ⎢ ⎢ ⎢ ∗ ∗ ∗ ⎢ ⎢ ⎢ ∗ ∗ ∗ ⎢ ⎢ ⎢ ∗ ∗ ⎢ ∗ ⎢ ⎢ ⎢ ∗ ∗ ∗ ⎢ ⎢ ⎢ ∗ ∗ ∗ ⎢ ⎢ ⎢ ∗ ∗ ⎢ ∗ ⎣ ∗ ∗ ∗

−ΨTA + Z¯2 0

⎤

X1 + T˜12

X1 D

X + T˜13

XD

∗

−DT XD − T˜21

∗

∗

0 −ΨTA1

⎥ ⎥ ⎥ XD ⎥ ⎥ < 0, ⎥ T ˜ −D XD − T22 ⎥ ⎦ −DT XD − T˜23

ΦB

ΦTG

ΦTA1

0

ΦTG1

0 − Z¯2

X1 D

−ΨTA2

0

0

0

ΦTG2

¯ + Z¯3 −Q

0

0

−ΨB

0

∗

−Z¯3

0

0

0

∗

∗

¯ −dR

0

0

∗

∗

∗

−γI

ΦTB1

∗

∗

∗

∗

−ΦX

∗

∗

∗

∗

∗

∗

∗

∗

∗

∗

(29)

dΦTA

τ ΦTA

dΦTA1

τ ΦTA1

⎤

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ T T dΦA2 τ ΦA 2 ⎥ ⎥ ⎥ ⎥ 0 0 ⎥ ⎥ ⎥ 0 0 ⎥ ⎥ < 0. ⎥ 0 0 ⎥ ⎥ ⎥ T T ⎥ dΦB τ ΦB ⎥ ⎥ ⎥ 0 0 ⎥ ⎥ ⎥ ¯ 0 −d(2ΦX − R) ⎥ ⎦ ¯1) ∗ −τ (2ΦX − R

(30)

14

And then, the desired filter gain matrices can be taken as Af = M −1 V2 X1−1 N −T , Bf = M −1 V1 , Cf = V3 X1−1 N −T .

(31)

where ¯ ¯ + Z¯1 + d2 S, [1, 1] = ΦA + ΦTA + Q [2, 2] = ΦA1 + ΦTA1 − Z¯1 , ⎤ ⎡ X1 A ⎥ X1 A ⎢ ΦA = ⎣ ⎦, XA + V1 C + V2 XA + V1 C ⎤ ⎡ X1 Ai ⎥ ⎢ X1 Ai ΦA i = ⎣ ⎦ (i = 1, 2) XAi + V1 Ci XAi + V1 Ci ⎤ ⎤ ⎡ ⎡ B B X X 1 1 1 ⎥ ⎥ ⎢ ⎢ ΦB = ⎣ ⎦ , ΦB1 = ⎣ ⎦, XB1 + V1 B3 XB + V1 B2 ⎤ ⎡ T T (XA + V C + V ) D (XA + V C) D 1 2 1 ⎥ ⎢ ΨA = ⎣ ⎦, DT (XA + V1 C + V2 ) DT (XA + V1 C) ⎤ ⎡ T T ⎢D (XAi + V1 Ci ) D (XAi + V1 Ci )⎥ Ψ Ai = ⎣ ⎦ , (i = 1, 2), DT (XAi + V1 Ci ) DT (XAi + V1 Ci ) ⎤ ⎤ ⎡ ⎡ T ⎢D (XB + V1 B2 )⎥ ⎢X1 X1 ⎥ ΨB = ⎣ ⎦ , ΦX = ⎣ ⎦, X1 X DT (XB + V1 B2 ) ⎤ ⎡ ⎡ ⎤ ⎢X1 G XG + V1 H ⎥ ⎢X1 Gi XGi + V1 Hi ⎥ ΦG = ⎣ ⎦ , (i = 1, 2). ⎦ , ΦG i = ⎣ X1 Gi XGi + V1 Hi X1 G XG + V1 H Proof: Let ⎤ ⎤ ⎡ Y N X M ⎥ ⎥ ⎢ ⎢ P =⎣ ⎦ , P −1 = ⎣ ⎦, T T V W N M ⎤ ⎡ ⎡ ⎡ ⎤ −1 I⎥ ⎢Y ⎢I X ⎥ ˆ ⎢Y F1 = ⎣ ⎦, Y = ⎣ ⎦ , F2 = ⎣ 0 NT 0 0 MT ⎡

It is not hard to see P F1 = F2 . M and N are nonsingular matrices satisfying M N T = I − XY.

⎤ 0⎥ ⎦. I

(32)

15

Therefore, the matrices F 1 and F2 are all nonsingular. Set T1 = diag{F1 , F1 , F1 , F1 , F1 , F1 , I, F1 , F1 , F1 }, T2 = diag{Yˆ , Yˆ , Yˆ , Yˆ , Yˆ , Yˆ , I, Yˆ , Yˆ , Yˆ }, ¯ Yˆ T F T Zi F1 Yˆ = Z¯i , (i = 1, 2, 3), Yˆ T F T RF1 Yˆ = R, ¯ Yˆ T F T SF1 Yˆ = S, ¯ Yˆ T F1T QF1 Yˆ = Q, 1 1 1 Y −1 = X1 , M Bf = V1 , M Af N T Y −1 = V2 , Cf N T Y −1 = V3 . Pre- and post-multiplying (23) in Theorem 2 by T 2T T1T and its transpose respectively, one yields the linear matrix inequaltiy (30) in Theorem 3. Similarly, set J 1 = diag{F1 , I}, J2 = diag{Yˆ , I}, J3 = diag{F1 , F1 }, J4 = diag{Yˆ , Yˆ }, pre- and post-multiplying (21) in Theorem 2 by J 2T J1T and its transpose respectively, and pre- and post-multiplying (22) in Theorem 2 by J4T J3T and its transpose respectively, we can obtain the linear matrix inequalities (28)-(29) in Theorem 3, respectively. Thus, this completes the proof. Remark 7: Theorem 3 gives the L 2 -L∞ filter design method. These conditions are in terms of LMIs and can be easily solved by using the Matlab LMI Control Toolbox. By solving LMIs (28)-(30), we can get the solution of positive definite matrices X1 and X. It follows from the expression in (32) and Y

−1

= X1 that I − XX1−1 = M N T . Then, the nonsingular matrices

M and N are derived by the LU decomposition of (I − XX 1−1 ). Therefore, the L 2 -L∞ filter gain matrices Af , Bf and Cf can be obtained from (31). Remark 8: The L2 -L∞ performance γ can be obtained and described as M inimize

¯ R, ¯R ¯ 1 ,S, ¯ Z,V ¯ 1 ,V2 ,V3 X1 ,X,Q,

γ

(33)

Subject to the linear matrix inequalities (28)-(30). IV. N UMERICAL EXAMPLES In this section, some numerical examples will be given to illustrate the effectiveness of the filter design method and the improvement this approach provided over previous method. Example 1. Assume the discrete delay d is equal to the neutral delay τ , and consider system (18) with ⎡ ⎡ ⎤ ⎤ ⎡ ⎤ 0 ⎥ ⎢−2 0 ⎥ ⎢−0.2 0.1 ⎥ ⎢−1.4 D=⎣ ⎦, ⎦,A = ⎣ ⎦ , A1 = ⎣ 0 −2 0 −0.2 0 −1.9 ⎤ ⎡ ⎤ ⎡ 0 ⎥ 0 ⎥ ⎢−1.9 ⎢−0.1 G=⎣ ⎦ , τ = d. ⎦ , G1 = ⎣ 0 −1.9 0 −0.1 The maximal allowable delays for exponential stability of the system (18) by [28], [29], [34] and Corollary 2 are listed in Table II. It shows that the proposed result in this paper is less conservative.

16

TABLE II M AXIMAL ADMISSIBLE DELAYS BY DIFFERENT METHODS .

Method

[29]

[28]

[34]

Corollary 2

dmax

0.1550

0.0053

0.1882

0.3058

TABLE III M AXIMAL ADMISSIBLE DELAYS BY DIFFERENT METHODS .

c

[29]

[28]

[34]

Corollary 2

2.0 1.8 1.6 1.4 1.2 1.0

0.1435 0.1459 0.1479 0.1493 0.1497 0.1491

-

0.1858 0.1966 0.2089 0.2219 0.2392 0.2586

0.3351 0.3523 0.3701 0.3871 0.3989 0.3915

Example 2. Consider system (18) with ⎡ ⎡ ⎤ ⎡ ⎤ 0 ⎥ 0 ⎥ ⎢−0.3 ⎢−c ⎢−2 D=⎣ ⎦,A = ⎣ ⎦ , A1 = ⎣ 0 −0.3 0 0 −1.9 ⎡

⎤

⎡

⎤ 0⎥ ⎦, −c

⎤

⎢ 2 0⎥ ⎢0.2 0 ⎥ G=⎣ ⎦ , τ = d. ⎦ , G1 = ⎣ 0 2 0 0.2 This system was studied in [28], [29], [34]. The comparisons of the maximum allowed delay d for various parameter c are listed in Table III. Also, consider the system in Example 2 in [34], the corresponding upper bounds of the time delay d by [28], [29], [34] are 0.1550, 0.0033, 0.1882, respectively. When our Corollary 2 is applied to this case, it has been found the corresponding upper bound of delay d is 0.3059. Then the reduction of the conservatism of our method is quite significant when compared with [28], [29], [34]. Example 3. Consider the neutral stochastic delay systems (1) with ⎡ ⎡ ⎡ ⎤ ⎤ ⎤ ⎡ ⎡ ⎤ ⎤ 0.1 0 −0.2 0 −0.5 0 −0.9 0 ⎢ ⎢ ⎢ ⎥ ⎥ ⎥ ⎢ ⎢0.1⎥ ⎥ D=⎣ ⎦,A = ⎣ ⎦ , A2 = ⎣ ⎦B = ⎣ ⎦, ⎦ , A1 = ⎣ 0 0.1 0.2 −0.2 0 0.5 0 −2 0.2 ⎡ ⎡ ⎤ ⎤ ⎡ ⎤ ⎤ 0.2 0 0.1 0.5 0 −0.2 0 ⎢ ⎢ ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ G=⎣ ⎦ , B1 = ⎣ ⎦,C = ⎣ ⎦ , C1 = C2 = 0, ⎦ , G1 = G2 = ⎣ 0 0.2 −0.1 0 0.3 0 −0.2 ⎡

⎡

⎡ ⎡ ⎤ ⎤ ⎡ ⎤ ⎢ 0.2 ⎥ ⎢0.1 0 ⎥ ⎢ 0.2 ⎥ ⎢0.1 0 ⎥ B2 = ⎣ ⎦ , H = H 1 = H2 = ⎣ ⎦ , B3 = ⎣ ⎦,L = ⎣ ⎦ , τ = 0.5, d = 0.3. −0.1 0 0.1 −0.1 0 0.1 ⎤

From Remark 8, by solving (33), we can obtain the minimum L 2 -L∞ disturbance attenuation level is γ min = 0.024. In

17

0.01 z1 z2 zf1 zf2

0.005

0

−0.005

−0.01

−0.015

0

1

2

3

4

5

time(sec)

Fig. 1. The trajectories of signal to be estimated z(t) and its estimation signal z f (t) special, taking γ = 0.2, the corresponding solutions of the determined matrices can be given as follows ⎡ ⎤ ⎡ ⎤ ⎢ 4.7209 −0.0490⎥ ⎢ 2.7916 −0.3036⎥ X =⎣ ⎦ , X1 = ⎣ ⎦, −0.0490 2.3546 −0.3036 1.3769 ⎡

⎡ ⎡ ⎤ ⎤ ⎤ ⎢−0.4238 1.4454⎥ ⎢ 3.2361 −0.1507⎥ ⎢0.0407 0.0060⎥ V1 = ⎣ ⎦ , V2 = ⎣ ⎦ , V3 = ⎣ ⎦. 0.6264 1.5618 −0.6103 2.8603 0.0123 0.0404 Do LU decomposition to the block matrix (I − XX 1−1 ), we can obtain the matrices M and N : ⎡ ⎤ ⎡ ⎤ 0 ⎢−0.8807 1.0000⎥ ⎢ 0.8274 ⎥ M =⎣ ⎦,N = ⎣ ⎦. 1.0000 0 −0.7481 −0.0045 Then, from (31), the desired full-order L 2 -L∞ filter gain matrices can be obtained ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ 0.0091 −458.9463 0.6264 1.5618 0.0186 −4.7764 ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ Af = ⎣ ⎦ , Bf = ⎣ ⎦ , Cf = ⎣ ⎦. 1.4287 −671.0113 0.1279 2.8209 0.0094 −8.3733

(34)

The initial condition and the exogenous disturbance input are given as φ(t) = −0.1 cos t

T 0.1 cos t

, t ∈ [−0.5, 0], v(t) = e−2t sin t.

Based on the above initial condition φ(t), we choose the initial state of the system (3) be ξ(0) = [−0.1, 0.1, 0, 0] T . With our designed filter (34), Figs. 1 and 2 show the curves of signal to be estimated z(t) and its estimation signal z f (t), and the curve of filtering error e(t), respectively. It follows from Fig. 3 that the maximum value of e T (t)e(t) is 2.05 × 10−4 . Meanwhile, from v(t) = e −2t sin t, we have v(t)22

 =

0

∞

 T

v (t)v(t)dt =

0

∞

e−4t sin2 tdt = 0.025 = 2.5 × 10−2 .

18

0.015 e1 e2 0.01

0.005

0

−0.005

−0.01

−0.015

0

1

2

3

4

5

time(sec)

Fig. 2. The trajectory of estimation error e(t) −4

2.5

x 10

|e(t)|2

2

1.5

1

0.5

0

0

1

2

3

4

5

time(sec)

Fig. 3. The trajectory of e T (t)e(t) Hence, 2.05 × 10−4 Ee(t)2∞ = = 0.0082 ≈ 0.092 , 2 v2 2.5 × 10−2 and E  e(t) ∞ < 0.09  v(t) 2 < 0.2  v(t) 2 . Then, the L2 -L∞ disturbance attenuation level γ = 0.2 is immediate. V. C ONCLUSIONS L2 -L∞ filtering for neutral stochastic systems with mixed delays has been investigated in this paper. A modified Lyapunovkrasovskii functional is established and an improved mean-square exponential stability criterion is derived without using model transformation method, cross-term technique and free-weighting matrix approach. Thus the method leads to a simple criterion and shows less conservatism. Furthermore, the L 2 -L∞ performance condition is obtained, different from most of the existing

19

results, this proposed performance condition is with the help of two slack matrices. And then a stochastic exponential L 2 -L∞ filter has been designed by solving three strict linear matrix inequalities. Some numerical examples including comparisons with some known results have been provided to show the validity and improvement of our proposed method. Note that the considered time delays in this paper are all time-invariant. Theorem 1 is not valid for neutral stochastic systems with timevarying delays, because the formulas (11), (12) and (27) may not be satisfied in this case. The problem of L 2 -L∞ filtering for neutral stochastic systems with mixed time-varying delays is the part of our future work. On the other hand, in order to achieve more industrial oriented results, we will consider the relate topics within, especially reliable control framework in the current and future work, and some future conclusions can be given.

Acknowledgments This work is supported by the National Natural Science Foundation of China (61203143,61374039,61374040), the Hujiang Foundation of China (C14002), Key Discipline of Shanghai (S30501), Scientific Innovation program(13ZZ115), Shanghai Natural Science Foundation of China under Grant 13ZR1428500. R EFERENCES [1] L. Dugard and E. I. Verriest, “Stability and Control of Time-Delay Systems,” Berlin: Springer-Verlag, 1998. [2] Y. He, M. Wu, J. She, and G. Liu, “Delay-dependent robust stability criteria for uncertain neutral systems with mixed delays,” Systems and Control Letters, vol. 51, no. 1, pp. 57–65, 2004. [3] S. I. Niculescu, “Delay Effects on Stability: A Robust Control Approach,” Heidelberg: Springer-Verlag, 2001. [4] V. Suplin, E. Fridman, and U. Shaked, “H∞ control of linear uncertain time-delay systems-a projection approach,” IEEE Transactions on Automatic Control, vol. 51, no. 4, pp. 680–685, 2006. [5] H. R. Karimi, “Robust delay-dependent H∞ control of uncertain time-delay systems with mixed neutral, discrete, and distributed time-delays and Markovian switching parameters,” IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 58, no. 8, pp. 1910–1923, 2011. [6] H. Shen, S. Xu, J. Zhou, and J. Lu, “Fuzzy H∞ filtering for nonlinear Markovian jump neutral systems,” International Journal of Systems and Science, vol. 42, no. 5, pp. 767–780, 2011. [7] S. Wen, Z. Zeng, and T. Huang, “H∞ filtering for neutral systems with mixed delays and multiplicative noises,” IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 59, no. 11, pp. 820–824, 2012. [8] P.-A. Bliman, “Lyapunov equation for the stability of linear delay systems of retarded and neutral type,” IEEE Transactions on Automatic Control, vol. 47, no. 2, pp. 327–335, 2002. [9] Q. Zhu and J. Cao, “Stability analysis for stochastic neural networks of neutral type with both Markovian jump parameters and mixed time delays,” Neurocomputing, vol. 73, no. 13-15, pp. 2671–2680, 2010. [10] O. M. Kwon, M. J. Park, J. H. Park, S. M. Lee, and E. J. Cha, “New delay-partitioning approaches to stability criteria for uncertain neutral systems with time-varying delays,” Journal of the Franklin Institute, vol. 349, no. 9, pp. 2799–2823, 2012. [11] Y. F. Song and Y. Shen, “New criteria on asymptotic behavior of neutral stochastic functional differential equations,” Automatica, vol. 49, no. 2, pp. 626–632, 2013. [12] M. Hua, H. Tan, J. Chen, and J. Fei, “Robust delay-range-dependentnon-fragile H∞ filtering for uncertain neutral stochastic systems with markovian switching and mode-dependent time delays,” Journal of the Franklin Institute, vol. 352, no. 3, pp. 1318–1341, 2015. [13] Y. Kao, J. Xie, C. Wang, and H. R. Karimi, “A sliding mode approach to H∞ non-fragile observer-based control design for uncertain Markovian neutral-type stochastic systems,” Automatica, vol. 52, no. 2, pp. 218–226, 2015. [14] L. Huang and X. Mao, “SMC design for robust H∞ control of uncertain stochastic delay systems,” Automatica, vol. 46, no. 2, pp. 405–412, 2010. [15] Z. Wang, Y. Liu, G. Wei, and X. Liu, “A note on control of a class of discrete-time stochastic systems with distributed delays and nonlinear disturbances,” Automatica, vol. 46, no. 3, pp. 543–548, 2010.

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