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Exponential stability results for uncertain neutral systems with interval time-varying delays and Markovian jumping parameters
Applied Mathematics and Computation 216 (2010) 3396–3407
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Exponential stability results for uncertain neutral systems with interval time-varying delays and Markovian jumping parameters q P. Balasubramaniam *, A. Manivannan, R. Rakkiyappan Department of Mathematics, Gandhigram Rural University, Gandhigram 624 302, Tamilnadu, India
a r t i c l e
i n f o
Keywords: Markovian jumping parameters Global exponential stability Linear matrix inequality Lyapunov–Krasovskii functional Generalized eigenvalue problems
a b s t r a c t This paper deals with the global exponential stability analysis of neutral systems with Markovian jumping parameters and interval time-varying delays. The time-varying delay is assumed to belong to an interval, which means that the lower and upper bounds of interval time-varying delays are available. A new global exponential stability condition is derived in terms of linear matrix inequality (LMI) by constructing new Lyapunov–Krasovskii functionals via generalized eigenvalue problems (GEVPs). The stability criteria are formulated in the form of LMIs, which can be easily checked in practice by Matlab LMI control toolbox. Two numerical examples are given to demonstrate the effectiveness and less conservativeness of the proposed methods. 2010 Elsevier Inc. All rights reserved.
1. Introduction It is well known that, the applications of Markovian jumping parameters are more comprehensive, for instance, in economic systems [1], solar thermal receiver systems [2] and communication systems [3], etc. Markovian jumping parameters are a special class of hybrid systems with two components of mode and state. The dynamics of the jump mode and continuous state are respectively modeled by finite state Markov jumping parameters and differential equations. These systems are usually appropriate to describe dynamic systems subject to abrupt variation in their structures and parameters, such as sudden environment changes, subsystem switching, system noises, executor faults and failures occurred in components or interconnections, etc. Time delay phenomenon is often encountered in many practical systems, for example, aircraft stabilization, biological systems, chemical systems, hydraulic systems, metallurgical processing systems, nuclear reactor and electrical networks. It is well known that the existence of time delays in a system may cause instability or bad system performance. Hence the problem of stability of time-delay systems has received considerable attention in the last two decades, see for example [4] and the references therein. A neutral time-delay system contain delays both in its state and in its derivatives of state. Such system can be found in such places as population ecology, distributed networks containing lossless transmission lines, heat exchangers, robots in contact with rigid environments, etc. Because of its wider application, the problem of the stability of delay-differential neutral system has received considerable attention by many scholars in the last two decades. Neutral delay systems constitute a more general class than those of the retarded type. Stability of these systems proves to be a more complex issue because the system involves the derivative of the delayed state. Especially, in the past few decades increased attention has been devoted to the problem of robust delay-independent stability or delay-dependent stability and
q The work was supported by UGC-SAP (DRS-II), New Delhi, India. The work of the third author was supported by CSIR-SRF under Grant No. 09/715(0013)/ 2009-EMR-I. * Corresponding author. E-mail addresses: [email protected], [email protected] (P. Balasubramaniam).
0096-3003/$ - see front matter 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.04.077
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stabilization via different approaches for linear neutral systems with delayed state and/or input and parameter uncertainties. Therefore, the problem of the stability and stabilization of neutral time-delay systems has attracted considerable attention during the past few years, see for example [5–23]. Markovian jump system (MJSs) which initially introduced by Krasovskii and Lidskii [24] in 1961 are a special class of hybrid systems [25–27]. These systems have two components in the state vector. The first one which varies continuously is referred to be the continuous state of the system and the second one which varies discretely is referred to be the mode of the system. For MJS, the random jumps in system parameters are governed by a Markov process which takes values in a finite set as well. MJS is a suitable mathematical model to represent a class of dynamic systems due to random abrupt variations in their structures, and has many applications such as manufacturing systems, communication systems [28], and fault-tolerant systems. The stability analysis and controller synthesis problems for MJS have been extensively studied see for example [27,29], and references therein. To the best of the authors’ knowledge the problem of delay-range dependent exponential stability results for Markovian jumping neutral systems with interval time-varying delays has not been fully investigated and it is very challenging. In this paper, we discuss the problem of global exponential stability for norm-bounded uncertain neutral Markovian jumping systems (MJSs) with interval time-varying delays. We aim at dynamics of the neutral stochastic stability for all admissible uncertainties as well as interval time-varying delays. By selecting appropriate Lyapunov–Krasovskii functionals, we give the sufficient conditions so that the analysis problem can be tackled in the form of LMIs, which can be easily calculated by MATLAB LMI control toolbox [30] and the associated synthesis problem can be dealt which by solving the generalized eigenvalue problem (GEVP). In order to illustrate the proposed results, we provide real-world examples, that is, partial element equivalent circuit (PEEC) model presented in [31–33]. With the abrupt variation in its structures and parameters, we can present the PEEC model as a stochastic jump one. Notations. Rn and Rnm denote the n-dimensional Euclidean space and the set of all the n m real matrices, AT(or xT) and A1(or x1) denote the transpose and the inverse of matrix A or scalar x,rmax(A) and rmin(A) denote the maximal and minimal eigenvalue of a real matrix A,kAk denotes the Euclidean norm of a matrix A,jaj denotes the absolute value of the scalar a,E{} denotes the mathematics statistical expectation of the stochastic process or vector, Ln2 ½d; 0 is the space of n-dimensional square integrable function vector over [d, 0],P > 0 stands for a positive-definite matrix, I is the unit matrix with appropriate dimensions, ‘‘*” means the symmetric terms in a symmetric matrix.
2. Problem description and preliminaries Given a probability space (X, F, P) where X is the sample space, F is the algebra of events and P is the probability measure defined on F. Let the random process {rt,t P 0} be the Markov stochastic process taking values on a finite set M = {1, 2, . . . , N} with transition rate matrix P = {pij, i,j 2 M} and define the following transition probability from mode i at time t to mode j at time t + Dt as:
Pðr tþDt ¼ jjr t ¼ iÞ ¼
pij Dt þ oðDtÞ; if i – j; 1 þ pii Dt þ oðDtÞ; if i ¼ j;
where Dt > 0 and limDt!0 oðDDttÞ ! 0 with transition probability rates pij P 0 for i, j 2 M, i – j and
ð1Þ PN
j¼1;j–i
pij ¼ pii .
Consider a class of continuous neutral MJSs with time-delays and uncertainties over the space (X, F, P):
_ hðtÞÞ; _ xðtÞ ¼ ½Aðr t Þ þ DAðr t ÞxðtÞ þ ½Bðr t Þ þ DBðrt Þxðt sðtÞÞ þ Cðrt Þxðt xðnÞ ¼ gðnÞ 8 n 2 ½d; 0;
ð2Þ
where xðtÞ 2 Rn is the state, gðnÞ 2 Ln2 ½d; 0 is a continuous vector-valued initial function, A(rt), B(rt), C(rt) are known modedependent constant matrices with appropriate dimensions. For notational simplicity, when rt = i, i 2 M, A(rt), DA(rt), B(rt), DB(rt), C(rt) are, respectively, denoted as Ai, DAi, Bi, DBi, Ci. s(t) and h(t) are positive time-varying differentiable bounded delays which can be described as
h1 < sðtÞ 6 h2 ; 0 < hðtÞ 6 h;
ð3Þ
¼ maxfhðtÞg; d ¼ maxfh2 ; hg. This assumption guarantees that we can apply the Lyapunov– where h2 ¼ maxfsðtÞg; h Krasovskii functional approach to analyzing the exponential stochastic stability of uncertain neutral MJSs (2) with timevarying delays. DAi and DBi are time-varying but norm bounded uncertainties satisfying
½DAi ; DBi ¼ M i F i ðtÞ½Ni ; Nhi ;
ð4Þ
where Mi, Ni, Nhi are known mode-dependent matrices with appropriate dimensions and Fi(t) is time-varying unknown matrix function with Lebesgue norm measurable elements satisfying F Ti ðtÞF i ðtÞ 6 I.
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Assumption 2.1. The system matrix Ai(" i 2 M) is Hurwitz matrix with all the eigenvalues having negative real parts for each modes. Assumption 2.2. The Markov process is irreducible and the system mode rt is available at time t. Assumption 2.3. The matrix Mi(" i 2 M) is chosen as a full row rank matrix. Definition 2.4. The uncertain neutral MJSs (2) is said to be exponentially stochastically stable if, for every system mode and every gðnÞ 2 L2 ½d; 0; there exist scalars a > 0 and k > 0, such that
EkxðtÞk 6 aekt EkgðnÞk;
ð5Þ
where k > 0 is called the degree of exponential stability. Definition 2.5. In the Euclidean space fRn M Rþ g; we introduce the stochastic Lyapunov–Krasovskii functional of system (2) as V(x(t), rt = i,t > 0) = V(x(t), i), the weak infinite small operator satisfies
CVðxðtÞ; iÞ ¼ lim
Dt!0
1 ½EfVðxðt þ DtÞ; r tþDt ; t þ DtÞg VðxðtÞ; i; tÞ: Dt
ð6Þ
The aim of this paper is to study the exponential stability problem of uncertain neutral MJSs (2). By selecting appropriate Lyapunov–Krasovskii functional, the main results will be given in the form of LMIs. Before proceeding with the study, we state following Lemmas will be useful for the proof of main theorem. Lemma 2.6. Stochastically stable means almost surely (asymptotically) stable. Lemma 2.7. Let T, H, F and N be real matrices of appropriate dimensions with FTF 6 I. Then for a scalar a > 0, we have
T þ HFN þ NT F T HT 6 T þ aHHT þ a1 NT N:
ð7Þ
Lemma 2.8. [34] (Schur complement). Given constant matrices X1, X2 and X3 with appropriate dimensions, where XT1 ¼ X1 and XT2 ¼ X2 > 0, then
X1 þ XT3 X1 2 X3 < 0 if and only if
"
XT3 X2
X1
# < 0;
or
X2
X3 < 0: X1
3. Main results In this section, we will discuss the global exponential stability for uncertain neutral MJSs with interval time-varying delays, then in terms of LMIs, we can obtain the sufficient condition for the globally exponential stability. Theorem 3.1. For uncertain neutral MJSs (2) and given positive scalar k > 0, if there exist mode-dependent symmetric positivedefinite matrices
2
3
P1i
P2i
P 3i
P 4i
6 6 Pi ¼ 6 4
P5i
P 6i
P 8i
P 7i 7 7 7 > 0; P 9i 5
P10i
symmetric positive-definite matrices Q1, Q2, Q3, R1, Z1, Z2 and a sequence {ai > 0, i 2 M}, satisfying the following LMI:
2
Xi
6 4
NT R1
ST
3
7 R1 M i 5 < 0; ai I
ð8Þ
P. Balasubramaniam et al. / Applied Mathematics and Computation 216 (2010) 3396–3407
3399
where Xi = (Xm,n,i)8 8i,
X1;1;i ¼ 2kP 1i þ P1i Ai þ ATi PT1i þ P2i þ PT2i þ
N X
pij P1j þ
X1;6;i ¼ kP2i þ ATi P2i þ PT5i þ
N X
Q m þ h2 Z 1 þ ðh2 h1 ÞZ 2 þ ai NTi Ni ;
m¼1
j¼1
X1;2;i ¼ P1i Bi k1 ðP2i P3i þ P4i Þ þ ai N Ti Nhi ;
3 X
X1;3;i ¼ P4i ;
pij P2j ; X1;7;i ¼ kP3i þ ATi P3i þ P6i þ
j¼1
X1;8;i ¼ kP4i þ ATi P4i þ P7i þ
X1;5;i ¼ P1i C i ;
X1;4;i ¼ P3i ; N X
pij P3j ;
j¼1
N X
pij P4j ; X2;2;i ¼ k1 e2kh2 Q 1 þ ai NThi Nhi ;
j¼1
X2;3;i ¼ X2;4;i ¼ X2;5;i ¼ 0; X2;6;i ¼ BTi P2i k1 ðPT5i PT6i þ PT7i Þ; X2;7;i ¼ BTi P3i k1 ðP6i P T8i þ PT9i Þ; X2;8;i ¼ BTi P4i k1 ðP7i P9i þ PT10i Þ; X3;6;i ¼ PT7i ;
X3;4;i ¼ X3;5;i ¼ 0; X4;5;i ¼ 0;
X4;6;i ¼ PT6i ;
X5;6;i ¼ C Ti P2i ;
X3;7;i ¼ PT9i ;
X4;7;i ¼ PT8i ;
X5;7;i ¼ C Ti P3i ;
X3;8;i ¼ PT10i ;
X4;8;i ¼ P9i ;
X5;8;i ¼ C Ti P4i ;
X4;4;i ¼ e2kh2 Q 3 ;
X5;5;i ¼ k2 e2kh R1 ;
X6;6;i ¼ 2kP 5i þ
N X
pij P5j
j¼1
X6;7;i ¼ kP6i þ
N X
pij P6j ; X6;8;i ¼ kP7i þ
j¼1 N X
X7;7;i ¼ 2kP 8i þ
pij P8j
N X
pij P7j ;
e2kðh2 h1 Þ ðZ 1 þ Z 2 Þ ; ðh2 h1 Þ
pij P10j
j¼1
X7;8;i ¼ kP 9i þ
N X
pij P9j ;
j¼1
e2kðh2 h1 Þ Z2; ðh2 h1 Þ
h PT ¼ R1 Ai
R1 B i
0 0 R1 C i
i 0 0 0 ;
N T ¼ ½ R 1 Ai
R1 B i
0 0 R1 C i
0 0 0;
h ST ¼ ðP1i M i ÞT
e2kh2 Z 1 ; h2
j¼1
j¼1
X8;8;i ¼ 2kP 10i þ
N X
X3;3;i ¼ e2kh1 Q 2 ;
0 0 0 0 ðP2i Mi ÞT
i ðP4i M i ÞT ;
ðP3i M i ÞT
then the origin of system (2) is globally exponentially stochastically stable in the mean square. Moreover,
Ekxðt; gÞk 6 e
kt
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðrP þ ðrQ 1 þ rQ 3 ÞS1 ÞM 1 þ rQ 2 M2 S2 þ rR1 M 3 S3 þ rZ 1 M 1 S4 þ rZ 2 M 4 S5
rp
;
where
rP ¼ max rmax ðPi Þ; rp ¼ min rmin ðPi Þ; rR1 ¼ rmax ðR1 Þ; rZl ¼ rmax ðZ l Þ; ðl ¼ 1; 2Þ; i2M
i2M
rQ j ¼ rmax ðQ j Þ; ðj ¼ 1; 2; 3Þ; M1 ¼ sup EkxðhÞk2 ; M2 ¼ sup EkxðfÞk2 ; h2 6h60
_ lÞk2 ; M3 ¼ sup Ekxð l60 h6
S1 ¼
1 e2kh2 ; 2k
M4 ¼
sup
h1 6f60
EkxðgÞk2 ;
k1 ¼ inf ð1 s_ ðtÞÞ; tP0
h2 6g6h1
S2 ¼
1 e2kh1 ; 2k
S3 ¼
1 e2kh ; 2k
S4 ¼
_ k2 ¼ inf ð1 hðtÞÞ;
2kh2 e2kh2 1 4k
2
tP0
;
ð9Þ
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P. Balasubramaniam et al. / Applied Mathematics and Computation 216 (2010) 3396–3407
S5 ¼
e2kh2 e2kh1 þ 2kh2 2kh1 2
4k
:
Proof. Consider the following Lyapunov–Krasovskii functional
VðxðtÞ; i; tÞ ¼ V 1 ðxðtÞ; i; tÞ þ V 2 ðxðtÞ; i; tÞ þ V 3 ðxðtÞ; i; tÞ þ V 4 ðxðtÞ; i; tÞ
V 1 ðxðtÞ; i; tÞ ¼ e2kt nT1 ðtÞPi n1 ðtÞ; Z t Z V 2 ðxðtÞ; i; tÞ ¼ e2ks xT ðsÞQ 1 xðsÞds þ V 3 ðxðtÞ; i; tÞ ¼
Z
tsðtÞ
t
e2ks xT ðsÞ Q 2 xðsÞds þ
Z
th1
t
e2ks xT ðsÞQ 3 xðsÞds;
th2
t
_ e2ks x_ T ðsÞR1 xðsÞds; thðtÞ
V 4 ðxðtÞ; i; tÞ ¼
Z
Z
0
t
e2ks xT ðsÞZ 1 xðsÞdsdh þ
Z
tþh
h2
h1
Z
t
e2ks xT ðsÞZ 2 xðsÞds dh
tþh
h2
with
2
Z
n ðtÞ ¼ 4x ðtÞ T
T
!T
t
xðsÞds
Z
tsðtÞ
T
tsðtÞ
xðsÞds
Z
th1
tsðtÞ
th2
!T 3 xðsÞds 5:
Considering Definition 2.5, make the corresponding time derivative of V(x(t), i, t), such that
CV 1 ðxðtÞ; i; tÞ ¼ 2ke2kt nT1 ðtÞPi n1 ðtÞ þ 2e2kt nT1 ðtÞPi n_ 1 ðtÞ þ e2kt nT1 ðtÞ
N X
pij Pj n1 ðtÞ;
j¼1
CV 2 ðxðtÞ; i; tÞ ¼ e2kt xT ðtÞQ 1 xðtÞ e2kðtsðtÞÞ xT ðt sðtÞÞQ 1 xðt sðtÞÞ þ e2kt xT ðtÞQ 2 xðtÞ e2kðth1 Þ xT ðt h1 ÞQ 2 xðt h1 Þ þ e2kt xT ðtÞQ 3 xðtÞ e2kðth2 Þ xT ðt h2 Þ Q 3 xðt h2 Þ; _ e2kðthðtÞÞ x_ T ðt hðtÞÞR1 xðt _ hðtÞÞ; CV 3 ðxðtÞ; i; tÞ ¼ e2kt x_ T ðtÞR1 xðtÞ
CV 4 ðxðtÞ; i; tÞ ¼ h2 e2kt xT ðtÞZ 1 xðtÞ
Z
Z
tsðtÞ
e2ks xT ðsÞZ 1 xðsÞds th2
tsðtÞ
e2ks xT ðsÞZ 2 xðsÞds
Z
Z
t
e2ks xT ðsÞZ 1 xðsÞds þ ðh2 h1 Þe2kt xT ðtÞZ 2 xðtÞ
tsðtÞ th1
e2ks xT ðsÞZ 2 xðsÞds:
tsðtÞ
th2
Recalling that the uncertain neutral MJSs (2) over the space (X, F, P), it is not difficult to get the following relation:
T _ _ hðtÞÞ R1 Ai xðtÞ þ Bi xðt sðtÞÞ þ C i xðt _ hðtÞÞ x_ T ðtÞR1 xðtÞ ¼ Ai xðtÞ þ Bi xðt sðtÞÞ þ C i xðt _ hðtÞÞ ¼ xT ðtÞATi R1 Ai xðtÞ þ 2xT ðtÞATi R1 Bi xðt sðtÞÞ þ 2xT ðtÞATi R1 C i xðt _ hðtÞÞ þ x_ T ðt hðtÞÞC Ti R1 C i xðt _ hðtÞÞ: þ xT ðt sðtÞÞBTi R1 Bi xðt sðtÞÞ þ 2xT ðt sðtÞÞBTi R1 C i xðt By letting
2 bðtÞ ¼ 4xT ðtÞ xT ðt sðtÞÞ xT ðt h1 Þ xT ðt h2 Þ x_ T ðt hðtÞÞ it follows that CV(x(t), i, t) = e2ktbT(t)(Ki + Ri)b(t), where Ki = (Km,n,i)8
K1;1;i ¼ 2kP 1i þ P1i Ai þ ATi PT1i þ P2i þ PT2i þ
N X
pij P1j þ
j¼1
K1;2;i ¼ P 1i Bi k1 ðP2i P3i þ P4i Þ; K1;6;i ¼ kP 2i þ ATi P 2i þ PT5i þ
N X j¼1
K1;3;i ¼ P4i ;
3 X
Z
!T Z
t
xðsÞds tsðtÞ
8i,
Q m þ h2 Z 1 þ ðh2 h1 ÞZ 2 ;
m¼1
K1;4;i ¼ P3i ;
K1;5;i ¼ P1i C i ;
pij P2j ; K1;7;i ¼ kP3i þ ATi P3i þ P6i þ
tsðtÞ
N X j¼1
pij P3j ;
th2
T xðsÞds
Z
th1
tsðtÞ
!T 3T xðsÞds 5 ;
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P. Balasubramaniam et al. / Applied Mathematics and Computation 216 (2010) 3396–3407 N X
K1;8;i ¼ kP4i þ ATi P4i þ P7i þ
pij P4j ; K2;2;i ¼ k1 e2kh2 Q 1 ; K2;3;i ¼ K2;4;i ¼ K2;5;i ¼ 0;
j¼1
K2;6;i ¼ BTi P2i k1 ðPT5i PT6i þ PT7i Þ;
K2;7;i ¼ BTi P3i k1 ðP6i PT8i þ PT9i Þ;
K2;8;i ¼ BTi P4i k1 ðP7i P9i þ PT10i Þ; K3;7;i ¼ PT9i ;
K3;8;i ¼ P T10i ;
K3;3;i ¼ e2kh1 Q 2 ;
K4;4;i ¼ e2kh2 Q 3 ;
N X
K6;6;i ¼ 2kP 5i þ
pij P5j
j¼1 N X
K7;7;i ¼ 2kP 8i þ
pij P8j
j¼1
N X
K8;8;i ¼ 2kP 10i þ
e2kh2 Z 1 ; h2
K6;7;i ¼ kP6i þ
j¼1
K4;6;i ¼ P T6i ;
K5;7;i ¼ C Ti P3i ; N X
K4;7;i ¼ PT8i ;
K5;8;i ¼ C Ti P4i ;
pij P6j ; K6;8;i ¼ kP7i þ
N X
j¼1
e2kðh2 h1 Þ ðZ 1 þ Z 2 Þ ; ðh2 h1 Þ
pij P10j
K4;5;i ¼ 0;
K5;6;i ¼ C Ti P2i ;
K5;5;i ¼ k2 e2kh R1 ;
K4;8;i ¼ P9i ;
K3;6;i ¼ PT7i ;
K3;4;i ¼ K3;5;i ¼ 0;
pij P7j ;
j¼1
K7;8;i ¼ kP 9i þ
N X
pij P9j ;
j¼1
e2kðh2 h1 Þ Z2; ðh2 h1 Þ
h
iT
h
i
Ri ¼ Ai Bi 0 0 C i 0 0 0 R1 Ai Bi 0 0 C i 0 0 0 : Applying Schur complement, condition Ki + Ri < 0 is equivalent to the following inequality:
"
Si ¼
Ki P T R1
#
< 0:
In order to deal with uncertainties described as the form in Eq. (4), we can use the following approach Si = Ti + DTi < 0, where
" Ti ¼
Pi
NT
R1
# ;
Pi ¼ ðPm;n;i Þ88i ; P1;1;i ¼ 2kP 1i þ P1i Ai þ ATi PT1i þ P2i þ PT2i þ
N X
pij P1j þ
P1;6;i ¼ kP2i þ ATi P2i þ PT5i þ
N X
P1;3;i ¼ P 4i ;
Q m þ h2 Z 1 þ ðh2 h1 ÞZ 2 ;
m¼1
j¼1
P1;2;i ¼ P1i Bi k1 ðP 2i P3i þ P4i Þ;
3 X
P1;4;i ¼ P3i ;
pij P2j ; P1;7;i ¼ kP3i þ ATi P3i þ P6i þ
j¼1
P1;8;i ¼ kP4i þ ATi P4i þ P7i þ
N X
P1;5;i ¼ P1i C i ; N X
pij P3j ;
j¼1
pij P4j ; P2;2;i ¼ k1 e2kh2 Q 1 ; P2;3;i ¼ P2;4;i ¼ P2;5;i ¼ 0;
j¼1
P2;6;i ¼ BTi P2i k1 ðPT5i P T6i þ PT7i Þ; P2;8;i ¼ BTi P4i k1 ðP7i P 9i þ PT10i Þ; P3;8;i ¼ PT10i ;
P4;4;i ¼ e2kh2 Q 3 ;
P5;5;i ¼ k2 e2kh R1 ; P6;6;i ¼ 2kP 5i þ
N X j¼1
P5;6;i ¼ C Ti P2i ;
pij P5j
e2kh2 Z 1 ; h2
P2;7;i ¼ BTi P3i k1 ðP6i PT8i þ PT9i Þ; P3;3;i ¼ e2kh1 Q 2 ; P4;5;i ¼ 0;
P3;4;i ¼ P3;5;i ¼ 0;
P4;6;i ¼ PT6i ;
P5;7;i ¼ C Ti P3i ; P6;7;i ¼ kP 6i þ
P4;7;i ¼ PT8i ;
P3;6;i ¼ PT7i ;
P4;8;i ¼ P 9i ;
P5;8;i ¼ C Ti P 4i ; N X j¼1
pij P6j ; P6;8;i ¼ kP7i þ
P3;7;i ¼ PT9i ;
N X j¼1
pij P7j ;
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P. Balasubramaniam et al. / Applied Mathematics and Computation 216 (2010) 3396–3407
P7;7;i ¼ 2kP 8i þ
N X
e2kðh2 h1 Þ ðZ 1 þ Z 2 Þ ; ðh2 h1 Þ
pij P8j
j¼1
P8;8;i ¼ 2kP 10i þ
N X
pij P10j
j¼1
2 6 6 6 6 6 6 6 6 DT i ¼ 6 6 6 6 6 6 6 6 4
P7;8;i ¼ kP 9i þ
N X
e2kðh2 h1 Þ Z 2 ; ðh2 h1 Þ
c11
P1i DBi
0 0 0 DATi PT2i
DATi PT3i
DATi PT4i
DATi R1
0
DBTi PT2i 0 0 0 0 0 0 0 0 0 0
DBTi PT3i 0 0 0 0 0
DBTi PT4i 0 0 0 0 0 0
0
where c11 ¼ P 1i DAi þ
0 0 0
DATi PT1i .
pij P9j ;
j¼1
3
7 DBTi R1 7 7 0 7 7 7 0 7 7 0 7 7; 7 0 7 7 0 7 7 7 0 5
According to Lemma 2.7, DTi can be presented as the following form:
T T DT i ¼ L11 F i L12 þ LT12 F Ti LT11 < a1 i L11 L11 þ ai L12 L12 ;
where
h LT11 ¼ MTi PT1i L12 ¼ ½Ni
0 0 0 MTi PT2i
Nhi
MTi PT3i
M Ti P T4i
i M Ti R1 ;
0 0 0 0 0 0 0:
Applying Schur complement again, it is obvious that CV(x(t), i, t) is negative if LMIs (8) holds. On the other hand, since CV(x(t), i, t) < 0, we can get
VðxðtÞ; i; tÞ 6 Vðxð0Þ; r 0 ; 0Þ ¼ xT ð0ÞPi xð0Þ þ þ
Z
Z
0
e2ks xT ðsÞQ 1 xðsÞds þ
h2 0
h
_ e2ks x_ T ðsÞR1 xðsÞds þ
Z
0 h2
Z
0
e2ks xT ðsÞQ 2 xðsÞds þ
h1
Z
0
e2ks xT ðsÞZ 1 xðsÞdsdh þ
Z
h1
h2
h
Z
Z
0
e2ks xT ðsÞQ 3 xðsÞds
h2 0
e2ks xT ðsÞZ 2 xðsÞdsdh;
h
we have
VðxðtÞ; i; tÞ 6
Z
rP þ ðrQ 1 þ rQ 3 Þ
þ rZ 2 M 4
Z
h1
Z
h2
0
0
h2
Z e2ks ds M 1 þ rQ 2 M2
0 h1
e2ks ds þ rR1 M3
Z
0 h
e2ks ds þ rZ1 M 1
Z
0
h2
Z
0
e2ks dsdh
h
e2ks dsdh:
h
Recalling the given Lyapunov–Krasovskii functional, we can also have
VðxðtÞ; i; tÞ P e2kt rp EkxðtÞk2 : Therefore, the convergence rates of the system states in the form of Eq. (9) can be obtained. From Definition 2.4, the uncertain neutral MJSs (2) is exponentially stochastically stable in the mean square. This completes the proof. h Theorem 3.2. The uncertain neutral MJSs (2) is globally exponentially stochastically stable and satisfy Eq. (8), if there exist modedependent symmetric positive-definite matrices
2
3
P1i
P2i
P 3i
P 4i
6 6 Pi ¼ 6 4
P5i
P 6i
P 8i
P 7i 7 7 7 > 0; P 9i 5
P10i
K1i, K2i, K3i, K4i, K5i, K6i, K7i, K8i, K9i, K10i(i 2 M), symmetric positive-definite matrices Q1, Q2, Q3, R1, Z1, Z2, K2, K3, K4, K5, K6, K7, K8, a sequence {ai > 0,i 2 M} and a positive scalar c = k1, satisfying the following generalized eigenvalue problem (GEVP):
P. Balasubramaniam et al. / Applied Mathematics and Computation 216 (2010) 3396–3407
min c
3403
ð10Þ
subject to
2
7 R1 Mi 5 < 0;
R1
3
ST
NT
i X 6 4
ð11Þ
ai I
2P 1i < cK 1i ;
ð12Þ
P2i < cK 2i ;
ð13Þ
P3i < cK 3i ;
ð14Þ
P4i < cK 4i ; 2P 5i < cK 5i ;
ð15Þ ð16Þ
P6i < cK 6i ;
ð17Þ
P7i < cK 7i ;
ð18Þ
2P 8i < cK 8i ;
ð19Þ
P9i < cK 9i ;
ð20Þ
2P 10i < cK 10i ;
ð21Þ
2h2 K 2 6 cðQ 1 K 2 Þ;
ð22Þ
2h1 K 3 6 cðQ 2 K 3 Þ; 2h2 K 4 6 cðQ 3 K 4 Þ; 6 cðR K Þ; 2hK
ð23Þ ð24Þ ð25Þ
2h2 K 6 6 cðZ 1 K 6 Þ;
ð26Þ
5
1
5
2ðh2 h1 ÞK 7 6
cððZ 1 þ Z 2 Þ K 7 Þ
ðh2 h1 Þ ðZ 2 K 8 Þ ; 2ðh2 h1 ÞK 8 6 c ðh2 h1 Þ
ð27Þ
;
ð28Þ
where
i ¼ ðXm;n;i Þ X 88i ;
i Þ – ðXm;n;i Þ ðX whenððm; nÞÞ ¼ ðð1; 1Þ; ð1; 6Þ; ð1; 7Þ; ð1; 8Þ; ð2; 2Þ; ð3; 3Þ; ð4; 4Þ; ð5; 5Þ; ð6; 6Þ; ð6; 7Þ; ð6; 8Þ; ð7; 7Þ; ð7; 8Þ; ð8; 8ÞÞ;
1;1;i ¼ K 1i þ P1i Ai þ AT PT þ P2i þ PT þ X i 1i 2i
N X
pij P1j þ
N X
Q m þ ai N Ti Ni þ h2 Z 1 þ ðh2 h1 ÞZ 2 ;
m¼1
j¼1
1;6;i ¼ K 2i þ AT P2i þ PT þ X i 5i
3 X
1;7;i ¼ K 3i þ AT P 3i þ P6i þ pij P2j ; X i
j¼1
1;8;i ¼ K 4i þ AT P4i þ P7i þ X i
N X
N X
pij P3j ;
j¼1
2;2;i ¼ k1 K 2 þ ai N T Nhi ; X 3;3;i ¼ K 3 ; X 4;4;i ¼ K 4 ; pij P4j ; X hi
j¼1
6;6;i ¼ K 5i þ X
5;5;i ¼ k2 K 5 ; X
N X
6;7;i ¼ K 6i þ pij P5j K 6 ; X
j¼1
6;8;i ¼ K 7i þ X
N X
7;7;i ¼ K 8i þ pij P7j ; X
j¼1
8;8;i ¼ K 10i þ X
N X
N X
pij P6j ;
j¼1 N X
pij P8j K 7 ; X 7;8;i ¼ K 9i þ
j¼1
N X
pij P9j ;
j¼1
pij P10j K 8 :
j¼1
Proof. By defining
K 1i > 2kP 1i ;
K 2i > kP 2i ;
K 3i > kP 3i ;
K 4i > kP 4i ;
K 5i > 2kP5i ;
K 7i > kP 7i ;
K 8i > 2kP 8i ;
K 9i > kP 9i ;
K 10i > 2kP 10i ;
K 6i > kP6i ;
0 < K 2 < e2kh2 Q 1 ;
3404
P. Balasubramaniam et al. / Applied Mathematics and Computation 216 (2010) 3396–3407
0 < K 3 < e2kh1 Q 2 ;
0 < K7 <
e2kðh2 h1 Þ ðZ 1 þ Z 2 Þ ; ðh2 h1 Þ
we have
0 < K 5 < e2kh R1 ;
0 < K 4 < e2kh2 Q 3 ;
0 < K8 <
0 < K6 <
e2kh2 Z 1 ; h2
e2kðh2 h1 Þ Z 1 ; ðh2 h1 Þ
1i þ Ri bðtÞ < 0; CVðxðtÞ; i; tÞ 6 e2kt bT ðtÞ K where
1i ¼ ðKm;n;i Þ K 88i ;
1i – ðKm;n;i Þ when ððm; nÞÞ K
¼ ðð1; 1Þ; ð1; 6Þ; ð1; 7Þ; ð1; 8Þ; ð2; 2Þ; ð3; 3Þ; ð4; 4Þ; ð5; 5Þ; ð6; 6Þ; ð6; 7Þ; ð6; 8Þ; ð7; 7Þ; ð7; 8Þ; ð8; 8ÞÞ; 1;1;i ¼ K 1i þ P 1i Ai þ AT P T þ P2i þ PT þ K i 1i 2i
N X j¼1
1;6;i ¼ K 2i þ AT P2i þ PT þ K i 5i 1;8;i ¼ K 4i þ AT P4i þ P7i þ K i
N X j¼1 N X
pij P1j þ
3 X
Q m þ h2 Z 1 þ ðh2 h1 ÞZ 2 ;
m¼1
pij P2j ; K 1;7;i ¼ K 3i þ ATi P3i þ P6i þ
N X
pij P3j ;
j¼1
pij P4j ; K 2;2;i ¼ k1 K 2 ; K 3;3;i ¼ K 3 ; K 4;4;i ¼ K 4 ;
j¼1
6;6;i ¼ K 5i þ K
5;5;i ¼ k2 K 5 ; K
N X
6;7;i ¼ K 6i þ pij P5j K 6 ; K
j¼1
6;8;i ¼ K 7i þ K
N X
pij P7j ; K 7;7;i ¼ K 8i þ
j¼1
8;8;i ¼ K 10i þ K
N X
N X
pij P6j ;
j¼1 N X
pij P8j K 7 ; K 7;8;i ¼ K 9i þ
j¼1
N X
pij P9j ;
j¼1
pij P10j K 8 :
j¼1
By applying Schur complement and using some changes of matrix variables, CV(x(t), i, t) < 0 can be modified as LMI (11). On the other hand, by letting c = k1, we can get
K 1i > 2kP1i ) 2P 1i < cK 1i ;
K 2i > kP 2i ) P2i < cK 2i ;
K 3i > kP 3i ) P3i < cK 3i ;
K 4i > kP 4i ) P4i < cK 4i ;
K 5i > 2kP5i ) 2P 5i < cK 5i ;
K 6i > kP 6i ) P6i < cK 6i ;
K 7i > kP 7i ) P7i < cK 7i ;
K 8i > 2kP8i ) 2P 8i < cK 8i ;
K 9i > kP 9i ) P9i < cK 9i ;
K 10i > 2kP 10i ) 2P10i < cK 10i ; 0 < K 2 < e2kh2 Q 1 ) ð1 þ 2kh2 ÞK 2 < e2kh2 K 2 < Q 1 ) 2h2 K 2 6 cðQ 1 K 2 Þ; 0 < K 3 < e2kh1 Q 2 ) ð1 þ 2kh1 ÞK 3 < e2kh1 K 3 < Q 2 ) 2h1 K 3 6 cðQ 2 K 3 Þ; 0 < K 4 < e2kh2 Q 3 ) ð1 þ 2kh2 ÞK 4 < e2kh2 K 4 < Q 3 ) 2h2 K 4 6 cðQ 3 K 4 Þ; 5 < e2kh K 5 < R1 ) 2hK 5 6 cðR1 K 5 Þ; 0 < K 5 < e2kh R1 ) ð1 þ 2khÞK
0 < K 6 < e2kh2
Z1 Z1 ) ð1 2kh2 ÞK 6 < e2kh2 K 6 < ) 2h2 K 6 6 cðZ 1 K 6 Þ; h2 h2
0 < K7 <
e2kðh2 h1 Þ ðZ 1 þ Z 2 Þ Z1 þ Z2 cððZ 1 þ Z 2 Þ K 7 Þ ) ð1 2kðh2 h1 ÞÞK 7 < e2kðh2 h1 Þ K 7 < ) 2ðh2 h1 ÞK 7 6 ; ðh2 h1 Þ ðh2 h1 Þ ðh2 h1 Þ
0 < K8 <
e2kðh2 h1 Þ Z 2 Z2 ðZ 2 K 8 Þ ) ð1 2kðh2 h1 ÞÞK 8 < e2kðh2 h1 Þ K 8 < ) 2ðh2 h1 ÞK 8 6 c : ðh2 h1 Þ ðh2 h1 Þ ðh2 h1 Þ
3405
P. Balasubramaniam et al. / Applied Mathematics and Computation 216 (2010) 3396–3407
Therefore, the quasi-convex optimization problem can be solved by using MATLAB LMI Control Toolbox. Obviously, GEVP (10)–(28) presents an optimization algorithm which guarantees that the degree of exponential stability k is maximal (or c is minimal). This completes the proof. h Remark 3.3. From Definition 2.4, we conclude that the equilibrium point of system (2) is globally exponentially stable. We hope that the degree of exponential stability k is maximal (or c is minimal) such that the system converges to the equilibrium point as fast as possible. It requires solving the generalized eigenvalue minimization problem (10)–(28), which is a quasi-convex optimization problem and can be solved by using the MATLAB LMI control toolbox [30]. Although the conservativeness of Theorem 3.2 is higher than Theorem 3.1, it provides a simple method to determine the exponential stability of neutral systems (2) and get the upper bound of the exponential convergence rate and will be widely applied to stability analysis. Remark 3.4. Theorem 3.2 provides delay-dependent stability criterion for neutral systems with interval time-varying delays and Markovian jumping parameters. Moreover, a new type of Markovian jumping matrix Pi is taken into account. These type of augmented matrix has not been used in any of the existing literatures, for the exponential stability of neutral systems with Markovian jumping parameters via GEVPs.
4. Numerical examples Example 1. Consider the stochastic neutral partial element equivalent circuit (PEEC) model described by uncertain neutral MJSs with time-varying delays
_ _ hðtÞÞ xðtÞ ¼ ½Ai þ DAi xðtÞ þ ½Bi þ DBi xðt sðtÞÞ þ C i xðt
ð29Þ
with
A1 ¼ A2 ¼
C2 ¼
2
0
0
3
0:1 0 ; 0 0:1
N2 ¼ ½ 0 0:1 ;
;
B1 ¼ B2 ¼
M1 ¼
0:1 ; 0:1
0
1
;
1 1
M2 ¼
N h1 ¼ ½ 0:1 0:1 ;
C1 ¼
0 ; 0:1
0:3
0
0
0:3
;
N1 ¼ ½ 0:1 0 ;
Nh2 ¼ ½ 0:1 0:1 ;
¼ 0:2. From Eq. where s(t) = 0.1jsin tj and h(t) = 0.2jcos tj. Since 0 6 jsin tj 6 1, 0 6 jcos tj 6 1, we can get h1 ¼ 0; h2 ¼ 0:1; h _ ¼ 0:8. By solving GEVP (10)–(28), we can get (3), we can also get k1 ¼ inf tP0 ð1 s_ ðtÞÞ ¼ 0:9; k2 ¼ inf tP0 ð1 hðtÞÞ c = 0.8454 and the presented matrices are as follows:
"
1:0121
0:2658
0:2658
0:4248
0:0298
0:0191
0:0191
0:0387
P11 ¼
Q2 ¼
Z1 ¼
0:2598 0:6666
K 12 ¼
K4 ¼
K7 ¼
1:4679 0:2598
# ;
;
" P 12 ¼
Z2 ¼
;
1:7916
0:5190
0:5190
0:8391
0:0174
0:0113
0:0113
0:0233
1:0609
0:1098
0:1098
0:7527
;
Q3 ¼
0:8909
0:2564
0:2564
0:4116
0:0271
0:0179
0:0179
0:0382
0:3095
0:1956
0:1956
0:4654
K2 ¼
;
K5 ¼
;
K8 ¼
# ;
;
" Q1 ¼
R1 ¼
K 11 ¼
;
0:5021
0:1552
0:1552
0:3912
0:1356
0:0398
0:0398
0:0598
0:2507
0:1487
0:1487
0:3392
;
; :
0:6076
0:1896
0:1896
0:4762
0:2027
0:0604
0:0604
0:0903
2:0345
0:5382
0:5382
0:8612
K3 ¼
K6 ¼
;
0:0142
0:0142
0:0289
0:3153 0:5414
;
;
0:0222
1:4387 0:3153
#
;
;
3406
P. Balasubramaniam et al. / Applied Mathematics and Computation 216 (2010) 3396–3407
Therefore, according to Theorem 3.2, the uncertain neutral PEEC system presented by MJSs (2) with time-varying delay is globally exponentially stochastically stable. For the above system, applying Theorem 2 in [31] it is found that the equilibrium solution of neutral MJSs (29) is robustly exponentially stable in the mean square for any c satisfying c = 2.7157. However, by Theorem 3.2 in this paper we can conclude that if c = 0.8454, system (29) is robustly exponentially stable in the mean square sense based on the GEVPs. Example 2. Consider the stochastic neutral partial element equivalent circuit (PEEC) model described by uncertain neutral MJSs with time-varying delays
_ _ hðtÞÞ xðtÞ ¼ ½Ai þ DAi xðtÞ þ ½Bi þ DBi xðt sðtÞÞ þ C i xðt
ð30Þ
with
A1 ¼
C1 ¼
5
0
0
6
;
A2 ¼
0:5 0 ; 0 0:5
C2 ¼
N2 ¼ ½ 0 0:2 ;
4
0
0
5
;
B1 ¼
0:3 0 ; 0 0:3
M1 ¼
N h1 ¼ ½ 0:3 0:3 ;
0
1:6
1:8 1:5
0:2 ; 0:2
B2 ¼
;
M2 ¼
2
0
0:9 1:2
0 ; 0:3
;
N1 ¼ ½ 0:2 0 ;
Nh2 ¼ ½ 0:2 0:2 ;
¼ 0:3. From Eq. where s(t) = 0.1jsin tj and h(t) = 0.3jcos tj. Since 0 6 j sin t j 6 1,0 6 jcos t j 6 1, we can get h1 ¼ 0; h2 ¼ 0:3; h _ ¼ 0:8. By solving GEVP (10)–(28), we can get c = 2.9340 (3), we can also get k1 ¼ inf tP0 ð1 s_ ðtÞÞ ¼ 0:9; k2 ¼ inf tP0 ð1 hðtÞÞ and the presented matrices are as follows:
"
2:5228
0:4801
0:4801
0:7965
0:1121
0:1487
0:1487
0:2711
0:5119
0:2654
0:2654
0:6831
5:6467
1:0191
1:0191
1:8867
P11 ¼ Q2 ¼ Z1 ¼
K 12 ¼
K4 ¼
K7 ¼
0:0702
0:0934
0:0934
0:1704
#
" ;
;
P12 ¼ Q3 ¼
;
;
;
0:6495 0:5322 ; 0:5322 1:1445
Z2 ¼
K2 ¼
K5 ¼
K8 ¼
2:7942
0:4622
0:4622
0:8583
0:1161
0:1564
0:1564
0:2869
Q1 ¼
;
R1 ¼
0:3280
0:3280
0:6651
3:7785
0:4677
0:4677
0:9659
;
"
0:2970
#
0:4141
0:0727
0:0727
0:1047
;
;
K 11 ¼
K3 ¼
K6 ¼
0:7786
1:5999
0:0938
0:1326
5:1159
0:9937
0:9937
1:6620
0:1057
0:1057
0:1935
0:1918
0:1918
0:5452
;
;
0:0806
0:4717
#
;
0:0938
0:7786
0:5068
;
6:0676
;
;
0:2410 0:2591 : 0:2591 0:5258
Therefore, according to Theorem 3.2, the uncertain neutral PEEC system presented by MJSs (2) with time-varying delay is globally exponentially stochastically stable. By Theorem 3.2 in this paper we can conclude that if c = 2.9340, system (29) is robustly exponentially stable in the mean square sense based on the GEVPs. 5. Conclusion A new sufficient condition is derived to guarantee the global exponential stability of the Markovian jumping neutral systems with interval time-varying delays. By selecting appropriate Lyapunov–Krasovskii functionals, we give the sufficient conditions so that the analysis problem can be tackled in the form of LMIs, which can be easily calculated by MATLAB LMI control toolbox and the associated synthesis problem can be dealt with by solving the generalized eigenvalue problem (GEVP). The result through the examples shown that the derived stability criteria are less conservative than some recent ones. Adding additional terms and more positive-definite matrices which would be appeared in the mode dependent form of L–K functional lead the system stability to be complicated which would improve the numerical results to some extend than the results presented in this paper. This issue is one of the future directions of the topic.
P. Balasubramaniam et al. / Applied Mathematics and Computation 216 (2010) 3396–3407
3407
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Ruehli, Methods for linear systems of circuit delay differential equations of neutral type, IEEE Transactions on Circuits and System 46 (1999) 212–215. [33] D. Yue, Q.L. Han, A delay-dependent stability criterion of neutral systems and its application to a partial element equivalent circuit model, in: Proceedings of the 2004 American Control Conference, IEEE Press, Piscataway, 2004, pp. 5438–5442. [34] S. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in Systems and Control Theory, SIAM Studies in Applied Mathematics, SIAM, Philadelphia, 1994.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Exponential stability results for uncertain neutral systems with interval time-varying delays and Markovian jumping parameters q P. Balasubramaniam *, A. Manivannan, R. Rakkiyappan Department of Mathematics, Gandhigram Rural University, Gandhigram 624 302, Tamilnadu, India
a r t i c l e
i n f o
Keywords: Markovian jumping parameters Global exponential stability Linear matrix inequality Lyapunov–Krasovskii functional Generalized eigenvalue problems
a b s t r a c t This paper deals with the global exponential stability analysis of neutral systems with Markovian jumping parameters and interval time-varying delays. The time-varying delay is assumed to belong to an interval, which means that the lower and upper bounds of interval time-varying delays are available. A new global exponential stability condition is derived in terms of linear matrix inequality (LMI) by constructing new Lyapunov–Krasovskii functionals via generalized eigenvalue problems (GEVPs). The stability criteria are formulated in the form of LMIs, which can be easily checked in practice by Matlab LMI control toolbox. Two numerical examples are given to demonstrate the effectiveness and less conservativeness of the proposed methods. 2010 Elsevier Inc. All rights reserved.
1. Introduction It is well known that, the applications of Markovian jumping parameters are more comprehensive, for instance, in economic systems [1], solar thermal receiver systems [2] and communication systems [3], etc. Markovian jumping parameters are a special class of hybrid systems with two components of mode and state. The dynamics of the jump mode and continuous state are respectively modeled by finite state Markov jumping parameters and differential equations. These systems are usually appropriate to describe dynamic systems subject to abrupt variation in their structures and parameters, such as sudden environment changes, subsystem switching, system noises, executor faults and failures occurred in components or interconnections, etc. Time delay phenomenon is often encountered in many practical systems, for example, aircraft stabilization, biological systems, chemical systems, hydraulic systems, metallurgical processing systems, nuclear reactor and electrical networks. It is well known that the existence of time delays in a system may cause instability or bad system performance. Hence the problem of stability of time-delay systems has received considerable attention in the last two decades, see for example [4] and the references therein. A neutral time-delay system contain delays both in its state and in its derivatives of state. Such system can be found in such places as population ecology, distributed networks containing lossless transmission lines, heat exchangers, robots in contact with rigid environments, etc. Because of its wider application, the problem of the stability of delay-differential neutral system has received considerable attention by many scholars in the last two decades. Neutral delay systems constitute a more general class than those of the retarded type. Stability of these systems proves to be a more complex issue because the system involves the derivative of the delayed state. Especially, in the past few decades increased attention has been devoted to the problem of robust delay-independent stability or delay-dependent stability and
q The work was supported by UGC-SAP (DRS-II), New Delhi, India. The work of the third author was supported by CSIR-SRF under Grant No. 09/715(0013)/ 2009-EMR-I. * Corresponding author. E-mail addresses: [email protected], [email protected] (P. Balasubramaniam).
0096-3003/$ - see front matter 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2010.04.077
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stabilization via different approaches for linear neutral systems with delayed state and/or input and parameter uncertainties. Therefore, the problem of the stability and stabilization of neutral time-delay systems has attracted considerable attention during the past few years, see for example [5–23]. Markovian jump system (MJSs) which initially introduced by Krasovskii and Lidskii [24] in 1961 are a special class of hybrid systems [25–27]. These systems have two components in the state vector. The first one which varies continuously is referred to be the continuous state of the system and the second one which varies discretely is referred to be the mode of the system. For MJS, the random jumps in system parameters are governed by a Markov process which takes values in a finite set as well. MJS is a suitable mathematical model to represent a class of dynamic systems due to random abrupt variations in their structures, and has many applications such as manufacturing systems, communication systems [28], and fault-tolerant systems. The stability analysis and controller synthesis problems for MJS have been extensively studied see for example [27,29], and references therein. To the best of the authors’ knowledge the problem of delay-range dependent exponential stability results for Markovian jumping neutral systems with interval time-varying delays has not been fully investigated and it is very challenging. In this paper, we discuss the problem of global exponential stability for norm-bounded uncertain neutral Markovian jumping systems (MJSs) with interval time-varying delays. We aim at dynamics of the neutral stochastic stability for all admissible uncertainties as well as interval time-varying delays. By selecting appropriate Lyapunov–Krasovskii functionals, we give the sufficient conditions so that the analysis problem can be tackled in the form of LMIs, which can be easily calculated by MATLAB LMI control toolbox [30] and the associated synthesis problem can be dealt which by solving the generalized eigenvalue problem (GEVP). In order to illustrate the proposed results, we provide real-world examples, that is, partial element equivalent circuit (PEEC) model presented in [31–33]. With the abrupt variation in its structures and parameters, we can present the PEEC model as a stochastic jump one. Notations. Rn and Rnm denote the n-dimensional Euclidean space and the set of all the n m real matrices, AT(or xT) and A1(or x1) denote the transpose and the inverse of matrix A or scalar x,rmax(A) and rmin(A) denote the maximal and minimal eigenvalue of a real matrix A,kAk denotes the Euclidean norm of a matrix A,jaj denotes the absolute value of the scalar a,E{} denotes the mathematics statistical expectation of the stochastic process or vector, Ln2 ½d; 0 is the space of n-dimensional square integrable function vector over [d, 0],P > 0 stands for a positive-definite matrix, I is the unit matrix with appropriate dimensions, ‘‘*” means the symmetric terms in a symmetric matrix.
2. Problem description and preliminaries Given a probability space (X, F, P) where X is the sample space, F is the algebra of events and P is the probability measure defined on F. Let the random process {rt,t P 0} be the Markov stochastic process taking values on a finite set M = {1, 2, . . . , N} with transition rate matrix P = {pij, i,j 2 M} and define the following transition probability from mode i at time t to mode j at time t + Dt as:
Pðr tþDt ¼ jjr t ¼ iÞ ¼
pij Dt þ oðDtÞ; if i – j; 1 þ pii Dt þ oðDtÞ; if i ¼ j;
where Dt > 0 and limDt!0 oðDDttÞ ! 0 with transition probability rates pij P 0 for i, j 2 M, i – j and
ð1Þ PN
j¼1;j–i
pij ¼ pii .
Consider a class of continuous neutral MJSs with time-delays and uncertainties over the space (X, F, P):
_ hðtÞÞ; _ xðtÞ ¼ ½Aðr t Þ þ DAðr t ÞxðtÞ þ ½Bðr t Þ þ DBðrt Þxðt sðtÞÞ þ Cðrt Þxðt xðnÞ ¼ gðnÞ 8 n 2 ½d; 0;
ð2Þ
where xðtÞ 2 Rn is the state, gðnÞ 2 Ln2 ½d; 0 is a continuous vector-valued initial function, A(rt), B(rt), C(rt) are known modedependent constant matrices with appropriate dimensions. For notational simplicity, when rt = i, i 2 M, A(rt), DA(rt), B(rt), DB(rt), C(rt) are, respectively, denoted as Ai, DAi, Bi, DBi, Ci. s(t) and h(t) are positive time-varying differentiable bounded delays which can be described as
h1 < sðtÞ 6 h2 ; 0 < hðtÞ 6 h;
ð3Þ
¼ maxfhðtÞg; d ¼ maxfh2 ; hg. This assumption guarantees that we can apply the Lyapunov– where h2 ¼ maxfsðtÞg; h Krasovskii functional approach to analyzing the exponential stochastic stability of uncertain neutral MJSs (2) with timevarying delays. DAi and DBi are time-varying but norm bounded uncertainties satisfying
½DAi ; DBi ¼ M i F i ðtÞ½Ni ; Nhi ;
ð4Þ
where Mi, Ni, Nhi are known mode-dependent matrices with appropriate dimensions and Fi(t) is time-varying unknown matrix function with Lebesgue norm measurable elements satisfying F Ti ðtÞF i ðtÞ 6 I.
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Assumption 2.1. The system matrix Ai(" i 2 M) is Hurwitz matrix with all the eigenvalues having negative real parts for each modes. Assumption 2.2. The Markov process is irreducible and the system mode rt is available at time t. Assumption 2.3. The matrix Mi(" i 2 M) is chosen as a full row rank matrix. Definition 2.4. The uncertain neutral MJSs (2) is said to be exponentially stochastically stable if, for every system mode and every gðnÞ 2 L2 ½d; 0; there exist scalars a > 0 and k > 0, such that
EkxðtÞk 6 aekt EkgðnÞk;
ð5Þ
where k > 0 is called the degree of exponential stability. Definition 2.5. In the Euclidean space fRn M Rþ g; we introduce the stochastic Lyapunov–Krasovskii functional of system (2) as V(x(t), rt = i,t > 0) = V(x(t), i), the weak infinite small operator satisfies
CVðxðtÞ; iÞ ¼ lim
Dt!0
1 ½EfVðxðt þ DtÞ; r tþDt ; t þ DtÞg VðxðtÞ; i; tÞ: Dt
ð6Þ
The aim of this paper is to study the exponential stability problem of uncertain neutral MJSs (2). By selecting appropriate Lyapunov–Krasovskii functional, the main results will be given in the form of LMIs. Before proceeding with the study, we state following Lemmas will be useful for the proof of main theorem. Lemma 2.6. Stochastically stable means almost surely (asymptotically) stable. Lemma 2.7. Let T, H, F and N be real matrices of appropriate dimensions with FTF 6 I. Then for a scalar a > 0, we have
T þ HFN þ NT F T HT 6 T þ aHHT þ a1 NT N:
ð7Þ
Lemma 2.8. [34] (Schur complement). Given constant matrices X1, X2 and X3 with appropriate dimensions, where XT1 ¼ X1 and XT2 ¼ X2 > 0, then
X1 þ XT3 X1 2 X3 < 0 if and only if
"
XT3 X2
X1
# < 0;
or
X2
X3 < 0: X1
3. Main results In this section, we will discuss the global exponential stability for uncertain neutral MJSs with interval time-varying delays, then in terms of LMIs, we can obtain the sufficient condition for the globally exponential stability. Theorem 3.1. For uncertain neutral MJSs (2) and given positive scalar k > 0, if there exist mode-dependent symmetric positivedefinite matrices
2
3
P1i
P2i
P 3i
P 4i
6 6 Pi ¼ 6 4
P5i
P 6i
P 8i
P 7i 7 7 7 > 0; P 9i 5
P10i
symmetric positive-definite matrices Q1, Q2, Q3, R1, Z1, Z2 and a sequence {ai > 0, i 2 M}, satisfying the following LMI:
2
Xi
6 4
NT R1
ST
3
7 R1 M i 5 < 0; ai I
ð8Þ
P. Balasubramaniam et al. / Applied Mathematics and Computation 216 (2010) 3396–3407
3399
where Xi = (Xm,n,i)8 8i,
X1;1;i ¼ 2kP 1i þ P1i Ai þ ATi PT1i þ P2i þ PT2i þ
N X
pij P1j þ
X1;6;i ¼ kP2i þ ATi P2i þ PT5i þ
N X
Q m þ h2 Z 1 þ ðh2 h1 ÞZ 2 þ ai NTi Ni ;
m¼1
j¼1
X1;2;i ¼ P1i Bi k1 ðP2i P3i þ P4i Þ þ ai N Ti Nhi ;
3 X
X1;3;i ¼ P4i ;
pij P2j ; X1;7;i ¼ kP3i þ ATi P3i þ P6i þ
j¼1
X1;8;i ¼ kP4i þ ATi P4i þ P7i þ
X1;5;i ¼ P1i C i ;
X1;4;i ¼ P3i ; N X
pij P3j ;
j¼1
N X
pij P4j ; X2;2;i ¼ k1 e2kh2 Q 1 þ ai NThi Nhi ;
j¼1
X2;3;i ¼ X2;4;i ¼ X2;5;i ¼ 0; X2;6;i ¼ BTi P2i k1 ðPT5i PT6i þ PT7i Þ; X2;7;i ¼ BTi P3i k1 ðP6i P T8i þ PT9i Þ; X2;8;i ¼ BTi P4i k1 ðP7i P9i þ PT10i Þ; X3;6;i ¼ PT7i ;
X3;4;i ¼ X3;5;i ¼ 0; X4;5;i ¼ 0;
X4;6;i ¼ PT6i ;
X5;6;i ¼ C Ti P2i ;
X3;7;i ¼ PT9i ;
X4;7;i ¼ PT8i ;
X5;7;i ¼ C Ti P3i ;
X3;8;i ¼ PT10i ;
X4;8;i ¼ P9i ;
X5;8;i ¼ C Ti P4i ;
X4;4;i ¼ e2kh2 Q 3 ;
X5;5;i ¼ k2 e2kh R1 ;
X6;6;i ¼ 2kP 5i þ
N X
pij P5j
j¼1
X6;7;i ¼ kP6i þ
N X
pij P6j ; X6;8;i ¼ kP7i þ
j¼1 N X
X7;7;i ¼ 2kP 8i þ
pij P8j
N X
pij P7j ;
e2kðh2 h1 Þ ðZ 1 þ Z 2 Þ ; ðh2 h1 Þ
pij P10j
j¼1
X7;8;i ¼ kP 9i þ
N X
pij P9j ;
j¼1
e2kðh2 h1 Þ Z2; ðh2 h1 Þ
h PT ¼ R1 Ai
R1 B i
0 0 R1 C i
i 0 0 0 ;
N T ¼ ½ R 1 Ai
R1 B i
0 0 R1 C i
0 0 0;
h ST ¼ ðP1i M i ÞT
e2kh2 Z 1 ; h2
j¼1
j¼1
X8;8;i ¼ 2kP 10i þ
N X
X3;3;i ¼ e2kh1 Q 2 ;
0 0 0 0 ðP2i Mi ÞT
i ðP4i M i ÞT ;
ðP3i M i ÞT
then the origin of system (2) is globally exponentially stochastically stable in the mean square. Moreover,
Ekxðt; gÞk 6 e
kt
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðrP þ ðrQ 1 þ rQ 3 ÞS1 ÞM 1 þ rQ 2 M2 S2 þ rR1 M 3 S3 þ rZ 1 M 1 S4 þ rZ 2 M 4 S5
rp
;
where
rP ¼ max rmax ðPi Þ; rp ¼ min rmin ðPi Þ; rR1 ¼ rmax ðR1 Þ; rZl ¼ rmax ðZ l Þ; ðl ¼ 1; 2Þ; i2M
i2M
rQ j ¼ rmax ðQ j Þ; ðj ¼ 1; 2; 3Þ; M1 ¼ sup EkxðhÞk2 ; M2 ¼ sup EkxðfÞk2 ; h2 6h60
_ lÞk2 ; M3 ¼ sup Ekxð l60 h6
S1 ¼
1 e2kh2 ; 2k
M4 ¼
sup
h1 6f60
EkxðgÞk2 ;
k1 ¼ inf ð1 s_ ðtÞÞ; tP0
h2 6g6h1
S2 ¼
1 e2kh1 ; 2k
S3 ¼
1 e2kh ; 2k
S4 ¼
_ k2 ¼ inf ð1 hðtÞÞ;
2kh2 e2kh2 1 4k
2
tP0
;
ð9Þ
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P. Balasubramaniam et al. / Applied Mathematics and Computation 216 (2010) 3396–3407
S5 ¼
e2kh2 e2kh1 þ 2kh2 2kh1 2
4k
:
Proof. Consider the following Lyapunov–Krasovskii functional
VðxðtÞ; i; tÞ ¼ V 1 ðxðtÞ; i; tÞ þ V 2 ðxðtÞ; i; tÞ þ V 3 ðxðtÞ; i; tÞ þ V 4 ðxðtÞ; i; tÞ
V 1 ðxðtÞ; i; tÞ ¼ e2kt nT1 ðtÞPi n1 ðtÞ; Z t Z V 2 ðxðtÞ; i; tÞ ¼ e2ks xT ðsÞQ 1 xðsÞds þ V 3 ðxðtÞ; i; tÞ ¼
Z
tsðtÞ
t
e2ks xT ðsÞ Q 2 xðsÞds þ
Z
th1
t
e2ks xT ðsÞQ 3 xðsÞds;
th2
t
_ e2ks x_ T ðsÞR1 xðsÞds; thðtÞ
V 4 ðxðtÞ; i; tÞ ¼
Z
Z
0
t
e2ks xT ðsÞZ 1 xðsÞdsdh þ
Z
tþh
h2
h1
Z
t
e2ks xT ðsÞZ 2 xðsÞds dh
tþh
h2
with
2
Z
n ðtÞ ¼ 4x ðtÞ T
T
!T
t
xðsÞds
Z
tsðtÞ
T
tsðtÞ
xðsÞds
Z
th1
tsðtÞ
th2
!T 3 xðsÞds 5:
Considering Definition 2.5, make the corresponding time derivative of V(x(t), i, t), such that
CV 1 ðxðtÞ; i; tÞ ¼ 2ke2kt nT1 ðtÞPi n1 ðtÞ þ 2e2kt nT1 ðtÞPi n_ 1 ðtÞ þ e2kt nT1 ðtÞ
N X
pij Pj n1 ðtÞ;
j¼1
CV 2 ðxðtÞ; i; tÞ ¼ e2kt xT ðtÞQ 1 xðtÞ e2kðtsðtÞÞ xT ðt sðtÞÞQ 1 xðt sðtÞÞ þ e2kt xT ðtÞQ 2 xðtÞ e2kðth1 Þ xT ðt h1 ÞQ 2 xðt h1 Þ þ e2kt xT ðtÞQ 3 xðtÞ e2kðth2 Þ xT ðt h2 Þ Q 3 xðt h2 Þ; _ e2kðthðtÞÞ x_ T ðt hðtÞÞR1 xðt _ hðtÞÞ; CV 3 ðxðtÞ; i; tÞ ¼ e2kt x_ T ðtÞR1 xðtÞ
CV 4 ðxðtÞ; i; tÞ ¼ h2 e2kt xT ðtÞZ 1 xðtÞ
Z
Z
tsðtÞ
e2ks xT ðsÞZ 1 xðsÞds th2
tsðtÞ
e2ks xT ðsÞZ 2 xðsÞds
Z
Z
t
e2ks xT ðsÞZ 1 xðsÞds þ ðh2 h1 Þe2kt xT ðtÞZ 2 xðtÞ
tsðtÞ th1
e2ks xT ðsÞZ 2 xðsÞds:
tsðtÞ
th2
Recalling that the uncertain neutral MJSs (2) over the space (X, F, P), it is not difficult to get the following relation:
T _ _ hðtÞÞ R1 Ai xðtÞ þ Bi xðt sðtÞÞ þ C i xðt _ hðtÞÞ x_ T ðtÞR1 xðtÞ ¼ Ai xðtÞ þ Bi xðt sðtÞÞ þ C i xðt _ hðtÞÞ ¼ xT ðtÞATi R1 Ai xðtÞ þ 2xT ðtÞATi R1 Bi xðt sðtÞÞ þ 2xT ðtÞATi R1 C i xðt _ hðtÞÞ þ x_ T ðt hðtÞÞC Ti R1 C i xðt _ hðtÞÞ: þ xT ðt sðtÞÞBTi R1 Bi xðt sðtÞÞ þ 2xT ðt sðtÞÞBTi R1 C i xðt By letting
2 bðtÞ ¼ 4xT ðtÞ xT ðt sðtÞÞ xT ðt h1 Þ xT ðt h2 Þ x_ T ðt hðtÞÞ it follows that CV(x(t), i, t) = e2ktbT(t)(Ki + Ri)b(t), where Ki = (Km,n,i)8
K1;1;i ¼ 2kP 1i þ P1i Ai þ ATi PT1i þ P2i þ PT2i þ
N X
pij P1j þ
j¼1
K1;2;i ¼ P 1i Bi k1 ðP2i P3i þ P4i Þ; K1;6;i ¼ kP 2i þ ATi P 2i þ PT5i þ
N X j¼1
K1;3;i ¼ P4i ;
3 X
Z
!T Z
t
xðsÞds tsðtÞ
8i,
Q m þ h2 Z 1 þ ðh2 h1 ÞZ 2 ;
m¼1
K1;4;i ¼ P3i ;
K1;5;i ¼ P1i C i ;
pij P2j ; K1;7;i ¼ kP3i þ ATi P3i þ P6i þ
tsðtÞ
N X j¼1
pij P3j ;
th2
T xðsÞds
Z
th1
tsðtÞ
!T 3T xðsÞds 5 ;
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P. Balasubramaniam et al. / Applied Mathematics and Computation 216 (2010) 3396–3407 N X
K1;8;i ¼ kP4i þ ATi P4i þ P7i þ
pij P4j ; K2;2;i ¼ k1 e2kh2 Q 1 ; K2;3;i ¼ K2;4;i ¼ K2;5;i ¼ 0;
j¼1
K2;6;i ¼ BTi P2i k1 ðPT5i PT6i þ PT7i Þ;
K2;7;i ¼ BTi P3i k1 ðP6i PT8i þ PT9i Þ;
K2;8;i ¼ BTi P4i k1 ðP7i P9i þ PT10i Þ; K3;7;i ¼ PT9i ;
K3;8;i ¼ P T10i ;
K3;3;i ¼ e2kh1 Q 2 ;
K4;4;i ¼ e2kh2 Q 3 ;
N X
K6;6;i ¼ 2kP 5i þ
pij P5j
j¼1 N X
K7;7;i ¼ 2kP 8i þ
pij P8j
j¼1
N X
K8;8;i ¼ 2kP 10i þ
e2kh2 Z 1 ; h2
K6;7;i ¼ kP6i þ
j¼1
K4;6;i ¼ P T6i ;
K5;7;i ¼ C Ti P3i ; N X
K4;7;i ¼ PT8i ;
K5;8;i ¼ C Ti P4i ;
pij P6j ; K6;8;i ¼ kP7i þ
N X
j¼1
e2kðh2 h1 Þ ðZ 1 þ Z 2 Þ ; ðh2 h1 Þ
pij P10j
K4;5;i ¼ 0;
K5;6;i ¼ C Ti P2i ;
K5;5;i ¼ k2 e2kh R1 ;
K4;8;i ¼ P9i ;
K3;6;i ¼ PT7i ;
K3;4;i ¼ K3;5;i ¼ 0;
pij P7j ;
j¼1
K7;8;i ¼ kP 9i þ
N X
pij P9j ;
j¼1
e2kðh2 h1 Þ Z2; ðh2 h1 Þ
h
iT
h
i
Ri ¼ Ai Bi 0 0 C i 0 0 0 R1 Ai Bi 0 0 C i 0 0 0 : Applying Schur complement, condition Ki + Ri < 0 is equivalent to the following inequality:
"
Si ¼
Ki P T R1
#
< 0:
In order to deal with uncertainties described as the form in Eq. (4), we can use the following approach Si = Ti + DTi < 0, where
" Ti ¼
Pi
NT
R1
# ;
Pi ¼ ðPm;n;i Þ88i ; P1;1;i ¼ 2kP 1i þ P1i Ai þ ATi PT1i þ P2i þ PT2i þ
N X
pij P1j þ
P1;6;i ¼ kP2i þ ATi P2i þ PT5i þ
N X
P1;3;i ¼ P 4i ;
Q m þ h2 Z 1 þ ðh2 h1 ÞZ 2 ;
m¼1
j¼1
P1;2;i ¼ P1i Bi k1 ðP 2i P3i þ P4i Þ;
3 X
P1;4;i ¼ P3i ;
pij P2j ; P1;7;i ¼ kP3i þ ATi P3i þ P6i þ
j¼1
P1;8;i ¼ kP4i þ ATi P4i þ P7i þ
N X
P1;5;i ¼ P1i C i ; N X
pij P3j ;
j¼1
pij P4j ; P2;2;i ¼ k1 e2kh2 Q 1 ; P2;3;i ¼ P2;4;i ¼ P2;5;i ¼ 0;
j¼1
P2;6;i ¼ BTi P2i k1 ðPT5i P T6i þ PT7i Þ; P2;8;i ¼ BTi P4i k1 ðP7i P 9i þ PT10i Þ; P3;8;i ¼ PT10i ;
P4;4;i ¼ e2kh2 Q 3 ;
P5;5;i ¼ k2 e2kh R1 ; P6;6;i ¼ 2kP 5i þ
N X j¼1
P5;6;i ¼ C Ti P2i ;
pij P5j
e2kh2 Z 1 ; h2
P2;7;i ¼ BTi P3i k1 ðP6i PT8i þ PT9i Þ; P3;3;i ¼ e2kh1 Q 2 ; P4;5;i ¼ 0;
P3;4;i ¼ P3;5;i ¼ 0;
P4;6;i ¼ PT6i ;
P5;7;i ¼ C Ti P3i ; P6;7;i ¼ kP 6i þ
P4;7;i ¼ PT8i ;
P3;6;i ¼ PT7i ;
P4;8;i ¼ P 9i ;
P5;8;i ¼ C Ti P 4i ; N X j¼1
pij P6j ; P6;8;i ¼ kP7i þ
P3;7;i ¼ PT9i ;
N X j¼1
pij P7j ;
3402
P. Balasubramaniam et al. / Applied Mathematics and Computation 216 (2010) 3396–3407
P7;7;i ¼ 2kP 8i þ
N X
e2kðh2 h1 Þ ðZ 1 þ Z 2 Þ ; ðh2 h1 Þ
pij P8j
j¼1
P8;8;i ¼ 2kP 10i þ
N X
pij P10j
j¼1
2 6 6 6 6 6 6 6 6 DT i ¼ 6 6 6 6 6 6 6 6 4
P7;8;i ¼ kP 9i þ
N X
e2kðh2 h1 Þ Z 2 ; ðh2 h1 Þ
c11
P1i DBi
0 0 0 DATi PT2i
DATi PT3i
DATi PT4i
DATi R1
0
DBTi PT2i 0 0 0 0 0 0 0 0 0 0
DBTi PT3i 0 0 0 0 0
DBTi PT4i 0 0 0 0 0 0
0
where c11 ¼ P 1i DAi þ
0 0 0
DATi PT1i .
pij P9j ;
j¼1
3
7 DBTi R1 7 7 0 7 7 7 0 7 7 0 7 7; 7 0 7 7 0 7 7 7 0 5
According to Lemma 2.7, DTi can be presented as the following form:
T T DT i ¼ L11 F i L12 þ LT12 F Ti LT11 < a1 i L11 L11 þ ai L12 L12 ;
where
h LT11 ¼ MTi PT1i L12 ¼ ½Ni
0 0 0 MTi PT2i
Nhi
MTi PT3i
M Ti P T4i
i M Ti R1 ;
0 0 0 0 0 0 0:
Applying Schur complement again, it is obvious that CV(x(t), i, t) is negative if LMIs (8) holds. On the other hand, since CV(x(t), i, t) < 0, we can get
VðxðtÞ; i; tÞ 6 Vðxð0Þ; r 0 ; 0Þ ¼ xT ð0ÞPi xð0Þ þ þ
Z
Z
0
e2ks xT ðsÞQ 1 xðsÞds þ
h2 0
h
_ e2ks x_ T ðsÞR1 xðsÞds þ
Z
0 h2
Z
0
e2ks xT ðsÞQ 2 xðsÞds þ
h1
Z
0
e2ks xT ðsÞZ 1 xðsÞdsdh þ
Z
h1
h2
h
Z
Z
0
e2ks xT ðsÞQ 3 xðsÞds
h2 0
e2ks xT ðsÞZ 2 xðsÞdsdh;
h
we have
VðxðtÞ; i; tÞ 6
Z
rP þ ðrQ 1 þ rQ 3 Þ
þ rZ 2 M 4
Z
h1
Z
h2
0
0
h2
Z e2ks ds M 1 þ rQ 2 M2
0 h1
e2ks ds þ rR1 M3
Z
0 h
e2ks ds þ rZ1 M 1
Z
0
h2
Z
0
e2ks dsdh
h
e2ks dsdh:
h
Recalling the given Lyapunov–Krasovskii functional, we can also have
VðxðtÞ; i; tÞ P e2kt rp EkxðtÞk2 : Therefore, the convergence rates of the system states in the form of Eq. (9) can be obtained. From Definition 2.4, the uncertain neutral MJSs (2) is exponentially stochastically stable in the mean square. This completes the proof. h Theorem 3.2. The uncertain neutral MJSs (2) is globally exponentially stochastically stable and satisfy Eq. (8), if there exist modedependent symmetric positive-definite matrices
2
3
P1i
P2i
P 3i
P 4i
6 6 Pi ¼ 6 4
P5i
P 6i
P 8i
P 7i 7 7 7 > 0; P 9i 5
P10i
K1i, K2i, K3i, K4i, K5i, K6i, K7i, K8i, K9i, K10i(i 2 M), symmetric positive-definite matrices Q1, Q2, Q3, R1, Z1, Z2, K2, K3, K4, K5, K6, K7, K8, a sequence {ai > 0,i 2 M} and a positive scalar c = k1, satisfying the following generalized eigenvalue problem (GEVP):
P. Balasubramaniam et al. / Applied Mathematics and Computation 216 (2010) 3396–3407
min c
3403
ð10Þ
subject to
2
7 R1 Mi 5 < 0;
R1
3
ST
NT
i X 6 4
ð11Þ
ai I
2P 1i < cK 1i ;
ð12Þ
P2i < cK 2i ;
ð13Þ
P3i < cK 3i ;
ð14Þ
P4i < cK 4i ; 2P 5i < cK 5i ;
ð15Þ ð16Þ
P6i < cK 6i ;
ð17Þ
P7i < cK 7i ;
ð18Þ
2P 8i < cK 8i ;
ð19Þ
P9i < cK 9i ;
ð20Þ
2P 10i < cK 10i ;
ð21Þ
2h2 K 2 6 cðQ 1 K 2 Þ;
ð22Þ
2h1 K 3 6 cðQ 2 K 3 Þ; 2h2 K 4 6 cðQ 3 K 4 Þ; 6 cðR K Þ; 2hK
ð23Þ ð24Þ ð25Þ
2h2 K 6 6 cðZ 1 K 6 Þ;
ð26Þ
5
1
5
2ðh2 h1 ÞK 7 6
cððZ 1 þ Z 2 Þ K 7 Þ
ðh2 h1 Þ ðZ 2 K 8 Þ ; 2ðh2 h1 ÞK 8 6 c ðh2 h1 Þ
ð27Þ
;
ð28Þ
where
i ¼ ðXm;n;i Þ X 88i ;
i Þ – ðXm;n;i Þ ðX whenððm; nÞÞ ¼ ðð1; 1Þ; ð1; 6Þ; ð1; 7Þ; ð1; 8Þ; ð2; 2Þ; ð3; 3Þ; ð4; 4Þ; ð5; 5Þ; ð6; 6Þ; ð6; 7Þ; ð6; 8Þ; ð7; 7Þ; ð7; 8Þ; ð8; 8ÞÞ;
1;1;i ¼ K 1i þ P1i Ai þ AT PT þ P2i þ PT þ X i 1i 2i
N X
pij P1j þ
N X
Q m þ ai N Ti Ni þ h2 Z 1 þ ðh2 h1 ÞZ 2 ;
m¼1
j¼1
1;6;i ¼ K 2i þ AT P2i þ PT þ X i 5i
3 X
1;7;i ¼ K 3i þ AT P 3i þ P6i þ pij P2j ; X i
j¼1
1;8;i ¼ K 4i þ AT P4i þ P7i þ X i
N X
N X
pij P3j ;
j¼1
2;2;i ¼ k1 K 2 þ ai N T Nhi ; X 3;3;i ¼ K 3 ; X 4;4;i ¼ K 4 ; pij P4j ; X hi
j¼1
6;6;i ¼ K 5i þ X
5;5;i ¼ k2 K 5 ; X
N X
6;7;i ¼ K 6i þ pij P5j K 6 ; X
j¼1
6;8;i ¼ K 7i þ X
N X
7;7;i ¼ K 8i þ pij P7j ; X
j¼1
8;8;i ¼ K 10i þ X
N X
N X
pij P6j ;
j¼1 N X
pij P8j K 7 ; X 7;8;i ¼ K 9i þ
j¼1
N X
pij P9j ;
j¼1
pij P10j K 8 :
j¼1
Proof. By defining
K 1i > 2kP 1i ;
K 2i > kP 2i ;
K 3i > kP 3i ;
K 4i > kP 4i ;
K 5i > 2kP5i ;
K 7i > kP 7i ;
K 8i > 2kP 8i ;
K 9i > kP 9i ;
K 10i > 2kP 10i ;
K 6i > kP6i ;
0 < K 2 < e2kh2 Q 1 ;
3404
P. Balasubramaniam et al. / Applied Mathematics and Computation 216 (2010) 3396–3407
0 < K 3 < e2kh1 Q 2 ;
0 < K7 <
e2kðh2 h1 Þ ðZ 1 þ Z 2 Þ ; ðh2 h1 Þ
we have
0 < K 5 < e2kh R1 ;
0 < K 4 < e2kh2 Q 3 ;
0 < K8 <
0 < K6 <
e2kh2 Z 1 ; h2
e2kðh2 h1 Þ Z 1 ; ðh2 h1 Þ
1i þ Ri bðtÞ < 0; CVðxðtÞ; i; tÞ 6 e2kt bT ðtÞ K where
1i ¼ ðKm;n;i Þ K 88i ;
1i – ðKm;n;i Þ when ððm; nÞÞ K
¼ ðð1; 1Þ; ð1; 6Þ; ð1; 7Þ; ð1; 8Þ; ð2; 2Þ; ð3; 3Þ; ð4; 4Þ; ð5; 5Þ; ð6; 6Þ; ð6; 7Þ; ð6; 8Þ; ð7; 7Þ; ð7; 8Þ; ð8; 8ÞÞ; 1;1;i ¼ K 1i þ P 1i Ai þ AT P T þ P2i þ PT þ K i 1i 2i
N X j¼1
1;6;i ¼ K 2i þ AT P2i þ PT þ K i 5i 1;8;i ¼ K 4i þ AT P4i þ P7i þ K i
N X j¼1 N X
pij P1j þ
3 X
Q m þ h2 Z 1 þ ðh2 h1 ÞZ 2 ;
m¼1
pij P2j ; K 1;7;i ¼ K 3i þ ATi P3i þ P6i þ
N X
pij P3j ;
j¼1
pij P4j ; K 2;2;i ¼ k1 K 2 ; K 3;3;i ¼ K 3 ; K 4;4;i ¼ K 4 ;
j¼1
6;6;i ¼ K 5i þ K
5;5;i ¼ k2 K 5 ; K
N X
6;7;i ¼ K 6i þ pij P5j K 6 ; K
j¼1
6;8;i ¼ K 7i þ K
N X
pij P7j ; K 7;7;i ¼ K 8i þ
j¼1
8;8;i ¼ K 10i þ K
N X
N X
pij P6j ;
j¼1 N X
pij P8j K 7 ; K 7;8;i ¼ K 9i þ
j¼1
N X
pij P9j ;
j¼1
pij P10j K 8 :
j¼1
By applying Schur complement and using some changes of matrix variables, CV(x(t), i, t) < 0 can be modified as LMI (11). On the other hand, by letting c = k1, we can get
K 1i > 2kP1i ) 2P 1i < cK 1i ;
K 2i > kP 2i ) P2i < cK 2i ;
K 3i > kP 3i ) P3i < cK 3i ;
K 4i > kP 4i ) P4i < cK 4i ;
K 5i > 2kP5i ) 2P 5i < cK 5i ;
K 6i > kP 6i ) P6i < cK 6i ;
K 7i > kP 7i ) P7i < cK 7i ;
K 8i > 2kP8i ) 2P 8i < cK 8i ;
K 9i > kP 9i ) P9i < cK 9i ;
K 10i > 2kP 10i ) 2P10i < cK 10i ; 0 < K 2 < e2kh2 Q 1 ) ð1 þ 2kh2 ÞK 2 < e2kh2 K 2 < Q 1 ) 2h2 K 2 6 cðQ 1 K 2 Þ; 0 < K 3 < e2kh1 Q 2 ) ð1 þ 2kh1 ÞK 3 < e2kh1 K 3 < Q 2 ) 2h1 K 3 6 cðQ 2 K 3 Þ; 0 < K 4 < e2kh2 Q 3 ) ð1 þ 2kh2 ÞK 4 < e2kh2 K 4 < Q 3 ) 2h2 K 4 6 cðQ 3 K 4 Þ; 5 < e2kh K 5 < R1 ) 2hK 5 6 cðR1 K 5 Þ; 0 < K 5 < e2kh R1 ) ð1 þ 2khÞK
0 < K 6 < e2kh2
Z1 Z1 ) ð1 2kh2 ÞK 6 < e2kh2 K 6 < ) 2h2 K 6 6 cðZ 1 K 6 Þ; h2 h2
0 < K7 <
e2kðh2 h1 Þ ðZ 1 þ Z 2 Þ Z1 þ Z2 cððZ 1 þ Z 2 Þ K 7 Þ ) ð1 2kðh2 h1 ÞÞK 7 < e2kðh2 h1 Þ K 7 < ) 2ðh2 h1 ÞK 7 6 ; ðh2 h1 Þ ðh2 h1 Þ ðh2 h1 Þ
0 < K8 <
e2kðh2 h1 Þ Z 2 Z2 ðZ 2 K 8 Þ ) ð1 2kðh2 h1 ÞÞK 8 < e2kðh2 h1 Þ K 8 < ) 2ðh2 h1 ÞK 8 6 c : ðh2 h1 Þ ðh2 h1 Þ ðh2 h1 Þ
3405
P. Balasubramaniam et al. / Applied Mathematics and Computation 216 (2010) 3396–3407
Therefore, the quasi-convex optimization problem can be solved by using MATLAB LMI Control Toolbox. Obviously, GEVP (10)–(28) presents an optimization algorithm which guarantees that the degree of exponential stability k is maximal (or c is minimal). This completes the proof. h Remark 3.3. From Definition 2.4, we conclude that the equilibrium point of system (2) is globally exponentially stable. We hope that the degree of exponential stability k is maximal (or c is minimal) such that the system converges to the equilibrium point as fast as possible. It requires solving the generalized eigenvalue minimization problem (10)–(28), which is a quasi-convex optimization problem and can be solved by using the MATLAB LMI control toolbox [30]. Although the conservativeness of Theorem 3.2 is higher than Theorem 3.1, it provides a simple method to determine the exponential stability of neutral systems (2) and get the upper bound of the exponential convergence rate and will be widely applied to stability analysis. Remark 3.4. Theorem 3.2 provides delay-dependent stability criterion for neutral systems with interval time-varying delays and Markovian jumping parameters. Moreover, a new type of Markovian jumping matrix Pi is taken into account. These type of augmented matrix has not been used in any of the existing literatures, for the exponential stability of neutral systems with Markovian jumping parameters via GEVPs.
4. Numerical examples Example 1. Consider the stochastic neutral partial element equivalent circuit (PEEC) model described by uncertain neutral MJSs with time-varying delays
_ _ hðtÞÞ xðtÞ ¼ ½Ai þ DAi xðtÞ þ ½Bi þ DBi xðt sðtÞÞ þ C i xðt
ð29Þ
with
A1 ¼ A2 ¼
C2 ¼
2
0
0
3
0:1 0 ; 0 0:1
N2 ¼ ½ 0 0:1 ;
;
B1 ¼ B2 ¼
M1 ¼
0:1 ; 0:1
0
1
;
1 1
M2 ¼
N h1 ¼ ½ 0:1 0:1 ;
C1 ¼
0 ; 0:1
0:3
0
0
0:3
;
N1 ¼ ½ 0:1 0 ;
Nh2 ¼ ½ 0:1 0:1 ;
¼ 0:2. From Eq. where s(t) = 0.1jsin tj and h(t) = 0.2jcos tj. Since 0 6 jsin tj 6 1, 0 6 jcos tj 6 1, we can get h1 ¼ 0; h2 ¼ 0:1; h _ ¼ 0:8. By solving GEVP (10)–(28), we can get (3), we can also get k1 ¼ inf tP0 ð1 s_ ðtÞÞ ¼ 0:9; k2 ¼ inf tP0 ð1 hðtÞÞ c = 0.8454 and the presented matrices are as follows:
"
1:0121
0:2658
0:2658
0:4248
0:0298
0:0191
0:0191
0:0387
P11 ¼
Q2 ¼
Z1 ¼
0:2598 0:6666
K 12 ¼
K4 ¼
K7 ¼
1:4679 0:2598
# ;
;
" P 12 ¼
Z2 ¼
;
1:7916
0:5190
0:5190
0:8391
0:0174
0:0113
0:0113
0:0233
1:0609
0:1098
0:1098
0:7527
;
Q3 ¼
0:8909
0:2564
0:2564
0:4116
0:0271
0:0179
0:0179
0:0382
0:3095
0:1956
0:1956
0:4654
K2 ¼
;
K5 ¼
;
K8 ¼
# ;
;
" Q1 ¼
R1 ¼
K 11 ¼
;
0:5021
0:1552
0:1552
0:3912
0:1356
0:0398
0:0398
0:0598
0:2507
0:1487
0:1487
0:3392
;
; :
0:6076
0:1896
0:1896
0:4762
0:2027
0:0604
0:0604
0:0903
2:0345
0:5382
0:5382
0:8612
K3 ¼
K6 ¼
;
0:0142
0:0142
0:0289
0:3153 0:5414
;
;
0:0222
1:4387 0:3153
#
;
;
3406
P. Balasubramaniam et al. / Applied Mathematics and Computation 216 (2010) 3396–3407
Therefore, according to Theorem 3.2, the uncertain neutral PEEC system presented by MJSs (2) with time-varying delay is globally exponentially stochastically stable. For the above system, applying Theorem 2 in [31] it is found that the equilibrium solution of neutral MJSs (29) is robustly exponentially stable in the mean square for any c satisfying c = 2.7157. However, by Theorem 3.2 in this paper we can conclude that if c = 0.8454, system (29) is robustly exponentially stable in the mean square sense based on the GEVPs. Example 2. Consider the stochastic neutral partial element equivalent circuit (PEEC) model described by uncertain neutral MJSs with time-varying delays
_ _ hðtÞÞ xðtÞ ¼ ½Ai þ DAi xðtÞ þ ½Bi þ DBi xðt sðtÞÞ þ C i xðt
ð30Þ
with
A1 ¼
C1 ¼
5
0
0
6
;
A2 ¼
0:5 0 ; 0 0:5
C2 ¼
N2 ¼ ½ 0 0:2 ;
4
0
0
5
;
B1 ¼
0:3 0 ; 0 0:3
M1 ¼
N h1 ¼ ½ 0:3 0:3 ;
0
1:6
1:8 1:5
0:2 ; 0:2
B2 ¼
;
M2 ¼
2
0
0:9 1:2
0 ; 0:3
;
N1 ¼ ½ 0:2 0 ;
Nh2 ¼ ½ 0:2 0:2 ;
¼ 0:3. From Eq. where s(t) = 0.1jsin tj and h(t) = 0.3jcos tj. Since 0 6 j sin t j 6 1,0 6 jcos t j 6 1, we can get h1 ¼ 0; h2 ¼ 0:3; h _ ¼ 0:8. By solving GEVP (10)–(28), we can get c = 2.9340 (3), we can also get k1 ¼ inf tP0 ð1 s_ ðtÞÞ ¼ 0:9; k2 ¼ inf tP0 ð1 hðtÞÞ and the presented matrices are as follows:
"
2:5228
0:4801
0:4801
0:7965
0:1121
0:1487
0:1487
0:2711
0:5119
0:2654
0:2654
0:6831
5:6467
1:0191
1:0191
1:8867
P11 ¼ Q2 ¼ Z1 ¼
K 12 ¼
K4 ¼
K7 ¼
0:0702
0:0934
0:0934
0:1704
#
" ;
;
P12 ¼ Q3 ¼
;
;
;
0:6495 0:5322 ; 0:5322 1:1445
Z2 ¼
K2 ¼
K5 ¼
K8 ¼
2:7942
0:4622
0:4622
0:8583
0:1161
0:1564
0:1564
0:2869
Q1 ¼
;
R1 ¼
0:3280
0:3280
0:6651
3:7785
0:4677
0:4677
0:9659
;
"
0:2970
#
0:4141
0:0727
0:0727
0:1047
;
;
K 11 ¼
K3 ¼
K6 ¼
0:7786
1:5999
0:0938
0:1326
5:1159
0:9937
0:9937
1:6620
0:1057
0:1057
0:1935
0:1918
0:1918
0:5452
;
;
0:0806
0:4717
#
;
0:0938
0:7786
0:5068
;
6:0676
;
;
0:2410 0:2591 : 0:2591 0:5258
Therefore, according to Theorem 3.2, the uncertain neutral PEEC system presented by MJSs (2) with time-varying delay is globally exponentially stochastically stable. By Theorem 3.2 in this paper we can conclude that if c = 2.9340, system (29) is robustly exponentially stable in the mean square sense based on the GEVPs. 5. Conclusion A new sufficient condition is derived to guarantee the global exponential stability of the Markovian jumping neutral systems with interval time-varying delays. By selecting appropriate Lyapunov–Krasovskii functionals, we give the sufficient conditions so that the analysis problem can be tackled in the form of LMIs, which can be easily calculated by MATLAB LMI control toolbox and the associated synthesis problem can be dealt with by solving the generalized eigenvalue problem (GEVP). The result through the examples shown that the derived stability criteria are less conservative than some recent ones. Adding additional terms and more positive-definite matrices which would be appeared in the mode dependent form of L–K functional lead the system stability to be complicated which would improve the numerical results to some extend than the results presented in this paper. This issue is one of the future directions of the topic.
P. Balasubramaniam et al. / Applied Mathematics and Computation 216 (2010) 3396–3407
3407
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