# Design ofH∞ Filter for a Class of Switched Linear Neutral Systems

Authors: Caiyun Wu; Yue-E. Wang
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

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 weighted H∞ performance is established. All the conditions obtained are delay-dependent. Finally, two numerical examples are given to illustrate the effectiveness of the proposed theory.


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## Body

## 1. Introduction

Switched time-delay systems have been attracting considerable attention during the recent years [1–9], due to the significance both in theory development and practical applications. 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, theH∞ 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 an H∞ filter of a Luenberger observer type have been provided. However, to the authors' best knowledge, the H∞ 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-dependentH∞ filtering problem 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 weighted H∞ 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. TheH∞ filtering problem for switched neutral systems is formulated in Section 2. Section 3 presents our main results. Numerical examples are given in Section 4, and we conclude this paper in Section 5.Notation. Throughout this paper, Rn denotes n-dimensional Euclidean space; Rn×m is the set of all n×m real matrices; P>0 means that P is positive definite; L2 denotes 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
(1)[XY*Z]=[XYYTZ].


## 2. Problem Statement

Consider the following switched linear neutral system:(2)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},
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 to L2[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 signal σ(t), we have the switching sequence {xt0:(i0,t0),…,(ik,tk),…,∣ik∈M,k=0,1,…}, which means that the ikth subsystem is active when t∈[tk,tk+1). The system coefficient matrices A0ik, A1ik, Fik, Bik, C0ik, C1ik, Dik, and Lik are known real constant matrices of appropriate dimensions. h(t) and τ(t) are time-varying delays satisfying
(3)0≤h(t)≤h,h˙(t)≤h¯<1,0≤τ(t)≤τ,τ˙(t)≤τ¯<1.The objective of this paper is to design a family of filters of Luenberger observer type as follows:(4)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},
where Kik are the filter parameters, which are to be determined.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:
(5)x˙e(t)=A~0σ(t)xe(t)+A~1σ(t)xe(t-h(t))+Fσ(t)x˙e(t-τ(t))+B~σ(t)ω(t),ze(t)=Lσ(t)xe(t),xe(θ)=ψ(θ)-ψ^(θ),∀θ∈[-H,0],H=max{h,τ},
where A~0σ=A0σ-KσC0σ, A~1σ=A1σ-KσC1σ, B~σ=Bσ-KσDσ.The following definitions are introduced, which will play key roles in deriving our main results.Definition 1 (see [26]).
The equilibriumxe*=0 of the filtering error system (5) is said to be exponentially stable under σ(t) if the solution xe(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),x˙e(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)/Ta holds for Ta>0, N0≥0, then Ta is called average dwell time. As commonly used in the literature, we choose N0=0.The filtering problem addressed in this paper is to seek for suitable filter gainKi such that the filtering error system (5) for any switching signal with average dwell time has a prescribed H∞ performance γ; that is,(1)
the error system (5) with ω(t)=0 is exponentially stable;(2)
under the zero initial conditions, that is,xe(θ)=0, for all θ∈[-H,0], the weighted H∞ performance ∫0∞‍e-αszeT(s)ze(s)ds≤γ2∫0∞‍ωT(s)ω(s)ds is guaranteed for all nonzero ω(t)∈L2[0,∞) and a prescribed level of noise attenuation γ>0.Before concluding this section, we introduce three lemmas which are essential for the development of the results.Lemma 3 (see [25]).
Letx(t)∈Rn be a vector-valued function with first-order continuous-derivative entries. Then, the following integral inequality holds for any matrices M1, M2∈Rn×n, and X=XT>0, and a scalar h≥0,
(6)-∫t-ht‍x˙T(s)Xx˙(s)ds≤ξT(t)[M1T+M1-M1T+M2*-M2T-M2]ξ(t)+hξT(t)[M1TM2T]X-1[M1M2]ξ(t),
where ξT(t)=[xT(t)xT(t-h)].Lemma 4 (see [27]).
For any constant matrix0<R=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.Lemma 5 (Schur complement).
For givenS=[S11S12*S22]<0, where S11=S11T and S22=S22T, the following is equivalent:
(7)(1)S11<0,S22-S12TS11-1S12<0;(2)S22<0,S11-S12S22-1S12T<0.


## 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 desired H∞ filters for switched neutral system (2).


### 3.1. Stability Analysis

Theorem 6.
Givenα>0, ∥Fik∥<1, for all ik∈M. If there exist matrices Pik>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,(8)Σ¯ik=[Σ¯11Σ¯12Σ¯13Σ¯14Σ¯15A~0ikTN6ikΣ¯17*Σ¯22Σ¯23-N4ikΣ¯25-N6ik-N7ik**Σ¯33A~1ikTN4ikΣ¯35A~1ikTN6ikA~1ikTN7ik***Σ¯44N4ikTFik0τT2ikT****Σ¯55FikTN6ikFikTN7ik*****-1he-αhMik0******-τeατNik]<0,(9)Pik≤μPij,Qik≤μQij,Rik≤μRij,Mik≤μMij,Nik≤μNij,∀ik,ij∈M,where
(10)Σ¯11=hMik+αPik+Qik+e-ατT1ikT+e-ατT1ik+N1ikTA~0ik+A~0ikTN1ik,Σ¯12=Pik-N1ikT+A~0ikTN2ik,Σ¯13=N1ikTA~1ik+A~0ikTN3ik,Σ¯14=-e-ατT1ikT+e-ατT2ik+A~0ikTN4ik,Σ¯15=N1ikTFik+A~0ikTN5ik,Σ¯17=τT1ikT+A~0ikTN7ik,Σ¯22=Rik+τNik-N2ikT-N2ik,Σ¯23=-N3ik+N2ikTA~1ik,Σ¯25=-N5ik+N2ikTFik,Σ¯33=-(1-h¯)e-αhQik+N3ikTA~1ik+A~1ikTN3ik,Σ¯35=N3ikTFik+A~1ikTN5ik,Σ¯44=-e-ατT2ikT-e-ατT2ik,Σ¯55=-(1-τ¯)e-ατRik+N5ikTFik+FikTN5ik,
then the error dynamic system (5) with ω(t)=0 is exponentially stable for any switching signal with average dwell time satisfying Ta>Ta*=lnμ/α.Proof.
Define the piecewise Lyapunov-Krasovskii functional candidate(11)V(t)=Vσ(t)(t)=∑j=15‍Vjσ(t)(t),
where
(12)V1ik(t)=xeT(t)Pikxe(t),V2ik(t)=∫t-h(t)t‍xeT(s)eα(s-t)Qikxe(s)ds,V3ik(t)=∫t-τ(t)t‍x˙eT(s)eα(s-t)Rikx˙e(s)ds,V4ik(t)=∫-h0‍∫t+θt‍xeT(s)eα(s-t)Mikxe(s)dsdθ,V5ik(t)=∫-τ0‍∫t+θt‍x˙eT(s)eα(s-t)Nikx˙e(s)dsdθ.
Now, taking the derivative ofVjik(t), j=1,2,…,5 with respect to t along the trajectory of the error system (5) with ω(t)=0, according to (3) and Lemma 4, we have
(13)V˙ik(t)+αVik(t)≤2xeT(t)Pikx˙e(t)+xeT(t)Qikxe(t)+x˙eT(t)Rikx˙e(t)-(1-h¯)xeT(t-h(t))e-αhQikxe(t-h(t))+hxeT(t)Mikxe(t)+τx˙eT(t)Nikx˙e(t)+αxeT(t)Pikxe(t)-(1-τ¯)x˙eT(t-τ(t))e-ατRikx˙e(t-τ(t))-e-αhh∫t-ht‍xeT(s)dsMik∫t-ht‍xe(s)ds-∫t-τt‍x˙eT(s)eα(s-t)Nikx˙e(s)ds.
From Lemma3, it holds
(14)-∫t-τt‍x˙eT(s)eα(s-t)Nikx˙e(s)ds≤-∫t-τt‍x˙eT(s)e-ατNikx˙e(s)ds≤e-ατ[xeT(t)xeT(t-τ)][T1ikT+T1ik-T1ikT+T2ik*-T2ikT-T2ik]×[xe(t)xe(t-τ)]+τe-ατ[xeT(t)xeT(t-τ)]×[T1ikTT2ikT]Nik-1[T1ikT2ik][xe(t)xe(t-τ)].
Define(15)ξeT(t)=[ζeT(t)x˙eT(t-τ(t))∫t-ht‍xeT(s)ds],
where ζeT(t)=[xeT(t)x˙eT(t)xeT(t-h(t))xeT(t-τ)].
By some algebraic manipulations, it is easy to show that(16)V˙ik(t)+αVik(t)≤ξeT(t)Σikξe(t),
where
(17)Σik=[Σ11Pik0Σ1400*Rik+τNik0000**-(1-h¯)e-αhQik000***Σ4400****Σ550*****Σ66],
where
(18)Σ11=hMik+αPik+Qik+e-ατT1ikT+e-ατT1ik+τe-ατT1ikTNik-1T1ik,Σ14=-e-ατT1ikT+e-ατT2ik+τe-ατT1ikTNik-1T2ik,Σ55=-(1-τ¯)e-ατRik,Σ44=-e-ατT2ikT-e-ατT2ik+τe-ατT2ikTNik-1T2ik,Σ66=-1he-αhMik.
From Lemma5, Σik<0 is equivalent to
(19)Σ~ik=[Σ~11Pik0Σ~1400τT1ikT*Rik+τNik00000**Σ~330000***Σ~4400τT2ikT****Σ~5500*****Σ~660******-τeατNik]<0,
where
(20)Σ~11=hMik+αPik+Qik+e-ατT1ikT+e-ατT1ik,Σ~14=-e-ατT1ikT+e-ατT2ik,Σ~33=-(1-h¯)e-αhQik,Σ~44=-e-ατT2ikT-e-ατT2ik,Σ~55=Σ55,Σ~66=Σ66.
In addition, the following is true from (5) with ω(t)=0:
(21)2ξ¯(t)eT[N1ikN2ikN3ikN4ikN5ikN6ikN7ik]T×[A~0ik-IA~1ik0Fik00]ξ¯e(t)=0,
where ξ¯(t)eT=[ξeT(t)0].
Then, (19) along with (21) gives Σ¯ik<0, which yields Σ¯ik<0; thus,
(22)V˙ik(t)+αVik(t)≤0.
Combining (9) with (22), for any t∈[tk,tk+1), we have
(23)V(t)=Vik(t)≤e-α(t-tk)Vik(tk)≤μe-α(t-tk)Vσ(tk-)(tk-)≤μe-α(t-tk)e-α(tk-tk-1)Vσ(tk-1)(tk-1)≤⋯≤μke-α(t-t0)V(t0)≤e-(α-(lnμ/Ta))(t-t0)V(t0).
According to (11), we have
(24)a∥xe(t)∥2≤V(t)≤b∥xe(t0)∥H2,
where
(25)a=min∀ik∈M{λmin(Pik)},b=max∀ik∈M{λmax(Pik)}+hmax∀ik∈M{λmax(Qik)}+τmax∀ik∈M{λmax(Rik)}+h22max∀ik∈M{λmax(Mik)}+τ22max∀ik∈M{λmax(Nik)}.
Considering (23) and (24), it holds ∥xe(t)∥≤(b/a)e-(1/2)(α-(lnμ/Ta))(t-t0)∥xe(t0)∥H.
Therefore, ifα-(lnμ/Ta)>0, that is Ta>(lnμ/α), then error dynamic system (5) is exponentially stable.Remark 7.
Whenμ=1, we have Ta*=0, which means that the switching signal σ(t) can be arbitrary. In this case, condition (9) implies that there exists a common Lyapunov functional for all subsystems. Moreover, setting α=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).Theorem 10.
Givenα>0, if ∥Fik∥<1, for all ik∈M, and if there exists matrices Pik>0, Qik>0, Rik>0, Mik>0, Nik>0, and T1ik, T2ik, Wik, Xik of appropriate dimensions, and μ≥1, such that, for any ik∈M,(26)Ωik=[Ω11Ω12Ω13Ω14WikTFik0τT1ikTΩ18*Ω22Ω230WikTFik00Ω28**Ω330WikTFik00Ω38***Ω4400τT2ikT0****Ω55000*****Ω6600******-τeατNik0*******-γ2I]<0,(27)Pik≤μPij,Qik≤μQij,Rik≤μRij,Mik≤μMij,Nik≤μNij,∀ik,ij∈M,where
(28)Ω11=hMik+αPik+Qik+e-ατT1ikT+e-ατT1ik+WikTA0ik+A0ikTWik-XikC0ik-C0ikTXikT+LikTLik,Ω12=Pik-WikT+A0ikTWik-C0ikTXikT,Ω13=WikTA1ik+A0ikTWik-XikC1ik-C0ikTXikT,Ω14=-e-ατT1ikT+e-ατT2ik,Ω18=WikTBik-XikDik,Ω22=Rik+τNik-WikT-Wik,Ω23=WikTA1ik-Wik-XikC1ik,Ω28=WikTBik-XikDik,Ω33=-(1-h¯)e-αhQik+WikTA1ik+A1ikTWik-XikC1ik-C1ikTXikT,Ω38=WikTBik-XikDik,Ω44=-e-ατT2ikT-e-ατT2ik,Ω55=-(1-τ¯)e-ατRik,Ω66=-1he-αhMik,
then the filter problem for the system (2) is solvable for any switching signal with average dwell time satisfying Ta>Ta*=lnμ/α.
Moreover, the filter gainKik are given by Kik=Wik-TXik.Proof.
Consider the piecewise Lyapunov-krasovskii functional candidate as (11) and introduce the vector ηeT(t)=[ξ¯(t)eTωT(t)], where ξ¯e(t) is defined in (21). Then, replace (21) with the following
(29)2ηeT(t)[WikWikWik00000]T×[A~0ik-IA~1ik0Fik00B~ik]ηe(t)=0.
LetXik=WikTKik and T(t)=zeT(t)ze(t)-γ2ωT(t)ω(t), similar to the proof of Theorem 6, we have
(30)V˙(t)+αV(t)+T(t)≤ηeT(t)Ωikηe(t).
IfΩik<0, it has
(31)V˙(t)≤-αV(t)-T(t).
Integrating both sides of (31) from tk to t, for any t∈[tk,tk+1), gives
(32)V(t)≤e-α(t-tk)V(tk)-∫tkt‍e-α(t-s)T(s)ds.
Therefore, similar to the proof method of Theory 2 in [13], we have
(33)V(t)≤e-α(t-t0)+Nσ(t0,t)lnμV(t0)-∫t0t‍e-α(t-s)+Nσ(s,t)lnμT(s)ds.
Under the zero initial condition, (33) gives
(34)0≤-∫0t‍e-α(t-s)+Nσ(s,t)lnμT(s)ds.
Multiplying both sides of (34) by e-Nσ(0,t)lnμ yields
(35)∫0t‍e-α(t-s)-Nσ(0,s)lnμzeT(s)ze(s)ds≤∫0t‍e-α(t-s)-Nσ(0,s)lnμγ2ωT(s)ω(s)ds.
Notice thatNσ(0,s)≤(s/Ta) and Ta>Ta*=(lnμ/α), we have Nσ(0,s)lnμ≤αs. Thus, (35) implies that
(36)∫0t‍e-α(t-s)-αszeT(s)ze(s)ds≤γ2∫0t‍e-α(t-s)ωT(s)ω(s)ds.
Integrating both sides of (36) from t=0 to t=∞, we have
(37)∫0∞‍∫0t‍e-αtzeT(s)ze(s)dsdt≤γ2∫0∞‍∫0t‍e-α(t-s)ωT(s)ω(s)dsdt.
Then, we can obtain(38)∫0∞‍1αe-αszeT(s)ze(s)ds≤γ2∫0∞‍1αe-αseαsωT(s)ω(s)ds.
Obviously, it follows from (38) that
(39)∫0∞‍e-αszeT(s)ze(s)ds≤γ2∫0∞‍ωT(s)ω(s)ds.Remark 11.
Theorem10 provides a sufficient condition for the solvability of the H∞ filtering problem for switched neutral system with time-varying delay. If the condition is satisfied, then matrices Wi are nonsingular.


## 3.1. Stability Analysis

Theorem 6.
Givenα>0, ∥Fik∥<1, for all ik∈M. If there exist matrices Pik>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,(8)Σ¯ik=[Σ¯11Σ¯12Σ¯13Σ¯14Σ¯15A~0ikTN6ikΣ¯17*Σ¯22Σ¯23-N4ikΣ¯25-N6ik-N7ik**Σ¯33A~1ikTN4ikΣ¯35A~1ikTN6ikA~1ikTN7ik***Σ¯44N4ikTFik0τT2ikT****Σ¯55FikTN6ikFikTN7ik*****-1he-αhMik0******-τeατNik]<0,(9)Pik≤μPij,Qik≤μQij,Rik≤μRij,Mik≤μMij,Nik≤μNij,∀ik,ij∈M,where
(10)Σ¯11=hMik+αPik+Qik+e-ατT1ikT+e-ατT1ik+N1ikTA~0ik+A~0ikTN1ik,Σ¯12=Pik-N1ikT+A~0ikTN2ik,Σ¯13=N1ikTA~1ik+A~0ikTN3ik,Σ¯14=-e-ατT1ikT+e-ατT2ik+A~0ikTN4ik,Σ¯15=N1ikTFik+A~0ikTN5ik,Σ¯17=τT1ikT+A~0ikTN7ik,Σ¯22=Rik+τNik-N2ikT-N2ik,Σ¯23=-N3ik+N2ikTA~1ik,Σ¯25=-N5ik+N2ikTFik,Σ¯33=-(1-h¯)e-αhQik+N3ikTA~1ik+A~1ikTN3ik,Σ¯35=N3ikTFik+A~1ikTN5ik,Σ¯44=-e-ατT2ikT-e-ατT2ik,Σ¯55=-(1-τ¯)e-ατRik+N5ikTFik+FikTN5ik,
then the error dynamic system (5) with ω(t)=0 is exponentially stable for any switching signal with average dwell time satisfying Ta>Ta*=lnμ/α.Proof.
Define the piecewise Lyapunov-Krasovskii functional candidate(11)V(t)=Vσ(t)(t)=∑j=15‍Vjσ(t)(t),
where
(12)V1ik(t)=xeT(t)Pikxe(t),V2ik(t)=∫t-h(t)t‍xeT(s)eα(s-t)Qikxe(s)ds,V3ik(t)=∫t-τ(t)t‍x˙eT(s)eα(s-t)Rikx˙e(s)ds,V4ik(t)=∫-h0‍∫t+θt‍xeT(s)eα(s-t)Mikxe(s)dsdθ,V5ik(t)=∫-τ0‍∫t+θt‍x˙eT(s)eα(s-t)Nikx˙e(s)dsdθ.
Now, taking the derivative ofVjik(t), j=1,2,…,5 with respect to t along the trajectory of the error system (5) with ω(t)=0, according to (3) and Lemma 4, we have
(13)V˙ik(t)+αVik(t)≤2xeT(t)Pikx˙e(t)+xeT(t)Qikxe(t)+x˙eT(t)Rikx˙e(t)-(1-h¯)xeT(t-h(t))e-αhQikxe(t-h(t))+hxeT(t)Mikxe(t)+τx˙eT(t)Nikx˙e(t)+αxeT(t)Pikxe(t)-(1-τ¯)x˙eT(t-τ(t))e-ατRikx˙e(t-τ(t))-e-αhh∫t-ht‍xeT(s)dsMik∫t-ht‍xe(s)ds-∫t-τt‍x˙eT(s)eα(s-t)Nikx˙e(s)ds.
From Lemma3, it holds
(14)-∫t-τt‍x˙eT(s)eα(s-t)Nikx˙e(s)ds≤-∫t-τt‍x˙eT(s)e-ατNikx˙e(s)ds≤e-ατ[xeT(t)xeT(t-τ)][T1ikT+T1ik-T1ikT+T2ik*-T2ikT-T2ik]×[xe(t)xe(t-τ)]+τe-ατ[xeT(t)xeT(t-τ)]×[T1ikTT2ikT]Nik-1[T1ikT2ik][xe(t)xe(t-τ)].
Define(15)ξeT(t)=[ζeT(t)x˙eT(t-τ(t))∫t-ht‍xeT(s)ds],
where ζeT(t)=[xeT(t)x˙eT(t)xeT(t-h(t))xeT(t-τ)].
By some algebraic manipulations, it is easy to show that(16)V˙ik(t)+αVik(t)≤ξeT(t)Σikξe(t),
where
(17)Σik=[Σ11Pik0Σ1400*Rik+τNik0000**-(1-h¯)e-αhQik000***Σ4400****Σ550*****Σ66],
where
(18)Σ11=hMik+αPik+Qik+e-ατT1ikT+e-ατT1ik+τe-ατT1ikTNik-1T1ik,Σ14=-e-ατT1ikT+e-ατT2ik+τe-ατT1ikTNik-1T2ik,Σ55=-(1-τ¯)e-ατRik,Σ44=-e-ατT2ikT-e-ατT2ik+τe-ατT2ikTNik-1T2ik,Σ66=-1he-αhMik.
From Lemma5, Σik<0 is equivalent to
(19)Σ~ik=[Σ~11Pik0Σ~1400τT1ikT*Rik+τNik00000**Σ~330000***Σ~4400τT2ikT****Σ~5500*****Σ~660******-τeατNik]<0,
where
(20)Σ~11=hMik+αPik+Qik+e-ατT1ikT+e-ατT1ik,Σ~14=-e-ατT1ikT+e-ατT2ik,Σ~33=-(1-h¯)e-αhQik,Σ~44=-e-ατT2ikT-e-ατT2ik,Σ~55=Σ55,Σ~66=Σ66.
In addition, the following is true from (5) with ω(t)=0:
(21)2ξ¯(t)eT[N1ikN2ikN3ikN4ikN5ikN6ikN7ik]T×[A~0ik-IA~1ik0Fik00]ξ¯e(t)=0,
where ξ¯(t)eT=[ξeT(t)0].
Then, (19) along with (21) gives Σ¯ik<0, which yields Σ¯ik<0; thus,
(22)V˙ik(t)+αVik(t)≤0.
Combining (9) with (22), for any t∈[tk,tk+1), we have
(23)V(t)=Vik(t)≤e-α(t-tk)Vik(tk)≤μe-α(t-tk)Vσ(tk-)(tk-)≤μe-α(t-tk)e-α(tk-tk-1)Vσ(tk-1)(tk-1)≤⋯≤μke-α(t-t0)V(t0)≤e-(α-(lnμ/Ta))(t-t0)V(t0).
According to (11), we have
(24)a∥xe(t)∥2≤V(t)≤b∥xe(t0)∥H2,
where
(25)a=min∀ik∈M{λmin(Pik)},b=max∀ik∈M{λmax(Pik)}+hmax∀ik∈M{λmax(Qik)}+τmax∀ik∈M{λmax(Rik)}+h22max∀ik∈M{λmax(Mik)}+τ22max∀ik∈M{λmax(Nik)}.
Considering (23) and (24), it holds ∥xe(t)∥≤(b/a)e-(1/2)(α-(lnμ/Ta))(t-t0)∥xe(t0)∥H.
Therefore, ifα-(lnμ/Ta)>0, that is Ta>(lnμ/α), then error dynamic system (5) is exponentially stable.Remark 7.
Whenμ=1, we have Ta*=0, which means that the switching signal σ(t) can be arbitrary. In this case, condition (9) implies that there exists a common Lyapunov functional for all subsystems. Moreover, setting α=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).Theorem 10.
Givenα>0, if ∥Fik∥<1, for all ik∈M, and if there exists matrices Pik>0, Qik>0, Rik>0, Mik>0, Nik>0, and T1ik, T2ik, Wik, Xik of appropriate dimensions, and μ≥1, such that, for any ik∈M,(26)Ωik=[Ω11Ω12Ω13Ω14WikTFik0τT1ikTΩ18*Ω22Ω230WikTFik00Ω28**Ω330WikTFik00Ω38***Ω4400τT2ikT0****Ω55000*****Ω6600******-τeατNik0*******-γ2I]<0,(27)Pik≤μPij,Qik≤μQij,Rik≤μRij,Mik≤μMij,Nik≤μNij,∀ik,ij∈M,where
(28)Ω11=hMik+αPik+Qik+e-ατT1ikT+e-ατT1ik+WikTA0ik+A0ikTWik-XikC0ik-C0ikTXikT+LikTLik,Ω12=Pik-WikT+A0ikTWik-C0ikTXikT,Ω13=WikTA1ik+A0ikTWik-XikC1ik-C0ikTXikT,Ω14=-e-ατT1ikT+e-ατT2ik,Ω18=WikTBik-XikDik,Ω22=Rik+τNik-WikT-Wik,Ω23=WikTA1ik-Wik-XikC1ik,Ω28=WikTBik-XikDik,Ω33=-(1-h¯)e-αhQik+WikTA1ik+A1ikTWik-XikC1ik-C1ikTXikT,Ω38=WikTBik-XikDik,Ω44=-e-ατT2ikT-e-ατT2ik,Ω55=-(1-τ¯)e-ατRik,Ω66=-1he-αhMik,
then the filter problem for the system (2) is solvable for any switching signal with average dwell time satisfying Ta>Ta*=lnμ/α.
Moreover, the filter gainKik are given by Kik=Wik-TXik.Proof.
Consider the piecewise Lyapunov-krasovskii functional candidate as (11) and introduce the vector ηeT(t)=[ξ¯(t)eTωT(t)], where ξ¯e(t) is defined in (21). Then, replace (21) with the following
(29)2ηeT(t)[WikWikWik00000]T×[A~0ik-IA~1ik0Fik00B~ik]ηe(t)=0.
LetXik=WikTKik and T(t)=zeT(t)ze(t)-γ2ωT(t)ω(t), similar to the proof of Theorem 6, we have
(30)V˙(t)+αV(t)+T(t)≤ηeT(t)Ωikηe(t).
IfΩik<0, it has
(31)V˙(t)≤-αV(t)-T(t).
Integrating both sides of (31) from tk to t, for any t∈[tk,tk+1), gives
(32)V(t)≤e-α(t-tk)V(tk)-∫tkt‍e-α(t-s)T(s)ds.
Therefore, similar to the proof method of Theory 2 in [13], we have
(33)V(t)≤e-α(t-t0)+Nσ(t0,t)lnμV(t0)-∫t0t‍e-α(t-s)+Nσ(s,t)lnμT(s)ds.
Under the zero initial condition, (33) gives
(34)0≤-∫0t‍e-α(t-s)+Nσ(s,t)lnμT(s)ds.
Multiplying both sides of (34) by e-Nσ(0,t)lnμ yields
(35)∫0t‍e-α(t-s)-Nσ(0,s)lnμzeT(s)ze(s)ds≤∫0t‍e-α(t-s)-Nσ(0,s)lnμγ2ωT(s)ω(s)ds.
Notice thatNσ(0,s)≤(s/Ta) and Ta>Ta*=(lnμ/α), we have Nσ(0,s)lnμ≤αs. Thus, (35) implies that
(36)∫0t‍e-α(t-s)-αszeT(s)ze(s)ds≤γ2∫0t‍e-α(t-s)ωT(s)ω(s)ds.
Integrating both sides of (36) from t=0 to t=∞, we have
(37)∫0∞‍∫0t‍e-αtzeT(s)ze(s)dsdt≤γ2∫0∞‍∫0t‍e-α(t-s)ωT(s)ω(s)dsdt.
Then, we can obtain(38)∫0∞‍1αe-αszeT(s)ze(s)ds≤γ2∫0∞‍1αe-αseαsωT(s)ω(s)ds.
Obviously, it follows from (38) that
(39)∫0∞‍e-αszeT(s)ze(s)ds≤γ2∫0∞‍ωT(s)ω(s)ds.Remark 11.
Theorem10 provides a sufficient condition for the solvability of the H∞ filtering problem for switched neutral system with time-varying delay. If the condition is satisfied, then matrices Wi are 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. 							(40)A01=[-1.5-0.20.2-1.3],A11=[-0.300.1-0.4],D1=0.03,F1=[0.1-0.600.09],C01=[-0.20.26],C11=[-0.20.1],B1=[0.2-0.3],L1=[0.5-0.19],
Subsystem 2. 							(41)A02=[-1.4-0.20.2-1.3],A12=[-0.200.1-0.4],D2=D1,F2=F1,C02=[-0.20.46],C12=C11,B2=B1,L2=[0.5-0.09],h(t)=0.3,τ(t)=0.3,α=0.0376.
By using the LMI toolbox, it can be checked that the conditions given in Theorem10 are satisfied. Therefore, the previously switched neutral system has the given H∞ performance γ, when Ta≥Ta*=lnμmin/α=2.6596e-004 (here, the allowable minimum of μ is μmin=1.00001).
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
(42)K1=[-5.07681.2264],K2=[-2.87240.9217].
In the following, we illustrate the effectiveness of the designedH∞ filter through simulation. Let the initial condition be xe(t)=[0.30.2], t∈[-0.3,0]. Figures 1 and 2 are, respectively, the simulation results on xe(t) and ze(t); we can see that the filtering error dynamic system with ω(t)=0 is stable. xe(t) and ze(t) of the filtering error dynamic system with w(t)=0.1e-0.5t are given in Figures 3 and 4. Figure 5 shows w(t).Figure 1
The state responses of the filtering error dynamic system withω(t)=0.Figure 2
z
e(t) of the filtering error dynamic system with ω(t)=0.Figure 3
The state responses of the filtering error dynamic system withω(t)=0.1e-0.5t.Figure 4
z
e
(
t
) of the filtering error dynamic system with ω(t)=0.1e-0.5t.Figure 5
The noise signalw(t).Example 2.
Consider the switched neutral system in Example1 with constant delays; that is, h¯=0, τ¯=0, α=0.0376, and μ=1.0001. We calculate the admissible maximum value hmax of h, τmax of τ, 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 gain Ki are listed in Table 1.Table 1
γ
h
max
τ
max
K
1
K
2
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


## 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 the H∞ filtering problem. The piecewise 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 significance which deserves further study.



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*Source: 537249-2013-10-27.xml*