# Robust Control for Uncertain Switched Systems with Interval Nondifferentiable Time-Varying Delays

Authors: T. La-inchua; P. Niamsup
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

This paper addresses the conditions for robust stabilization of a class of uncertain
switched systems with delay. The system to be considered is autonomous and the state
delay is time-varying. Using Lyapunov functional approach, restriction on the derivative
of time-delay function is not required to design switching rule for the robust stabilization
of switched systems with time-varying delays. The delay-dependent stability conditions
are presented in terms of the solution of LMIs which can be solved by various available
algorithms. A numerical example is given to illustrate the effectiveness of theoretical
results.


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## Body

## 1. Introduction

The switched system is a type of hybrid systems that consist of a family of a differential or difference equations and a switching rule to indicate which subsystem will be activated at a specific interval of time. For applications, switched systems can be used to describe several physical or chemical processes which are concerned by more than one dynamics: some systems work at some interval time then stop and other systems take over such as the automatic system in airplane, car energy system, traffic system, and machine industrial system, see [1, 2]. Thus, a switching strategy must be designed in the study of stability of switched systems, also see [3–5].The delay system has been considered in many research, especially the real processes in our world often involve time-delay; that is, the present state depends on the past states which brings more difficulty to investigate the stability of the system, especially time varying delay system, see [6–10]. In general, the following assumption on the derivative of the delay is made, namely h˙(t)<1, see [11, 12] and references cited therein. This assumption may leads to conservativeness; for example, it might not be used when the delay is a fast or a nondifferential time varying function.Moreover, in study of real world applications, the systems are in general influenced by disturbances which might cause inaccuracy of the data. The system can become unstable or less capable because of disturbance. Consequently, the study of robust stability of switched systems with time varying delay becomes important and has been studied by many researchers, see [7, 12–17].In this paper, we study the problem of robust stability for a class of switched systems with time-varying delay. Compared with existing results in the literature, the novelty of our results is twofold. Firstly, the state delay is time-varying in which the restriction on the derivative of the time-delay function is not required to design switching rule in term of a dwell time for the robust stability of the system. Secondly, the obtained conditions for the robust stability are delay-dependent and formulated in terms of the solution of standard LMIs which can be solved by various available algorithms [18]. The paper is organized as follows. Section 2 presents notations, definitions, and auxiliary propositions required for the proof of the main results. Switching design for the robust stability of the system with illustrative examples is presented in Sections 3 and 4, respectively. The paper ends with a conclusion followed by cited references.


## 2. Problem Formulation and Preliminaries

Throughout this paper, the following notations will be used:ℝn—thendimensional Euclidean space;ℝn×n—the  set  of  alln×nreal  matrices;ℕ—the  set  of  all  positive  integers;∥x∥—the  Euclidean  norm  of  vectorx∈ℝn;diag{·}—the  block  diagonal  matrix;I—the  identity  matrix;AT—the  transpose  of  matrixA;A-1—the  inverse  of  matrixA;[AB*C]—* the symmetric form of matrix, namely,*=BT;∥A∥=sup{∥Ax∥:x∈ℝn,∥x∥=1}foranyA∈ℝn×n;Ch2—space  of  continuous  vector-valued  function  defined  on[-h2,0];xt(θ)=x(t+θ),-h2≤θ≤0,wherext∈Ch2;∥xt∥ch2=sup-h2≤θ≤0∥x(t+θ)∥;λmax(A)=max{Re(λ):λiseigenvalueofA};λmin(A)=min{Re(λ):λ is eigenvalue of A};M={1,2,…,k}.Consider the following uncertain switched system with time varying delay:
(2.1)x˙(t)=(Aσ(t)+ΔAσ(t)(t))x(t)+(Dσ(t)+ΔDσ(t)(t))x(t-hσ(t)(t))+(Bσ(t)+ΔBσ(t)(t))uσ(t)(t),t>0,x(t)=ϕ(t),t∈[-h2,0],
where x(t)=[x1(t),x2(t),…,xn(t)]T∈ℝn is the state vector. Ai,Di,Bi, i∈M are known constant matrices, ΔAi,ΔDi,ΔBi are uncertainty matrices which are of the form
(2.2)ΔAi=E1,iF1,i(t)H1,i,ΔDi=E2,iF2,i(t)H2,i,ΔBi=E3,iF3,i(t)H3,i,Fj,iT(t)Fj,i(t)≤I,j=1,2,3,
where F1,i(t),F2,i(t),F3,i(t) are unknown matrices, I is the identity matrix of appropriate dimension. hi(t) is the delay function for an ith subsystem which satisfies the following condition:
(2.3)0≤h1≤h1,i≤hi(t)≤h2,i≤h2.
Let ϕ(t) be an initial condition. σ(t):R+∪{0}→M, t∈[tk,tk+1),k∈ℕ, σ(t) is called the switching signal; we have the switching sequence {xt0;(i0,t0),…∣ik∈M,k=0,1,2,…}, which means that when t∈[tk,tk+1), the ikth subsystem is activated.Definition 2.1.
T
*
=
inf
{
t
l
-
t
l
-
1
} is called the dwell time of switched system.Definition 2.2.
The system (2.1) is said to be stabilizabled if there exists a feedback controller u(t)∈ℝm such that the closed loop switched systems (without uncertainties) is asymptotically stable.Definition 2.3.
The system (2.1) is said to be robustly stabilizable if there exists a feedback controller u(t)∈ℝm such that the closed loop uncertain switched systems are robustly stable.The following lemmas will be used throughout this paper.Lemma 2.4 (Schur complement lemma).
Given constant symmetric matricesQ,SandR∈ℝn×n, where R>0,Q=QT, and R=RT, one has
(2.4)[QSST-R]<0⟺Q+SR-1ST<0.Lemma 2.5 (see [13]).
Givenε>0 and matrices D, E, and F with FTF≤I, one has
(2.5)DEF+ETFTDT≤εDDT+ε-1ETE.Lemma 2.6 (see [13]).
Given a positive definite matrixP∈ℝn×n, any symmetric matrix Q∈ℝn×n and x∈ℝn, then
(2.6)λmin⁡(P-1Q)xTPx≤xTQx≤λmax⁡(P-1Q)xTPx.Lemma 2.7 (see [13]).
For any positive semidefinite matrixM∈ℝn×n, a scalar γ>0, and a vector function ω:[0,γ]→ℝn×n such that the integrals concerned are well-defined, one has
(2.7)(∫0γ‍ω(s)ds)TM(∫0γ‍ω(s)ds)≤γ∫0γ‍ωT(s)Mω(s)ds.Lemma 2.8 (Cauchy inequality).
For any symmetric positive definite matrixN∈Mm×m and x,y∈ℝn, one has
(2.8)±2xTy≤xTNx+yTN-1y.
For simplicity of later presentation, one gives the following notations.(2.9)Σ11,i=PiAiT+AiPi-BiBiT+(2+h12+h22)Qi-Ri;Σ12,i=PiAiT-0.5BiBiT;Σ13,i=0.5Ri;Σ14,i=0.5Ri;Σ15,i=DiPi;Σ22,i=(h1,i2+h2,i2)Ri+δi2Qi-2Pi;Σ25,i=DiPi;Σ33,i=-Qi-0.5Ri-0.5Ui;Σ35,i=0.5Ui;Σ44,i=-Qi-0.5Ri-0.5Ui;Σ45,i=0.5Ui;Σ55,i=-Ui;λ1=max⁡i∈M⁡{λmax⁡(Pi-1)};λ2=min⁡i∈M⁡{λmin⁡(Pi-1)};λ3=max⁡i∈M⁡{λmax⁡(Pi)};λ4=min⁡i∈Ik⁡{λmin⁡(-Ω1,i)};λ5=min⁡i∈M⁡{λmin⁡(-Ω2,i)};λ*=min⁡⁡{λ4,λ5};ha,i=12(h1,i+h2,i);δi=(h2,i-h1,i);h*=min⁡i∈M⁡{h2,i-ha,i};δ=h2-h1.


## 3. Main Results

### 3.1. Asymptotical Stabilization for Nominal Switched Systems with Interval Time-Varying Delay

The nominal switched systems are given by(3.1)x˙(t)=Aσ(t)x(t)+Dσ(t)x(t-hσ(t)(t))+Bσ(t)uσ(t)(t),t>0x(t)=ϕ(t),t∈[-h2,0].
We now state the main result on sufficient condition for stabilization of the switched systems (3.1).Theorem 3.1.
Givenα∈(0,1). If there exists symmetric positive definite matrices Pi,Qi,Ri,Ui such that the following conditions hold:
(3.2)Ω1,i=Ωi-[000-II]TUi[000-II]<0,Ω2,i=Ωi-[00I0-I]TUi[00I0-I]<0,Ωi=[Σ11,iΣ12,iΣ13,iΣ14,iΣ15,i*Σ22,i00Σ25,i**Σ33,i0Σ35,i***Σ44,iΣ45,i****Σ55,i],
and if
(3.3)T*≥1ρln⁡(λ1αλ2),
where ρ=min{h*λ*/δλ3,1/2h2,1/2}, T* is the dwell time, then for any switching rule satisfying (3.3) the switched system (3.1) is stabilizable under the feedback controller
(3.4)ui(t)=-12BiTPi-1x(t),t≥0.Proof.
LetYi=Pi-1,y(t)=Yix(t). Using the feedback controller (3.4), we choose a Lyapunov-Krasovskii functional candidates as
(3.5)Vi(xt)=V1,i(xt)+⋯+V8,i(xt),
where
(3.6)V1,i(xt)=xT(t)Yix(t),V2,i(xt)=∫t-h1,it‍xT(s)YiQiYix(s)ds,V3,i(xt)=∫t-h2,it‍xT(s)YiQiYix(s)ds,V4,i(xt)=h1,i∫-h1,i0‍∫t+st‍x˙T(θ)YiRiYix˙(θ)dθds,V5,i(xt)=h2,i∫-h2,i0‍∫t+st‍x˙T(θ)YiRiYix˙(θ)dθds,V6,i(xt)=δi∫-h2,i-h1,i‍∫t+st‍x˙T(θ)YiUiYix˙(θ)dθds,V7,i(xt)=h1,i∫-h1,i0‍∫t+st‍xT(θ)YiQiYix(θ)dθds,V8,i(xt)=h2,i∫-h2,i0‍∫t+st‍xT(θ)YiQiYix(θ)dθds.
It is easy to see that
(3.7)Vi(xt)≥c1‖x(t)‖2,
for some c1>0. Taking the derivative of Vi(xt) with respect to t along any trajectory of solution of (3.1) yields
(3.8)V˙1,i(xt)=2xT(t)Yix˙(t),=yT(t)[PiAiT+AiPi]y(t)-yT(t)BiTBiy(t)+2yT(t)DiPiy(t-hi(t)),(3.9)V˙2,i(xt)=yT(t)Qiy(t)-yT(t-h1,i)Qiy(t-h1,i),(3.10)V˙3,i(xt)=yT(t)Qiy(t)-yT(t-h2,i)Qiy(t-h2,i),(3.11)V˙4,i(xt)=h1,i2y˙T(t)Riy˙(t)-h1,i2∫t-h1,it‍y˙T(s)Riy˙(s)ds-h1,i2∫t-h1,it‍y˙T(s)Riy˙(s)ds,(3.12)V˙5,i(xt)=h2,i2y˙T(t)Riy˙(t)-h2,i2∫t-h2,it‍y˙T(s)Riy˙(s)ds-h2,i2∫t-h2,it‍y˙T(s)Riy˙(s)ds,(3.13)V˙6,i(xt)=δi2y˙T(t)Uiy˙(t)-δi2∫t-h2,it-h1,i‍y˙T(s)Uiy˙(s)ds-δi2∫t-h2,it-h1,i‍y˙T(s)Riy˙(s)ds,(3.14)V˙7,i(xt)=h1,i2yT(t)Qiy(t)-h1,i∫t-h1,it‍xT(s)YiQiYix(s)ds,(3.15)V˙8,i(xt)=h2,i2yT(t)Qiy(t)-h2,i∫t-h2,it‍xT(s)YiQiYix(s)ds.
Then by applying Lemma 2.7 and Leibniz-Newton formular, we have
(3.16)-h1,i2∫t-h1,it‍y˙T(s)Riy˙(s)ds≤yT(t)(-Ri2)y(t)+yT(t)Riy(t-h1,i)+yT(t-h1,i)(-Ri2)y(t-h1,i),-h2,i2∫t-h2,it‍y˙T(s)Riy˙(s)ds≤yT(t)(-Ri2)y(t)+yT(t)Riy(t-h2,i)+yT(t-h2,i)(-Ri2)y(t-h2,i).
Note that
(3.17)-δi2∫t-h2,it-h1,i‍y˙T(s)Uiy˙(s)ds=-δi2∫t-h2,it-hi(t)‍y˙T(s)Uiy˙(s)ds-δi2∫t-hi(t)t-h1,i‍y˙T(s)Uiy˙(s)ds=-h2,i-hi(t)2∫t-h2,it-h(t)‍y˙T(s)Uiy˙(s)ds-h(t)-h1,i2×∫t-h2,it-hi(t)‍y˙T(s)Uiy˙(s)ds-hi(t)-h1,i2∫t-hi(t)t-h1,i‍y˙T(s)Uiy˙(s)ds-h2,i-hi(t)2∫t-h(t)t-h1,i‍y˙T(s)Uiy˙(s)ds.
Using Lemma 2.7 yields
(3.18)-h2,i-hi(t)2∫t-h2,it-hi(t)‍y˙T(s)Uiy˙(s)ds≤[y(t-hi(t))-y(t-h2,i)]T(-Ui2)[y(t-hi(t))-y(t-h2,i)],-hi(t)-h1,i2∫t-hi(t)t-h1,i‍y˙T(s)Uiy˙(s)ds≤[y(t-h1,i)-y(t-hi(t))]T(-Ui2)[y(t-h1,i)-y(t-hi(t))].
Let β=(h2,i-hi(t))/(h2,i-h1,i)≤1. Then
(3.19)-h2,i-hi(t)2∫t-hi(t)t-h1,i‍y˙T(s)Uiy˙(s)ds=-β∫t-hi(t)t-h1,i‍(h2,i-h1,i)y˙T(s)(Ui2)y˙(s)ds≤-β∫t-hi(t)t-h1,i‍(hi(t)-h1,i)y˙T(s)(Ui2)y˙(s)ds≤-β[y(t-h1,i)-y(t-hi(t))]T(Ui2)×[y(t-h1,i)-y(t-hi(t))].-hi(t)-h1,i2∫t-h2,it-hi(t)‍y˙T(s)Uiy˙(s)ds=-(1-β)∫t-h2,it-hi(t)‍(h2,i-h1,i)y˙T(s)(Ui2)y˙(s)ds≤-(1-β)∫t-h2,it-hi(t)‍(h2,i-hi(t))y˙T(s)(Ui2)y˙(s)ds≤-(1-β)[y(t-hi(t))-y(t-h2,i)]T(Ui2)×[y(t-hi(t))-y(t-h2,i)].
Therefore from (3.18)-(3.19), we have
(3.20)-δi2∫t-h2,it-h1,i‍y˙T(s)Uiy˙(s)ds≤[y(t-hi(t))-y(t-h2,i)]T(-Ui2)[y(t-hi(t))-y(t-h2,i)]+[y(t-hi(t))-y(t-h2,i)]T(-Ui2)[y(t-hi(t))-y(t-h2,i)]-β[y(t-h1,i)-y(t-hi(t))]T(Ui2)[y(t-h1,i)-y(t-hi(t))]-(1-β)[y(t-hi(t))-y(t-h2,i)]T(Ui2)×[y(t-hi(t))-y(t-h2,i)].
Furthermore, from the following zero equation
(3.21)-Piy˙(t)+AiPiy(t)+DiPiy(t-hi(t))-0.5BiBiTy(t)=0,
we obtain
(3.22)-2y˙TPiy˙(t)+2y˙TAiPiy(t)+2y˙TDiPiy(t-hi(t))-2y˙T0.5BiBiTy(t)=0.
Hence, from (3.5),(3.8)–(3.16),(3.20), and (3.22), we can get
V˙i(xt)≤ξT(t)Ωiξ(t)-β[y(t-h1,i)-y(t-h(t))]T(Ui2)[y(t-h1,i)-y(t-hi(t))]-(1-β)[y(t-hi(t))-y(t-h2,i)]T(Ui2)[y(t-hi(t))-y(t-h2,i)]-h1,i2∫t-h1,it‍y˙T(s)Riy˙(s)ds-h2,i2∫t-h2,it‍y˙T(s)Riy˙(s)ds-δi2∫t-h2,it-h1,i‍y˙T(s)Riy˙(s)ds-h1,i∫t-h1,it‍xT(s)YiQiYix(s)ds-h2,i∫t-h2,it‍xT(s)YiQiYix(s)ds=ξT(t)[(1-β)Ω1,i+βΩ2,i]ξ(t)-h1,i2∫t-h1,it‍y˙T(s)Riy˙(s)ds-h2,i2∫t-h2,it‍y˙T(s)Riy˙(s)ds-δi2∫t-h2,it-h1,i‍y˙T(s)Riy˙(s)ds,-h1,i∫t-h1,it‍xT(s)YiQiYix(s)ds-h2,i∫t-h2,it‍xT(s)YiQiYix(s)ds,
where ξT(t)=[yT(t)y˙T(t)yT(t-h1,i)yT(t-h2,i)yT(t-hi(t))].
Supposeτ is the time when the system switches from state j to state i, that is I(τ+) = i and I(τ-) = j, where τ+ and τ- are the right and the left limit of the time τ, respectively. According to Lemma 2.6, we obtain
(3.24)V1,i(xτ+)=xT(τ)Yix(τ)≤λ1λ2λ2xT(τ)x(τ)≤λ1λ2xT(τ)Pjx(τ)=λ1λ2V1,j(xτ-).
We can apply this argument to integral terms in the Lypunov-Krasovskii function, so we get
(3.25)Vi(xτ+)≤λ1λ2Vj(xτ-).
Now let ν be the time when the system switches from state k to state j, that is, I(ν+)=j and I(ν-)=k, where ν+ and ν- are a right and a left limit of the time ν, respectively. In order to show that the switched system is stable, we need to compare Vi(xτ+) with Vj(xν+) and estimate the upper bound for the term ξT(t)[(1-β)Ω1,i+βΩ2,i]ξ(t) in the inequality (3.23). Hence we consider two following possible cases.
Case  1 (0≤h1≤h1,i≤h(t)≤ha,i≤h2,i≤h2). Since 0≤β≤1,Ω1,i<0 and Ω2,i<0, we have
(3.26)ξT(t)[(1-β)Ω1,i+βΩ2,i]ξ(t)≤ξT(t)[βΩ2,i]ξ(t)≤-(h2,i-ha,iδi)λ5(‖y(t)‖2+‖y˙(t)‖2+‖y(t-h1,i)‖2+‖y(t-h2,i)‖2+‖y(t-hi(t))‖2)≤-(h2,i-ha,iδi)λ5(‖y(t)‖2)=-(h2,i-ha,iδi)λ5λ3λ3(‖y(t)‖2)≤-(h2,i-ha,iδ)λ5λ3xT(t)Pi-1PiPi-1x(t)≤-(h2,i-ha,iδ)λ5λ3V1,i≤-(h2,i-ha,iδ)λ*λ3V1,i≤-(h*δ)λ*λ3V1,i.
Case  2 (0≤h1≤h1,i≤ha,i≤h(t)≤h2,i≤h2). Since 0≤β≤1,Ω1,i<0 and Ω2,i<0, we get
ξT(t)[(1-β)Ω1,i+βΩ2,i]ξ(t)≤ξT(t)[(1-β)Ω1,i]ξ(t)≤-(ha,i-h1,iδi)λ4(‖y(t)‖2+‖y˙(t)‖2+‖y(t-h1,i)‖2+‖y(t-h2,i)‖2+‖y(t-hi(t))‖2)≤-(ha,i-h1,iδi)λ4(‖y(t)‖2)=-(ha,i-h1,iδi)λ4λ3λ3(∥y(t)∥2)≤-(ha,i-h1,iδ)λ4λ3xT(t)Pi-1PiPi-1x(t)≤-(ha,i-h1,iδ)λ4λ3V1,i.
Note that h2,i-ha,i=ha,i-h1,i, so we obtain
(3.28)ξT(t)[(1-β)Ω1,i+βΩ2,i]ξ(t)≤-(h2,i-ha,iδ)λ*λ3V1,i≤-(h*δ)λ*λ3V1,i.
Moreover, from the definition of V2,i, …, V8,i, we can get
(3.29)-h1,i2∫t-h1,it‍y˙T(s)Riy˙(s)ds≤-12∫-h1,i0‍∫t+st‍x˙T(θ)YiRiYix˙(θ)dθds≤-12h2,iV4,i,-h2,i2∫t-h2,it‍y˙T(s)Riy˙(s)ds≤-12∫-h2,i0‍∫t+st‍x˙T(θ)YiRiYix˙(θ)dθds≤-12h1,iV5,i,-δi2∫t-h2,it-h1,i‍y˙T(s)Uiy˙(s)ds≤-12∫-h2,i-h1,i‍∫t+st‍x˙T(θ)YiUiYix˙(θ)dθds≤-12δiV6,i.
Since
(3.30)-h1,i∫t-h1,it‍xT(s)YiQiYix(s)ds≤-∫-h1,i0‍∫t+st‍xT(θ)YiQiYix(θ)dθds≤-1h1,iV7,i,
we can get
(3.31)-h1,i∫t-h1,it‍xT(s)YiQiYix(s)ds-h1,i∫t-h1,it‍xT(s)YiQiYix(s)ds≤-1h1,iV7,i-h1,i∫t-h1,it‍xT(s)YiQiYix(s)ds.
So we obtain
(3.32)-h1,i∫t-h1,it‍xT(s)YiQiYix(s)ds≤-12h1,iV7,i-12V2,i.
Similar to (3.32), we have
(3.33)-h2,i∫t-h2,it‍xT(s)YiQiYix(s)ds≤-12h2,iV3,i-12V8,i.
According to (3.23), (3.28)–(3.33), we have
(3.34)V˙i(xt)≤-min⁡⁡{h*λ*δλ3,12h1,i,12h2,i,12δi,12}Vi(xt)≤-min⁡⁡{h*λ*δλ3,12h2,12}Vi(xt)=-ρVi(xt).
As a result, we get
(3.35)V˙j(xt)≤-ρVj(xt),
which yields
(3.36)1Vj(xt)dVj(xt)≤-ρdt.
Integrating (3.36) on (ν,τ), we obtain
(3.37)Vj(xτ-)≤Vj(xν+)e-ρ(τ-ν).
From (3.25) and (3.37), we have
(3.38)Vi(xτ+)≤λ1λ2Vj(xν+)e-ρ(τ-ν).
Since
(3.39)τ-ν≥T0≥1ρln⁡(λ1αλ2),
we obtain from (3.38) that
(3.40)Vi(xτ+)≤αVj(xν+).
Let N(t) be the number of time the system is switched on [0,t). Since the switched system (3.1) undergoes infinite times of switches on [0,∞), we obtain limt→∞N(t)=∞. Assume that I(t)=i and I(0)=i0. Then, according to (3.7) and (3.40), we have
(3.41)c1∥x(t)∥2≤Vi(xt)≤αN(t)Vi0(x0).
Since 0<α<1 and limt→∞N(t)=∞, it follows from (3.41) that ∥x(t)∥→0 as t→∞. We conclude that the zero solution of (3.1) is stabilizable.


### 3.2. Robust Stabilization for Switched Systems with Interval Time-Varying Delay

Theorem 3.2.
Givenα∈(0, 1). If there exists symmetric positive definite matrices Pi,Qi,Ri,Ui such that the following conditions hold:
(3.42)Ω1,i=Ωi-[000-II]TUi[000-II]<0,(3.43)Ω2,i=Ωi-[00I0-I]TUi[00I0-I]<0,(3.44)Ω3,i=[Ω311,iPiH1,iTPiH1,iTBiH3,iT2BiH3,iT2*-ϵ1I000**-ϵ2I00***-ϵ5I20****-ϵ6I2]<0,(3.45)Ω4,i=[Ω411,iPiH2,iTPiH2,iT*-ϵ3I0**-ϵ4I]<0,(3.46)Ωi=[Λ11,iΛ12,iΛ13,iΛ14,iΛ15,i*Λ22,i00Λ25,i**Λ33,i0Λ35,i***Λ44,iΛ45,i****Λ55,i],
where
(3.47)Λ11,i=PiAiT+AiPi-0.5BiBiT+(2+h1,i2+h2,i2)Qi-0.5Ri+ϵ1E1TE1+ϵ3E2TE2+ϵ52E3TE3;Λ12,i=PiAiT-0.5BiBiT;Λ13,i=0.5Ri;Λ14,i=0.5Ri;Λ15,i=DiPi;Λ22,i=(h1,i2+h2,i2+δi2)Qi-2Ri+ϵ2E1TE1+ϵ4E2TE2+ϵ62E3TE3;Λ25,i=DiPi;Λ33,i=-Qi-0.5Ri-0.5Ui;Λ35,i=0.5Ui;Λ44,i=-Qi-0.5Ri-0.5Ui;Λ45,i=0.5Ui;Λ55,i=-0.5Ui;Ω311,i=-0.5BiBiT-0.5Ri;Ω411,i=-0.5Ui.
Then for any switching rule satisfying (3.3) the switched system (2.1) is robustly stabilizable under the feedback controller given by (3.4).Proof.
Construct Lyapunov-Krasovskii functional as in (3.5), we can proof this theorem by using a similar argument as in the proof of Theorem 3.1. By replacing Ai,Di,Bi in (3.8) with Ai+E1,iF1,i(t)H1,i, Di+E2,iF2,i(t)H2,i, Bi+E3,iF3,i(t)H3,i, respectively, we obtain
V˙i(xt)≤yT(t)[Pi(Ai+E1,iF1,i(t)H1,i)T+(Ai+E1,iF1,i(t)H1,i)Pi]y(t)+2yT(t)(Di+E2,iF2,i(t)H2,i)Piy(t-hi(t))-yT(t)BiTBiy(t)-yT(t)E3,iF3,i(t)H3,iBiTy(t)+yT(t)(2+h1,i2+h2,i2)Qiy(t)-yT(t-h1)Qiy(t-h1)-yT(t-h2)Qiy(t-h2)+y˙T(t)[(h1,i2+h1,i2)Ri+δi2Ui]y˙(t)+yT(t)(-Ri2)y(t)+yT(t)Riy(t-h1,i)+yT(t-h1,i)(-Ri2)y(t-h1,i)+yT(t)(-Ri2)y(t)+yT(t)Riy(t-h2,i)+yT(t-h2,i)(-Ri2)y(t-h2,i)+[y(t-hi(t))-y(t-h2,i)]T(-Ui2)[y(t-hi(t))-y(t-h2,i)]+[y(t-hi(t))-y(t-h2,i)]T(-Ui2)[y(t-hi(t))-y(t-h2,i)]-β[y(t-h1,i)-y(t-hi(t))]T(-Ui2)[y(t-h1,i)-y(t-hi(t))]-(1-β)[y(t-hi(t))-y(t-h2,i)]T(-Ui2)[y(t-hi(t))-y(t-h2,i)]-2y˙TPiy˙(t)+2y˙T(Ai+E1,iF1,i(t)H1,i)Piy(t)+2y˙T(Di+E2,iF2,i(t)H2,i)Pi×y(t-hi(t))-2y˙T0.5BiBiTy(t)-y˙TE3,iF3,i(t)H3,iBiTy(t)-h1,i2∫t-h1,it‍y˙T(s)Riy˙(s)ds-h2,i2∫t-h2,it‍y˙T(s)Riy˙(s)ds-δi2∫t-h2,it-h1,i‍y˙T(s)Riy˙(s)ds-h1,i∫t-h1,it‍xT(s)YiQiYix(s)ds-h2,i∫t-h2,it‍xT(s)YiQiYix(s)ds.
And using Lemma 2.5, we can get the following upper bounds for the uncertain terms in (3.48):
(3.49)2yT(t)E1,iF1,i(t)H1,iPiy(t)≤ϵ1,iyT(t)E1,iTE1,iy(t)+ϵ1,i-1yT(t)PiH1,iTH1,iPiy(t),2y˙T(t)E1,iF1,i(t)H1,iPiy(t)≤ϵ2,iy˙T(t)E1,iTE1,iy˙(t)+ϵ2,i-1yT(t)PiH1,iTH1,iPiy(t),2yT(t)E2,iF2,i(t)H2,iPiy(t-hi(t))≤ϵ3,iyT(t)E2,iTE2,iy(t)+ϵ3,i-1yT(t-hi(t))PiH2,iTH2,iPiy(t-hi(t)),2y˙T(t)E2,iF2,i(t)H2,iPiy(t-hi(t))≤ϵ4,iy˙T(t)E2,iTE2,iy˙(t)+ϵ4,i-1yT(t-hi(t))PiH2,iTH2,iPiy(t-hi(t)),-yT(t)E3,iF3,i(t)H3,iBiTy(t)≤ϵ5,i2yT(t)E3,iTE3,iy(t)+ϵ5,i-12yT(t)BiH3,iTH3,iBiy(t),-y˙T(t)E3,iF3,i(t)H3,iBiTy(t)≤ϵ6,i2y˙T(t)E3,iTE3,iy˙(t)+ϵ6,i-12yT(t)BiH3,iTH3,iBiy(t).
According to (3.48)-(3.49), it follows that
(3.50)V˙i(xt)≤ξT(t)[(1-β)Ω1,i+βΩ2,i]ξ(t)+yT(t)Ψ3,iy(t)+yT(t-hi(t))Ψ4,iy(t-hi(t))-h1,i2∫t-h1,it‍y˙T(s)Riy˙(s)ds-h2,i2∫t-h2,it‍y˙T(s)Riy˙(s)ds-δi2∫t-h2,it-h1,i‍y˙T(s)Riy˙(s)ds,-h1,i∫t-h1,it‍xT(s)YiQiYix(s)ds-h2,i∫t-h2,it‍xT(s)YiQiYix(s)ds,
where
(3.51)Ψ3,i=Ω311,i+ε1,i-1PiH1,iTH1,iPi+ϵ2,i-1PiH1,iTH1,iPi+ϵ5,i-12BiH3,iTH3,iBiT+ϵ6,i-12BiH3,iTH3,iBiT,Ψ4,i=Ω311,i+ϵ3,i-1PiH2,iTH2,iPi+ϵ4,i-1PiH2,iTH2,iPi.
Applying Lemma 2.4, the LMIs Ψ3,i<0 and Ψ4,i<0 are equivalent to Ω3,i<0 and Ω4,i<0, respectively. Similarly to the previous theorem we can conclude that the switched system (2.1) is robustly stabilizable.Remark 3.3.
Our proposed method can remove the conservative restrictions on the derivatives of the time-varying delays, meanwhile traditional design methods require the conditionh˙(t)≤hD (see [14, 19, 20]). So our method can deal with fast time-varying delays. Moreover, the improved method need not to introduce any free-weighting matrix variables which turn out to be less conservative than results in [14, 20, 21].


## 3.1. Asymptotical Stabilization for Nominal Switched Systems with Interval Time-Varying Delay

The nominal switched systems are given by(3.1)x˙(t)=Aσ(t)x(t)+Dσ(t)x(t-hσ(t)(t))+Bσ(t)uσ(t)(t),t>0x(t)=ϕ(t),t∈[-h2,0].
We now state the main result on sufficient condition for stabilization of the switched systems (3.1).Theorem 3.1.
Givenα∈(0,1). If there exists symmetric positive definite matrices Pi,Qi,Ri,Ui such that the following conditions hold:
(3.2)Ω1,i=Ωi-[000-II]TUi[000-II]<0,Ω2,i=Ωi-[00I0-I]TUi[00I0-I]<0,Ωi=[Σ11,iΣ12,iΣ13,iΣ14,iΣ15,i*Σ22,i00Σ25,i**Σ33,i0Σ35,i***Σ44,iΣ45,i****Σ55,i],
and if
(3.3)T*≥1ρln⁡(λ1αλ2),
where ρ=min{h*λ*/δλ3,1/2h2,1/2}, T* is the dwell time, then for any switching rule satisfying (3.3) the switched system (3.1) is stabilizable under the feedback controller
(3.4)ui(t)=-12BiTPi-1x(t),t≥0.Proof.
LetYi=Pi-1,y(t)=Yix(t). Using the feedback controller (3.4), we choose a Lyapunov-Krasovskii functional candidates as
(3.5)Vi(xt)=V1,i(xt)+⋯+V8,i(xt),
where
(3.6)V1,i(xt)=xT(t)Yix(t),V2,i(xt)=∫t-h1,it‍xT(s)YiQiYix(s)ds,V3,i(xt)=∫t-h2,it‍xT(s)YiQiYix(s)ds,V4,i(xt)=h1,i∫-h1,i0‍∫t+st‍x˙T(θ)YiRiYix˙(θ)dθds,V5,i(xt)=h2,i∫-h2,i0‍∫t+st‍x˙T(θ)YiRiYix˙(θ)dθds,V6,i(xt)=δi∫-h2,i-h1,i‍∫t+st‍x˙T(θ)YiUiYix˙(θ)dθds,V7,i(xt)=h1,i∫-h1,i0‍∫t+st‍xT(θ)YiQiYix(θ)dθds,V8,i(xt)=h2,i∫-h2,i0‍∫t+st‍xT(θ)YiQiYix(θ)dθds.
It is easy to see that
(3.7)Vi(xt)≥c1‖x(t)‖2,
for some c1>0. Taking the derivative of Vi(xt) with respect to t along any trajectory of solution of (3.1) yields
(3.8)V˙1,i(xt)=2xT(t)Yix˙(t),=yT(t)[PiAiT+AiPi]y(t)-yT(t)BiTBiy(t)+2yT(t)DiPiy(t-hi(t)),(3.9)V˙2,i(xt)=yT(t)Qiy(t)-yT(t-h1,i)Qiy(t-h1,i),(3.10)V˙3,i(xt)=yT(t)Qiy(t)-yT(t-h2,i)Qiy(t-h2,i),(3.11)V˙4,i(xt)=h1,i2y˙T(t)Riy˙(t)-h1,i2∫t-h1,it‍y˙T(s)Riy˙(s)ds-h1,i2∫t-h1,it‍y˙T(s)Riy˙(s)ds,(3.12)V˙5,i(xt)=h2,i2y˙T(t)Riy˙(t)-h2,i2∫t-h2,it‍y˙T(s)Riy˙(s)ds-h2,i2∫t-h2,it‍y˙T(s)Riy˙(s)ds,(3.13)V˙6,i(xt)=δi2y˙T(t)Uiy˙(t)-δi2∫t-h2,it-h1,i‍y˙T(s)Uiy˙(s)ds-δi2∫t-h2,it-h1,i‍y˙T(s)Riy˙(s)ds,(3.14)V˙7,i(xt)=h1,i2yT(t)Qiy(t)-h1,i∫t-h1,it‍xT(s)YiQiYix(s)ds,(3.15)V˙8,i(xt)=h2,i2yT(t)Qiy(t)-h2,i∫t-h2,it‍xT(s)YiQiYix(s)ds.
Then by applying Lemma 2.7 and Leibniz-Newton formular, we have
(3.16)-h1,i2∫t-h1,it‍y˙T(s)Riy˙(s)ds≤yT(t)(-Ri2)y(t)+yT(t)Riy(t-h1,i)+yT(t-h1,i)(-Ri2)y(t-h1,i),-h2,i2∫t-h2,it‍y˙T(s)Riy˙(s)ds≤yT(t)(-Ri2)y(t)+yT(t)Riy(t-h2,i)+yT(t-h2,i)(-Ri2)y(t-h2,i).
Note that
(3.17)-δi2∫t-h2,it-h1,i‍y˙T(s)Uiy˙(s)ds=-δi2∫t-h2,it-hi(t)‍y˙T(s)Uiy˙(s)ds-δi2∫t-hi(t)t-h1,i‍y˙T(s)Uiy˙(s)ds=-h2,i-hi(t)2∫t-h2,it-h(t)‍y˙T(s)Uiy˙(s)ds-h(t)-h1,i2×∫t-h2,it-hi(t)‍y˙T(s)Uiy˙(s)ds-hi(t)-h1,i2∫t-hi(t)t-h1,i‍y˙T(s)Uiy˙(s)ds-h2,i-hi(t)2∫t-h(t)t-h1,i‍y˙T(s)Uiy˙(s)ds.
Using Lemma 2.7 yields
(3.18)-h2,i-hi(t)2∫t-h2,it-hi(t)‍y˙T(s)Uiy˙(s)ds≤[y(t-hi(t))-y(t-h2,i)]T(-Ui2)[y(t-hi(t))-y(t-h2,i)],-hi(t)-h1,i2∫t-hi(t)t-h1,i‍y˙T(s)Uiy˙(s)ds≤[y(t-h1,i)-y(t-hi(t))]T(-Ui2)[y(t-h1,i)-y(t-hi(t))].
Let β=(h2,i-hi(t))/(h2,i-h1,i)≤1. Then
(3.19)-h2,i-hi(t)2∫t-hi(t)t-h1,i‍y˙T(s)Uiy˙(s)ds=-β∫t-hi(t)t-h1,i‍(h2,i-h1,i)y˙T(s)(Ui2)y˙(s)ds≤-β∫t-hi(t)t-h1,i‍(hi(t)-h1,i)y˙T(s)(Ui2)y˙(s)ds≤-β[y(t-h1,i)-y(t-hi(t))]T(Ui2)×[y(t-h1,i)-y(t-hi(t))].-hi(t)-h1,i2∫t-h2,it-hi(t)‍y˙T(s)Uiy˙(s)ds=-(1-β)∫t-h2,it-hi(t)‍(h2,i-h1,i)y˙T(s)(Ui2)y˙(s)ds≤-(1-β)∫t-h2,it-hi(t)‍(h2,i-hi(t))y˙T(s)(Ui2)y˙(s)ds≤-(1-β)[y(t-hi(t))-y(t-h2,i)]T(Ui2)×[y(t-hi(t))-y(t-h2,i)].
Therefore from (3.18)-(3.19), we have
(3.20)-δi2∫t-h2,it-h1,i‍y˙T(s)Uiy˙(s)ds≤[y(t-hi(t))-y(t-h2,i)]T(-Ui2)[y(t-hi(t))-y(t-h2,i)]+[y(t-hi(t))-y(t-h2,i)]T(-Ui2)[y(t-hi(t))-y(t-h2,i)]-β[y(t-h1,i)-y(t-hi(t))]T(Ui2)[y(t-h1,i)-y(t-hi(t))]-(1-β)[y(t-hi(t))-y(t-h2,i)]T(Ui2)×[y(t-hi(t))-y(t-h2,i)].
Furthermore, from the following zero equation
(3.21)-Piy˙(t)+AiPiy(t)+DiPiy(t-hi(t))-0.5BiBiTy(t)=0,
we obtain
(3.22)-2y˙TPiy˙(t)+2y˙TAiPiy(t)+2y˙TDiPiy(t-hi(t))-2y˙T0.5BiBiTy(t)=0.
Hence, from (3.5),(3.8)–(3.16),(3.20), and (3.22), we can get
V˙i(xt)≤ξT(t)Ωiξ(t)-β[y(t-h1,i)-y(t-h(t))]T(Ui2)[y(t-h1,i)-y(t-hi(t))]-(1-β)[y(t-hi(t))-y(t-h2,i)]T(Ui2)[y(t-hi(t))-y(t-h2,i)]-h1,i2∫t-h1,it‍y˙T(s)Riy˙(s)ds-h2,i2∫t-h2,it‍y˙T(s)Riy˙(s)ds-δi2∫t-h2,it-h1,i‍y˙T(s)Riy˙(s)ds-h1,i∫t-h1,it‍xT(s)YiQiYix(s)ds-h2,i∫t-h2,it‍xT(s)YiQiYix(s)ds=ξT(t)[(1-β)Ω1,i+βΩ2,i]ξ(t)-h1,i2∫t-h1,it‍y˙T(s)Riy˙(s)ds-h2,i2∫t-h2,it‍y˙T(s)Riy˙(s)ds-δi2∫t-h2,it-h1,i‍y˙T(s)Riy˙(s)ds,-h1,i∫t-h1,it‍xT(s)YiQiYix(s)ds-h2,i∫t-h2,it‍xT(s)YiQiYix(s)ds,
where ξT(t)=[yT(t)y˙T(t)yT(t-h1,i)yT(t-h2,i)yT(t-hi(t))].
Supposeτ is the time when the system switches from state j to state i, that is I(τ+) = i and I(τ-) = j, where τ+ and τ- are the right and the left limit of the time τ, respectively. According to Lemma 2.6, we obtain
(3.24)V1,i(xτ+)=xT(τ)Yix(τ)≤λ1λ2λ2xT(τ)x(τ)≤λ1λ2xT(τ)Pjx(τ)=λ1λ2V1,j(xτ-).
We can apply this argument to integral terms in the Lypunov-Krasovskii function, so we get
(3.25)Vi(xτ+)≤λ1λ2Vj(xτ-).
Now let ν be the time when the system switches from state k to state j, that is, I(ν+)=j and I(ν-)=k, where ν+ and ν- are a right and a left limit of the time ν, respectively. In order to show that the switched system is stable, we need to compare Vi(xτ+) with Vj(xν+) and estimate the upper bound for the term ξT(t)[(1-β)Ω1,i+βΩ2,i]ξ(t) in the inequality (3.23). Hence we consider two following possible cases.
Case  1 (0≤h1≤h1,i≤h(t)≤ha,i≤h2,i≤h2). Since 0≤β≤1,Ω1,i<0 and Ω2,i<0, we have
(3.26)ξT(t)[(1-β)Ω1,i+βΩ2,i]ξ(t)≤ξT(t)[βΩ2,i]ξ(t)≤-(h2,i-ha,iδi)λ5(‖y(t)‖2+‖y˙(t)‖2+‖y(t-h1,i)‖2+‖y(t-h2,i)‖2+‖y(t-hi(t))‖2)≤-(h2,i-ha,iδi)λ5(‖y(t)‖2)=-(h2,i-ha,iδi)λ5λ3λ3(‖y(t)‖2)≤-(h2,i-ha,iδ)λ5λ3xT(t)Pi-1PiPi-1x(t)≤-(h2,i-ha,iδ)λ5λ3V1,i≤-(h2,i-ha,iδ)λ*λ3V1,i≤-(h*δ)λ*λ3V1,i.
Case  2 (0≤h1≤h1,i≤ha,i≤h(t)≤h2,i≤h2). Since 0≤β≤1,Ω1,i<0 and Ω2,i<0, we get
ξT(t)[(1-β)Ω1,i+βΩ2,i]ξ(t)≤ξT(t)[(1-β)Ω1,i]ξ(t)≤-(ha,i-h1,iδi)λ4(‖y(t)‖2+‖y˙(t)‖2+‖y(t-h1,i)‖2+‖y(t-h2,i)‖2+‖y(t-hi(t))‖2)≤-(ha,i-h1,iδi)λ4(‖y(t)‖2)=-(ha,i-h1,iδi)λ4λ3λ3(∥y(t)∥2)≤-(ha,i-h1,iδ)λ4λ3xT(t)Pi-1PiPi-1x(t)≤-(ha,i-h1,iδ)λ4λ3V1,i.
Note that h2,i-ha,i=ha,i-h1,i, so we obtain
(3.28)ξT(t)[(1-β)Ω1,i+βΩ2,i]ξ(t)≤-(h2,i-ha,iδ)λ*λ3V1,i≤-(h*δ)λ*λ3V1,i.
Moreover, from the definition of V2,i, …, V8,i, we can get
(3.29)-h1,i2∫t-h1,it‍y˙T(s)Riy˙(s)ds≤-12∫-h1,i0‍∫t+st‍x˙T(θ)YiRiYix˙(θ)dθds≤-12h2,iV4,i,-h2,i2∫t-h2,it‍y˙T(s)Riy˙(s)ds≤-12∫-h2,i0‍∫t+st‍x˙T(θ)YiRiYix˙(θ)dθds≤-12h1,iV5,i,-δi2∫t-h2,it-h1,i‍y˙T(s)Uiy˙(s)ds≤-12∫-h2,i-h1,i‍∫t+st‍x˙T(θ)YiUiYix˙(θ)dθds≤-12δiV6,i.
Since
(3.30)-h1,i∫t-h1,it‍xT(s)YiQiYix(s)ds≤-∫-h1,i0‍∫t+st‍xT(θ)YiQiYix(θ)dθds≤-1h1,iV7,i,
we can get
(3.31)-h1,i∫t-h1,it‍xT(s)YiQiYix(s)ds-h1,i∫t-h1,it‍xT(s)YiQiYix(s)ds≤-1h1,iV7,i-h1,i∫t-h1,it‍xT(s)YiQiYix(s)ds.
So we obtain
(3.32)-h1,i∫t-h1,it‍xT(s)YiQiYix(s)ds≤-12h1,iV7,i-12V2,i.
Similar to (3.32), we have
(3.33)-h2,i∫t-h2,it‍xT(s)YiQiYix(s)ds≤-12h2,iV3,i-12V8,i.
According to (3.23), (3.28)–(3.33), we have
(3.34)V˙i(xt)≤-min⁡⁡{h*λ*δλ3,12h1,i,12h2,i,12δi,12}Vi(xt)≤-min⁡⁡{h*λ*δλ3,12h2,12}Vi(xt)=-ρVi(xt).
As a result, we get
(3.35)V˙j(xt)≤-ρVj(xt),
which yields
(3.36)1Vj(xt)dVj(xt)≤-ρdt.
Integrating (3.36) on (ν,τ), we obtain
(3.37)Vj(xτ-)≤Vj(xν+)e-ρ(τ-ν).
From (3.25) and (3.37), we have
(3.38)Vi(xτ+)≤λ1λ2Vj(xν+)e-ρ(τ-ν).
Since
(3.39)τ-ν≥T0≥1ρln⁡(λ1αλ2),
we obtain from (3.38) that
(3.40)Vi(xτ+)≤αVj(xν+).
Let N(t) be the number of time the system is switched on [0,t). Since the switched system (3.1) undergoes infinite times of switches on [0,∞), we obtain limt→∞N(t)=∞. Assume that I(t)=i and I(0)=i0. Then, according to (3.7) and (3.40), we have
(3.41)c1∥x(t)∥2≤Vi(xt)≤αN(t)Vi0(x0).
Since 0<α<1 and limt→∞N(t)=∞, it follows from (3.41) that ∥x(t)∥→0 as t→∞. We conclude that the zero solution of (3.1) is stabilizable.


## 3.2. Robust Stabilization for Switched Systems with Interval Time-Varying Delay

Theorem 3.2.
Givenα∈(0, 1). If there exists symmetric positive definite matrices Pi,Qi,Ri,Ui such that the following conditions hold:
(3.42)Ω1,i=Ωi-[000-II]TUi[000-II]<0,(3.43)Ω2,i=Ωi-[00I0-I]TUi[00I0-I]<0,(3.44)Ω3,i=[Ω311,iPiH1,iTPiH1,iTBiH3,iT2BiH3,iT2*-ϵ1I000**-ϵ2I00***-ϵ5I20****-ϵ6I2]<0,(3.45)Ω4,i=[Ω411,iPiH2,iTPiH2,iT*-ϵ3I0**-ϵ4I]<0,(3.46)Ωi=[Λ11,iΛ12,iΛ13,iΛ14,iΛ15,i*Λ22,i00Λ25,i**Λ33,i0Λ35,i***Λ44,iΛ45,i****Λ55,i],
where
(3.47)Λ11,i=PiAiT+AiPi-0.5BiBiT+(2+h1,i2+h2,i2)Qi-0.5Ri+ϵ1E1TE1+ϵ3E2TE2+ϵ52E3TE3;Λ12,i=PiAiT-0.5BiBiT;Λ13,i=0.5Ri;Λ14,i=0.5Ri;Λ15,i=DiPi;Λ22,i=(h1,i2+h2,i2+δi2)Qi-2Ri+ϵ2E1TE1+ϵ4E2TE2+ϵ62E3TE3;Λ25,i=DiPi;Λ33,i=-Qi-0.5Ri-0.5Ui;Λ35,i=0.5Ui;Λ44,i=-Qi-0.5Ri-0.5Ui;Λ45,i=0.5Ui;Λ55,i=-0.5Ui;Ω311,i=-0.5BiBiT-0.5Ri;Ω411,i=-0.5Ui.
Then for any switching rule satisfying (3.3) the switched system (2.1) is robustly stabilizable under the feedback controller given by (3.4).Proof.
Construct Lyapunov-Krasovskii functional as in (3.5), we can proof this theorem by using a similar argument as in the proof of Theorem 3.1. By replacing Ai,Di,Bi in (3.8) with Ai+E1,iF1,i(t)H1,i, Di+E2,iF2,i(t)H2,i, Bi+E3,iF3,i(t)H3,i, respectively, we obtain
V˙i(xt)≤yT(t)[Pi(Ai+E1,iF1,i(t)H1,i)T+(Ai+E1,iF1,i(t)H1,i)Pi]y(t)+2yT(t)(Di+E2,iF2,i(t)H2,i)Piy(t-hi(t))-yT(t)BiTBiy(t)-yT(t)E3,iF3,i(t)H3,iBiTy(t)+yT(t)(2+h1,i2+h2,i2)Qiy(t)-yT(t-h1)Qiy(t-h1)-yT(t-h2)Qiy(t-h2)+y˙T(t)[(h1,i2+h1,i2)Ri+δi2Ui]y˙(t)+yT(t)(-Ri2)y(t)+yT(t)Riy(t-h1,i)+yT(t-h1,i)(-Ri2)y(t-h1,i)+yT(t)(-Ri2)y(t)+yT(t)Riy(t-h2,i)+yT(t-h2,i)(-Ri2)y(t-h2,i)+[y(t-hi(t))-y(t-h2,i)]T(-Ui2)[y(t-hi(t))-y(t-h2,i)]+[y(t-hi(t))-y(t-h2,i)]T(-Ui2)[y(t-hi(t))-y(t-h2,i)]-β[y(t-h1,i)-y(t-hi(t))]T(-Ui2)[y(t-h1,i)-y(t-hi(t))]-(1-β)[y(t-hi(t))-y(t-h2,i)]T(-Ui2)[y(t-hi(t))-y(t-h2,i)]-2y˙TPiy˙(t)+2y˙T(Ai+E1,iF1,i(t)H1,i)Piy(t)+2y˙T(Di+E2,iF2,i(t)H2,i)Pi×y(t-hi(t))-2y˙T0.5BiBiTy(t)-y˙TE3,iF3,i(t)H3,iBiTy(t)-h1,i2∫t-h1,it‍y˙T(s)Riy˙(s)ds-h2,i2∫t-h2,it‍y˙T(s)Riy˙(s)ds-δi2∫t-h2,it-h1,i‍y˙T(s)Riy˙(s)ds-h1,i∫t-h1,it‍xT(s)YiQiYix(s)ds-h2,i∫t-h2,it‍xT(s)YiQiYix(s)ds.
And using Lemma 2.5, we can get the following upper bounds for the uncertain terms in (3.48):
(3.49)2yT(t)E1,iF1,i(t)H1,iPiy(t)≤ϵ1,iyT(t)E1,iTE1,iy(t)+ϵ1,i-1yT(t)PiH1,iTH1,iPiy(t),2y˙T(t)E1,iF1,i(t)H1,iPiy(t)≤ϵ2,iy˙T(t)E1,iTE1,iy˙(t)+ϵ2,i-1yT(t)PiH1,iTH1,iPiy(t),2yT(t)E2,iF2,i(t)H2,iPiy(t-hi(t))≤ϵ3,iyT(t)E2,iTE2,iy(t)+ϵ3,i-1yT(t-hi(t))PiH2,iTH2,iPiy(t-hi(t)),2y˙T(t)E2,iF2,i(t)H2,iPiy(t-hi(t))≤ϵ4,iy˙T(t)E2,iTE2,iy˙(t)+ϵ4,i-1yT(t-hi(t))PiH2,iTH2,iPiy(t-hi(t)),-yT(t)E3,iF3,i(t)H3,iBiTy(t)≤ϵ5,i2yT(t)E3,iTE3,iy(t)+ϵ5,i-12yT(t)BiH3,iTH3,iBiy(t),-y˙T(t)E3,iF3,i(t)H3,iBiTy(t)≤ϵ6,i2y˙T(t)E3,iTE3,iy˙(t)+ϵ6,i-12yT(t)BiH3,iTH3,iBiy(t).
According to (3.48)-(3.49), it follows that
(3.50)V˙i(xt)≤ξT(t)[(1-β)Ω1,i+βΩ2,i]ξ(t)+yT(t)Ψ3,iy(t)+yT(t-hi(t))Ψ4,iy(t-hi(t))-h1,i2∫t-h1,it‍y˙T(s)Riy˙(s)ds-h2,i2∫t-h2,it‍y˙T(s)Riy˙(s)ds-δi2∫t-h2,it-h1,i‍y˙T(s)Riy˙(s)ds,-h1,i∫t-h1,it‍xT(s)YiQiYix(s)ds-h2,i∫t-h2,it‍xT(s)YiQiYix(s)ds,
where
(3.51)Ψ3,i=Ω311,i+ε1,i-1PiH1,iTH1,iPi+ϵ2,i-1PiH1,iTH1,iPi+ϵ5,i-12BiH3,iTH3,iBiT+ϵ6,i-12BiH3,iTH3,iBiT,Ψ4,i=Ω311,i+ϵ3,i-1PiH2,iTH2,iPi+ϵ4,i-1PiH2,iTH2,iPi.
Applying Lemma 2.4, the LMIs Ψ3,i<0 and Ψ4,i<0 are equivalent to Ω3,i<0 and Ω4,i<0, respectively. Similarly to the previous theorem we can conclude that the switched system (2.1) is robustly stabilizable.Remark 3.3.
Our proposed method can remove the conservative restrictions on the derivatives of the time-varying delays, meanwhile traditional design methods require the conditionh˙(t)≤hD (see [14, 19, 20]). So our method can deal with fast time-varying delays. Moreover, the improved method need not to introduce any free-weighting matrix variables which turn out to be less conservative than results in [14, 20, 21].


## 4. Numerical Examples

In this section, we provide some examples to illustrate the effectiveness of our results in Theorems3.1 and 3.2.Example 4.1.
Consider the nominal switched systems with interval time-varying delay(4.1)x˙(t)=Aix(t)+Dix(t-hi(t))+Biui(t),i=1,2,
with
(4.2)A1=[-2000.5],D1=[-0.20.10.1-0.1],B1=[01],A2=[-3100.1],D2=[-0.10.10.1-0.1],B2=[01].
And we choose α=0.9,h1(t)=0.1+|sint|,h2(t)=0.1+|cost|, so we get h1=0.1,h2=1.1. Then, by using the LMI control toolbox in Matlab, solutions of LMIs (3.2) are given by
(4.3)P1=[0.8273-0.0052-0.00520.8167],Q1=[0.24450.00400.00400.0665],R1=[0.12380.00300.00300.5581],U1=[0.2249-0.0084-0.00840.3873],P2=[3.47351.24441.24444.2337],Q2=[0.71320.15630.15630.1691],R2=[0.44290.97770.97772.9654],U2=[0.56840.42590.42591.8023],
with stabilizing controllers
(4.4)u1(t)=[-0.0039-0.6122]x(t),u2(t)=[0.0473-0.1320]x(t).
By computation, we obtain λ*=0.0461, λ1=1.2276, λ2=0.1940, λ3=5.1547, h*=0.5. Thus, ρ=0.0054 and from (3.3) we obtain
(4.5)T0≥361.1694.
Hence, from Theorem 3.1, the switched systems (4.1) with arbitrary switching law subject to (4.5) are stabilizable under the feedback controllers which are shown in (4.4).
By choosing initial condition asϕ(t)=[sin(t)sin(t)],t∈[-1.1,0], the trajectories of solutions of the switched system and the trajectories of solutions of subsystems 1 and 2 for this example are shown in Figures 1, 2, 3, and 4.
In Tables1 and 2 we give comparison of maximum allowable value of h2 of asymptotic stability of nominal switched systems obtained in [14, 22] and Theorem 3.1. As we can see that for some of h1, the maximum allowable bounds for h2 obtained from Theorem 3.1 are greater than that obtained in [14, 22]. More important, the differentiability of the time delay h(t) is not required in our theorem.Table 1
Maximum allowable upper boundsh2 of the time-varying delay for different values of the lower bounds h1 in example 1 of  [8].
h
2
Result
h
1
Our results
[8]
[4]
No restriction onh˙(t)
h
d
=
0.4
0
1.2644
—
1.2843
h
d
≥
1 or unknown
0
0.9339
—
1.2843
h
d
≥
1 or unknown
0.3
—
0.9514
1.3120
h
d
≥
1 or unknown
0.6
—
1.04
1.3913Table 2
Maximum allowable upper boundsh2 of the time-varying delay for different values of the lower bounds h1 in example 2 of [4].
h
2
Result
h
1
Our results
[4]
No restriction onh˙(t)
h
d
≥
1 or unknown
0
0.687
0.9871
h
d
≥
1 or unknown
0.4
0.856
1.0227Figure 1
The trajectory of solution of systemi=1.Figure 2
The trajectory of solution of systemi=2.Figure 3
The trajectory of solution of systemi= 1 under the feedback controller (4.4).Figure 4
The trajectory of solution of systemi= 2 under the feedback controller (4.4).Example 4.2.
Consider the following uncertain switched systems with interval time-varying delay(4.6)x˙(t)=(Ai+ΔAi)x(t)+(Di+ΔDi)x(t-hi(t))+(Bi+ΔBi)ui(t),i=1,2,
with
(4.7)A1=[-2100.1],D1=[0.10.10.1-0.1],B1=[00.1],A2=[-3000.1],D2=[-0.10.10.1-0.1],B2=[00.1],(4.8)E1,i=E2,i=E3,i=[0.020.01-0.010.01],H1,i=H2,i=[0.02000.01],H3,i=[0.010],i=1,2.
As an illustration, we choose α=0.9,ϵj,i=1 for j=1,2,…6,i=1,2,
(4.9)h(t)=0.1+|sin⁡t|,h2(t)=0.1+|cos⁡t|,F1,1(t)=F1,2(t)=F2,1(t)=F2,2(t)=[sin⁡t00sin⁡t].
In this case, we can take h1=0.1,hM=1.1. Then, by using the LMI control toolbox in Matlab, solutions of LMIs (3.42) and (3.45) are given by
(4.10)P1=[0.04380.00460.00460.0081],Q1=[0.01530.00050.00050.0001],R1=[0.00370.00050.00050.0010],U1=[0.0122-0.0005-0.00050.0006],P2=[0.0272-0.0010-0.00100.0088],Q2=[0.00590.00010.00010.0001],R2=[0.0006-0.0002-0.00020.0022],U2=[0.0038-0.0010-0.00100.0012],
with stabilizing controllers
(4.11)u1(t)=[0.6907-6.5520]x(t),u2(t)=[-0.1997-5.6729]x(t).
By computation, we obtain λ*=0.000007,λ1=132.7976,λ2=22.5048,λ3=0.0444,h*=0.5 Thus, ρ=7.8829×10-5, and from (3.3) we obtain
(4.12)T0≥2.3855×104.
Hence, from Theorem 3.2, the switched systems (4.6) with arbitrary switching law subject to (4.12) are robustly stabilizable under the feedback controllers which are shown in (4.11).
By choosing the same initial condition as in Example4.1, the trajectories of solutions of the switched system and the trajectories of solutions of subsystems 1 and 2 for this example are shown in Figures 5, 6, 7, and 8.Figure 5
The trajectory of solution of systemi=1.Figure 6
The trajectory of solution of systemi=2.Figure 7
The trajectory of solution of systemi=1 under the feedback controller (4.12).Figure 8
The trajectory of solution of systemi=2 under the feedback controller (4.12).


## 5. Conclusion

In this paper, we study the problem of robust stabilization for a class of switched systems with time-varying delay. Comparing with some existing results in the literature, the novelty of our results is twofold. Firstly, the state delay is time-varying in which the restriction on the derivative of the time-delay function is not required to design switching rule for the robust stability of the system. Secondly, the obtained conditions for the robust stability are delay-dependent and formulated in terms of the solution of standard LMIs which can be solved by various available algorithms. Numerical example is given to illustrate the effectiveness of the theoretical result.



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*Source: 718618-2012-08-15.xml*