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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
[email protected] [email protected]
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.
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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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