# Exponential Stability and Guaranteed Cost of Switched Linear

Authors: El Houssaine Tissir
Journal: ISRN Applied Mathematics
 (2011)

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## Abstract

This paper deals with the problems of exponential stability and guaranteed cost of switched linear systems with mixed time delays. Based on Lyapunov functional method, we present new delay-dependent conditions that guarantee both the exponential stability and an upper bound for some performance index. The criteria are delay-dependent conditions and are given in terms of linear matrix inequalities. Numerical examples are provided to illustrate the effectiveness of the results.


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## Body

## 1. Introduction

A switched system is a dynamical system that includes several subsystems and a logical rule that orchestrates switching between these subsystems at each instant of time [1]. The logical rule generates switching signals to determine which subsystem will be activated on a certain time interval. In fact, switching systems arise in many practical processes that cannot be described by exclusively continuous or exclusively discrete models in manufacturing, communication networks automotive engine control, chemical processes, and so on see, for example, [2, 3]. There have been many studies on stability of switching systems [4–6].The stability analysis of switched time delay has attracted a lot of attention [7–9]. The main approach for stability analysis relies on the use of Lyapunov Krasovskii functionals and LMI approach for constructing common Lyapunov function and switching rules [10–12]. In [13], the asymptotic stability of switched linear time delay symmetric systems has been studied. In [11], a switching system composed by a finite number of linear point time delay differential equations has been studied, and it has been shown that the asymptotic stability may be achieved by using a common Lyapunov function method and minimum switching rule. The results of [11] have been extended in [14] to linear switching system with discrete and distributed delays. The delays considered are time invariant, and the conditions are given in terms of maximal and minimal eigenvalues of certain matrices. The exponential stability problem was considered in [15] for switching linear systems with impulsive effects by using the matrix measure concept. The exponential stability of time delay systems has been considered in [16, 17], and switching linear systems with mixed delays have been studied in [18]. In [19], sufficient stability conditions have been obtained by using a generalisation of Halanay’s inequality.In this paper, we will focus on exponential stability of switched linear systems with mixed time delays. The subsystems considered are continuous with time-varying delays. The derived conditions are delay dependent and are given in terms of LMIs. The conditions guarantee both the exponential stability and an upper bound of a performance index. The approach is based on Lyapunov Krasovskii functional and allows us to avoid explicitly substitution of the dynamic system equation in the derivative of the Lyapunov function. Furthermore, we use an extended variable to completely avoid bounding treatment of the weighted cross-product of the instantaneous state and the delayed state. As a consequence, the conservatism of the conditions is expected to be reduced with respect to earlier works (e.g. [10, 16, 20]). The conditions allow us to compute easily both the stability factor and the decay rate of the solution. Numerical computations are performed for illustration.


## 2. Preliminaries

Consider a class of switched linear systems with time-varying delay of the form(2.1)ẋ(t)=Aσx(t)+Dσx(t-h(t))+Eσ∫t-τ(t)tx(s)ds,t>0,x(t)=ϕ(t),t∈[-r,0],
where x∈Rn is the state, σ(x):Rn→I:={1,2,…,N} is the switching rule which is piece-wise constant function depending on the state in each time, ϕ∈C([-h,0],Rn) is the initial function, with norm ∥ϕ∥=supθ∈[-h̅,0]∥ϕ(θ)∥, Aσ,Dσ,Eσ∈{[Ai,Di,Ei],i=1,2,…,N}, Ai, Di, and Ei are given matrices. Moreover σ(x)=i implies that the ith subsystem is activated, and we have the following subsystem:(2.2)ẋ(t)=Aix(t)+Dix(t-h(t))+Ei∫t-τ(t)tx(s)ds,t>0.h(t) and τ(t) are unknown time-varying delay terms, but bounded by(2.3)0≤h(t)≤h̅,ḣ(t)≤μh,(2.4)0≤τ(t)≤τ̅,τ(t)≤μτ,
where h̅, τ̅, μh, and μτ are given nonnegative constants, r=max(h̅,τ̅).In this paper, we are interested in establishing conditions guaranteeing the asymptotic stability of switched system (2.1) and finding the least upper bound for the cost function given by(2.5)J=∫0+∞xT(t)Yx(t)dt,
where Y>0. Assume that there exists a Hurwitz linear convex combination Ã of Ai, that is,(2.6)Ã=∑i=1NλiAi,0<λi<1,∑i=1Nλi=1.
Then for a positive definite matrix Z1, there exists a positive definite matrix R11 satisfying(2.7)ÃTR11+R11Ã=-Z1.
Hence for x≠0, we can write(2.8)xT(ÃTR11+R11Ã)x=-xTZ1x<0.
Which implies that(2.9)λixT(AiTR11+R11Ai)x<0.
for at least one i. So for a positive definite matrix Z, construct N region of Rn as follows:(2.10)Ωi={x∈Rn:xT(AiTR11+R11Ai)x<-xTZx},i∈I.
It is clear that ⋃i=1NΩi=Rn∖{0}.Denote(2.11)Ω̅1=Ω1,Ω̅i=Ωi∖⋃j=1i-1Ωj,i=2,…,N.
It follows that ⋃i=1NΩ̅i=Rn∖{0}, Ω̅i∩Ω̅j=Ø,i≠j. The switching rule is chosen as follows:[SR]:
Step 0: letx(t)=ϕ(t).Step 1: setσ(x)=argmin{xT(AiTR11+R11Ai)x}.Step 2: stay in theith mode as long as x(t) in Ω̅i.Step 3: ifx(t) hits the boundary of Ω̅i, go to Step 1 to determine the next mode.Definition 2.1.
Givenα>0, the system (2.1) is α-exponentially stable if there exist a switching rule σ and a constant β≥1 such that every solution x(t,ϕ) of the system satisfies the following inequality:
(2.12)‖x(t,ϕ)‖≤βe-αt‖ϕ‖,t≥0.The following lemmas will be useful.Lemma 2.2.
For anyX,Y∈Rn, matrices M>0,FTF≤I, one has
(2.13)2XTY≤XTM-1X+YTMY.Lemma 2.3.
For any constant matrixM=MT∈Rn×n, M>0, scalar γ≥η(t)>0, vector function ω:[0,γ]→Rn such that the integrations in the following are well defined, then
(2.14)η(t)∫0η(t)ωT(β)Mω(β)dβ≥[∫0η(t)ω(β)dβ]TM[∫0η(t)ω(β)dβ].


## 3. Main Results

Let,(3.1)β1=λmin⁡(R11),(3.2)β2=(1+h̅2)λmax⁡(R)+1-e-2αh̅2αλmax⁡(P)+e-2αh̅+2αh̅-14α2λmax⁡(Q)+e-2ατ̅+2ατ̅-14α2λmax⁡(S),(3.3)β=β2β1,Li=R11Ai+AiTR11+Z,Z̅=(μhR1200μhR22000000),Ẑ=diag⁡⁡{μhZ1,μhZ2},Gi=(G11iG12iG13iR11Ei+AiTF3T-F0+AiTF4T*G22iG23iDiTF3T-F1+DiTF4T**G33iR12TEi+EiTF3T-F2+EiTF4T***-1-μττ̅e-2ατ̅-F3****-F4-F4T),
where “*” denotes the symmetric part in a symmetric matrix and,(3.4)G11i=R12+R12T+P+2αR11+h̅Q+τ̅S+F0Ai+AiTF0T-Z,G12i=-R12+R11Di+F0Di+AiTF1T,G13i=2αR12+R22+AiTF2T+F0Ei+AiTR12,G22i=-(1-μh)e-2αh̅P+F1Di+DiTF1T+μhZ1,G23i=-R22+DiTF2T+DiR12+F1Ei,G33i=-1h̅e-2αh̅Q+2αR22+EiTF2T+F2Ei+μhZ2.Theorem 3.1.
For givenα>0, switched linear system (2.1) is α-exponentially stable if there exist symmetric positive definite matrices Z, Z1, Z2, S, P, Q, R11, R22, Y, matrices R12, F0, F1, F2, F3, and F4 satisfying. (i)
There exist0<λi<1,i=1,2,…,N such that ∑i=1Nλi=1 and
(3.5)∑i=1NλiLi<-Y.(ii)
(3.6)(GiZ̅Z̅T-Ẑ)<0,i=1,…,N.The switching rule is given by [SR], and the solution x(t,ϕ) of the system satisfies
(3.7)‖x(t,ϕ)‖≤βe-αt‖ϕ‖,t≥0.
Furthermore, the cost function in (2.5) satisfies
(3.8)J≤J*=(ϕ(0)∫0-h(0)0ϕ(s)ds)TR(ϕ(0)∫0-h(0)0ϕ(s)ds)+∫-h̅0e2αθϕT(θ)Pϕ(θ)dθ+∫-h̅0∫s0e2αθϕT(θ)Qϕ(θ)dθds+∫-τ(0)0∫s0e2αθϕT(θ)Sϕ(θ)dθds.Proof.
Letxt:={x(t+s),s∈[-r,0]} and consider the following Lyapunov Krasovskii functional:
(3.9)V(xt)=V1(xt)+V2(xt)+V3(xt)+V4(xt).
With
(3.10)V1(xt)=(x(t)∫t-h(t)tx(s)ds)TR(x(t)∫t-h(t)tx(s)ds),V2(xt)=∫-h̅0e2αsxT(t+s)Px(t+s)ds,V3(xt)=∫-h̅0∫s0e2αθxT(t+θ)Qx(t+θ)dθds,V4(xt)=∫-τ(t)0∫s0e2αθxT(t+θ)Sx(t+θ)dθds.
It is easy to verify that
(3.11)0<β1‖x(t)‖2≤V(xt)≤β2‖xt‖2,
where β1 and β2 are defined by (3.1) and (3.2). Computing the first time derivative of V(xt), we obtain
(3.12)V̇1(xt)=2(x(t)∫t-h(t)tx(s)ds)TR(ẋ(t)x(t)-(1-ḣ(t))x(t-h(t)))=2(x(t)∫t-h(t)tx(s)ds)TR(Aix(t)+Dix(t-h(t))+Ei∫t-τ(t)tx(s)ds,x(t)-(1-ḣ(t))x(t-h(t))).
Letting ξ(t)=[xT(t)xT(t-h(t))∫t-h(t)txT(s)ds∫t-τ(t)txT(s)ds]T, taking account of (2.3), and using the fact that for some positive definite matrices Z1 and Z2, the following inequalities hold
(3.13)2ḣ(t)xT(t)R12x(t-h(t))≤μhxT(t)R12Z1-1R12Tx(t)+μhxT(t-h(t))Z1x(t-h(t)),2ḣ(t)xT(t-h(t))R22∫t-h(t)tx(s)ds≤μhxT(t-h(t))R22Z2-1R22x(t-h(t))+μh[∫t-h(t)tx(s)ds]TZ2[∫t-h(t)tx(s)ds].
We can write
(3.14)V̇1(xt)≤xT(t)⌊R11Ai+AiTR11⌋x(t)+ξT(t)T1ξ(t)-2αV1(xt),
where
(3.15)T1=(R12+R12T+2αR11+μhR12Z1-1R12T***DiTR11-R12TμhZ1+μhR22Z2-1R22T**R22+2αR12T+R12TAiR12TDi-R222αR22*EiTR110EiTR120).
And “*” denotes the symmetric part in a symmetric matrix
(3.16)V̇2(xt)=xT(t)Px(t)-(1-ḣ(t))e-2αh(t)xT(t-h(t))Px(t-h(t))-2αV2(xt)≤xT(t)Px(t)-(1-μh)e-2αh̅xT(t-h(t))Px(t-h(t))-2αV2(xt),V̇3(xt)=h̅xT(t)Qx(t)-∫t-h̅te2α(θ-t)xT(θ)Qx(θ)dθ-2αV3(xt)≤h̅xT(t)Qx(t)-e-2αh̅h̅[∫t-h(t)tx(θ)dθ]TQ[∫t-h(t)tx(θ)dθ]-2αV3(xt).
Taking account of (2.4) and applying Lemma 2.3, we obtain
(3.17)V̇4(xt)=τ(t)xT(t)Qx(t)-(1-τ̇(t))∫t-τ(t)te2α(θ-t)xT(θ)Sx(θ)dθ-2αV4(xt)≤τ̅xT(t)Qx(t)-(1-μτ)τ̅e-2ατ̅[∫t-τ(t)tx(θ)dθ]TS[∫t-τ(t)tx(θ)dθ]-2αV4(xt).
Now let Bi=[AiDiEi0-I] and F=[F0TF1TF2TF3TF4T]T. We can easily verify that Biχ=0, where χ(t)=[ξT(t)ẋT(t)]T, and
(3.18)χT(t)⌊FBi+BiTFT⌋χ(t)=0,i=1,2…,N.
Taking account of (3.18), and adding and subtracting the term xT(t)Zx(t) with Z a positive definite matrix, we get
(3.19)V̇(xt)+2αV(xt)≤xT(t)(R11Ai+AiTR11+Z)x(t)+χT(t)(Gi+Z̅Ẑ-1Z̅T)χ(t).
Since the condition (3.6) holds we have χT(t)(Gi+Z̅Ẑ-1Z̅T)χ(t)<0, it follows that
(3.20)V̇(xt)+2αV(xt)≤xT(t)Lix(t).
From condition (i), we have ∑i=1NλiLi<-Y, where 1>λi>0, i=1,2,…,N and ∑i=1Nλi=1. So
(3.21)∑i=1Nλimin⁡i=1,…,N⁡xT(t)Lix(t)≤∑i=1NλixT(t)Lix(t)<-xT(t)Yx(t)<0.
By choosing the switching rule as
(3.22)σ(x)=argmin⁡i=1,...,NxT(t)Lix(t).
We have
(3.23)V̇(xt)+2αV(xt)≤xT(t)Lix(t)≤∑i=1NλixT(t)Lix(t)<-xT(t)Yx(t)<0.
This implies that V(xt)≤V(ϕ)e-2αt,t≥0. Taking account of (3.11), we obtain
(3.24)β1‖x(t)‖2≤V(xt)≤V(ϕ)e-2αt≤β2e-2αt‖ϕ‖2.
And then, ∥x(t)∥≤βe-αt∥ϕ∥,t≥0. Furthermore, since (3.11) holds, we have
(3.25)V̇(xt)≤-xT(t)Yx(t).
Integrating both sides of (3.25) from 0 to T and using the initial conditions, we obtain
(3.26)-∫0TxT(t)Yx(t)dt≥(x(T)∫T-h(t)Tx(s)ds)TR(x(T)∫T-h(T)Tx(s)ds)-(x(0)∫0-h(0)0x(s)ds)TR(x(0)∫0-h(0)0x(s)ds)+∫T-h̅Te-2αTe2αθxT(θ)Px(θ)dθ-∫-h̅0e2αθxT(θ)Px(θ)dθ+∫-h̅0∫T+sTe-2αTe2αθxT(θ)Qx(θ)dθds-∫-h̅0∫s0e2αθxT(θ)Qx(θ)dθds+∫-τ(T)0∫T+sTe-2αTe2αθxT(θ)Sx(θ)dθds-∫-τ(0)0∫s0e2αθxT(θ)Sx(θ)dθds.
As the system is asymptotically stable, when T→∞ we have
(3.27)(x(T)∫T-h(t)Tx(s)ds)TR(x(T)∫T-h(T)Tx(s)ds)⟶0,∫-h̅0∫T+sTe-2αTe2αθxT(θ)Qx(θ)dθds⟶0,∫T-h̅Te-2αTe2αθxT(θ)Px(θ)dθ⟶0,∫-τ(T)0∫T+sTe-2αTe2αθxT(θ)Sx(θ)dθds⟶0.
Hence, we get
(3.28)∫0TxT(t)Yx(t)dt≤(ϕ(0)∫0-h(0)0ϕ(s)ds)TR(ϕ(0)∫0-h(0)0ϕ(s)ds)+∫-h̅0e2αθϕT(θ)Pϕ(θ)dθ+∫-h̅0∫s0e2αθϕT(θ)Qϕ(θ)dθds+∫-τ(0)0∫s0e2αθϕT(θ)Sϕ(θ)dθds
which concludes the proof.Remark 3.2.
In order to improve the results, we can use instead of (3.18) the relation
(3.29)χT(t)⌊FiBi+BiTFiT⌋χ(t)=0,i=1,2,…,N,
where Fi is given by Fi=[F0iTF1iTF2iTF3iTF4iT]T. Then we can state the following result.Theorem 3.3.
For givenα>0, switched linear system (2.1) is α-exponentially stable if there exist symmetric positive definite matrices Z, Z1, Z2, S, P, Q, R11, R22, Y, matrices R12, F0i, F1i, F2i, F3i and F4i satisfying the conditions (3.5) and (3.6), where the matrices Fk,k=0,1,…,4 are replaced with Fki,k=0,1,…,4, i=1,…,N. The switching rule is given by [SR], the solution x(t,ϕ) of the system satisfies (3.7), and the cost function in (2.5) satisfies (3.8).Ifh(t)≠0 and τ(t)=0, the system (2.2) is reduced to the system as follows:(3.30)ẋ(t)=Aix(t)+Dix(t-h(t)),t>0.In this case, we have the following corollary.Corollary 3.4.
For givenα>0, switched linear system (3.30) is α-exponentially stable if there exist symmetric positive definite matrices Z, Z1, Z2, S, P, Q, R11, R22, Y, matrices R12, Fki, k=0,…,4, i=1,…,N satisfying condition (3.5) and the following LMI:
(3.31)(Ii℘℘T-Ẑ)<0,i=1,…,N,
where,
(3.32)Ii=(I11iI12i2αR12+R22+AiTF2iT+AiTR12-F0i+AiTF3iT*I22i-R22+DiTF2iT+DiR12-F1i+DiTF3iT**-1h̅e-2αh̅Q+2αR22+μhZ2-F2i***-F3i-F3iT)<0,℘=(μhR1200μhR220000).
And where “*” denotes the symmetric part and
(3.33)I11i=R12+R12T+P+2αR11+h̅Q+F0iAi+AiTF0iT-Z,I12i=-R12+R11Di+F0iDi+AiTF1iT,I22i=-(1-μh)e-2αh̅P+F1iDi+DiTF1iT+μhZ1.Remark 3.5.
In [16], the results are given in terms of a set of generalized Lyapunov equations type, and in [18], the results are expressed in terms of generalized algebraic Riccati equation type, while in this paper, the results are expressed in terms of linear matrix inequalities.


## 4. Examples

In this section, two examples will be presented for illustration and comparison.Example 4.1 (see [18]).
Consider the uncertain switched linear systems (2.1), where N=2, h=1, τ=1 and
(4.1)A1=(-201-32),D1=(1-11-1),E1=(111-3),A2=(4-11-32),D2=(1-13-4),E2=(-11-14).
We can see that each subsystem is unstable. Applying the results of [18], it is found that the system is asymptotically stable with decay rate α=0.5. Applying the results of [14], by choosing Q=8*I, and applying Theorem 3.1, the critical delay of asymptotic stability (without decay rate) is found as 0.5722. Therefore, with the results of [14], we can't conclude about stability of the considered system. Note that both [14, 18] study the systems with constant delays. It is well known that the results for time varying delays are more conservative than those with constant delays. Applying our results, we let μh=0 and μτ=0, by Theorem 3.1, with λ1=λ2=0.5, we obtain a decay rate α=0.984 with the following solutions:
(4.2)P=(3.1461-3.6923-3.69235.1228),R=10-2(112.4135-41.2719-0.1118-3.2593-41.271973.7166-0.20171.6277-1.0907-0.20170.0337-0.0079-3.25931.6277-0.00790.1106),Z=(17.153-8.754-8.75422.110),Q=(0.250.00240.00240.6492),F0=(0.15460.0216-0.11530.3648),F1=10-2(-0.87093.37592.7474-4.8734),F2=10-2(0.14750.53182.27160.1949),S=(2.1245-0.4151-0.41517.8435),F3=10-2(1.4000-0.0729-3.5071-0.2284),F4=10-2(0.78300.2873-0.20200.0708),Z1=(2.975002.975),Z2=(2.975002.975),Y=10-2(0.4050-0.0579-0.05790.2367).
And the matrices
(4.3)L1=(-25.336-2.413-2.41324.233),L2=(25.3202.4152.415-24.243).
Satisfy λ1L1+λ2L2=10-3(-811-5)<-Y.
The setsΩ1 and Ω2 are given by
(4.4)Ω1={(x1,x2)∈R2:-25.336x12-4.826x1x2+24.233x22<0},Ω2={(x1,x2)∈R2:25.32x12+4.83x1x2-24.243x22<0}.
It can be seen that Ω1∪Ω2=R2∖{0}, therefore, the switching regions are given by: Ω̅1=Ω1,Ω̅2=Ω2∖Ω̅1, and the switching rule is given by
(4.5)σ={1ifx(t)∈Ω̅1,2ifx(t)∈Ω̅2.
Moreover the solution of the system satisfies ∥x(t,ϕ)∥≤3.6297e-0.984t∥ϕ∥,t≥0. For the initial conditions x1(t)=et+1 and x2(t)=0 for t∈[-1,0], we obtain a guaranteed cost J*=14.5632. When we apply Theorem 3.3, we obtain the stability with decay rate α=1.186, stability factor β=2.3345, and the guaranteed cost J*=19.244.Example 4.2.
Consider the following system:(4.6)A01=(-302-54),A02=(2-22-34),Ad1=(2-2-2-3.9),Ad2=(6-2515.31).
Applying the results of [20], when the time delay is constant, that is, h(t)=h̅, we obtain the stability bound h̅=0.063 with the parameters β1=15 and β2=0.8. Applying our results, for comparison, we set μh=0 and α=0, then we apply Corollary 3.4 with λ1=λ2=0.5, it is found that the system is asymptotically stable independent of delay. This shows the improvements of our approach. For h̅=250 s, we have the following results:
(4.7)Y=(51.57348.32158.32153.6640),L1=103(-1.00450.00400.00400.6358),L2=(798.1735-37.3230-37.3230-650.4835),λ1L1+λ2L2=(-103.1467-16.6431-16.6431-7.3280)<-Y.
The switching regions Ω̅1 and Ω̅1 are given by
(4.8)Ω̅1={(x1,x2)∈R2:-1004.5x12+8x1x2+635.8x22<0},Ω̅2={(x1,x2)∈R2:-1004.5x12+8x1x2+635.8x22≥0},(x1,x2)≠(0,0).
The state trajectories are depicted in Figure 1 for h(t)=0.5 s and in Figure 2 for h(t)=250 s. It is clear that as the delay increases as the states require much time for convergence to zeros. Now, letting  h̅=0.5 s, α=0.4, λ1=0.3, λ2=0.7, and the initial conditions x1(t)=et+1 and x2(t)=0 for t∈[-0.5,0], we obtain a guaranteed cost J*=80.004.Figure 1
State trajectories forh=0.5 s.Figure 2
State trajectories forh=250 s.


## 5. Conclusion

By using Lyapunov Krasovskii approach, the problems of exponential stability and guaranteed cost are investigated for switched linear systems with mixed time delays. Via a designed switching rule, the results allow to the bounds that characterise the exponential stability, that is, the stability factor and the decay rate for the solution. The results are delay dependent and are expressed in terms of linear matrix inequalities. Some numerical examples are given to illustrate these results presented in this paper that have significant improvement over existing ones.



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