# Exponential Stabilization of Impulsive Switched Systems with Time Delays Using Guaranteed Cost Control

Authors: Xiang Xie; Honglei Xu; Rong Zhang
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

This paper investigates the stabilization problem for impulsive switched systems with time delays. First, exponential stability criteria of the delayed impulsive switched systems are established by use of the Lyapunov-Krasovskii functional method. Based on these results, sufficient conditions for the existence of a guaranteed cost control are also given. Subject to these sufficient conditions, the closed-loop impulsive switched system under the guaranteed cost control law will be exponentially stable with a guaranteed cost value.


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## Body

## 1. Introduction

During the evolution process of a variety of disciplines such as engineering, economics, and biology, there often exists an impressive characteristic that the system state changes rapidly due to state jumping and the coexistence of continuous dynamics and discrete events. These jumping effects and switching phenomena can be modeled as an impulsive switched system. The system stability problem is one of main problems for such impulsive switched systems.Recently, there are many results on stability analysis of impulsive switched systems. For instance, the robust stabilization problem for a class of impulsive switched systems under the LQ guaranteed cost control is studied in [1]. In addition, sufficient conditions for the existence of a guaranteed cost control law are given as well. In [2], sufficient conditions, independent of time delays and impulsive switching intervals, are derived by using a Lyapunov-Krasovskii’s technique, ensuring asymptotical stability of impulsive switched systems with time invariant delays. Moreover, stability criteria problems for uncertain impulsive switched systems with input delay are studied by using the receding horizon method in [3]. Appropriate switching controllers are designed and linear matrix inequality conditions are derived to guarantee the closed-loop uncertain impulsive switched systems under a designed delayed controller. Other relevant references can be found in [4–9].On the other hand, time delay has become a common phenomenon frequently occurred in science and engineering. Stability of time delay systems has received increasing attention among the applied mathematics and control community; for example, see [10–12] and the references therein. Most of these results are derived by use of either the Razumikhin method or the Lyapunov-Krasovskii method.Since both impulses and time delays are ubiquitous in the real world, it is necessary to analyze their stability performance of dynamical systems involving impulses, time delays, and switchings. There already existed some effective results for the stability analysis of either impulsive switched systems or time delay systems mentioned above. However, there are few results on impulsive switched systems with time delays. This motivates our research in this paper. Our aim is to apply a set of improved Lyapunov functions to analyze the exponential stability performance of the impulsive switched systems with time delays, to derive sufficient conditions to ensure the exponential stability, and to design guaranteed cost controllers based on the linear matrix inequality approach. The designed guaranteed cost controller can ensure not only exponential stability of the closed-loop system but also the guaranteed cost performance.The remaining sections of the paper proceed as follows. Section2 provides necessary propositions and definitions. In Section 3, sufficient conditions for ensuring the closed-loop system exponentially stable are presented, and the designed guaranteed cost controller for impulsive switched systems with time delays are presented. Finally, conclusion remarks are presented in Section 4.


## 2. Problem Statement

Consider the following impulsive switched system with time delay:(1a)x˙(t)=A^ikx(t)+B^ikx(t-h(t))+Ciku(t)t≠tk,(1b)Δx(t)=Dkx(t)t=tk,(1c)x(t)=φ(t)-1≤t≤0,(1d)A^ik=Aik+ΔAik,B^ik=Bik+ΔBik,where x(t)∈Rn is the state, u(t)∈RP is the control input, n,p∈N. Aik, Bik, Cik, and Dk are constant real matrices with appropriate dimensions. The delay function h(t) satisfies the condition
(2)0≤h1≤h(t)≤h2.Δx(t)=x(t+)-x(t-), x(t+)=lim⁡v→0+⁡x(t+v), lim⁡v→0+⁡x(t-v)=x(tk-)=x(tk), where ik∈{1,2,3,…,m}, k∈N, m∈N. tk is an impulsive switching point, t0<t1<t2<⋯<tk<⋯<t∞. ΔAik(·) and ΔBik(·) are unknown real norm-bounded matrix valued functions, representing time-varying parameter uncertainties. The admissible uncertainties are assumed to be of the form
(3)[ΔAik(t)ΔBik(t)]=EikΨik(t)[HikJik].Eik, Hik, Jik are known real constant matrices, Ψik(t) is an unknown real time-varying matrix satisfying ΨikT(t)Ψik(t)<I, and I is the identity matrix with appropriate dimension.Consider the following cost function:(4)J=∑k=0∞∫tktk+1[xT(t)Rikx(t)+uT(t)Qiku(t)]dt,
where Rik>0 and Qik>0 are positive definite symmetric matrices.Definition 1.
Suppose that there exist a control lawu(t)=Fikx(t) and a positive constant J* such that, for all admissible uncertainties, the closed-loop system ((1a), (1b), (1c), and (1d)) is exponentially stable and the cost function (4) satisfies J≤J*, and then u(t) is the exponential guaranteed cost control law for the impulsive switched system with time delay. Here, J* is the guaranteed cost value.In addition, the following lemmas are necessary to proceed.Lemma 2 (see [13, Proposition 2.2]).
LetA, D, E, and F be real matrices of appropriate dimensions with ∥F∥≤1. Then, for any matrix P=PT>0 and scalar ε>0,
(5)εI-EPET>0,(A+DFE)P(A+DFE)T≤APAT+APET(εI-EPET)-1EPAT+εDDT.Lemma 3 (Schur complement lemma (see [14])).
Given constant matricesX, Y, and Z with appropriate dimensions satisfying X=XT and Y=YT>0. then X+ZTY-1Z<0 if and only if
(6)(XZTZ-Y)<0.Lemma 4 (integral inequality [15]).
For any constant matrixZ=ZT>0 and a scalar b such that the following integrations are well defined,
(7)b∫t-btx(s)TZx(s)ds≥(∫t-btx(s)ds)TZ(∫t-btx(s)ds).


## 3. Main Results

First, we will present sufficient conditions for exponential stability of the impulsive switched system ((1a), (1b), (1c), and (1d)) without control input.Theorem 5.
Consider the time-delayed impulsive switched system ((1a), (1b), (1c), and (1d)) with the cost function (4). Suppose there exist α>0, ε1>0, Pik=PikT>0, such that the following conditions are satisfied: (i)
(8)[k110λ2S1λ3S20*k22λ3S3λ3S30**k3300***k440****k55]<0,(ii)
(9)[Pik-1(I+Dk)TPikPik(I+Dk)Pik]>0,
where,(10)k11=A^ikTPik+PikA^ik+U1+U2-λ2S1-λ3S2+2αPik+Rik,k22=B^ikTPik+PikB^ik-2λ3S3,k33=-λ2U1-λ2S1-λ3S3,k44=-λ3U2-λ3S2-λ3S3,k55=h12S1+h22S2+(h2-h1)2S3.
Then, the impulsive switched system ((1a), (1b), (1c), and (1d)) is exponentially stable.Proof.
Whent∈(tk,tk+1], consider the Lyapunov-Krasovskii function candidate
(11)V(t,xt)=∑i=14Vi(t,xt),
where
(12)V1(t,xt)=xT(t)Pikx(t),V2(t,xt)=∫t-h1te2α(s-t)xT(s)U1x(s)ds+∫t-h2te2α(s-t)xT(s)U2x(s)ds,V3(t,xt)=h1∫-h10∫t+ste2α(τ-s)x˙T(τ)S1x˙(τ)dτds+h2∫-h20∫t+ste2α(τ-s)x˙T(τ)S2x˙(τ)dτds,V4(t,xt)=(h2-h1)×∫-h2-h1∫t+ste2α(τ-s)x˙T(τ)S3x˙(τ)dτds.
Let
(13)Ω=λmax⁡(Pik)+h1λmax⁡(U1)+h2λmax⁡(U2)+0.5h13λmax⁡(S1)+0.5h23λmax⁡(S2)+(h22-h12)(h2-h1)λmax⁡(S3);
then, we can easily check that
(14)λ∥x(t)∥2≤V(t,xt)≤Ω∥x(t)∥2,t∈R+.
Taking the derivative of Vi(t,xt), i=1,2,3,4, along the trajectory of the closed-loop system, we obtain
(15)V1˙(t,xt)=[A^ikx(t)+B^ikx(t-h(t))]TPikx(t)+xT(t)Pik[A^ikx(t)+B^ikx(t-h(t))]=xT(t)[A^ikTPik+PikA^ik]x(t)+xT(t-h(t))B^ikTPikx(t)+xT(t)PikB^ikTx(t-h(t)),(16)V2˙(t,xt)=-2αe-2αt[∫t-h1te2αsxT(s)U1x(s)ds+∫t-h2te2αsxT(s)U2x(s)ds]+e-2αt[e2αtxT(t)U1x(t)-e2α(t-h1)xT(t-h1)×U1x(t-h1)+e2αtxT(t)U2x(t)-e2α(t-h2)xT(t-h2)U1x(t-h2)]=-2αV2+xT(t)(U1+U2)x(t)-λ2xT(t-h1)U1x(t-h1)-λ3xT(t-h2)U1x(t-h2),(17)V3˙(t,xt)=-2αV3+e-2αtddt[h1∫-h10∫t+ste2ατx˙T(τ)S1x˙(τ)dτds+h2∫-h20∫t+ste2ατx˙T(τ)S2x˙(τ)dτds]=-2αV3+x˙T(t)(h12S1+h22S2)x˙(t)-h1∫-h10e2αsx˙T(t+s)S1x˙(t+s)ds-h2∫-h20e2αsx˙T(t+s)S2x˙(t+s)ds.
From Lemma 4, we have
(18)V3˙(t,xt)≤-2αV3+x˙T(t)(h12S1+h22S2)x˙(t)-xT(t)(λ2S1+λ3S2)x(t)+2λ2xT(t)S1x(t-h1)-λ2xT(t-h1)S1x(t-h1)+2λ3xT(t)S2x(t-h2)-λ3xT(t-h2)S2x(t-h2),(19)V4˙(t,xt)=-2αV4+e-2αt(h2-h1)×∫-h2-h1[e2αtx˙T(t)S3x˙(t)-e2α(t+s)x˙T(t+s)S3x˙(t+s)]ds=-2αV4+(h2-h1)2x˙T(t)S3x˙(t)-(h2-h1)∫-h2-h1e2αsx˙T(t+s)S3x˙(t+s)ds≤-2αV4+(h2-h1)2x˙T(t)S3x˙(t)-(h2-h1)e-2αh2[∫-h2-hx˙T(t+s)S3x˙(t+s)ds+∫-h-h1x˙T(t+s)S3x˙(t+s)ds]≤-2αV4+(h2-h1)2x˙T(t)S3x˙(t)-λ3[2xT(t-h(t))S3x(t-h(t))-2xT(t-h(t))S3x(t-h2)+xT(t-h2)S3x(t-h2)+xT(t-h1)S3x(t-h1)-2xT(t-h(t))S3x(t-h1)].
Then, we add (15) to (19) together to yield
(20)V˙(t,xt)+2αV(t,xt)≤xT(t)[A^ikTPik+PikA^ik+U1+U2+2αPik-λ2S1-λ3S2+RikA^ikT]x(t)+xT(t-h(t))[B^ikTPik+PikB^ikT-2λ3S3]×x(t-h(t))+xT(t-h1)[-λ2U1-λ2S1-λ3S3]x(t-h1)+xT(t-h2)[-λ3U2-λ3S2-λ3S3]x(t-h2)+2λ3xT(t)S2x(t-h2)+2λ2xT(t)S1x(t-h1)+2λ3xT(t-h(t))S3x(t-h2)+2λ3xT(t-h(t))S3x(t-h1)+x˙T(t)[h12S1+h22S2+(h2-h1)2S3]x˙(t)-xT(t)Rikx(t).
Let ξT(t)=[xT(t),xT(t-h(t)),xT(t-h1),xT(t-h2),x˙T(t)],
(21)k=[k110λ2S1λ3S20*k22λ3S3λ3S30**k3300***k440****k55].
We have
(22)V˙(t,xt)+2αV(t,xt)≤ξT(t)kξ(t)-xT(t)Rikx(t).
Since k<0 from ((1a), (1b), (1c), and (1d)), it is clear that V˙(t,xt)+2αV(t,xt)<0 or, equivalently, V(t,xt)≤V(0,x0)e-2αt, ∀t∈R+.
Taking (14) into account, we have
(23)λ∥x(t)∥2≤V(t,xt)≤V(0,x0)e-2αt≤Ωe-2αt∥x(0)∥2,
and hence
(24)∥x(t)∥≤Ωλ∥x(0)∥e-αt,
which shows that the impulsive switched system ((1a), (1b), (1c), and (1d)) is exponentially stable when t∈(tk,tk+1].
Next, let us consider the case at the impulsive and switching time pointtk,tk=1,2,3,…,
(25)V(tk+)-V(tk-)=x(tk+)TPikx(tk+)-x(tk-)TPik-1x(tk-).
Noting that Δx(tk)=x(tk+)-x(tk-), x(tk-)=x(tk), we have
(26)V(tk+)-V(tk-)=x(tk+)TPikx(tk+)-x(tk)TPik-1x(tk)=x(tk)T[(I+Dk)TPik(I+Dk)-Pik-1]x(tk).
From (ii), we know that [Pik-1(I+Dk)TPikPik(I+Dk)Pik]>0, which is equivalent to (I+Dk)TPik(I+Dk)-Pik-1<0.
So,V(tk+)-V(tk-)<0. Therefore, we can conclude that the impulsive switched system ((1a), (1b), (1c), and (1d)) is exponentially stable.
Theorem5 develops sufficient conditions for exponential stability of the impulsive switched system ((1a), (1b), (1c), and (1d)) under the case that u(t)=0. In the next theorem, we will design the guaranteed cost controller in the form of u(t)=Fikx(t) and sufficient conditions for the existence of a guaranteed cost control are also presented as well.Theorem 6.
Consider the impulsive switched system with time delay ((1a), (1b), (1c), and (1d)) with the cost function (4) and suppose there exist α>0, ε1>0, Pik=PikT>0. If there exist symmetric positive definite matrices U1, U2, S1, S2, S3, and K such that the following conditions are satisfied:(i)
(27)[ε1IEikTPikPikEikPik]>0,(ii)
(28)[Π1Π2*Π3]<0,(iii)
(29)[Pik-1(I+Dk)TPikPik(I+Dk)Pik]>0,
where(30)Π1=[w11~0λ2S1λ3S20*w22~λ3S3λ3S30**w3300***w440****w55],Π2=[AikTEikKK(Cik-1)THikT00000(Bik+ΔBik)T000000000000000],Π3=diag⁡(EikTEik-ε1I,-I,-Qik-1,-1ε1I,-I),w11~=AikTAik+3Pik2+U1+U2-λ2S1-λ3S2+2αPik+Rik,w22~=-2λ3S3,w33=-λ2U1-λ2S1-λ3S3,w44=-λ3U2-λ3S2-λ3S3,w55=h12S1+h22S2+(h2-h1)2S3,λ1=e-2α(h1+h2),λ2=e-2αh1,λ3=e-2αh2,λ=λmin⁡(Pik),
then the guaranteed cost control law u(t)=Cik-1KTx(t) can ensure that the impulsive switched system ((1a), (1b), (1c), and (1d)) is exponentially stable. Moreover, the guaranteed cost value is
(31)J*=V(0,x0).Proof.
Whent∈(tk,tk+1], choose the Lyapunov-Krasovskii function candidate to be similar to Theorem 5(32)V(t,xt)=∑i=14Vi(t,xt),
where
(33)V1(t,xt)=xT(t)Pikx(t),V2(t,xt)=∫t-h1te2α(s-t)xT(s)U1x(s)ds+∫t-h2te2α(s-t)xT(s)U2x(s)ds,V3(t,xt)=h1∫-h10∫t+ste2α(τ-s)x˙T(τ)S1x˙(τ)dτds+h2∫-h20∫t+ste2α(τ-s)x˙T(τ)S2x˙(τ)dτds,V4(t,xt)=(h2-h1)×∫-h2-h1∫t+ste2α(τ-s)x˙T(τ)S3x˙(τ)dτds.
Taking the derivative of Vi(t,xt), i=1,2,3,4, along the trajectory of the closed-loop system, we obtain
(34)V1˙(t,xt)=[A^ikx(t)+B^ikx(t-h(t))+Ciku(t)]TPikx(t)+xT(t)Pik[A^ikx(t)+B^ikx(t-h(t))+Ciku(t)].
Put (1d) and u(t)=Fikx(t) into it and then we can get
(35)V1˙(t,xt)=xT(t)[(Aik+ΔAik)TPik+Pik(Aik+ΔAik)+FikTCikTPik+PikCikFik(Aik+ΔAik)T]x(t)+xT(t-h(t))(Bik+ΔBik)TPikx(t)+xT(t)Pik(Bik+ΔBik)x(t-h(t)).
Next, it follows from Lemma 2 that
(36)(Aik+ΔAik)TPik+Pik(Aik+ΔAik)≤(Aik+ΔAik)T(Aik+ΔAik)+Pik2≤AikTAik+AikTEik(ε1I-EikTEik)-1EikTAik+ε1HikTHik+Pik2,xT(t-h(t))(Bik+ΔBik)TPikx(t)+xT(t)Pik(Bik+ΔBik)x(t-h(t))≤xT(t)Pik2x(t)+xT(t-h(t))×(Bik+ΔBik)T(Bik+ΔBik)x(t-h(t)).
Substituting (36) into (35) yields
(37)V1˙(t,xt)≤xT(t)[AikTAik+AikTEik(ε1I-EikTEik)-1EikTAik+ε1HikTHik+3Pik2+FikTCikTCikFik(ε1I-EikTEik)-1]x(t)+xT(t-h(t))[(Bik+ΔBik)T(Bik+ΔBik)]×x(t-h(t)).
The derivatives of V2(t,xt), V3(t,xt), and V4(t,xt) are the same as those in Theorem 5. Then, we can put (37), (16), (18), and (19) together to obtain
(38)V˙(t,xt)+2αV(t,xt)≤xT(t)[AikTAik+AikTEik(ε1I-EikTEik)-1EikTAik+ε1HikTHik+3Pik2+FikTCikTCikFik+U1+U2-λ2S1-λ3S2+2αPik+Rik+FikTQikFik(ε1I-EikTEik)-1]x(t)+xT(t-h(t))[(Bik+ΔBik)T(Bik+ΔBik)-2λ3S3]×x(t-h(t))+xT(t-h1)[-λ2U1-λ2S1-λ3S3]x(t-h1)+xT(t-h2)[-λ3U2-λ3S2-λ3S3]x(t-h2)+2λ3xT(t)S2x(t-h2)+2λ2xT(t)S1x(t-h1)+2λ3xT(t-h(t))S3x(t-h2)+2λ3xT(t-h(t))S3x(t-h1)+x˙T(t)[h12S1+h22S2+(h2-h1)2S3]x˙(t)-xT(t)[Rik+FikTQikFik]x(t).
Let
(39)w=[w110λ2S1λ3S20*w22λ3S3λ3S30**w3300***w440****w55],w11=AikTAik+AikTEik(ε1I-EikTEik)-1EikTAik+ε1HikTHik+3Pik2+FikTCikTCikFik+U1+U2-λ2S1-λ3S2+2αPik+Rik+FikTQikFik,w22=(Bik+ΔBik)T(Bik+ΔBik)-2λ3S3,w33=-λ2U1-λ2S1-λ3S3,w44=-λ3U2-λ3S2-λ3S3,w55=h12S1+h22S2+(h2-h1)2S3.
Then, we have
(40)V˙(t,xt)+2αV(t,xt)≤ξT(t)wξ(t)-xT(t)[Rik+FikTQikFik]x(t).
By Lemma 3, it proves that w<0 holds if
(41)[w1w2*w3]<0,
where
(42)w1=[w11~0λ2S1λ3S20*w22~λ3S3λ3S30**w3300***w440****w55],w2=[AikTEikFikTCikTFikTHikT00000(Bik+ΔBik)T000000000000000],w3=diag⁡(EikTEik-ε1I,-I,-Qik-1,-1ε1I,-I),w11~=AikTAik+3Pik2+U1+U2-λ2S1-λ3S2+2αPik+Rik,w22~=-2λ3S3.
Letting FikTCikT=K, we get Fik=Cik-1KT. Then, we have
(43)[w1w2*w3]=[Π1Π2*Π3]<0,Π1=w1,Π2=[AikTEikKK(Cik-1)THikT00000(Bik+ΔBik)T000000000000000]Π3=w3.
In addition, since [Π1Π2*Π3]<0 (see Theorem 6 (ii)), [w1w2*w3]<0, then we plug it into (40) and obtain
(44)V˙(t,xt)+2αV(t,xt)<0.
Now, we can easily conclude that the impulsive switched system ((1a), (1b), (1c), and (1d)) is exponentially stable. Looking back to (40), we can get
(45)V˙(t,xt)≤-xT(t)[Rik+FikTQikFik]x(t),(46)J=∑k=0∞∫tktk+1[xT(t)Rikx(t)+uT(t)Qiku(t)]dt≤-lim⁡k→∞⁡∑j=0k∫tjtj+1dV(t,xt)dtdt=-lim⁡k→∞{-V(0,x0)+V(t1-,xt1-)-V(t1+,xt1+)+V(t2-,xt2-)-V(t2+,xt2+)+⋯-V(tk-,xtk-)+V(tk+,xtk+)-V(tk+1-,xtk+1-)}=lim⁡k→∞{V(0,x0)+∑j=1k{V(tj+,xtj+)-V(tj-,xtj-)}+V(tk+1-,xtk+1-)∑j=1k{V(tj+,xtj+)-V(tj-,xtj-)}}=V(0,x0)+lim⁡k→∞⁡∑j=1kx(tj)T{(I+Dj)TPij(I+Dj)-Pij-1}x(tj)+lim⁡k→∞V(tk+1-,xtk+1-).
We know that
(47)lim⁡k→∞V(tk+1-,xtk+1-)=0,lim⁡k→∞⁡∑j=1kx(tj)T{(I+Dj)TPij(I+Dj)-Pij-1}x(tj)<0.
Thus, we can get
(48)J<V(0,x0)=J*.
This completes the proof.


## 4. Conclusion

The exponential stabilization problem for impulsive switched systems with time delays is considered in this paper. Sufficient conditions for exponential stability of the impulsive switched systems without control input are derived via the Lyapunov-Krasovskii functional method. Moreover, sufficient conditions for the exponential stability of the impulsive switched systems under the designed guaranteed cost control law are established and the corresponding guaranteed cost value can be obtained as well.



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