Systems & Control Letters 58 (2009) 703–708

Contents lists available at ScienceDirect

Systems & Control Letters

journal homepage: www.elsevier.com/locate/sysconle

On stabilization of switched nonlinear systems with unstable modesI

Hao Yang a,b, Vincent Cocquempot b,∗, Bin Jiang aa College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, 29 YuDao Street, Nanjing, 210016, PR Chinab LAGIS-CNRS, UMR 8146, Université de Lille1: Sciences et Technologies, 59655 Villeneuve d’Ascq cedex, France

a r t i c l e i n f o

Article history:Received 12 November 2008Received in revised form22 June 2009Accepted 24 June 2009Available online 24 July 2009

Keywords:Switched systemsUnstable modesStabilizationStability

a b s t r a c t

This paper addresses stabilization issue of switched nonlinear systems where some modes are stableand others may be unstable. A new stabilizing switching law that determines the initial states and theswitching instants for any given switching sequence is proposed. The developed technique relies on thetradeoff among the functions’ gains of continuous modes, and does not depend on the constant ratiocondition required in ‘‘dwell-time scheme’’.

© 2009 Elsevier B.V. All rights reserved.

1. Introduction

Switched systems arise in engineering practice where severaldynamical models are required to represent a system due tovarious jumping parameters and changing environmental factors.Several stability notions have been introduced and investigatedfor switched systems [1–17]. In most of these literatures, theindividual stability of each mode is assumed. These works canbe traced back to two main stability methods: multiple Lyapunovfunctions (MLFs) technique [2–7] and dwell-time scheme [10–14].MLFs method can be applied to switched systems with all

stable modes. Refs. [2] and [4–6] claim that the stability can beachieved if the value of eachmode’s Lyapunov function (1) does notincrease when the mode works and (2) is non-increasing over theconsecutive time sequence when the corresponding mode is justswitched on. In [3] the MLFs condition is relaxed by introducing toeach mode a Lyapunov-like function which is allowed to increasewithin an upper bound during activating period of the mode. Therecent work [7] extends MLFs condition [3] to a more general casethat allows a bounded increase on Lyapunov-like function overthe ‘‘switched on’’ time sequence of the corresponding mode. The

I This work is partially supported by National Natural Science Foundation ofChina (60874051, 60811120024), Natural Science Foundation of Jiangsu Province(BK2007195) and Graduate innovation research funding of Jiangsu Province(CX07B-112z).∗ Corresponding author. Tel.: +33 (0)3 20 43 62 43; fax: +33 (0)3 20 33 71 89.E-mail addresses: younghao82@yahoo.com.cn (H. Yang),

vincent.cocquempot@wanadoo.fr, vincent.cocquempot@univ-lille1.fr(V. Cocquempot), binjiang@nuaa.edu.cn (B. Jiang).

0167-6911/$ – see front matter© 2009 Elsevier B.V. All rights reserved.doi:10.1016/j.sysconle.2009.06.007

relations among different Lyapunov functions are not consideredsince the state trajectories in each activating mode are bounded.Another idea is to achieve the stability via dwell-time scheme[10,12,15], which introduces a minimum time interval called‘‘dwell time’’, and claims that the system is stable if the intervalbetween any two consecutive switching instants is not smallerthan the ‘‘dwell time’’. In such scheme, a ‘‘µ’’ condition is oftenimposed where a maximal global constant ratio is required amongthe functions, i.e. Vp ≤ µVq for µ ≥ 1, p, q are the numbers ofmodes. However, µ is not always easy to find. Ref. [10] considersthe state trajectory directly rather than the Lyapunov functions.Thus the ‘‘µ’’ condition disappears.The stabilization of switched systems with unstable modes

deserves further investigation. This is motivated by two facts:

(1) A switched system is given where some modes are unstabledue to disturbance, unmodeled dynamics or possible faults[18–20]. Inmany situations, eachmode has its own usefulness,the switching sequence is not expected to be changedarbitrarily even some unstable modes would be activated.

(2) In a switching control problem, according to the certainswitching law, the system often takes finite number ofswitchings among several destabilizing controllers until thecorrected controller is selected and applied [1].

Two natural ways follow to deal with such switched systems.One way is to design a robust or a reliable control law ineach unstable mode such that it becomes stable, then apply thestandard stability results [19]. Another way is to research directlythe stabilization of a switched system without reconfiguring thecontroller in each mode. The latter case is addressed in this paper.

704 H. Yang et al. / Systems & Control Letters 58 (2009) 703–708

Some contributions have been devoted to switched systemswith unstablemodes. Note thatMLFs techniques developed in [3,7]are difficult to apply if the state trajectories in an unstable modeare not bounded, since such amode does not have a correspondingLyapunov-like function. In this case, the dwell-time scheme withthe constant ratio constraint becomes a useful tool. Ref. [21] provesthat if the total activating period of unstablemodes is small enoughcompared with that of stable modes, the stability of switchedlinear systems is guaranteed. Similar result can be seen in [22] forstochastic cases and [20] for switchednonlinear systems. However,the existence of a constant ratioµ restricts the application of theseresults. Ref. [23] addresses this issue without the ‘‘µ’’ condition,but a common Lyapunov function is considered. For other relatedinteresting results, we refer to [24] and [25], where the linear caseis considered and geometrical tools are applied.In this work, we focus on the stabilization problem of switched

nonlinear systems with some unstable modes, where the ‘‘µ’’condition is not imposed. To the best of our knowledge, the resultalong this direction has not been reported prior to this work.The main contribution of this paper is to establish a stabilizingswitching law for the considered switched system. The developedswitching law determines the initial states and the switchinginstants without changing the prescribed switching sequence suchthat the stability is achieved in spite of any given sequence and thegiven region that the states are within.The rest of the paper is organized as follows: Section 2 gives

some preliminaries. Section 3 addresses the stabilization issueand gives the main results. An example is taken to illustratethe theoretical results in Section 4, followed by some concludingremarks in Section 5.

2. Preliminaries

Let R denote the field of real numbers, Rr the r-dimensionalreal vector space. | · | the Euclidean norm. Class K is a class ofstrictly increasing and continuous functions [0,∞) → [0,∞)which are zero at zero. ClassK∞ is the subset ofK consisting ofall those functions that are unbounded. β : [0,∞) × [0,∞) →[0,∞) belongs to class KL if β(·, t) is of class K for each fixedt ≥ 0 and β(s, t) decreases to 0 as t →∞ for each fixed s ≥ 0. t−denotes the left limit time instant of t . (·)> is the transposition.The considered switched system takes the general form

x(t) = fσ(t)(x(t)) (1)

where x ∈ X ⊂ Rn are the states. Define M = 1, 2, . . . ,N,where N is the number of modes. σ(t) : [0,∞) → M denotesthe switching function, which is assumed to be a piecewise constantfunction continuous from the right. fi, i ∈M are smooth functionswith fi(0) = 0. Denote by tj, j = 1, 2, . . . the jth switching instant,t0 = 0. It follows thatmode σ(ti) is activated in [ti, ti+1), andmodeσ(ti+1) is switched into at ti+1, for i = 0, 1, 2 . . . . Let tik, i ∈ M,k = 1, 2, . . . be the kth time when mode i is switched on. Nσ(ts,tf )represents the number of switchings in [ts, tf ). In this work, weonly consider non-Zeno sequences (i.e., sequences that switch atmost a finite number of times in any finite time interval). However,our theory allows infinite switchings in infinite time interval. Wealso assume that the states do not jump at the switching instants.Specially, we define a class GKL function γ : [0,∞)× [0,∞)

→ [0,∞) if γ (·, t) is of classK for each fixed t ≥ 0 and γ (s, t)increases as t →∞ for each fixed s ≥ 0.DenoteMs ⊂ M as the set of stable modes andMus ⊂ M the

set of unstable ones.M = Ms ∪Mus,Ms ∩Mus = ∅ andMs 6= ∅.Suppose that there exist continuous non-negative functions Vp :

Rn → R≥0, αp1 , α

p2 ∈ K∞, ∀p ∈ M, and φp ∈ KL∀p ∈ Ms,

φp ∈ GKL∀p ∈Mus that satisfy

αp1(|x|) ≤ Vp(x) ≤ α

p2(|x|) ∀p ∈M (2)

Vp(x(t)) ≤ φp(Vp(x(tpk)), t − tpk),

∀p ∈Ms, φp ∈ KL, t ≥ tpk, k = 1, 2, . . . (3)Vp(x(t)) ≤ φp(Vp(x(tpk)), t − tpk),

∀p ∈Mus, φp ∈ GKL, t ≥ tpk, k = 1, 2, . . . . (4)

Formulations (2)–(4) include various converging and divergingforms (e.g., the exponential decay form [12], the constant gainform [7]). For unstable modes, inequality (4) implies that Vp mayincrease infinitely as described by a GKL function if t → ∞.GKL function is more general than the Lyapunov-like functionin [3] since we do not impose an upper bound on Vp. Note that (3)and (4) are properties satisfied by functions of each mode, and donot depend on the switching sequence. Vp(∀p ∈M) is not requiredto be differentiable.

Definition 1. Given a switching function σ(t), the origin of aswitched system (1) is said to be stable under σ if for any ε > 0,there exists a δ > 0 such that |x(t)| ≤ ε, t ≥ 0, whenever|x(0)| ≤ δ.

The objective of this work is to propose a switching law thatstabilizes the system (1) satisfying (2)–(4) and guarantees |x(t)| ≤ ε∀t ≥ 0 by determining the initial states x(0) and switching instantsin spite of any given ε > 0 and any switching sequence.

3. Stabilization of switched systems

For the sake of simplification, we will denote φt−tiσ(ti), φσ(ti)

(Vσ(ti)(x(ti)), t − ti), Vtσ(ti)

, Vσ(ti)(x(t)) if there is no confusion.Specially, φtσ(0) , φσ(0)(Vσ(0)(x(0)), t).In the following, we first establish a stability condition for the

considered switched systems in the finite time interval with finitenumber of switchings (Lemma 1). Based on such stability criterion,a stabilizing switching law will be constructed (Theorem 1).

Lemma 1. Consider a switched system (1) satisfying (2)–(4). Underσ(t), if there exists a constant β > 0 such that

Nσ(ts,t)∑k=0

(Nσ(ts,t)∏i=k

φti+1−tiσ(ti)

V tiσ(ti)

)≤ β, t > ts ≥ 0,

where tNσ(ts,t)+1 , t,Nσ(ts,t) is finite . (5)

Then x is bounded in [ts, t). Moreover, for any bounded x(ts), the upperbound of |x(t)| can be estimated.

Remark 1. Note thatφt−tiσ(ti)

Vtiσ(ti)for t ≥ ti is the bound of the gain of

function Vσ(ti) when mode σ(ti) is activated. Condition (5) gives arelation among the gains of each activated mode and its activatingperiod. More precisely, x is bounded in [ts, t) if the product of gainsfrom each activatedmode to the terminatedmode is bounded, andthe sum of these products values is also bounded. It deserves topoint out that for a switched system with unstable modes, even inthe finite time interval with finite switching times, x may escapeto infinity under inappropriate switching law.

Proof of Lemma 1. For the sake of clearness, suppose that ts =t0 = 0. Denote Nσ(t) , Nσ(0,t).

H. Yang et al. / Systems & Control Letters 58 (2009) 703–708 705

Consider t ∈ [0, t1), we have V tσ(0) ≤φtσ(0)

V0σ(0)V 0σ(0). Condition (5)

ensures thatφtσ(0)

V0σ(0)≤ β . It follows from (2)–(4) that

|x(t1)| ≤ (ασ(0)1 )−1 β α

σ(0)2︸ ︷︷ ︸

ϑt1

(|x(0)|) (6)

for ϑt1 ∈ K∞. According to (2), one has

V t1σ(t1) ≤ Vt1σ(t−1 )+ α

σ(t1)2 (ϑt1(|x(0)|))− α

σ(t−1 )1 (ϑt1(|x(0)|)). (7)

Define αt1 = max[ασ(t1)2 ϑt1 , α

σ(t−1 )1 ϑt1 ]. Since α

σ(t1)2 , α

σ(t−1 )1 ,

ϑt1 ∈ K∞, it is clear that αt1 ∈ K∞ and

αt1(|x(0)|) ≥ ασ(t1)2 (ϑt1(|x(0)|))− α

σ(t−1 )1 (ϑt1(|x(0)|)). (8)

Substituting (8) into (7) results in

V t1σ(t1) ≤ Vt1σ(t−1 )+ αt1(|x(0)|). (9)

For t ∈ [t1, t2), we have

V tσ(t) ≤φt−t1σ(t1)

V t1σ(t1)V t1σ(t1) ≤

φt−t1σ(t1)

V t1σ(t1)

[Vt−1σ(t−1 )+ αt1(|x(0)|)

]

≤φt−t1σ(t1)

V t1σ(t1)

φt1σ(0)

V 0σ(0)V 0σ(0) +

φt−t1σ(t1)

V t1σ(t1)αt1(|x(0)|). (10)

Note that V 0σ(0) is bounded and αt1 ∈ K∞. Condition (5) ensures

thatφt−t1σ(t1)

Vt1σ(t1)

φt1σ(0)

V0σ(0)≤ β and

φt−t1σ(t1)

Vt1σ(t1)≤ β . It follows from (2)–(4) and (10)

that

|x(t2)| ≤ (ασ(0)1 )−1 β

(ασ(0)2 (|x(0)|)+ αt1(|x(0)|)

)︸ ︷︷ ︸

ϑt2 (|x(0)|)

(11)

for ϑt2 ∈ K∞. One further has

V t2σ(t2) ≤ Vt2σ(t−2 )+ α

σ(t2)2 (ϑt2(|x(0)|))− α

σ(t−2 )1 (ϑt2(|x(0)|)). (12)

Define αt2 = max[ασ(t2)2 ϑt2 , α

σ(t−2 )1 ϑt2 ]. Since α

σ(t2)2 , α

σ(t−2 )1 , ϑt2

∈ K∞, it follows that αt2 ∈ K∞ and

αt2(|x(0)|) ≥ ασ(t2)2 (ϑt2(|x(0)|))− α

σ(t−2 )1 (ϑt2(|x(0)|)). (13)

Substituting (13) into (12) results in

V t2σ(t2) ≤ Vt2σ(t−2 )+ αt2(|x(0)|) (14)

for αt2 ∈ K∞.By induction, we find that under condition (5) there exists a

function α ∈ K∞ such that at each switching instant ti > 0, i =1, 2, . . . ,Nσ(t)

Vσ(ti)(x(ti)) ≤ Vσ(t−i )(x(ti))+ α(|x(0)|) (15)

where α(|x(0)|) , supi=1,2,...,Nσ(t) [αti(|x(0)|)].Denote j = Nσ(t) for t ≥ 0, j ≥ 0, it follows from (3)–(4) that

Vσ(t)(x(t)) ≤ φt−tjσ(tj)=

φt−tjσ(tj)

Vtjσ(tj)

Vtjσ(tj)

≤

φt−tjσ(tj)

Vtjσ(tj)

[Vt−jσ(t−j )+ α(|x(0)|)

]

≤

φt−tjσ(tj)

Vtjσ(tj)

φtj−tj−1σ(tj−1)

+

φt−tjσ(tj)

Vtjσ(tj)

α(|x(0)|)

≤

φt−tjσ(tj)

Vtjσ(tj)

φtj−tj−1σ(tj−1)

Vtj−1σ(tj−1)

Vt−j−1σ(t−j−1)

+

φt−tjσ(tj)

Vtjσ(tj)

φtj−tj−1σ(tj−1)

Vtj−1σ(tj−1)

+

φt−tjσ(tj)

Vtjσ(tj)

α(|x(0)|)...

≤

Nσ(t)∏s=0

φts+1−tsσ(ts)

V tsσ(ts)Vσ(0)(x(0))+

Nσ(t)∑k=1

(Nσ(t)∏i=k

φti+1−tiσ(ti)

V tiσ(ti)

)α(|x(0)|). (16)

Based on (2) and (15), since α ∈ K∞, there exists aK∞ functionα such that

α(|x(0)|) = max[ασ(0)2 (|x(0)|), α(|x(0)|)

]. (17)

Substituting (17) into (16), together with (5), yields

Vσ(t)(x(t)) ≤Nσ(t)∑k=0

(Nσ(t)∏i=k

φti+1−tiσ(ti)

V tiσ(ti)

)α(|x(0)|) ≤ βα(|x(0)|). (18)

From (2), we finally obtain

|x(t)| ≤ (ασ(t)1 )−1βα(|x(0)|). (19)

Since β > 0 is a constant, ασ(t)1 , α ∈ K∞, the stability resultfollows.From the above procedures, one can find that under condition

(5), given any x(ts), β and switching sequence, each αti(|x(ts)|) canbe calculated which is independent from the switching instants.Thus, for any bounded x(ts), we can find a functionΩ(·) such that|x(t)| ≤ Ω(|x(ts)|). This completes the proof.

Remark 2. The main contributions of Lemma 1 are twofold:(1) Both stable and unstable modes are allowed in the switchednonlinear system; (2) The ‘‘µ’’ condition is removed by introducinga difference α(|x(0)|) among functions Vp ∀p ∈ M. However, thecondition (5) is independent from α(|x(0)|). (3) The upper boundof |x(t)| can be estimated without the information of switchinginstants in [0, t). This property will be very useful in switching lawdesign.

Remark 3. The condition (5) is valid since Vσ is a non-negativefunction and is impossible to become zero unless a stronger finitetime stability [26] is achieved. For the case that finite time stabilityis achieved, (5) is available if we take j instead of Nσ(t) whereV tσ(t) > 0 for t < tj+1.

Remark 4. It is often not easy to verify (5) on-line, which relieson the solutions of the system. However, this condition can helpto construct a stabilizing switching law as shown below. Theproposed stabilization scheme will automatically guarantee thevalidation of (5).

Unlike the usual designmethods that adjust both the switchingsequence and switching instants [7,16], we only redesign theswitching instants such that the origin of switched system isalways stable under any given switching sequence where eachprescribed mode can be activated.

706 H. Yang et al. / Systems & Control Letters 58 (2009) 703–708

...mode 6mode 5

stable

mode 4mode 3mode 1 mode 2

unstablestableunstableunstable stable

Fig. 1. Switching sequence.

Assumption 1. There exists a known constant χ ≥ 1 such that

χ = maxj∈M,k=1,2...

φj(Vj(x(tjk)), 0)Vj(x(tjk))

. (20)

Remark 5. Assumption 1 means that the initial gain of function Vjis bounded when the corresponding mode j is just switched on att = tjk. In some situations, φj(Vj(x(tjk)), 0) is affine w.r.t. Vj(x(tjk)),e.g. the exponential decay form [12], the constant gain form [7]. Inthese cases, χ can be easily obtained a priori.

Without loss of generality, suppose that for a given sequence,at most m unstable modes (m is finite) are activated one by onewithout being interrupted by stable modes as shown in Fig. 1(where 2 unstable modes are activated one by one).Choose a constant β > max[m(1 + χ)χm,m(m + 1)χm+1],

where χ is defined in (20). Given any required upper bound ε of|x(t)| and switching sequence, the switching law is designed as:

Switching law S (with a given ε and a switchingsequence)

1. Let i = 0, choose x(0) such that (ασ(0)1 )−1φσ(0)(Vσ(0)(x(0), 0)) ≤ ε

2. If (C1 ) mode σ(ti) is stable and mode σ(ti+1) isstable, then go to 3;Else, go to 5.

3. Choose ti+1 such that (ασ(ti+1)1 )−1φσ(ti+1)(Vσ(ti+1)(x(ti+1),

0)) ≤ ε.4. Let i = i+ 1, go to 2.5. If (C2 ) mode σ(ti) is stable and mode σ(ti+1) is

unstable, and there exist h − 1 unstable modes(h ≤ m ) activated successively after modeσ(ti+1), then go to 6;Else, go to 9.

6. Determine the boundΩ(|x(ti+1)|)satisfying |x(ti+h+1)|≤ Ω(|x(ti+1)|) using (19) in Lemma 1, choose ti+1 such

that (ασ(ti+h+1)1 )−1φσ(ti+h+1)(α

σ(ti+h+1)2 (Ω(|x(ti+1)|)), 0)) ≤ ε,

let s = 0.

7. Choose ti+2+s such that∑i+1+sk=0

(∏i+1+sj=k

φtj+1−tjσ(tj)

Vtjσ(tj)

)≤

β

(h+1−s)χh+1−s− 1

8. Let s = s + 1; If s 6= h, then go to 7; Else, leti = i+ h, go to 2.

9. If (C3 ) the initial mode σ(0) is unstable,and there exist h − 1 unstable modes (h ≤ m )activated successively after mode σ(0), then goto 10.

10. Determine the bound Ω(|x(0)|) satisfying |x(th)| ≤Ω(|x(0)|) using (19) in Lemma 1, choose x(0) such that(ασ(th)1 )−1φσ(th)(α

σ(th)2 (Ω(|x(0)|)), 0)) ≤ ε, let s = 0.

11. Choose t1+s such that∑sk=0

(∏sj=k

φtj+1−tjσ(tj)

Vtjσ(tj)

)≤

β

(h+1−s)χh+1−s

− 1.12. Let s = s + 1; If s 6= h, then go to 11; Else, leti = h, go to 2.

The main idea behind S is that for current stable mode σ(ti),if next mode σ(ti+1) is stable, we let mode σ(ti) be activateduntil ti+1 such that x(ti+1) results in |x(t)| ≤ ε during modeσ(ti+1)’s working period [ti+1, ti+2) (step 3). When we predict thath unstable modes will be activated after stable mode σ(ti), we letmode σ(ti) be activated long enough until ti+1 such that x(ti+1)results in |x(t)| ≤ ε for t ∈ [ti+1, ti+h+2), i.e. the total activatingperiods of all h unstable modes and stable mode σ(ti+h+1) (step6). This can be achieved because the upper bound Ω(|x(ti+1)|)can be obtained without the information of switching instantsti+1, . . . , ti+h+1 (see Lemma 1 and Remark 2). The switchingscheme among unstable modes is based on Lemma 1 (steps 7, 8,11, 12). For initial stable/unstable modes, the initial states x(0)are also chosen in different ways (steps 1 and 10). The switchingdetermined by S depends on the states which can also be regardedas a state-dependent switching [13].

Theorem 1. Consider a switched system (1) satisfying (2)–(4) andAssumption 1. For any given ε > 0 and any switching sequencewhere at most m unstable modes are activated one by one, underthe switching law S, there exist an initial states x(0) and a series ofswitching instants satisfy 0 < t1 < t2 < · · ·, such that the origin isstable and |x(t)| ≤ ε∀t ≥ 0.

Proof. In the step 1 of S, choosing x(0) satisfying (ασ(0)1 )−1φσ(0)(Vσ(0)(x(0), 0)) ≤ ε leads to |x(0)| ≤ ε when mode σ(0) is justactivated. If mode σ(0) is stable, we have from (2) and (3) that|x(t)| ≤ ε for t ∈ [0, t1). We will consider respectively three casesC1–C3 in S.For C1, since mode σ(ti) is stable, it follows from (2) and

(3) that there always exists a time instant ti+1 > ti satisfying(ασ(ti+1)1 )−1φσ(ti+1)(Vσ(ti+1)(x(ti+1), 0)) ≤ ε, this implies that|x(ti+1)| ≤ ε when mode σ(ti+1) is just activated. Since modeσ(ti+1) is also stable, we have |x(t)| ≤ ε for t ∈ [ti+1, ti+2).For C2, switching on mode σ(ti+2) at t = ti+2 results in

φσ(ti+2)(Vti+2σ(ti+2)

, 0)

V ti+2σ(ti+2)

( i+1∑k=0

( i+1∏j=k

φtj+1−tjσ(tj)

Vtjσ(tj)

)+ 1

)≤

β

(h+ 1)χh.

Since β > m(m + 1)χm+1, h ≤ m, we have β

(h+1)χh<

β

hχh− 1.

Thus we can choose ti+3 > ti+2 such that

φti+3−ti+2σ(ti+2)

V ti+2σ(ti+3)

( i+1∑k=0

( i+1∏j=k

φtj+1−tjσ(tj)

Vtjσ(tj)

)+ 1

)≤

β

hχh− 1.

By induction, for s = 1, 2, . . . , h − 1 we have β

(h+1−s)χh−s<

β

(h−s)χh−s− 1. Choose ti+3+s as S, we obtain

φti+3+s−ti+2+sσ(ti+2+s)

V ti+2+sσ(ti+2+s)

(i+1+s∑k=0

(i+1+s∏j=k

φtj+1−tjσ(tj)

Vtjσ(tj)

)+ 1

)≤

β

(h− s)χh−s− 1.

Finally, we verify condition (5) with t = ti+1+h and ts = ti+1.There are finite number of switchings occurring in (ti+1, ti+1+h],it follows from Lemma 1 that we can find a bound Ω(|x(ti+1)|)satisfying |x(ti+h+1)| ≤ Ω(|x(ti+1)|) using (19). Since this boundis independent from the switching instants, we can determine itbefore h unstable modes are switched into.Note that mode σ(ti) is stable, we can find a time instant ti+1 >

ti such that

(ασ(ti+h+1)1 )−1φσ(ti+h+1)(α

σ(ti+h+1)2 (Ω(|x(ti+1)|)), 0)) ≤ ε.

This guarantees that |x(t)| ≤ ε for t ∈ [ti+1, ti+h+1]. Modeσ(ti+h+1) is also stable, we further have |x(t)| ≤ ε for t ∈[ti+h+1, ti+h+2).

H. Yang et al. / Systems & Control Letters 58 (2009) 703–708 707

For C3, note that β > m(1+ χ)χm and χ ≥ 1, which result in

χ <β

hχh− 1. We can choose t1 such that

φt1σ(0)

V0σ(0)≤

β

hχh− 1, the rest

of the proof follows the same procedure as in C2, thus is omittedhere. We finally obtain (5) with t = th and ts = 0.Based on the above analysis, one finds that for a switched

system with any given switching sequence, finite or infinitenumber of switchings and both stable and unstable modes, theswitching law Smaintains the stability of the origin, and |x(t)| ≤ εfor t ≥ 0. This completes the proof.

Remark 6. Roughly speaking,S lets the activating periods of stablemodes large enough and lets the activating periods of unstablemodes small enough such that the state trajectory is boundedunder a given switching sequence. Such an idea is similar to thatof dwell-time schemes in [20–22] where an aggregated systemis considered including stable modes and consequently activatedunstable ones. This aggregated system would be stable if the totalactivating periods of stable modes are sufficiently large. However,S provides an alternative way to approach stability in the absenceof the ‘‘µ’’ condition.

Remark 7. The switching law in [10] guarantees that for each

activating mode, the gainφtj+1−tjσ(tj)

Vtjσ(tj)

≤ λ, 0 < λ < 1. In the case of all

stable modes, inequality (5) can be satisfied if the above inequalityholds with some λ. However, for the case of unstable modes,φtj+1−tjσ(tj)

Vtjσ(tj)

is impossible to be less than 1. The proposed switching

scheme is effective for the system with unstable modes.

4. An illustrative example

This section presents a networked control system (NCS)example to illustrate the application of the theoretical results. Theconsidered NCS can be modeled as a switched system, where thedifferentmodes represent different control operations through thenetwork. In some control operations, unreliable communicationchannels often transmit data, which leads to unstable modes.However, the prescribed switching sequence is not expected to bechanged such that the pre-specified task can be completed.Suppose thatM = 1, 2, 3, x = [x1, x2]>, the modes take the

following forms

f1 =[−x1 + 4x32−x1 − 0.5x2

], f2 =

[x1 − x2x2 + x31

], f3 =

[x1 − x2x1 + x2

].

For mode 1, it is not easy to find a quadratic Lyapunovfunction. However the origin is still stable, we choose a polynomialLyapunov function V1 = x21 + 2x

42, this results in V1(x(t)) <

e−2tV1(x(0)) for t ≥ 0. Both mode 2 and mode 3 are unstable,applying V1 to modes 2 and 3 yields

dV1(x)dx

f2(x) ≤ V 0.51 (x)+ 7V1(x)+ 4V 1.51 (x)+ 4V 31 (x) (21)

dV1(x)dx

f3(x) ≤ V 0.51 (x)+ 11V1(x)+ 2V 1.51 (x). (22)

It can be seen that a common Lyapunov function is hard to imposehere because inequalities (21)–(22) do not satisfy the generalLyapunov function formulation in dwell-time scheme [1]. Themethod in [23] is also not easy to be implemented since the rightsides of (21) and (22) are polynomial forms of V1 rather thanaVm1 (x) for a,m > 0 in [23], and the exponents larger and smallerthan 1 exist simultaneously.

4 3 2 1 0 1 2 3 4 4

3

2

1

0

1

2

3

4

x1

x2

x (0)

|x| = 4

Fig. 2. State trajectory.

We choose V2 = x41 + 2x22, V3 = x21 + x

22. It follows that

V2(x(t)) < e4tV2(x(0)), V3(x(t)) < e2tV3(x(0)), for t ≥ 0. Notethat MLFs techniques are difficult to be applied since the statetrajectories in unstable modes are not bounded and Lyapunov-likefunctions are not easy to find. The ‘‘µ’’ condition is also hard toimpose here, because V1 and V2 are non-quadratic.Set ε = 4 which means that |x(t)| ≤ 4 must hold for all t ≥ 0.

The prescribed switching sequence ismode 1→ mode 2→ mode 3→ mode 1→ · · · · · ·Nowwe design the switching instants according to S. Mode 1 is

stable, choose x(0) = [1, 2]> from step 1 of S such that |x(t)| ≤ 4for t ∈ [0, t1). Since both mode 2 and mode 3 are unstable, theswitching scheme based on Lemma 1 is applied after t1. It can beobtained from (20) that χ = 1.m = 2 due to two unstable modes.Choose β = 6.3 > 2(2 + 1). The activating periods of modes2 and 3 can be calculated from step 7 of S: 0.0059 s for mode 2;0.2602 s for mode 3. Choose t1 = 0.9 s from step 6 of S such that|x(t)| ≤ 4 for t ∈ [0, t4). Consequently, choose t2 = 0.9059 s,t3 = 1.1661 s. The activating period of mode 1 is set to be 0.9 s inthe following switching process, i.e., t4 = 2.0661 s. Although ourtheory allows infinite switchings in infinite time interval, in thenumerical simulation, a finite time interval [0 s, 4 s] is considered.Other switching instants can be obtained straightly. Fig. 2 showsthe state trajectory, from which we can see that the stability isachieved and |x| ≤ 4 always holds.

5. Conclusion

This paper provides a stabilization method for switchednonlinear systemswith unstablemodes. The obtained results couldbe the basis of some future works as follows:1. The obtained stability condition (5) deserves extension tothe analysis of other stability notions for switched systems,e.g., asymptotic stability, input-to-state stability, etc. Moreover,various switchings can be considered, e.g. impulsive switching,stochastic switching.

2. The proposed switching laws S can be potentially appliedto the switching control design for nonlinear systems withuncertainties and faults.

Acknowledgements

The author would like to thank the anonymous reviewersand the AE for their helpful and insightful comments for furtherimproving the quality of this work.

708 H. Yang et al. / Systems & Control Letters 58 (2009) 703–708

References

[1] D. Liberzon, Switching in Systems and Control, Birkhauser, Boston, MA, 2003.[2] M.S. Branicky, Multiple Lyapunov functions and other analysis tools forswitched and hybrid systems, IEEE Transactions on Automatic Control 43 (1)(1998) 475–482.

[3] H. Ye, A.N. Michel, L. Hou, Stability theory for hybrid dynamical systems, IEEETransactions on Automatic Control 43 (4) (1998) 461–474.

[4] N.H. EI-Farra, P. Mhaskar, P.D. Christofides, Output feedback control ofswitched nonlinear systems using multiple lyapunov functions, System andControl Letters 54 (12) (2005) 1163–1182.

[5] P. Mhaskar, N.H. El-Farra, P.D. Christofides, Predictive control of switchednonlinear systems with scheduled mode transitions, IEEE Transactions onAutomatic Control 50 (11) (2005) 1670–1680.

[6] P. Mhaskar, N.H. El-Farra, P.D. Christofides, Robust predictive control ofhybrid systems: satisfying uncertain schedules subject to state and controlconstraints, International Journal of Adaptive Control and Signal Processing22 (2008) 161–179.

[7] J. Zhao, D.J. Hill, On stability, L2 gain and H∞ control for switched systems,Automatica 44 (5) (2008) 1220–1232.

[8] J. Zhao, D.J. Hill, Dissipativity theory for switched systems, IEEE Transactionson Automatic Control 36 (11) (2008) 1228–1240.

[9] J. Zhao, D.J. Hill, Passivity and stability of switched systems: Amultiple storagefunction method, System and Control Letters 57 (2) (2008) 158–164.

[10] W.X. Xie, C.Y. Wen, Z.G. Li, Input-to-state stabilization of switched nonlinearsystems, IEEE Transactions on Automatic Control 46 (7) (2001) 1111–1116.

[11] P. Colaneri, J.C. Geromel, A. Astolfi, Stabilization of continuous-time switchednonlinear systems, System and Control Letters 57 (1) (2008) 95–103.

[12] J.P. Hespanha, A.S. Morse, Stability of switched systems with average dwelltime. in: Proceedings of the 38th IEEE Conference on Decision and Control,Phoenix, USA, 1999, pp. 2655–2660.

[13] J.P. Hespanha, Stabilization through hybrid control, in: Control Systems,Robotics, and Automation, in: Encyclopedia of Life Support Systems (EOLSS),2004.

[14] L. Vu, D. Chatterjee, D. Liberzon, Input-to-state stability of switched systemsand switching adaptive control, Automatica 43 (4) (2007) 639–646.

[15] D.V. Efimov, E. Panteley, A. Loria, On input-to-output stability of switchednonlinear systems. in: Proceedings of the 17th IFAC World congress, Seoul,Korea, 2008, pp. 3647–3652.

[16] X. Xu, G. Zhai, Practical stability and stabilization of hybrid and switchedsystems, IEEE Transactions on Automatic Control 50 (11) (2005) 1897–1903.

[17] Z. Sun, A note on marginal stability of switched systems, IEEE Transactions onAutomatic Control 53 (2) (2008) 625–631.

[18] B. Jiang, M. Staroswiecki, V. Cocquempot, Fault accommodation for nonlineardynamic systems, IEEE Transactions on Automatic Control 51 (9) (2006)1578–1583.

[19] H. Yang, B. Jiang, V. Cocquempot, A fault tolerant control framework forperiodic switched nonlinear systems, International Journal of Control 82 (1)(2009) 117–129.

[20] H. Yang, B. Jiang, Cocquempot, Observer based fault tolerant control for a classof hybrid impulsive systems, in: International Journal of Robust and NonlinearControl, 2009 (in press).

[21] G. Zhai, B. Hu, K. Yasuda, A.N. Michel, Stability analysis of switched systemswith stable and unstable subsystems: An average dwell time approach,International Journal of System Science 32 (8) (2001) 1055–1061.

[22] W. Feng, J.F. Zhang, Stability analysis and stabilization control ofmulti-variableswitched stochastic systems, Automatica 42 (1) (2006) 169–176.

[23] D. Munoz de la Pena, P.D. Christofides, Stability of nonlinear asynchronoussystems, in: Proceedings of 46th IEEEConference onDecision andControl, NewOrleans, LA, USA, 2007, pp. 4576–4583.

[24] X. Xu, P. Antsaklis, Stabilization of second-order LTI switched system,International Journal of Control 73 (14) (2000) 1261–1279.

[25] C. Yfoulis, R. Shorten, A computational technique characterizing the asymp-totic stabilizability of planar linear switched systems with unstable modes,in: Proceedings of 43rd IEEE Conference on Decision and Control, Atlantis, Par-adise Island, Bahamas, 2004, pp. 2792–2797.

[26] S. Bhat, D Bernstein, Finite-time stability of continuous autonomous systems,SIAM Journal on Control and Optimization 38 (3) (2000) 751–766.