Applied Mathematics and Computation 238 (2014) 358–369

Contents lists available at ScienceDirect

Applied Mathematics and Computation

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

New stability and stabilization results for discrete-timeswitched systems

http://dx.doi.org/10.1016/j.amc.2014.04.0160096-3003/� 2014 Elsevier Inc. All rights reserved.

⇑ Corresponding author.E-mail address: 123.wrh@163.com (R. Wang).

Ruihua Wang ⇑, Shumin FeiSchool of Automation, Southeast University, Nanjing 210096, Jiangsu, People’s Republic of China

a r t i c l e i n f o

Keywords:Switched systemsExponential stabilityForward average dwell timeForward mode-dependent average dwelltimeMode-dependent controllers

a b s t r a c t

In this paper, the stability and stabilization problems are considered for discrete-timeswitched systems in both nonlinear and linear contexts. By introducing the concept offorward average dwell time and using multiple-sample Lyapunov-like functions variation,the extended stability results for discrete-time switched systems in the nonlinear settingare first derived. Then, the criteria for stability and stabilization of linear switched systemsare obtained via a newly constructed switching strategy, which allows us to obtainexponential stability results. A new kind of mode-dependent controller is designed torealize the stability conditions expressed by the multiple-sample Lyapunov-like functionsvariation. Based on this controller, the stabilization conditions can be given in terms oflinear matrix inequalities (LMIs), which are easy to be checked by using recently developedalgorithms in solving LMIs. Finally, a numerical example is provided to show the validityand potential of the results.

� 2014 Elsevier Inc. All rights reserved.

1. Introduction

Switched systems are composed of continuous-time or discrete-time systems with (isolated) discrete switching events,which are triggered by switching mechanisms. Switched systems have strong engineering background in various areas andare often used as a unified modeling tool for a great number of real-world systems such as power electronics, chemicalprocesses, mechanical systems, automotive industry, aircraft and air traffic control and many other fields. Therefore, lotsof efficient methodologies have been proposed in the literature to deal with the stability and stabilization problems forswitched systems [1,2,4–7,9,10,13,16]. One way is ‘‘dwell time’’, which, then, is extended to the concept of ‘‘average dwelltime (ADT)’’ for more flexibility and availability in system analysis and control synthesis; see [3,4,10,12,13,15,16] and thereferences therein. By allowing the Lyapunov-like function of every subsystem to increase with a finite increase rate andin less than a constant time during its running time, the extended stability results were obtained for continuous-timeand discrete-time switched systems with ADT in nonlinear setting [12]. Recently, a new concept, ‘‘mode-dependent ADT’’,has been introduced by Zhao et al. [16], where each mode in the underlying switched system has its own ADT. The increasecoefficient of the Lyapunov-like function at switching instants and the decay rate of the Lyapunov-like function during therunning time of subsystems are set in a mode-dependent manner, which reduces the conservativeness in the results.

R. Wang, S. Fei / Applied Mathematics and Computation 238 (2014) 358–369 359

Multiple-sample Lyapunov functions variation has been used to investigate stability and stabilization for fuzzy systemsand switched systems [17,18,20,21]. In [17], the stabilization problem for a class of uncertain discrete-time Takagi–Sugenofuzzy systems was studied through a nonquadratic Lyapunov function. k-sample Lyapunov functions variation, i.e.,DkVðxðtÞÞ ¼ Vðxðt þ kÞÞ � VðxðtÞÞ, is put in use to design a robust control law such that the close-loop system is globallyasymptotically stable. This paper [18] proposed a new approach for the stability analysis and controller synthesis of dis-crete-time Takagi–Sugeno fuzzy dynamic systems. In this paper non-monotonic Lyapunov function is utilized to relax themonotonic requirement of Lyapunov theorem which renders a larger class of functions to provide stability, whichdecreases every few steps; however, can be increased locally. The authors [20] gave two new sufficient conditions of glo-bal asymptotic stability for nonlinear systems that allow the Lyapunov functions to increase locally, but guarantee anaverage decrease every few steps. In order to demonstrate the strength of their methodology, it was shown that tighterbounds on the joint spectral radius can be obtained using common non-monotonic Lyapunov functions for discrete-timelinear switched systems. In [21], the stability problem was studied for discrete time linear switched systems based onminimum dwell time by means of a family of quadratic Lyapunov functions. The inequality ðAMi Þ

TPjA

M

i � Pi, where M isthe lower bound for dwell time, was introduced to obtain asymptotic stability for linear switched systems. However,the inequality AT

i PiAi � Pi < 0 is also necessary therein, which denotes that every subsystem is required to be asymptoti-cally stable. Besides, because ðAMi Þ

TPjA

M

i � Pi is nonconvex in Ai, the authors didn’t find efficient controller design methods.Inspired by these results, we will employ multiple-sample Lyapunov-like functions variation to establish new mode-dependent controllers (multiple controllers for each mode in one switching interval), guaranteeing that the closed-loopsystem is exponentially stable.

The remaining of the paper is organized as follows. In Section 2, necessary preliminary knowledge is given and theconsidered problem is formulated. Section 3 is devoted to the main results of the paper. By using multiple-sampleLyapunov-like functions variation and a new concept–forward ADT, the sufficient conditions ensuring the asymptoticstability of nonlinear discrete-time switched systems are firstly derived. Then, new mode-dependent controllers are con-structed to guarantee the exponential stability of the resulting closed-loop system for linear switched systems. In Section 4,a powerful example is provided to show the potential and the validity of the obtained results. In the end, concludingremarks are given in Section 5.

Notations. The notations in this paper are fairly standard. We use A > 0 (A < 0) to stand for a positive definite (negativedefinite) matrix A. AT denotes the transpose of a matrix A, and kmaxðAÞ (respectively, kminðAÞ) represents the maximum(respectively, minimum) eigenvalue of A. The set of symmetric matrices of size n is denoted by Sn. Let ZP0 denote the setof nonnegative integers, and Rn mean the n-dimensional Euclidean space. As is commonly used in other literature, � denotesthe elements below the main diagonal of a symmetric matrix, and max and min, respectively, stands for the maximum andminimum. C1 denotes the space of continuously differentiable functions, and a function q : ½0;1Þ ! ½0;1Þ is said to be ofclass K1 if it is continuous, strictly increasing, unbounded, and qð0Þ ¼ 0. In addition, k � k is used to denote the vectorEuclidean norm, and I refers to an identity matrix of the appropriate dimension. Matrices, if not explicitly stated, areassumed to have compatible dimensions for algebraic operations.

2. Problem formulation and preliminaries

Consider a class of discrete-time switched linear systems described by

xðkþ 1Þ ¼ ArðkÞxðkÞ þ BrðkÞuðkÞ; k P k0; ð1Þ

where xðkÞ 2 Rnx is the state, and uðkÞ 2 Rnu is the control input. rðkÞ is a piecewise constant function of time, called a switch-ing signal, which takes its values in a finite set N ¼ f1;2; . . . ;Ng; N > 1 is the number of subsystems. Ai and Bi are constantreal matrices with appropriate dimensions for any i 2 N.

For a piecewise constant switching signal rðsÞ, let k1 < k2 < � � � < km < � � � denote the switching instants of rðsÞ andfxðk0Þ; ði0; k0Þ; ði1; k1Þ; . . . ; ðim; kmÞ; . . .g denote the switching sequence, which means that the imth subsystem is activatedwhen km 6 s < kmþ1, or equivalently, rðsÞ ¼ im when km 6 s < kmþ1.

For the purpose of the paper, the definitions of the ADT property used to restrict a class of switching signals are recalled asfollows.

Definition 1 [13]. For any k P k0 P 0 and any switching signal r, let Nr denote the number of switchings of r over theinterval ½k0; kÞ. For given constants Ta > 0; N0 P 0, if the inequality Nr 6 N0 þ ðk� k0Þ=Ta holds, then the positive constantTa is called the ADT and N0 is called the chattering bound.

Definition 2 [16]. For a switching signal r and any k P k0 P 0, let Nriðk; k0Þ be the number that the ith subsystem is acti-vated over the interval ½k0; kÞ and Tiðk; k0Þ denote the total running time of the ith subsystem over the interval ½k0; kÞ; 8i 2 N.For given constants Tai > 0; N0i P 1, if the inequalities Nriðk; k0Þ 6 N0i þ Tiðk; k0Þ=Tai hold, then the positive constant Tai iscalled the mode-dependent ADT and N0i is called the mode-dependent chattering bound.

360 R. Wang, S. Fei / Applied Mathematics and Computation 238 (2014) 358–369

Remark 1. In the above definition, Nriðk; k0Þ cannot be simply explained as the number of switchings from subsystem j – i tosubsystem i over the interval ½k0; kÞ. For subsystem i – rðk0Þ; Nriðk; k0Þ can be the number of switchings from any othersubsystem to the ith subsystem; however, as far as subsystem rðk0Þ is concerned, the number of switchings from any othersubsystem to subsystem rðk0Þ over the interval ½k0; kÞ equals to Nrrðk0Þðk0; kÞ � 1. After all, Nrrðk0Þ also includes the initialswitching.

In the following, we will introduce the definitions of forward ADT and forward mode-dependent ADT for discrete-timeswitched systems.

Definition 3. For any k P k0 and any switching signal r, let Nr denote the number of switchings of r over the interval ½k0; kÞ.If there exist constants T1

a > 0; N1 P 1 such that the inequality

Nr Pk� k0

T1a

� N1 ð2Þ

hold, then the positive constant T1a is called the forward ADT and N1 is called the forward chattering bound.

Remark 2. About Definition 3, when N1 > 1 is a positive integer, Nr P ðk� k0Þ=T1a � N1 implies that even if the length of

½k0; kÞ has exceeded N1T1a , it is possible that only one switching happens. In general, if N1 switchings (more precisely, the

smallest integer greater than N1) are added, then the average time between consecutive switchings is at most T1a .

Remark 3. In effect, it is in order to specify the switching signal that we introduce forward ADT for switched systems. Toelaborate, the forward ADT switching denotes that the number of switchings in a finite interval cannot be too small. In otherwords, the average time between consecutive switchings must be proper. In ADT technique, ADT is generally required to benot less than a fixed constant by researchers; however, about forward ADT, we don’t restrict forward ADT to satisfy similarrequirements, i.e., T1

a only needs to be assigned an appropriate positive constant, which can be found in Theorem 1 and theproof of it.

Definition 4. For any k P k0 and any switching signal r, let Nriðk; k0Þ denote the number that the ith subsystem is activatedover the interval ½k0; kÞ and Tiðk; k0Þ denote the total running time of the ith subsystem over the interval ½k0; kÞ; 8i 2 N. Ifthere exist constants T1

ai > 0 and N1i P 0 such that the inequalities

Nriðk; k0ÞPTiðk; k0Þ

T1ai

� N1i ð3Þ

hold, then the positive constant T1ai is called the forward mode-dependent ADT and N1i is called the forward mode-dependent

chattering bound.

Remark 4. For a switching signal, if the length of every switching interval is not larger than a positive constant, it must carryforward ADT, satisfying Definition 3 or Definition 4. However, that is not enough for practice. It is of interest to allow thepossibility that the length of some time interval exceeds the constant when necessary.

Remark 5. For the switching instants k1; k2; . . . ; kNr over the interval ½k0; kÞ, one can get Nr þ 1 ¼PN

i¼1Nriðk; k0Þ according tothe definitions of Nr and Nriðk; k0Þ.

The definition of exponential stability and three important lemmas are expressly stated below for the derivation of themain results and later discussion.

Definition 5 [11]. System (1) with u ¼ 0 is said to be exponentially stable if its solution satisfies

kxðkÞk 6 ck�ðk�k0Þkxðk0Þk; 8k P k0;

where c > 0 is the decay coefficient, and k > 1 is the decay rate.

Lemma 1 [17]. Let P 2 Sn; C 2 Sp; U and W be matrices of appropriate dimensions. The two statements are equivalent:

Find P > 0 such that UT PU� C < 0; Find P > 0 and W such that�C �WU �W�WT þ P

� �< 0.

Lemma 2 [8]. Consider a vector x 2 Rn and two matrices Q ¼ Q T 2 Rn�n and R 2 Rm�n such that rankðRÞ < n. The two followingexpressions are equivalent:

1. xT Qx < 0; x 2 fx 2 Rn; x – 0; Rx ¼ 0g;2. 9M 2 Rn�m such that Q þMRþ RT MT < 0.

R. Wang, S. Fei / Applied Mathematics and Computation 238 (2014) 358–369 361

Lemma 3 [12]. Consider the discrete-time switched system xðkþ 1Þ ¼ frðkÞðxðkÞÞ; r 2 N and let 0 < c < 1; ~g P 1 and l > 1 begiven constants. Suppose that there exist C1 functions VrðkÞ : Rn ! R; rðkÞ 2 N, and two class K1 functions j1 and j2 such that8rðkÞ ¼ i 2 N,

j1ðkxðkÞkÞ 6 ViðkÞ 6 j2ðkxðkÞkÞ;

Viðkþ 1Þ 6cViðkÞ; 8k 2 T #ðkm; kmþ1Þ~gViðkÞ; 8k 2 T "ðkm; kmþ1Þ

�; ð4Þ

and 8ðrðkmÞ ¼ i; rðkm � 1Þ ¼ jÞ 2 N � N; i – j,

ViðkmÞ 6 lVjðkmÞ; ð5Þ

then the system is asymptotically stable for any switching signal with ADT

Ta > �fT max½ln ~g� ln c� þ ln lg= ln c; ð6Þ

where T "ðkm; kmþ1Þ and T #ðkm; kmþ1Þ represent the unions of the dispersed intervals during which Lyapunov function is increasingand decreasing within the interval ½km; kmþ1Þ respectively, and T "ðkmþ1 � kmÞ is used to denote the length ofT "ðkm; kmþ1Þ; T max ,max8m2ZP0T "ðkmþ1 � kmÞ.

A mode-dependent controller with the form uðkÞ ¼ KrðkÞxðkÞ, where Ki; i 2 N are controller gains, is designed in a greatdeal of past literature. For linear switched systems, the k-sample Lyapunov functions variation, i.e.,

DkViðxðtÞÞ ¼ Viðxðt þ kÞÞ � ViðxðtÞÞ < 0 can be expressed as ðAki Þ

TPiA

ki � Pi < 0, which is nonconvex in Ai and hence difficult

to be applied for designing mode-dependent controllers. Therefore, the objective of this paper is to construct a set ofmode-dependent state feedback controllers by applying a new switching strategy (specified by the forward ADT) such thatthe resulting closed-loop system is ensured to be exponentially stable.

3. Main results

3.1. Stability analysis for nonlinear switched systems

This subsection will present the asymptotic stability results for switched nonlinear systems with forward ADT.In multiple Lyapunov-like function theory, the Lyapunov-like function constructed for each active subsystem is generally

considered to be decreasing and as one-sample form. Herein, we use the multiple-sample Lyapunov-like functions variation,for example, m-sample variation, i.e., DmVðkÞ ¼ VðkþmÞ � VðkÞ, where m > 1, to develop our asymptotic stability results forswitched systems without requiring the condition that every subsystem is stable.

Theorem 1. Consider the discrete-time switched system xðkþ 1Þ ¼ frðkÞðxðkÞÞ; r 2 N. Suppose that there exist C1 functionsVrðkÞ : Rn ! R; rðkÞ 2 N and two class K1 functions j1 and j2 such that 8ðrðkmÞ ¼ i; rðkmþ1Þ ¼ jÞ 2 N � N; i – j,

j1ðkxðkÞkÞ 6 ViðkÞ 6 j2ðkxðkÞkÞ; ð7Þ

Vjðkmþ1Þ 6 aimViðkmÞ; ð8Þ

where aim are positive constants. Under the switching signal r with the forward ADT T1a ðor the forward mode-dependent ADT T1

aiÞ,if there exist constants 0 < a < 1 and M > 0 such that the following inequality holds

YNr

r¼1

arðkNr�rÞðNr�rÞ 6 MaNr ; ð9Þ

then the nonlinear switched system is asymptotically stable.

Proof. Before proving the above theorem, a weak assumption is first given: for 8k 2 ½kl; klþ1Þ, there must exist a class K1function q such that VrðkÞðkÞ 6 qðVrðklÞðklÞÞ. By combining (8) and (9) with Definition 3, we can obtain

VrðklÞðklÞ 6 arðkl�1Þðl�1ÞVrðkl�1Þðkl�1Þ

6 arðkl�1Þðl�1Þarðkl�2Þðl�2ÞVrðkl�2Þðkl�2Þ

6 � � � 6Yl

r¼1

arðkl�rÞðl�rÞVrðk0Þðk0Þ 6 MaNr Vrðk0Þðk0Þ

6 Ma�N1 ða1

T1a Þ

k�k0

Vrðk0Þðk0Þ:

362 R. Wang, S. Fei / Applied Mathematics and Computation 238 (2014) 358–369

Thus, VrðkÞðkÞ 6 q Ma�N1 ða1

T1a Þ

k�k0

Vrðk0Þðk0Þ !

.

For r with T1ai, we have aNr ¼ a�1QN

i¼1aNriðk;k0Þ.

According to (3), it follows that

VrðklÞðklÞ 6 Ma�1YN

i¼1

aNriðk;k0ÞVrðk0Þðk0Þ 6 Ma�1YN

i¼1

a�N1i maxi2N

a1

T1ai

� �k�k0

Vrðk0Þðk0Þ: ð10Þ

Therefore,

VrðkÞðkÞ 6 q Ma�1YN

i¼1

a�N1i maxi2N

a1

T1ai

� �k�k0

Vrðk0Þðk0Þ !

: ð11Þ

In the end, from a < 1, it follows that a1

T1ai < 1. Thus,we can conclude that VrðkÞðkÞ converges to zero as k!1. Then the

asymptotic stability can be deduced with the aid of (7). h

Remark 6. In fact, if the state xðkÞ is bounded with a nonzero lower bound and an upper bound, i.e., m 6 kxðkÞk 6 M, wherem;M > 0, before the state achieves stability, the weak assumption mentioned above can be guaranteed to hold.

Remark 7. Assume that the switching signal r with ADT (6) also satisfies Definition 3 (or Definition 4), i.e., there exist con-stants T1

a and N1 (or T1ai and N1i) such that the inequality (2) (or (3)) holds, and then we can apply Theorem 1 to prove Lemma

3 as follows:By (4) and (5), we can obtain 8ðrðkmÞ ¼ i; rðkmþ1Þ ¼ jÞ 2 N � N; i – j,

Vjðkmþ1Þ 6 lð1� cÞT #ðkm ;kmþ1Þð1þ gÞT "ðkm ;kmþ1ÞViðkmÞ:

Noting Nr � 1 ¼ NrðkNr ; k0Þ 6 ðkNr � k0Þ=Ta þ N0, we have

YNr

r¼1ð1� cÞT #ðkr�1 ;krÞð1þ gÞT "ðkr�1 ;krÞlNr ¼ ð1� cÞT #ðkNr ;k0Þð1þ gÞT "ðkNr ;k0ÞlNr

6 ð1� cÞkNr�k01þ g1� c

� �NrT max

lNr

6 ð1� cÞð�1�N0ÞTa ð1� cÞTa 1þ g1� c

� �T max

l" #Nr

:

Set M ¼ ð1� cÞð�1�N0ÞTa and a ¼ ð1� cÞTa 1þg1�c

� �T max

l. From (6), we get 0 < a < 1. Up to this point, all the conditions of the

above theorem are satisfied. In all, if the switching signal in Lemma 3 also carries forward ADT, Lemma 3 can be viewedas the corollary of Theorem 1. Besides, we can prove that Lemma 1 of [14] and Lemma 4 of [16] can also be the corollariesof Theorem 1, if the switching signals therein carry forward ADT.

For brevity, we will simplify Theorem 1 in the following.

Corollary 1. Consider the discrete time switched system xðkþ 1Þ ¼ frðkÞðxðkÞÞ; r 2 N, and let ai > 0; 8i 2 N be given constants.Suppose that there exist C1 functions VrðkÞ : Rn ! R; rðkÞ 2 N and two class K1 functions j1 and j2 such that8ðrðkmÞ ¼ i; rðkmþ1Þ ¼ jÞ 2 N � N; i – j,

j1ðkxðkÞkÞ 6 ViðkÞ 6 j2ðkxðkÞkÞ;

Vjðkmþ1Þ 6 aiV iðkmÞ: ð12Þ

If the switching signal r has the forward mode-dependent ADT T1ai and there exist constants M and 0 < a < 1 such that the follow-

ing inequality holds

YNi¼1

aNriðkNr ;k0Þi 6 MaNr ; ð13Þ

then the nonlinear switched system is asymptotically stable.

Remark 8. In the model prediction control, one could establish controllers by using the rolling optimization technique.Herein, the switching signal can be designed as well via the rolling strategy so as to realize (9) and (13) in reality.

R. Wang, S. Fei / Applied Mathematics and Computation 238 (2014) 358–369 363

3.2. Stability and stabilization for linear switched systems

Now, we are in a position to give the stability condition of system (1) based on Corollary 1. For simplicity, let Ti; 8i 2 Ndenote the running time of the ith subsystem when it is activated every time. Obviously, the switching signal carries mode-dependent ADT T1

ai; 8i 2 N.

Theorem 2. Consider system (1) with u ¼ 0 and let ai > 0; ai P 1; 8i 2 N be given constants. Suppose that there exist positivedefinite matrices Pi; 8i 2 N such that

ðATii Þ

TPjA

Tii 6 aiPi; 8i – j; ð14Þ

ATi PiAi 6 aiPi; 8i 2 N: ð15Þ

If there exist constants 0 < a < 1 and M > 0 such that the inequality (13) is satisfied, then the closed-loop system is exponentiallystable.

Proof. Using the Lyapunov-like function ViðkÞ ¼ xTðkÞPixðkÞ; 8i 2 N, we have

Vjðkþ TiÞ ¼ xTðkÞðATii Þ

TPjA

Tii xðkÞ; 8j – i:

By (14), it follows that Vjðkþ TiÞ 6 aiV iðkÞ.8k P k0, there exists a constant l 2 ZP0 such that k 2 ½kl; klþ1Þ. Together with (13), we can obtain the inequality (10).

Besides, 8k 2 ðkl; klþ1Þ, there exists a positive integer 1 6 j 6 Ti � 1 such that k ¼ kl þ j.(15) yields

VrðklÞðkl þ jÞ 6 ajrðklÞ

VrðklÞðklÞ:

Then, we can get

VrðkÞðkÞ 6maxi2N

aTi�1i Ma�1

YN

i¼1

a�N1i maxi2N

a1

T1ai

� �k�k0

Vrðk0Þðk0Þ:

Therefore,

kxðkÞk 6 k1

k2max

i2NaTi�1

i Ma�1YN

i¼1

a�N1i

!12

maxi2N

a1

T1ai

� �k�k02

kxðk0Þk; ð16Þ

where k1 ¼maxi2NfkmaxfPigg; k2 ¼mini2NfkminfPigg. This ends the proof. h

Remark 9. In fact, similar inequalities to (14) have appeared in Theorem 1 of the paper [21]. However, therein the conditionsAT

i PiAi � Pi < 0 are included in its theorem, which mean all the subsystems are required to be asymptotically stable. And thetheorem is only for asymptotic stability, not exponential stability. In addition, the authors therein didn’t give easy controllerdesign ways due to the nonconvexity in Ai as shown in (14).

Next, the stabilization problem of system (1) with control input u is considered. With respect to the general mode-depen-dent controller, it is easily seen that only one admissible controller is applied on each switching interval. However, it is notabsolute in effect, for example, in [12] the Eq. (31) subsumes two controllers for every subsystem because of asynchronousswitching, one for ½kl; kl þ smaxÞ, and the other for ½kl þ smax; klþ1Þ; in [19], the Eq. (12) indicates

xðkþ nþ 1Þ ¼Xr

i¼1

hiðzðkþ nÞÞAi �Xr

i¼1

hiðzðkþ nÞÞBi

Xr

i¼1

hiðzðkþ nÞÞFi

Xr

i¼1

hiðzðkþ nÞÞHi

!�10@

1Axðkþ nÞ;

n ¼ 0;1; . . . ; t � 1; t 2 ZP0; k 2 ZP0;

where uðkÞ ¼ �Pr

i¼1hiðzðkÞÞFiPr

i¼1hiðzðkÞÞHi �1xðkÞ for 8k 2 ½k; kþ tÞ. The two facts enlighten us to endow every subsystem

with more than one admissible controller on its every switching interval.Suppose that the ith subsystem is activated at the switching instant k. Then, the new mode-dependent controller can be

described:

uðkþ r � 1Þ ¼ Girxðkþ r � 1Þ; r ¼ 1; . . . ; Ti; ð17Þ

i.e.,

xðkþ 1Þ ¼ ðAi þ BiGi1ÞxðkÞ;xðkþ 2Þ ¼ ðAi þ BiGi2Þxðkþ 1Þ;� � �xðkþ TiÞ ¼ ðAi þ BiGiTi

Þxðkþ Ti � 1Þ;

where Gir 2 Rnu�nx ; r ¼ 1; . . . ; Ti will be determined in the following.

364 R. Wang, S. Fei / Applied Mathematics and Computation 238 (2014) 358–369

Theorem 3. Consider system (1) and let ai > 0; 8i 2 N be given constants. Suppose that there exist positive definite matrices Pi,invertible matrices Mi1; . . . ;MiTi

, and matrices Gi1; . . . ;GiTi; 8i 2 N, such that

�aiPi � 0 � � � 0 0

ð2;1Þ H1 � . ..

0 P1 H2. .

. ... ..

.

P2. .

.� 0 0

..

. . .. . .

.HTi�1 � 0

0 � � � 0 PTi�1 HTi�

0 � � � 0 0 MiTi�2I þ Pj

26666666666666664

37777777777777775

< 0; 8i – j; ð18Þ

The admissible controller can be given by

Gi1 ¼ Gi1; Gi2 ¼ Gi2M�1i1 ; . . . ;GiTi

¼ GiTiM�1

iðTi�1Þ:

For positive definite matrices Pi, and controller gain matrices Gi1; . . . ;GiTi, there also exist positive constants ai P 1; 8i 2 N such

that

ðAi þ BiGirÞT PiðAi þ BiGirÞ 6 aiPi; 8r 2 f1; . . . ; Ti � 1g: ð19Þ

If there exist constants 0 < a < 1 and M > 0 such that the inequality (13) is satisfied, then the closed-loop system is expo-

nentially stable, and the decay rate is ðmaxi2Na1

T1ai Þ�1

2

, where

ð2;1Þ ¼ Ai þ BiGi1; Hs ¼ �Mis �MTis; s ¼ 1; . . . ; Ti;

Pr ¼ AiMir þ BiGiðrþ1Þ; r ¼ 1; . . . ; Ti � 1:

Proof. Firstly, we have

Vjðkþ TiÞ � aiV iðkÞ ¼

xðkÞxðkþ 1Þ

..

.

xðkþ TiÞ

266664

377775

T �aiPi 0 � � � 00 0 � � � 0... ..

.� � � ..

.

0 0 � � � Pj

266664

377775

xðkÞxðkþ 1Þ

..

.

xðkþ TiÞ

266664

377775:

Note that, under the controller (17), the subsystem dynamics for Ti consecutive samples should be

Ai1 �I 0 � � � 0 00 Ai2 �I � � � 0 0

..

. ... ..

.� � � ..

. ...

0 0 0 � � � AiTi�I

2666664

3777775

xðkÞxðkþ 1Þ

..

.

xðkþ TiÞ

266664

377775 ¼ 0;

where Air ¼ Ai þ BiGir; r ¼ 1; . . . ; Ti.To obtain LMI conditions and to be able to recover classical conditions, one can choose

Mi ¼

0 0 0 � � � 0 0MT

i1 0 0 � � � 0 0

0 MTi2 0 � � � 0 0

..

. ... ..

. ... ..

.

0 0 0 � � � MTiðTi�1Þ 0

0 0 0 � � � 0 MTiTi

266666666664

377777777775;

where Mis ¼ M�1is ; s ¼ 1; . . . ; Ti.

After employing Lemma 2, congruence with matrix diagðI M�1i1 � � �M

�1iTiÞ and applying Lemma 1, we have from (18)

Vjðkþ TiÞ 6 aiV iðkÞ: ð20Þ

8k P k0, there exists a constant l 2 ZP0 such that k 2 ½kl; klþ1Þ. Considering (13), we can obtain the inequality (10).

R. Wang, S. Fei / Applied Mathematics and Computation 238 (2014) 358–369 365

Besides, 8k 2 ðkl; klþ1Þ, there exists a positive integer 1 6 j 6 Ti � 1 such that k ¼ kl þ j.(19) yields

VrðklÞðkl þ jÞ 6 ajrðklÞ

VrðklÞðklÞ: ð21Þ

Combining (20) with (21), we get (16). This completes the proof of Theorem 3. h

If the switched system is with many subsystems, the number of the LMI conditions (18) will become very big. For easyoperation, we will give the following theorem.

Theorem 4. Consider system (1) and let ai; ~ai; bi with ~aibi 6 ai; 8i 2 N be given positive constants. Suppose that there exist

positive definite matrices Pi, invertible matrices Mi1; . . . ;MiTi, and matrices Gi1; . . . ;GiTi

; 8i 2 N, such that

�~aiPi � 0 � � � 0 0

ð2;1Þ H1 � . ..

0 P1 H2. .

. ... ..

.

P2. .

.� 0 0

..

. . .. . .

.HTi�1 � 0

0 � � � 0 PTi�1 HTi�

0 � � � 0 0 MiTi�2I þ Pið Þ

26666666666666664

37777777777777775

< 0; ð22Þ

Pi 6 bjPj; 8i; j 2 N: ð23Þ

If the inequalities (19) hold for controller gain matrices Gi1 ¼ Gi1; Gi2 ¼ Gi2M�1i1 ; . . . ;GiTi

¼ GiTiM�1

iðTi�1Þ and there exist constants0 < a < 1 and M > 0 such that the inequality (13) is satisfied, then the closed-loop system is exponentially stable, and the decay

rate is ðmaxi2Na1

T1ai Þ�1

2

, where ð2;1Þ;Hs;Pr are the same as defined in Theorem 3.

Proof. 8ðrðkmÞ ¼ i;rðkmþ1Þ ¼ jÞ 2 N � N, from (22) and (23), we can obtain Vjðkmþ1Þ 6 biV iðkmþ1Þ 6 bi ~aiV iðkmÞ. Then, throughthe similar proof procedure to Theorem 3, we can prove the theorem. h

Remark 10. From the above proofs, it is indicated that the LMI conditions (18) and (22) imply ð�ÞT PjAiTiAiðTi�1Þ � � �

Ai1 � aiPi 6 0 and ð�ÞT PiAiTiAiðTi�1Þ � � �Ai1 � ~aiPi 6 0 respectively. Even though we can get Gi1 ¼ Gi2 ¼ � � � ¼ GiTi

by solving LMIs(18) or (22), they are substantially different from general mode-dependent controllers. Because the general mode-dependentcontrollers are designed based on the one-sample variation, i.e., the general mode-dependent controller gains Ki are

expected to meet ðAi þ BiKiÞT PiðAi þ BiKiÞ � Pi < 0 in the past literature such that there holds Viðkþ 1Þ < ViðkÞ. We will illus-trate this point in the numerical example.

4. Numerical example

In this section, we will provide an example to show the effectiveness of the main results in the paper.

Example 1. Consider the discrete-time switched system (1) with parameters given below:

A1 ¼�1 2 12:5 �1 1:52 1 �1

264

375; A2 ¼

2 1 26:5 �2 11 0 1

264

375; A3 ¼

1 �1 01 1 0:50 1 1

264

375;

B1 ¼�1 �31 2:50 1

264

375; B2 ¼

�0:15 01 0:051 0:5

264

375; B3 ¼

�1 �0:10 11 2

264

375:

Based on Theorem 4, the mode-dependent stabilizing controllers as shown in (17) are designed to guarantee that theresulting closed-loop system is exponentially stable. Assigning b1 ¼ 1:12; b2 ¼ 1:1; b3 ¼ 0:91; ~a1 ¼ 0:8; ~a2 ¼ 1:5;~a3 ¼ 0:55; a1 ¼ 0:896; a2 ¼ 1:65; a3 ¼ 0:5005; M ¼ 1; a ¼ 0:905, we can obtain

0 5 10 15 20 25 30 35

1

1.5

2

2.5

3

k

σ(k)

Fig. 1. Switching signal rðkÞ.

Table 1Eigenvalues of Closed-loop Subsystems.

A1 þ B1G11 ½0:6076; �0:0055; �0:1355�T

A1 þ B1G12 ½0:1984; �0:0653; �0:3300�T

A1 þ B1G13 ½2:1905; �0:0018; �0:0383�T

Ac11 ½81:6110; �1:6184; �1:5938�T

Ac12 ½111:6193; �1:6131; �1:3569�T

Ac13 ½7:42; �1:6254; �1:6082�T

A2 þ B2G21 ½132:8þ 56:8i; 132:8� 56:8i; �120�T � 10�3

A2 þ B2G22 ½3:5þ 44:7i; 3:5� 44:7i; �441:7�T � 10�3

A2 þ B2G23 ½3049:5; �13:7; 10:5�T � 10�3

Ac21 ½77:226; �1:2567; �1:6295�T

Ac22 ½106:7765; �1:2231; �1:6299�T

Ac23 ½21:3849; �1:6392; �1:6235�T

A3 þ B3G31 ½0:4085; 0:0594; 0:0085�T

A3 þ B3G32 ½13:4þ 93i; 13:4� 93i; 302:1�T � 10�3

A3 þ B3G33 ½0:4930; 0:0122; 0:0016�T

Ac31 ½�1:809; �1:7889; �0:3502�T

Ac32 ½0:2326; �1:4932; �1:8039�T

Ac33 ½�0:3873; �1:7906; �1:8091�T

⁄Acir , ðAi þ BiGirÞT PiðAi þ BiGirÞ � Pi; i; r ¼ 1;2;3.

366 R. Wang, S. Fei / Applied Mathematics and Computation 238 (2014) 358–369

P1 ¼1:6201 0:0044 �0:00060:0044 1:6137 0:0078�0:0006 0:0078 1:6299

264

375; P2 ¼

1:6342 0:0045 �0:00070:0045 1:6278 0:0079�0:0007 0:0079 1:6441

264

375;

P3 ¼1:7968 0:0050 �0:00070:0050 1:7897 0:0087�0:0007 0:0087 1:8077

264

375; G11 ¼

7:7227 6:5721 0:4111�2:2224 �1:1058 0:7144

� �;

G12 ¼9:6646 7:7278 2:3403�2:6938 �1:3798 0:1079

� �; G13 ¼

2:4079 3:4168 �3:9699�1:5069 �0:6950 1:3583

� �;

G21 ¼�2:6857 2:6280 1:4428�4:4940 �6:6262 �9:9936

� �; G22 ¼

�1:6393 2:8565 2:2057�7:8607 �7:6172 �12:7239

� �;

G23 ¼�6:5663 2:3044 �0:661211:0115 �4:5936 �0:7044

� �; G31 ¼

1:1985 �0:6238 0:0395�0:7100 �0:4140 �0:5108

� �;

0 5 10 15 20 25 30 35

−2

0

2

4

6

k

x

1. Evolutions of state variables x1,x2,x3

x1x2x3

−3−2

−10

1−202468

−0.5

0

0.5

1

1.5

x2

2. Trajectory of state x

x1

x 3

Fig. 2. State x.

0 5 10 15 20 25 30 350

5

10

15

20

25

k

||x(k)||s(k)

Fig. 3. kxk and sðkÞ.

R. Wang, S. Fei / Applied Mathematics and Computation 238 (2014) 358–369 367

G32 ¼1:2661 �0:1504 0:0847�0:9149 �0:6287 �0:4762

� �; G33 ¼

1:2216 �0:6281 0:0419�0:6890 �0:3506 �0:5160

� �:

If let N1i ¼ 1; T1ai ¼ 3; 8i 2 N, we can see that (3) and (13) are satisfied via the switching signal pictured in Fig. 1. The relevant

eigenvalues of the closed-loop system under the controller (17) are listed in Table 1. We have pinned down in italics theeigenvalues of Ai þ BiGir whose amplitudes are larger than 1, and the positive eigenvalues of Ac

ir . Set the initial value of

the closed-loop system xð0Þ ¼ ½�0:5 1:5 1:3�T . Fig. 2-1 and 2-2 and show the time histories of the state variables x1; x2; x3

and the trajectory of the state x in the three-dimensional space respectively. Let sðkÞ , k1k2

maxi2NaTi�1i Ma�1QN

i¼1a�N1i

� �12

maxi2Na1

T1ai

� �k2

kxð0Þk. It is easily seen from Fig. 3 that kxðkÞk 6 sðkÞ, which implies the resulting closed-loop system is expo-

nentially stable. Fig. 4 illustrates the evolution of the Lyapunov-like function VrðkÞ on different time intervals.In Table 1, the eigenvalues of Ai þ BiGir in italics suggest that the corresponding system matrices aren’t stable, while the

eigenvalues of Acir in italics insinuate that the corresponding matrices are not negative definite. It is remarkable that even if a

6 7 8 90

0.5

1

1.5

2x 10−5

kV σ

a

v

9 10 11 120

0.2

0.4

0.6

0.8

1x 10−9

k

V σ

b

v

15 16 17 180

0.2

0.4

0.6

0.8

1x 10−16

k

V σ

c

v

0 5 10 15 20 25 30 350

20

40

60

80

k

V σ d

k

v

Fig. 4. Lyapunov-like function Vr .

368 R. Wang, S. Fei / Applied Mathematics and Computation 238 (2014) 358–369

matrix Ai þ BiGir is stable, for example, A1 þ B1G11; Viðkþ 1Þ may be still larger than ViðkÞ as is demonstrated in Fig. 4-b,d.The reason lies in that P1 is not the positive definite matrix P expected to satisfy ðA1 þ B1G11ÞT PðA1 þ B1G11Þ � P < 0. Besides,from Fig. 4-a,c, we can see V2ð7Þ > 1:5V2ð6Þ and V2ð16Þ > 1:5V2ð15Þ, which further confirm Remark 10.

5. Conclusions

In this paper, we have investigated the stabilizing control problem for a class of discrete-time switched linear systems viathe multiple-sample Lyapunov-like functions variation. The new mode-dependent controllers are designed to realize the sta-bility conditions in the form of multiple-sample Lyapunov-like functions variation. As a result, the LMI conditions for expo-nential stabilization can be obtained with the aid of the new concept–forward ADT. A numerical example has been providedto demonstrate the effectiveness of the main results.

Acknowledgement

This work is supported by National Natural Science Foundation (NNSF) of China under Grant 61273119.

References

[1] M.S. Branicky, Multiple Lyapunov functions and other analysis tools for switched and hybrid systems, IEEE Trans. Automat. Control 43 (4)(1998) 475–782.

[2] J. Daffouz, P. Riedinger, C. Iung, Stability analysis and control synthesis for switched systems: a switched Lyapunov function approach, IEEE Trans.Automat. Control 47 (11) (2002) 1883–1887.

[3] J.P. Hespanha, A.S. Morse, Stability of switched systems with average dwell time, in: Proc. 38th Conf. Decision Control, Phoenix, AZ, 1999,pp. 2655–2660.

[4] D. Liberzon, Switching in Systems and Control, Birkhauser, Berlin, 2003.[5] D. Liberzon, J.P. Hespanha, A.S. Morse, Stability of switched systems: a Lie-algebraic condition, Syst. Control Lett. 37 (3) (1999) 117–122.[6] H. Lin, P.J. Antsaklis, Stability and stabilizability of switched linear systems: a survey of recent results, IEEE Trans. Automat. Control 54 (2)

(2009) 308–322.[7] P. Peleties, R.A. DeCarlo, Asymptotic stability of m-switched systems using Lyapunov-like functions, in: Proc. 1991 American Control Conf., Boston, MA,

1991, pp. 1679–1684.[8] M.C. Oliveira, J. Bernussou, J.C. Geromel, A new discrete-time robust stability condition, Syst. Control Lett. 37 (4) (1999) 261–265.[9] A. Trofino, D. Assmann, C.C. Scharlau, D.F. Coutinho, Switching rule design for switched dynamic systems with affine vector fields, IEEE Trans. Automat.

Control 54 (9) (2009) 2215–2222.[10] G.S. Zhai, B. Hu, K. Yasuda, A.N. Michel, Stability analysis of switched systems with stable and unstable subsystems: an average dwell time approach,

in: Proc. 2000 American Control Conf., Chicago, IL, 2000, pp. 200–204.[11] G.S. Zhai, B. Hu, K. Yasuda, A.N. Michel, Qualitative analysis of discrete-time switched systems, in: Proc. 2002 American Control Conf., Anchorage, AK,

2002, pp. 1880–1885.[12] L.X. Zhang, H.J. Gao, Asynchronously switched control of switched linear system swith average dwell time, Automatica 46 (5) (2010) 953–958.[13] W.A. Zhang, L. Yu, Stability analysis for discrete-time switched time-delay systems, Automatica 45 (10) (2009) 2265–2271.[14] L.X. Zhang, P. Shi, l2–l1 model reduction for switched LPV systems with average dwell time, IEEE Trans. Automat. Control 53 (10) (2008) 2443–2448.

R. Wang, S. Fei / Applied Mathematics and Computation 238 (2014) 358–369 369

[15] X.D. Zhao, P. Shi, L.X. Zhang, Asynchronous switched control of a class of slowly switched linear systems, Syst. Control Lett. 61 (12) (2012) 1151–1156.[16] X.D. Zhao, L.X. Zhang, P. Shi, M. Liu, Stability and stabilization of switched linear systems with mode-dependent average dwell time, IEEE Trans.

Automat. Control 57 (7) (2012) 1809–1815.[17] A. Kruszewski, R. Wang, T.M. Guerra, Nonquadratic stabilization conditions for a class of uncertain nonlinear discrete time TS fuzzy models: a new

approach, IEEE Trans. Automat. Control 53 (2) (2008) 606–611.[18] S.F. Derakhshan, A. Fatehi, Non-monotonic Lyapunov functions for stability analysis and stabilization of discrete time Takagi–Sugeno fuzzy systems,

Int. J. Innov. Comput. I. (2012).[19] Zs. Lendek, T.M. Guerra, J. Lauber, Construction of extended Lyapunov functions and control laws for discrete-time TS systems, in: WCCI 2012 IEEE

World Congress on Computational Intelligence, Brisbane, Australia, 2012, pp. 1–6.[20] A.A. Ahmadi, P.A. Parrilo, Non-monotonic Lyapunov functions for stability of discrete time nonlinear and switched Systems, in: Proc. 47th IEEE CDC,

Cancun, Mexico, 2008, pp. 614–621.[21] J.C. Geromel, P. Colaneri, Stability and stabilization of discrete time switched systems, Int. J. Control 79 (7) (2006) 719–728.