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A new Lyapunov-like functional approach to dwell-time dependent stability for impulsive systems
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A new Lyapunov-like functional approach to dwell-time dependent stability for impulsive systems Hanyong Shao , Lin Shao PII: DOI: Reference:
S0016-0032(19)30829-4 https://doi.org/10.1016/j.jfranklin.2019.11.040 FI 4275
To appear in:
Journal of the Franklin Institute
Received date: Revised date: Accepted date:
8 April 2019 23 September 2019 13 November 2019
Please cite this article as: Hanyong Shao , Lin Shao , A new Lyapunov-like functional approach to dwell-time dependent stability for impulsive systems, Journal of the Franklin Institute (2019), doi: https://doi.org/10.1016/j.jfranklin.2019.11.040
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A new Lyapunov-like functional approach to dwell-time dependent stability for impulsive systems Hanyong Shao1 and Lin Shao2 1 The Institute of Automation, Qufu Normal University, Rizhao 276826, Shandong Province, China 2 The College of Electronics, Communication and Physics, Shandong University of Science and Technology, Qingdao 266590, Shandong Province, China
Abstract: This paper investigates the dwell-time dependent stability for impulsive systems by employing a new Lyapunov-like functional that is of the second order in time t . In contrary to those built on [tk , t ] , a part of the impulsive interval [tk , tk 1 ] , the Lyapunov-like functional is two-sided in the sense of employing the system information on [t , tk 1 ] as well as [tk , t ] . To deal with the derivative of the two-sided Lyapunov-like functional, which involves integrals of the state and integrals coupled by [t , tk 1 ] and [tk , t ] , integral equations of the impulsive systems are introduced and an advanced inequality is employed. By the Lyapunov-like functional theory, new dwell-time dependent stability results with ranged dwell-time, maximal dwell-time and minimal dwell-time are derived for periodic or aperiodic impulsive systems. The stability results turn out to be less conservative than some existing ones, which is illustrated by numerical examples. Keywords: impulsive systems, Lyapunov-like functional, dwell-time dependent stability, linear matrix inequalities (LMIs).
1. Introduction An impulsive system is modeled after many real world processes that undergo abrupt state changes, and it is comprised of three parts; a continuous-time dynamical equation which governs the evolution of the system between two successive impulsive instants; a difference equation which describes how the system state is changed at impulsive instants; and finally a criterion for determining the impulsive instant sequence [1-7]. Stability with minimal dwell-time or maximal dwell-time for impulsive systems was defined and discussed based on the characters of the system matrices in [4]. The stability of impulsive systems has been paid attention due to their applications in many fields such as epidemiology [8], impulsive control [9,10], power
electronics [11], networked control systems [12,13], sampled-data systems [14-18]. For example, in [16], the sampled-data systems were formulated as impulsive systems, and robust stability of sampled-data systems was investigated. The discrete time method is a basic approach to the stability of impulsive systems, by which the impulsive system is transformed into discrete time systems, and then the eigenvalue analysis is conducted. This method, however, encounters difficulties when impulsive systems involve uncertainties or aperiodic impulses. To deal with this problem, a Lyapunov functional method was proposed in [16] to address the stability for impulsive systems, and the stability results were applied to sampled-data systems with uncertainties. To improve the Lyapunov functional method further, there are two methods to deal with the stability for impulsive systems. One is the Lyapunov-like functional approach where the Lyapunov-like functional is not required positive definite or continuous [17-18]. In [17] stability results were derived for periodic impulsive systems with or without polytope uncertainties. Less conservative stability results were obtained by constructing a different Lyapunov-like functional in [18]. The other approach to improve the Lyapunov functional method is the looped-functional method [19-22], where stability results with ranged dwell-time, maximal dwell-time and minimal dwell-time were derived. Convex dwell-time stability characterizations were given for the linear impulsive systems with uncertainties in [19, 20], where the stability analysis was formulated as an infinite dimensional feasibility problem, which was hard to solve. By contrast, the stability results obtained in [21, 22] were in the form of linear matrix inequalities (LMIs) and could be checked easily. Recently the stability for impulsive systems was studied by using the Lyapunov-like functional approach [23], where stability results with ranged dwell-time, maximal dwell-time and minimal dwell-time were derived. The stability results obtained in [23] have improved over those in [21, 22]. However, the Lyapunov-like functional in [23] as well as [17], and the looped-functional in [21] or [22] only used the system information on the interval (tk , t ] , failed to make use of that on [t , tk 1 ] . Therefore, the existing results can be expected to further improve if the Lyapunov-like functional or the looped-functional is expanded by making full use of the system information on the whole impulsive interval [tk , tk 1 ] .It is challenging how to employ the information on [t , tk 1 ] and (tk , t ] to construct a two-sided Lyapunov functional and develop new techniques to deal with the derivative of the two-sided Lyapunov functional. It is worth noting that recently some scholars have turned towards impulsive networks [30,31] and impulsive stochastic functional differential systems [32,33]. For instance, in [33] the stability problem was addressed for time-varying stochastic functional differential systems with distributed-delay dependent impulsive effects. Based on stochastic theory, some stability results were derived by employing a Lyapunov approach. Nevertheless, in this paper we will further study stability for the
linear impulsive systems in [17-23] by a new Lyapunov-like functional. The study features: The Lyapunov-like functional is second order in time t , and two-sided in the sense of its including the system information on both [t , tk 1 ] and (tk , t ] ; An advanced inequality and integral equations of the impulsive system are employed when estimating the derivative of the two-sided Lyapunov-like functional; Improved dwell-time dependent stability results, including those with ranged dwell-time, maximal dwell-time and minimal dwell-time for impulsive systems with periodic or aperiodic impulses, are derived. Numerical examples are given to illustrate the less conservatism of the dwell-time dependent stability results. Notations: Throughout this paper, X T and X 1 denote the transposition and the inverse of the matrix X , respectively. * in the symmetric matrix stands for the symmetric terms. For a symmetric matrix X , X 0 0 denotes X is positive definite (negative definite). I
denotes the identity matrix with appropriate
dimensions. | | is the Euclidean norm for a vector and is the induced matrix norm.
n
,
nm
refer to the set of n dimension vectors, n m matrices and
,
non-negative integers, respectively. min () , max () and () denote the smallest eigenvalue, largest eigenvalue and the spectral radius of a real symmetric matrix. For a given matrix A , He{ A} stands for A AT . For the impulsive instant t k ,
x(tk ) lim x( s) , x(tk ) lim x( s) . s t k
s tk
n
n
and
are the set of symmetric and
symmetric positive definite matrices of size n n respectively.
2. Problem formulation Consider the following linear impulsive system
x(t ) Ax(t ),t x(t ) Jx(t ),t where x(t )
n
is the state vector, A
nn
\
,J
(1) nn
are constant matrices，
{t0 , t1 , t2 ,tk ,} is an increasing sequence of impulse instants. The dwell- time hk tk 1 tk satisfies
0 h hk h ,
(2)
where h and h are known constants. In the case of periodic impulses, the dwell-time hk h , where h is the period. In this paper, we will study the relationship between dwell-time and stability, namely the dwell-time dependent stability for the impulsive system (1)-(2). At first, we give the following definition: Definition 1. Consider the following nonlinear system: x(t ) f ( x(t ), t ), t (tk , tk 1 ], k x(tk ) Jx(tk )
where hk
(3)
tk 1 tk satisfies (2), f ( x(t ), t ) satisfies f (0,0) 0 ,and for x(t ), y(t )
n
there exists L 0 such that |f ( x(t ), t ) f ( y(t ), t )| L|x(t ) y(t )| . Then, the trivial solution to system (3) is: stable if for any 0 , there exists ( ) 0 such that x(0) implies x(t ) for any t 0 .
asymptotically stable if it is stable and there exists lim x(t ) 0 whenever x
0 such
that
x(0) .
Based on the definition we have the following lemma: Lemma 1 [23]. Assume that there exist positive scalars c1 ,c2 , a piecewise continuous functional
Va ( x(t ))
and another time-varying one
Vb ( x(t ), t ) ,
t (tk , tk 1 ] along the trajectory of system (3) satisfying i)
c1|x(tk )|2 Va ( x(tk )) c2 |x(tk )|2 ;
ii) Vb ( x(tk ), tk ) Vb ( x(tk 1 ), tk 1 ); iii) For W ( x(t ), t ) (tk 1 tk )Va ( x(t )) (t tk )(Va ( x(tk )) Va ( x(tk ))) Vb ( x(t ), t ) ,
W ( x(t ), t ) 0 , t (tk , tk 1 ) . Then the trivial solution to impulsive system (3) is asymptotically stable. Remark 1: The piecewise functional W ( x(t ), t ) in Lemma 1 is time-varying and decreasing, but not imposed positive definite or continuous. It is different from a usual Lyapunov functional; in the following we refer to it as a Lyapunov-like
functional. A new Lyapunov-like functional will be constructed in the paper. To deal with the derivative of the Lyapunov-like functional, we need an equality that is relevant to
1 Z M ( , Z1 , Z 2 ) 1 0 where Z1
n1
Lemma 2.
n2
, Z2
For U
0 , Z 2
1
and (0,1) . For M ( , Z1 , Z 2 ) we have ( n1 n2 )( n1 n2 )
, the following inequality holds:
0 Z11 T 0 T U U U U , (0,1) . Z2 (1 ) Z 21 0
Z M ( , Z1 , Z 2 ) 1 0 Proof : Since
Z11 0 0, 1 (1 ) Z 2 0 1 Z1 U 0
0 Z11 1 Z1 0 U 0 1 (1 ) Z 21 1 Z 2 0
T
0 0 . 1 1 Z 2
That is,
1 Z1 0
0 Z11 T 0 T U U U U 0 1 1 Z 0 (1 ) Z 2 1 2
Noting
1 Z1 0
0 Z M ( , Z1 , Z 2 ) 1 Z2 0
1 1
0 , Z 2
it follows that
Z M ( , Z1 , Z 2 ) 1 0 Corollary 1.
0 Z11 T 0 T U U U U . Z 2 (1 ) Z 21 0
If there exist X1
n1
, X2
n2
and Y1 , Y2
n1 n2
0, 1 Z1 0
0 X Y 0 T1 1 (1 ) T Z2 Y1 0 Y2
Y2 0, X 2
the following inequality holds:
(1 ) X 1 Y1 (1 )Y2 M ( , Z1 , Z 2 ) , (0,1) . X2 Proof : Let
such that for
Z U T T1 Y1
Y2 , Z 2
and then from Lemma 2,
0 Z11 T 0 Z T M ( , Z1 , Z 2 ) 1 U U U U 1 (1 ) Z 2 0 Z2 0 0 Z1 Z 1 T 0 Z 2 Y1 Z T1 Y1
Y 2 Z11 Z 2 0 T
T
Y2 Z1 Z 2 Y1T
Y2 Z 2
Z 1 Y 2 T ( 1 Z) Y 1 Z 2 0
1 2
(1 ) Xˆ 1 Y1 (1 )Y2 Xˆ 2 where Xˆ 1 Z1 Y1Z 21Y1T , Xˆ 2 Z 2 Y2T Z11Y2 . On the other hand, from the condition of Corollary 1 it follows
X 1 Z1 Y1Z 21Y1T , X 2 Z 2 Y2T Z11Y2 . So,
(1 ) X 1 Y1 (1 )Y2 (1 ) Xˆ 1 Y1 (1 )Y2 . X2 Xˆ 2 This means Corollary 1 holds. Corollary 2.
For Y
n1 n2
Z1 such that T Y
0 M ( , Z1 , Z 2 ) T Y
Y 0 , then for (0,1) Z 2 Y . 0
Proof: Let X 1 0 , X 2 0 and Y1 Y2 Y , and then Corollary 1 reduces to Corollary 2. Remark 2. Recently some inequalities were employed in [24,25] to estimate the derivative of a Lyapunov functional. When n1 n2 n , Corollary 1 reduces to the inequality in [24], while Corollary 2 reduces to the inequality in [25]. So Lemma 1 is less conservative than the inequality in [24], even less than the inequality in [25]. In this paper we will employ Lemma 1 as well as the following lemmas to deal with the derivative of the Lyapunov-like functional to be constructed: Lemma 3 [15]. For a given matrix R 0 , the following inequality holds for all continuously differentiable function in [a, b]
n
1 3 T R T R , ba ba b 2 where (b) (a) , (b) (a) (u)du. b a a
b
T (u ) R (u )du
a
f ( x) ax2 bx c with x
Lemma 4. For the quadratic function
a, b, c
n,
and
then f ( x) 0 , x [ x1 , x2 ] if a 0 , f ( x1 ) 0 and f ( x2 ) 0 .
Proof: For 0 y
n
, consider yT f ( x) y ( yT ay) x2 ( yT by) x ( yT cy) . Since a 0 , we
have yT ay 0 .When yT ay 0 , by yT f ( x1 ) y 0 and yT f ( x2 ) y 0 , it is obvious yT f ( x) y 0 x [ x1 , x2 ] ;
when yT ay 0 , by yT f ( x1 ) y 0 and yT f ( x2 ) y 0 we can also
conclude yT f ( x) y 0 x [ x1 , x2 ] . Therefore, f ( x) 0 x [ x1 , x2 ] .
3. Main results For t (tk , tk 1 ] , a Lyapunov-like functional in the form of W ( x(t ), t ) , as in Lemma 1, is constructed for system (1): W ( x(t ), t ) (tk 1 tk )Va ( x(t )) (t tk )(Va ( x(tk )) Va ( x(tk ))) Vb ( x(t ), t ) ,
with Va ( x(t )) xT (t ) Px(t ) and Vb ( x(t ), t ) (tk 1 tk ) xT (t ) P0 x(t ) (t tk )( xT (tk ) P0 x(tk ) xT (tk 1 ) P0 x(tk 1 ))
x(tk ) (tk 1 t )( x(t ) x(tk ))T Q( x(t ) x(tk )) 2 R x(tk 1 ) t
(t tk )( x(t ) x(tk 1 ))T S ( x(t ) x(tk 1 )) (tk 1 t ) xT ( s) Zx( s)ds tk
T
x( s ) t (tk 1 t ) x(tk ) tk x(t ) k 1
(t tk )
tk 1
t
(t tk )
tk 1
t
2
tk 1
t
Q1 * *
M1 Q2 *
M 2 x( s ) M 3 x(tk ) ds Q3 x(tk 1 )
x(tk ) xT ( s) Z1 x( s)ds 2(t tk )( x(t ) x(tk 1))T R 1 x(tk 1 )
xT (s)Yx(s)ds 2(t tk )
tk 1
t
t
xT ( s)dsL3 x( s)ds tk
xT (s)ds L1 x(tk ) L2 x(tk 1 )
(4)
(tk 1 t )2 2
t
tk
xT ( s)W1 x( s)ds
(t tk )2 2
tk 1
t
xT ( s)W2 x( s)ds
where the related matrices will be defined in the following Theorem 1. Remark 3: Lyapunov-like functional (4) extends recently reported ones in [23] as well as [21-22] by adding the second order function in t
(tk 1 t )2 2
t
tk
xT ( s)W1 x( s)ds
(t tk )2 2
tk 1
t
xT ( s)W2 x( s)ds
and the four following terms:
(t tk )
tk 1
t
xT ( s) Z1 x(s)ds ,
x(tk ) 2(t tk )( x(t ) x(tk 1 ))T R1 , x(tk 1 )
(t tk )
tk 1
t
2
tk 1
t
xT (s)Yx(s)ds 2(t tk )
tk 1
t
xT (s)ds L1 x(tk ) L2 x(tk 1 ) ,
t
xT ( s)dsL3 x( s)ds , tk
which are based on the system information on [t , tk 1 ] .
The first one is about the
integral of the derivative of the state on [t , tk 1 ] , the second one involves the difference of the state on [t , tk 1 ] , the third one introduces
tk 1
t
x( s)ds , the integral of
the state on [t , tk 1 ] , and the cross terms among this integral and the impulsive state, the fourth one is a cross term of the integral of the state on [t , tk 1 ] and that on (tk , t ] . Having employed the system information on both (tk , t ] and [t , tk 1 ] , the Lyapunov-like functional (4) is two-sided. Now using two-sided Lyapunov-like functional (4), we are in a position to propose a dwell-time dependent stability result for system (1)-(2) as follows. Theorem 1. For given h h 0 ，system (1)-(2) is asymptotically stable, if there exist P , Y , Z , Z1 , W1 , W2 , Q1 R , R1
n2 n
, Ui
2nn
such that for h {h, h }
n
, P0 , S , Q , Qi
(i 1, 2) , Li , M i
nn
n
(i 2,3) , matrices
(i 1, 2,3) , N j R5nn ( j 1, 2,
,6)
0 h(1 3 ) (h 2 / 2)e1T AT W1 Ae1 * * * *
hN 4 3hN 5 hZ1 0 * 3hZ1 * * *
*
0 h( 2 3 ) (h 2 / 2)e1T AT W2 Ae1 * * * * where
hN1 hZ * *
3hN 2 0 3hZ *
E1TU1 0 0 W1
*
*
*
E2TU 2 0 0 0, 0 1 W2 3
E1T U 2 0 0 W2 *
E2TU1 0 0 0, 0 1 W1 3
T 0 he1T He{PA P0 A}e1 e2T ( J T ( P P0 ) J P)e2 e3T P0e3 e12 Qe12 T T He{e12 RG} e13 Se13 He{N1e12 3N2e124 N3e12 } T He{N6e13} He{e13 R1G} He{N4e13 3N5e135}
E1T He{U1 U 2 }E1 3E2T He{U1 U 2 }E2 W 0 W 0 T 1 E1T 1 E1 3E2 E2 , 0 W2 0 W2 T 1 He{e12 QAe1 e1T AT RG} e1T AT ZAe1 He{e1T M1e2 e1T M 2e3} e1T Q1e1
He{N6 Ae5} e5T Ye5 He{e5T L1e2 e5T L2e3} He{e5T L3e1} , T 2 He{N3 Ae4 e4T M1e2 e4T M 2e3} e4T Q1e4 He{e13 SAe1}
e1T AT Z1 Ae1 He{( Ae1 )T R1G} He{e1T L3e4 } e1T Ye1 He{e1T L1e2 e1T L2e3} , 3 e2T Q2e2 He{e2T M 3e3} e3T Q3e3 ,
with ei 0n(i 1) n
In
0n(5i ) n (i 1, 2,
,5) , e12 e1 Je2 , e13 e1 e3 ,
e124 e1 Je2 2e4 , e135 e1 e3 2e5 ,
e12 e124 e G 2 , E1 , E2 . e13 e135 e3
(5)
(6)
Proof: System (1)-(2) is a special case of system (3), stated in Lemma 1, and Lyapunov-like functional (4) satisfies (i ) and (ii ) of Lemma 1. In the following we show the Lyapunov-like functional also satisfies (iii) of Lemma 1. At first, for t (tk , tk 1 ] define
1 (t )
2 (t )
1 t tk
t
tk
x( s)ds ,
tk 1 1 x( s)ds , tk 1 t t
and
(t ) xT (t ) xT (tk ) xT (tk 1 ) 1T (t ) 2T (t ) . T
For tk t tk 1 (k ) , computing the time derivative of Lyapunov-like functional (4) along the trajectory of system (1), and using the notations in Theorem 1, we represent the derivative as W ( x(t ), t ) (tk 1 tk )Va ( x(t )) Va ( x(tk )) Va ( x(tk )) Vb ( x(t ), t ) ,
(7)
with Va ( x(t )) T (t )(e1T PAe1 e1T AT Pe1 ) (t ) , Vb ( x(t ), t ) (tk 1 tk ) T (t ) He{e1T P0 Ae1} (t ) T (t )(e2T J T P0 Je2 e3T P0e3 ) (t ) , T T T T (t )(e12 Qe12 He{e12 RG}) (t ) (tk 1 t ) T (t ) He{e12 QAe1 e1T AT RG} (t ) T T (t )e1T3Se 13 t( ) t ( tk )Tt He ( )e SAe 1t } t(k ) t (1x tT Zx) t ( ) 1{3
t
xT (s)Zx(s)ds (tk 1 t ) T (t )(e1T Q1e1 He{e1T M1e2 e1T M 2e3} 3 ) (t ) tk t
xT ( s) Q d s ( kt T)t 1 x( s) tk
(t tk ) xT (t )Z1 x(t )
tk 1
t
( )t ( 4TH {e1 e2 MT4 e
2
e3M} e3
xT ( s) Z1 x( s)ds
T (t ) He{(e13 )T R1G} (t ) (t tk ) T (t ) He{( Ae1 )T R1G} (t ) (t tk ) T (t )(e1T Ye1 He{e1T L1e2 e1T L2e3}) (t )
tk 1
t
xT ( s)Yx( s)ds (tk 1 t ) T (t )( He{e5T L 1e 2 e5T L e32}) (t )
(tk 1 t ) T (t )He e{T5L e3 1} (t tk )He e{T L1 e 3}4t( )
) ( ) t
( )
t (tk 1 t )2 T x (t )W1 x(t ) (tk 1 t ) xT (s)W1 x(s)ds tk 2
tk 1 (t tk )2 T x (t )W2 x(t ) (t tk ) xT (s)W2 x(s)ds . t 2
Now we deal with the integrals in the time derivative of Lyapunov-like functional (4). By defining
(t ) xT (t ) xT (tk ) xT (tk 1 ) 1T (t ) . T
and Lemma 3 we have t
xT ( s) Zx( s)ds tk
1 T (t )(e1 e2 )T Z (e1 e2 ) (t ) t tk
3 T (t )(e1 e2 2e4 )T Z (e1 e2 2e4 ) (t ) . t tk
Noting x(tk ) Jx(tk ) it follows t
xT ( s) Zx( s)ds tk
1 T 3 T T (t )e12T Ze12 (t ) (t )e124 Ze124 (t ) , t tk t tk
where e12 and e124 are defined in Theorem 1. Since there exist N1 , N2
5 nn
such
that [15]
1 T T e12 Ze12 (t tk ) N1Z 1 N1T N1e12 e12 N1T t tk
3 T T e124 Ze124 3(t tk ) N 2 Z 1 N 2T 3N 2e124 3e124 N 2T . t tk
Therefore t
T N1T (t ) xT ( s) Zx( s)ds T (t ) (t tk ) N1Z 1 N1T N1e12 e12
tk
T T (t ) 3(t tk ) N2 Z 1 N2T 3N2e124 3e124 N2T (t ) . (8)
Similarly there exist N 4 , N5
tk 1
t
5 nn
such that
xT ( s) Z1 x(s)ds T (t ) (tk 1 t ) N4 Z11 N4T N4e13 e1T3 N 4T (t ) T T (t ) 3(tk 1 t ) N5 Z11 N5T 3N5e135 3e135 N5T (t ) .
By Lemma 3
(9)
t
tk 1
tk
t
(tk 1 t ) xT (s)W1 x(s)ds (t tk )
xT (s)W2 x(s)ds
tk 1 t T t tk T (t )(e1 Je2 )T W1 (e1 Je2 ) (t ) (t )(e1 e3 )T W2 (e1 e3 ) (t ) t tk tk 1 t
3
tk 1 t T (t )(2e4 e1 Je2 )T W1 (2e4 e1 Je2 ) (t ) t tk
3
t tk T (t )(2e5 e1 e3 )T W2 (2e5 e1 e3 ) (t ) tk 1 t
Let
t tk t tk tk 1 t 1 and then , . By lemma 2 it follows tk 1 tk tk 1 t 1 t tk t
tk 1
tk
t
(tk 1 t ) xT (s)W1 x(s)ds (t tk )
xT (s)W2 x(s)ds
W 0 t tk t t T (t ) E1T 1 U1W11U1T k 1 U 2W21U 2T E1 (t ) He{U1 U 2 } tk 1 tk tk 1 tk 0 W2 W 0 t tk t t 3 T (t ) E2T 1 U1W11U1T k 1 U 2W21U 2T E2 (t ) He{U1 U 2 } tk 1 tk tk 1 tk 0 W2 (10)
Based on Jensen inequality [26] it is derived that t
tk 1
tk
t
xT ( s)Q1 x(s)ds (t tk ) 1T (t )Q1 1 (t ) ,
xT (s)Yx( s)ds (tk 1 t ) 2T (t )Y 2 (t ) .
(11) To make full use of the integral of the state, integrating both sides of system (1) from
t k to t (tk , tk 1 ] , we obtain the integral equation t
x(t ) x(tk ) A x( s)ds . tk
Therefore, there exists N3
5 nn
(12)
such that
0 2 T (t ) N3 ( x(t ) x(tk )) 2(t tk ) T (t ) N3 A1 (t ) .
(13)
Analogously, integrating both sides of system (1) from t (tk , tk 1 ] to tk 1 , we obtain the integral equation x( tk 1 ) x( t)
So there exists N6
5 nn
such that
tk 1
A t
x( s). d s
(14)
0 2 T (t ) N6 ( x(t ) x(tk 1 )) 2(tk 1 t ) T (t ) N6 A 2 (t ) .
(15)
From (7)-(11), (13) and (15) it follows that
W ( x(t ), t ) T (t )Π(t ) (t ) ,
(16)
where
Π(t ) 0 (t tk )(2 3 ) (tk 1 t )(1 3 )
(tk 1 t )2 T T (t tk )2 T T e1 A W1 Ae1 e1 A W2 Ae1 2 2
(t tk )( N1Z 1 N1T 3N2 Z 1 N2T ) (tk 1 t )( N4 Z11 N4T 3N5 Z11N5T )
E1T (
t tk t t U1W11U1T k 1 U 2W21U 2T ) E1 tk 1 tk tk 1 tk
3E2T (
t tk t t U1W11U1T k 1 U 2W21U 2T ) E2 , tk 1 tk tk 1 tk
(17)
with i (i 0,1, 2,3) given in Theorem 1. On the other hand, from (5)-(6), by Schur complement lemma, it is derived that for hk {h, h }
0 E1TU 2W21U 2T E1 3E2TU 2W21U 2T E2 hk (1 3 ) hk N 4 Z11 N 4T 3hk N5 Z11 N5T
hk2 T T e1 A W1 Ae1 0 , 2
and
0 E1TU1W11U1T E1 3E2TU1W11U1T E2 hk (2 3 ) hk N1Z 1 N1T 3hk N 2 Z 1 N 2T
hk2 T T e1 A W2 Ae1 0 . 2
Noting e1T ATW1 Ae1 0 , e1T ATW2 Ae1 0 , by Lemma 4 we have for hk [ h, h ]
0 E1TU 2W21U 2T E1 3E2TU 2W21U 2T E2 hk (1 3 ) hk N 4 Z11 N 4T 3hk N5 Z11 N5T and
0 E1TU1W11U1T E1 3E2TU1W11U1T E2
hk2 T T e1 A W1 Ae1 0 , 2
hk (2 3 ) hk N1Z 1 N1T 3hk N 2 Z 1 N 2T
hk2 T T e1 A W2 Ae1 0 . 2
By the definition of Π(t ) in (17), it follows that Π(tk ) 0 and Π(tk 1 ) 0 . Noting
d 2(t ) e1T AT (W1 W2 ) Ae1 0 , Using Lemma 4 leads to Π(t ) 0 for t (tk , tk 1 ) . So 2 dt from (16) it is concluded W ( x(t ), t ) 0 for t (tk , tk 1 ) . By Lemma 1, system (1)-(2) is asymptotically stable. This completes the proof. Remark 4: Note that the derivative of Lyapunov-like functional (4) includes integrals of the state and integrals coupled by [t , tk 1 ] and [tk , t ] . To deal with the coupled integrals, instead of the inequalities recently reported in [24,25], an advanced one of Lemma 2 is employed, as shown in (10). As for the integrals of the state, the integral equations (12) and (14) are introduced to make full use of the system information. Both of the two techniques aim to derive less conservative results, as demonstrated in Section 4. Remark 5: Recently some inequalities were reported in [27-29]. If they are employed to deal with the derivative of Lyapunov-like functional (4), even less conservative stability result may be obtained but at cost of more complexity. However, the stability results in this paper leave little room to improve, as seen from Examples 2 & 3 in Section 4. Remark 6: In the form of LMIs, Theorem 1 provides a stability criterion for impulsive system (1)-(2), and has a polynomial-time complexity. For the LMIs, the total number of scalar decision variables is M 50n2 6n while the row size L 25n , so the numerical complexity of Theorem 1 is proportional to LM 3 [34].
When R1 0 , L1 L2 L3 0 , N4 N5 N6 0 , U1 U 2 0 , it is easy to verify that Theorem 1 can lead to the stability result in [23], which is stated as follows: Corollary 1 [23]. For given h h 0 ，system (1)-(2) is asymptotically stable, if there exist P , Z , Q1
n
, P0 , S , Q , Qi
n
(i 2,3) , matrices R
Ni R 4nn (i 1, 2,3) such that
0 h(1 3 ) 0 ,
n2 n
, Mi
nn
,
0 h(2 3 ) hN1 * hZ * *
3hN 2 0 0, 3hZ
where T 0 he1T He{PA P0 A}e1 e2T ( J T ( P P0 ) J P)e2 e3T P0e3 e12 Qe12 T T He{e12 RG} e13 Se13 He{N1e12 3N2e124 N3e12 } , T 1 He{e12 QAe1 e1T AT RG} e1T AT ZAe1 He{e1T M1e2 e1T M 2e3} e1T Q1e1 , T 2 He{N3 Ae4 e4T M1e2 e4T M 2e3} e4T Q1e4 He{e13 SAe1},
3 e2T Q2e2 He{e2T M 3e3} e3T Q3e3 ,
e1 I
0 0 0 ,e2 0 I
0 0 ,e3 0 0 I
0 ,
e4 0 0 0 I , e12 e1 Je2 , e13 e1 e3 , e124 e1 Je2 2e4 , G [e2T e3T ]T . When h h h , by Theorem 1 we can get a stability result for impulsive systems with periodic impulse: Corollary 2 (Periodic impulses case). For given h 0 , system (1) with the period
h
is
W1
,
R , R1
asymptotically
W2 n2 n
,
Q1
, Ui
n
2nn
stable, ,
P0
,
if
there
S
,
(i 1, 2) , Li , M i
Q nn
exist ,
Qi
, Y
, Z
, Z1 ,
(i 2,3)
,
matrices
P n
(i 1, 2,3) , N j R5nn ( j 1, 2,
,6)
satisfying (5)-(6). If matrix A is Hurwitz or anti-Hurwitz (there exists P
n
such that PA AT P 0
or PA AT P 0 ), impulsive system (1)-(2) will be asymptotically stable with the minimal dwell-time or the maximal dwell-time. The stability result is stated as follows: Theorem 2 (Maximal dwell-time). For given h 0 , assume that there exist P
, Y , Z , Z1 , W1 , W2 , Q1
R , R1
n2 n
, Ui
2nn
n
, P0 , S , Q , Qi
(i 1, 2) , Li , M i
nn
n
(i 2,3) , matrices
(i 1, 2,3) , N j R5nn ( j 1, 2,
,6)
satisfying (5)-(6) and PA AT P 0 . Then, for any 0 h and any impulsive sequence {tk }k satisfying tk 1 tk h , impulsive system (1) is asymptotically
stable. Proof: The proof is similar to that of [23], so it is omitted. Theorem 3 (Minimal dwell-time). For given h 0 , assume that there exist P
, Y , Z , Z1 , W1 , W2 , Q1
R , R1
n2 n
, Ui
2nn
n
, P0 , S , Q , Qi
(i 1, 2) , Li , M i
nn
n
(i 2,3) , matrices
(i 1, 2,3) , N j R5nn ( j 1, 2,
,6)
satisfying (5)-(6) and PA AT P 0 . Then, for any impulsive sequence {tk }k satisfying h tk 1 tk , impulsive system (1) is asymptotically stable. Being similar to [23], the proof is omitted.
4. Examples In this section, examples are given to demonstrate the proposed results have less conservatism than some existing ones. Example 1. Consider the following impulsive system with
3.5 0.2 1.3 0.1 A ,J . 0.1 0.6 0.1 0.01 The object is to find the maximum dwell-time range for which the aperiodic impulsive system is stable. This can be accomplished from two optimization problems: One is （i）Find a h1 that satisfies LMIs of Theorem 1; （ii） max h subject to h h1 and LMIs of Theorem 1. The other is （i）Find a h2 that satisfies LMIs of Theorem 1; （ii） min h subject to h h2 and LMIs of Theorem 1. To solve those problems, we can turn to Matlab LMI Tool Box for help. For this example, the maximum dwell-time range obtained is [0.0780, 5.2968]. For the periodic impulsive case, we aim to find the maximum periodic range for which the system is stable. This can be achieved from the following optimization problems: max h subject to LMIs of Corollary 2, and min h subject to LMIs of Corollary 2. The optimization problems can be solved via Matlab LMI Tool Box. For this example, the maximum periodic range computed is [0.0780, 5.6374]. The maximum dwell-time range or periodic range in this paper and those in [17,18,
21-23] are listed in Table 1. Table 1. The maximum dwell-time range or periodic range Methods
dwell-time range
periodic range
[0.0802,1.2082] [0.0780,4.0867] [21] [0.0780,2.3214] [0.0780,4.6337] [22] [0.0780, 2.6417] [0.0780, 5.2789] [23] [0.0780, 5.2789] [17] [0.0780, 5.4697] [18] [0.0780, 5.2968] [0.0780, 5.6374] This paper As is shown in Table 1, the dwell-time ranges obtained by Theorem 1 and the periodic ranges obtained by Corollary 2 in this paper cover those by the stability results in [17,18, 21-23]. In this sense, the stability results Theorem 1 and Corollary 2 in this paper have less conservatism than those in [17,18,21-23].
Choose the period h 5 and the initial state x(0) 1 1T , and then the state trajectory for the impulsive system is plotted in Fig. 1:
Fig. 1 The trajectory of the impulse system
As seen from the figure, the impulse system is stable. Example 2. Consider impulsive system (1) with
1 3 0.5 0 A ,J . 1 2 0 0.5 Since A is anti-Hurwitz and J is Schur, the dwell-time hk should be sufficiently small for the impulsive system to be stable. The impulsive system is expected to be asymptotically stable with the dwell-time hk h , where h is referred to the maximal dwell-time. By Theorem 2 and the stability results in [21-23], the maximal dwell-time h can be obtained, which is given in Table 2. Table 2. The maximal dwell-time for aperiodic impulsive systems
Methods [21] [22] [23] Theorem 2 in this paper h 0.4471 0.4483 0.4618 0.4620 As is shown in Table 2, the maximal dwell-time h obtained by Theorem 2 in this paper is larger than those by the stability results in [21-23]. In term of the maximal dwell-time, Theorem 2 in this paper has less conservatism than the stability results in [21-23]. Note that for this example the maximal dwell-time h obtained by Theorem 2 is identical with the analytical result 0.4620 coincidently [21]. Example 3. Consider impulsive system (1) with
1 0 2 1 A ,J . 1 2 1 3 Contrary to example 2, in this example A is Hurwitz and J is anti-Schur, the dwell-time hk should be sufficiently large for the impulsive system to be stable. The impulsive system is expected to be asymptotically stable with the dwell-time hk h , where h is referred to the minimal dwell-time. By Theorem 3 and the stability results in [21-23], the minimal dwell-time h can be obtained as in Table 3. Table 3. The minimal dwell-time for aperiodic impulsive systems Methods [21] [22] [23] Theorem 3 in this paper h 1.2323 1.232 1.1417 1.1410 As seen from Table 3, the minimal dwell-time h given by Theorem 3 in this paper is smaller than those in [21-23]. Therefore, the stability result Theorem 3 in this paper has less conservatism than those in [21-23]. Note that the minimal dwell-time 1.1410 obtained in this paper is very close to the analytical one 1.1405 [21].
5. Conclusion In this paper, the dwell-time dependent stability for impulsive systems has been investigated by a Lyapunov-like functional approach. A second order Lyapunov-like functional of time was constructed based on the system information on the interval
[t , tk 1 ] as well as (tk , t ] , where the integrals on [t , tk 1 ] as well as (tk , t ] was introduced. Integral equations of the system and an advanced inequality were used to estimate the derivative of the Lyapunov-like functional. New stability results with ranged dwell-time, maximal dwell-time and minimal dwell-time were obtained for periodic or aperiodic impulsive systems. In the future, the stability for impulsive systems with delays will be considered by employing the Lyapunov-like functional approach.
Declaration of interest statement The authors declared that they have no conflicts of interest to this work. We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted
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A new Lyapunov-like functional approach to dwell-time dependent stability for impulsive systems Hanyong Shao , Lin Shao PII: DOI: Reference:
S0016-0032(19)30829-4 https://doi.org/10.1016/j.jfranklin.2019.11.040 FI 4275
To appear in:
Journal of the Franklin Institute
Received date: Revised date: Accepted date:
8 April 2019 23 September 2019 13 November 2019
Please cite this article as: Hanyong Shao , Lin Shao , A new Lyapunov-like functional approach to dwell-time dependent stability for impulsive systems, Journal of the Franklin Institute (2019), doi: https://doi.org/10.1016/j.jfranklin.2019.11.040
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A new Lyapunov-like functional approach to dwell-time dependent stability for impulsive systems Hanyong Shao1 and Lin Shao2 1 The Institute of Automation, Qufu Normal University, Rizhao 276826, Shandong Province, China 2 The College of Electronics, Communication and Physics, Shandong University of Science and Technology, Qingdao 266590, Shandong Province, China
Abstract: This paper investigates the dwell-time dependent stability for impulsive systems by employing a new Lyapunov-like functional that is of the second order in time t . In contrary to those built on [tk , t ] , a part of the impulsive interval [tk , tk 1 ] , the Lyapunov-like functional is two-sided in the sense of employing the system information on [t , tk 1 ] as well as [tk , t ] . To deal with the derivative of the two-sided Lyapunov-like functional, which involves integrals of the state and integrals coupled by [t , tk 1 ] and [tk , t ] , integral equations of the impulsive systems are introduced and an advanced inequality is employed. By the Lyapunov-like functional theory, new dwell-time dependent stability results with ranged dwell-time, maximal dwell-time and minimal dwell-time are derived for periodic or aperiodic impulsive systems. The stability results turn out to be less conservative than some existing ones, which is illustrated by numerical examples. Keywords: impulsive systems, Lyapunov-like functional, dwell-time dependent stability, linear matrix inequalities (LMIs).
1. Introduction An impulsive system is modeled after many real world processes that undergo abrupt state changes, and it is comprised of three parts; a continuous-time dynamical equation which governs the evolution of the system between two successive impulsive instants; a difference equation which describes how the system state is changed at impulsive instants; and finally a criterion for determining the impulsive instant sequence [1-7]. Stability with minimal dwell-time or maximal dwell-time for impulsive systems was defined and discussed based on the characters of the system matrices in [4]. The stability of impulsive systems has been paid attention due to their applications in many fields such as epidemiology [8], impulsive control [9,10], power
electronics [11], networked control systems [12,13], sampled-data systems [14-18]. For example, in [16], the sampled-data systems were formulated as impulsive systems, and robust stability of sampled-data systems was investigated. The discrete time method is a basic approach to the stability of impulsive systems, by which the impulsive system is transformed into discrete time systems, and then the eigenvalue analysis is conducted. This method, however, encounters difficulties when impulsive systems involve uncertainties or aperiodic impulses. To deal with this problem, a Lyapunov functional method was proposed in [16] to address the stability for impulsive systems, and the stability results were applied to sampled-data systems with uncertainties. To improve the Lyapunov functional method further, there are two methods to deal with the stability for impulsive systems. One is the Lyapunov-like functional approach where the Lyapunov-like functional is not required positive definite or continuous [17-18]. In [17] stability results were derived for periodic impulsive systems with or without polytope uncertainties. Less conservative stability results were obtained by constructing a different Lyapunov-like functional in [18]. The other approach to improve the Lyapunov functional method is the looped-functional method [19-22], where stability results with ranged dwell-time, maximal dwell-time and minimal dwell-time were derived. Convex dwell-time stability characterizations were given for the linear impulsive systems with uncertainties in [19, 20], where the stability analysis was formulated as an infinite dimensional feasibility problem, which was hard to solve. By contrast, the stability results obtained in [21, 22] were in the form of linear matrix inequalities (LMIs) and could be checked easily. Recently the stability for impulsive systems was studied by using the Lyapunov-like functional approach [23], where stability results with ranged dwell-time, maximal dwell-time and minimal dwell-time were derived. The stability results obtained in [23] have improved over those in [21, 22]. However, the Lyapunov-like functional in [23] as well as [17], and the looped-functional in [21] or [22] only used the system information on the interval (tk , t ] , failed to make use of that on [t , tk 1 ] . Therefore, the existing results can be expected to further improve if the Lyapunov-like functional or the looped-functional is expanded by making full use of the system information on the whole impulsive interval [tk , tk 1 ] .It is challenging how to employ the information on [t , tk 1 ] and (tk , t ] to construct a two-sided Lyapunov functional and develop new techniques to deal with the derivative of the two-sided Lyapunov functional. It is worth noting that recently some scholars have turned towards impulsive networks [30,31] and impulsive stochastic functional differential systems [32,33]. For instance, in [33] the stability problem was addressed for time-varying stochastic functional differential systems with distributed-delay dependent impulsive effects. Based on stochastic theory, some stability results were derived by employing a Lyapunov approach. Nevertheless, in this paper we will further study stability for the
linear impulsive systems in [17-23] by a new Lyapunov-like functional. The study features: The Lyapunov-like functional is second order in time t , and two-sided in the sense of its including the system information on both [t , tk 1 ] and (tk , t ] ; An advanced inequality and integral equations of the impulsive system are employed when estimating the derivative of the two-sided Lyapunov-like functional; Improved dwell-time dependent stability results, including those with ranged dwell-time, maximal dwell-time and minimal dwell-time for impulsive systems with periodic or aperiodic impulses, are derived. Numerical examples are given to illustrate the less conservatism of the dwell-time dependent stability results. Notations: Throughout this paper, X T and X 1 denote the transposition and the inverse of the matrix X , respectively. * in the symmetric matrix stands for the symmetric terms. For a symmetric matrix X , X 0 0 denotes X is positive definite (negative definite). I
denotes the identity matrix with appropriate
dimensions. | | is the Euclidean norm for a vector and is the induced matrix norm.
n
,
nm
refer to the set of n dimension vectors, n m matrices and
,
non-negative integers, respectively. min () , max () and () denote the smallest eigenvalue, largest eigenvalue and the spectral radius of a real symmetric matrix. For a given matrix A , He{ A} stands for A AT . For the impulsive instant t k ,
x(tk ) lim x( s) , x(tk ) lim x( s) . s t k
s tk
n
n
and
are the set of symmetric and
symmetric positive definite matrices of size n n respectively.
2. Problem formulation Consider the following linear impulsive system
x(t ) Ax(t ),t x(t ) Jx(t ),t where x(t )
n
is the state vector, A
nn
\
,J
(1) nn
are constant matrices，
{t0 , t1 , t2 ,tk ,} is an increasing sequence of impulse instants. The dwell- time hk tk 1 tk satisfies
0 h hk h ,
(2)
where h and h are known constants. In the case of periodic impulses, the dwell-time hk h , where h is the period. In this paper, we will study the relationship between dwell-time and stability, namely the dwell-time dependent stability for the impulsive system (1)-(2). At first, we give the following definition: Definition 1. Consider the following nonlinear system: x(t ) f ( x(t ), t ), t (tk , tk 1 ], k x(tk ) Jx(tk )
where hk
(3)
tk 1 tk satisfies (2), f ( x(t ), t ) satisfies f (0,0) 0 ,and for x(t ), y(t )
n
there exists L 0 such that |f ( x(t ), t ) f ( y(t ), t )| L|x(t ) y(t )| . Then, the trivial solution to system (3) is: stable if for any 0 , there exists ( ) 0 such that x(0) implies x(t ) for any t 0 .
asymptotically stable if it is stable and there exists lim x(t ) 0 whenever x
0 such
that
x(0) .
Based on the definition we have the following lemma: Lemma 1 [23]. Assume that there exist positive scalars c1 ,c2 , a piecewise continuous functional
Va ( x(t ))
and another time-varying one
Vb ( x(t ), t ) ,
t (tk , tk 1 ] along the trajectory of system (3) satisfying i)
c1|x(tk )|2 Va ( x(tk )) c2 |x(tk )|2 ;
ii) Vb ( x(tk ), tk ) Vb ( x(tk 1 ), tk 1 ); iii) For W ( x(t ), t ) (tk 1 tk )Va ( x(t )) (t tk )(Va ( x(tk )) Va ( x(tk ))) Vb ( x(t ), t ) ,
W ( x(t ), t ) 0 , t (tk , tk 1 ) . Then the trivial solution to impulsive system (3) is asymptotically stable. Remark 1: The piecewise functional W ( x(t ), t ) in Lemma 1 is time-varying and decreasing, but not imposed positive definite or continuous. It is different from a usual Lyapunov functional; in the following we refer to it as a Lyapunov-like
functional. A new Lyapunov-like functional will be constructed in the paper. To deal with the derivative of the Lyapunov-like functional, we need an equality that is relevant to
1 Z M ( , Z1 , Z 2 ) 1 0 where Z1
n1
Lemma 2.
n2
, Z2
For U
0 , Z 2
1
and (0,1) . For M ( , Z1 , Z 2 ) we have ( n1 n2 )( n1 n2 )
, the following inequality holds:
0 Z11 T 0 T U U U U , (0,1) . Z2 (1 ) Z 21 0
Z M ( , Z1 , Z 2 ) 1 0 Proof : Since
Z11 0 0, 1 (1 ) Z 2 0 1 Z1 U 0
0 Z11 1 Z1 0 U 0 1 (1 ) Z 21 1 Z 2 0
T
0 0 . 1 1 Z 2
That is,
1 Z1 0
0 Z11 T 0 T U U U U 0 1 1 Z 0 (1 ) Z 2 1 2
Noting
1 Z1 0
0 Z M ( , Z1 , Z 2 ) 1 Z2 0
1 1
0 , Z 2
it follows that
Z M ( , Z1 , Z 2 ) 1 0 Corollary 1.
0 Z11 T 0 T U U U U . Z 2 (1 ) Z 21 0
If there exist X1
n1
, X2
n2
and Y1 , Y2
n1 n2
0, 1 Z1 0
0 X Y 0 T1 1 (1 ) T Z2 Y1 0 Y2
Y2 0, X 2
the following inequality holds:
(1 ) X 1 Y1 (1 )Y2 M ( , Z1 , Z 2 ) , (0,1) . X2 Proof : Let
such that for
Z U T T1 Y1
Y2 , Z 2
and then from Lemma 2,
0 Z11 T 0 Z T M ( , Z1 , Z 2 ) 1 U U U U 1 (1 ) Z 2 0 Z2 0 0 Z1 Z 1 T 0 Z 2 Y1 Z T1 Y1
Y 2 Z11 Z 2 0 T
T
Y2 Z1 Z 2 Y1T
Y2 Z 2
Z 1 Y 2 T ( 1 Z) Y 1 Z 2 0
1 2
(1 ) Xˆ 1 Y1 (1 )Y2 Xˆ 2 where Xˆ 1 Z1 Y1Z 21Y1T , Xˆ 2 Z 2 Y2T Z11Y2 . On the other hand, from the condition of Corollary 1 it follows
X 1 Z1 Y1Z 21Y1T , X 2 Z 2 Y2T Z11Y2 . So,
(1 ) X 1 Y1 (1 )Y2 (1 ) Xˆ 1 Y1 (1 )Y2 . X2 Xˆ 2 This means Corollary 1 holds. Corollary 2.
For Y
n1 n2
Z1 such that T Y
0 M ( , Z1 , Z 2 ) T Y
Y 0 , then for (0,1) Z 2 Y . 0
Proof: Let X 1 0 , X 2 0 and Y1 Y2 Y , and then Corollary 1 reduces to Corollary 2. Remark 2. Recently some inequalities were employed in [24,25] to estimate the derivative of a Lyapunov functional. When n1 n2 n , Corollary 1 reduces to the inequality in [24], while Corollary 2 reduces to the inequality in [25]. So Lemma 1 is less conservative than the inequality in [24], even less than the inequality in [25]. In this paper we will employ Lemma 1 as well as the following lemmas to deal with the derivative of the Lyapunov-like functional to be constructed: Lemma 3 [15]. For a given matrix R 0 , the following inequality holds for all continuously differentiable function in [a, b]
n
1 3 T R T R , ba ba b 2 where (b) (a) , (b) (a) (u)du. b a a
b
T (u ) R (u )du
a
f ( x) ax2 bx c with x
Lemma 4. For the quadratic function
a, b, c
n,
and
then f ( x) 0 , x [ x1 , x2 ] if a 0 , f ( x1 ) 0 and f ( x2 ) 0 .
Proof: For 0 y
n
, consider yT f ( x) y ( yT ay) x2 ( yT by) x ( yT cy) . Since a 0 , we
have yT ay 0 .When yT ay 0 , by yT f ( x1 ) y 0 and yT f ( x2 ) y 0 , it is obvious yT f ( x) y 0 x [ x1 , x2 ] ;
when yT ay 0 , by yT f ( x1 ) y 0 and yT f ( x2 ) y 0 we can also
conclude yT f ( x) y 0 x [ x1 , x2 ] . Therefore, f ( x) 0 x [ x1 , x2 ] .
3. Main results For t (tk , tk 1 ] , a Lyapunov-like functional in the form of W ( x(t ), t ) , as in Lemma 1, is constructed for system (1): W ( x(t ), t ) (tk 1 tk )Va ( x(t )) (t tk )(Va ( x(tk )) Va ( x(tk ))) Vb ( x(t ), t ) ,
with Va ( x(t )) xT (t ) Px(t ) and Vb ( x(t ), t ) (tk 1 tk ) xT (t ) P0 x(t ) (t tk )( xT (tk ) P0 x(tk ) xT (tk 1 ) P0 x(tk 1 ))
x(tk ) (tk 1 t )( x(t ) x(tk ))T Q( x(t ) x(tk )) 2 R x(tk 1 ) t
(t tk )( x(t ) x(tk 1 ))T S ( x(t ) x(tk 1 )) (tk 1 t ) xT ( s) Zx( s)ds tk
T
x( s ) t (tk 1 t ) x(tk ) tk x(t ) k 1
(t tk )
tk 1
t
(t tk )
tk 1
t
2
tk 1
t
Q1 * *
M1 Q2 *
M 2 x( s ) M 3 x(tk ) ds Q3 x(tk 1 )
x(tk ) xT ( s) Z1 x( s)ds 2(t tk )( x(t ) x(tk 1))T R 1 x(tk 1 )
xT (s)Yx(s)ds 2(t tk )
tk 1
t
t
xT ( s)dsL3 x( s)ds tk
xT (s)ds L1 x(tk ) L2 x(tk 1 )
(4)
(tk 1 t )2 2
t
tk
xT ( s)W1 x( s)ds
(t tk )2 2
tk 1
t
xT ( s)W2 x( s)ds
where the related matrices will be defined in the following Theorem 1. Remark 3: Lyapunov-like functional (4) extends recently reported ones in [23] as well as [21-22] by adding the second order function in t
(tk 1 t )2 2
t
tk
xT ( s)W1 x( s)ds
(t tk )2 2
tk 1
t
xT ( s)W2 x( s)ds
and the four following terms:
(t tk )
tk 1
t
xT ( s) Z1 x(s)ds ,
x(tk ) 2(t tk )( x(t ) x(tk 1 ))T R1 , x(tk 1 )
(t tk )
tk 1
t
2
tk 1
t
xT (s)Yx(s)ds 2(t tk )
tk 1
t
xT (s)ds L1 x(tk ) L2 x(tk 1 ) ,
t
xT ( s)dsL3 x( s)ds , tk
which are based on the system information on [t , tk 1 ] .
The first one is about the
integral of the derivative of the state on [t , tk 1 ] , the second one involves the difference of the state on [t , tk 1 ] , the third one introduces
tk 1
t
x( s)ds , the integral of
the state on [t , tk 1 ] , and the cross terms among this integral and the impulsive state, the fourth one is a cross term of the integral of the state on [t , tk 1 ] and that on (tk , t ] . Having employed the system information on both (tk , t ] and [t , tk 1 ] , the Lyapunov-like functional (4) is two-sided. Now using two-sided Lyapunov-like functional (4), we are in a position to propose a dwell-time dependent stability result for system (1)-(2) as follows. Theorem 1. For given h h 0 ，system (1)-(2) is asymptotically stable, if there exist P , Y , Z , Z1 , W1 , W2 , Q1 R , R1
n2 n
, Ui
2nn
such that for h {h, h }
n
, P0 , S , Q , Qi
(i 1, 2) , Li , M i
nn
n
(i 2,3) , matrices
(i 1, 2,3) , N j R5nn ( j 1, 2,
,6)
0 h(1 3 ) (h 2 / 2)e1T AT W1 Ae1 * * * *
hN 4 3hN 5 hZ1 0 * 3hZ1 * * *
*
0 h( 2 3 ) (h 2 / 2)e1T AT W2 Ae1 * * * * where
hN1 hZ * *
3hN 2 0 3hZ *
E1TU1 0 0 W1
*
*
*
E2TU 2 0 0 0, 0 1 W2 3
E1T U 2 0 0 W2 *
E2TU1 0 0 0, 0 1 W1 3
T 0 he1T He{PA P0 A}e1 e2T ( J T ( P P0 ) J P)e2 e3T P0e3 e12 Qe12 T T He{e12 RG} e13 Se13 He{N1e12 3N2e124 N3e12 } T He{N6e13} He{e13 R1G} He{N4e13 3N5e135}
E1T He{U1 U 2 }E1 3E2T He{U1 U 2 }E2 W 0 W 0 T 1 E1T 1 E1 3E2 E2 , 0 W2 0 W2 T 1 He{e12 QAe1 e1T AT RG} e1T AT ZAe1 He{e1T M1e2 e1T M 2e3} e1T Q1e1
He{N6 Ae5} e5T Ye5 He{e5T L1e2 e5T L2e3} He{e5T L3e1} , T 2 He{N3 Ae4 e4T M1e2 e4T M 2e3} e4T Q1e4 He{e13 SAe1}
e1T AT Z1 Ae1 He{( Ae1 )T R1G} He{e1T L3e4 } e1T Ye1 He{e1T L1e2 e1T L2e3} , 3 e2T Q2e2 He{e2T M 3e3} e3T Q3e3 ,
with ei 0n(i 1) n
In
0n(5i ) n (i 1, 2,
,5) , e12 e1 Je2 , e13 e1 e3 ,
e124 e1 Je2 2e4 , e135 e1 e3 2e5 ,
e12 e124 e G 2 , E1 , E2 . e13 e135 e3
(5)
(6)
Proof: System (1)-(2) is a special case of system (3), stated in Lemma 1, and Lyapunov-like functional (4) satisfies (i ) and (ii ) of Lemma 1. In the following we show the Lyapunov-like functional also satisfies (iii) of Lemma 1. At first, for t (tk , tk 1 ] define
1 (t )
2 (t )
1 t tk
t
tk
x( s)ds ,
tk 1 1 x( s)ds , tk 1 t t
and
(t ) xT (t ) xT (tk ) xT (tk 1 ) 1T (t ) 2T (t ) . T
For tk t tk 1 (k ) , computing the time derivative of Lyapunov-like functional (4) along the trajectory of system (1), and using the notations in Theorem 1, we represent the derivative as W ( x(t ), t ) (tk 1 tk )Va ( x(t )) Va ( x(tk )) Va ( x(tk )) Vb ( x(t ), t ) ,
(7)
with Va ( x(t )) T (t )(e1T PAe1 e1T AT Pe1 ) (t ) , Vb ( x(t ), t ) (tk 1 tk ) T (t ) He{e1T P0 Ae1} (t ) T (t )(e2T J T P0 Je2 e3T P0e3 ) (t ) , T T T T (t )(e12 Qe12 He{e12 RG}) (t ) (tk 1 t ) T (t ) He{e12 QAe1 e1T AT RG} (t ) T T (t )e1T3Se 13 t( ) t ( tk )Tt He ( )e SAe 1t } t(k ) t (1x tT Zx) t ( ) 1{3
t
xT (s)Zx(s)ds (tk 1 t ) T (t )(e1T Q1e1 He{e1T M1e2 e1T M 2e3} 3 ) (t ) tk t
xT ( s) Q d s ( kt T)t 1 x( s) tk
(t tk ) xT (t )Z1 x(t )
tk 1
t
( )t ( 4TH {e1 e2 MT4 e
2
e3M} e3
xT ( s) Z1 x( s)ds
T (t ) He{(e13 )T R1G} (t ) (t tk ) T (t ) He{( Ae1 )T R1G} (t ) (t tk ) T (t )(e1T Ye1 He{e1T L1e2 e1T L2e3}) (t )
tk 1
t
xT ( s)Yx( s)ds (tk 1 t ) T (t )( He{e5T L 1e 2 e5T L e32}) (t )
(tk 1 t ) T (t )He e{T5L e3 1} (t tk )He e{T L1 e 3}4t( )
) ( ) t
( )
t (tk 1 t )2 T x (t )W1 x(t ) (tk 1 t ) xT (s)W1 x(s)ds tk 2
tk 1 (t tk )2 T x (t )W2 x(t ) (t tk ) xT (s)W2 x(s)ds . t 2
Now we deal with the integrals in the time derivative of Lyapunov-like functional (4). By defining
(t ) xT (t ) xT (tk ) xT (tk 1 ) 1T (t ) . T
and Lemma 3 we have t
xT ( s) Zx( s)ds tk
1 T (t )(e1 e2 )T Z (e1 e2 ) (t ) t tk
3 T (t )(e1 e2 2e4 )T Z (e1 e2 2e4 ) (t ) . t tk
Noting x(tk ) Jx(tk ) it follows t
xT ( s) Zx( s)ds tk
1 T 3 T T (t )e12T Ze12 (t ) (t )e124 Ze124 (t ) , t tk t tk
where e12 and e124 are defined in Theorem 1. Since there exist N1 , N2
5 nn
such
that [15]
1 T T e12 Ze12 (t tk ) N1Z 1 N1T N1e12 e12 N1T t tk
3 T T e124 Ze124 3(t tk ) N 2 Z 1 N 2T 3N 2e124 3e124 N 2T . t tk
Therefore t
T N1T (t ) xT ( s) Zx( s)ds T (t ) (t tk ) N1Z 1 N1T N1e12 e12
tk
T T (t ) 3(t tk ) N2 Z 1 N2T 3N2e124 3e124 N2T (t ) . (8)
Similarly there exist N 4 , N5
tk 1
t
5 nn
such that
xT ( s) Z1 x(s)ds T (t ) (tk 1 t ) N4 Z11 N4T N4e13 e1T3 N 4T (t ) T T (t ) 3(tk 1 t ) N5 Z11 N5T 3N5e135 3e135 N5T (t ) .
By Lemma 3
(9)
t
tk 1
tk
t
(tk 1 t ) xT (s)W1 x(s)ds (t tk )
xT (s)W2 x(s)ds
tk 1 t T t tk T (t )(e1 Je2 )T W1 (e1 Je2 ) (t ) (t )(e1 e3 )T W2 (e1 e3 ) (t ) t tk tk 1 t
3
tk 1 t T (t )(2e4 e1 Je2 )T W1 (2e4 e1 Je2 ) (t ) t tk
3
t tk T (t )(2e5 e1 e3 )T W2 (2e5 e1 e3 ) (t ) tk 1 t
Let
t tk t tk tk 1 t 1 and then , . By lemma 2 it follows tk 1 tk tk 1 t 1 t tk t
tk 1
tk
t
(tk 1 t ) xT (s)W1 x(s)ds (t tk )
xT (s)W2 x(s)ds
W 0 t tk t t T (t ) E1T 1 U1W11U1T k 1 U 2W21U 2T E1 (t ) He{U1 U 2 } tk 1 tk tk 1 tk 0 W2 W 0 t tk t t 3 T (t ) E2T 1 U1W11U1T k 1 U 2W21U 2T E2 (t ) He{U1 U 2 } tk 1 tk tk 1 tk 0 W2 (10)
Based on Jensen inequality [26] it is derived that t
tk 1
tk
t
xT ( s)Q1 x(s)ds (t tk ) 1T (t )Q1 1 (t ) ,
xT (s)Yx( s)ds (tk 1 t ) 2T (t )Y 2 (t ) .
(11) To make full use of the integral of the state, integrating both sides of system (1) from
t k to t (tk , tk 1 ] , we obtain the integral equation t
x(t ) x(tk ) A x( s)ds . tk
Therefore, there exists N3
5 nn
(12)
such that
0 2 T (t ) N3 ( x(t ) x(tk )) 2(t tk ) T (t ) N3 A1 (t ) .
(13)
Analogously, integrating both sides of system (1) from t (tk , tk 1 ] to tk 1 , we obtain the integral equation x( tk 1 ) x( t)
So there exists N6
5 nn
such that
tk 1
A t
x( s). d s
(14)
0 2 T (t ) N6 ( x(t ) x(tk 1 )) 2(tk 1 t ) T (t ) N6 A 2 (t ) .
(15)
From (7)-(11), (13) and (15) it follows that
W ( x(t ), t ) T (t )Π(t ) (t ) ,
(16)
where
Π(t ) 0 (t tk )(2 3 ) (tk 1 t )(1 3 )
(tk 1 t )2 T T (t tk )2 T T e1 A W1 Ae1 e1 A W2 Ae1 2 2
(t tk )( N1Z 1 N1T 3N2 Z 1 N2T ) (tk 1 t )( N4 Z11 N4T 3N5 Z11N5T )
E1T (
t tk t t U1W11U1T k 1 U 2W21U 2T ) E1 tk 1 tk tk 1 tk
3E2T (
t tk t t U1W11U1T k 1 U 2W21U 2T ) E2 , tk 1 tk tk 1 tk
(17)
with i (i 0,1, 2,3) given in Theorem 1. On the other hand, from (5)-(6), by Schur complement lemma, it is derived that for hk {h, h }
0 E1TU 2W21U 2T E1 3E2TU 2W21U 2T E2 hk (1 3 ) hk N 4 Z11 N 4T 3hk N5 Z11 N5T
hk2 T T e1 A W1 Ae1 0 , 2
and
0 E1TU1W11U1T E1 3E2TU1W11U1T E2 hk (2 3 ) hk N1Z 1 N1T 3hk N 2 Z 1 N 2T
hk2 T T e1 A W2 Ae1 0 . 2
Noting e1T ATW1 Ae1 0 , e1T ATW2 Ae1 0 , by Lemma 4 we have for hk [ h, h ]
0 E1TU 2W21U 2T E1 3E2TU 2W21U 2T E2 hk (1 3 ) hk N 4 Z11 N 4T 3hk N5 Z11 N5T and
0 E1TU1W11U1T E1 3E2TU1W11U1T E2
hk2 T T e1 A W1 Ae1 0 , 2
hk (2 3 ) hk N1Z 1 N1T 3hk N 2 Z 1 N 2T
hk2 T T e1 A W2 Ae1 0 . 2
By the definition of Π(t ) in (17), it follows that Π(tk ) 0 and Π(tk 1 ) 0 . Noting
d 2(t ) e1T AT (W1 W2 ) Ae1 0 , Using Lemma 4 leads to Π(t ) 0 for t (tk , tk 1 ) . So 2 dt from (16) it is concluded W ( x(t ), t ) 0 for t (tk , tk 1 ) . By Lemma 1, system (1)-(2) is asymptotically stable. This completes the proof. Remark 4: Note that the derivative of Lyapunov-like functional (4) includes integrals of the state and integrals coupled by [t , tk 1 ] and [tk , t ] . To deal with the coupled integrals, instead of the inequalities recently reported in [24,25], an advanced one of Lemma 2 is employed, as shown in (10). As for the integrals of the state, the integral equations (12) and (14) are introduced to make full use of the system information. Both of the two techniques aim to derive less conservative results, as demonstrated in Section 4. Remark 5: Recently some inequalities were reported in [27-29]. If they are employed to deal with the derivative of Lyapunov-like functional (4), even less conservative stability result may be obtained but at cost of more complexity. However, the stability results in this paper leave little room to improve, as seen from Examples 2 & 3 in Section 4. Remark 6: In the form of LMIs, Theorem 1 provides a stability criterion for impulsive system (1)-(2), and has a polynomial-time complexity. For the LMIs, the total number of scalar decision variables is M 50n2 6n while the row size L 25n , so the numerical complexity of Theorem 1 is proportional to LM 3 [34].
When R1 0 , L1 L2 L3 0 , N4 N5 N6 0 , U1 U 2 0 , it is easy to verify that Theorem 1 can lead to the stability result in [23], which is stated as follows: Corollary 1 [23]. For given h h 0 ，system (1)-(2) is asymptotically stable, if there exist P , Z , Q1
n
, P0 , S , Q , Qi
n
(i 2,3) , matrices R
Ni R 4nn (i 1, 2,3) such that
0 h(1 3 ) 0 ,
n2 n
, Mi
nn
,
0 h(2 3 ) hN1 * hZ * *
3hN 2 0 0, 3hZ
where T 0 he1T He{PA P0 A}e1 e2T ( J T ( P P0 ) J P)e2 e3T P0e3 e12 Qe12 T T He{e12 RG} e13 Se13 He{N1e12 3N2e124 N3e12 } , T 1 He{e12 QAe1 e1T AT RG} e1T AT ZAe1 He{e1T M1e2 e1T M 2e3} e1T Q1e1 , T 2 He{N3 Ae4 e4T M1e2 e4T M 2e3} e4T Q1e4 He{e13 SAe1},
3 e2T Q2e2 He{e2T M 3e3} e3T Q3e3 ,
e1 I
0 0 0 ,e2 0 I
0 0 ,e3 0 0 I
0 ,
e4 0 0 0 I , e12 e1 Je2 , e13 e1 e3 , e124 e1 Je2 2e4 , G [e2T e3T ]T . When h h h , by Theorem 1 we can get a stability result for impulsive systems with periodic impulse: Corollary 2 (Periodic impulses case). For given h 0 , system (1) with the period
h
is
W1
,
R , R1
asymptotically
W2 n2 n
,
Q1
, Ui
n
2nn
stable, ,
P0
,
if
there
S
,
(i 1, 2) , Li , M i
Q nn
exist ,
Qi
, Y
, Z
, Z1 ,
(i 2,3)
,
matrices
P n
(i 1, 2,3) , N j R5nn ( j 1, 2,
,6)
satisfying (5)-(6). If matrix A is Hurwitz or anti-Hurwitz (there exists P
n
such that PA AT P 0
or PA AT P 0 ), impulsive system (1)-(2) will be asymptotically stable with the minimal dwell-time or the maximal dwell-time. The stability result is stated as follows: Theorem 2 (Maximal dwell-time). For given h 0 , assume that there exist P
, Y , Z , Z1 , W1 , W2 , Q1
R , R1
n2 n
, Ui
2nn
n
, P0 , S , Q , Qi
(i 1, 2) , Li , M i
nn
n
(i 2,3) , matrices
(i 1, 2,3) , N j R5nn ( j 1, 2,
,6)
satisfying (5)-(6) and PA AT P 0 . Then, for any 0 h and any impulsive sequence {tk }k satisfying tk 1 tk h , impulsive system (1) is asymptotically
stable. Proof: The proof is similar to that of [23], so it is omitted. Theorem 3 (Minimal dwell-time). For given h 0 , assume that there exist P
, Y , Z , Z1 , W1 , W2 , Q1
R , R1
n2 n
, Ui
2nn
n
, P0 , S , Q , Qi
(i 1, 2) , Li , M i
nn
n
(i 2,3) , matrices
(i 1, 2,3) , N j R5nn ( j 1, 2,
,6)
satisfying (5)-(6) and PA AT P 0 . Then, for any impulsive sequence {tk }k satisfying h tk 1 tk , impulsive system (1) is asymptotically stable. Being similar to [23], the proof is omitted.
4. Examples In this section, examples are given to demonstrate the proposed results have less conservatism than some existing ones. Example 1. Consider the following impulsive system with
3.5 0.2 1.3 0.1 A ,J . 0.1 0.6 0.1 0.01 The object is to find the maximum dwell-time range for which the aperiodic impulsive system is stable. This can be accomplished from two optimization problems: One is （i）Find a h1 that satisfies LMIs of Theorem 1; （ii） max h subject to h h1 and LMIs of Theorem 1. The other is （i）Find a h2 that satisfies LMIs of Theorem 1; （ii） min h subject to h h2 and LMIs of Theorem 1. To solve those problems, we can turn to Matlab LMI Tool Box for help. For this example, the maximum dwell-time range obtained is [0.0780, 5.2968]. For the periodic impulsive case, we aim to find the maximum periodic range for which the system is stable. This can be achieved from the following optimization problems: max h subject to LMIs of Corollary 2, and min h subject to LMIs of Corollary 2. The optimization problems can be solved via Matlab LMI Tool Box. For this example, the maximum periodic range computed is [0.0780, 5.6374]. The maximum dwell-time range or periodic range in this paper and those in [17,18,
21-23] are listed in Table 1. Table 1. The maximum dwell-time range or periodic range Methods
dwell-time range
periodic range
[0.0802,1.2082] [0.0780,4.0867] [21] [0.0780,2.3214] [0.0780,4.6337] [22] [0.0780, 2.6417] [0.0780, 5.2789] [23] [0.0780, 5.2789] [17] [0.0780, 5.4697] [18] [0.0780, 5.2968] [0.0780, 5.6374] This paper As is shown in Table 1, the dwell-time ranges obtained by Theorem 1 and the periodic ranges obtained by Corollary 2 in this paper cover those by the stability results in [17,18, 21-23]. In this sense, the stability results Theorem 1 and Corollary 2 in this paper have less conservatism than those in [17,18,21-23].
Choose the period h 5 and the initial state x(0) 1 1T , and then the state trajectory for the impulsive system is plotted in Fig. 1:
Fig. 1 The trajectory of the impulse system
As seen from the figure, the impulse system is stable. Example 2. Consider impulsive system (1) with
1 3 0.5 0 A ,J . 1 2 0 0.5 Since A is anti-Hurwitz and J is Schur, the dwell-time hk should be sufficiently small for the impulsive system to be stable. The impulsive system is expected to be asymptotically stable with the dwell-time hk h , where h is referred to the maximal dwell-time. By Theorem 2 and the stability results in [21-23], the maximal dwell-time h can be obtained, which is given in Table 2. Table 2. The maximal dwell-time for aperiodic impulsive systems
Methods [21] [22] [23] Theorem 2 in this paper h 0.4471 0.4483 0.4618 0.4620 As is shown in Table 2, the maximal dwell-time h obtained by Theorem 2 in this paper is larger than those by the stability results in [21-23]. In term of the maximal dwell-time, Theorem 2 in this paper has less conservatism than the stability results in [21-23]. Note that for this example the maximal dwell-time h obtained by Theorem 2 is identical with the analytical result 0.4620 coincidently [21]. Example 3. Consider impulsive system (1) with
1 0 2 1 A ,J . 1 2 1 3 Contrary to example 2, in this example A is Hurwitz and J is anti-Schur, the dwell-time hk should be sufficiently large for the impulsive system to be stable. The impulsive system is expected to be asymptotically stable with the dwell-time hk h , where h is referred to the minimal dwell-time. By Theorem 3 and the stability results in [21-23], the minimal dwell-time h can be obtained as in Table 3. Table 3. The minimal dwell-time for aperiodic impulsive systems Methods [21] [22] [23] Theorem 3 in this paper h 1.2323 1.232 1.1417 1.1410 As seen from Table 3, the minimal dwell-time h given by Theorem 3 in this paper is smaller than those in [21-23]. Therefore, the stability result Theorem 3 in this paper has less conservatism than those in [21-23]. Note that the minimal dwell-time 1.1410 obtained in this paper is very close to the analytical one 1.1405 [21].
5. Conclusion In this paper, the dwell-time dependent stability for impulsive systems has been investigated by a Lyapunov-like functional approach. A second order Lyapunov-like functional of time was constructed based on the system information on the interval
[t , tk 1 ] as well as (tk , t ] , where the integrals on [t , tk 1 ] as well as (tk , t ] was introduced. Integral equations of the system and an advanced inequality were used to estimate the derivative of the Lyapunov-like functional. New stability results with ranged dwell-time, maximal dwell-time and minimal dwell-time were obtained for periodic or aperiodic impulsive systems. In the future, the stability for impulsive systems with delays will be considered by employing the Lyapunov-like functional approach.
Declaration of interest statement The authors declared that they have no conflicts of interest to this work. We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted
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