Stability analysis of impulsive control systems

J.T. Sun and Y.P. Zhang

Abstract: A comparison theorem for the asymptotic stability of impulsive differential systems ispresented. Based on this result, less conservative conditions for the asymptotic stability ofimpulsive control systems with impulses at fixed times are derived. The results are used to design animpulsive control for a class of nonlinear systems, that improves and extends the existing results.The class of nonlinear systems considered is also enlarged.

1 Introduction

Many dynamic systems in physics, chemistry, biology, andinformation science have an impulsive dynamic behavioursdue to abrupt jumps at certain instants during the dynamicprocesses. These complex dynamic behaviours can bemodelled by impulsive differential systems. Some practicalexamples are given in [1–5]. In recent years, manyresearchers have studied impulsive systems and impulsivecontrol [1–15]. In [2–4, 6–9], the stability of impulsivesystems by using Lyapunov functions that are required to benon-increasing along the whole sequence or subsequence ofthe switchings is considered. Li et al. [8] and Sun et al. [9]have given less conservative conditions for the asymptoticstability of impulsive systems with impulses at fixed times.The results were used to design an impulsive control for aclass of nonlinear systems. Interesting applications ofimpulsive control in chaotic systems and chaotic spreadspectrum communications, were presented in [5, 10–14].Panas et al. [15] have given methods for the experimentalsettings to achieve the impulsive controls.

We shall also consider the impulsive control of nonlinearsystems as in [9]. We first obtain a comparison theorem.Based on this theory of an impulsive differential system, wederive sufficient conditions for the stability of the impulsivesystems with impulses at fixed times. These results are thenused to design impulsive control laws for a class ofnonlinear systems, and improve and extend the results in[8] and [9]. Our method can be applied to a wider class ofnonlinear systems and is helpful in the improvement ofexisting technologies used in chaotic systems control andchaotic communications systems [5, 10–14].

2 Stability of impulsive differential systems

An impulsive differential system with impulses at fixedtimes is described by:

_XXðtÞ ¼ f ðt;XðtÞÞ t 6¼ �k

DXðtÞ ¼D

XðtþÞ � Xðt�Þ ¼ IkðXÞ t ¼ �k; k ¼ 1; 2; . . .

8<:

ð1Þ

where f : Rþ � Rn ! Rn is continuous; Ik : Rn ! Rn iscontinuous; X 2 Rn is the state variable; f�kg ðk ¼ 1; 2; . . .Þthat satisfies 0<�1 <�2 < . . . <�k <�kþ1< . . . ; �k ! 1as k ! 1:

Definition 1 [2]: Let V: Rþ � Rn ! Rþ; then V is said tobelong to class V0 if:

1. V is continuous in ð�k�1; �k � Rn and for eachX 2 Rn; k ¼ 1; 2; . . . :

limðt;YÞ!ð�þ

k;XÞ

Vðt;YÞ ¼ Vð�þk ; XÞ

exists;2. V is locally Lipschitzian in X.

Definition 2 [2]: For ðt;XÞ 2 ð�i�1; �i � Rn we define:

DþVðt;XÞ ¼D

limh!0þ

sup1

h½Vðt þ h;X þ hf ðt;XÞÞ � Vðt;XÞ

Definition 3 [3]: Comparison system: Let V 2 V0 andassume that:

DþVðt;XÞ4gðt;Vðt;XÞÞ t 6¼ �k

Vðt;X þ IkðXÞÞ4CkðVðt;XÞÞ t ¼ �k

�where g: Rþ � Rþ ! R is continuous and Ck: Rþ ! Rþ isnon-decreasing. Then the following system:

_!! ¼ gðt; !Þ t 6¼ �k

!ð�þk Þ ¼ Ckð!ð�kÞÞ

!ðtþ0 Þ ¼ !050

8<: ð2Þ

is the comparison system of (1).

We obtain the following comparison theorem

Lemma 1: Let V: Rþ � Rn ! Rþ; V 2 V0;K: Rþ ! ð0;1Þand assume there exists a integer number �51 such that

KðtÞDþVðt;X�ÞþVðt;X�ÞDþKðtÞ4gðt;KðtÞVðt;X�ÞÞ t 6¼ �k

Kð�þk ÞVð�þk ;ðXþIkðXÞÞ�Þ4CkðKð�kÞVð�k;X�ÞÞ t¼ �k; k¼1;2; . . .

8<:then

KðtÞVðt;X�ðt; t0;X0ÞÞ4rðt; t0; r0Þ

for t5t0 if Kðtþ0 ÞVðtþ0 ;X�0 Þ4r0

where rðt; t0; r0Þ is the maximal solution of (2) on½t0;1Þ; gðt; 0Þ ¼ 0; and g is continuous in ð�k�1; �k � Rn

q IEE, 2003

IEE Proceedings online no. 20030599

doi: 10.1049/ip-cta:20030599

The authors are with the Department of Applied Mathematics, TongjiUniversity, Shanghai, 200092, Peoples Republic of China

Paper first received 19th November 2002 and in revised form 23rd April2003

IEE Proc.-Control Theory Appl., Vol. 150, No. 4, July 2003 331

for each X 2 Rn; k ¼ 1; 2; . . . ; also

limðt;YÞ!ð�þ

k;XÞ

gðt;YÞ ¼ gð�þk ;XÞ

exists and Ck is continuous and non-decreasing, X� ¼ðx

�1 ; x

�2 ; . . . ; x�n Þ

T :

Proof: Let XðtÞ ¼D

Xðt; t0; x0Þ is a any solution ofsystem (1) on ½t0; TÞ and satisfy Kðtþ0 ÞVðtþ0 ;X�

0 Þ4!0;mðtÞ ¼

DKðtÞVðt; X�ðtÞÞ: Since V 2 V0; for t 6¼ tk; we have:

DþmðtÞ ¼ limh!0þ

sup1

h½mðtþhÞ�mðtÞ

¼ limh!0þ

sup1

h½KðtþhÞVðtþh;X�ðtþhÞÞ

�KðtÞVðt;X�ðtÞÞ

¼ limh!0þ

sup1

h½KðtþhÞVðtþh;X�ðtþhÞÞ

�KðtÞVðtþh;X�ðtþhÞÞ

þKðtÞVðtþh;X�ðtþhÞÞ

�KðtÞVðtþh;ðXþhf ðt;XðtÞÞÞ�Þ

þKðtÞVðtþh;ðXþhf ðt;XðtÞÞÞ�Þ

�KðtÞVðt;X�ðtÞÞ

4 limh!0þ

sup1

hf½KðtþhÞ�KðtÞ Vðtþh;ðXðtÞ

þhf ðt;XðtÞÞþoðhÞÞ�ÞþKðtÞLjðXðtÞ

þhf ðt;XðtÞÞþoðhÞÞ� �ðXðtÞþhf ðt;XðtÞÞÞ�j

þKðtÞ½Vðtþh; ðXðtÞþhf ðt;XðtÞÞÞ�Þ

�Vðt;X�ðtÞÞ g ¼Vðt;X�ðtÞÞDþKðtÞ

þKðtÞDþVðt;X�ðtÞÞ4gðt;mðtÞÞ

Hence, one can obtain an inequality system as follows:

DþmðtÞ4gðt;mðtÞÞ t 6¼ �k

mð�þk Þ4Ckðmð�kÞÞ; k¼ 1;2; . . .

mðt0Þ4!0

8><>:

From the classical comparison theorem and the inductivemethod, we obtain mðtÞ4!ðt; t0;!0Þ which is a maximalsolution on ½t0;TÞ of system (2). The proof of lemma 1 iscompleted. A

Remark 1: If � ¼ 1 in lemma 1, we can obtain result of [6].

Theorem 1: Assume that the following three conditions:(i) V: Rþ � Rn ! Rþ; V 2 V0; then there exists a integer

number �51; such that:

KðtÞDþVðt;X�Þ þ DþKðtÞVðt;X�Þ

4gðt;KðtÞVðt;X�ÞÞ; t 6¼ �k

where g(x) is as in lemma 1, also

KðtÞ5m > 0; limt!��

k

KðtÞ ¼ Kð�kÞ

i.e. K(t) is left continuous at t ¼ �k also:

limt!�þ

k

KðtÞ exists; k ¼ 1; 2; . . . ;

and

DþKðtÞ ¼ limh!0þ

sup1

h½Kðt þ hÞ � KðtÞ ;

(ii)

Kð�þk ÞVð�þk ; ðX þ IkðXÞÞ�Þ

4CkðKð�kÞVð�k;X�ÞÞ k ¼ 1; 2; . . . ;

(iii) Vðt; 0Þ ¼ 0 and �ðkXk�Þ4Vðt;X�Þ on Rþ � Rn;where �ð�Þ 2 K (class of continuous strictly increasingfunctions � : Rþ ! Rþ such that �ð0Þ ¼ 0Þ are satisfied.Then, the global asymptotic stability of the trivialsolution ! ¼ 0 of the comparison system implies globalasymptotic stability of the trivial solution of impulsivesystem (1).

Proof: We begin by providing that the trivial solutionX ¼ 0 of system (1) is stable. System (2) is stable and hence,for any given t0 2 Rþ; " > 0 there exists a 1 ¼ 1ðt0; "Þsuch that:

rðt; t0; r0Þ<m�ð"�Þ for 0<r0<1; t5t0:

Since Vðt0; 0Þ ¼ 0; then there exists a 2 ¼ 2ðt0; 1Þ > 0;such that:

Kðtþ0 ÞVðtþ0 ;X�0 Þ4r0<1 for kX0k<2

Let ¼ minð1; 2Þ; from lemma 1 and the conditions oftheorem 1, we have:

m�ðkXk�Þ4KðtÞVðt;X�ðt; t0;X0ÞÞ4 rðt; t0; r0Þ<m�ð"�Þ

for kX0k<

Hence kXk<"; i.e. system (1) is stable.Now, we prove that trivial solution X ¼ 0 of system (1) is

attractive.For any given X0 2 Rþ; t0 2 Rþ; " > 0; there exists a

r0 that satisfies Kðtþ0 ÞVðtþ0 ;X�0 Þ4r0; since rðt; t0; r0Þ

is global asymptotically stable, hence, there exists aTð"; t0; r0ðX0ÞÞ such that:

rðt; t0; r0Þ<m�ð"�Þ for t 5 t0 þ T

From lemma 1 and the conditions of theorem 1, weobtain:

m�ðkXk�Þ4KðtÞV ðt; X�ðt; t0; X0ÞÞ4rðt; t0; r0Þ<m�ð"�Þ

for t5t0 þT

i.e. kXk<"; hence, the trivial solution X ¼ 0 of system(1) is globally attractive. Therefore, the origin ofimpulsive system (1) is globally asymptotically stable.A

Remark 2: If we let � ¼ 1 in theorem 1, then we can obtaintheorem in [9]. If � ¼ 1 and KðtÞ � 1 in theorem 1, then wecan obtain theorem in [8].

In order to design an impulsive control for a class ofnonlinear systems, from theorem 1 and the result in [9],we can obtain following stability result.

Theorem 2: Let gðt; !Þ ¼ _��ðtÞ!; � 2 C1½Rþ;Rþ ;Ckð!Þ ¼dk!; dk50 for all k51: Then, the origin of system (1) isglobally asymptotically stable if the conditions of theorem 1and the following conditions hold:

IEE Proc.-Control Theory Appl., Vol. 150, No. 4, July 2003332

(i) l(t) is non-decreasing:

limt!�k

�ðtÞ ¼ �ð�kÞ;

i.e. l(t) is left continuous at t ¼ �k, also

limt!�þ

k

�ðtÞ ¼ �ð�þk Þ

exists, for all k ¼ 1; 2; . . . ;(ii) supifdi expð�ð�iþ1Þ � �ð�þi ÞÞg ¼ �0 <1;

(iii) there exists an r > 1 such that �ð�2kþ3Þ þ �ð�2kþ2Þ þlnðrd2kþ2d2kþ1Þ4�ð�þ2kþ2Þ þ �ð�þ2kþ1Þ holds forall d2kþ1d2kþ2 6¼ 0; k ¼ 1; 2; . . . ; or there exists anr > 1 such that �ð�kþ1Þ þ lnðrdkÞ4�ð�þk Þ for all k;

(iv) Vðt; 0Þ ¼ 0 and there exists an a(·) in class K suchthat �ðkXkÞ4Vðt; XÞ:

3 Impulsive control of a nonlinear system

In this Section, we will use the stability results obtainedin Section 2 to design an impulsive control for a class ofnonlinear systems. A formal definition of impulsivecontrol, which is slightly modified from [1], can begiven as:

Definition 4 [8]: Consider a plant P whose state variableis denoted by X 2 Rn; a set of control instantsT ¼ f�kg; �k 2 Rþ; �k <�kþ1; k ¼ 1; 2; . . . ; and controllaws Uðk; XÞ 2 Rn; k ¼ 1; 2; . . . :: At each �k; X ischanged impulsively, i.e., Xð�þk Þ ¼ Xð��k Þ þ Uðk;XÞ;such that the system is stable and certain specificationsare achieved.

Now, we consider the impulsive control design for thefollowing nonlinear systems:

_XXðtÞ ¼ AXðtÞ þFðXðtÞÞ

YðtÞ ¼ CXðtÞ

(ð3Þ

where X 2 Rn is state vector, A is an n £ n constant matrix,Y 2 Rm is output, C is an m £ n constant matrix, F : Rn !Rn is a nonlinear function satisfying kFðXÞk4LkXk with Lbeing a positive number.

The control instant is defined by 0<�1 <�2 < . . . <�k

< tkþ1 < . . . �k ! 1; as k ! 1 and the time varying controlUðk;XÞ is given by Uðk; Xð�kÞÞ ¼ BkYð�kÞ; k ¼ 1; 2; . . . ;then, we can obtain a nonlinear impulsive control system asfollows:

_XXðtÞ ¼ AXðtÞ þFðXðtÞÞ

YðtÞ ¼ CXðtÞ t 6¼ �k

Xð�þk Þ ¼ Xð��k Þ þ Uðk;XÞ

Uðk;Xð�kÞÞ ¼ BkYð�kÞ k ¼ 1; 2; . . .

8>>>><>>>>:

To use the results obtained in Section 2, the above systemis rewritten as:

_XXðtÞ ¼ AXðtÞ þFðXðtÞÞ t 6¼ �k

DXðtÞ ¼ Uðk;XðtÞÞ ¼ BkCXðtÞ; t ¼ �k; k ¼ 1; 2; . . .

Xðtþ0 Þ ¼ X0

8>><>>:

ð4Þ

where U(k, X(t)) corresponds to IkðXÞ defined in (1).Then, we can obtain the result on the design of impulsive

control as follows

Theorem 3: Let q be the largest eigenvalue of ðA þ AT Þ;then the origin of the impulsive control system (4) isasymptotically stable if the following conditions hold:

(i) ðI þ BkCÞT ðI þ BkCÞ4 ðkÞI; where I is the identitymatrix, ðkÞ ðk ¼ 1; 2; . . .Þ are positive constants;

(ii) there exists an r > 1; integer number �51 and adifferentiable at t 6¼ �k and non-increasing function K (t)which satisfies condition of theorem 1, such that:

�ðq þ 2LÞð�2kþ3 � �2kþ1Þ þ lnKð�2kþ3Þ

Kð�þ2kþ2Þþ ln

Kð�2kþ2Þ

Kð�þ2kþ1Þ

4 � lnðr �ð2k þ 2Þ �ð2k þ 1ÞÞ

or

�ðq þ 2LÞð�kþ1 � �kÞ þ lnKð�kþ1Þ

Kð�þk Þ

4 � lnðr �ðk þ 1Þ �ðkÞÞ;

(iii)

�ðq þ 2LÞ þDþKðtÞ

KðtÞ50;

and

supi

�ðiÞ exp ðq þ 2LÞð�iþ1 � �iÞ þ lnKð�iþ1Þ

Kð�þi Þ

" #( )

¼ "0 <1:

Proof: Let VðXÞ ¼ XT X; since K(t) is differentiable att 6¼ �k; for t 6¼ �k; we have:

KðtÞDþVðX�Þ þ DþKðtÞVðX�Þ

4KðtÞ½2�ðXT XÞ��1XT ðAX þFðXÞÞ þ _KKðtÞVðX�Þ

¼ �KðtÞðXT XÞ��1½XT ðAT þ AÞX þ 2XTFðXÞ

þ _KKðtÞVðX�Þ4 �ðq þ 2LÞþ_KKðtÞ

KðtÞ

( )KðtÞVðX�Þ

and

Kð�þk ÞVððX þ Uðk;XÞÞ�Þ4Kð�kÞ �ðkÞVðX�Þ; for t ¼ �k

Then, we can obtain the following comparison system:

_!!ðtÞ ¼ �ðq þ 2LÞ þ_KKðtÞKðtÞ

h i!ðtÞ t 6¼ �k

!ð�þk Þ ¼ �ðkÞ!ð�kÞ t ¼ �k; k ¼ 1; 2; . . .

!ðtþ0 Þ ¼ !050

8>><>>:

It follows from theorem 2 that if:

Z�2kþ3

�2kþ2

�ðq þ 2LÞ þ_KKðtÞ

KðtÞ

" #dt þ

Z�2kþ2

�2kþ1

�ðq þ 2LÞ þ_KKðtÞ

KðtÞ

" #dt

þ lnðr �ð2k þ 2Þ �ð2k þ 1ÞÞ40;

i.e.

�ðq þ 2LÞð�2kþ3 � �2kþ1Þ þ lnKð�2kþ3Þ

Kð�þ2kþ2Þþ ln

Kð�2kþ2Þ

Kð�þ2kþ1Þ

4 � lnðr �ð2k þ 2Þ �ð2k þ 1ÞÞ

and

IEE Proc.-Control Theory Appl., Vol. 150, No. 4, July 2003 333

�ðq þ 2LÞ þDþKðtÞ

KðtÞ50

are satisfied, then the origin of (4) is asymptotically stable.

A

4 Conclusions

We have obtained sufficient conditions for asymptoticstability of impulsive differential equations. The results areapplied to design an impulsive control for a class ofnonlinear systems. The method can be applied to improvethe technologies currently used in chaotic systems controland chaotic communications systems.

5 References

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