Finite-time stability of switched linear systems with subsystems which are not finite-time stable

www.ietdl.org

Published in IET Control Theory and ApplicationsReceived on 12th July 2013Revised on 10th January 2014Accepted on 26th January 2014doi: 10.1049/iet-cta.2013.0648

ISSN 1751-8644

Finite-time stability of switched linear systems withsubsystems which are not finite-time stableXiangze Lin1, Shihua Li2,Yun Zou3

1College of Engineering, Nanjing Agricultural University/Jiangsu Key Laboratory for Intelligent Agricultural Equipment,Nanjing 210031, People’s Republic of China2School of Automation, Southeast University, Nanjing 210096, People’s Republic of China3 School of Automation, Nanjing University of Science andTechnology, Nanjing 210094, People’s Republic of ChinaE-mail: [email protected]

Abstract: Up to now, the potential assumption of most existing results for finite-time stability and finite-time boundednessof switched linear systems is that each subsystem should be finite-time stable or finite-time bounded. If any one subsystem ofswitched systems is not finite-time stable or finite-time bounded, the previous results may not be true anymore. In this paper,finite-time stability and finite-time boundedness of switched linear systems with subsystems that are not finite-time stableor finite-time bounded are discussed. Sufficient conditions are given under which switched linear systems with subsystemsthat are not finite-time stable or finite-time bounded is guaranteed to be still finite-time stable or finite-time bounded. Theresults also show the effect of the switching signals on finite-time stability and finite-time boundedness of switched linearsystems. Moreover, finite-time L2-gain of switched linear systems with subsystems which are not finite-time bounded is alsogiven to measure its disturbance tolerance capability in the fixed time interval. A numerical example is employed to verifythe efficiency of the proposed method.

1 Introduction

Switched systems have been attracting considerable attentionduring the last decades because of their success in prac-tical applications and importance in theory development.Lyapunov stability of switched systems, which is definedover an infinite-time interval to show the qualitativebehaviour of a dynamical system, is a basic topic in theresearch. Up to now, most of existing literatures related tostability of switched systems focused on Lyapunov asymp-totic stability, such as [1–5] and the references therein.

However, a system could be Lyapunov stable but mayshow poor dynamic characteristics if it possesses undesirabletransient performances. Moreover, in practice, not only thequalitative behaviour of a dynamical system but also quan-titative information [6] (e.g. specific estimates of trajectorybounds over a fixed short time) is of great interest. In thesecases, finite-time stability of a system, which focuses itsattention on the system behaviour over a finite-time interval,can give quantitative information and improve the transientperformance of the system. Hence, an asymptotically stablesystem can own prescribed dynamic characteristics if it isalso finite-time stable.

Finite-time stability of a system means that given a boundon the initial condition, its state remains within prescribedbounds in the fixed time interval. It should be noted thatfinite-time stability and Lyapunov asymptotical stability areindependent concepts: a system could be finite-time sta-ble but not Lyapunov asymptotically stable, and vice versa.

IET Control Theory Appl., 2014, Vol. 8, Iss. 12, pp. 1137–1146doi: 10.1049/iet-cta.2013.0648

Some early results on finite-time stability problems dateback to the sixties of the last century [7, 8]. Recently,the concept of finite-time stability has been revisited inthe light of linear matrix inequality theory. Many valuableresults have been obtained for this type of stability [9–16].In [9–11, 16], sufficient conditions for finite-time stabilityand stabilisation of continuous-time systems or discrete-timesystems have been provided. The authors of [12–15] haveextended the definition of finite-time stability to the systemwith impulsive effects or singular systems with impulsiveeffects, respectively, and derived some sufficient conditionsfor finite-time stability and stabilisation problem. In addition,it should be pointed out, the authors of [17–19] presentedsome results of finite-time stability for different systems, butthe finite-time stability which consists of Lyapunov stabil-ity and finite-time convergence is different from that in thispaper and [7–16].

Recently, finite-time stability problem of switched sys-tems has attracted more and more attention, such as [20–29].Based on the linear matrix inequalities (LMIs), finite-timestability and stabilisation conditions for the switched systemwithout or with time-delay were developed in [22, 27, 30],respectively. Finite-time stability of a switched discrete-timesystem were presented in [26], and also in its references. Thecontrollers, if it have in those papers, are mainly linear statefeedback.

It should be pointed out that the potential assumptionof most existing results, such as those in [20–22, 26–30], is that each subsystem should be finite-time stable or

1137© The Institution of Engineering and Technology 2014

www.ietdl.org

finite-time bounded. If any one subsystem of the switchedsystem is not finite-time stable or finite-time bounded, theresults in the previous papers may not be true. In thispaper, finite-time stability and finite-time boundedness ofswitched linear systems with subsystems that are not finite-time stable or finite-time bounded are discussed. Sufficientconditions are given under which a switched linear systemwith subsystems which are not finite-time stable or finite-time bounded is guaranteed to be still finite-time stable orfinite-time bounded. The results also show the effect of theswitching signals on finite-time stability of switched linearsystems. Moreover, finite-time L2-gain problem is discussedto measure its disturbance tolerance capability. A prelim-inary attempt to tackle this problem was reported in ourrecent conference paper [31].

The paper is organised as follows. In Section 2, somenotations and problem formulation are presented. The defi-nitions of finite-time stability and finite-time boundednessfor switched linear systems are reviewed. In Section 3,finite-time stability problem of switched linear systems withsubsystems which are not finite-time stable is dealt with.Sufficient conditions of the finite-time stability of switchedlinear systems with subsystems that are not finite-time sta-ble are presented and the effect of switching signals onfinite-time stability on this class of switched systems is dis-cussed. Finite-time boundedness problem for switched linearsystems with subsystems that are not finite-time boundedis discussed in Section 4. In Section 5, finite-time L2-gainanalysis is presented to measure the disturbance tolerancecapability of the system. Simulation results are presented toillustrate the efficiency of our approach in Section 6. Finally,conclusions are given in Section 7.

2 Preliminaries and problem formulation

In this paper, for any symmetric positive definite (orpositive-semidefinite) matrix P, λmax(P) and λmin(P) denotethe maximum and minimum eigenvalues of matrix P,respectively. The identity matrix of order n is denoted asIn (or, simply, I if no confusion arises).

Consider a class of switched linear systems with the formas follows

{x(t) = Aσ(t)x(t), x(0) = x0

y = Cσ(t)x(1)

where x(t) ∈ Rn is the state, y(t) ∈ Rl is the output, σ(t) :[0, ∞) → M = {1, 2, . . . , m} is the switching signal which isa piecewise constant function depending on time t or statex(t), Ai and Ci are constant real matrices for i ∈ M , the two-matrix pair (Ai, Ci), i ∈ M represents the ith subsystems of(1), m denotes the number of the subsystems.

The following assumptions are common for the switchedlinear system (1) as that in [1, 2].

Assumption 1: The state of switched linear systems doesnot jump at switching instants, that is, the trajectory x(t)is everywhere continuous.

Assumption 2: Switching signal σ(t) has finite switchingnumber in any finite-time interval.


Corresponding to the switching signal σ(t), we have thefollowing switching sequence

{x0; (i0, t0), . . . , (ik , tk), . . . , |ik ∈ M , k = 0, 1, . . .}which means that ik th subsystem is activated when t ∈[tk , tk+1). Let S denote the set of switching signal whichhas finite number of switchings in any finite-time interval.In this paper, we only consider the switching signal whichbelongs to S, that is, σ(t) ∈ S.

First, let us review the definition of finite-time stability ofthe switched linear system (1) which is extended from [9].

Definition 1: Given three positive constants c1, c2, Tf , withc1 < c2, a positive definite matrix R, and a given switchingsignal σ(t) ∈ S, the switched linear system (1) is said to befinite-time stable with respect to (c1, c2, Tf , R, σ), if

xT0 Rx0 ≤ c1 ⇒ x(t)TRx(t) < c2, ∀t ∈ (0, Tf ]

Definition 2: Given three positive constants c1, c2, Tf , withc1 < c2, a positive definite matrix R, the switched linearsystem (1) is said to be uniformly finite-time stable withrespect to (c1, c2, Tf , R, S), if for any switching signalσ(t) ∈ S,

xT0 Rx0 ≤ c1 ⇒ x(t)TRx(t) < c2, ∀t ∈ (0, Tf ]

Consider a switched linear system with disturbances{x(t) = Aσ(t)x(t) + Gσ(t)ω(t), x(0) = x0

y = Cσ(t)x + Dσ(t)ω(t)(2)

where x(t) ∈ Rn is the state, y(t) ∈ Rl is the output, ω(t) ∈Rp is the disturbance, Aσ(t) and Cσ(t) are the same as thoseof the system (1), Gσ(t) and Dσ(t) are constant matrices withcorresponding dimensions.

Assumption 3: The external disturbance ω(t) is time-varyingand satisfies the constraint

∫Tf

0 ωT(t)ω(t) dt ≤ d, d ≥ 0.

In [9], the concept of finite-time boundedness was pro-posed to study the transient behaviour of a linear system withexternal disturbances in a fixed time interval. In the sequel,let us extend the definition of finite-time boundedness to theswitched linear system (2).

Definition 3: Given five positive constants c1, c2, Tf , d withc1 < c2, d ≥ 0 and a positive definite matrix R, the switchedlinear system (2) is said to be finite-time bounded withrespect to (c1, c2, Tf , R, σ(t), d), if

x(t0)TRx(t0) ≤ c1 ⇒ x(t)TRx(t) < c2, ∀t ∈ [0, Tf ],

∀σ(t) ∈ S, ∀ω(t) :∫ Tf

0

ωT(t)ω(t) dt ≤ d (3)

If equation (3) holds under an arbitrary switching signalσ(t), the switched linear system (2) is said to be uniformlyfinite-time bounded with respect to (c1, c2, Tf , R, S, d)

Remark 1: The meaning of ‘uniformity’ in Definitions 2and 3, is with respect to the switching signal rather thanthe time, which is identical to that of [1, 4].


www.ietdl.org

Disturbance attenuation properties for switched systemshave been widely studied. The disturbance tolerance capa-bility can be measured by the largest bound on the energyof the disturbance, say L2-gain. In [32], weighted L2-gainproblem was introduced and some sufficient conditions weregiven. Then, in [33], weighted L2-gain analysis for a classof switched linear systems with time-delay was discussed.Finite-time L2-gain problem of the switched system whosesubsystems are finite-time bounded has been discussed in[22, 26–28]. Here, in this paper, we investigate finite-timeL2-gain of the switched system whose subsystems are notfinite-time bounded in a fixed time interval.

Definition 4: For Tf > 0 and the given parameter γ > 0, theswitched linear system (2) is said to have finite-time L2-gainless than γ , if under zero initial condition x(t0) = 0, it holdsthat ∫ Tf

0

yT(s)y(s) ds ≤ γ 2

∫ Tf

0

ωT(s)ω(s) ds (4)

Although finite-time stability and finite-time bounded-ness of switched linear systems has attracted much moreattentions, it should be pointed out that the potential assump-tion of most existing results is that each subsystem shouldbe finite-time stable or finite-time bounded. If any one sub-system of the switched system is not finite-time bounded orfinite-time stable, the results in those paper may not be true.The aim of this paper is to deal with the problem of finite-time stability and finite-time boundedness of the switchedlinear systems (1) and (2) with subsystems which are notfinite-time stable or finite-time bounded, and find sufficientconditions which can guarantee the switched linear systems(1) and (2) still finite-time stable or finite-time bounded.

Without loss of generality, we assume that A1, . . . , Ar ,(1 ≤ r < m) in the system (1) are not finite-time stable andthe remaining subsystems are finite-time stable. Under thisassumption, there exist scalars ai ≥ 0, αi > 0, βi > 0 suchthat

‖eAit‖ ≤ eai+αi t , 1 ≤ i ≤ r

‖eAit‖ ≤ eai+βi t , r + 1 ≤ i ≤ m(5)

The scalars ai ≥ 0, αi > 0, βi > 0 are easy to be com-puted, for example, if Ai has distinct positive eigenvaluesλj(Ai) (j = 1, 2, . . . , m), there always exists a non-singularmatrix Pi such that P−1

i AiPi = diag{λ1(Ai), . . . , λn(Ai)} andthus ai ≥ 0 and αi > 0, βi > 0 in (5) can be chosen aslog[σmax(Pi)/σmin(Pi)] and maxi[λj(Ai)], respectively, whereσmax(Pi)(σmin(Pi)) denotes the maximum (minimum) singu-lar value of Pi.

Switching signals play important roles in the researchof stability of switched systems. The average dwell timeswitching means that the number of switches in a finiteinterval is bounded and the average time between consecu-tive switching is not less than a constant. The average dwelltime switching can cover the dwell time switching, and itsextreme case is actually the case of arbitrary switching. The


definition of the average dwell time of switching signals iscited as follows.

Definition 5 [3]: For any T ≥ t ≥ 0, let Nσ (t, T ) denote theswitching number of σ(t) over (t, T ). If

Nσ (t, T ) ≤ N0 + T − t

τa

holds for τa > 0 and an integer N0 ≥ 0, then τa is called anaverage dwell-time and N0 is called the chattering bound.

For simplicity, here we let N0 = 0 as that in [22, 33].It should be pointed out that the original definition of theaverage dwell-time, which is defined in the whole inter-val [0, ∞), actually indicates the switching number occursduring [t, T ). In a finite-time interval, if we take N0 = 0,then for any finite-time interval T − t < τa, no switchingcan occur. In this note, we assume that T − t > τa in thefinite-time interval, that is, switching does happen.

In order to prove the main results, the well-knownCauchy–Schwarz inequality is cited as follows [34].

Lemma 1 (Cauchy–Schwarz inequality): If f (x) and g(x) iscontinuous on [a, b], then

[∫ b

a

f (x)g(x) dx

]2

≤∫ b

a

f 2(x) dx ·∫ b

a

g2(x) dx (6)

3 Finite-time stability of switched linearsystems

In this section, finite-time stability of switched linearsystems is discussed. Sufficient conditions under whichswitched linear systems with subsystems that are not finite-time stable are still finite-time stable are given.

Theorem 1: For the switched linear system (1), there are twocases of its finite-time stability:

(a) when a = 0, if the total dwell time T + of the subsystemswhich are not finite-time stable satisfies the inequality

T +

Tf≤ ln [c2λmin(R)] − ln [c1λmax(R)] − 2βTf

2(α − β)Tf(7)

then the switched linear system (1) is uniformly finite-timestable with respect to (c1, c2, Tf , R);(b) when a > 0, if the following inequality (see (8))

is satisfied, then the switched linear system (1) is finite-time stable with respect to (c1, c2, T , R, σ). Moreover, if theswitching signal satisfies the average dwell time condition,the inequality (8) can be rewritten by virtue of average dwelltime τa with N0 = 0 as follows (see (9))

N (0, Tf ) ≤ ln [c2λmin(R)] − ln [c1λmax(R)] − 2[(α − β)T + + βTf ] − 2a

2a(8)

τa ≥ 2aTf

ln [c2λmin(R)] − ln [c1λmax(R)] − 2[(α − β)T + + βTf ] − 2a(9)


www.ietdl.org

where T + denotes the total dwell time of the subsys-tems which are not finite-time stable, a = max1≤i≤m{ai},α = max1≤i≤m{αi}, β = max1≤i≤m{βi}.Proof: Let t1, t2 . . . denote the switching instants and pj forthe value of σ on [tj−1, tj). Then, for any t satisfying t0 <· · · < ti ≤ t < ti+1 ≤ T , one obtains

x(t) = eApi+1 (t−ti)eApi (ti−ti−1) · · · eAp1 (t1−t0)x0 (10)

From the inequality (5), we obtain the following estimate ofthe norm of the state

‖x(t)‖ ≤(

i+1∏q=1

eapq

)eαT++βT−‖x(0)‖

≤ e(i+1)a eαT++βT−‖x(0)‖ (11)

where T − = Tf − T + denotes the total dwell time of finite-time stable subsystems of the switched linear system (1).Then, we have

xTRx ≤ λmax(R)‖x‖2

≤ λmax(R)e2(i+1)a+2(αT++βT−)‖x(0)‖2

≤ λmax(R)e2a e2ia e2(αT++βT−)‖x(0)‖2

≤ λmax(R)e2a e2aN (0,Tf )e2[αT++β(Tf −T+)]‖x(0)‖2

≤ λmax(R)

λmin(R)e2a e2aN (0,Tf )e2[(α−β)T++βTf ]xT

0 Rx0

≤ λmax(R)

λmin(R)e2a e2aN (0,Tf )e2[(α−β)T++βTf ]c1 (12)

Case 1: if a = 0, one obtains

xTRx ≤ λmax(R)

λmin(R)e2[(α−β)T++βTf ]c1 (13)

If inequality (7) is satisfied, we obtain

xTRx ≤ c2

From inequality (13), we can see that finite-time stability ofthe switched linear system (1) is independent on the switch-ing number N [0, T ], but is only dependent on the total dwell


time T + of the subsystems which are not finite-time sta-ble. Therefore the switched linear system (1) are uniformlyfinite-time stable.

Case 2: if a > 0

xTRx ≤ λmax(R)

λmin(R)e2a e2aN (0,Tf )e2[(α−β)T++βTf ]c1

If the inequality (8) is satisfied, one obtains (see (14))

If the switching signal satisfies the average dwell timecondition, and N0 = 0, the inequality equals to (see (15))

then we can obtain (see (16)) �

Remark 2: From Theorem 1, we can see that if a = 0, andthe ratio of the total dwell time T + satisfies inequality (7),then finite-time stability of the switched linear system (1)can be guaranteed under arbitrarily switching signals. How-ever, if a>0, the ratio of the total dwell time T + and thenumber of switching on time interval [0, Tf ] should be con-sidered simultaneously to guarantee the finite-time stabilityof the switched linear system (1). From equation (7), wesee that switching signal is an important factor which influ-ences the finite-time stability of the switched system (1)when a > 0. Therefore we should not ignore the effect ofswitching signal on finite-time stability of switched linearsystems.

Remark 3: If T + = 0, then the problem is degenerated tothe case that all the subsystems are finite-time stable. In thiscase, from equation (10), we can see that the state transitionof the switched system (1) is in the following form

σ = eApi+1 (t−ti)eApi (ti−ti−1) · · · eAp1 (t1−t0)

Sufficient conditions can obtain from the results in [30].

Remark 4: It should be pointed out that Lyapunov asymp-totical stability of switched systems with unstable subsys-tems has been attracted much attention, and many resultshave been presented, such as [5, 27, 35, 36, 38–43]. Allthese results investigated Lyapunov asymptotical stabilityof switched systems. However, a system could be Lya-punov stable but have poor dynamic characteristics when

xTRx ≤ λmax(R)

λmin(R)e2a(1+N (0,Tf ))+2[(α−β)T++βTf ]c1

≤ λmax(R)

λmin(R)e2a

(1+ ln[c2λmin (R)]−ln[c1λmax (R)]−2[(α−β)T++βTf ]−2a

2a

)e2[(α−β)T++βTf ]c1

≤ λmax(R)

λmin(R)eln[c2λmin(R)]−ln[c1λmax(R)]c1

= c2 (14)

Tf

τa≤ ln [c2λmin(R)] − ln [c1λmax(R)] − 2[(α − β)T + + βTf ] − 2a

2a(15)

τa ≥ 2aTf

ln [c2λmin(R)] − ln [c1λmax(R)] − 2[(α − β)T + + βTf ] − 2a(16)


www.ietdl.org

it possesses undesirable transient performances. Thereforeit is an interesting work to discuss the finite-time stabil-ity of switched systems with finite-time unstable subsys-tems borrowing the thought and methods from references[5, 27, 35, 36, 38–43].

4 Finite-time boundedness of switchedlinear systems

In this section, finite-time boundedness of switched lin-ear systems is discussed. Sufficient conditions under whichswitched linear systems with subsystems that are not finite-time bounded are still finite-time bounded are given.

Theorem 2: For the switched linear system (2), there are twocases of its finite-time boundedness:

(a) when a = 0, if the total dwell time T + of the subsystemswhich are not finite-time stable satisfies the inequality (see(17))

then the switched linear system (2) is uniformly finite-timebounded with respect to (c1, c2, Tf , R);(b) when a > 0, if the following inequality (see (18))

is satisfied, then the switched linear system (2) is finite-timebounded with respect to (c1, c2, T , R, σ). Moreover, if theswitching signal satisfies the average dwell time condition,inequality (18) can be rewritten by virtue of average dwelltime τa with N0 = 0 as follows (see (19))

where T + denotes the total dwell time of the subsystemswhich are not finite-time stable, a = max

1≤i≤m{ai}, α = max

1≤i≤m{αi},

β = max1≤i≤m

{βi}, ‖G‖ = max1≤i≤m

{Gi}.

Proof: Let t1, t2 . . . denote the switching instants and pj forthe value of σ on [tj−1, tj). Then, for any t satisfying t0 <· · · < ti ≤ t < ti+1 ≤ T , one obtains

x(t) = eApi+1 (t−ti)eApi (ti−ti−1) · · · eAp1 (t1−t0)x0

+∫ t1

t0

eApi+1 (t−ti)eApi (ti−ti−1) · · · eAp1 (t1−τ)Gp1ω(τ) dτ


+∫ t2

t1

eApi+1 (t−ti)eApi (ti−ti−1)

· · · eAp2 (t2−τ)Gp2ω(τ) dτ + · · ·

+∫ ti

ti−1

eApi+1 (t−ti)eApi (ti−τ)Gpiω(τ) dτ

+∫ t

ti

eApi+1 (t−τ)Gpi+1ω(τ) dτ (20)

In the right-hand side of the above equation, the firstterm eApi+1 (t−ti)eApi (ti−ti−1) · · · eAp1 (t1−t0)x0 corresponds to thecase of switchings between linear subsystems with-out perturbations during [t0, t), whereas the integralterm eApi+1 (t−ti)eApi (ti−ti−1) · · · eApj (tj−τ), (1 ≤ j ≤ i) correspondsto the switchings during [τ , t), it is not difficult to see that

‖eApi+1 (t−ti)eApi (ti−ti−1) · · · eAp1 (t1−t0)‖

≤(

i+1∏q=1

eapq

)eαT+(t0,t)+βT−(t0,t) (21)

‖eApi+1 (t−ti)eApi (ti−ti−1) · · · eApj (tj−τ)‖

≤(

i+1∏q=j

eapq

)eαT+(τ ,t)+βT−(τ ,t)

≤(

i+1∏q=1

eapq

)eαT+(t0,t)+βT−(t0,t)

≤ e(i+1)a eαT++βT−(22)

From the above inequalities, we obtain the following esti-mate of the norm of the state

‖x(t)‖ ≤(

i+1∏q=1

eapq

)eαT++βT−‖x(0) +

∫ t

t0

‖G‖ω(τ) dτ‖

≤ e(i+1)a eαT++βT−‖x(0) +∫ t

t0

‖G‖ω(τ) dτ‖ (23)

where T − = Tf − T + denotes the total dwell time of finite-time stable subsystems of the switched linear system (2).

T +

Tf≤

ln [c2] − ln

[2λmax(R)

(1

λmin(R)c1 + Tf ‖G‖2d

)]− 2βTf

2(α − β)Tf(17)

N (0, Tf ) ≤ln [c2] − ln

[2λmax(R)

(1


)]− 2[(α − β)T + + βTf ] − 2a

2a(18)

τa ≥ 2aTf

ln [c2] − ln

[2λmax(R)

(1


)]− 2[(α − β)T + + βTf ] − 2a

(19)


www.ietdl.org

Then, by virtue of inequality (23) and Lemma 1, we have(see (24))

where t0 = 0.Case 1: if a = 0, one obtains

xTRx ≤ 2λmax(R)e2[(α−β)T++βTf ]

×{

1


}(25)

If inequality (17) is satisfied, we obtain

xTRx ≤ c2

From inequality (25), we can see that the finite-time bound-edness of the switched linear system (2) is independent onthe switching number N [0, T ], but is only dependent on thetotal dwell time T + of the subsystems which are not finite-time bounded. Therefore the switched linear system (2) areuniformly finite-time bounded.

Case 2: if a > 0

xTRx ≤ 2λmax(R)e2a e2aN (0,Tf )e2[(α−β)T++βTf ]

×{

1


}

If the inequality (18) is satisfied, one obtains (see (26))

If the switching signal satisfies the average dwell timecondition, and N0 = 0, the inequality (18) equals to (see(27))

then we can obtain (see (28)) �

Remark 5: From Theorem 2, we can see that in addition togiving the ratio of the total dwell time T + and the numberof switching on time interval [0, Tf ], the parameter a = 0or a > 0 also provides the condition for the switched linearsystem (2) uniformly finite-time bounded or not.

xTRx ≤ λmax(R)‖x‖2

≤ λmax(R)e2(i+1)a+2(αT++βT−)‖x(0) +∫ t

t0

‖G‖ω(τ) dτ‖2

≤ λmax(R)e2a e2ia e2(αT++βT−)

(2‖x(0)‖2 + 2‖

∫ t

t0


)

≤ λmax(R)e2a e2aN (0,Tf )e2[αT++β(Tf −T+)](

2‖x(0)‖2 + 2‖∫ t

t0


)

≤ 2λmax(R)e2a e2aN (0,Tf )e2[(α−β)T++βTf ]{

1

λmin(R)xT

0 Rx0 + (t − t0)‖G‖2

∫ t

t0

ω(τ)Tω(τ) dτ

}


1

λmin(R)xT

0 Rx0 + (Tf − t0)‖G‖2

∫ Tf

t0

ω(τ)Tω(τ) dτ

}


1


}(24)

xTRx ≤ 2λmax(R)e2a(1+N (0,Tf ))+2[(α−β)T++βTf ]{

1


}

≤ 2λmax(R)e2a

⎛⎝1+

ln[c2]−ln

[2λmax (R)

(1

λmin (R)c1+Tf ‖G‖2d

)]−2[(α−β)T++βTf ]−2a

2a +2[(α−β)T++βTf ]⎞⎠

×{

1


}

≤ 2λmax(R)e(

ln[c2]−ln[2λmax(R)

(1

λmin (R)c1+Tf ‖G‖2d

)]) {1


}= c2 (26)

Tf

τa≤

ln [c2] − ln[2λmax(R)

(1


)]− 2[(α − β)T + + βTf ] − 2a

2a(27)

τa ≥ 2aTf


(1


)]− 2[(α − β)T + + βTf ] − 2a

(28)

1142 IET Control Theory Appl., 2014, Vol. 8, Iss. 12, pp. 1137–1146© The Institution of Engineering and Technology 2014 doi: 10.1049/iet-cta.2013.0648

www.ietdl.org

5 Finite-time L2-gain analysis

In this subsection, a restricted L2 gain analysis of switchedlinear systems with subsystem not finite-time bounded, thatis, finite-time L2-gain with zero initial conditions analysis,is discussed in the following theorem.

Theorem 3: For the switched linear system (2), there are twocases of its finite-time boundedness and finite-time L2-gainless than the given parameter γ > 0:

(a) when a = 0, if the total dwell time T + of the subsystemswhich are not finite-time bounded satisfies the inequality(see (29))

then the switched linear system (2) is uniformly finite-timebounded with respect to (c1, c2, Tf , R) and have finite-timeL2-gain less than γ ;

(b) when a > 0, if the following inequality (see (30))

is satisfied, then the switched linear system (2) is finite-time bounded with respect to (c1, c2, T , R, σ) and have finite-time L2-gain less than γ . Moreover, if the switching signalsatisfies the average dwell time condition, the inequality (30)can be rewritten by virtue of average dwell time τa withN0 = 0 as follows (see (31))

where T + denotes the total dwell time of the subsys-tems which are not finite-time bounded, a = max1≤i≤n{ai},α = max1≤i≤n{αi}, β = max1≤i≤n{βi}, ‖G‖ = max1≤i≤m{Gi},λc2 = max1≥i≤1 λmax(CT

i Ci), λd2 = max1≥i≤1 λmax(DTi Di).

Proof: From Theorem 2, we can obtain that the switchedlinear system (2) are finite-time bounded with respect to(c1, c2, T , R, σ).

From the proof of Theorem 2, we obtain the followingestimate of the norm of the state

‖x(t)‖ ≤(

i+1∏q=1

eapq

)eαT++βT−‖x(0) +

∫ t

t0

‖G‖ω(τ) dτ‖

≤ e(i+1)a eαT++βT−‖x(0) +∫ t

t0

‖G‖ω(τ) dτ‖ (32)

where T − = Tf − T + denotes the total dwell time of finite-time stable subsystems of the switched system (2).

Under zero initial condition x(t0) = 0, we have

‖x‖2 ≤ e2(i+1)a+2(αT++βT−)

∥∥∥∥∫ t

t0

‖G‖ω(τ) dτ

∥∥∥∥2

≤ e2a e2aN (0,Tf )e2[(α−β)T++βTf ]

×{(t − t0)‖G‖2

∫ t

t0

ω(τ)Tω(τ) dτ

}

≤ e2a e2aN (0,Tf )e2[(α−β)T++βTf ]

×{(Tf − t0)‖G‖2

∫ Tf

t0

ω(τ)Tω(τ) dτ

}

≤ Tf ‖G‖2de2a e2aN (0,Tf )e2[(α−β)T++βTf ] (33)

Furthermore, we can reach that

∫ t

0

(y(τ )Ty(τ ) − γ 2ω(τ)Tω(τ)) dτ

=∫ t

0

[(CTi x + Diω)T(CT

i x + Diω) − γ 2ωTω] dτ

=∫ t

0

[xTCTi Cix + xTCT

i Diω + ωTDTi Cix

+ ωTDTi Diω − γ 2ωTω] dτ

≤∫ t

0

[2xTCTi Cix + 2ωTDT

i Diω − γ 2ωTω] dτ

≤ 2λc2

∫ t

0

xTxdτ + (2λd2 − γ 2)

∫ t

0

ωTω dτ

≤ 2λc2

∫ Tf

0

xTxdτ + (2λd2 − γ 2)

∫ Tf

0

ωTω dτ

≤ 2T 2f ‖G‖2de2a e2aN (0,Tf )e2[(α−β)T++βTf ]

+ (2λd2 − γ 2)d (34)

T +

Tf≤ min

⎧⎨⎩


(1


)]− 2βTf

2(α − β)Tf,

ln((γ 2 − 2λd2)d) − ln(2T 2f ‖G‖2d) − 2βTf

2(α − β)Tf

⎫⎬⎭ (29)

N (0, Tf ) ≤ min

{ ln [c2] − ln[2λmax(R)

(1


)]− 2[(α − β)T + + βTf ] − 2a

2a,

ln[(γ 2 − 2λd2)d] − ln[2T 2f ‖G‖2d] − 2[(α − β)T + + βTf ] − 2a

2a

}(30)

τa ≥ max

{2aTf


(1


)]− 2[(α − β)T + + βTf ] − 2a

,

2aTf

ln[(γ 2 − 2λd2)d] − ln[(2T 2f ‖G‖)2d] − 2[(α − β)T + + βTf ] − 2a

}(31)

IET Control Theory Appl., 2014, Vol. 8, Iss. 12, pp. 1137–1146 1143doi: 10.1049/iet-cta.2013.0648 © The Institution of Engineering and Technology 2014

www.ietdl.org

Case 1: if a = 0, one obtains

∫ t

0


≤ 2T 2f ‖G‖2de2[(α−β)T++βTf ] + (2λd2 − γ 2)d (35)

Since

T +

Tf≤ ln((γ 2 − 2λd2)d) − ln(2T 2

f ‖G‖2d) − 2βTf

2(α − β)Tf(36)

then ∫ t

0

(y(τ )Ty(τ ) − γ 2ω(τ)Tω(τ)) dτ ≤ 0 (37)

that is∫ t

0

(y(τ )Ty(τ )τ ) ≤ γ 2

∫ t

0

(ω(τ)Tω(τ)) dτ (38)

Case 2: if a > 0, we have

∫ t

0


≤ 2T 2f ‖G‖2de2a e2aN (0,Tf )e2[(α−β)T++βTf ]

+ (2λd2 − γ 2)d (39)

Since (see (40))

one obtains∫ t

0

(y(τ )Ty(τ ) − γ 2ω(τ)Tω(τ)) dτ ≤ 0 (41)

that is∫ t

0

(y(τ )Ty(τ )τ ) ≤ γ 2

∫ t

0

(ω(τ)Tω(τ)) dτ (42)

If the switching signal satisfies the average dwell timecondition, and N0 = 0, the inequality (30) equals to (see(43)) �

then we can obtain (see (44))

Remark 6: In Theorem 3, parameter γ > 0 is given inadvance as that in [22, 26, 33] to predetermine the distur-bance tolerance capability of switched linear systems.


Remark 7: It is really an important point should be clarified.In references [22, 32, 44], weighted L2 gain of switchedsystems has been discussed by virtue of LMIs based onmultiple Lyapunov-like functions. It is also pointed out inthose papers that if parameter μ, which is the bound of theratio of two-pair Lyapunov-like functions, equals to 1, thenweighted L2 gain degenerates into the normal L2 gain. Inthis case, the damp of the energy can be bounded by onelittle larger function, and so the norm of the trajectory forthe quadratic Lyapunov functions have been used. In thisnote, the norm of the trajectory of switched linear systemsis estimated by the same boundedness. Therefore the normalL2 gain is discussed in this note.

6 Numerical simulation

In this section, a numerical example is presented as follows,which also shows the reason why we are interested in dis-cussing finite-time stability of switched linear systems withsubsystems which are not finite-time stable. Similar simula-tions can be done for finite-time boundedness of switchedlinear systems with subsystems which are not finite-timebounded.

Example 1: Consider the finite-time stability problem for aswitched linear system as follows

x(t) = Aσ(t)x(t) (45)

The corresponding parameters are specified as follows

A1 =(

3 30 1

)

A2 =(

0.01 0−1 1

)c1 = 1, c2 = 10, T = 1, R = I , σ(t) = {1, 2}, ∀t

It is not difficult to verify that the 1st subsystem isnot finite-time stable because the value of x(t)TRx(t) hasevidently larger than c2 when t > 0.6 and the second sub-system is finite-time stable for x(t)TRx(t) < c2, ∀t ∈ [0, T ]. Simulation results are presented in Figs. 1 and 2. Fig. 1shows the value of x(t)TRx(t) along the trajectory of the firstsubsystem in time interval [0, T ] and Fig. 2 shows the valueof x(t)TRx(t) along the trajectory of the second subsystemin time interval [0, T ].

N (0, Tf ) ≤ ln[(γ 2 − 2λd2)d] − ln[2T 2f ‖G‖2d] − 2[(α − β)T + + βTf ] − 2a

2a(40)

Tf

τa≤

ln[(γ 2 − 2λd2)d] − ln[4λc2

{1


}Tf

]− 2[(α − β)T + + βTf ] − 2a

2a(43)

τa ≥ 2aTf

ln[(γ 2 − 2λd2)d] − ln[2T 2f ‖G‖2d] − 2[(α − β)T + + βTf ] − 2a

(44)


www.ietdl.org

0 0.2 0.4 0.6 0.8 10

100

200

300

400

500

600

700

Time(s)

xT(t

)Rx(

t)

Fig. 1 Time response of x(t)T Rx(t)

Calculating the parameters in Theorem 1, we have

P1 =(

1 − 32

0 1

)

P2 =(

1 00.99 1

)

and the eigenvalues of the Pi, 1 ≤ i ≤ 2 are

λ(P1) = 1, 1, λ(P2) = 1, 1

Therefore

a = max1≤i≤2

{ai} = max1≤i≤2

{log[σmax(Pi)/σmin(Pi)]} = 0

α = max1≤i≤r

{αi} = max1≤i≤r

{max1≤k≤n

{λj(Ak)}} = 3

β = maxr+1≤i≤n

{βi} = maxr+1≤i≤n

{max1≤k≤n

{λj(Ak)}} = 1

By virtue of the inequality (7), calculating the total dwelltime T + of the subsystems which are not finite-time stable,

0

2

4

6

8

10

12

14

16

18

20

0 0.2 0.4 0.6 0.8 1Time(s)

xT(t

)Rx(

t)


one obtains

T + ≤ ln [c2λmin(R)] − ln [c1λmax(R)] − 2βTf

2(α − β)

= ln 50 − ln 1 − 2

2(3 − 1)= 0.4780 (46)

Let T + = 0.45 < 0.478, the switching signal is periodic andT = 0.05. For t ≤ 0.8, σ(0) = 2 and the switching signalis periodic. If 0.8 < t ≤ 1, then σ(t) = 2. The simulationresults are presented as follows, see Fig. 3. It is not difficultto verify that the switched linear system is finite-time stablebecause x(t)TRx(t) < 20 = c2, ∀t ∈ [0, T ].

7 Conclusion

This note has investigated the finite-time stability and finite-time boundedness problem of switched linear systems withsubsystems which are not finite-time stable or finite-time

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 12

2.001

2.002

2.003

2.004

2.005

2.006

2.007

2.008

2.009

Time(s)

xT(t

)Rx(

t)


IET Control Theory Appl., 2014, Vol. 8, Iss. 12, pp. 1137–1146 1145doi: 10.1049/iet-cta.2013.0648 © The Institution of Engineering and Technology 2014

www.ietdl.org

bounded. Some sufficient conditions have been provided forfinite-time stability and finite-time boundedness of switchedlinear systems. The results in this note have removed thepotential limits of the results in most existing literatures thateach subsystem should be finite-time stable or finite-timebounded. If some subsystems of switched linear systems arenot finite-time stable or finite-time bounded, the results inthis note still work. In this sense, the results in this note isless conservative.

8 Acknowledgments

The authors thank the anonymous reviewers for several com-ments that found their way into the final version of thepaper. This work was supported by Natural Science Founda-tion of China (61074013 and 61174038), and PostdoctoralScience Foundation of China (2013M531372) and JiangsuPostdoctoral Research Grants Program (1301118C).

9 References

1 Liberzon, D., Morse, A.: ‘Basic problems in stability and design ofswitched systems’, IEEE Control Syst. Mag., 1999, 19, (5), pp. 59–70

2 Branicky, M.: ‘Multiple Lyapunov functions and other analysis toolsfor switched and hybrid systems’, IEEE Trans. Autom. Control, 1998,43, (4), pp. 475–482

3 Hespanha, J., Morse, A.: ‘Stability of switched systems with averagedwell time’. Proc. 38th Conf. Decision and Control, 1999, pp. 2655–2660

4 Sun, Z., Ge, S.: ‘Analysis and synthesis of switched linear controlsystems’, Automatica, 2005, 41, (2), pp. 181–195

5 Zhang, L., Shi, P.: ‘Stability, l2-gain and asynchronous H∞ control ofdiscrete-time switched systems with average dwell time’. IEEE Trans.Autom. Control, 2009, 54, pp. 2193–2200

6 Michel, A., Hou, L.: ‘Finit-time and practical stability of a class ofstochastic dynamical systems’. Forty-seventh IEEE Conf. Decision andControl, pp. 3452–3456, Cancun, Mexico, December 2008

7 Dorato, P.: ‘Short time stability in linear time-varying systems’. Proc.IRE Int. Convention Record, Part 4, New York, 1961, pp. 83–87

8 Weiss, L., Infante, E.: ‘Finite-time stability under perturbing forcesand on product spaces’, IEEE Trans. Autom. Control, 1967, 12,pp. 54–59

9 Amato, F., Ariola, M., Dorato, P.: ‘Finite-time control of linear sys-tems subject to parametric uncertainties and disturbances’, Automatica,2001, 37, pp. 1459–1463

10 Amato, F., Ariola, M.: ‘Finite-time control of discrete-time linearsystem’, IEEE Trans. Autom. Control, 2005, 50, (5), pp. 724–729

11 Amato, F., Ariola, M., Dorato, P.: ‘Finite-time stabilization viadynamic output feedback’, Automatica, 2006, 42, pp. 337–342

12 Liu, L., Sun, J.: ‘Finite-time stabilization of linear systems viaimpulsive control’, Int. J. Control, 2008, 81, pp. 905–909

13 Zhao, S., Sun, J., Liu, L.: ‘Finite-time stability of linear time-varyingsingular systems with impulsive effects’, Int. J. Control, 2008, 81,pp. 1824–1829

14 Amato, F., Ambrosino, R., Cosentino, C., Tommasi, G.: ‘Finite-timestabilization of impulsive dynamical linear systems’, Nonlinear Anal.Hybrid Syst., 2011, 5, pp. 89–101

15 Amato, F., Tommasi, F., Pironti, A.: ‘Necessary and sufficient condi-tions for finite-time stability of impulsive dynamcial linear systems’Automatica, 2013, 49, (8), pp. 2546–2550

16 ElBsat, M., Yaz, E.: ‘Robust and resilient finite-time bounded controlof discrete-time uncertain nonlinear systems’, Automatica, 2013, 49,pp. 2292–2296

17 Rosier, L.: ‘Homogeneous Lyapunov function for homogeneous con-tinuous vector field’, Syst. Control Lett., 1992, 19, (4), pp. 467–473

18 Bhat, S., Bernstein, D.: ‘Finite-time stability of homogeneous sys-tems’. Proc. American Control Conf., 1997, pp. 2513–2514

19 Hong, Y., Jiang, Z.: ‘Finite-time stabilization of nonlinear systems withparametric and dynamic uncertainties’, IEEE Trans. Autom. Control,2006, 51, pp. 1950–1956


20 Amato, F., Ambrosino, R., Ariola, M., Cosentino, C.: ‘Finite-time sta-bility of linear time-varying systems with jumps’, Automatica, 2009,45, (5), pp. 1354–1358

21 Du, H., Lin, X., Li, S.: ‘Finite-time boundedness and stabilization ofswitched linear systems’, Kybernetika, 2010, 46, (5), pp. 970–889

22 Lin, X., Du, H., Li, S.: ‘Finite-time boundedness and L2-gain analy-sis for switched delay systems with norm-bounded disturbance’, Appl.Math. Comput., 2011, 217, pp. 5982–5993

23 Ambrosino R, Calabrese, F., Cosentino, C., De Tommasi, G.:‘Sufficient conditions for finite-time stability of impulsive dynamicalsystems’, IEEE Trans. Autom. Control, 2009, 54, (4), pp. 861–865

24 Amato, F., Ambrosino, R., Ariola, M., De Tommasi, G.: ‘Finite-timestabilization of impulsive dynamical linear systems’, Nonlinear Anal.Hybrid Syst., 2011, 5, (1), pp. 89–101

25 Amato, F., Ambrosino, R., Ariola, M., De Tommasi, G.: ‘Robust finite-time stability of impulsive dynamical linear systems subject to norm-bounded uncertainties’, Int. J. Robust Nonlinear Control, 2011, 21,(10), pp. 1080–1092

26 Lin, X., Du, H., Li, S., Zou, Y.: ‘Finite-time boundedness and finite-time l2 gain analysis of discrete-time switched linear systems withaverage dwell time’, J. Franklin Inst., 2013, 350, pp. 911–928

27 Lin, X., Du, H., Li, S., Zou, Y.: ‘Finite time stability and finite-time weighted L2-gain analysis for switched systems with time-varyingdelay’, IET Control Theory Appl., 2013, 7, (7), pp. 1058–1069

28 Zhao, G., Wang, J.: ‘Finite time stability and L2-gain analysis forswitched linear systems with state-dependent switching’, J. FranklinInst., 2013, 350, pp. 1075–1092

29 Liu, H., Shen, Y., Zhao, X.: ‘Finite-time stabilization and boundednessof switched linear system under state-dependent switching’, J. FranklinInst., 2013, 350, pp. 541–555

30 Du, H., Lin, X., Li, S.: ‘Finite-time stability and stabilizaitonof switched linear systems’. Joint 48th IEEE Conf. on Decisionand Control and 28th Chinese Control Conf. Shanghai, P.R. China,December 16–18, 2009, pp. 1938–1943

31 Lin, X., Li, X., Li, S., Zou, Y.: ‘Finite-time stability of switched linearsystems with subsystems which are not finite-time stable’. The 32ndChinese Control Conf., July 26–28, 2013, Xi’an, China

32 Zhai, G., Hu, B., Yasuda, K., Michel, A.: ‘Disturbance attenuationproperties of time-controlled switched systems’, J. Franklin Inst.,2001, 338, pp. 765–779

33 Sun, X., Zhao, J., Hill, D.: ‘Stability and L2 gain analysis for switcheddelay systems: a delay-dependent method’, Automatica, 2006, 42, (10),pp. 1769–1774

34 Berberian, S.: ‘Fundamentals of real analysis’ (Springer-Verlag, NewYork, Inc., 1999)

35 Zhai, G., Hu, B., Yasuda, K., et al.: ‘Stability analysis ofswitched systems with stable and unstable subsystems: an aver-age dwell time approach’. Proc. the American Control Conf., 2000,pp. 200–204

36 Sun, X., G. Liu, P., Rees, D., Wang, W.: ‘Stability of systemswith controller failure and time-varying delay’, IEEE Trans. Autom.Control, 53, (10), 2008, pp. 2391–2396

37 Sun, X., Wang, W.: ‘Input-to-sate stability for hybrid delayed sys-tems with unstable continuous dynamics’, Automatica, 48, (9), 2012,pp. 2359–2364

38 Zhang, L., Gao, H.: ‘Asynchronously switched control of switchedlinear systems with average dwell time, Automatica 46, (5), 2010,pp. 953–958

39 Zhao, X., Shi, P., Zhang, L.X.: ‘Asynchronously switched control of aclass of slowly switched linear systems Syst. Control Lett., 2012, 61,(12), pp. 1151–1156

40 Wang, R., Shi, P., Wu, Z., Sun, Y.: ‘Stabilization of switched delaysystems with polytopic uncertainties under asynchronous switching’,J. Franklin Inst., 2013, 350, (8), pp. 2028–2043

41 Wu, L., Zheng, W.X.: ‘Weighted H∞ model reduction for linearswitched systems with time-varying delay’, Automatica, 2009, 45, (1),pp. 186–193

42 Zhao, X., Liu, H., Zhang, J., Li, H.: ‘Multiple-Mode Observer Designfor a Class of Switched Linear Systems Linear Systems’, IEEE Trans.Autom. Sci. Eng., doi: 10.1109/TASE.2013.2281466

43 Zhao, X., Zhang, L., Shi, P., Liu, M.: ‘Stability of switched positivelinear systems with average dwell time switching’, Automatica, 2012,48, (6), pp. 1132–1137

44 Zhao, X., H Liu, Wang, Z.: ‘Weighted H∞ performance analysis ofswitched linear systems with mode-dependent average dwell time’, Int.J. Syst. Sci., 2013, 44, (11), pp. 2130–2139


Finite-time stability of switched linear systems with subsystems which are not finite-time stable

Documents

Transcript of Finite-time stability of switched linear systems with subsystems which are not finite-time stable