Robust L∞ reliable control for uncertain switched nonlinear systems with time delay under asynchronous switching

Applied Mathematics and Computation 222 (2013) 658–670

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Applied Mathematics and Computation

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Robust L1 reliable control for uncertain switched nonlinearsystems with time delay under asynchronous switching

0096-3003/$ - see front matter � 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.amc.2013.07.084

⇑ Corresponding author.E-mail address: [email protected] (Z. Xiang).

Shipei Huang, Zhengrong Xiang ⇑School of Automation, Nanjing University of Science and Technology, Nanjing 210094, People’s Republic of China

a r t i c l e i n f o

Keywords:Time delayL1 reliable controlSwitched nonlinear systemsAsynchronous switching

a b s t r a c t

This paper investigates the problem of robust L1 reliable control for uncertain switchednonlinear systems with time delay and actuator failures, where the switching instants ofthe controller experience delays with respect to those of the system. A state feedback con-troller is proposed to guarantee the exponential stability with L1 performance of theresulting closed-loop system. The dwell time approach is utilized for the stability analysisand controller design. Finally, a numerical example is given to illustrate the effectiveness ofthe proposed method.

� 2013 Elsevier Inc. All rights reserved.

1. Introduction

Switched systems are a class of important hybrid systems consisting of subsystems and a switching law which defines aspecific subsystem being activated during a certain interval of time [1]. Due to the theoretical development as well as prac-tical applications, analysis and synthesis of switched systems have gained considerable attention during the last decades,and many results on stability analysis and control synthesis for switched systems have been developed (see [2–11]). It is wellknown that time delay phenomenon is very common in many kinds of engineering systems and often causes undesirableperformance, so many scholars have devoted their energies to the study of time delay systems and many valuable resultson switched systems with time delay have been obtained (see [12–21]).

On the other hand, the actuators may be subjected to failures in actual operation. When controlling a real plant with fail-ures of control components, classical control methods may not achieve satisfactory performance. To overcome this problem,reliable control which is a kind of effective control approach to improve system reliability was proposed. Recently, severalapproaches to the design of reliable controllers have been developed for linear systems and nonlinear systems (see [22–24]),and some of them have been extended to investigate the problem of reliable control for switched systems (see [25–29]). Forinstance, an H1 reliable controller was designed for switched nonlinear systems in [25]. Xiang and Wang [27] proposed anL1 reliable controller for switched nonlinear systems with time delay.

It is worth pointing out that the aforementioned results are based on an assumption that the switching instants of thecontrollers coincide with those of system modes. However, as pointed out in [30], it inevitably takes some time to identifythe system modes and apply the matched controller, the switching instants of the controller always lag behind those of thesystem. Thus, it is necessary to consider asynchronous switching for efficient control design. Some results on switched sys-tems under asynchronous switching have been proposed in [31–34]. To the best of our knowledge, L1 reliable control forswitched nonlinear systems under asynchronous switching has not been fully investigated, which constitutes the main moti-vation of the present study.

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S. Huang, Z. Xiang / Applied Mathematics and Computation 222 (2013) 658–670 659

In this paper, we deal with the problem of robust L1 reliable control for a class of uncertain switched nonlinear systemshaving the following properties: (1) the uncertainties are considered in the input matrix, which is different from most of theexisting results that consider uncertainties in the system matrix only, e.g. [27,28]; (2) each model has a sector-bounded non-linearity; (3) each model may have the actuator faults; and (4) both time varying delay and asynchronous switching exist.The main contributions of this paper are as follows: (1) the proposed method dealing with a class of uncertain matrices mul-tiplication phenomena in [29] is introduced for stability analysis; (2) a sufficient condition for the existence of a switchedstate feedback controller that achieves robust L1 reliable control goal is provided in terms of linear matrix inequalities(LMIs).

The remainder of the paper is organized as follows. In Section 2, problem formulation and some necessary lemmas aregiven. In Section 3, the main results are developed. A numerical example is given to illustrate the effectiveness of the pro-posed approach in Section 4. Concluding remarks are given in Section 5.

Notation: Throughout this paper, AT denotes the transpose of matrix A, L1 denotes the space of functions with boundedamplitude. k � k and k � k1 denote the Euclidean norm and1-norm, kmaxðPÞ and kminðPÞ denote the maximum and minimumeigenvalues of matrix P, respectively. I is the identity matrix. The notation X P Y (X > Y) means that X � Y is positive semi-definite (positive definite). diagfaig denotes a diagonal matrix with the diagonal elements ai; i ¼ 1;2; . . . ;n. The asterisk � in amatrix is used to denote a term that is induced by symmetry.

2. Problem formulation and preliminaries

Consider the following switched nonlinear systems with time delay and actuator failures:

_xðtÞ ¼ ArðtÞxðtÞ þ AdrðtÞxðt � dðtÞÞ þ BrðtÞuf ðtÞ þ DrðtÞfrðtÞðxðtÞÞ þ GrðtÞwðtÞ ð1aÞ

xðt0 þ hÞ ¼ uðhÞ; h 2 ½�s;0� ð1bÞ

where xðtÞ 2 Rn is the state vector, uf ðtÞ 2 Rl is the control input of actuator fault. dðtÞ denotes the time delay which is every-where time-differentiable and satisfies 0 < dðtÞ 6 s and _dðtÞ 6 g < 1 for known constants s and g. uðhÞ is a continuous vec-tor-valued initial function. wðtÞ 2 Rq is the disturbance input which belongs to L1. t0 ¼ 0 is the initial time, and tk denotes thekth switching instant. rðtÞ : ½t0;1Þ ! N ¼ f1;2; . . . Ng is the switching signal, and N denotes the number of subsystems. Di

and Gi ði 2 NÞ are known constant matrices. Ai, Bi and Adi ði 2 NÞ are uncertain real-valued matrices with appropriate dimen-sions and have the following forms:

Ai ¼ Ai þ HiFiðtÞE1i; Bi ¼ Bi þ HiFiðtÞE2i; Adi ¼ Adi þ HiFiðtÞEdi ð2Þ

where Ai;Bi;Adi;Hi; E1i; E2i and Edi are known real constant matrices with appropriate dimensions, and FiðtÞ is an unknowntime-varying matrix satisfying

FTi ðtÞFiðtÞ 6 I ð3Þ

For each ði 2 NÞ, fiðxðtÞÞ with fið0Þ ¼ 0 is a nonlinear function satisfying the following sector-bounded condition

½fiðxðtÞÞ �Pi1xðtÞ�T ½fiðxðtÞÞ �Pi2xðtÞ� 6 0 ð4Þ

where Pi1 and Pi2 are known real constant matrices with Pi2 �Pi1 P 0.

Remark 1. The nonlinear function fiðxðtÞÞ satisfying (4) is said to be belong to the sector ½Pi1;Pi2�. It should be pointed outthat such a nonlinear condition is more general than the usual Lipschitz conditions that have been widely used in [27,28].

The nonlinear function fiðxðtÞÞ can be rewritten as

fiðxðtÞÞ ¼ Pi1xðtÞ þ giðxðtÞÞ

where giðxðtÞÞ satisfies

giðxðtÞÞT ½giðxðtÞÞ �Pi3xðtÞ� 6 0; Pi3 ¼ Pi2 �Pi1 ð5Þ

The control input of actuator fault uf ðtÞ can be described as

uf ðtÞ ¼ MrðtÞuðtÞ ð6Þ

where uðtÞ ¼ KrðtÞxðtÞ is the control input to be designed, and Mi ði 2 NÞ is the actuator fault matrix of the form

Mi ¼ diagfmi1;mi2; . . . ;milg; 0 6 mij 6 mij 6 mij; �mij 6 1; j ¼ 1;2; . . . l

For simplicity, we introduce the following notations

Mi ¼ diagf �mi1; �mi2; . . . ; �milg ð7Þ

Mi ¼ diagfmi1;mi2; . . . ;milg ð8Þ

660 S. Huang, Z. Xiang / Applied Mathematics and Computation 222 (2013) 658–670

Mi0 ¼12ðMi þMiÞ ð9Þ

Mi1 ¼12ðMi �MiÞ ð10Þ

From (6)–(10), we have

Mi ¼ Mi0 þMi1Li ð11Þ

where Li ¼ diagfvi1;vi2; . . . ;vilg;�1 6 vij 6 1;j ¼ 1;2; . . . ; l.

Remark 2. mij ¼ 1 means the jth actuator of the ith subsystem is nominal. mij ¼ 0 denotes the jth actuator of the ithsubsystem is outage. When mij > 0 and mij–1, it corresponds to the case of partial failure of the jth actuator of the ithsubsystem.

The switching instances of system (1) can be described as

R : fðt0;rðt0ÞÞ; ðt1;rðt1ÞÞ; . . . ; ðtk;rðtkÞÞ; . . .g

However, in actual operation, it needs to take some time to identify the system modes and apply the matched con-troller. The switching instants of the controller often experience delays with respect to those of the system. For con-venience, let r0ðtÞ denote the switching signal of the controller. Then the switching instances of the controller can bewritten as

R0 : fðt0;r0ðt0ÞÞ; ðt1 þ D1;r0ðt1 þ D1ÞÞ; . . . ; ðtk þ Dk;r0ðtk þ DkÞÞ; . . .g;

where 0 < Dk < infkP1ðtkþ1 � tkÞ.In practice, the interval rests with the identification and scheduling process among all the candidates of stabilizing con-

trollers, which may be different in different environments. Here we assume that the maximal delay of asynchronous switch-ing, Tmax, is known a priori without loss of generality.

The real control input will become uðtÞ ¼ Kr0 ðtÞxðtÞ, and the corresponding closed-loop system can be written as

_xðtÞ ¼ ðArðtÞ þ BrðtÞðMrðtÞ0 þMrðtÞ1LrðtÞÞKr0 ðtÞÞxðtÞ þ AdrðtÞxðt � dðtÞÞ þ DrðtÞfrðtÞðxðtÞÞ þ GrðtÞwðtÞ ð12Þ

When the ith subsystem is activated at the switching instant tk�1, and the jth subsystem is activated at the switching in-stant tk, due to the existence of asynchronous switching, the corresponding switches of the controller occur at the instantstk�1 þ Dk�1 and tk þ Dk, respectively.

Definition 1 [17]. System (12) with wðtÞ � 0 is said to be exponentially stable under the switching signal rðtÞ if the solutionxðtÞ of system (12) satisfies

kxðtÞk 6 fkxðt0Þkce�kðt�t0Þ; 8t P t0

for constants f > 0 and k > 0, where kxðt0Þkc ¼ sup�s6h60

kxðt0 þ hÞk.

Definition 2 [1]. For any T2 > T1 P t0, let NrðT1; T2Þ denote the switching number of rðtÞ on the interval ½T1; T2Þ. If

NrðT1; T2Þ 6 N0 þT2 � T1

Tað13Þ

holds for given N0 P 0 and Ta > 0, then the constant Ta is called the average dwell time. As commonly used in the literature,we choose N0 ¼ 0 in this paper.

Definition 3 [27]. Let c be a positive constant. For system (1) with uðtÞ � 0, under zero initial conditions, i.e.,xðtÞ ¼ 0; t 2 ½�s;0�, if the following inequality holds:

kxðtÞk1 6 ckwðtÞk1; 8wðtÞ 2 L1 ð14Þ

then the system is said to have an L1 performance level c.For system (1), if there exists a controller uðtÞ ¼ KrðtÞxðtÞ such that the closed-loop system (12) satisfies (14), then

uðtÞ ¼ KrðtÞxðtÞ is said to be a robust L1 reliable controller.The following lemmas will be essential for our later development.

Lemma 1 [35]. Let U;V ;W and X be real matrices of appropriate dimensions with X satisfying X ¼ XT , then for all VT V 6 I,

X þ UVW þWT VT UT < 0

if and only if there exists a scalar e > 0 such that

X þ eUUT þ e�1WT W < 0


Lemma 2 [29]. Let E;G;X and Y be matrices of appropriate dimensions, F be an uncertain matrix such that FT F 6 I, and L be adiagonal uncertain matrix satisfying LT L 6 I. Then, there exist a positive definite diagonal matrix U and a positive scalar d such that

dI � GUGT > 0

and

EFX þ XT FT ET þ EFGLY þ YT LT GT FT ET < YT U�1Y þ XTðdI � GUGTÞ�1X þ dEET

Remark 3. In the closed-loop system (12), the uncertain matrices Fi and Li are multiplied together. Lemma 2 will be used todeal with the phenomenon. As we all know, the uncertainties of a system universally exist, and the faults appear at sometime. Thus, it is necessary to introduce the proposed result in Lemma 2.

The objective of this paper is to design a robust L1 reliable controller for system (1) with actuator failures under asyn-chronous switching.

3. Main results

3.1. Stability analysis

To obtain the main results, we first consider the stability of system (12) with wðtÞ ¼ 0, and the result is presented in thefollowing theorem.

Theorem 1. Consider system (12) with wðtÞ ¼ 0, for given positive scalars a and b, if there exist positive scalars e1i; e1ij; e2i; e2ij; di

and dij, positive definite matrices Xi and Qi, and positive definite diagonal matrices Ui and Uij with appropriate dimensions, suchthat, 8i; j 2 N, i–j,

~Hi11 AdiXi~Hi13 e1iHi ðE1iXiÞT e2iBiMi1 ðKiXiÞT ðKiXiÞT ~Hi19

� �~ge�asQ i 0 0 ðEdiXiÞT 0 0 0 0� � �I 0 0 0 0 0 0� � � �e1iI 0 0 0 0 0� � � � �e1iI 0 0 0 0� � � � � �e2iI 0 0 0� � � � � � �e2iI 0 0� � � � � � � �Ui 0� � � � � � � � ~Hi99

266666666666666664

377777777777777775

< 0; ð15Þ

~Hij11 AdjXi~Hij13 e1ijHj ðE1jXiÞT e2ijBjMj1 ðKiXiÞT ðKiXiÞT ~Hij19

� �~gQ i 0 0 ðEdjXiÞT 0 0 0 0� � �I 0 0 0 0 0 0� � � �e1ijI 0 0 0 0 0� � � � �e1ijI 0 0 0 0� � � � � �e2ijI 0 0 0� � � � � � �e2ijI 0 0� � � � � � � �Uij 0

� � � � � � � � ~Hij99

2666666666666666664

3777777777777777775

< 0; ð16Þ

where

~Hi11 ¼ AiXi þ XiATi þ BiMi0KiXi þ ðBiMi0KiXiÞT þ DiPi1Xi þ XiP

Ti1DT

i þ Q i þ aXi þ diHiHTi

~Hi13 ¼ Di þ ðXiPTi3Þ=2; ~Hij13 ¼ Dj þ ðXiP

Tj3Þ=2; ~g ¼ 1� g;

~Hi19 ¼ ðE2iMi0KiXiÞT ; ~Hi99 ¼ �ðdiI � E2iMi1UiðE2iMi1ÞTÞ;

~Hij19 ¼ ðE2jMj0KiXiÞT ; ~Hij99 ¼ �ðdijI � E2jMj1UijðE2jMj1ÞTÞ

~Hij11 ¼ AjXi þ XiATj þ BjMj0KiXi þ ðBjMj0KiXiÞT þ DjPj1Xi þ XiP

Tj1DT

j þ Q i � bXi þ dijHjHTj ;


then under the following average dwell time scheme

Ta > T�a ¼ðaþ bÞTmax þ lnðl1l2Þ

að17Þ

where l2 ¼ eðaþbÞs and l1 satisfies

X�1j 6 l1X�1

i ;X�1j Q jX

�1j 6 l1X�1

i Q iX�1i ; 8i; j 2 N; i–j ð18Þ

system (12) with wðtÞ ¼ 0 is exponentially stable.

Proof. When t 2 ½tk�1 þ Dk�1; tkÞ, system (12) can be written as

_xðtÞ ¼ ðAi þ BiðMi0 þMi1LiÞKiÞxðtÞ þ Adixðt � dðtÞÞ þ DifiðxðtÞÞ þ GiwðtÞ ð19Þ

We consider a Lyapunov functional for system (19) as follows

ViðtÞ ¼ xTðtÞPixðtÞ þZ t

t�dðtÞe�aðt�sÞxTðsÞQ ixðsÞds ð20Þ

Along the trajectory of system (19), we have

_ViðtÞ 6 2xTðtÞPiððAi þ BiðMi0 þMi1LiÞKiÞxðtÞ þ Adixðt � dðtÞÞ þ DifiðxðtÞÞ þ GiwðtÞÞ þ xTðtÞQ ixðtÞ

� aZ t

t�dðtÞe�aðt�sÞxTðsÞQ ixðsÞds� e�asð1� gÞxTðt � dðtÞÞQ ixðt � dðtÞÞ:

When wðtÞ ¼ 0, it can be obtained from (2) and (5) that

_ViðtÞ 6 2xTðtÞPiðAi þ HiFiðtÞE1i þ ðBi þ HiFiðtÞE2iÞðMi0 þMi1LiÞKi þ DiPi1ÞxðtÞ þ 2xTðtÞPiðAdi þ HiFiðtÞEdiÞxðt � dðtÞÞ

þ 2xTðtÞPiDigiðxðtÞÞ þ xTðtÞQ ixðtÞ � aZ t

t�dðtÞe�aðt�sÞxTðsÞQ ixðsÞds� e�asð1� gÞxTðt � dðtÞÞQ ixðt � dðtÞÞ

� giðxðtÞÞT ½giðxðtÞÞ �Pi3xðtÞ�

¼ nTðtÞHinðtÞ � aViðtÞ

where

nTðtÞ ¼ xTðtÞ xTðt � dðtÞÞ gTi ðxðtÞÞ

� �

Hi ¼Hi11 PiðAdi þ HiFiðtÞEdiÞ PiDi þPT

i3=2� �ð1� gÞe�asQ i 0� � �I

264

375;

Hi11 ¼ PiAi þ PiHiFiðtÞE1i þ PiðBi þ HiFiðtÞE2iÞðMi0 þMi1LiÞKi þ PiDiPi1 þ Q i þ aPi þ ATi Pi þ ðPiHiFiðtÞE1iÞT

þ ðPiðBi þ HiFiðtÞE2iÞðMi0 þMi1LiÞKiÞT þ ðPiDiPi1ÞT :

Denote Hi ¼ Hi1 þHi2 þHi3, where

Hi1 ¼Hi11 PiAdi PiDi þPT

i3=2� �ð1� gÞe�asQ i 0� � �I

264

375;

Hi11 ¼ PiAi þ ATi Pi þ PiBiMi0Ki þ ðPiBiMi0KiÞT þ PiDiPi1þiP

Ti1DT

i Pi þ Qi þ aPi þ PiHiFiðtÞE2iMi0Ki þ ðPiHiFiðtÞE2iMi0KiÞT

þ PiHiFiðtÞE2iMi1LiKi þ ðPiHiFiðtÞE2iMi1LiKiÞT ;

Hi2 ¼PiHi

00

264

375FiðtÞ E1i Edi 0½ � þ E1i Edi 0½ �T FiðtÞT

PiHi

00

264

375

T

Hi3 ¼PiBiMi1

00

264

375Li Ki 0 0½ � þ Ki 0 0½ �T LT

i

PiBiMi1

00

264

375

T


According to Lemma 1 and Schur complement lemma, one obtains that Hi < 0 is equivalent to

Hi11 PiAdi Hi13 e1iPiHi ET1i e2iPiBiMi1 KT

i

� �ð1� gÞe�asQ i 0 0 ETdi 0 0

� � �I 0 0 0 0� � � �e1iI 0 0 0� � � � �e1iI 0 0� � � � � �e2iI 0� � � � � � �e2iI

2666666666664

3777777777775< 0; ð21Þ

where Hi13 ¼ PiDi þPTi3=2.

Denote Xi ¼ P�1i and Q i ¼ XiQiXi. Using diagfXi;Xi; I; I; I; I; Ig to pre- and post- multiply the left term of (21), one obtains

that

H_

i11 AdiXi H_

i13 e1iHi ðE1iXiÞT e2iBiMi1 ðKiXiÞT

� �ð1� gÞe�asQ i 0 0 ðEdiXiÞT 0 0� � �I 0 0 0 0� � � �e1iI 0 0 0� � � � �e1iI 0 0� � � � � �e2iI 0� � � � � � �e2iI

26666666666664

37777777777775< 0; ð22Þ

where

H_

i11 ¼ AiXi þ XiATi þ BiMi0KiXi þ ðBiMi0KiXiÞT þ DiPi1Xi þ XiP

Ti1DT

i þ Q i þ aXi þ HiFiðtÞE2iMi0KiXi þ ðHiFiðtÞE2iMi0KiXiÞT

þ HiFiðtÞE2iMi1LiKiXi þ ðHiFiðtÞE2iMi1LiKiXiÞT ;

H_

i13 ¼ Di þ ðXiPTi3Þ=2

By Lemma 2, there exist positive definite diagonal matrices Ui and positive scalars di such that

diI � E2iMi1UiðE2iMi1ÞT > 0 ð23Þ

HiFiðtÞE2iMi0KiXi þ ðHiFiðtÞE2iMi0KiXiÞT þ HiFiðtÞE2iMi1LiKiXi þ ðHiFiðtÞE2iMi1LiKiXiÞT

< ðKiXiÞT U�1i KiXi þ diHiH

Ti þ ðE2iMi0KiXiÞTðdiI � E2iMi1UiðE2iMi1ÞTÞ

�1E2iMi0KiXi ð24Þ

Thus (21) holds if (15) is satisfied. Then from (15), the following inequality can be obtained

_Vr0ðtÞðtÞ < �aVr0 ðtÞðtÞ; t 2 ½tk�1 þ Dk�1; tkÞ ð25Þ

It follows that

Vr0ðtk�1þDk�1ÞðtÞ < e�aðt�tk�1�Dk�1ÞVr0 ðtk�1þDk�1Þðtk�1 þ Dk�1Þ; t 2 ½tk�1 þ Dk�1; tkÞ

When t 2 ½tk; tk þ DkÞ, system (12) can be written as

_xðtÞ ¼ ðAj þ BjðMj0 þMj1LjÞKiÞxðtÞ þ Adjxðt � dðtÞÞ þ DjfjðxðtÞÞ þ GjwðtÞ ð26Þ

Consider a Lyapunov functional for system (26) as follows

ViðtÞ ¼ xTðtÞPixðtÞ þZ t

t�dðtÞebðt�sÞxTðsÞQ ixðsÞds ð27Þ

Similarly, we can obtain from (16) that

_Vr0ðtkÞðtÞ < bVr0ðtkÞðtÞ; t 2 ½tk; tk þ DkÞ

It follows that

Vr0ðtkÞðtÞ < ebðt�tkÞVr0 ðtkÞðtkÞ; t 2 ½tk; tk þ DkÞ ð28Þ

From (18), (20), and (27), we have

Vr0ðtkÞðtkÞ 6 l2Vr0ðt�kÞðt�k Þ; l2 ¼ eðaþbÞs

Vr0ðtk�1þDk�1Þðtk�1 þ Dk�1Þ 6 l1Vr0 ððtk�1þDk�1Þ�Þððtk�1 þ Dk�1Þ�Þ


For t 2 ½tk�1 þ Dk�1; tkÞ, one obtains

Vr0ðtÞðtÞ < e�aðt�tk�1�Dk�1ÞVr0ðtk�1þDk�1Þðtk�1 þ Dk�1Þ< l1e�aðt�tk�1�Dk�1ÞebDk�1 Vr0 ðtk�1Þðtk�1Þ< l1l2e�aðt�tk�1�Dk�1þtk�1�tk�2�Dk�2ÞebDk�1 Vr0ðtk�2þDk�2Þðtk�2 þ Dk�2Þ< l1l1l2e�aðt�tk�1�Dk�1þtk�1�tk�2�Dk�2ÞebðDk�1þDk�2ÞVr0 ðtk�2Þðtk�2Þ< � � � < ðl1l2Þ

Nrðt0 ;tÞe�aðt�t0�Dk�1�Dk�2��D1ÞebðDk�1þDk�2þ��þD1ÞVr0ðt0Þðt0Þ< ðl1l2Þ

Nrðt0 ;tÞeðaþbÞNrðt0 ;tÞTmax e�aðt�t0ÞVr0ðt0Þðt0Þ ð29Þ

Similarly, for t 2 ½tk; tk þ DkÞ, one has

Vr0ðtÞðtÞ < ebðt�tkÞVr0 ðtkÞðtkÞ< l2ebðt�tkÞe�aðtk�tk�1�Dk�1ÞVr0ðtk�1þDk�1Þðtk�1 þ Dk�1Þ< l1l2e�aðtk�tk�1�Dk�1ÞebDk�1 ebðt�tkÞVr0 ðtk�1Þðtk�1Þ< l2l1l2e�aðtk�tk�1�Dk�1þtk�1�tk�2�Dk�2ÞebDk�1 ebðt�tkÞVr0ðtk�2þDk�2Þðtk�2 þ Dk�2Þ< ðl1l2Þ

2e�aðtk�tk�1�Dk�1þtk�1�tk�2�Dk�2ÞebðDk�1þDk�2Þebðt�tkÞVr0 ðtk�2Þðtk�2Þ< � � � < l�1

1 ðl1l2ÞNrðt0 ;tÞe�aðtk�t0�Dk�1�Dk�2��D1ÞebðDk�1þDk�2þ��þD1Þebðt�tkÞVr0 ðt0Þðt0Þ

< l�11 ðl1l2Þ

Nrðt0 ;tÞeðaþbÞNrðt0 ;tÞTmax e�aðt�t0ÞVr0ðt0Þðt0Þ ð30Þ

Combining (29), (30), we have

Vr0ðtÞðtÞ < ðl1l2ÞNrðt0 ;tÞeðaþbÞNrðt0 ;tÞTmax e�aðt�t0ÞVr0ðt0Þðt0Þ ð31Þ

According to the definition of Nrðt0; tÞ, we get

Vr0ðtÞðtÞ < eðððaþbÞTmaxþlnðl1l2ÞÞ=Ta�aÞðt�t0ÞVr0 ðt0Þðt0Þ ð32Þ

It follows from (17) that

kxðtÞk < j1e�12qðt�t0Þkxðt0Þkc;

where q ¼ a� ðaþbÞTmaxþlnðl1l2ÞTa

> 0, j1 ¼maxi2NfkmaxðPiÞþebskmaxðQiÞg

mini2NfkminðPiÞg

.

Thus according to Definition 1, system (12) with wðtÞ ¼ 0 is exponentially stable.The proof is completed. h

3.2. L1 reliable controller design

In this section, we consider the L1 reliable controller design for system (1). A solvability condition for the L1 reliable con-trol problem is provided in the following theorem.

Theorem 2. Consider system (1), for given positive scalars a, b and c, if there exist positive scalars e1i; e1ij; e2i; e2ij; di and dij,positive definite matrices Xi and Qi, positive definite diagonal matrices Ui and Uij, and any matrices Yi with appropriatedimensions, such that, 8i; j 2 N, i–j,

Hi11 AdiXi Hi13 Gi e1iHi ðE1iXiÞT e2iBiMi1 YTi YT

i Hi19

� �~ge�asQ i 0 0 0 ðEdiXiÞT 0 0 0 0� � �I 0 0 0 0 0 0 0� � � �c2I 0 0 0 0 0 0� � � � �e1iI 0 0 0 0 0� � � � � �e1iI 0 0 0 0� � � � � � �e2iI 0 0 0� � � � � � � �e2iI 0 0� � � � � � � � �Ui 0� � � � � � � � � Hi99

266666666666666666664

377777777777777777775

< 0; ð33Þ


Hij11 AdjXi Hij13 Gj e1ijHj ðE1jXiÞT e2ijBjMj1 YTi YT

i Hij19

� �~gQ i 0 0 0 ðEdjXiÞT 0 0 0 0� � �I 0 0 0 0 0 0 0� � � �c2I 0 0 0 0 0 0� � � � �e1ijI 0 0 0 0 0� � � � � �e1ijI 0 0 0 0� � � � � � �e2ijI 0 0 0� � � � � � � �e2ijI 0 0� � � � � � � � �Uij 0

� � � � � � � � � Hij99

266666666666666666664

377777777777777777775

< 0; ð34Þ

where

Hi11 ¼ AiXi þ XiATi þ BiMi0Yi þ ðBiMi0YiÞT þ DiPi1Xi þ XiP

Ti1DT

i þ Q i þ aXi þ diHiHTi

Hi13 ¼ Di þ ðXiPTi3Þ=2; Hij13 ¼ Dj þ ðXiP

Tj3Þ=2; ~g ¼ 1� g

Hi19 ¼ ðE2iMi0YiÞT ; Hi99 ¼ �ðdiI � E2iMi1UiðE2iMi1ÞTÞ;

Hij19 ¼ ðE2jMj0YiÞT ; Hij99 ¼ �ðdijI � E2jMj1UijðE2jMj1ÞTÞ

Hij11 ¼ AjXi þ XiATj þ BjMj0Yi þ ðBjMj0YiÞT þ DjPj1Xi þ XiP

Tj1DT

j þ Qi � bXi þ dijHjHTj ;

then under the following reliable controller

uðtÞ ¼ Kr0 ðtÞxðtÞ; Ki ¼ YiX�1i ð35Þ

and the average dwell time scheme (17), where l2 ¼ eðaþbÞs and l1 satisfies (18), the corresponding closed-loop system is expo-nentially stable with an L1 performance level ~c, where

~c ¼ c

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil1eðaþbÞTmax maxi2NfkmaxðXiÞg

q

sð36Þ

q ¼ a� ðaþ bÞTmax þ lnðl1eðaþbÞsÞTa

ð37Þ

Proof. Denoting Yi ¼ KiXi, (15) and (16) can be deduced from (33) and (34). Thus by Theorem 1, system (12) with wðtÞ ¼ 0 isexponentially stable.

Now we are in a position to establish the L1 performance. From (20), (27), (33), and (34), we obtain

ddtðeatVr0 ðtÞðtÞÞ < c2eatwTðtÞwðtÞ; t 2 ½tk�1 þ Dk�1; tkÞ ð38Þ

ddtðe�btVr0 ðtÞðtÞÞ < c2e�btwTðtÞwðtÞ; t 2 ½tk; tk þ DkÞ ð39Þ

Integrating both sides of (38) and (39), we have

Vr0ðtÞðtÞ < e�aðt�tk�1�Dk�1ÞVr0ðtk�1þDk�1Þðtk�1 þ Dk�1Þ þZ t

tk�1þDk�1

e�aðt�sÞc2wTðsÞwðsÞds; t 2 ½tk�1 þ Dk�1; tkÞ ð40Þ

Vr0ðtÞðtÞ < ebðt�tkÞVr0 ðtkÞðtkÞ þZ t

tk

ebðt�sÞc2wTðsÞwðsÞds; t 2 ½tk; tk þ DkÞ ð41Þ

Then, according to (18) and Nrðt0; tÞ in Definition 2, one obtains, for t 2 ½tk�1 þ Dk�1; tkÞ,

Vr0 ðtÞðtÞ < e�aðt�tk�1�Dk�1ÞVr0 ðtk�1þDk�1Þðtk�1 þ Dk�1Þ þZ t

tk�1þDk�1

e�aðt�sÞc2wTðsÞwðsÞds

< l1e�aðt�tk�1�Dk�1ÞebDk�1 Vr0 ðtk�1Þðtk�1Þ þZ t

tk�1þDk�1


þZ tk�1þDk�1

tk�1

l1e�aðt�tk�1�Dk�1Þebðtk�1þDk�1�sÞc2wTðsÞwðsÞds


< l1l2e�aðt�tk�1�Dk�1þtk�1�tk�2�Dk�2ÞebDk�1 Vr0 ðtk�2þDk�2Þðtk�2 þ Dk�2Þ

þZ tk�1

tk�2þDk�2

l1l2e�aðt�s�Dk�1ÞebDk�1c2wTðsÞwðsÞdsþZ t

tk�1þDk�1


þZ tk�1þDk�1

tk�1


< l2l21e�aðt�tk�1�Dk�1þtk�1�tk�2�Dk�2ÞebðDk�1þDk�2ÞVr0 ðtk�2Þðtk�2Þ þ

Z tk�1

tk�2þDk�2

l1l2e�aðt�s�Dk�1ÞebDk�1c2wTðsÞwðsÞds

þZ t

tk�1þDk�1

e�aðt�sÞc2wTðsÞwðsÞdsþZ tk�1þDk�1

tk�1


þZ tk�2þDk�2

tk�2

l2l21e�aðt�Dk�1�tk�2�Dk�2ÞebðDk�1þDk�2þtk�2�sÞc2wTðsÞwðsÞds

< � � � < ðl1l2ÞNrðt0 ;tÞe�aðt�sÞeðaþbÞTmaxNrðt0 ;tÞVr0 ðt0Þðt0Þ þ

Z t

t0

l1eðaþbÞTmax ðl1l2ÞNrðs;tÞe�aðt�sÞeðaþbÞTmaxNrðs;tÞc2wTðsÞwðsÞds

ð42Þ

On the other hand, for t 2 ½tk; tk þ DkÞ, we have

Vr0 ðtÞðtÞ < ebðt�tkÞVr0 ðtkÞðtkÞ þZ t

tk

ebðt�sÞc2wTðsÞwðsÞds < l2ebðt�tkÞe�aðtk�tk�1�Dk�1ÞVr0 ðtk�1þDk�1Þðtk�1 þ Dk�1Þ

þZ t

tk

ebðt�sÞc2wTðsÞwðsÞdsþZ tk

tk�1þDk�1

l2ebðt�tkÞe�aðtk�sÞc2wTðsÞwðsÞds

< l1l2e�aðtk�tk�1�Dk�1ÞebDk�1 ebðt�tkÞVr0 ðtk�1Þðtk�1Þ þZ t

tk

ebðt�sÞc2wTðsÞwðsÞds

þZ tk

tk�1þDk�1

l2ebðt�tkÞe�aðtk�sÞc2wTðsÞwðsÞdsþZ tk�1þDk�1

tk�1

l1l2e�aðtk�tk�1�Dk�1Þebðt�tkþtk�1þDk�1�sÞc2wTðsÞwðsÞds

< l2l1l2e�aðtk�tk�1�Dk�1þtk�1�tk�2�Dk�2ÞebDk�1 ebðt�tkÞVr0 ðtk�2þDk�2Þðtk�2 þ Dk�2Þ þZ t

tk

ebðt�sÞc2wTðsÞwðsÞds

þZ tk

tk�1þDk�1

l2ebðt�tkÞe�aðtk�sÞc2wTðsÞwðsÞdsþZ tk�1þDk�1

tk�1

l1l2e�aðtk�tk�1�Dk�1Þebðt�tkþtk�1þDk�1�sÞc2wTðsÞwðsÞds

þZ tk�1

tk�2þDk�2

l2l1l2e�aðtk�Dk�1�sÞebDk�1 ebðt�tkÞc2wTðsÞwðsÞds

< � � � < l�11 ðl1l2Þ

Nrðt0 ;tÞe�aðt�sÞeðaþbÞTmaxNrðt0 ;tÞVr0 ðt0Þðt0Þ þZ t

t0

eðaþbÞTmax ðl1l2ÞNrðs;tÞe�aðt�sÞeðaþbÞTmaxNrðs;tÞc2wTðsÞwðsÞds

ð43Þ

Under the zero initial condition, we obtain from (42) and (43) that

Vr0ðtÞðtÞ <Z t

t0

l1eðaþbÞTmax e�qðt�sÞc2wTðsÞwðsÞds ð44Þ

Notice that

Vr0ðtÞðtÞP mini2NfkminðPiÞgkxðtÞk2�; kwðtÞk 6 sup

t2½t0 ;1ÞkwðtÞk

It follows from (44) that

mini2NfkminðPiÞg xðtÞk k2

< l1eðaþbÞTmax supt2½t0 ;1Þ

wðtÞk k !2

c2Z t

t0

e�qðt�sÞds 61q

l1eðaþbÞTmax supt2½t0 ;1Þ

wðtÞk k !2

c2:

That is

kxðtÞk < c

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil1eðaþbÞTmax

qmini2NfkminðPiÞg

ssup

t2½t0 ;1ÞkwðtÞk;8t P t0;

which implies that


supt2½t0 ;1ÞkxðtÞksupt2½t0 ;1ÞkwðtÞk

< c


q

s: ð45Þ

By the definition of L1-norm, we can obtain

kxðtÞk1 < c


q

skwðtÞk1:

This completes the proof. h

Remark 4. It is noted that the uncertainties exist in the system matrix and the input matrix. In [23,24,27,28], theuncertainties only exist in the system matrix. Comparing with the result in [22,24,25,29], we get sufficient conditions of L1performance instead of H1 performance. Also, the result is suitable for sector-bounded nonlinearities, which cover Lipschitznonlinearities of [27,28] as a special case. In addition, this paper takes the asynchronous switching into consideration, whichalso leads to different results from the ones proposed in [27,28], where the asynchronous switching is not considered.

In the absence of asynchronous switching, i.e., Tmax ¼ 0 in Theorem 2, we can easily get the following corollary.

Corollary 1. Consider system (1), for given positive scalars a and c, if there exist positive scalars e1i; e2i, and di, positive definitematrices Xi and Qi, positive definite diagonal matrices Ui, and any matrices Yi with appropriate dimensions, such that, 8i 2 N, (33)holds, then under the reliable controller

uðtÞ ¼ KrðtÞxðtÞ; Ki ¼ YiX�1i ; ð46Þ

and the following average dwell time scheme

Ta > T�a ¼ln l1

a; ð47Þ

where l1 satisfies (18), the corresponding closed-loop system is exponentially stable with an L1 performance level �c, where

�c ¼ c

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffil1maxi2NfkmaxðXiÞg

�q

sð48Þ

�q ¼ a� ln l1

Tað49Þ

4. Numerical example

In this section we present an example to illustrate the effectiveness of the proposed method. Consider system (1) withparameters as follows:

A1 ¼�0:6 0:3

0 0:5

" #; Ad1 ¼

�0:1 0

0 �0:12

" #; B1 ¼

0:5 0

0:3 0:6

" #;

D1 ¼0:3 �0:20 �0:1

� �; H1 ¼

0:1 0:10 0:03

� �; E11 ¼

0:6 0:20:1 0:2

� �;

E21 ¼0:7 0:30:3 0:4

� �; Ed1 ¼

0:3 00:1 0:4

� �; G1 ¼

�0:1 0�0:1 0

� �;

A2 ¼�0:7 0:40:1 0:3

� �; Ad2 ¼

�0:2 0:30 �0:1

� �; B2 ¼

0:3 0:10:6 0:2

� �;

D2 ¼�0:1 0:1�0:1 0:2

� �; H2 ¼

0:12 00:12 0

� �; E12 ¼

0:5 0:30:5 0:3

� �;

E22 ¼0:4 0:60:5 0:2

� �; Ed2 ¼

0:2 0:40:4 0:3

� �; G2 ¼

0 �0:20 �0:3

� �:


The nonlinear functions are

f1ðxðtÞÞ ¼0:4 tanð�x1Þ þ 0:2x1 þ 0:1x2

0:1x1 � 0:4 tanðx2Þ þ 0:2x2

� �; f 2ðxðtÞÞ ¼

0:1x1 � 0:4 tanðx2Þ þ 0:2x2

0:2x1 þ 0:4 tanð�x1Þ þ 0:1x2

� �

It is easy to get that

P11 ¼�0:3 0:10:1 �0:3

� �; P12 ¼

0:2 0:10:1 0:2

� �; P21 ¼

0:1 �0:3�0:3 0:1

� �; P22 ¼

0:1 0:20:2 0:1

� �:

The fault matrices are

0:1 6 m11 6 0:5; 0:2 6 m12 6 0:8

0:2 6 m21 6 0:4; 0:3 6 m22 6 0:9

By (9) and (10), we obtain

M10 ¼0:3 00 0:5

� �; M11 ¼

0:2 00 0:3

� �; M20 ¼

0:3 00 0:6

� �; M21 ¼

0:1 00 0:3

� �:

Choose a ¼ 0:1; b ¼ 0:01 and c ¼ 1. By solving the LMIs in Theorem 2, we have

X1 ¼7:5467 �0:5287�0:5287 0:2156

� �; X2 ¼

6:4999 �0:4700�0:4700 0:2118

� �;

Y1 ¼�8:4914 �8:9446�0:9611 12:0331

� �; Y2 ¼

�8:8012 �8:9857�1:7550 �11:8443

� �:

Then the controller gain matrices can be obtained by (35)

K1 ¼�4:8667 �53:4113�4:8733 �67:7509

� �; K2 ¼

�5:2657 �54:1005�5:1367 �67:3087

� �:

Letting Tmax ¼ 0:5, one can obtain from (17) and (18) that l1 ¼ 1:2143 and T�a ¼ 2:9313.Selecting Ta ¼ 4, one can get ~c ¼ 19:08 by (36). Switching signals are plotted in Fig. 1 and state response of the closed-

loop system is shown in Fig. 2, where the initial conditions are

xðtÞ ¼ 0; t 2 ½�0:4; 0Þ; xð0Þ ¼ ½0:5 �0:5 �T ;

and the disturbance input is

wðtÞ ¼ ½0:5e�0:02t sinðtÞ 0:5e�0:02t sinðtÞ �T

It can be seen from Figs. 1 and 2 that the closed-loop system is exponentially stable, which demonstrates the effectivenessof the proposed method.

0 2 4 6 8 10 12 14 160

1

2

3

time(s)

mod

es

system modecontroller mode

Fig. 1. Switching signals.

0 2 4 6 8 10 12 14 16-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

time(s)

stat

e x

x1x2

Fig. 2. State response of the closed-loop system.


5. Conclusions

This paper has presented a solution to the problem of robust L1 reliable control for uncertain switched nonlinear systemswith time delay and actuator failures under asynchronous switching. A sufficient condition for the existence of a robust L1reliable controller is derived by using the average dwell time approach. A numerical example is provided to demonstrate theapplicability of the proposed approach.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant No. 61273120.

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Robust L∞ reliable control for uncertain switched nonlinear systems with time delay under asynchronous switching