# Switching Signal Design for Global Exponential Stability of Uncertain Switched Neutral Systems

Authors: Ker-Wei Yu
Journal: Mathematical Problems in Engineering
 (2009)

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## Abstract

The switching signal design for global exponential stability of switched neutral systems is investigated in this paper. LMI-based delay-dependent and delay-independent criteria are proposed to guarantee the global stability via the constructed switching signal. Razumikhin-like approach is used to find the stability results. Finally, some numerical examples are illustrated to show the main results.


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## Body

## 1. Introduction

It is well known that the existence of delay in a system may cause instability or bad system performance in control systems.  Time-delay phenomenon appears in many practical systems, such as AIDS epidemic, aircraft stabilization, chemical engineering systems, inferred grinding model, manual control, neural network, nuclear reactor, population dynamic model, rolling mill, ship stabilization, and systems with lossless transmission lines.  Hence stability analysis for time-delay systems has been considered in the  recent years [1–3].  Neutral systems are described by functional differential equations which depend on the delays of state and state derivative.  Some practical examples of neutral systems include distributed networks, heat exchanges, and processes including steam [4].Switched system is a class of hybrid systems which is consisting of several subsystems and uses the switching signal to specify which subsystem is activated to the system trajectories at each instant of time.   Some examples for switched systems are automated highway systems, constrained robotics, power systems and power electronics, transmission and stepper motors [5].  Stability analysis of switched time-delay systems has been an attractive research topic [6–13].  It is interesting to note that the stability for each subsystem cannot imply that of the overall system under arbitrary switching signal [9].  Another interesting fact is that the stability of a switched system can be achieved by choosing the switching signal even when each subsystem is unstable [6, 7, 10].  In this paper, the switching signal design will be considered for uncertain switched neutral systems with mixed delays.  The switching signal will be proposed to guarantee the stability of switched system even when each subsystem is unstable.  Based on Razumikhin-like approach [11], delay-dependent and delay-independent results are provided.  New and flexible LMI conditions are proposed to design the switching signal which guarantees the global exponential and asymptotic stability of uncertain switched neutral systems.  Some numerical examples are provided to demonstrate the use of our results.The notation used throughout this paper is as follows.  For a matrixA, we denote the transpose by AT, spectral norm by ∥A∥, symmetric positive (negative) definite by A>0(A<0), maximal eigenvalue by λmax(A), and minimal eigenvalue by λmin(A).  A≤B means that matrix B-A is symmetric positive semidefinite.  For two sets X and Y,  X-Y means that the set of all points in X that are not in Y.  For a vector x, we denote the Euclidean norm by ∥x∥ and ∥xt∥s=sup-H≤θ≤0∥x(t+θ)∥2+∥ẋ(t+θ)∥2.  I denotes the identity matrix.  ℜn denotes n-dimensional real space.


## 2. Problem Formulations and Main Results

Consider the following switched neutral system with mixed time delays:(2.1)ẋ(t)-Dẋ(t-τ)=A0σx(t)+A1σx(t-h(t)),t≥0,x(t)=ϕ(t),t∈[-H,0],
where x∈ℜn, xt is state at time t defined by xt(θ)≔x(t+θ), ∀θ∈[-H,0], σ is a switching signal which is a piecewise constant function and may depend on t or x, σ, taking its values in the finite set {1,2,…,N}, and time-varying delay satisfies 0≤h(t)≤hM, ḣ(t)≤hD, hM>0, τ>0, H=max{hM,τ}.  Matrices D, A0i, and  A1i∈ℜn×n,  i=1,2,…,N, are constant, and the initial vector ϕ∈C1, where C1 is the set of differentiable functions from [-H,0]  to ℜn.Now we define some functionsλi(t),  i=1,2,…,N, that will be used to represent our system:
(2.2)λi(t)={1,σ=i,0,otherwise,i=1,2,…,N.
The switched system in (2.1) can be rewritten as follows:
(2.3)ẋ(t)-Dẋ(t-τ)=∑i=1Nλi(t){A0ix(t)+A1ix(t-h(t))},t≥0,x(t)=ϕ(t),t∈[-H,0],
where λi(t) is defined in (2.2) and ∑i=1Nλi(t)=1,∀t≥0.Lemma 2.1 (see [14]).
LetU, V, W, and M be real matrices of appropriate dimensions with M satisfying M=MT, then
(2.4)M+UVW+WTVTUT<0∀VTV≤I,
if and only if there exists a scalar ε>0 such that
(2.5)M+ε-1·UUT+ε·WTW=M+ε-1·UUT+ε-1·(εW)T(εW)<0.Lemma 2.2 (Schur complement of [15]).
For a given matrixS=[S11S12S12TS22] with S11=S11T,  S22=S22T, the following conditions are equivalent:(1)
S<0,(2)
S22<0,  S11-S12S22-1S12T<0.Assumption 2.3.
Assume that there exists a convex combinationF=∑i=1NαiA0i such that F is Hurwitz, where 0≤αi≤1 and ∑i=1Nαi=1.SinceF is Hurwitz, there exist positive definite matrices P and Q satisfying
(2.6)FTP=PF≤-Q.
Define some domains(2.7)Ωi={x∈ℜn:xT(A0iTP+PA0i)x≤-xTQx},i∈{1,2,…,N}.
From the similar proof of [7], it is easy to show ⋃i=1NΩi=ℜn.  Construct some domains(2.8)Ω¯1=Ω1,Ω¯2=Ω2-Ω¯1,Ω¯3=Ω3-Ω¯1-Ω¯2,…,Ω¯N=ΩN-Ω̅1-⋯-Ω̅N-1.
We can obtain ⋃i=1NΩ̅i=ℜn and Ω̅i∩Ω̅j=Φ,  i≠j, where Φ is an empty set.  If  Assumption 2.3 is satisfied, then the following results can be derived:
(2.9)xT(A0iTP+PA0i)x≤-xTQx,x(t)∈Ω̅i.
Define the following switching function:
(2.10)σ=i,∀x∈Ω̅i.Definition 2.4 (see [14]).
The system (2.1) with the designed switching signal is said to be the globally exponentially stabilizable with convergence rate α>0 by the designed switching signal, if there are two positive constants α and Ψ such that
(2.11)∥x(t)∥≤Ψ·e-αt,t≥0.Now we present a result to design the switching signal that guarantees global exponential stability of system (2.1).Theorem 2.5.
Assume that for∥D∥<1, 0<α<-(ln∥D∥)/τ,  0≤αi≤1,  i=1,2,…,N, and ∑i=1Nαi=1, there exist some n×n matrices P,Q,R1,R2>0, such that the following LMI conditions hold for all i=1,2,…,N:
(2.12)Ξi=[Ξ11iΞ12iΞ13i*Ξ22iΞ23i**Ξ33i]<0,FTP+PF+Q<0,F=∑i=1NαiA0i,
where
(2.13)Ξ11i=2α·P+R1+R2-Q,Ξ12i=-2α·PD-A0iTPD,Ξ13i=PA1i,Ξ22i=2α·DTPD-e-2ατ·R1,Ξ23i=-DTPA1i,Ξ33i=-(1-hD)·e-2αhM·R2.
Then the system (2.1) is globally exponentially stabilizable with convergence rate α by the switching signal given in (2.10).Proof.
Define the Lyapunov functional(2.14)V(xt)=e2αt·(x(t)-Dx(t-τ))TP(x(t)-Dx(t-τ))+∫t-τte2αs·xT(s)R1x(s)ds+∫t-h(t)te2αs·xT(s)R2x(s)ds,
where P,R1,R2>0.  The time derivatives of V(xt) along the trajectories of system (2.3) under the switching function (2.10) satisfy
(2.15)V̇(xt)=e2αt·[2α·(x(t)-Dx(t-τ))TP(x(t)-Dx(t-τ))]+e2αt·∑i=1Nλi(t)[(A0ix(t)+A1ix(t-h(t)))TP(x(t)-Dx(t-τ))]+e2αt·∑i=1Nλi(t)[(x(t)-Dx(t-τ))TP(A0ix(t)+A1ix(t-h(t)))]+e2αt·[xT(t)R1x(t)-e-2ατ·xT(t-τ)R1x(t-τ)]+e2αt·[xT(t)R2x(t)-(1-ḣ(t))·e-2αh(t)·xT(t-h(t))R2x(t-h(t))].
By the condition (2.9) and switching function (2.10), we obtain
(2.16)V̇(xt)≤e2αt·∑i=1Nλi(t)·[XT·Ξi·X],
where Ξi,  i=1,2,…,N, are defined in (2.12), XT=[xT(t)xT(t-τ)xT(t-h(t))].  From (2.16) with Ξi<0, we have
(2.17)V(xt)≤V(x0),t≥0,
where
(2.18)V(x0)≤δ1·∥x0∥s2,δ1=λmax(P)(1+∥D∥)2+τ·λmax(R1)+hM·λmax(R2).
From (2.14), we have
(2.19)λmin(P)·e2αt·∥℘(t)∥2≤e2αt·℘T(t)P℘(t)≤V(xt)≤δ1·∥x0∥s2,
where ℘(t)=x(t)-Dx(t-τ).  From (2.19), we can obtain
(2.20)∥x(t)∥=∥℘(t)+Dx(t-τ)∥≤∥D∥·∥x(t-τ)∥+∥℘(t)∥≤∥D∥·∥x(t-τ)∥+δ2·e-αt,t≥0,
where δ2=δ1/λmin(P)·∥x0∥s.  Since ∥D∥<1 and τ>0, we can choose a sufficiently small positive constant ξ=α<-(ln∥D∥)/τ satisfying ∥D∥·eξτ<1.  By the Razumikhin-like approach of [14], we have
(2.21)∥x(t)∥≤[∥x0∥s+δ21-∥D∥eξh]·e-ξt,t≥0.
This completes the proof.Consider the following uncertain switched neutral system with mixed time delays:(2.22a)ẋ(t)-Dẋ(t-τ)=[A0σ+ΔA0σ(t)]x(t)+[A1σ+ΔA1σ(t)]x(t-h(t)),t≥0,(2.22b)x(t)=ϕ(t),t∈[-H,0],
where ΔA0i(t) and ΔA1i(t) are some perturbed matrices and satisfy the following condition:
(2.22c)[ΔA0i(t)ΔA1i(t)]=MiFi(t)[N0iN1i],∀i∈{1,2,…,N},t≥0,
where Mi, N0i, and N1i,  i=1,2,…,N, are some given constant matrices with appropriate dimensions, and Fi(t),  i=1,2,…,N, are unknown matrices representing the parameter perturbation which satisfy
(2.23)FiT(t)Fi(t)≤I,∀i∈{1,2,…,N},t≥0.
The uncertain switched system in (2.22a)–(2.22c) can be rewritten as follows:
(2.24)ẋ(t)-Dẋ(t-τ)=∑i=1Nλi(t){[A0i+ΔA0i(t)]x(t)+[A1i+ΔA1i(t)]x(t-h(t))},t≥0,x(t)=ϕ(t),t∈[-H,0],
where λi(t) is defined in (2.2) and ∑i=1Nλi(t)=1,∀t≥0.Now we consider the exponential stability for uncertain switched system (2.22a)–(2.22c).Theorem 2.6.
Assume that for∥D∥<1,  0<α<-(ln∥D∥)/τ, 0≤αi≤1,  i=1,2,…,N, and ∑i=1Nαi=1, there exist constants εi>0,  i=1,2,…,N, and some n×n matrices P,Q,R1,R2>0, such that the following LMI conditions hold for all i=1,2,…,N:
(2.25)Ξ¯i=[Ξ11iΞ12iΞ13iΞ̅14iΞ̅15i*Ξ22iΞ23iΞ̅24i0**Ξ33i0Ξ̅35i***Ξ̅44i0****Ξ̅55i]<0,FTP+PF+Q<0,F=∑i=1NαiA0i,
where Ξjki,  j,k=1,2,3, are defined in (2.12)
(2.26)Ξ¯14i=PMi,Ξ¯15i=εi·N0iT,Ξ¯24i=-DTPMi,Ξ¯35i=εi·N1iT,Ξ¯44i=Ξ¯55i=-εi·I.
Then the system (2.22a)–(2.22c) is globally exponentially stabilizable with convergence rate α by the switching signal given in (2.10).Proof.
The time derivatives ofV(xt) in (2.14) along the trajectories of system (2.22a)–(2.22c) under the switching function (2.9) satisfy
(2.27)V̇(xt)≤e2αt·∑i=1Nλi(t)·[XT·Ξ̂i·X],
where
(2.28)Ξ̂i=Ξi+[PMi-DTPMi0]Fi(t)[N0i0N1i]+[N0iT0N1iT]FiT(t)[MiTP-MiTPD0].
By Lemmas 2.1 and 2.2, the condition Ξ¯i<0 in (2.25) is equivalent to Ξ̂i<0.  By the same derivation of Theorem 2.5, this proof can be completed.If we choose the convergence rateα=0, we can obtain the following delay-independent condition for the global asymptotic stability of system (2.22a)–(2.22c).Corollary 2.7.
Assume that for∥D∥<1,  0≤αi≤1,  i=1,2,…,N, and ∑i=1Nαi=1, there exist constants εi>0, i=1,2,…,N, some n×n matrices P,Q,R1,R2>0, such that the following LMI conditions hold for all i=1,2,…,N:
(2.29)Ξ̃i=[Ξ̃11iΞ̃12iΞ̃13iΞ̃14iΞ̃15i*Ξ̃22iΞ̃23iΞ̃24i0**Ξ̃33i0Ξ̃35i***Ξ̃44i0****Ξ̃55i]<0,FTP+PF+Q<0,F=∑i=1NαiA0i,
where
(2.30)Ξ̃11i=R1+R2-Q,Ξ̃12i=-A0iTPD,Ξ̃13i=PA1i,Ξ̃14i=PMi,Ξ̃15i=εi·N0iT,Ξ̃22i=-R1,Ξ̃23i=-DTPA1i,Ξ̃24i=-DTPMi,Ξ̃33i=-(1-hD)·R2,Ξ̃35i=εi·N1iT,Ξ̃44i=Ξ̃55i=-εi·I.
Then the system (2.22a)–(2.22c) is globally asymptotically stabilizable by the switching signal given in (2.10).IfD=0, Corollary 2.7 can be reduced to the following corollary.Corollary 2.8.
Assume that for some constantsαi,  i=1,2,…,N, there exist constants εi>0,  i=1,2,…,N, some n×n matrices P,Q,R2>0, such that the following LMI conditions hold for all i=1,2,…,N:
(2.31)Ξ̃̃i=[Ξ̃̃11iΞ̃̃12iΞ̃̃13iΞ̃̃14i*Ξ̃̃22i0Ξ̃̃24i**Ξ̃̃33i0***Ξ̃̃44i]<0,FTP+PF+Q<0,F=∑i=1NαiA0i,
where
(2.32)Ξ̃̃11i=R2-Q,Ξ̃̃12i=PA1i,Ξ̃̃13i=PMi,Ξ̃̃14i=εi·N0iT,Ξ̃̃22i=-(1-hD)·R2,Ξ̃̃24i=εi·N1iT,Ξ̃̃33i=Ξ̃̃44i=-εi·I.
Then the system (2.22a)–(2.22c) is globally asymptotically stabilizable by the switching signal given in (2.10).Assumption 2.9.
Assume that there exists a convex combinationF=∑i=1NαiA0i, some positive definite matrices P and Q, some matrices Ŝi,  i=1,2,…,N, such that
(2.33)FTP+PF+∑i=1Nαi·(Ŝi+ŜiT)<-Q,
where 0≤αi≤1 and ∑i=1Nαi=1.Define some domains(2.34)Ω̂i={x∈ℜn:xT(A0iTP+PA0i+Ŝi+ŜiT)x≤-xTQx},i∈{1,2,…,N}.
From the similar proof of [7], it is easy to show ⋃i=1NΩ̂i=ℜn.  Construct some domains
(2.35)Ω̃1=Ω̂1,Ω̃2=Ω̂2-Ω̃1,Ω̃3=Ω̂3-Ω̃1-Ω̃2,…,Ω̃N=Ω̂N-Ω̃1-⋯-Ω̃N-1.
We can obtain ⋃i=1NΩ̃i=ℜn and Ω̃i∩Ω̃j=Φ,  i≠j, where Φ is an empty set.  If Assumption 2.9 is satisfied, then the following results can be derived:
(2.36)xT(A0iTP+PA0i+Ŝi+ŜiT)x≤-xTQx,x(t)∈Ω̃i.
Define the following switching function:
(2.37)σ=i,∀x∈Ω̃i.Remark 2.10.
In [6, 7, 10], their assumption is given by
(2.38)F1TP+PF1=-Q,
where F1=∑i=1Nαi·(A0i+A1i).  We can see that our Assumption 2.9 is more flexible with Ŝi=PA1i,  ∀i=1,2,…,N.  The main difference of Assumptions 2.3 and 2.9 is that some matrices Ŝi are introduced in Assumption 2.9.  These matrices play a key role to derive the delay-dependent results.Theorem 2.11.
Assume that for∥D∥<1, 0<α<-(ln∥D∥)/τ, 0≤αi≤1,  i=1,2,…,N, and ∑i=1Nαi=1, there exist constants εi>0,  i=1,2,…,N, some n×n matrices P,Q,R1,R2,R3,R4>0, and some n×n matrices R12, U, Ŝi, V1i, V2i, and V3i,  i=1,2,…,N, such that the following LMI conditions hold for all i=1,2,…,N:
(2.39)Σ̅i=[Σ11iΣ12iΣ13iΣ14iΣ15iΣ16iΣ17iΣ18i*Σ22i0Σ24iΣ25i0Σ27i0**Σ33iΣ34iΣ35iΣ36iΣ37i0***Σ44iΣ45i0Σ47i0****Σ55iΣ56i0Σ58i*****Σ66i00******Σ77i0*******Σ88i]<0,FTP+PF+∑i=1Nαi·(Ŝi+ŜiT)+Q<0,F=∑i=1NαiA0i,[R1R12*R2]>0,
where
(2.40)Σ11i=2α·P-Q+R1+R3-V1i-V1iT,Σ12i=R12+A0iTUT,Σ13i=-2αPD-A0iTPD-ŜiTD,Σ14i=-A0iTUTD,Σ15i=PA1i-Ŝi-V2i+V1iT,Σ16i=-Ŝi-V3i+V1iT,Σ17i=PMi,Σ18i=εi·N0iT,Σ22i=R2+hM2·R4-U-UT,Σ24i=(U+UT)D,Σ25i=UA1i,Σ27i=UMi,Σ33i=2αDTPD-e-2ατ·R1,Σ34i=-e-2ατ·R12,Σ35i=-DTPA1i+DTŜi,Σ36i=DTŜi,Σ37i=-DTPMi,Σ44i=-e-2ατ·R2-DT(U+UT)D,Σ45i=-DTUA1i,Σ47i=-DTUMi,Σ55i=-(1-hD)·e-2αhM·R3+V2i+V2iT,Σ56i=V3i+V2iT,Σ58i=εi·N1iT,Σ66i=-e-2αhM·R4+V3i+V3iT,Σ77i=Σ88i=-εi·I.
Then the system (2.22a)–(2.22c) is globally exponentially stabilizable with convergence rate α by the switching signal given in (2.37).Proof.
Define the Lyapunov functional(2.41)V(xt)=e2αt·(x(t)-Dx(t-τ))TP(x(t)-Dx(t-τ))+∫t-τte2αs·[xT(s)ẋT(s)][R1R12*R2][x(s)ẋ(s)]ds+∫t-h(t)te2αs·xT(s)R3x(s)ds+hM·∫t-hMte2αs·(s-(t-hM))ẋT(s)R4ẋ(s)ds,
where P,R3,R4>0, [R1R12*R2]>0.  The time derivatives of V(xt) along the trajectories of system (2.24) satisfy
(2.42)V̇(xt)=e2αt·[2α·(x(t)-Dx(t-τ))TP(x(t)-Dx(t-τ))]+e2αt·∑i=1Nλi(t)[((A0i+Si+ΔA0i(t))x(t)-Si·∫t-h(t)tẋ(s)ds+(A1i+ΔA1i(t)-Si)x(t-h(t))∫t-h(t)tẋ(s)ds)TP(x(t)-Dx(t-τ))∫t-h(t)tẋ(s)ds]+e2αt·∑i=1Nλi(t)[(x(t)-Dx(t-τ))TP((A0i+Si+ΔA0i(t))x(t)-Si·∫t-h(t)tẋ(s)ds+(A1i+ΔA1i(t)-Si)x(t-h(t))∫t-h(t)tẋ(s)ds)∫t-h(t)tẋ(s)ds]+e2αt·[[xT(t)ẋT(t)][R1R12*R2][x(t)ẋ(t)]-e-2ατ·[xT(t-τ)ẋT(t-τ)][R1R12*R2][x(t-τ)ẋ(t-τ)]]+e2αt·[xT(t)R3x(t)-(1-ḣ(t))·e-2αh(t)·xT(t-h(t))R3x(t-h(t))]+e2αt·[hM2·ẋT(t)R4ẋ(t)-hM·∫t-hMte2α(s-t)·ẋT(s)R4ẋ(s)ds],
where  Ŝi=PSi.  By the inequality in [1, page 322], we have
(2.43)-hM·∫t-hMte2αs·ẋT(s)R4ẋ(s)ds≤-h(t)·e2αt·e-2αhM·∫t-h(t)tẋT(s)R4ẋ(s)ds≤-e2αt·e-2αhM·[∫t-h(t)tẋ(s)ds]TR4[∫t-h(t)tẋ(s)ds].
By system (2.24) and Leibniz-Newton formula, we have
(2.44)∑i=1Nλi(t)·e2αt·[∫t-h(t)tẋ(s)ds-x(t)+x(t-h(t))]T[V1ix(t)+V2ix(t-h(t))+V3i∫t-h(t)tẋ(s)ds]+∑i=1Nλi(t)·e2αt·[V1ix(t)+V2ix(t-h(t))+V3i∫t-h(t)tẋ(s)ds]T×[∫t-h(t)tẋ(s)ds-x(t)+x(t-h(t))]=0,-e2αt·(ẋ(t)-Dẋ(t-τ))T(U+UT)(ẋ(t)-Dẋ(t-τ))+∑i=1Nλi(t)·e2αt·(ẋ(t)-Dẋ(t-τ))TU{[A0i+ΔA0i(t)]x(t)+[A1i+ΔA1i(t)]x(t-h(t))}+∑i=1Nλi(t)·e2αt·{[A0i+ΔA0i(t)]x(t)+[A1i+ΔA1i(t)]x(t-h(t))}TUT(ẋ(t)-Dẋ(t-τ))=0.
By the conditions (2.42)–(2.44), we obtain the following result:
(2.45)V̇(xt)≤e2αt·∑i=1Nλi(t)·[XT·Σi·X],
where
(2.46)XT=[xT(t)ẋT(t)xT(t-τ)ẋT(t-τ)xT(t-h(t))∫t-h(t)tẋT(s)ds],Σi=[Σ11iΣ12iΣ13iΣ14iΣ15iΣ16i*Σ22i0Σ24iΣ25i0**Σ33iΣ34iΣ35iΣ36i***Σ44iΣ45i0****Σ55iΣ56i*****Σ66i]+[PMiUMi-DTPMi-DTUMi00]Fi(t)[N0iT000N1iT0]T+[N0iT000N1iT0]FiT(t)[PMiUMi-DTPMi-DTUMi00]T.
By Lemmas 2.1 and 2.2, the condition Σ̅i<0 in (2.39) is equivalent to Σi<0 in (2.45).  From Σi<0 and by the similar derivation of Theorem 2.5, the proof can be completed.IfD=0, Theorem 2.11 can be reduced to the following corollary.Corollary 2.12.
Assume that forα>0, 0≤αi≤1,  i=1,2,…,N, there exist constants εi>0,  i=1,2,…,N, some n×n matrices P,Q,R3,R4>0, some n×n matrices U, Ŝi, V1i, V2i, and V3i,  i=1,2,…,N, such that the following LMI conditions hold for all i=1,2,…,N:
(2.47)Σ̂i=[Σ̂11iΣ̂12iΣ̂13iΣ̂14iΣ̂15iΣ̂16i*Σ̂22iΣ̂23i0Σ̂25i0**Σ̂33iΣ̂34i0Σ̂36i***Σ̂44i00****Σ̂55i0*****Σ̂66i]<0,FTP+PF+∑i=1Nαi·(Ŝi+ŜiT)+Q<0,F=∑i=1NαiA0i,
where
(2.48)Σ̂11i=2α·P-Q+R3-V1i-V1iT,Σ̂12i=A0iTUT,Σ̂13i=PA1i-Ŝi-V2i+V1iT,Σ̂14i=-Ŝi-V3i+V1iT,Σ̂15i=PMi,Σ̂16i=εi·N0iT,Σ̂22i=hM2·R4-U-UT,Σ̂23i=UA1i,Σ̂25i=UMi,Σ̂33i=-(1-hD)·e-2αhM·R3+V2i+V2iT,Σ̂34i=V3i+V2iT,Σ̂36i=εi·N1iT,Σ̂44i=-e-2αhM·R4+V3i+V3iT,Σ̂55i=Σ̂66i=-εi·I.
Then the system (2.22a)–(2.22c) with D=0 is globally exponentially stabilizable with convergence rate α by the switching signal given in (2.37).IfD=Mi=N0i=N1i=0,  i=1,2,…,N, Corollary 2.12 can be reduced to the following corollary.Corollary 2.13.
Assume that forα>0, 0≤αi≤1, i=1,2,…,N, there exist some n×n matrices P,Q,R3,R4>0, some n×n matrices U, Ŝi, V1i, V2i, and V3i,  i=1,2,…,N, such that the following LMI conditions hold for all i=1,2,…,N:
(2.49)Σ̂i=[Σ̂11iΣ̂12iΣ̂13iΣ̂14i*Σ̂22iΣ̂23i0**Σ̂33iΣ̂34i***Σ̂44i]<0,FTP+PF+∑i=1Nαi·(Ŝi+ŜiT)+Q<0,F=∑i=1NαiA0i,
where Σ̂jki,  j,k=1,2,3,4,  i=1,2,…,N, are defined in Corollary 2.12.  Then the system (2.22a)–(2.22c) with D=Mi=N0i=N1i=0 is globally exponentially stabilizable with convergence rate α by the switching signal given in (2.37).Remark 2.14.
By settingα=0 in Theorems 2.5–2.11 and Corollaries 2.7–2.13, the global asymptotic stability for system (2.22a)–(2.22c) can be guaranteed.


## 3. Numerical Examples

Example 3.1.
Consider the system (2.22a)–(2.22c) and the following parameters:
(3.1)N=2,D=[0.100.1-0.1],A01=[-4210],A11=[100.51],A02=[0-1-1-4],A12=[0.500.51],M1=[0.10.2],M2=[0.20.1],N01=N02=[0.10.1],N11=N12=[0.20.1].
By Corollary2.7, a feasible solution of LMI (2.29) with (3.1), α=0, hD=0.2, and α1=α2=0.5 is given by
(3.3)P=[169.08086.96916.9691147.4925],Q=[618.4486-21.7634-21.7634538.3595],R1=[120.4512-58.0284-58.028490.4631],R2=[275.026225.670225.6702180.9190],ε1=145.7753,ε2=180.2272.
Select the switching signal by
(3.4)σ={1,x∈Ω̅1,2,x∈Ω̅2,
where Ω¯1=Ω1, Ω¯2=Ω2-Ω¯1,
(3.5)Ω1={x∈ℜn:-720.2596x12+872.0285x1x2+566.2359x22≤0},Ω2={x∈ℜn:604.5104x12-732.4262x1x2-655.5187x22≤0}.
The switching regions Ω1 and Ω2 are sketched in Figure 1.  The system (2.22a)–(2.22c) with hD=0.2 and (3.1) is globally asymptotically stabilizable by the switching signal (3.4).  Some comparisons are made in Table 1.  The result of this paper provides a major improvement to guarantee the global asymptotic stability of system (2.22a)–(2.22c) with (3.1).Table 1
Comparison with other previous results.
Allowable time-varying delay bounds retaining global asymptotic stability (α=0)of the system (2.22a)–(2.22c) with (3.1) under switching signal (3.4)ResultshD=0hD=0.2hD=0.3Reference [10] (without perturbations and D=0)hM=0.9999998hM=0.9999997hM=0.9999996Our results (with perturbations)hM≥0hM≥0hM≥0Figure 1
Switching regions.Example 3.2.
Consider the system (2.22a)–(2.22c) and the following parameters  [7]:
(3.6)N=2,D=Mi=N0i=N1i=0,A01=[-22-20-2],A11=[-1-7236],A02=[-210-4-2],A12=[4-51-8].
By Corollary2.13, some comparisons with the obtained results for switched system (2.22a)–(2.22c) with (3.6) are made in Table 2.  The results of this paper provide a larger allowable upper bound for time delay to guarantee the global asymptotic stability of system (2.22a)–(2.22c) with (3.6) by the switching signal (2.37).Table 2
Comparison with other previous results.
Allowable time-varying delay bounds retaining global asymptotic stability (α=0)of the system (2.22a)–(2.22c) with (3.6)ResultshD=0hD=0.1hD=0.5hD=0.9hD=1Reference  [7]hM=0.001573No resultsReference  [6]hM=0.00392No resultsReference  [10]hM=0.0202hM=0.0179hM=0.0176hM=0.0176hM=0.0176Our resultshM=0.0309hM=0.0271hM=0.0193hM=0.0181hM=0.0180Example 3.3.
Consider the following switched system with input time delay   [7]:
(3.7)ẋ(t)=A0σx(t)+Bσu(t-h(t)),t≥0,
where
(3.8)N=2,A01=[32-5-1],A02=[-120-22],B1=[-101-1],B2=[110-1].
The feedback control is given by u(t)=Kσ(t)x(t) with
(3.9)K1=[50201],K2=[-3-1424].
For the given feedback control (3.9), system (3.7) can be rewritten as
(3.10)ẋ(t)=A0σx(t)+A1σx(t-h(t)),t≥0,
where A1σ=BσKσ.  As shown in Table 3, the results obtained in this paper provide larger allowable time delay bounds guaranteeing the global stability of system (3.7) with (3.9) by switching signal (2.37).  In [7, 10], the convex combination parameters are chosen by α1=1/3 and α2=2/3.  The convex combination parameters of our results are chosen by α1=0.1 and α2=0.9.Table 3
Comparison with other previous results.
Allowable time-varying delay bounds retaining global asymptoticand exponential stability of the system (3.7) with (3.9)ResultshD=0, α=0hD=0.1, α=0hD=0, α=0.1hD=0, α=0.2Reference  [7]hM=0.008723No resultsReference  [10]hM=0.0189hM=0.0182No resultsOur resultshM=0.03112hM=0.0294hM=0.0296hM=0.0281


## 4. Conclusions

In this paper, the switching signal design for global exponential stability of uncertain switched neutral systems with mixed time delays has been considered.  LMI and Razumikhin-like approaches are used to derive delay-dependent and delay-independent stability criteria.  The results obtained in this paper are less conservative than the previous ones for the numerical examples investigated in this paper.



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*Source: 191760-2009-08-27.xml*