design of filter for a class of switched linear neutral systems

This paper is concerned with the?∞ filtering problem for a class of switched linear neutral systems with time-varying delays.. First, we address the delay-dependent ?∞ filtering prob-lem

Mathematical Problems in Engineering

Volume 2013, Article ID 537249, 9 pages

http://dx.doi.org/10.1155/2013/537249

Research Article

Linear Neutral Systems

Caiyun Wu1and Yue-E Wang2

1 School of Equipment Engineering, Shenyang Ligong University, Shenyang 110159, China

2 College of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710062, China

Correspondence should be addressed to Caiyun Wu; wu cai yun@126.com

Received 15 July 2013; Revised 15 September 2013; Accepted 15 September 2013

Academic Editor: Hossein Jafari

Copyright © 2013 C Wu and Yue-E Wang This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

This paper is concerned with the𝐻∞ filtering problem for a class of switched linear neutral systems with time-varying delays The time-varying delays appear not only in the state but also in the state derivatives Based on the average dwell time approach and the piecewise Lyapunov functional technique, sufficient conditions are proposed for the exponential stability of the filtering error dynamic system Then, the corresponding solvability condition for a desired filter satisfying a weighted𝐻∞performance is established All the conditions obtained are delay-dependent Finally, two numerical examples are given to illustrate the effectiveness

of the proposed theory

1 Introduction

Switched time-delay systems have been attracting

consid-erable attention during the recent years [1–9], due to the

significance both in theory development and practical

appli-cations However, it is worth noting that only the state

time delay is considered, and the time delay in the state

derivatives is largely ignored in the existing literature If

each subsystem of a switched system has time delay in

the state derivatives, then the switched system is called

switched neutral system [10] Switched neutral systems exist

widely in engineering and social systems, many physical

plants can be modeled as switched neutral systems, such as

distributed networks and heat exchanges For example, in

[11], a switched neutral type delay equation with nonlinear

perturbations was exploited to model the drilling system

Compared with the switched systems with state time delay,

switched neutral systems are much more complicated [12–15]

As effective tools, the common Lyapunov function method,

dwell time approaches, and average dwell time approaches

have been extended to study the switched neutral systems,

and many valuable results have been obtained for switched

neutral systems On the research of stability analysis for

switched neutral systems, the asymptotically stable problem

of switched neutral systems was considered in [16] If there exists a Hurwitz linear convex combination of state matrices, and subsystems are not necessarily stable, switching rules can

be designed to guarantee the asymptotical stability of the switched neutral system The method of Lyapunov-Metzler linear matrix inequalities in [17] was extended to switched neutral systems [18], and some less conservative stability results were obtained

In contrast with the traditional Kalman filtering, the

𝐻∞ filtering does not require the exact knowledge of the statistics of the external noise signals, and it is insensitive

to the uncertainties both in the exogenous signal statistics and in dynamic models [19,20] Because of these advantages, the 𝐻∞ filtering has attracted much attention in the past decade for nonswitched neutral systems [21–24] In [22], some sufficient conditions for the existence of an𝐻∞filter

of a Luenberger observer type have been provided However,

to the authors’ best knowledge, the𝐻∞filtering for switched neutral systems has been rarely investigated and still remains challenging This motivates our research

The contribution of this paper lies in three aspects First, we address the delay-dependent 𝐻∞ filtering prob-lem for switched linear neutral systems with time-varying delays, which appear not only in the state, but also in the

state derivatives The resulting filter is of the

Luenberger-observer type Second, by using average dwell time approach

and the piecewise Lyapunov function technique, we derive

a delay-dependent sufficient condition, which guarantees

exponential stability of the filtering error system Then,

the corresponding solvability condition for a desired filter

satisfying a weighted𝐻∞ performance is established Here,

to reduce the conservatism of the delay-dependent condition,

we introduce some slack matrix variables and a new integral

inequality recently proposed in [25] Finally, we succeed in

transforming the filter design problem into the feasibility

problem of some linear matrix inequalities To show the

efficiency of the obtained results, we present two relevant

examples

The remainder of this paper is organized as follows

The 𝐻∞ filtering problem for switched neutral systems is

formulated inSection 2.Section 3presents our main results

Numerical examples are given inSection 4, and we conclude

this paper inSection 5

Notation Throughout this paper,𝑅𝑛 denotes𝑛-dimensional

Euclidean space;𝑅𝑛×𝑚 is the set of all𝑛 × 𝑚 real matrices;

𝑃 > 0 means that 𝑃 is positive definite; 𝐿2denotes the space

of square integrable vector functions on[0, ∞) with norm

‖ ⋅ ‖ = (∫0∞‖ ⋅ ‖2𝑑𝑡)1/2, where‖⋅‖ denotes the Euclidean vector

norm;𝐼 is the identity matrix with appropriate dimensions;

the symmetric terms in a symmetric matrix are denoted by∗

as for example

[𝑋 𝑌∗ 𝑍] = [𝑌𝑋 𝑌𝑇 𝑍] (1)

2 Problem Statement

Consider the following switched linear neutral system:

̇𝑥 (𝑡) = 𝐴0𝜎(𝑡)𝑥 (𝑡) + 𝐴1𝜎(𝑡)𝑥 (𝑡 − ℎ (𝑡))

+ 𝐹𝜎(𝑡) ̇𝑥 (𝑡 − 𝜏 (𝑡)) + 𝐵𝜎(𝑡)𝜔 (𝑡) ,

𝑦 (𝑡) = 𝐶0𝜎(𝑡)𝑥 (𝑡) + 𝐶1𝜎(𝑡)𝑥 (𝑡 − ℎ (𝑡)) + 𝐷𝜎(𝑡)𝜔 (𝑡) ,

𝑧 (𝑡) = 𝐿𝜎(𝑡)𝑥 (𝑡) ,

𝑥 (𝜃) = 𝜓 (𝜃) , ∀𝜃 ∈ [−𝐻, 0] , 𝐻 = max {ℎ, 𝜏} ,

(2)

where 𝑥(𝑡) ∈ 𝑅𝑛 is the state vector; 𝑦(𝑡) ∈ 𝑅𝑚 is

the measurements vector; 𝜔(𝑡) ∈ 𝑅𝑝 is the noise signal

vector, which belongs to𝐿2[0, ∞); 𝑧(𝑡) ∈ 𝑅𝑞 is the signal

to be estimated; 𝜓(𝑡) is the initial vector function that is

continuously differentiable on[−𝐻, 0]; 𝜎(𝑡) : [0, ∞) → 𝑀 =

{1, 2, , 𝑚} is a piecewise constant function of time 𝑡 called

switching signal Corresponding to the switching signal𝜎(𝑡),

we have the switching sequence{𝑥𝑡0: (𝑖0, 𝑡0), , (𝑖𝑘, 𝑡𝑘), , |

𝑖𝑘 ∈ 𝑀, 𝑘 = 0, 1, }, which means that the 𝑖𝑘th subsystem

is active when𝑡 ∈ [𝑡𝑘, 𝑡𝑘+1) The system coefficient matrices

𝐴0𝑖, 𝐴1𝑖, 𝐹𝑖 , 𝐵𝑖, 𝐶0𝑖 , 𝐶1𝑖, 𝐷𝑖, and 𝐿𝑖 are known real

constant matrices of appropriate dimensions.ℎ(𝑡) and 𝜏(𝑡) are time-varying delays satisfying

0 ≤ ℎ (𝑡) ≤ ℎ, ̇ℎ (𝑡) ≤ ℎ < 1,

0 ≤ 𝜏 (𝑡) ≤ 𝜏, ̇𝜏 (𝑡) ≤ 𝜏 < 1 (3) The objective of this paper is to design a family of filters

of Luenberger observer type as follows:

̇̂𝑥 (𝑡) = 𝐴0𝜎(𝑡)̂𝑥 (𝑡) + 𝐴1𝜎(𝑡)̂𝑥 (𝑡 − ℎ (𝑡)) + 𝐹𝜎(𝑡) ̇̂𝑥 (𝑡 − 𝜏 (𝑡)) + 𝐾𝜎(𝑡)[𝑦 (𝑡) − 𝐶0𝜎(𝑡)̂𝑥 (𝑡) − 𝐶1𝜎(𝑡)̂𝑥 (𝑡 − ℎ (𝑡))] ,

̂𝑧 (𝑡) = 𝐿𝜎(𝑡)̂𝑥 (𝑡) ,

̂𝑥 (𝜃) = ̂𝜓 (𝜃) , ∀𝜃 ∈ [−𝐻, 0] , 𝐻 = max {ℎ, 𝜏} ,

(4) where 𝐾𝑖𝑘 are the filter parameters, which are to be deter-mined

Now, we introduce the estimation errors:𝑥𝑒(𝑡) = 𝑥(𝑡) −

̂𝑥(𝑡), 𝑧𝑒(𝑡) = 𝑧(𝑡) − ̂𝑧(𝑡)

Combining (2) with (4) gives the following filtering error dynamic system:

𝑒(𝑡) = ̃𝐴0𝜎(𝑡)𝑥𝑒(𝑡) + ̃𝐴1𝜎(𝑡)𝑥𝑒(𝑡 − ℎ (𝑡)) + 𝐹𝜎(𝑡) ̇𝑥𝑒(𝑡 − 𝜏 (𝑡)) + ̃𝐵𝜎(𝑡)𝜔 (𝑡) ,

𝑧𝑒(𝑡) = 𝐿𝜎(𝑡)𝑥𝑒(𝑡) ,

𝑥𝑒(𝜃) = 𝜓 (𝜃) − ̂𝜓 (𝜃) , ∀𝜃 ∈ [−𝐻, 0] , 𝐻 = max {ℎ, 𝜏} ,

(5)

where ̃𝐴0𝜎 = 𝐴0𝜎− 𝐾𝜎𝐶0𝜎, ̃𝐴1𝜎 = 𝐴1𝜎− 𝐾𝜎𝐶1𝜎, ̃𝐵𝜎 = 𝐵𝜎−

𝐾𝜎𝐷𝜎 The following definitions are introduced, which will play key roles in deriving our main results

Definition 1 (see [26]) The equilibrium𝑥∗

𝑒 = 0 of the filtering error system (5) is said to be exponentially stable under𝜎(𝑡)

if the solution𝑥𝑒(𝑡) of system (5) with 𝜔(𝑡) = 0 satisfies

‖𝑥𝑒(𝑡)‖ ≤ Γ𝑒−𝜆(𝑡−𝑡0 )‖𝑥𝑒(𝑡0)‖𝐻, for all 𝑡 ≥ 𝑡0 for constants

Γ > 0 and 𝜆 > 0, where ‖ ⋅ ‖ denotes the Euclidean norm, and‖𝑥𝑒(𝑡)‖𝐻= sup−𝐻≤𝜃≤0{𝑥𝑒(𝑡 + 𝜃), ̇𝑥𝑒(𝑡 + 𝜃)}

Definition 2 (see [26]) For any𝑇2 > 𝑇1 ≥ 0, let 𝑁𝜎(𝑇1, 𝑇2) denote the number of switching of 𝜎(𝑡) over (𝑇1, 𝑇2) If

𝑁𝜎(𝑇1, 𝑇2) ≤ 𝑁0+ (𝑇2− 𝑇1)/𝑇𝑎holds for𝑇𝑎 > 0, 𝑁0 ≥ 0, then𝑇𝑎 is called average dwell time As commonly used in the literature, we choose𝑁0= 0

The filtering problem addressed in this paper is to seek for suitable filter gain𝐾𝑖such that the filtering error system (5) for any switching signal with average dwell time has a prescribed𝐻∞performance𝛾; that is,

(1) the error system (5) with𝜔(𝑡) = 0 is exponentially stable;

(2) under the zero initial conditions, that is,𝑥𝑒(𝜃) = 0,

for all𝜃 ∈ [−𝐻, 0], the weighted 𝐻∞performance

∫0∞𝑒−𝛼𝑠𝑧𝑇

𝑒(𝑠)𝑧𝑒(𝑠)𝑑𝑠 ≤ 𝛾2∫0∞𝜔𝑇(𝑠)𝜔(𝑠)𝑑𝑠 is

guaran-teed for all nonzero𝜔(𝑡) ∈ 𝐿2[0, ∞) and a prescribed

level of noise attenuation𝛾 > 0

Before concluding this section, we introduce three

lem-mas which are essential for the development of the results

function with first-order continuous-derivative entries Then,

the following integral inequality holds for any matrices 𝑀1,

𝑀2∈ 𝑅𝑛×𝑛, and𝑋 = 𝑋𝑇> 0, and a scalar ℎ ≥ 0,

− ∫𝑡

𝑡−ℎ ̇𝑥𝑇(𝑠) 𝑋 ̇𝑥 (𝑠) 𝑑𝑠

≤ 𝜉𝑇(𝑡) [

[

𝑀𝑇

1 + 𝑀1 −𝑀𝑇

1 + 𝑀2

∗ −𝑀𝑇

2 − 𝑀2] ]

𝜉 (𝑡)

+ ℎ𝜉𝑇(𝑡) [

[

𝑀𝑇 1

𝑀𝑇 2

] ]

𝑋−1[𝑀1 𝑀2] 𝜉 (𝑡) ,

(6)

where𝜉𝑇(𝑡) = [𝑥𝑇(𝑡) 𝑥𝑇(𝑡 − ℎ)].

𝑅𝑇 ∈ 𝑅𝑛×𝑛, scalar 𝑟 > 0, vector function 𝜔 : [0, 𝑟] → 𝑅𝑛

such that the integrations concerned are well defined; then,

(∫0𝑟𝜔(𝑠)𝑑𝑠)𝑇𝑅(∫0𝑟𝜔(𝑠)𝑑𝑠) ≤ 𝑟 ∫0𝑟𝜔𝑇(𝑠)𝑅𝜔(𝑠)𝑑𝑠.

∗ 𝑆 22] < 0,

where𝑆11= 𝑆𝑇

11and𝑆22= 𝑆𝑇

22, the following is equivalent:

(1) 𝑆11< 0, 𝑆22− 𝑆𝑇12𝑆−111𝑆12< 0;

(2) 𝑆22< 0, 𝑆11− 𝑆12𝑆−122𝑆𝑇12< 0 (7)

3 Main Results

In this section, we first present a sufficient condition for exponential stability of the filtering error system (5) with 𝜔(𝑡) = 0 Then, it is applied to formulate an approach to design the desired𝐻∞filters for switched neutral system (2)

3.1 Stability Analysis

exist matrices𝑃𝑖𝑘 > 0, 𝑄𝑖𝑘 > 0, 𝑅𝑖𝑘 > 0, 𝑀𝑖𝑘 > 0, 𝑁𝑖𝑘 > 0, and

𝑇1𝑖𝑘,𝑇2𝑖𝑘,𝑁𝑔𝑖𝑘( 𝑔 = 1, 2, , 7) of appropriate dimensions, and

𝜇 ≥ 1, such that for all 𝑖𝑘 ∈ 𝑀,

Σ𝑖𝑘=

[ [ [ [ [ [ [ [ [ [ [ [

Σ11 Σ12 Σ13 Σ14 Σ15 𝐴̃𝑇

0𝑖𝑘𝑁6𝑖𝑘 Σ17

∗ Σ22 Σ23 −𝑁4𝑖𝑘 Σ25 −𝑁6𝑖𝑘 −𝑁7𝑖𝑘

∗ ∗ Σ33 𝐴̃𝑇1𝑖𝑘𝑁4𝑖𝑘 Σ35 𝐴̃𝑇1𝑖𝑘𝑁6𝑖𝑘 𝐴̃𝑇1𝑖𝑘𝑁7𝑖𝑘

∗ ∗ ∗ Σ44 𝑁𝑇

4𝑖 𝑘𝐹𝑖𝑘 0 𝜏𝑇𝑇

2𝑖 𝑘

∗ ∗ ∗ ∗ Σ55 𝐹𝑇

𝑖 𝑘𝑁6𝑖𝑘 𝐹𝑇

𝑖 𝑘𝑁7𝑖𝑘

∗ ∗ ∗ ∗ ∗ −1

ℎ𝑒−𝛼ℎ𝑀𝑖𝑘 0

∗ ∗ ∗ ∗ ∗ ∗ −𝜏𝑒𝛼𝜏𝑁𝑖𝑘

] ] ] ] ] ] ] ] ] ] ] ]

< 0, (8)

𝑃𝑖𝑘≤ 𝜇𝑃𝑖𝑗, 𝑄𝑖𝑘 ≤ 𝜇𝑄𝑖𝑗, 𝑅𝑖𝑘 ≤ 𝜇𝑅𝑖𝑗, 𝑀𝑖𝑘≤ 𝜇𝑀𝑖𝑗, 𝑁𝑖𝑘≤ 𝜇𝑁𝑖𝑗, ∀𝑖𝑘, 𝑖𝑗∈ 𝑀, (9)

where

Σ11= ℎ𝑀𝑖𝑘+ 𝛼𝑃𝑖𝑘+ 𝑄𝑖𝑘+ 𝑒−𝛼𝜏𝑇𝑇

1𝑖 𝑘+ 𝑒−𝛼𝜏𝑇1𝑖𝑘 + 𝑁1𝑖𝑇𝑘𝐴̃0𝑖𝑘+ ̃𝐴𝑇0𝑖𝑘𝑁1𝑖𝑘,

Σ12= 𝑃𝑖𝑘− 𝑁1𝑖𝑇𝑘+ ̃𝐴𝑇0𝑖𝑘𝑁2𝑖𝑘,

Σ13= 𝑁1𝑖𝑇𝑘𝐴̃1𝑖𝑘+ ̃𝐴𝑇0𝑖𝑘𝑁3𝑖𝑘,

Σ14= −𝑒−𝛼𝜏𝑇1𝑖𝑇𝑘+ 𝑒−𝛼𝜏𝑇2𝑖𝑘+ ̃𝐴𝑇0𝑖𝑘𝑁4𝑖𝑘,

Σ15= 𝑁1𝑖𝑇 𝐹𝑖𝑘+ ̃𝐴𝑇0𝑖𝑁5𝑖𝑘,

Σ17= 𝜏𝑇1𝑖𝑇𝑘+ ̃𝐴𝑇0𝑖𝑘𝑁7𝑖𝑘,

Σ22= 𝑅𝑖𝑘+ 𝜏𝑁𝑖𝑘− 𝑁2𝑖𝑇𝑘− 𝑁2𝑖𝑘,

Σ23 = −𝑁3𝑖𝑘+ 𝑁2𝑖𝑇𝑘𝐴̃1𝑖𝑘,

Σ25= −𝑁5𝑖𝑘+ 𝑁2𝑖𝑇𝑘𝐹𝑖𝑘,

Σ33= − (1 − ℎ) 𝑒−𝛼ℎ𝑄𝑖𝑘+ 𝑁3𝑖𝑇𝑘𝐴̃1𝑖𝑘+ ̃𝐴𝑇1𝑖𝑘𝑁3𝑖𝑘,

Σ35= 𝑁3𝑖𝑇𝑘𝐹𝑖𝑘+ ̃𝐴𝑇1𝑖𝑘𝑁5𝑖𝑘,

Σ44= −𝑒−𝛼𝜏𝑇2𝑖𝑇𝑘− 𝑒−𝛼𝜏𝑇2𝑖𝑘,

Σ55= − (1 − 𝜏) 𝑒−𝛼𝜏𝑅𝑖𝑘+ 𝑁5𝑖𝑇𝑘𝐹𝑖𝑘+ 𝐹𝑖𝑇𝑘𝑁5𝑖𝑘,

(10)

then the error dynamic system (5) with 𝜔(𝑡) = 0 is

exponen-tially stable for any switching signal with average dwell time

satisfying𝑇𝑎 > 𝑇∗

𝑎 = ln 𝜇/𝛼.

Proof Define the piecewise Lyapunov-Krasovskii functional

candidate

𝑉 (𝑡) = 𝑉𝜎(𝑡)(𝑡) =∑5

𝑗=1

𝑉𝑗𝜎(𝑡)(𝑡) , (11)

where

𝑉1𝑖𝑘(𝑡) = 𝑥𝑇𝑒 (𝑡) 𝑃𝑖𝑘𝑥𝑒(𝑡) ,

𝑉2𝑖𝑘(𝑡) = ∫𝑡

𝑡−ℎ(𝑡)𝑥𝑇𝑒 (𝑠) 𝑒𝛼(𝑠−𝑡)𝑄𝑖𝑘𝑥𝑒(𝑠) 𝑑𝑠,

𝑉3𝑖𝑘(𝑡) = ∫𝑡

𝑡−𝜏(𝑡) ̇𝑥𝑇𝑒 (𝑠) 𝑒𝛼(𝑠−𝑡)𝑅𝑖𝑘 ̇𝑥𝑒(𝑠) 𝑑𝑠,

𝑉4𝑖𝑘(𝑡) = ∫0

−ℎ∫𝑡

𝑡+𝜃𝑥𝑇𝑒 (𝑠) 𝑒𝛼(𝑠−𝑡)𝑀𝑖𝑘𝑥𝑒(𝑠) 𝑑𝑠 𝑑𝜃,

𝑉5𝑖𝑘(𝑡) = ∫0

−𝜏∫𝑡

𝑡+𝜃 ̇𝑥𝑇

𝑒 (𝑠) 𝑒𝛼(𝑠−𝑡)𝑁𝑖𝑘 ̇𝑥𝑒(𝑠) 𝑑𝑠 𝑑𝜃

(12)

Now, taking the derivative of𝑉𝑗𝑖𝑘(𝑡), 𝑗 = 1, 2, , 5 with

respect to𝑡 along the trajectory of the error system (5) with

𝜔(𝑡) = 0, according to (3) andLemma 4, we have

̇𝑉

𝑖 𝑘(𝑡) + 𝛼𝑉𝑖𝑘(𝑡)

≤ 2𝑥𝑇𝑒 (𝑡) 𝑃𝑖𝑘 ̇𝑥𝑒(𝑡) + 𝑥𝑇𝑒 (𝑡) 𝑄𝑖𝑘𝑥𝑒(𝑡) + ̇𝑥𝑇𝑒 (𝑡) 𝑅𝑖𝑘 ̇𝑥𝑒(𝑡)

− (1 − ℎ) 𝑥𝑇𝑒 (𝑡 − ℎ (𝑡)) 𝑒−𝛼ℎ𝑄𝑖𝑘𝑥𝑒(𝑡 − ℎ (𝑡))

+ ℎ𝑥𝑇𝑒 (𝑡) 𝑀𝑖𝑘𝑥𝑒(𝑡) + 𝜏 ̇𝑥𝑇𝑒 (𝑡) 𝑁𝑖𝑘 ̇𝑥𝑒(𝑡) + 𝛼𝑥𝑇𝑒 (𝑡) 𝑃𝑖𝑘𝑥𝑒(𝑡)

− (1 − 𝜏) ̇𝑥𝑇𝑒 (𝑡 − 𝜏 (𝑡)) 𝑒−𝛼𝜏𝑅𝑖𝑘 ̇𝑥𝑒(𝑡 − 𝜏 (𝑡))

−𝑒−𝛼ℎℎ ∫𝑡

𝑡−ℎ𝑥𝑇𝑒 (𝑠) 𝑑𝑠𝑀𝑖𝑘∫𝑡

𝑡−ℎ𝑥𝑒(𝑠) 𝑑𝑠

− ∫𝑡

𝑡−𝜏 ̇𝑥𝑇

𝑒 (𝑠) 𝑒𝛼(𝑠−𝑡)𝑁𝑖𝑘 ̇𝑥𝑒(𝑠) 𝑑𝑠

(13)

FromLemma 3, it holds

− ∫𝑡

𝑡−𝜏 ̇𝑥𝑇𝑒 (𝑠) 𝑒𝛼(𝑠−𝑡)𝑁𝑖𝑘 ̇𝑥𝑒(𝑠) 𝑑𝑠

≤ − ∫𝑡

𝑡−𝜏 ̇𝑥𝑇

𝑒 (𝑠) 𝑒−𝛼𝜏𝑁𝑖𝑘 ̇𝑥𝑒(𝑠) 𝑑𝑠

≤ 𝑒−𝛼𝜏[𝑥𝑇

𝑒 (𝑡) 𝑥𝑇𝑒 (𝑡 − 𝜏)] [

[

𝑇𝑇 1𝑖 𝑘+ 𝑇1𝑖𝑘 −𝑇𝑇

1𝑖 𝑘+ 𝑇2𝑖𝑘

∗ −𝑇𝑇

2𝑖 𝑘− 𝑇2𝑖𝑘] ]

× [ 𝑥𝑒(𝑡)

𝑥𝑒(𝑡 − 𝜏)] + 𝜏𝑒−𝛼𝜏[𝑥𝑇

𝑒 (𝑡) 𝑥𝑇𝑒 (𝑡 − 𝜏)]

× [ [

𝑇𝑇 1𝑖 𝑘

𝑇𝑇 2𝑖 𝑘

] ]

𝑁𝑖−1𝑘 [𝑇1𝑖𝑘 𝑇2𝑖𝑘] [

[

𝑥𝑒(𝑡)

𝑥𝑒(𝑡 − 𝜏)]

]

(14)

Define

𝜉𝑒𝑇(𝑡) = [𝜁𝑇

𝑒 (𝑡) ̇𝑥𝑇𝑒 (𝑡 − 𝜏 (𝑡)) ∫𝑡

𝑡−ℎ𝑥𝑇

𝑒 (𝑠) 𝑑𝑠] , (15)

where𝜁𝑇𝑒(𝑡) = [𝑥𝑇

𝑒(𝑡) ̇𝑥𝑇

𝑒(𝑡) 𝑥𝑇

𝑒(𝑡 − ℎ(𝑡)) 𝑥𝑇

𝑒(𝑡 − 𝜏)]

By some algebraic manipulations, it is easy to show that

̇𝑉

𝑖 𝑘(𝑡) + 𝛼𝑉𝑖𝑘(𝑡) ≤ 𝜉𝑇𝑒 (𝑡) Σ𝑖𝑘𝜉𝑒(𝑡) , (16) where

Σ𝑖𝑘 =

[ [ [ [ [

Σ11 𝑃𝑖𝑘 0 Σ14 0 0

∗ 𝑅𝑖𝑘+ 𝜏𝑁𝑖𝑘 0 0 0 0

∗ ∗ − (1 − ℎ) 𝑒−𝛼ℎ𝑄𝑖𝑘 0 0 0

] ] ] ] ] ,

(17)

where

Σ11= ℎ𝑀𝑖𝑘+ 𝛼𝑃𝑖𝑘+ 𝑄𝑖𝑘+ 𝑒−𝛼𝜏𝑇1𝑖𝑇𝑘+ 𝑒−𝛼𝜏𝑇1𝑖𝑘 + 𝜏𝑒−𝛼𝜏𝑇1𝑖𝑇𝑘𝑁𝑖−1𝑘 𝑇1𝑖𝑘,

Σ14= −𝑒−𝛼𝜏𝑇𝑇

1𝑖 𝑘+ 𝑒−𝛼𝜏𝑇2𝑖𝑘+ 𝜏𝑒−𝛼𝜏𝑇𝑇

1𝑖 𝑘𝑁−1

𝑖 𝑘 𝑇2𝑖𝑘,

Σ55= − (1 − 𝜏) 𝑒−𝛼𝜏𝑅𝑖𝑘,

Σ44= −𝑒−𝛼𝜏𝑇2𝑖𝑇𝑘− 𝑒−𝛼𝜏𝑇2𝑖𝑘+ 𝜏𝑒−𝛼𝜏𝑇2𝑖𝑇𝑘𝑁𝑖−1𝑘 𝑇2𝑖𝑘,

Σ66= −1ℎ𝑒−𝛼ℎ𝑀𝑖𝑘

(18)

0 500 1000 1500 2000 2500 3000

−0.05

0

Time

xe

0.3

0.25

0.2

0.15

0.1

0.05

Figure 1: The state responses of the filtering error dynamic system

with𝜔(𝑡) = 0

FromLemma 5,Σ𝑖𝑘< 0 is equivalent to

̃Σ𝑖𝑘 =

[

[

[

[

[

[

̃Σ11 𝑃𝑖𝑘 0 ̃Σ14 0 0 𝜏𝑇𝑇

1𝑖 𝑘

∗ 𝑅𝑖𝑘+ 𝜏𝑁𝑖𝑘 0 0 0 0 0

∗ ∗ ̃Σ33 0 0 0 0

∗ ∗ ∗ ̃Σ44 0 0 𝜏𝑇𝑇

2𝑖 𝑘

∗ ∗ ∗ ∗ ̃Σ55 0 0

∗ ∗ ∗ ∗ ∗ ̃Σ66 0

∗ ∗ ∗ ∗ ∗ ∗ −𝜏𝑒𝛼𝜏𝑁𝑖𝑘

] ] ] ] ] ]

< 0,

(19)

where

̃Σ11= ℎ𝑀𝑖𝑘+ 𝛼𝑃𝑖𝑘+ 𝑄𝑖𝑘+ 𝑒−𝛼𝜏𝑇1𝑖𝑇𝑘+ 𝑒−𝛼𝜏𝑇1𝑖𝑘,

̃Σ14= −𝑒−𝛼𝜏𝑇1𝑖𝑇𝑘+ 𝑒−𝛼𝜏𝑇2𝑖𝑘, ̃Σ33= − (1 − ℎ) 𝑒−𝛼ℎ𝑄𝑖𝑘,

̃Σ44= −𝑒−𝛼𝜏𝑇2𝑖𝑇𝑘− 𝑒−𝛼𝜏𝑇2𝑖𝑘, ̃Σ55= Σ55, ̃Σ66= Σ66

(20)

In addition, the following is true from (5) with𝜔(𝑡) = 0:

2𝜉𝑇𝑒(𝑡)[𝑁1𝑖𝑘 𝑁2𝑖𝑘 𝑁3𝑖𝑘 𝑁4𝑖𝑘 𝑁5𝑖𝑘 𝑁6𝑖𝑘 𝑁7𝑖𝑘]𝑇

× [ ̃𝐴0𝑖𝑘 −𝐼 ̃𝐴1𝑖𝑘 0 𝐹𝑖𝑘 0 0] 𝜉𝑒(𝑡) = 0, (21)

where𝜉𝑇𝑒(𝑡) = [𝜉𝑇

𝑒(𝑡) 0]

Then, (19) along with (21) givesΣ𝑖𝑘 < 0, which yields Σ𝑖𝑘<

0; thus,

̇𝑉

𝑖 𝑘(𝑡) + 𝛼𝑉𝑖𝑘(𝑡) ≤ 0 (22)

Combining (9) with (22), for any𝑡 ∈ [𝑡𝑘, 𝑡𝑘+1), we have

𝑉 (𝑡) = 𝑉𝑖𝑘(𝑡)

≤ 𝑒−𝛼(𝑡−𝑡𝑘 )𝑉𝑖𝑘(𝑡𝑘)

≤ 𝜇𝑒−𝛼(𝑡−𝑡𝑘 )𝑉𝜎(𝑡−

𝑘 )(𝑡−𝑘)

≤ 𝜇𝑒−𝛼(𝑡−𝑡𝑘 )𝑒−𝛼(𝑡𝑘 −𝑡 𝑘−1 )𝑉𝜎(𝑡𝑘−1)(𝑡𝑘−1)

≤ ⋅ ⋅ ⋅

≤ 𝜇𝑘𝑒−𝛼(𝑡−𝑡0 )𝑉 (𝑡0)

≤ 𝑒−(𝛼−(ln 𝜇/𝑇𝑎 ))(𝑡−𝑡 0 )𝑉 (𝑡0)

(23)

According to (11), we have

𝑎󵄩󵄩󵄩󵄩𝑥𝑒(𝑡)󵄩󵄩󵄩󵄩2≤ 𝑉 (𝑡) ≤ 𝑏󵄩󵄩󵄩󵄩𝑥𝑒(𝑡0)󵄩󵄩󵄩󵄩2𝐻, (24) where

𝑎 = min

∀𝑖 𝑘 ∈𝑀{𝜆min(𝑃𝑖𝑘)} ,

𝑏 = max

∀𝑖 𝑘 ∈𝑀{𝜆max(𝑃𝑖𝑘)} + ℎ max

∀𝑖 𝑘 ∈𝑀{𝜆max(𝑄𝑖𝑘)}

+ 𝜏 max

∀𝑖 𝑘 ∈𝑀{𝜆max(𝑅𝑖𝑘)} + ℎ2

2∀𝑖max𝑘 ∈𝑀{𝜆max(𝑀𝑖𝑘)} +𝜏2

2∀𝑖max𝑘 ∈𝑀{𝜆max(𝑁𝑖𝑘)}

(25)

Considering (23) and (24), it holds ‖𝑥𝑒(𝑡)‖ ≤

√(𝑏/𝑎)𝑒−(1/2)(𝛼−(ln 𝜇/𝑇 𝑎 ))(𝑡−𝑡 0 )‖𝑥𝑒(𝑡0)‖𝐻 Therefore, if𝛼 − (ln 𝜇/𝑇𝑎) > 0, that is 𝑇𝑎> (ln 𝜇/𝛼), then error dynamic system (5) is exponentially stable

Remark 7 When 𝜇 = 1, we have 𝑇∗

𝑎 = 0, which means that the switching signal𝜎(𝑡) can be arbitrary In this case, condition (9) implies that there exists a common Lyapunov functional for all subsystems Moreover, setting𝛼 = 0 in (8) gives asymptotic stability of the filtering error system (5) under arbitrary switching

Remark 8 The filters of Luenberger observer type has been

adopted in the literatures, see [17] The Luenberger-type observer can produce an approximation to the system state that is independent of the system trajectory, and it only depends on the initial value of the system state

Remark 9 The condition‖𝐹𝑖𝑘‖ < 1 guarantees that Lipschitz constant for the right hand of (2) with respect to ̇𝑥(𝑡 − 𝜏(𝑡))

is less than one

3.2 Filter Design Now, we design the desired𝐻∞filter for the switched neutral system (2)

if there exists matrices𝑃𝑖𝑘 > 0, 𝑄𝑖𝑘 > 0, 𝑅𝑖𝑘 > 0, 𝑀𝑖𝑘 > 0,

𝑁𝑖𝑘 > 0, and 𝑇1𝑖𝑘,𝑇2𝑖𝑘,𝑊𝑖𝑘,𝑋𝑖𝑘of appropriate dimensions, and

𝜇 ≥ 1, such that, for any 𝑖𝑘∈ 𝑀,

Ω𝑖𝑘 =

[ [ [ [ [ [ [ [ [ [ [ [ [

Ω11 Ω12 Ω13 Ω14 𝑊𝑇

𝑖 𝑘𝐹𝑖𝑘 0 𝜏𝑇𝑇

1𝑖 𝑘 Ω18

∗ Ω22 Ω23 0 𝑊𝑇

𝑖 𝑘𝐹𝑖𝑘 0 0 Ω28

∗ ∗ Ω33 0 𝑊𝑇

𝑖 𝑘𝐹𝑖𝑘 0 0 Ω38

∗ ∗ ∗ Ω44 0 0 𝜏𝑇2𝑖𝑇𝑘 0

∗ ∗ ∗ ∗ Ω55 0 0 0

∗ ∗ ∗ ∗ ∗ Ω66 0 0

∗ ∗ ∗ ∗ ∗ ∗ −𝜏𝑒𝛼𝜏𝑁𝑖𝑘 0

∗ ∗ ∗ ∗ ∗ ∗ ∗ −𝛾2𝐼

] ] ] ] ] ] ] ] ] ] ] ] ]

< 0, (26)

𝑃𝑖𝑘≤ 𝜇𝑃𝑖𝑗, 𝑄𝑖𝑘 ≤ 𝜇𝑄𝑖𝑗, 𝑅𝑖𝑘 ≤ 𝜇𝑅𝑖𝑗, 𝑀𝑖𝑘≤ 𝜇𝑀𝑖𝑗, 𝑁𝑖𝑘≤ 𝜇𝑁𝑖𝑗, ∀𝑖𝑘, 𝑖𝑗∈ 𝑀, (27)

where

Ω11 = ℎ𝑀𝑖𝑘+ 𝛼𝑃𝑖𝑘+ 𝑄𝑖𝑘+ 𝑒−𝛼𝜏𝑇1𝑖𝑇𝑘+ 𝑒−𝛼𝜏𝑇1𝑖𝑘

+ 𝑊𝑖𝑇𝑘𝐴0𝑖𝑘+ 𝐴𝑇0𝑖𝑘𝑊𝑖𝑘− 𝑋𝑖𝑘𝐶0𝑖𝑘− 𝐶𝑇0𝑖𝑘𝑋𝑇𝑖𝑘+ 𝐿𝑇𝑖𝑘𝐿𝑖𝑘,

Ω12= 𝑃𝑖𝑘− 𝑊𝑖𝑇𝑘 + 𝐴𝑇0𝑖𝑘𝑊𝑖𝑘− 𝐶𝑇0𝑖𝑘𝑋𝑇𝑖𝑘,

Ω13= 𝑊𝑖𝑇𝑘𝐴1𝑖𝑘+ 𝐴𝑇0𝑖𝑘𝑊𝑖𝑘− 𝑋𝑖𝑘𝐶1𝑖𝑘− 𝐶𝑇0𝑖𝑘𝑋𝑇𝑖𝑘,

Ω14= −𝑒−𝛼𝜏𝑇1𝑖𝑇𝑘+ 𝑒−𝛼𝜏𝑇2𝑖𝑘,

Ω18= 𝑊𝑖𝑇𝑘𝐵𝑖𝑘− 𝑋𝑖𝑘𝐷𝑖𝑘,

Ω22= 𝑅𝑖𝑘+ 𝜏𝑁𝑖𝑘− 𝑊𝑖𝑇𝑘 − 𝑊𝑖𝑘,

Ω23 = 𝑊𝑖𝑇𝑘𝐴1𝑖𝑘− 𝑊𝑖𝑘− 𝑋𝑖𝑘𝐶1𝑖𝑘,

Ω28= 𝑊𝑇

𝑖 𝑘𝐵𝑖𝑘− 𝑋𝑖𝑘𝐷𝑖𝑘,

Ω33= − (1 − ℎ) 𝑒−𝛼ℎ𝑄𝑖𝑘+ 𝑊𝑖𝑇𝑘𝐴1𝑖𝑘+ 𝐴𝑇1𝑖𝑘𝑊𝑖𝑘

− 𝑋𝑖𝑘𝐶1𝑖𝑘− 𝐶1𝑖𝑇𝑘𝑋𝑇𝑖𝑘,

Ω38= 𝑊𝑖𝑇𝑘𝐵𝑖𝑘− 𝑋𝑖𝑘𝐷𝑖𝑘,

Ω44= −𝑒−𝛼𝜏𝑇2𝑖𝑇𝑘− 𝑒−𝛼𝜏𝑇2𝑖𝑘,

Ω55= − (1 − 𝜏) 𝑒−𝛼𝜏𝑅𝑖𝑘, Ω66= −1

ℎ𝑒−𝛼ℎ𝑀𝑖𝑘,

(28)

then the filter problem for the system (2) is solvable for any

switching signal with average dwell time satisfying𝑇𝑎 > 𝑇𝑎∗ =

ln𝜇/𝛼.

Moreover, the filter gain𝐾𝑖𝑘are given by𝐾𝑖𝑘 = 𝑊𝑖−𝑇𝑘 𝑋𝑖𝑘.

Proof Consider the piecewise Lyapunov-krasovskii

func-tional candidate as (11) and introduce the vector 𝜂𝑒𝑇(𝑡) = [𝜉𝑇

𝑒(𝑡) 𝜔𝑇(𝑡)], where 𝜉𝑒(𝑡) is defined in (21) Then, replace (21) with the following

2𝜂𝑇𝑒 (𝑡)[𝑊𝑖𝑘 𝑊𝑖𝑘 𝑊𝑖𝑘 0 0 0 0 0]𝑇

× [ ̃𝐴0𝑖𝑘 −𝐼 ̃𝐴1𝑖𝑘 0 𝐹𝑖𝑘 0 0 ̃𝐵𝑖𝑘] 𝜂𝑒(𝑡) = 0 (29) Let𝑋𝑖𝑘 = 𝑊𝑖𝑇𝑘𝐾𝑖𝑘 and𝑇(𝑡) = 𝑧𝑇𝑒(𝑡)𝑧𝑒(𝑡) − 𝛾2𝜔𝑇(𝑡)𝜔(𝑡), similar to the proof ofTheorem 6, we have

̇𝑉 (𝑡) + 𝛼𝑉 (𝑡) + 𝑇 (𝑡) ≤ 𝜂𝑇

𝑒 (𝑡) Ω𝑖𝑘𝜂𝑒(𝑡) (30)

IfΩ𝑖𝑘 < 0, it has

̇𝑉 (𝑡) ≤ −𝛼𝑉 (𝑡) − 𝑇 (𝑡) (31) Integrating both sides of (31) from𝑡𝑘 to 𝑡, for any 𝑡 ∈ [𝑡𝑘, 𝑡𝑘+1), gives

𝑉 (𝑡) ≤ 𝑒−𝛼(𝑡−𝑡𝑘 )𝑉 (𝑡𝑘) − ∫𝑡

𝑡 𝑘

𝑒−𝛼(𝑡−𝑠)𝑇 (𝑠) 𝑑𝑠 (32)

Therefore, similar to the proof method of Theory 2 in [13],

we have

𝑉 (𝑡) ≤ 𝑒−𝛼(𝑡−𝑡0 )+𝑁 𝜎 (𝑡 0 ,𝑡) ln 𝜇𝑉 (𝑡0)

− ∫𝑡

𝑡0𝑒−𝛼(𝑡−𝑠)+𝑁𝜎 (𝑠,𝑡) ln 𝜇𝑇 (𝑠) 𝑑𝑠 (33) Under the zero initial condition, (33) gives

0 ≤ − ∫𝑡

0𝑒−𝛼(𝑡−𝑠)+𝑁𝜎 (𝑠,𝑡) ln 𝜇𝑇 (𝑠) 𝑑𝑠 (34)

Multiplying both sides of (34) by𝑒−𝑁 𝜎 (0,𝑡) ln 𝜇yields

∫𝑡

0𝑒−𝛼(𝑡−𝑠)−𝑁𝜎 (0,𝑠) ln 𝜇𝑧𝑒𝑇(𝑠) 𝑧𝑒(𝑠) 𝑑𝑠

≤ ∫𝑡

0𝑒−𝛼(𝑡−𝑠)−𝑁𝜎 (0,𝑠) ln 𝜇𝛾2𝜔𝑇(𝑠) 𝜔 (𝑠) 𝑑𝑠

(35)

Notice that𝑁𝜎(0, 𝑠) ≤ (𝑠/𝑇𝑎) and 𝑇𝑎 > 𝑇∗

𝑎 = (ln 𝜇/𝛼), we have𝑁𝜎(0, 𝑠) ln 𝜇 ≤ 𝛼𝑠 Thus, (35) implies that

∫𝑡

0𝑒−𝛼(𝑡−𝑠)−𝛼𝑠𝑧𝑇𝑒 (𝑠) 𝑧𝑒(𝑠) 𝑑𝑠 ≤ 𝛾2∫𝑡

0𝑒−𝛼(𝑡−𝑠)𝜔𝑇(𝑠) 𝜔 (𝑠) 𝑑𝑠

(36) Integrating both sides of (36) from𝑡 = 0 to 𝑡 = ∞, we

have

∫∞

0 ∫𝑡

0𝑒−𝛼𝑡𝑧𝑇𝑒 (𝑠) 𝑧𝑒(𝑠) 𝑑𝑠 𝑑𝑡

≤ 𝛾2∫∞

0 ∫𝑡

0𝑒−𝛼(𝑡−𝑠)𝜔𝑇(𝑠) 𝜔 (𝑠) 𝑑𝑠 𝑑𝑡

(37)

Then, we can obtain

∫∞

0

1

𝛼𝑒−𝛼𝑠𝑧𝑇𝑒 (𝑠) 𝑧𝑒(𝑠) 𝑑𝑠 ≤ 𝛾2∫∞

0

1

𝛼𝑒−𝛼𝑠𝑒𝛼𝑠𝜔𝑇(𝑠) 𝜔 (𝑠) 𝑑𝑠.

(38) Obviously, it follows from (38) that

∫∞

0 𝑒−𝛼𝑠𝑧𝑇𝑒 (𝑠) 𝑧𝑒(𝑠) 𝑑𝑠 ≤ 𝛾2∫∞

0 𝜔𝑇(𝑠) 𝜔 (𝑠) 𝑑𝑠 (39)

Remark 11. Theorem 10provides a sufficient condition for the

solvability of the𝐻∞filtering problem for switched neutral

system with time-varying delay If the condition is satisfied,

then matrices𝑊𝑖are nonsingular

4 Numerical Examples

In this section, we present two numerical examples to

illustrate the effectiveness of the results presented previously

Example 1 Consider the switched neutral system (2) with

two subsystems

Subsystem 1.

𝐴01= [−1.5 −0.20.2 −1.3] , 𝐴11= [−0.3 00.1 −0.4] ,

𝐷1= 0.03, 𝐹1= [0.1 −0.60 0.09] ,

𝐶01 = [−0.2 0.26] , 𝐶11= [−0.2 0.1] ,

𝐵1= [ 0.2−0.3] , 𝐿1= [0.5 −0.19] ,

(40)

−0.05 0 0.05

0.15

0.1

Time

ze

Figure 2:𝑧𝑒(t) of the filtering error dynamic system with𝜔(𝑡) = 0

0 0.05 0.15 0.25 0.35

0.1 0.2 0.3

Time

xe

Figure 3: The state responses of the filtering error dynamic system with𝜔(𝑡) = 0.1𝑒−0.5𝑡

Subsystem 2.

𝐴02= [−1.4 −0.20.2 −1.3] , 𝐴12= [−0.2 00.1 −0.4] ,

𝐷2= 𝐷1, 𝐹2= 𝐹1,

𝐶02= [−0.2 0.46] , 𝐶12= 𝐶11,

𝐵2= 𝐵1, 𝐿2= [0.5 −0.09] ,

ℎ (𝑡) = 0.3, 𝜏 (𝑡) = 0.3, 𝛼 = 0.0376

(41)

By using the LMI toolbox, it can be checked that the conditions given inTheorem 10are satisfied Therefore, the previously switched neutral system has the given𝐻∞ perfor-mance𝛾, when 𝑇𝑎≥ 𝑇𝑎∗= ln 𝜇min/𝛼 = 2.6596𝑒−004(here, the allowable minimum of𝜇 is 𝜇min= 1.00001)

0

0.05

0.15

0.1

Time

ze

Figure 4:𝑧𝑒(𝑡) of the filtering error dynamic system with 𝜔(𝑡) =

0.1𝑒−0.5𝑡

1

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

Time

Figure 5: The noise signal𝑤(𝑡)

Setting𝜇 = 1.01 and solving LMIs (26) using the LMI

Toolbox in MATLAB, it follows that the minimized feasible

𝛾 is 𝛾∗ = 2.0, 𝑇𝑎 = 0.2646, and the corresponding filter

parameters are computed as

𝐾1= [−5.07681.2264 ] , 𝐾2= [−2.87240.9217 ] (42)

In the following, we illustrate the effectiveness of the

designed𝐻∞filter through simulation Let the initial

con-dition be𝑥𝑒(𝑡) = [0.3], 𝑡 ∈ [−0.3, 0] Figures1and 2are,

respectively, the simulation results on𝑥𝑒(𝑡) and 𝑧𝑒(𝑡); we can

see that the filtering error dynamic system with𝜔(𝑡) = 0 is

stable.𝑥𝑒(𝑡) and 𝑧𝑒(𝑡) of the filtering error dynamic system

with𝑤(𝑡) = 0.1𝑒−0.5𝑡are given in Figures3and4.Figure 5

shows𝑤(𝑡)

Table 1

0.80 2.0 3.2 [−4.7398 1.1340]𝑇 [−2.8595 0.9824]𝑇

0.54 1.1 1.9 [−4.1874 1.1606]𝑇 [−2.5316 0.9494]𝑇

0.48 0.7 0.2 [−3.9220 1.1308]𝑇 [−2.3354 0.8774]𝑇

0.46 0.5 1.3 [−3.8076 1.0948]𝑇 [−2.2459 0.8288]𝑇

0.44 0.3 1.1 [−3.6949 1.0686]𝑇 [−2.1560 0.7834]𝑇

Example 2 Consider the switched neutral system in

Example 1 with constant delays; that is, ℎ = 0, 𝜏 = 0,

𝛼 = 0.0376, and 𝜇 = 1.0001 We calculate the admissible maximum valueℎmaxofℎ, 𝜏maxof𝜏, which ensures that the resulting filtering error system is exponentially stable with

a prescribed level 𝛾 of noise attenuation For the different values,𝛾, the obtained ℎmax,𝜏max, and the filter gain𝐾𝑖are listed inTable 1

5 Conclusions

We have addressed the𝐻∞ filtering problem for a class of switched neutral systems with time-varying delays which appear in both the state and the state derivatives For switched neutral systems with average dwell time scheme, we have provided a condition, in terms of upper bounds on the delays and in terms of a lower bound on the average dwell time, for the solvability of the𝐻∞ filtering problem The piece-wise Lyapunov functional technique has been used, which makes the proposed conditions are both delay-dependent and neutral delay-dependent The design of filters for switched neutral systems is a difficult issue that is far from being well explored Since multiple Lyapunov functional approach is commonly considered less conservative, the design of filters for switching neutral system with an appropriate switching law using multiple Lyapunov functionals is of great signifi-cance which deserves further study

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