# L

Authors: Ronghao Wang; Jianchun Xing; Ping Wang; Qiliang Yang; Zhengrong Xiang
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

This paper is concerned with the problem of controller design for switched systems under asynchronous switching with exogenous disturbances. The attention is focused on designing the feedback controller that guarantees the finite-time bounded andL∞ finite-time stability of the dynamic system. Firstly, when there exists asynchronous switching between the controller and the system, a sufficient condition for the existence of stabilizing switching law for the addressed switched system is derived. It is proved that the switched system is finite-time stabilizable under asynchronous switching satisfying the average dwell-time condition. Furthermore, the problem of L∞ control for switched systems under asynchronous switching is also investigated. Finally, a numerical example is given to illustrate the effectiveness of the proposed method.


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## Body

## 1. Introduction

Switched systems are a class of hybrid systems consisting of subsystems and a switching law, which defines a specific subsystem being activated during a certain interval of time. Many real-world processes and systems can be modeled as switched systems such as chemical processes and computer controlled systems. Besides, switched systems are widely applied in many domains, including mechanical systems, automotive industry, aircraft and air traffic control, and many other fields [1–3].At early time, the issue of stability of switched systems which has attracted most of the attention is one basic research topic. Lyapunov stability theory and its variations or generalizations had played an important role in this research field. Common Lyapunov function method and multiple Lyapunov functions method for switched system are presented by researchers [4–8]. For most switched systems, it is hard to find a common Lyapunov function; however, we can guarantee the switched system is still stable under some properly chosen switching signals which are found by using the multiple Lyapunov functions technique. In addition, more researchers pay attention to average dwell-time control of switched systems [9, 10]. In particular, the average dwell-time approach is employed to deal with the control, observe, and filtering problem of switched delay systems or network control systems [11–14].As we know, a large number of literatures related to stability of switched systems focus on Lyapunov asymptotic stability, which is defined over an infinite time interval. In many practical applications, however, the main concern is the behavior of the system over a fixed finite-time interval, for instance to avoid saturations or the excitation of nonlinear dynamics. It should be clear that a finite-time stable system may not be Lyapunov asymptotical stable, and a Lyapunov asymptotical stable system may not be finite-time stable since the transient of a system response may exceed the bound. Recently, there have been some literatures discussing the finite-time stability analysis of switched systems [15–17]. In [18], finite-time bounded and finite-time weighted L2-gain for a class of switched delay systems with time-varying external disturbances is addressed. Reference [19] investigated finite-time control for switched discrete-time system. Considering the potential faults in a system, [20] studied fault-tolerant control with finite-time stability for switched linear systems. Delay-dependent observer-based H∞ finite-time control for switched systems with time-varying delay was studied in [21]. In [22], the problems of finite-time stability analysis and stabilization for switched nonlinear discrete-time systems are investigated, and then the results are extended to H∞ finite-time bounded. However, in many applications, external disturbance is always persistent bounded with infinite energy. H∞ control cannot be employed to deal with a system with persistent bounded disturbance. In this situation, it is more appealing to develop L∞ control for switched systems with disturbances of this type. So far, however, compared with research results on H∞ finite-time stability, few results on L∞ finite-time stability of switched systems have been given in the literature.Additionally, in actual operation, there inevitably exists asynchronous switching between the controllers and the practical subsystems, that is, the real switching time of controllers exceeds or lags behind that of the practical subsystems, which will deteriorate performance of systems, even makes system out of control. Up to now, there have been a number of literatures on asynchronous switching control research of switched system [23–28]. But it is worth to point that all of these studies focus on designing the controller to guarantee the Lyapunov asymptotical stable or exponential stable of the system. To the best of our knowledge, the finite-time stabilization issue of switched system under asynchronous switching has not been fully investigated, which is quite an important issue for the switched system. This motivates us to carry out present work. In this paper, we deal with the problem of L∞ finite-time stabilization for switched systems under asynchronous switching.The main contributions of this paper are that several sufficient conditions ensuring the finite-time bounded andL∞  finite-time stability are proposed with asynchronous switching between the controllers and the practical subsystems. The result shows that it is unnecessary to guarantee each subsystem can be finite-time stabilizable with L∞ performance by the designed asynchronous switching controller. During the finite-time interval, the switching frequency only needs to be limited in some value, then the switched system is finite-time stable with L∞ performance by the designed controller despite of the asynchronous switching between the controllers and the practical subsystems.This paper is organized as follows. In Section2, some preliminary definitions are provided, and the problem we deal with is precisely stated. Section 3 provides, the main results of this paper: a sufficient condition for the existence of a state feedback controller guaranteeing the finite-time stability under asynchronous switching between the controllers and the practical subsystems. Moreover, L∞ control with finite-time stability for switched systems under asynchronous switching is provided in Section 4. Finally, a numerical example is presented by using LMI toolbox to illustrate the efficiency of the proposed method in Section 5. Our conclusions are drawn in Section 6.Notation. Throughout this paper, AT denotes transpose of matrix A, L∞ denotes space of functions with bounded amplitude, ∥x(t)∥ denotes the usually 2-norm. λmax(P), and λmin(P) denote the maximum and minimum eigenvalues of matrix P, respectively, I is an identity matrix with appropriate dimension.  S>0  denotes S is a positive definite symmetric matrix. Z denotes the integer set and Z+ denotes the positive integer set.


## 2. Problem Formulation and Preliminary

A switched system is considered as follows:(2.1)x˙(t)=Aσ(t)x(t)+Bσ(t)u(t)+Gσ(t)w(t),
where  x(t)∈Rn is the system state. u(t)∈Rp is the control input, x(t0)=x0 is the initial state of the system. w(t)∈Rq is the measurement noise over the interval [t0,Tf], which satisfies supt∈[t0,Tf]∥w(t)∥<∞,σ(t):Z+→N_={1,2,…,N} is a switching signal which is a piecewise constant function depending on time t or state x(t), and  N denotes the number of subsystems. Moreover, σ(t)=i means that the ith subsystem is activated. Ai∈Rn×n,Bi∈Rn×p,Gi∈Rn×q for i∈N_ are real-valued matrices with appropriate dimensions.Assume that the state of the switched system (2.1) does not jump at the switching instants, that is, the trajectory x(t) is everywhere continuous. The switching law σ(t):Z+→N_={1,2,…,N} discussed in this paper is time dependent, that is, σ(t):{(t0,σ(t0)),(t1,σ(t1)),…,(tk,σ(tk))}, k∈Z, where t0 is the initial switching instant, and tk denotes the kth switching instant.Owing to asynchronous switching, the practical switching instant of controller is different from that of systems. For convenience,σ'(t) is used to denote the practical switching signal of controller, σ'(t) can be written as σ'(t):{(t0+Δ0,σ(t0)),(t1+Δ1,σ(t1)),…,(tk+Δk,σ(tk))}, k∈Z, where |Δk|<infk≥0(tk+1-tk),Δk>0 (or |Δk|<infk≥0(tk-tk-1),Δk<0); Δk represents the delayed period of the controller switching (or the exceeded period of the controller switching). In both cases, the period Δk is said to be the mismatched period between the controller and the system.Remark 2.1.
Mismatched periodΔk guarantees that there always exists a period that the controller and the system operate synchronously, which makes it possible to design the stabilizable controller for the system.
Under the asynchronous switching, the switched controller can be written as(2.2)u(t)=Kσ′(t)x(t).
If we substitute the u(t)=Kσ'(t)x(t) into system (2.1), we can obtain that
(2.3)x˙(t)=(Aσ(t)+Bσ(t)Kσ′(t))x(t)+Gσ(t)w(t).
The following lemma will be useful for the design of controller.Lemma 2.2 (see [29]).
If a real scalar functionφ(t),v(t)  satisfies the following differential inequality:
(2.4)φ˙(t)≤ςφ(t)+κv(t),
then we have
(2.5)φ(t)≤eς(t-t0)φ(t0)+κ∫0t-t0eςτv(t-τ)dτ,
where ς∈R,κ∈R,t≥t0.Let us review the definition of average dwell-time, which will be useful in designing the stabilization controller to guarantee the system finite-time stable.Definition 2.3 (see [30]).
For anyT2>T1≥0, let Nσ(T1,T2) denote the switching number of σ(t) on an interval (T1,T2), if
(2.6)Nσ(T1,T2)≤N0+T2-T1τa
holds for given N0≥0,τa>0. Then the constant τa is called the average dwell time, and N0 is the chatter bound.
For switched system, the general conception of finite-time stability concerns the boundness of continuous statex(t) over finite-time interval [t0,Tf] with respect to given initial condition x0. This conception can be formulized through following definition.Definition 2.4.
The switched linear system (2.1) with Gσ(t)≡0 is said to be finite-time stabilizable under the asynchronous switching control mode with respect to (c1,c2,Tf,σ(t),σ'(t)) with c1<c2 and a given switching signal σ(t), if ∥x(t)∥≤c2, for all t∈[t0,Tf], whenever ∥x0∥≤c1.Definition 2.5.
Switched system (2.1) is said to be L∞ finite-time stabilizable with respect to (c1,c2,Tf,σ(t),σ'(t)) where c1<c2, σ(t) is a switching signal of the system, and σ′(t) is a switching signal of the controller, the following conditions should be satisfied.(i)
Switched linear system (2.1) with Gσ(t)≡0 is finite-time stabilizable.(ii)
Under zero-initial conditionx(t0)=0, the following inequality holds:
(2.7)supt∈[t0,Tf]∥x(t)∥≤γsupt∈[t0,Tf]∥w(t)∥,∀w(t):supt∈[t0,Tf]∥w(t)∥<∞.
The main issue in this paper is given as follows.
Given switched system (2.1), find a sufficient condition ensuring the finite-time stability with respect to (c1,c2,Tf,σ(t),σ′(t)) under the asynchronous switching control mode, then the result will be extended to the L∞ controller design for system (2.1).


## 3. Finite-Time Stabilization under the Asynchronous Switching

It is assumed that theith subsystem switched to the jth subsystem at the switching instant tk. Owing to asynchronous switching, the switching instant of ith controller is tk+Δk, then there exists mismatched period at time interval [tk,tk+Δk), Δk>0(or (tk+Δk,tk), Δk<0). In this period, the controller Ki affected the jth subsystem (or the controller Kj affected the ith subsystem).Remark 3.1.
We consider the case ofΔk>0, that is to say, the switching time of the controller is lag of the switching time of the system. Figure 1, illustrates the asynchronous switching mode between the controller and the subsystems. From Figure 1, we can see that the controller Ki of the ith subsystem affects the ith subsystem in the matched period [tk-1+Δk-1,tk) and affects the jth subsystem in the mismatched period [tk,tk+Δk).Figure 1
Asynchronous switching mode.The following theorem presents the finite-time stabilization design method of the system (2.1) under asynchronous switching.Theorem 3.2.
If there exist matricesPi>0, Pij>0,Ki and scalars μ1>1,μ2>1,λ+>0,λ->0 such that
(3.1)Pi<μ1Pij,Pij<μ2Pi,(3.2)(Ai+BiKi)TPi+Pi(Ai+BiKi)<λ-Pi,(3.3)(Aj+BjKi)TPij+Pij(Aj+BjKi)<λ+Pij,(3.4)τa>(Tf-t0)ln(μ1μ2)ln((ε2/δ2)·ℬ·(μ2/(μ1μ2)N0))-λ+T+(t0,Tf)-λ-T-(t0,Tf),
where  ℬdenotesinfi,j∈N_{λmin(Pi),λmin(Pij)}/supi,j∈N_{λmax(Pi),λmax(Pij)}, then switched system (2.1) is finite-time stabilizable with respect to (δ,ε,Tf,σ(t),σ'(t)) under the feedback controller u(t)=Kσ'(t)x(t), where T-(t0,Tf) and T+(t0,Tf) denote the matched period and the mismatched period in finite-time interval [t0,Tf], respectively.Proof.
Here, we only discuss the situation ofΔk>0. For Δk<0, the proof method is similar, and we can reach the same conclusion.
Whent∈[tk-1+Δk-1,tk), for the ith subsystem, the state feedback controller u(t)=Kix(t). So the state equation of closed-loop system can be written as
(3.5)x˙(t)=(Ai+BiKi)x(t).
Choose a switching Lyapunov function as follows:
(3.6)Vi(t)=xT(t)Pix(t).
By (3.2), it implies that
(3.7)V˙i(t)<λ-Vi(t).
When t∈[tk,tk+Δk), for the jth subsystem, the state feedback controller is still u(t)=Kix(t). So the closed-loop system can be described as
(3.8)x˙(t)=(Aj+BjKi)x(t).
Consider the Lyapunov function candidate as follows:
(3.9)Vij(t)=xT(t)Pijx(t).
By (3.3), we can obtain that
(3.10)V˙ij(t)<λ+Vij(t).
Notice that the Lyapunov function (3.6) and (3.9) can be rewritten as
(3.11)Vi(t)=xT(t)Pix(t),t∈[tk-1+Δk-1,tk),k=1,2,…,Vi(t)=xT(t)Pijx(t),t∈[tk,tk+Δk),k=0,1,….
Let t0<t1<t2<⋯<tk=Tf is the switching time in the period [t0,Tf], we define the following piecewise Lyapunov function:
(3.12)V(t)={xT(t)Pix(t),t∈[tr+Δr,tr+1),r=0,1,…,k-1,xT(t)Pijx(t),t∈[tr,tr+Δr),r=0,1,…,k-1.
By (3.7) and (3.10), we can obtain that
(3.13)V(t)<eλ-(t-tk-1-Δk-1)V(tk-1+Δk-1)<μ1eλ-(t-tk-1-Δk-1)V((tk-1+Δk-1)-)<μ1eλ+Δk-1eλ-(t-tk-1-Δk-1)V(tk-1)<μ1μ2eλ+Δk-1eλ-(t-tk-1-Δk-1)V(tk-1-)<μ1μ2eλ+Δk-1eλ-(t-tk-1-Δk-1)eλ-(tk-1-tk-2-Δk-2)V(tk-2+Δk-2)<μ12μ2eλ+Δk-1eλ-(t-tk-1-Δk-1)eλ-(tk-1-tk-2-Δk-2)V((tk-2+Δk-2)-)<μ12μ2eλ+Δk-1eλ+Δk-2eλ-(t-tk-1-Δk-1)eλ-(tk-1-tk-2-Δk-2)V(tk-2)=μ12μ2eλ+(Δk-1+Δk-2)+λ-[(t-tk-1-Δk-1)+(tk-1-tk-2-Δk-2)]V(tk-2)⋯<μ1kμ2k-1eλ+(Δk-1+⋯+Δ0)+λ-[(t-tk-1-Δk-1)+(tk-1-tk-2-Δk-2)+⋯+(t1-t0-Δ0)]V(t0)<μ2-1(μ1μ2)k[t0,Tf]eλ+T+(t0,Tf)+λ-T-(t0,Tf)V(t0),
where T+(t0,Tf) denotes the sum of the mismatched period between the controllers and subsystem in (t0,Tf). T-(t0,Tf) denotes the sum of the matched period between the controllers and subsystem in [t0,Tf].
And from (3.12) we have
(3.14)V(t)≥infi,j∈N_{λmin(Pi),λmin(Pij)}∥x(t)∥2.
On the other hand, for i∈N_, we have
(3.15)V(t0)≤supi,j∈N_{λmax(Pi),λmax(Pij)}∥x(t0)∥2.
Using the fact
(3.16)∥x(t0)∥≤δ,
we get
(3.17)V(t0)≤supi,j∈N_{λmax(Pi),λmax(Pij)}δ2.
Altogether (3.13)–(3.17), the following inequality can be derived
(3.18)∥x(t)∥2≤μ2-1(μ1μ2)k[t0,Tf]eλ+T+(t0,Tf)+λ-T-(t0,Tf)supi,j∈N_{λmax(Pi),λmax(Pij)}infi,j∈N_{λmin(Pi),λmin(Pij)}δ2.
From the Definition 2.3, we know that k[t0,Tf]=Nσ, then we have the relation
(3.19)k[t0,Tf]≤N0+Tf-t0τa.
From (3.4) and (3.19), we get
(3.20)μ2-1(μ1μ2)k[t0,Tf]eλ+T+(t0,Tf)+λ-T-(t0,Tf)supi,j∈N_{λmax(Pi),λmax(Pij)}infi,j∈N_{λmin(Pi),λmin(Pij)}δ2<ε2.
According to (3.18) and (3.20), we have
(3.21)∥x(t)∥<ε.
The proof is completed.Remark 3.3.
From (3.2) and (3.3), we know that for finite-time stabilization issue, the subsystem needs not to be stabilized in finite-time interval, that is to say, the designed asynchronous switching controller needs not to stabilize the subsystem in the matched period and the mismatched period in finite-time interval [t0,Tf], but the whole system is finite-time stabilizable. Reference [31] gives the exponential stabilization condition under asynchronous switching, which requests that the subsystem can be exponentially stabilized in the matched period. But as to the problem of finite-time stabilization, it is unnecessary to request that the subsystem can be stabilized in the matched period or mismatched period. In particular, when λ+=λ-=λ in (3.2) and (3.3), (3.4) becomes
(3.22)τa>(Tf-t0)ln(μ1μ2)ln((ε2/δ2)·ℬ·(μ2/(μ1μ2)N0))-λ(Tf-t0)
which is independent of  T+(t0,Tf)  and  T-(t0,Tf).Remark 3.4.
In fact, (3.4) in Theorem 3.2 implies that if switching sequence σ(t):{(t0,σ(t0)),(t1,σ(t1)),…,(tk,σ(tk))}of the system can be prespecified,thatis,τa is a known constant, the matched period T-(t0,Tf) and the mismatched period T+(t0,Tf) should satisfy the following relation:
(3.23)λ+T+(t0,Tf)+λ-T-(t0,Tf)<ln(ε2δ2·infi,j∈N_{λmin(Pi),λmin(Pij)}supi,j∈N_{λmax(Pi),λmax(Pij)}·μ2(μ1μ2)N0)-(Tf-t0)ln(μ1μ2)τa.Remark 3.5.
Reference [31] gives the design method of exponential stabilization controller under asynchronous switching. The condition implies that the ratio of the mismatched period and the matched period should be less than some value which means that the matched period should be large enough to stabilize the subsystem. However, from the condition of Theorem 3.2, we know that when the switching sequence is unknown, the ratio of the mismatched period and the matched period can be designed freely to guarantee the finite-time stability of the system by the asynchronous switched controller. But if switching sequence of the system is prespecified, the ratio of the mismatched period and the matched period may need to be limited. On the other hand, the average dwell-time scheme with Lyapunov stability limits the dwell-time τa and the ratio of T+(t0,Tf) and T-(t0,Tf) to satisfy the proposed condition in [31] at the same time. But for the average dwell-time scheme with finite-time stability, we can predetermine one value among two parameters of the dwell-time τa and the ratio of T+(t0,Tf) and T-(t0,Tf), then the other value can be determined by the condition (3.4).Remark 3.6.
In order to get the solution of the asynchronous switched controllerKi, we denote Xi=Pi-1,Xij=Pij-1,Wi=KiPi-1, then (3.1) to (3.3) can be written as
(3.24)μ1Xi>Xij,μ2Xij>Xi,(3.25)(AiXi+BiWi)T+(AiXi+BiWi)<λ-Xi,(3.26)Xij(Aj+BjWiXi-1)T+(Aj+BjWiXi-1)Xij<λ+Xij.
It is noticed that the matrix inequalities (3.24), (3.25), and(3.26) are coupled. Therefore, we can firstly solve the linear matrix inequality (3.25) to obtain the solution to matrices Xi and Wi. Then we solve the matrix inequality (3.24), (3.26) by substituting Xi and Wi into (3.24), (3.26). By adjusting the parameter μ1,μ2, and  λ+ appropriately, we seek the feasible solutions Xi,Wi and Xij such that the matrix inequalities (3.24) and (3.26) hold. If the chosen parameters μ1, μ2, and λ+ have no feasible solution, we can adjust μ1,μ2, or λ+ to be larger. Following this guideline, the solution to the matrix inequalities (3.24)to (3.26) will be found.


## 4.L∞ Finite-Time Stabilization under the Asynchronous Switching

Now, we are in a position to investigateL∞ finite-time stabilization design method of the system (2.1) under asynchronous switching.Theorem 4.1.
If there exist matricesPi>0,Pij>0,Ki and scalars μ1>1,μ2>1,λ+>0,λ->0 such that
(4.1)Pi<μ1Pij,Pij<μ2Pi,(4.2)[(Ai+BiKi)TPi+Pi(Ai+BiKi)-λ-PiPiGiGiTPi-εiI]<0,(4.3)[(Aj+BjKi)TPij+Pij(Aj+BjKi)-λ+PijPijGjGjTPij-εijI]<0,(4.4)τa>(Tf-t0)ln(μ1μ2)ln((ε2/δ2)·ℬ·(μ2/(μ1μ2)N0))-λ+T+(t0,Tf)-λ-T-(t0,Tf).L∞ disturbance attenuation performance γ-≤εi(eλ-T-(t0,Tf)-1)/λ-λmin(Pi) during the matched period and γ+≤εij(eλ+T+(t0,Tf)-1)/λ+λmin(Pij) during the mismatched period, then switched system (2.1) is finite-time stabilizable of L∞ disturbance attenuation performance with respect to (δ,ε,Tf,σ(t),σ'(t)) under the feedback controller u(t)=Kσ′(t)x(t), where T-(t0,Tf) and T+(t0,Tf) denote the matched period and the mismatched period in finite-time interval [t0,Tf], respectively.Proof.
It can be concluded from Theorem4.1 that system (2.1) is finite-time stable under the feedback controller  u(t)=Kσ′(t)x(t).
Whent∈[tk-1+Δk-1,tk), for the ith subsystem, the state feedback controller u(t)=Kix(t). So the state equation of closed-loop system can be written as
(4.5)x˙(t)=(Ai+BiKi)x(t)+Giw(t).
Choose a switching Lyapunov function as follows:
(4.6)Vi(t)=xT(t)Pix(t),t∈[tk-1+Δk-1,tk),k=1,2,….
By (4.2), it implies that
(4.7)V˙i(t)≤λ-Vi(t)+εiwT(t)w(t).
With zero initial conditions, by Lemma 2.2, we have
(4.8)Vi(t)≤εi∫0t-tk-1-Δk-1eλ-τwT(t-τ)w(t-τ)dτ.
Note that
(4.9)Vi(t)≥λmin(Pi)∥x(t)∥2.
From (4.8) and (4.9), we can obtain
(4.10)λmin(Pi)supt∈[tk-1+Δk-1,tk)∥x(t)∥2≤εi(eλ-T-(t0,Tf)-1)λ-supt∈[tk-1+Δk-1,tk)∥w(t)∥2.
From (4.10), we have
(4.11)supt∈[tk-1+Δk-1,tk)∥x(t)∥supt∈[tk-1+Δk-1,tk)∥w(t)∥≤εi(eλ-T-(t0,Tf)-1)λ-λmin(Pi).
When t∈[tk,tk+Δk), for the jth subsystem, the state feedback controller is still u(t)=Kix(t). So the closed-loop system can be described as
(4.12)x˙(t)=(Aj+BjKi)x(t)+Gjw(t).
Consider the Lyapunov function candidate as follows:
(4.13)Vij(t)=xT(t)Pijx(t),t∈[tk,tk+Δk),k=0,1,….
By (4.3), it implies that
(4.14)V˙ij(t)≤λ+Vij(t)+εijwT(t)w(t).
With zero initial conditions, by Lemma 2.2, we have
(4.15)Vij(t)≤εij∫0t-tkeλ+τwT(t-τ)w(t-τ)dτ.
Notice that
(4.16)Vij(t)≥λmin(Pij)∥x(t)∥2.
From (4.15) and (4.16), we can obtain
(4.17)λmin(Pij)supt∈[tk,tk+Δk)∥x(t)∥2≤εij(eλ+T+(t0,Tf)-1)λ+supt∈[tk,tk+Δk)∥w(t)∥2.
From (4.17), we have
(4.18)supt∈[tk,tk+Δk)∥x(t)∥supt∈[tk,tk+Δk)∥w(t)∥≤εij(eλ+T+(t0,Tf)-1)λ+λmin(Pij).
By (4.11) and (4.18), during the finite-time [t0,Tf]=⋃r=0k-1[tr,tr+Δr)∪[tr+Δr,tr+1), we can obtain
(4.19)supt∈[t0,Tf)∥x(t)∥supt∈[t0,Tf)∥w(t)∥≤max(eλ-T-(t0,Tf)-1λ-maxi∈N_(εiλmin(Pi)),eλ+T+(t0,Tf)-1λ+maxi,j∈N_(εijλmin(Pij)))
By the definition of L∞ finite-time stabilization, we can obtain that the designed controller u(t)=Kσ′(t)x(t) can guarantee the finite-time stability of L∞ disturbance attenuation performance. This completes the proof.Remark 4.2.
Theorem4.1 represents that if each subsystem satisfies L∞ disturbance attenuation performance during the mismatched period and the matched period, the designed asynchronous switched controller u(t)=Kσ′(t)x(t) can guarantee the whole system has L∞ disturbance attenuation performance. However, the condition of each subsystem satisfying L∞ disturbance attenuation performance during the mismatched period and the matched period seems to be more  conservative, and in fact through the following theorem, this condition is not essential.Remark 4.3.
Although Theorem4.1 gives the method of finite-time stabilization with L∞ disturbance attenuation performance, the matched period T-(t0,Tf) and the mismatched period T+(t0,Tf) need to be prespecified in order to obtain L∞ disturbance attenuation performance of the system. However, in practical engineering it is difficult to obtain the matched period T-(t0,Tf) and the mismatched period T+(t0,Tf) before designing the controller. Based on these, the following result can be derived.Theorem 4.4.
If there exist matricesPi>0, Pij>0,Ki and scalars μ1>1,μ2>1,λ+>0,λ->0 such that
(4.20)Pi<μ1Pij,Pij<μ2Pi,(4.21)[(Ai+BiKi)TPi+Pi(Ai+BiKi)-λ-PiPiGiGiTPi-εiI]<0,(4.22)[(Aj+BjKi)TPij+Pij(Aj+BjKi)-λ+PijPijGjGjTPij-εijI]<0,(4.23)τa>(Tf-t0)ln(μ1μ2)ln((ε2/δ2)·ℬ·(μ2/(μ1μ2)N0))-λ+T+(t0,Tf)-λ-T-(t0,Tf)
and in finite-time interval [t0,Tf] the measurement noise w(t) satisfies supt∈[t0,Tf]∥w(t)∥<∞, then switched system (2.1) is finite-time stabilizable of L∞ disturbance attenuation performance γ=maxi,j∈N_(εi,εij)(emax(λ+,λ-)(Tf-t0)-1)/max(λ+,λ-)mini,j∈N_(λmin(Pi),λmin(Pij)) with respect to (δ,ε,Tf,σ(t),σ'(t)) under the feedback controller u(t)=Kσ'(t)x(t), where T-(t0,Tf) and T+(t0,Tf) denote the matched period and the mismatched period in finite-time interval [t0,Tf], respectively.Proof.
At first, from Theorem4.4, system (2.1) is finite-time stable under the feedback controller u(t)=Kσ′(t)x(t).
Then following the proof line of Theorem4.1 and considering (4.6) and (4.13), we can define piecewise Lyapunov function
(4.24)V(t)={xT(t)Pix(t),t∈[tr+Δr,tr+1),r=0,1,…,k-1,xT(t)Pijx(t),t∈[tr,tr+Δr),r=0,1,…,k-1.
By (4.21) and (4.22), it implies that
(4.25)V˙(t)≤max(λ+,λ-)V(t)+maxi,j∈N_(εi,εij)wT(t)w(t).
With zero initial conditions, by Lemma 2.2, we have
(4.26)V(t)≤maxi,j∈N_(εi,εij)∫0Tf-t0emax(λ+,λ-)τwT(t-τ)w(t-τ)dτ.
Notice that
(4.27)V(t)≥mini,j∈N_(λmin(Pi),λmin(Pij))∥x(t)∥2.
From (4.26) and (4.27), we can obtain
(4.28)mini,j∈N_(λmin(Pi),λmin(Pij))supt∈[t0,Tf]∥x(t)∥2≤maxi,j∈N_(εi,εij)(emax(λ+,λ-)(Tf-t0)-1)max(λ+,λ-)supt∈[t0,Tf]∥w(t)∥2.
From (4.28), we have
(4.29)supt∈[t0,Tf]∥x(t)∥supt∈[t0,Tf]∥w(t)∥≤maxi,j∈N_(εi,εij)(emax(λ+,λ-)(Tf-t0)-1)max(λ+,λ-)mini,j∈N_(λmin(Pi),λmin(Pij)).
By the definition of L∞ finite-time stabilization, we can obtain that the designed controller u(t)=Kσ'(t)x(t) can guarantee the finite-time stability of L∞ disturbance attenuation performance. This completes the proof.Remark 4.5.
It should be pointed out that the conditions in Theorems4.4 are not standard LMIs conditions. However, through the variable substitution, (4.20) to (4.22) can be solved following the method proposed in Remark 3.6.Remark 4.6.
Theorem4.4 presents that if the measurement noise w(t) is magnitude bounded during finite-time interval [t0,Tf], then we can design the asynchronous switching controller such that the system has L∞ disturbance attenuation performance. However, it is unnecessary to guarantee L∞ disturbance attenuation performance during the mismatched period and the matched period by the designed controller which is less conservative than Theorem 4.1.


## 5. Numerical Example

We consider an example to illustrate the main result. Consider the switched linear system given by the system (2.1) with u(t)=Kσ'(t)x(t),
(5.1)x˙(t)=(Aσ(t)+Bσ(t)Kσ′(t))x(t)+Gσ(t)w(t),
where A1=[-1000.1], A2=[2.1100.3], B1=[0.20.1402],B2=[100.30.1], G1=[0.200.30.1], G2=[0.10.20.40].Applying Theorem4.4 and solving corresponding matrix inequalities lead to feasible solutions, when δ=0.1,ε=10,ε1=ε2=100,ε12=ε21=10,μ1=μ2=20,λ+=100,λ-=10,Tf=0.005,t0=0,N0=0,τa=0.00375. 						(5.2)K1=[9.63641.4424-10.35390.4207],K2=[2.13370.4083-0.66233.5807],X1=[8.2146-14.6028-14.602886.9322],X2=[92.656914.602814.602813.9393],X12=[7.9844-0.3851-0.38519.9854],X21=[10.17660.14610.14618.7611].
Then from (3.23), we know that the matched period T-(t0,Tf) and the mismatched period T+(t0,Tf) satisfy the following relation:
(5.3)100T+(t0,Tf)+10T-(t0,Tf)<0.36.
Notice that  T+(t0,Tf)+T-(t0,Tf)=0.005, then we have
(5.4)T+(t0,Tf)<0.003,0.003<T-(t0,Tf)<0.005.
the L∞ state feedback controller K1, K2 can guarantee that system (5.1) is finite-time stabilizable with respect to (0.1,10,0.005,σ(t),σ'(t)) under the asynchronous switching where L∞ disturbance attenuation performance γ=7.8.


## 6. Conclusions

TheL∞ finite-time stabilization problems for switched linear system are addressed in this paper. When there exists asynchronous switching between the controller and the system, a sufficient condition for the existence of stabilizing switching law for the addressed switched system is derived. It is proved that the switched system is finite-time stabilizable under asynchronous switching satisfying the average dwell-time condition. Furthermore, the problem of L∞ control for switched systems under asynchronous switching is also investigated. At last, a numerical example is given to illustrate the effectiveness of the proposed method.



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*Source: 929503-2012-10-18.xml*