The exponential ISS stability for nonlinear impulsive hybrid systems

Lanping Chen 1,2, Zhenghua Ma1* ,Suolin Duan1

1School of information science & engineering, Jiangsu Polytech University, ChangZhou, China 213164 2Shool of electronic, information and electrical engineering, Shanghai Jiaotong University ,Shanghai, China, 200240

Lanping.chen@gmail.com zhengh.ma@gmail.com

Abstract

This paper introduces appropriate concepts of input-to

state stability (ISS) for nonlinear impulsive hybrid systems

(NIHS). First we study the case in which one kind of dynamics

system without external inputs has a stability property, then

by applying the ISS-Lyapunov function theory, several

sufficient conditions are derived for the ISS property of the

whole NIHS with inputs. The results of ISS are used to study

the robustly globally uniformly exponential stability for imp

ulsive hybrid systems. The impulsive hybrid system can keep

ISS no matter how often the impulses occur. Our proposed

results are evaluated using illustrative example to show their

effectiveness.

Keywords: ISS, Exponential stability, Impulsive hybrid systems,

Nonlinear.

. Introduction In many fields of applications, continuous and

discrete dynamics are interacting strongly, thus the use of techniques for the analysis and design of hybrid dynamical systems has become more and more necessary. In recent years, impulsive hybrid system are receiving considerable attention in the literature as illustrated by ref. [1][2][3].With the growing importance of hybrid models, a number of algorithmic approaches to analyze, design, and optimize for different types of hybrid systems have emerged ,as reported in[4][5][6].

Since Sontag proposed the notion of input-to-state stability( ISS) in the late 1980s[7], ISS property analysis has quickly become more and more actively , and been successfully employed in nonlinear stability analysis and control design for nonlinear systems with external disturbances as shown by[8]. ISS analysis of nonlinear

systems aims to investigate how external disturbances affect the system stability. In this paper we mainly focus on the study of the ISS property related to nonlinear impulsive hybrid systems. Applying the Lyapunov function theory, several sufficient conditions are established for the ISS property of the whole nonlinear impulsive hybrid systems. The rest of the paper is organized as follows. In Section2, we provides some notations and definitions. In Section 3, we firstly analyze stability property for impulsive hybrid systems without external inputs, and then develop a ISS-Lyapunov formulation and establish several sufficient conditions for general dynamical hybrid systems satisfying ISS property. In Section 4, we employ example to show the implementation of the main result. Section 5 presents some conclusions.

. Notations and definitions

Notation: A function RR: is of class k if it is

continuous, zero at zero and strictly increasing. It is of class

k if it is of class k and is unbounded. A continuous

function RRR: is of class kL if ),( t

is of class k for 0t and ),(s is monotonically

decreasing to zero for 0s .Consider the impulsive system with inputs

kc ttuxftx ),()(

kd ttuxgx ),( (1)

Where mc Ru is a locally bounded external input,

2010 International Conference on Intelligent Computation Technology and Automation

978-0-7695-4077-1/10 $26.00 © 2010 IEEE

DOI 10.1109/ICICTA.2010.864

807

nd Ru is the impulsive disturbance input.

Definition For the prescribed sequence kt the

impulsive system (1) is said to e input-to-state stable (ISS)

if there exist functions kL and k , such that

for every initial condition and every inputs, the corresponding solution to (1) exists globally and satisfies

)()(),)(()(],[2],[100

00 ttdttc uutttxtx

0tt (2)

where denotes the supremum norm on an interval .

. Main results In this section, we will state a theorem to guarantee

that the system (1) is ISS over the class S by employing a Lyapunov function with discontinuities at the impulse times. Now we first consider the general nonlinear impulsive hybrid system without inputs of the form:

kttxtAxxtftx )()(),()(

,...3,2,1,)(),()(

i

kkK

NktttBxxtgtx

(3)Where the nonlinear function )(x satisfy the Lipschitz

condition Lxx)( .

We make the following Assumptions throughout the paper.

(A1) Let 1 and k be the largest eigenvalues of

)( PAPAT and )()( kT

K BIBI , respectively.

(A2)There exists positive constants c and d such that

cL21 (4)

dkk

k

jj

0

0ln

sup (5)

Where iNkkNk 0,,...3,2,1 .

Let the Lyapunov function be in the form of

PxxxV T)( , then using (A1), the derivative of a

Lyapunov function with respect to system (3) as follows.

PxLxxPAPAx

xPxPxxtxVTTT

TT

2)())((

)(]2)([ max xVLPAPAT

)(]2[ 1 xVL ))(( txcV (6)

On the other hand, it follows from the second equation of system (3) that

)()]()[()(

)()())((

kkT

kT

k

kT

kk

txBIBItx

tPxtxtxV

))(( kk txV (7)

Combining (6) with (7) together, we get

)(210

0))(())(( ttck etxVtxV

)(ln

000))(( ttceetxV

k

jj

By the inequality (5) of assumption (A2), we derive

)()(0

00))(())(( ttckkdetxVtxV

)(0

0))(( ttcdNietxV (8)

It is shown that under these assumptions, the system (3)

exist a unique solution ))(,( 0txtx that is robustly globally

exponential stability.

808

Then we will establish one theorem which provides sufficient conditions for ISS of system (1).Theorem 1. Assume that there exist a candidate exponential ISS-Lyapunov function RRV n: fordynamic systems (1) which is locally Lipschitz, functions

,, 21 k2 and positive scalars 0,0 ddcc

such that

(B1) )()()( 21 xxVx

(B2) )()(),()( 1 cc uxcVuxfxVD

(B3) )()()),(( 2 dd

d uxVeuxgV

Then the system (1) is uniformly ISS over the impulsive time sequences

kt .

Proof: By (B2), )()(),()( 1 cc uxcVuxfxVD .

By transformation, we have )())(( 1 ccc ueVeD ,

then integrating the inequality on both sides from kt to t

gives

)(1)()( 1)(

ckttc u

ctVetV k (9)

By (B2) hold:

)()()()()( 1 cuxVccxVcxV

This means that

))(())(()()()( 1 txVctxVuxVcc c (10)

)(1))(( 1 cucc

txV (11)

Similarly, from (B3) we get

))(())(( kd

k txVetxV (12)

)(1))(( 2 dddk uee

txV (13)

From (9), combining (12) with (13), when it take ktt ,

we conclude that

)(1)(1)( 12)(

cddddttc u

cu

eeeetV k

)()(],[1],[2

00 ttcttdd uue (14)

Where let 0)1,1min(:dd eecc

. From (8) and

(14), we obtain

))((

))(())((()(

],[11

1

],[21

102)(1

1

0

0

0

ttc

ttddttcdN

u

uetxetx

)()(),)((],[2],[100

00 ttdttc uutttx

(15)

Where it is easy to verify that kL and k21, ,

which are defined as follows:

))((:),( 21

1 retr ctdN (16)

))((:)( 11

11 rr (17)

))((:)( 21

12 rer d (18)

Therefore, we complete the proof of Theorem 1. Hence, the conditions that (B1),(B2) and (B3)hold for all

),[ 0tt imply that the system (1) is uniformly ISS over the impulsive time sequences

kt .

. Simulation Consider a NIHS with external inputs:

0)0()()(

)()()(

xxutBxtx

uxtAxtx

dkk

c

(19)

Where

8.001.01.04.00

008.0A ,

Txxxxx 2131 sinsin0)( , IB 1.0 , the state

initial value Tx 2.04.08.00 . Firstly, let 0cu

and 0du that is no external disturbance inputs, then the

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numerical simulation is given in Fig. 1, which show that the system is exponential stability. Next we take random inputs:

Tc randrandrandu )1()1()1(1.0 and

Td randrandrandu )1()1()1(2.0 ,where

1)1(0 rand , then the exponential ISS property of the

whole system is shown in Fig. 2.

Fig 1. Exponential stability of NIHS without external inputs

Fig 2. Exponential stability of NIHS with external inputs

The results of the simulations means that the whole impulsive hybrid system (19) will be exponential ISS if and only if a candidate ISS-Lyapunov function with positive

rate coefficients satisfying the assumptions (B1),(B2) and (B3). We now conclude from theorem that the NIHS system is strongly exponential ISS that means that there are no constraints posed on the frequency of impulses.

. Conclusion We derived an ISS theorem for nonlinear impulsive

hybrid systems by employing Lyapunov functions with discontinuity at the impulse times. Then we applied the theorem to the analysis stability of NIHS. The ISS property of the whole impulsive hybrid system can be achieved provided that sufficient conditions concerning the Lyapunov function of NIHS is satisfied. Illustrative example shows the effectiveness.

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