# Finite-Time Stability Analysis for a Class of Continuous Switched Descriptor Systems

Authors: Pan Tinglong; Yang Kun; Shen Yanxia; Gao Zairui; Ji Zhicheng
Journal: Mathematical Problems in Engineering
 (2014)

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## Abstract

Finite-time stability has more practical application values than the classical Lyapunov asymptotic stability over a fixed finite-time interval. The problems of finite-time stability and finite-time boundedness for a class of continuous switched descriptor systems are considered in this paper. Based on the average dwell time approach and the multiple Lyapunov functions technique, the concepts of finite-time stability and boundedness are extended to continuous switched descriptor systems. In addition, sufficient conditions for the existence of state feedback controllers in terms of linear matrix inequalities (LMIs) are obtained with arbitrary switching rules, which guarantee that the switched descriptor system is finite-time stable and finite-time bounded, respectively. Finally, two numerical examples are presented to illustrate the reasonableness and effectiveness of the proposed results.


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## Body

## 1. Introduction

Switched systems are a special class of hybrid systems, which consist of a collection of continuous or discrete-time subsystems together with a switching rule that orchestrates switching between these subsystems to achieve the control objectives [1]. Descriptor systems are also referred to as singular systems, implicit systems, or differential-algebraic systems, which are also a natural representation of dynamic systems and can describe physical systems better than the normal linear systems. Descriptor systems have been widely applied in many practical systems such as networks, power systems, electrical circuits, economics mathematical modeling, and many other fields [2, 3]. In actual control systems, switching phenomenon of descriptor systems is ubiquitous. Nevertheless, because of the switching between multiple descriptor subsystems and the algebraic constraints in descriptor model, it is inevitably difficult to analyze and synthesize for switched descriptor systems.Up to now, much attention has been mainly focused on system stability and reliability [2–5], H∞ control [6–8], cost-guaranteed control [9–11], system controllability, and reachability [12, 13] for switched descriptor systems. Generally, most of existing results related to stability and performance criteria of switched descriptor systems are based on the classical Lyapunov asymptotic stability, which is defined as an infinite time interval. However, in many actual systems, such as network control systems, the practical system state does not exceed some bound during some time interval and we need to avoid saturations and the excitation of nonlinear dynamics. In this case, the asymptotic stability is not enough for practical applications, because the system could be Lyapunov asymptotically stable but it possesses undesirable transient performances. Then the concept of finite-time stability was firstly put forward in [14], which concerns the boundedness of the system state over a fixed finite-time interval. To a certain degree, the development of the finite-time stability theory is parallel with the development of Lyapunov asymptotic stability.In recent years, the abundant studies on finite-time stability of switched systems [15–20] and descriptor systems [21–24] have been obtained. Lin et al. [15] gave some results on finite-time boundedness and finite-time weighted L2-gain for a class of switched delay systems with time-varying exogenous disturbances. Sufficient conditions which ensure that the switched system with time-time is finite-time bounded and has finite-time weighted L2-gain were proposed based on the average dwell time approach and the multiple Lyapunov functions technique. In [23], the issue of robust finite-time stabilization of descriptor stochastic systems with time-varying norm-bounded disturbance and parametric uncertainties via static output feedback was discussed. Suppose that the state vector is not available for feedback; a static output feedback controller in terms of restricted LMIs was provided to guarantee the underlying closed-loop descriptor stochastic system finite-time stabilization with a prescribed H∞r-disturbance attenuation over the given finite-time interval. Meanwhile, an illustrative example was employed to verify the efficiency of the proposed method. In view of the importance of practical application, we need to pay great attention to the research on finite-time stability for switched descriptor systems compared with Lyapunov asymptotic stability. Few works that deal with the finite-time stability for this type of systems have been reported. Based on the different multiple Lyapunov functions, the papers [25, 26] focused on the discrete-time switched descriptor systems and switched descriptor systems with time-varying delay, respectively.The paper is organized as follows. Firstly, the concepts of finite-time stability and finite-time boundedness for normal systems are expanded to continuous switched descriptor systems. Then, based on the state transfer matrix method, the sufficient and necessary condition of finite-time stability for this kind of system is given. Moreover, we tackle the problems of state feedback finite-time stabilization and finite-time boundedness; the sufficient conditions for the existence of controllers are obtained with arbitrary switching rules, which guarantee that the closed-loop systems are finite-time stable and finite-time bounded, respectively. Detailed proofs are presented by using the multiple Lyapunov functions and the average dwell-time approach. Finally, two examples are presented to show the validity of the developed methodology. Our research results are totally different from those previous results and important supplements for stability study for switched descriptor systems.


## 2. Problem Description and Preliminaries

Consider a class of switched descriptor system as follows:(1)Ex˙(t)=Aσ(t)x(t)+Bσ(t)uσ(t)(t)+Gσ(t)w(t),w˙(t)=Fσ(t)w(t),
where x(t)∈Rn is the system state, uσ(t)∈Rp is the control input, and w(t)∈Rq is the exogenous disturbance signal and satisfies the constraint wT(0)w(0)≤d, d≥0; the switching signal σ(t):Z+→{1,2,…,m} is a piecewise constant and right continuous function; m is the number of subsystems; σ(t)=i represents that the ith subsystem is activated.E, Ai, Bi, Gi, and Fi are known constant matrices with appropriate dimension, and it is assumed that rankE=r≤n.Remark 1.
In view of the special structure of descriptor systems, the initial condition is given asEx(t0)=Ex0. Corresponding to the switching signal σ(t), the switching sequence is defined as follows:
(2){x0;(i0,t0),…,(ik,tk),…∣ik∈{1,2,…,m},k=0,1,…},
which means that the ikth subsystem is activated when t∈[tk,tk+1) and x0, t0 are the initial state and initial time, respectively. In addition, we can transform the descriptor matrix of different form into the singular matrix “E” of system (1) by nonsingular transformation.Assumption 2.
The initial state of system (1) discussed is the consistent initial state, and the system state does not jump at the switching moment.
Now, we give the definitions of finite-time stability and finite-time boundedness for the continuous switched descriptor systems.Definition 3.
Continuous switched descriptor system(3)Ex˙(t)=Aσ(t)x(t)+Gσ(t)w(t),w˙(t)=Fσ(t)w(t),wT(0)w(0)≤d
is said to be finite-time bounded with respect to (c1,c2,d,R,Tf,σ), if the following formula:
(4)xT(0)ETREx(0)≤c1⟹xT(t)ETREx(t)≤c2,∀t∈[0,Tf],
holds, where c2>c1>0, d≥0, Tf>0, and R>0.Definition 4.
Continuous switched descriptor system(5)Ex˙(t)=Aσ(t)x(t)
is said to be finite-time stable with respect to (c1,c2,R,Tf,σ), if the following formula:
(6)xT(0)ETREx(0)≤c1⟹xT(t)ETREx(t)≤c2,∀t∈[0,Tf],
holds, where c2>c1>0, Tf>0, and R>0.Definition 5 (see [15]).
For anytf≥t0≥0, if Nσ(t)(t0,tf)≤N0+(tf-t0)/τa holds for given N0≥0, τa>0, where Nσ(t)(t0,tf) denotes the switching number of σ(t) over [t0,tf), then the constant τa is called the average dwell time and N0 is the flutter bound. As commonly used in the previous literature, we choose N0=0.Remark 6.
Finite-time stability for norm switched descriptor systems refers to the fact that the state of slow subsystem is less than a given upper bound. According to regularity of systems, the state of fast subsystem is also less than a given upper bound.


## 3. Main Results

Firstly, the sufficient and necessary condition of finite-time stability for system (5) is given by applying the state transition matrix method.Theorem 7.
Given positive constantsc1, c2, and Tf, the system (5) is finite-time stable with respect to (c1,c2,R,Tf,σ), if and only if
(7)ΦσT(t,0)ETREΦσ(t,0)≤c2c1ETRE,∀t∈[0,Tf),
where Φσ(t,0)=Φik(t,tk)Φik-1(tk,tk-1)⋯Φi0(t1,0) is the state transition matrix.Proof.
The following proof can be divided into two cases.
(a) Sufficiency. Since Φσ(t,0) is the state transition matrix of system (5), then
(8)x(t)=Φσ(t,0)x(0),t∈[0,Tf),
where Φσ(t,0)=Φik(t,tk)Φik-1(tk,tk-1)⋯Φi0(t1,0). In view of xT(0)ETREx(0)≤c1, one obtains
(9)xT(t)ETREx(t)=xT(0)ΦσT(t,0)ETREΦσ(t,0)x(0).
On the other hand, from (7) and xT(0)ETREx(0)≤c1, we have
(10)xT(t)ETREx(t)≤xT(0)c2c1ETREx(0)≤c2.
Thus, the system (5) is finite-time stable according to Definition 4.
(b) Necessity. Suppose that the system (5) is finite-time stable with respect to (c1,c2,R,Tf,σ). By using the reduction to absurdity, if there exists Ex(t*)≠0,  t*∈[0,Tf), such that
(11)xT(t*)ΦσT(t*,0)ETREΦσ(t*,0)x(t*)≥xT(t*)c2c1ETREx(t*)>0.
Let λ=c1/xT(t*)ETREx(t*), x0=λx(t*); then we get x0TETREx0=c1. By virtue of (11), we have
(12)x0TΦσT(t*,0)ETREΦσ(t*,0)x0≥x0Tc2c1ETREx0.
Meanwhile, according to (8), we can obtain
(13)xT(t*)ETREx(t*)=x0TΦσT(t*,0)ETREΦσ(t*,0)x0≥c2.
Noticing that it is inconsistent with the hypothesis that the system (5) is finite-time stable with respect to (c1,c2,R,Tf,σ), the proof is completed.Remark 8.
From Theorem7 we can obtain the sufficient and necessary condition, which guarantees that the switched descriptor system (5) is finite-time stable. However, it is difficult and inconvenient to calculate state transition matrix and design controller. Thus, it is difficult to apply in the actual systems.Theorem 9.
For anyi,j∈{1,2,…,m}, if there exist nonsingular  matrix Pi, matrices Qi>0, Zi>0, and scalars α≥0, μ≥1 such that
(14)PiE=ETPiT≥0,(15)[PiAi+AiTPiT-αPiEPiGiGiTPiTFiTQi+QiFi-αQi]<0,(16)PiE=ETR1/2ZiR1/2E,(17)PiE≤μPjE,Qi≤μQj,(18)λ1c1+λ3dλ2c2<e-αTf.
And the average dwell time of the switching signal σ satisfies
(19)τa>τa*=Tflnμln(λ2c2)-ln(λ1c1+λ3d)-αTf,
where λ1=maxi=1,2,...,m{λmax(Zi)}, λ2=mini=1,2,...,m{λmin(Zi)}, and λ3=maxi=1,2,...,m{λmax(Qi)}; then system (3) is regular, pulse-free, and finite-time bounded with respect to (c1,c2,d,R,Tf,σ).Proof.
First, from (15) we have
(20)PiAi+AiTPiT-αPiE<0.
ConsideringrankE=r and assuming that there exist invertible matrices M, N, such that E-=MEN=diag{Ir,0}, then it follows from (14) and (20) that
(21)P-iE-=E-TP-iT≥0,(22)P-iA-i+A-iTP-iT-αP-iE-<0,
where E-=MEN=diag{Ir,0}, A-i=MAiN, P-i=NTPiM-1.
According to (21), we can obtain P-i as follows:
(23)P-i=[Pi1Pi20Pi3],Pi1>0,det(Pi3)≠0.
Correspondingly, suppose that A-i=[Ai11Ai12Ai21Ai22], and it follows from (22) that
(24)[###Pi3Ai22+Ai22TPi3T]<0,
where we do not need to know the expression of # and it does not affect the following discussion. It follows from (24) that Pi3Ai22+Ai22TPi3T<0; namely, Ai22 is nonsingular. Then, there exists a scalar s such that det(sE--A-i)≠0 and, for ∀s, degdet(sE--A-i)=rankE- holds. Thus, the system (3) is regular and pulse-free.
In the following, we will prove that system (3) is finite-time bounded with respect to (c1,c2,d,R,Tf,σ). Choose the multiple Lyapunov function as
(25)V(t,x(t),w(t))=Vσ(t,x(t),w(t))=xT(t)PσEx(t)+wT(t)Qσw(t).
Then, let V(t,x(t),w(t))=V(t) and Vσ(t,x(t),w(t))=Vσ(t). By virtue of (14) and switched sequence (2), when t∈[tk,tk+1), taking derivative of V(t) with respect to t along the trajectory of the system (3) yields
(26)V˙(t)=V˙σ(tk)(t)=[x(t)w(t)]T×[Pσ(tk)Aσ(tk)+Aσ(tk)TPσ(tk)TPσ(tk)Gσ(tk)Gσ(tk)TPσ(tk)TFσ(tk)TQσ(tk)+Qσ(tk)Fσ(tk)]×[x(t)w(t)].
Then, it follows from (15) that
(27)V˙<αV=αVσ(tk)(t).
Integrating both sides of (27) from tk to t gives
(28)V(t)<eα(t-tk)Vσ(tk)(tk).
Then, together with (17), at the switched moment tkwe derive
(29)V(t)<eα(t-tk)μVσ(tk-)(tk-).
For anyt∈[0,Tf), assuming that N is the switched number of systems between [0,Tf), one can obtain Nσ(0,t)≤N. Considering μ≥1 and using (17) together with (29), based on iterative method, we have
(30)V(t)<eαtμNσ(0,t)Vσ(0)(0)≤eαtμNVσ(0)(0).
Noticing that N≤Tf/τa, then
(31)V(t)<eαtμTf/τaVσ(0)(0).
On the other hand, it follows from (16) that
(32)V(t)≥xT(t)PσEx(t)≥λmin(Zσ)xT(t)ETREx(t)≥λ2xT(t)ETREx(t),Vσ(0)(0)=xT(0)Pσ(0)Ex(0)+wT(0)Qσ(0)w(0)≤λmax(Zσ(0))xT(0)ETREx(0)+λmax(Qσ(0))d≤λ1xT(0)ETREx(0)+λ3d,
when xT(0)ETREx(0)≤c1; putting together (31)-(32), it can easily be verified that
(33)xT(t)ETREx(t)<eαtμTf/τa(λ1c1+λ3dλ2).
According to (18), we get
(34)ln(λ2c2)-ln(λ1c1+λ3d)-αTf>0.
It follows from (19) and (34) that
(35)μTf/τa(λ1c1+λ3dλ2)<e-αTfc2.
By virtue of (33) and (35), we can obtain
(36)xT(t)ETREx(t)<eαte-αTfc2=eα(t-Tf)c2;
then, considering t<Tf and α≥0, we have
(37)xT(t)ETREx(t)<c2.
Thus, the system (3) is finite-time bounded with respect to (c1,c2,d,R,Tf,σ), and the proof is completed.Remark 10.
Since different Lyapunov functions can be constructed for different subsystems, so the multiple Lyapunov functions method is an effective and flexible design tool. Now, the multiple Lyapunov functions have been employed and discussed to study the stability and performance of switched or hybrid systems such as [27–29]. In this paper, the function V(t) in the proof of Theorem 9 is taken as the multiple Lyapunov functions. Compared with the classical Lyapunov function for switched systems of asymptotical stability, there is really no requirement of negative definiteness or negative semidefiniteness on V(t). Actually, if we limit the constants α<0 and the exogenous disturbance w(t)=0, then V(t) will be a negative definite function. In this case, the system (1) is asymptotically stable on the infinite interval [0,+∞). We can find the detailed proof in [29].
In order to design controller conveniently, the following conclusion is given to satisfy the condition of Theorem9.Theorem 11.
For anyi,j∈{1,2,…,m}, if there exist nonsingular matrix Xi, matrices Qi>0, Zi>0, and scalars α≥0, μ≥1 such that (18) and (19) hold and
(38)EXi=XiTET≥0,(39)[AiXi+XiTAiT-αEXiGiGiTFiTQi+QiFi-αQi]<0,(40)Xi-TE=ETR1/2ZiR1/2E,(41)(1-2μ)EXi+μEXj≤0,Qi≤αQj,
then system (3) is regular, pulse-free, and finite-time bounded with respect to (c1,c2,d,R,Tf,σ).Proof.
Multiply both sides of (38) separately by Xi-T and Xi-1. Let Xi-T=P; we can obtain that (14) holds. Similarly, multiply both sides of (39) separately by diag{Xi-T,I} and diag{Xi-1,I}. Let Xi-T=P; then (15) holds. On the other hand, let Xi=Pi-T, it follows from (38) and (41) that
(42)EPi-T-αEPi-T-αPi-1ET+αEPj-T≤0,Qi≤αQj.
Now, by virtue of -αPi-1PjEPi-T≤-αEPi-T-αPi-1ET+αEPj-T, it follows from (42) that EPi-T-αPi-1PjEPi-T≤0,  Qi≤αQj. Obviously, the previous equation is equivalent to (17), so (41) can ensure that (17) holds. From Theorem 9, it is easy to obtain that system (3) is regular, pulse-free, and finite-time bounded with respect to (c1,c2,d,R,Tf,σ).In the following, we give the following conclusion about the finite-time boundedness problem of system (1) via the action of the state feedback controller uσ(t)=Kσx(t).Theorem 12.
For anyi,j∈{1,2,…,m}, if there exist nonsingular matrices Xi, Li,  matrices Qi>0, Zi>0, and scalars α≥0, μ≥1 such that (18), (19), and (38)–(41) hold and
(43)[AiXi+XiTAiT+BiLi+LiTBiT-αEXiGiGiTFiTQi+QiFi-αQi]<0,
then the system (1) is regular, pulse-free, and finite-time bounded with respect to (c1,c2,d,R,Tf,σ) via the action of the state feedback controller ui(t)=LiXi-1x(t).Proof.
According to the proof of Theorem11, first replace Ai with Ai+BiKi and let KiXi=Li, and then it is easy to obtain the condition of Theorem 12.
Now, in order to solve by means of the LMI toolbox conveniently, we will process (38) and (40). According to rankE=r, there exist invertible matrices M, N such that E-=MEN=diag{Ir,0}. Let X-i=N-1XiMT and from (38) X-i is given as
(44)X-i=[Xi10Xi3Xi4],
where Xi1>0, det(Xi4)≠0, and Xi3 is a matrix with appropriate dimension. In addition, let Ψ=N[0In-r] and we can obtain EΨ=0. Based on the above discussion, the following equation holds:
(45)Xi=N[Xi10Xi3Xi4]M-T=N([Xi1000]+[00Xi3Xi4])M-T=NΩiNTET+ΨYiM-T,
where Ωi=diag{Xi1,#}>0 and Yi=[Xi3Xi4]. It is obvious that Xi=NΩiNTET+ΨYiM-T satisfies (41) and (43). Let Zi=R-1/2MTΩi-1MR-1/2, and then one obtains that (40) holds.
Equation (18) can be guaranteed by the following LMIs. For any i,j∈{1,2,…,m}, there exist scalars ε1, ε2 such that
(46)ε1I<R1/2M-1ΩiM-TR1/2<I,Qi<ε2I,[-eαTfc2+ε2d-c1-c1-ε1]<0.Theorem 13.
For anyi,j∈{1,2,…,m}, if there exist matrices Ωi=diag{Xi1,#}>0, Qi>0, Li, Yi and scalars α≥0, μ≥1 such that (19) and (46) hold and
(47)(1-2μ)ENΩiNTET+μENΩjNTET≤0,Qi≤αQj,[ΞiGiGiTFiTQi+QiFi-αQi]<0,
then the switched descriptor system (1) is regular, pulse-free, and finite-time bounded with respect to (c1,c2,d,R,Tf,σ) via the action of the state feedback controller ui(t)=Liβ-1(Ωi,Yi)x(t), where
(48)Ξi=Aiβ(Ωi,Yi)+βT(Ωi,Yi)AiT+BiLi+LiTBiT-αEβ(Ωi,Yi),
and the matrix β(Ωi,Yi)=NΩiNTET+ΨYiM-T is nonsingular; matrices M, N satisfy MEN=diag{Ir,0}, Ψ=N[0In-r]T.Remark 14.
If there existsE=I, Fi=0, we can obtain the sufficient condition of finite-time boundedness for switched systems with constant disturbance based on the above conclusion. If there exists w(t)=0, and V(t) is taken as common Lyapunov function, then the system is of consistent finite-time boundedness with arbitrary switching signals.Theorem 15.
Consider the switched descriptor system as follows:(49)Ex˙(t)=Aσ(t)x(t)+Bσ(t)uσ(t)(t).
For any i,j∈{1,2,…,m}, if there exist matrices Ωi=diag{Xi1,#}>0, Li, Yi and scalars α≥0, μ≥1 such that
(50)(1-2μ)ENΩiNTET+μENΩjNTET≤0,Aiβ(Ωi,Yi)+βT(Ωi,Yi)AiT+BiLi+LiTBiT-αEβ(Ωi,Yi)<0,ε1I<R1/2M-1ΩiM-TR1/2<ε2I,ε2c1<ε1c2
and the average dwell time satisfies
(51)τa>τa*=Tflnμln(λ2c2)-ln(λ1c1)-αTf,
then the switched descriptor system (49) is regular, pulse-free, and finite-time stable with respect to (c1,c2,d,R,Tf,σ) via the action of the state feedback controller ui(t)=Liβ-1(Ωi,Yi)x(t), where β(Ωi,Yi)=NΩiNTET+ΨYiM-T is nonsingular and matrices M, N satisfy MEN=diag{Ir,0}, Ψ=N[0In-r]T.


## 4. Numerical Simulations

Example 16.
Consider the continuous switched descriptor system (1) with parameters as follows:
(52)A1=[-2.11.0-0.3-0.8-0.40.40.8-0.61.0],B1=[1-11],G1=[0.20.1-0.10.30.1-0.1],F1=[00.1-0.10],A2=[-2.6-0.50.50.60.5-0.8-0.50.21.0],B2=[-111],G2=[-0.10.20.2-0.1-0.10.3],F2=[00.1-0.10].
The values of E, α, μ, c1, c2, d, Tf, and R are given as follows: α=0.3, μ=1.2, E=diag{1,1,0}, let c1=2, c2=10, d=3, Tf=1, and R=I.
Then, according to Theorem13, we get
(53)Ω1=[0.52490.026600.02660.46350000.4749],Ω2=[0.53530.027500.02750.47870000.4749],Y1=[-0.5388-0.2418-0.5585],Y2=[1.2591-0.2093-1.1868],L1=[0.22300.4453-0.2745],L2=[-0.0427-1.17100.3543],Q1=[2.1001002.1001],Q2=[2.1001002.1001].
In the following, we have by calculating
(54)β(Ω1,Y1)=[0.54530.026600.02660.46350-0.5388-0.2418-0.5585],β(Ω2,Y2)=[0.53530.027500.02750.478701.2591-0.2093-1.1868].
Then we can obtain the designed state feedback controllers as follows:
(55)u1(t)=L1β-1(Ω1,Y1)x(t)=[0.83761.16910.4914]x(t),u2(t)=L2β-1(Ω2,Y2)x(t)=[0.7566-2.6204-0.2985]x(t).
Furthermore, we can obtain λ1=2.1949,  λ2=1.8078,  λ3=2.1001, and τa>τ*=0.2517. For any switching signal σ(t) with average dwell time τa>0.2517,  the switched descriptor system (1) is finite-time stable.
Whenw(t)=[sin(0.1t)cos(0.1t)], the trajectory of xT(t)ETREx(t) is presented in Figure 1 and the switching signal is shown in Figure 2.Figure 1
Trajectory ofxT(t)ETREx(t).Figure 2
Switched signalσ(t).Example 17.
Consider the continuous switched descriptor system (49) with parameters as follows:
(56)A1=[-1-212-1301-2],B1=[-1-91],A2=[-21-11-102-2-1],B1=[3-8-1].
Choosing the matrix E=diag{1,1,0} and letting α=0.1,  μ=1.2, c1=1, c2=5, Tf=6, and R=I, by virtue of Theorem 15, the feasible solutions are given as
(57)Ω1=[33.4670-2.15490-2.154937.288400036.9815],Ω2=[31.46900.543500.543538.903100036.9815],Y1=[6.844421.344915.4656],Y2=[-1.3833-31.484332.2021],L1=[3.70615.11685.1099],L2=[11.9850-1.8186-5.5726].
Now, it can be obtained as
(58)β(Ω1,Y1)=[33.4670-2.15490-2.154937.284406.844421.344915.4656],β(Ω2,Y2)=[31.46900.543500.543538.90310-1.3833-31.484332.2021].
Then we can obtain the designed controllers as follows:
(59)u1(t)=L1β-1(Ω1,Y1)x(t)=[0.0400-0.04960.3304]x(t),u2(t)=L2β-1(Ω2,Y2)x(t)=[0.3766-0.1921-0.1731]x(t).
By calculating, we have λ1=0.0318,  λ2=0.0257, and τa>τ*=1.3758. For any switching signal σ(t) with average dwell time τa>1.3758, the system (49) is finite-time stable with respect to (c1,c2,d,R,Tf,σ) via the action of the state feedback controller. Figure 3 shows the trajectory of  xT(t)ETREx(t), and Figure 4 shows the switching signal.
If the switching is too frequent, it is possible that the whole system is not finite-time stable. For instance, given the switching signal as follows:(60)σ(t)={1,x2x3-x1x2<0,2,x2x3-x1x2≥0,
the whole system is not finite-time stable any more. The trajectory of xT(t)ETREx(t) is shown in Figure 5 and the switching signal σ(t) is shown in Figure 6.Figure 3
Trajectory ofxT(t)ETREx(t).Figure 4
Switched signalσ(t).Figure 5
Trajectory ofxT(t)ETREx(t).Figure 6
Switching signalσ(t).


## 5. Conclusion

In this paper, the issues of finite-time stability and finite-time boundedness for a class of continuous switched descriptor systems have been studied. The sufficient and necessary condition of finite-time stability for switched descriptor systems is presented by applying the state transition matrix method. The obtained condition has certain theoretical value, but it also has two disadvantages in practical application. First, it is difficult to calculate the state transition matrix; on the other hand, it is inconvenient to design controller. In order to solve these problems, based on the average dwell time approach and the multiple Lyapunov function technique, the existence of state feedback controllers is proposed with arbitrary switching rules, which guarantee that the switched descriptor systems are finite-time stable and finite-time bounded, respectively. More possible future works are to consider output feedback stabilization for the uncertain switched descriptor systems with time-varying delay.



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*Source: 979130-2014-01-08.xml*