J Dyn Control Syst (2015) 21:285–305 DOI 10.1007/s10883-014-9265-0

Stochastic Stability for Uncertain Neutral Markovian Jump Systems with Nonlinear Perturbations Xinghua Liu · Hongsheng Xi

Received: 12 April 2013 / Revised: 8 September 2014 / Published online: 22 January 2015 © Springer Science+Business Media New York 2015

Abstract The delay-range-dependent stochastic stability for uncertain neutral Markovian jump systems with interval time-varying delays and nonlinear perturbations is investigated. The perturbations under consideration are time-varying and norm-bounded. By delay interval dividing, a novel augmented Lyapunov functional which contains triple-integral terms to reduce the conservativeness is introduced. Based on the Lyapunov functional approach and the nature of convex combination, some improved delay-range-dependent stochastic stability criteria are obtained in terms of linear matrix inequalities without introducing any free-weighting matrices. Finally, numerical examples are given to illustrate the effectiveness of the developed techniques. Keywords Neutral Markovian jump systems · Nonlinear perturbations · Stochastic stability · Delay-range-dependent stability Mathematics Subject Classfication (2010) 93E15

1 Introduction As is well-known, neutral delay systems constitute a more general class than those of the retarded type, which have found many applications in population ecology [39], distributed networks containing lossless transmission lines [26], heat exchangers, robots in contact with rigid environments [28], etc. In the past few decades, attention has been devoted to the robust delay-independent stability or delay-dependent stability and stabilization via different approaches for linear neutral systems with delayed state or input. Most results are obtained based on the Lyapunov-Krasovskii(L-K) method, see for example [4, 18, 20, 29] and the references therein. In recent years, the problem of robust stability of neutral systems

X. Liu () · H. Xi Department of Auto, School of Information Science and Technology, University of Science and Technology of China, 230027, Anhui, China e-mail: [email protected]

286

Xinghua Liu and Hongsheng Xi

with nonlinear perturbations has also received considerable attention, since nonlinear perturbations may cause instability and poor performance of practical systems as well as delays. Not surprisingly, the stability analysis of these systems is more difficult because the system involves the derivative of the delayed state and perturbations. Various techniques have been proposed by many researchers to derive the delay-dependent stability criteria for these classes of systems, for example, model transformation techniques, the improved bounding techniques, and matrix decomposition approaches, which can be seen in [3, 7, 9, 12, 13, 25, 32, 37, 40, 43], and references therein for more details. In another line, the applications of Markovian jumping parameters are popular in, e.g., economic systems [34], solar thermal receiver systems [5] and communication systems, 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 Markovion 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. In recent years, researchers have made considerable progress on Markovian jump control theory. The robust stability analysis and controller synthesis problems for Markovian jump systems (MJSs) have been extensively studied [2, 16, 22, 30]. However, to the best of our knowledge, few results have been reported in the literature concerning the problem of stochastic stability for neutral Markovian jumping systems with time-varying delays and nonlinear perturbations. It has not been fully investigated and is very challenging. This motivates our present work. For simplicity, we first investigate the nominal systems and then extend the results to the uncertain case. A new augmented Lyapunov functional containing triple-integral terms is introduced for less conservative stochastic stability criteria for the considered system. Besides, the information on the lower bound of the delay is fully utilized in the functional. Then, the improved criteria are obtained by the nature of convex combination and expressed in linear matrix inequalities (LMIs), which can be easily checked numerically. The remainder of this paper is organized as follows. Section 2 gives the problem statement and preliminaries, Section 3 presents the main results, Section 4 provides a numerical example to verify the effectiveness of the results, and Section 5 concludes the paper. Notations. The following notations are used in this paper. Rn is the n dimensional Euclidean space and Rm×n is the set of all m × n matrices. X < Y (X > Y ), where X and Y are both symmetric matrices, means that X − Y is negative (positive) definite. I denotes the identity matrix with proper dimensions. λmax(min) (A) is the eigenvalue of matrix A with maximum(minimum) real part. For a symmetric block matrix, we use ∗ to denote the terms introduced by symmetry. E stands for the mathematical expectation, v is the 1 Euclidean norm of vector v, v = (v T v) 2 , while A is spectral norm of matrix A, 1 A = [λmax (AT A)] 2 .

2 Problem Statement and Preliminaries Given a probability space {, F , P} where  is the sample space, F is the algebra of events, and P is the probability measure defined on F . {rt , t ≥ 0} is a homogeneous, finite-state Markovian process with right continuous trajectories taking values in a finite set

Stochastic Stability Analysis for Neutral Markovian Jump

287

S = {1, 2, 3, · · · , N }, with the mode transition probability matrix being  πij t + o(t) i  = j P (rt+t = j |rt = i) = 1 + πii t + o(t) i = j where t > 0, limt→0 o(t) t = 0 and πij ≥ 0(i, j ∈ S, i  = j ) is the transition rate from mode i to j and for any state or mode i ∈ S, πii = −

N 

πij

j =1,j =i

Consider the following uncertain neutral Markovian jump systems with interval timevarying delays and nonlinear perturbations over the space {, F , P}, ˙ − τ ) = [A(rt ) + A(rt )]x(t) + [B(rt ) + B(rt )]x(t − d(t)) x(t) ˙ − C(rt )x(t +f1 (x(t), t) + f2 (x(t − d(t)), t) + f3 (x(t ˙ − τ ), t) x(s) = ϕ(s), rs = r0 , s ∈ [−ρ, 0]

(1) (2)

Rn

is the system state and τ > 0 is a constant neutral delay. The time-varying where x(t) ∈ delay d(t) satisfies ˙ ≤μ (3) 0 < d1 ≤ d(t) ≤ d2 , d(t) where d1 < d2 and μ ≥ 0 are constant real values. The initial condition ϕ(s) is a continuously differentiable vector-valued function. The continuous norm of ϕ(s) is defined as ϕc = max |ϕ(s)|, ρ = max{τ, d2 } s∈[−ρ,0]

Rn ,

f2 (x(t − d(t)), t) ∈ Rn and f3 (x(t ˙ − τ ), t) ∈ Rn are unknown nonlinear f1 (x(t), t) ∈ perturbations which with respect to the current state x(t), the delayed state x(t − d(t)) and the neutral delay state x(t ˙ − τ ), respectively. For all t, they are assumed to be bounded in magnitude as  f1 (x(t), t) ≤ αx(t)  f2 (x(t − d(t)), t) ≤ βx(t − d(t))  f3 (x(t ˙ − τ ), t) ≤ γ x(t ˙ − τ )

(4)

where α ≥ 0, β ≥ 0 and γ ≥ 0 are given constants, for simplicity, f1  f1 (x(t), t), f2  f2 (x(t − d(t)), t) and f3  f3 (x(t ˙ − τ ), t). A(rt ) ∈ Rn×n , B(rt ) ∈ Rn×n , C(rt ) ∈ Rn×n are known mode-dependent constant matrices, and A(rt ) ∈ Rn×n , B(rt ) ∈ Rn×n are uncertainties. For notational simplicity, A(rt ), A(rt ), B(rt ), B(rt ), and C(rt ) are, respectively, denoted as Ai , Ai , Bi , Bi , Ci for rt = i ∈ S. Throughout this paper, the parametric matrix Ci  < 1 and the admissible parametric uncertainties are assumed to satisfy the following condition, [Ai (t) Bi (t)] = Hi Fi (t) [EAi EBi ]

(5)

where Hi , EAi , EBi are known mode-dependent constant matrices with appropriate dimensions and Fi (t) is an unknown and time-varying matrix satisfying, FiT (t)Fi (t) ≤ I, ∀t

(6)

Particularly, the following nominal system is obtained for Fi (t) = 0, ˙ − τ ) = Ai x(t) + Bi x(t − d(t)) + f1 + f2 + f3 x(t) ˙ − Ci x(t

(7)

288

Xinghua Liu and Hongsheng Xi

In order to obtain the main results, the following assumptions, definitions, and lemmas are needed. Assumption 2.1 The system matrix Ai , (∀i ∈ S) is Hurwitz matrix with all the eigenvalues having negative real parts for each mode. The matrix Hi , (∀i ∈ S) is chosen as a full row rank matrix. Assumption 2.2 The Markov process is irreducible and the system mode rt is available at time t. Definition 2.1 ([14]) Define operator D : C([−ρ, 0], Rn ) → Rn as D(xt ) = x(t)−Cx(t − τ ). D is said to be stable if the homogeneous difference equation   D(xt ) = 0, t ≥ 0, x0 = ψ ∈ φ ∈ C([−ρ, 0], Rn ) : Dφ = 0 is uniformly asymptotically stable. Obviously, in this paper, we need Ci  + γ < 1 to guarantee the stability of operator D. Definition 2.2 ([42]) The systems which are described by Eq. 1 are said to be stochastically stable if there exists a positive constant ϒ such that  ∞  E x(rt , t)2 dt|ϕ(s), s ∈ [−ρ, 0], r0 < ϒ 0

Definition 2.3 ([36]) In the Euclidean space {Rn × S × R+ }, we introduce the stochastic Lyapunov-Krasovskii function of system (1) as V (x(t), rt = i, t > 0) = V (xt , i, t), the infinitesimal generator(an operator LV from Rn × S × R+ to R) satisfying 1  LV (x(t), i, t)= lim E {V (x(t + t), rt+t , t + t)|x(t)=x, rt = i} − V (x(t), i, t) t→0 t N  ∂ ∂ πij V (x(t), j, t) = V (x(t), i, t) + V (x(t), i, t)x(t) ˙ + ∂t ∂x j =1

Lemma 2.1 ([11, 15]) For any constant matrix H = H T > 0 and scalars τ2 > τ1 > 0 such that the following integrations are well defined, then (a)

 −(τ2 − τ1 )

t−τ1

 x T (s)H x(s)ds ≤ −

t−τ2

(b) 1 − (τ22 −τ12 ) 2



−τ1  t

−τ2

t+θ

t−τ1

  x T (s)ds H

t−τ2

 x T (s)H x(s)ds ≤ −

x(s)ds

t−τ2

−τ1  t

−τ2



t−τ1

  x T (s)ds H

t+θ



−τ1  t

−τ2

x(s)ds t+θ

Lemma 2.2 ([17]) For given matrices Q = QT , M and N with appropriate dimensions, then Q + MF (t)N + N T F T (t)M T < 0 for all F (t) satisfying F T (t)F (t) ≤ I , if and only if there exists a scalar δ > 0, such that Q + δ −1 MM T + δN N T < 0

Stochastic Stability Analysis for Neutral Markovian Jump

289

Lemma 2.3 ([8]) Given constant matrices 1 , 2 , and 3 , where 1 = T1 and 2 = T2 > 0, then 1 + T3 −1 2 3 < 0 if and only if



 1 T3 −2 T3 < 0 or <0 ∗ −2 ∗ 1 3 Main Results In this section, we will perform the stochastic stability analysis of neutral Markovian jump systems with interval time-varying delays and nonlinear perturbations described by Eqs. 7 and 1. 3.1 Stochastic Stability for the Nominal Systems Theorem 3.1 For given scalars α, β, γ , τ , d1 , d2 , μ, and constant scalar dm satisfying d1 < dm < d2 , the neutral Markovian jump systems with delays and perturbations as described in Eq. 7 are stochastically stable if Ci  + γ < 1 and there exist symmetric positive matrices Pi > 0, (i ∈ S), Qj > 0, (j = 1, 2, · · · , 5), Rk > 0, (k = 1, 2, · · · , 7), Yl > 0, (l = 1, 2, · · · , 6), Zm > 0, (m = 1, 2, 3), and scalars ε1 , ε2 , ε3 , such that the following linear matrix inequalities hold, i1 +  T M < 0

(8)

i2 +  T M < 0

(9)

i3 +  T M < 0

(10)

i4 +  T M < 0

(11)

where





T i1 = i0 − e13 Y5 e13 − (e4 − e5 ) Y6 e4T − e5T − 2 (e2 − e4 ) Y4 e2T − e4T



T T T − e10 − e10 Y3 e10 − (e3 − e2 ) Y4 e3T − e2T − 2(e12 − e10 )Y3 e12



T i2 = i0 − e13 Y5 e13 − (e4 − e5 )Y6 e4T − e5T − (e2 − e4 )Y4 e2T − e4T



T T T − e10 − 2e10 Y3 e10 −2(e3 − e2 )Y4 e3T − e2T − (e12 − e10 )Y3 e12



T i3 = i0 − e12 Y3 e12 − (e3 − e4 )Y4 e3T − e4T − 2(e2 − e5 )Y6 e2T − e5T



T T T − (e13 − e11 )Y5 e13 − e11 −(e4 − e2 )Y6 e4T − e2T − 2e11 Y5 e11



T − (e3 − e4 )Y4 e3T − e4T − (e2 − e5 )Y6 e2T − e5T i3 = i0 − e12 Y3 e12



T T T −2(e4 − e2 )Y6 e4T − e2T − e11 Y5 e11 − 2(e13 − e11 )Y5 e13 − e11

290

Xinghua Liu and Hongsheng Xi

where T T i0 = e1 W e1T + e1 Pi Bi e2T + e2 BiT Pi e1T + e1 Pi Ci e15 + e15 CiT Pi e1T + e1 Pi e17 + e17 Pi e1T   T T +e1 Pi e18 + e18 Pi e1T + e1 Pi e19 + e19 Pi e1T + e2 ε2 β 2 I − (1 − μ)R7 e2T

+e3 (R3 + R7 − R2 )e3T + e4 (R5 − R3 )e4T − e5 R5 e5T + e6 (R4 − R2 )e6T

T +e7 (R6 − R4 )e7T − e8 R6 e8T − e9 Y1 e9T − (e1 − e14 )Q4 e1T − e14





T −(e1 − e3 )Y2 e1T − e3T − (τ e1 − e16 )Q5 τ e1T − e16 − (d1 e1 − e9 )Z1 d1 e1T − e9T



T T T −(1 e1 − e12 )Z2 1 e1T − e12 − (2 e1 − e13 )Z3 2 e1T − e13 − e14 Q1 e14   T T T T T +e15 ε3 γ 2 I − Q2 e15 − e16 Q3 e16 − ε1 e17 e17 − ε2 e18 e18 − ε3 e19 e19

where ei {i = 1, 2, · · · , 19} are block entry matrices, e.g., e2T = [0 I 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0] W = ATi Pi + Pi Ai +

N 

πij pj + Q1 + τ 2 Q3 + R1 + d12 Y1 + 12 Y3 + 22 Y5 + ε1 α 2 I

j =1

M = Q2 + τ 2 Q4 +

τ4

Q5 + R2 + d12 Y2 + 12 Y4 + 22 Y6 +

4  = [Ai Bi 0 0 0 0 0 0 0 0 0 0 0 0 Ci 0 I I I ]

d14 Z1 + 32 Z2 + 42 Z3 4

1 = dm − d1 , 2 = d2 − dm

1 2 1 2 2 3 = dm − d12 , 4 = d2 − dm 2 2 Proof Construct the following Lyapunov functional, V (x(t), i, t) = Vr (xt , i) + Vτ (xt , i) + Vd1 (xt , i) + Vd2 (xt , i) + Vd3 (xt , i)

(12)

where Vr (xt , i), Vτ (xt , i), Vd1 (xt , i), Vd2 (xt , i), Vd3 (xt , i) denote Vr (x(t), i, t), Vτ (x(t), i, t), Vd1 (x(t), i, t), Vd2 (x(t), i, t), Vd3 (x(t), i, t) respectively and Vr (xt , i) = x T (t)Pi x(t) 



t

Vτ (xt , i) =

t−τ

 +



t

x˙ T (s)Q2 x(s)ds ˙ +

t−τ

0 t

−τ t+θ

Vd1 (xt , i) =



t

x T (s)Q1 x(s)ds +



t−d1  t−d1

+

t−dm  t−d1

+

t−d(t)



x˙ T (s) [τ Q4 ] x(s)dsdθ ˙ +

x T (s)R1 x(s)ds +

−τ θ

−τ t+θ

x T (s) [τ Q3 ] x(s)dsdθ



x˙ T (s)

t+λ

x˙ T (s)R2 x(s)ds ˙ + 

x˙ T (s)R4 x(s)ds ˙ +

0  0 t

0 t



t

t−d1

 τ2 ˙ (14) Q5 x(s)dsdλdθ 2

t−d1

x T (s)R3 x(s)ds

t−dm t−dm

t−d2

x T (s)R7 x(s)ds

(13)



x T (s)R5 x(s)ds +

t−dm

x˙ T (s)R6 x(s)ds ˙

t−d2

(15)

Stochastic Stability Analysis for Neutral Markovian Jump

 Vd2 (xt , i)=

0



t

 x T (s)[d1 Y1 ]x(s)dsdθ +

−d1 t+θ  −d1  t

+

−dm t+θ  −dm  t

+

−d2

291 0

t

x˙ T (s)[d1 Y2 ]x(s)dsdθ ˙

−d1 t+θ  −d1  t

x T (s)[1 Y3 ]x(s)dsdθ +

x T (s)[2 Y5 ]x(s)dsdθ +

t+θ



−dm t+θ  −dm  t −d2

x˙ T (s)[1 Y4 ]x(s)dsdθ ˙ x˙ T (s)[2 Y6 ]x(s)dsdθ ˙

(16)

t+θ



 d12 x˙ (s) Z1 x(s)dsdλdθ Vd3 (xt , i) = ˙ 2 −d1 θ t+λ  −d1  0  t x˙ T (s)[3 Z2 ]x(s)dsdλdθ ˙ + 

0

 0

t

T

−dm θ t+λ −dm  0  t

 +

−d2

θ

x˙ T (s)[4 Z3 ]x(s)dsdλdθ ˙

(17)

t+λ

From Definition 2.3, according to Eqs. 12–17, the infinitesimal generators or the operators LV (x(t), i, t), LVr (xt , i), LVτ (xt , i), LVd1 (xt , i), LVd2 (xt , i), LVd3 (xt , i) along the trajectory of system (7) are given in the following LV (x(t), i, t)=LVr (xt , i) + LVτ (xt , i) + LVd1 (xt , i) + LVd2 (xt , i) + LVd3 (xt , i) (18)   LVr (xt , i)=2 x T (t)ATi +x T (t − d(t))BiT + x˙ T (t − τ )CiT + f1T + f2T + f3T Pi x(t) +

N 

πij x T (t)Pj x(t)

(19)

j =1

    τ4 2 T 2 ˙ LVτ (xt , i)=x (t) Q1 + τ Q3 x(t) + x˙ (t) Q2 + τ Q4 + Q5 x(t) 4  t ˙ − τ) − x T (s)[τ Q3 ]x(s)ds −x T (t − τ )Q1 x(t − τ ) − x˙ T (t − τ )Q2 x(t t−τ    0 t  t τ2 T T Q5 x(s)dsdθ ˙ (20) x˙ (s)[τ Q4 ]x(s)ds ˙ − x˙ (s) − 2 t−τ −τ t+θ T

T ˙ ˙ − (1 − d(t))x (t − d(t))R7 x(t − d(t)) LVd1 (xt , i)=x T (t)R1 x(t) + x˙ T (t)R2 x(t)

+x T (t − d1 )[R3 + R7 − R2 ]x(t − d1 ) + x˙ T (t − d1 )[R4 − R2 ]x(t ˙ − d1 ) +x T (t − dm )[R5 − R3 ]x(t − dm ) + x˙ T (t − dm )[R6 − R4 ]x(t ˙ − dm ) ˙ − d2 ) −x T (t − d2 )R5 x(t − d2 ) − x˙ T (t − d2 )R6 x(t

(21)

292

Xinghua Liu and Hongsheng Xi

    LVd2 (xt , i)=x T (t) d12 Y1 + 12 Y3 + 22 Y5 x(t) + x˙ T (t) d12 Y2 + 12 Y4 + 22 Y6 x(t) ˙  t  t x T (s)[d1 Y1 ]x(s)ds − x˙ T (s)[d1 Y2 ]x(s)ds ˙ − t−d1 t−d1

 −

x T (s)[1 Y3 ]x(s)ds −

t−dm  t−dm

−

t−d1  t−d1

x˙ T (s)[1 Y4 ]x(s)ds ˙

t−dm  t−dm

x T (s)[2 Y5 ]x(s)ds −

t−d2

x˙ T (s)[2 Y6 ]x(s)ds ˙

(22)

t−d2

    0  t d14 d12 2 2 T Z1 + 3 Z2 + 4 Z3 x(t) Z1 x(s)dsdθ ˙ − ˙ x˙ (s) LVd3 (xt , i)= x˙ (t) 4 2 −d1 t+θ  −dm  t  −d1  t x˙ T (s)[3 Z2 ]x(s)dsdθ− ˙ x˙ T (s)[4 Z3 ]x(s)dsdθ ˙ (23) − 

T

−dm

−d2

t+θ

t+θ

In view of Eq. 4, the following inequalities hold for any scalars ε1 > 0, ε2 > 0 and ε3 > 0   ε1 α 2 x T (t)x(t) − f1T (x(t), t)f1 (x(t), t) ≥ 0   ε2 β 2 x T (t − d(t))x(t − d(t)) − f2T (x(t − d(t)), t)f2 (x(t − d(t)), t) ≥ 0   ˙ − τ ) − f3T (x(t ˙ − τ ), t)f3 (x(t ˙ − τ ), t) ≥ 0 (24) ε3 γ 2 x˙ T (t − τ )x(t From Eqs. 18 and 24, we have LV (x(t), i, t) ≤ LVr (xt , i) + LVτ (xt , i) + LVd1 (xt , i) + LVd2 (xt , i) + LVd3 (xt , i)     +ε1 α 2 x T (t)x(t) − f1T f1 + ε2 β 2 x T (t − d(t))x(t − d(t)) − f2T f2   ˙ − τ ) − f3T f3 (25) +ε3 γ 2 x˙ T (t − τ )x(t Then define ˙ − d1 ) x(t ˙ − dm ) x(t ˙ − d2 ) ξ(t) = col{x(t) x(t − d(t)) x(t − d1 ) x(t − dm ) x(t − d2 ) x(t  t−d1  t−d(t)  t−d1  t x(s)ds x(s)ds x(s)ds x(s)ds t−d1 t−dm

t−d(t)



t−d2

x(s)ds x(t − τ ) x(t ˙ − τ)



t−dm t

x(s)ds f1 f2 f3 }

t−τ

t−d2

Applying (a) of Lemma 2.1, we obtain  t T x T (s)[τ Q3 ]x(s)ds ≤ −ξ T (t)e16 Q3 e16 ξ(t) −

(26)

t−τ

 −

x T (s)[d1 Y1 ]x(s)ds ≤ −ξ T (t)e9 Y1 e9T ξ(t)

(27)



T ξ(t) x˙ T (s)[τ Q4 ]x(s)ds ˙ ≤ −ξ T (t)(e1 − e14 )Q4 e1T − e14

(28)



x˙ T (s)[d1 Y2 ]x(s)ds ˙ ≤ −ξ T (t)(e1 − e3 )Y2 e1T − e3T ξ(t)

(29)

t−d1



t

−

t−τ

 −

t

t

t−d1

Stochastic Stability Analysis for Neutral Markovian Jump

293

Applying (b) of Lemma 2.1, we have    0 t

τ2 T T Q5 x(s)dsdθ ˙ ≤ −ξ T (t)(τ e1 − e16 )Q5 τ e1T − e16 x˙ (s) ξ(t) − 2 −τ t+θ    0  t

d12 T x˙ (s) Z1 x(s)dsdθ ˙ ≤ −ξ T (t)(d1 e1 − e9 )Z1 d1 e1T − e9T ξ(t) − 2 −d1 t+θ  −d1  t

T ξ(t) x˙ T (s)[3 Z2 ]x(s)dsdθ ˙ ≤ −ξ T (t)(1 e1 − e12 )Z2 1 e1T − e12 − −dm

t+θ

−d2

t+θ

 −

−dm  t



T ξ(t) x˙ T (s)[4 Z3 ]x(s)dsdθ ˙ ≤ −ξ T (t)(2 e1 − e13 )Z3 2 e1T − e13

For d(t) ∈ [d1 , dm ], let λ =  t−d1 x˙ T (s)[1 Y4 ]x(s)ds ˙ −

d(t)−d1 1

t−dm



= −1

t−d(t)

t−dm



= −(dm − d(t))

t−d1

x˙ T (s)Y4 x(s)ds ˙ − (d(t) − d1 )

t−dm  t−d1

(32) (33)

x˙ T (s)Y4 x(s)ds ˙

t−d(t) t−d(t)

(31)

and use (a) of Lemma 2.1 again, then we have

 x˙ T (s)Y4 x(s)ds ˙ − 1

(30)



t−d(t)

x˙ T (s)Y4 x(s)ds ˙

t−dm t−d1

 x˙ T (s)Y4 x(s)ds ˙ − (dm − d(t)) x˙ T (s)Y4 x(s)ds ˙ t−d(t) t−d(t)



≤−(1 + λ)ξ T (t)(e2 −e4 )Y4 e2T −e4T ξ(t)−(2 − λ)ξ T (t)(e3 −e2 )Y4 e3T −e2T ξ(t) (34) −(d(t) − d1 )

and

 −

t−d1

x T (s)[1 Y3 ]x(s)ds

t−dm



T T T − e10 ξ(t) − (2 − λ)ξ T (t) e10 Y3 e10 ξ(t)(35) ≤ −(1 + λ)ξ T (t)(e12 − e10 )Y3 e12  t−d  t−d ˙ are directly estimated where − t−d2m x T (s)[2 Y5 ]x(s)ds and − t−d2m x˙ T (s)[2 Y6 ]x(s)ds by (a) of Lemma 2.1, i.e.  t−dm T x T (s)[2 Y5 ]x(s)ds ≤ −ξ T (t)e13 Y5 e13 ξ(t) (36) − t−d2

and

 −

t−dm

t−d2



x˙ T (s)[2 Y6 ]x(s)ds ˙ ≤ −ξ T (t)(e4 − e5 )Y6 e4T − e5T ξ(t)

(37)

Considering ˙ = ξ T (t) T Mξ(t) x˙ T (t)M x(t)

(38)

where M,  have been defined as before. We take the above equalities (19)–(23), inequalities (26)–(37), and (38) into (25), thus we finally get   LV (x(t), i, t) ≤ ξ T (t) λi1 + (1 − λ)i2 +  T M ξ(t)

294

Xinghua Liu and Hongsheng Xi

Since 0 ≤ λ ≤ 1, λi1 + (1 − λ)i2 +  T M is a convex combination of λ(i1 +  T M) and (1 − λ)(i2 +  T M). Therefore, λi1 + (1 − λ)i2 +  T M < 0 is equivalent to Eqs. 8 and 9. For another situation, when the time-delay d(t) ∈ [dm , d2 ], following the same m and finally obtain procedure, we define  λ = d(t)−d 2   LV (x(t), i, t) ≤ ξ T (t)  λi3 + (1 −  λ)i4 +  T M ξ(t) Obviously,  λi3 + (1 −  λ)i4 +  T M < 0 is equivalent to Eqs. 10 and 11. So we choose ϑ = max{ϑ1 , ϑ2 }, where   ϑ1 = max λmax λi1 + (1 − λ)i2 +  T M i∈S   λ)i2 +  T M ϑ2 = max λmax  λi1 + (1 −  i∈S

Then, we have LV (x(t), i, t) ≤ ϑξ(t)2 ≤ ϑx(t)2

(39)

According to Eq. 39, from the Dynkin’s Formula, we obtain   t E {V (x(t), i, t)} − V (x0 , r0 ) ≤ ϑ E x(s)2 ds 0

Let t → ∞, then we have



lim E

t→∞

t

 x(s)2 ds ≤ (−ϑ)−1 V (x0 , r0 )

0

From Definition 2.2, we know that the systems described by (7) are stochastically stable. This completes the proof. Remark 3.1 Theorem 3.1 provides a delay-range-dependent stochastic stability criterion for nominal neutral Markovian jump systems with delays and perturbations as described by Eq. 7. By utilizing a new Lyapunov functional, the less conservative criterion is obtained in terms of LMIs and it can be verified in Section 4. Remark 3.2 With regard to the stochastic stability of neutral Markovian jump systems with delays and nonlinear perturbations, the type of augmented Lyapunov functional has not been used in any of the existing literatures. Compared with the existing Lyapunov functional, the proposed one (12) contains some triple-integral terms, which is very effective in the reduction of conservatism in [15]. Besides, the information on the lower bound of the delay is sufficiently used in the Lyapunov functional by introducing the terms, such as  t−d1 T  t−dm T  t−dm T  t−d1 T ˙ ˙ t−dm x (s)R3 x(s)ds, t−dm x˙ (s)R4 x(s)ds, t−d2 x (s)R5 x(s)ds, t−d2 x˙ (s)R6 x(s)ds  t−d1 T and t−d(t) x (s)R7 x(s)ds. In many circumstances, the information on the delays derivative may not be available. So we give the following result as a corollary which can be obtained from Theorem 3.1 by setting R7 = 0. Corollary 3.1 For given scalars α, β, γ , τ, d1 , d2 , and constant scalar dm satisfying d1 < dm < d2 , the neutral Markovian jump systems with delays and perturbations as described in Eq. 7 are stochastically stable if Ci  + γ < 1 and there exist symmetric positive matrices

Stochastic Stability Analysis for Neutral Markovian Jump

295

Pi > 0, (i ∈ S), Qj > 0, (j = 1, 2, · · · , 5), Rk > 0, (k = 1, 2, · · · , 6), Yl > 0, (l = 1, 2, · · · , 6), Zm > 0, (m = 1, 2, 3), and scalars ε1 , ε2 , ε3 , such that the following linear matrix inequalities hold,  i1 +  T M < 0 (40)   i2 +  T M < 0 

(41)

 i3 +  T M < 0 

(42)

 i4 +  M < 0 

(43)

T

where





 i1 =   i0 − e13 Y5 eT − (e4 − e5 )Y6 eT − eT − 2(e2 − e4 )Y4 eT − eT  13 4 5 2 4



T T T T T −(e3 − e2 )Y4 e3 − e2 − 2(e12 − e10 )Y3 e12 − e10 − e10 Y3 e10



 i0 − e13 Y5 eT − (e4 − e5 )Y6 eT − eT − (e2 − e4 )Y4 eT − eT  i2 =   13 4 5 2 4



T T T − e10 − 2e10 Y3 e10 −2(e3 − e2 )Y4 e3T − e2T − (e12 − e10 )Y3 e12



 i0 − e12 Y3 eT − (e3 − e4 )Y4 eT − eT − 2(e2 − e5 )Y6 eT − eT  i3 =   12 3 4 2 5



T T T −(e4 − e2 )Y6 e4T − e2T − 2e11 Y5 e11 − (e13 − e11 )Y5 e13 − e11



 i0 − e12 Y3 eT − (e3 − e4 )Y4 eT − eT − (e2 − e5 )Y6 eT − eT  i4 =   12 3 4 2 5



T T T − (e13 − 2e11 )Y5 e13 − e11 −2(e4 − e2 )Y6 e4T − e2T − e11 Y5 e11

where  i0 =e1 W eT + e1 Pi Bi eT + e2 B T Pi eT + e1 Pi Ci eT + e15 C T Pi eT + e1 Pi eT + e17 Pi eT  i i 1 2 1 15 1 17 1 T T +e1 Pi e18 + e18 Pi e1T + e1 Pi e19 + e19 Pi e1T + ε2 β 2 e2 e2T

+e3 (R3 − R2 )e3T + e4 (R5 − R3 )e4T − e5 R5 e5T + e6 (R4 − R2 )e6T

T +e7 (R6 − R4 )e7T − e8 R6 e8T − e9 Y1 e9T − (e1 − e14 )Q4 e1T − e14





T −(d1 e1 − e9 )Z1 d1 e1T − e9T −(e1 −e3 )Y2 e1T −e3T −(τ e1 −e16 )Q5 τ e1T −e16



T T T − (2 e1 − e13 )Z3 2 e1T − e13 − e14 Q1 e14 −(1 e1 − e12 )Z2 1 e1T − e12   T T T T T +e15 ε3 γ 2 I − Q2 e15 − e16 Q3 e16 − ε1 e17 e17 − ε2 e18 e18 − ε3 e19 e19 and other notations are the same as Theorem 3.1. If Ci = 0 and γ = 0, then the system (7) reduces to the following Markovian jump system with nonlinear perturbations: x(t) ˙ = Ai x(t) + Bi x(t − d(t)) + f1 (x(t), t) + f2 (x(t − d(t)), t) x(s) = ϕ(s), rs = r0 , s ∈ [−d2 , 0]

(44)

According to Theorem 3.1, we have the following corollary for the delay-dependent stochastic stability of system (44).

296

Xinghua Liu and Hongsheng Xi

Corollary 3.2 For given scalars α, β, d1 , d2 , and constant scalar dm satisfying d1 < dm < d2 , the neutral Markovian jump systems with delays and perturbations as described in Eq. 44 are stochastically stable if there exist symmetric positive matrices Pi > 0, (i ∈ S), Rk > 0, (k = 1, 2, · · · , 7), Yl > 0, (l = 1, 2, · · · , 6), Zm > 0, (m = 1, 2, 3), and scalars ε1 , ε2 , such that the following linear matrix inequalities hold,

where

i1 +  T M < 0

(45)

i2 +  T M < 0

(46)

i3 +  T M < 0

(47)

i4 +  T M < 0

(48)





T i1 = i0 − e13 Y5 e13 − (e4 − e5 )Y6 e4T − e5T − 2(e2 − e4 )Y4 e2T − e4T



T T T − e10 − e10 Y3 e10 −(e3 − e2 )Y4 e3T − e2T − 2(e12 − e10 )Y3 e12



T i2 = i0 − e13 Y5 e13 − (e4 − e5 )Y6 e4T − e5T − (e2 − e4 )Y4 e2T − e4T



T T T − e10 − 2e10 Y3 e10 −2(e3 − e2 )Y4 e3T − e2T − (e12 − e10 )Y3 e12



T i3 = i0 − e12 Y3 e12 − (e3 − e4 )Y4 e3T − e4T − 2(e2 − e5 )Y6 e2T − e5T



T T T − (e13 − e11 )Y5 e13 − e11 −(e4 − e2 )Y6 e4T − e2T − 2e11 Y5 e11



T i4 = i0 − e12 Y3 e12 − (e3 − e4 )Y4 e3T − e4T − (e2 − e5 )Y6 e2T − e5T



T T T − (e13 − 2e11 )Y5 e13 − e11 −2(e4 − e2 )Y6 e4T − e2T − e11 Y5 e11

where T i0 = e1 W e1T + e1 Pi Bi e2T + e2 BiT Pi e1T + e1 Pi e17 + e17 Pi e1T   T +e1 Pi e18 + e18 Pi e1T + e2 ε2 β 2 I − (1 − μ)R7 e2T + e3 (R3 + R7 − R2 )e3T

+e4 (R5 −R3 )e4T −e5 R5 e5T +e6 (R4 −R2 )e6T +e7 (R6 − R4 )e7T − e8 R6 e8T −e9 Y1 e9T





T −(e1 −e3 )Y2 e1T −e3T −(d1 e1 −e9 )Z1 d1 e1T −e9T −(1 e1 −e12 )Z2 1 e1T −e12

T T T − ε2 e18 e18 − ε1 e17 e17 −(2 e1 − e13 )Z3 2 e1T − e13 W = ATi Pi + Pi Ai +

N 

πij pj + R1 + d12 Y1 + 12 Y3 + 22 Y5 + ε1 α 2 I

j =1

M = R2 + d12 Y2 + 12 Y4 + 22 Y6 +

d14 Z1 + 32 Z2 + 42 Z3 4

and other notations are the same as Theorem 3.1.

Stochastic Stability Analysis for Neutral Markovian Jump

297

3.2 Stochastic Stability for the Uncertain Neutral Markovian Jump Systems with Delays and Perturbations In this subsection, we consider the uncertain case which can be described by Eq. 1. Based on Theorem 3.1, we obtain the following theorem to guarantee the stochastic stability for the uncertain neutral Markovian jump systems with delays and perturbations. Theorem 3.2 For given scalars α, β, γ , τ , d1 , d2 , μ, and constant scalar dm satisfying d1 < dm < d2 , the neutral Markovian jump systems with delays and perturbations as described in Eq. 1 are stochastically stable if Ci  + γ < 1 and there exist symmetric positive matrices Pi > 0, (i ∈ S), Qj > 0, (j = 1, 2, · · · , 5), Rk > 0, (k = 1, 2, · · · , 7), Yl > 0, (l = 1, 2, · · · , 6), Zm > 0, (m = 1, 2, 3), and scalars ε1 , ε2 , ε3 , δn , (n = 1, 2, 3, 4) such that the following linear matrix inequalities hold,

  i1 + δ11 e1 Pi Hi HiT Pi e1T + δ1 εε T  T + δ11 e1 Pi Hi HiT M <0 (49) 1 T ∗ δ1 MHi Hi M − M

  i2 + δ12 e1 Pi Hi HiT Pi e1T + δ2 εε T  T + δ12 e1 Pi Hi HiT M <0 (50) 1 T ∗ δ2 MHi Hi M − M

  i3 + δ13 e1 Pi Hi HiT Pi e1T + δ3 εε T  T + δ13 e1 Pi Hi HiT M <0 (51) 1 T ∗ δ3 MHi Hi M − M

  i4 + δ14 e1 Pi Hi HiT Pi e1T + δ4 εε T  T + δ14 e1 Pi Hi HiT M <0 (52) 1 T ∗ δ4 MHi Hi M − M where ε T = EAi e1T +EBi e2T , and the remaining notations have been defined in Theorem 3.1. Proof On the basis of Theorem 3.1, we directly replace Ai , Bi with Ai + Ai (t), Bi + Bi (t) and obtain (53) i1 (t) +  T (t)M(t) < 0 i2 (t) +  T (t)M(t) < 0

(54)

i3 (t) +  T (t)M(t) < 0

(55)

i4 (t) +  T (t)M(t) < 0

(56)

where

  i1 (t) = i1 + e1 ATi (t)Pi + Pi Ai (t) e1T + e1 Pi Bi (t)e2T + e2 BiT (t)Pi e1T   i2 (t) = i2 + e1 ATi (t)Pi + Pi Ai (t) e1T + e1 Pi Bi (t)e2T + e2 BiT (t)Pi e1T   i3 (t) = i3 + e1 ATi (t)Pi + Pi Ai (t) e1T + e1 Pi Bi (t)e2T + e2 BiT (t)Pi e1T   i4 (t) = i4 + e1 ATi (t)Pi + Pi Ai (t) e1T + e1 Pi Bi (t)e2T + e2 BiT (t)Pi e1T

298

Xinghua Liu and Hongsheng Xi

Considering (53) and combining the uncertainties condition (5) by Lemma 2.3 we have

 

e1 Pi 0 i1  T M Hi Fi (t)ε T + M 0 M −M 

Pi e1T M <0 (57) + εFiT (t)HiT 0 0 Because of Eq. 6, by Lemma 2.2 from Eq. 57, we obtain  



T 

 1 e1 Pi 0 εε 0 i1  T M Pi e1T M Hi HiT + δ1 + <0 M 0 M −M 0 0 0 0 δ1

(58)

Obviously, Eq. 58 is equivalent to Eq. 49. Similarly, following the same procedure, from Eqs. 54, 55, and 56, we can get Eqs. 50, 51, and 52. Finally, following from the latter proof of Theorem 3.1, we know that the uncertain neutral Markovian jump systems with interval time-varying delays and nonlinear perturbations as described by Eq. 1 are stochastically stable if Eqs. 49, 50, 51, and 52 are satisfied. This completes the proof. Remark 3.3 It should be mentioned that Theorem 3.2 is an extension of Theorem 3.1 to uncertain neutral Markovian jump systems with interval time-varying delays and perturbations. It provides a stochastic delay-range-dependent stochastic stability criterion for Eq. 1 and it will be verified to be effective in Section 4. Similarly, in the uncertain case, the following corollaries are given, for unknown information on the delay derivative and reduced system with Ci = 0, γ = 0. Corollary 3.3 For given scalars α, β, γ , τ , d1 , d2 , and constant scalar dm satisfying d1 < dm < d2 , the neutral Markovian jump systems with delays and perturbations as described in Eq. 1 are stochastically stable if Ci  + γ < 1 and there exist symmetric positive matrices Pi > 0, (i ∈ S), Qj > 0, (j = 1, 2, · · · , 5), Rk > 0, (k = 1, 2, · · · , 6), Yl > 0, (l = 1, 2, · · · , 6), Zm > 0, (m = 1, 2, 3), and scalars ε1 , ε2 , ε3 , δn , (n = 1, 2, 3, 4) such that the following linear matrix inequalities hold,

   i1 + 1 e1 Pi Hi H T Pi eT + δ1 εε T  T + 1 e1 Pi Hi H T M  i i 1 δ1 δ1 <0 (59) 1 T ∗ δ1 MHi Hi M − M

   i2 + 1 e1 Pi Hi H T Pi eT + δ2 εε T  T + 1 e1 Pi Hi H T M  i i 1 δ2 δ2 <0 (60) 1 T ∗ δ2 MHi Hi M − M

   i3 + 1 e1 Pi Hi H T Pi eT + δ3 εε T  T + 1 e1 Pi Hi H T M  i i 1 δ3 δ3 <0 (61) 1 TM −M ∗ MH H i i δ3

   i4 + 1 e1 Pi Hi H T Pi eT + δ4 εε T  T + 1 e1 Pi Hi H T M  i i 1 δ4 δ4 <0 (62) 1 TM −M ∗ MH H i i δ4  i1 ,   i2 ,   i3 , and   i4 have been defined in Corollary 3.1, the remaining notations where  are the same as Theorem 3.2. Corollary 3.4 For given scalars α, β, γ , τ , d1 , d2 , μ, and constant scalar dm satisfying d1 < dm < d2 , the neutral Markovian jump systems with delays and perturbations as described in Eq. 1 are stochastically stable if there exist symmetric positive matrices Pi > 0, (i ∈ S),

Stochastic Stability Analysis for Neutral Markovian Jump

299

Rk > 0, (k = 1, 2, · · · , 7), Yl > 0, (l = 1, 2, · · · , 6), Zm > 0, (m = 1, 2, 3), and scalars ε1 , ε2 , δn , (n = 1, 2, 3, 4) such that the following linear matrix inequalities hold,

  i1 + δ11 e1 Pi Hi HiT Pi e1T + δ1 εε T  T + δ11 e1 Pi Hi HiT M <0 (63) 1 T ∗ δ1 MHi Hi M − M

  i2 + δ12 e1 Pi Hi HiT Pi e1T + δ2 εε T  T + δ12 e1 Pi Hi HiT M <0 (64) 1 T ∗ δ2 MHi Hi M − M

  i3 + δ13 e1 Pi Hi HiT Pi e1T + δ3 εε T  T + δ13 e1 Pi Hi HiT M <0 (65) 1 T ∗ δ3 MHi Hi M − M

  i4 + δ14 e1 Pi Hi HiT Pi e1T + δ4 εε T  T + δ14 e1 Pi Hi HiT M <0 (66) 1 T ∗ δ4 MHi Hi M − M where i1 , i2 , i3 , and i4 have been defined in Corollary 3.1, the remaining notations are the same as Theorem 3.2.

4 Numerical Examples In this section, numerical examples are given to show that the proposed theoretical results in this paper are less conservative than some previous ones. 4.1 Systems with No Perturbations In this subsection, we consider examples of the systems described by Eqs. 1, 7, and 44 with α = 0, β = 0, γ = 0. Example 4.1 To compare the stability result in this paper with those in [30, 33, 41], we consider the nominal system (44) with d1 = 0. For given (44) with the following parameters:   



−3.49 0.81 −2.49 0.29 −0.86 −1.29 A1 = , A2 = B1 = −0.65 −3.27 1.34 −0.02 −0.68 −2.07

 −2.83 0.50 , C1 = C2 = 0, Pij = [πij ]2×2 , i, j ∈ S = {1, 2} B2 = −0.84 −1.01 Given π22 = −0.8, different values of π11 and choose dm = d22 , by Corollary 3.2, the maximum d2 , which satisfies the LMIs in Eqs. 45, 46, 47, and 48, can be calculated by solving a quasi-convex optimization problem. Table 1 Maximum upper bound of d2 with μ = 0 and different parameter π11 Methods

π11 = −0.10

π11 = −0.50

π11 = −0.80

π11 = −1 0.4903

Cao et al. [41]

0.5012

0.4941

0.4915

Chen et al. [33]

0.5012

0.4941

0.4915

0.4903

Xu et al. [30]

0.6797

0.5794

0.5562

0.5465

Theorem 3.2

0.6874

0.5897

0.5649

0.5598

300

Xinghua Liu and Hongsheng Xi

Table 2 Maximum upper bound of d2 with μ = 1.5 and different parameter π11 π11 = −0.10

Methods

π11 = −0.50

π11 = −0.80

π11 = −1

Cao et al. [41]

–

–

–

–

Chen et al. [33]

–

–

–

–

Xu et al. [30]

0.3860

0.3656

0.3487

0.3378

Theorem 3.2

0.3977

0.3764

0.3516

0.3476

Table 1 gives the contrastive results, which show that the stochastic stability result in Theorem 3.1 is less conservative than those results. Table 2 shows that Theorem 3.1 in this paper can be applied to the time-varying delay case without the requirement on μ ¡ 1. Example 4.2 To compare the stability result in this paper with those in [10, 19, 21], we consider the nominal system (44) with the following parameters:

Ai =

 

−2 0 −1 0 , Bi = , i ∈ S = {1} 0 −0.9 −1 −1

For given d1 and dm , by Corollary 3.1, the maximum d2 , which satisfies the LMIs in Eqs. 40, 41, 42, and 43, can be calculated by solving a quasi-convex optimization problem. The results are shown in Table 3 . Table 3 gives the contrastive results and the last line shows the selected value of dm corresponding to different values of d1 in our paper and [19]. It also shows that the stability result in Corollary 3.1 is less conservative than those results. Example 4.3 As said in the literature [27], with the abrupt variation in its structures and parameters, we can present the PEEC model as a stochastic jump one. Consider the stochastic neutral partial element equivalent circuit(PEEC) model described by the following equation: ˙ − 0.3) = (Ai + A(t))x(t) + (Bi + B(t))x(t − d(t)) x(t) ˙ − Ci x(t

(67)

Table 3 Maximum upper bound of d2 with unknown μ and different parameter d1 Methods

d1

0

1

2

3

Shao [10]

d2

1.5296

1.8737

2.5049

3.2591

Park et al. [21]

d2

1.8680

2.0665

2.6181

3.3173

Xiao et al. [19]

d2

2.0656

2.1684

2.6461

3.3215

Corollary 3.1

d2

2.1347

2.2356

2.7148

3.3978

1.45

1.67

2.34

3.16

Parameter dm

Stochastic Stability Analysis for Neutral Markovian Jump

301

2.5

2

1.5

1

0.5

0

50

100

150

200

Fig. 1 The mode switching of rt

where i ∈ S = {1, 2} and the mode switching is governed by the rate matrix

 −1 1 , 2 −2

which is described by the following Fig.1.   



−5 0 −4 0 −1.6 0 , A2 = , B1 = A1 = 0 −6 0 −5 −1.8 −1.5   



−2 0 0.2 0 , C1 = 0.5I, C2 = 0.3I, H1 = , H2 = B2 = −0.9 −1.2 0.2 −0.3     EA1 = 0.2 0 , EA2 = 0 0.2 , EB1 = −0.3 0.3 , EB2 = 0.2 0.2 Given Fi (t) < 1 and the time-varying delay d(t) = 0.1(1 + sin2 t), from the graph of d(t) with t ∈ [0, 2π ] in the following Fig. 2, we easily obtain d1 = 0.1 and d2 = 0.2. In 0.2 0.19 0.18 0.17 0.16 0.15 0.14 0.13 0.12 0.11 0.1

0

1

2

Fig. 2 The graph of time-varying delay d(t)

3

4

5

6

7

302

Xinghua Liu and Hongsheng Xi

Table 4 Maximum upper bound of d2 with α = 0 and different parameter γ Methods

γ

0

0.1

0.2

0.3

Han et al. [25]

d2

0.9328

0.7402

0.5637

0.4042

Zhang et al. [32]

d2

0.9488

0.7695

0.6087

0.4667

Qiu et al. [7]

d2

0.9839

0.8024

0.6392

0.4941

Theorem 3.1

d2

1.0576

0.9458

0.8765

0.6732

Table 5 Maximum upper bound of d2 with α = 0.1 and different parameter γ Methods

γ

0

0.1

0.2

0.3

Han et al. [25]

d2

0.8148

0.6439

0.4864

0.3433

Zhang et al. [32]

d2

0.8408

0.6841

0.5420

0.4144

Qiu et al. [7]

d2

0.8752

0.7166

0.5727

0.4438

Theorem 3.1

d2

0.9136

0.8324

0.6835

0.5069

Table 6 Maximum upper bound of d2 with α = 0 and different parameter μ Methods

μ

0

0.5

1.2

Cao et al. [40]

d2

0.6811

0.5467

–

Han et al. [25]

d2

2.7424

1.1365

–

Zhang et al. [13]

d2

2.7420

1.1420

–

Zou et al. [43]

d2

2.7422

1.1424

–

Chen et al. [37]

d2

2.7423

1.1425

0.7355

Qiu et al. [7]

d2

2.7757

1.1849

0.9284

Theorem 3.2

d2

2.8034

1.2347

1.0875

Table 7 Maximum upper bound of d2 with α = 0.1 and different parameter μ Methods

μ

0

0.5

1.2

Cao et al. [40]

d2

0.6129

0.4950

–

Han et al. [25]

d2

1.8753

0.9952

–

Zhang et al. [13]

d2

1.8750

1.0090

–

Zou et al. [43]

d2

1.8753

1.0097

–

Chen et al. [37]

d2

1.8753

1.0097

0.7147

Qiu et al. [7]

d2

1.8959

1.0512

0.8865

Theorem 3.2

d2

2.4157

1.1044

1.0013

Stochastic Stability Analysis for Neutral Markovian Jump

303

˙ = 0.1 sin 2t and μ = 0.1. For simplicity, we choose dm = 0.15. Then, addition, we have d(t) the delay-range-dependent stochastic stability can be readily established by Theorem 3.2. 4.2 Systems with Delays and Perturbations Example 4.4 Consider the system (7) with the following parameters   



−1.2 0.1 −0.6 0.7 0.1 0 , Bi = , Ci = −0.1 −1 −1 −0.8 0 0.1 i ∈ S = {1}, β = 0.1, d1 = 0, μ = 0.5, τ = 1

Ai =

Given values of α, γ , and choose dm = d22 , by Theorem 3.1, the maximum d2 , which satisfies the LMIs in Eqs. 8, 9, 10, and 11, can be calculated by solving a quasi-convex optimization problem. This neutral system with perturbations was considered in references [7, 25, 32]. The results on the maximum upper bound of d2 are compared in Tables 6 and 7. Tables 4 and 5 illustrate the maximum d2 for different γ , α = 0, and α = 0.1, respectively. It can be seen that our proposed stability criterion gives less conservative result than those results. Example 4.5 Consider the system (44) with the following parameters

Ai =

 

−1.2 0.1 −0.6 0.7 , Bi = , i ∈ S = {1}, β = 0.1, d1 = 0 −0.1 −1 −1 −0.8

Given values of α, μ, and choose dm = d22 , by Corollary 3.2, the maximum d2 , which satisfies the LMIs in Eqs. 45, 46, 47, and 48, can be calculated by solving a quasi-convex optimization problem. The results on the maximum upper bound of d2 are compared in Tables 6 and 7. Tables 6 and 7 illustrate the maximum d2 for different μ, α = 0, and α = 0.1, respectively. It also can be seen that our proposed stability criterion gives less conservative result than those results in [7, 13, 25, 37, 40, 43].

5 Conclusions In this paper, the stochastic stability for neutral Markovian jumping systems with interval time-varying delays and nonlinear perturbations has been considered. To begin with the nominal systems, by dividing the time-varying delay interval into two subintervals, we construct a new Lyapunov-Krasovskii functional containing some triple-integral terms and obtain some less conservative results to guarantee the stochastic stability of the system. Meanwhile, we expand the results to the uncertain case. Numerical examples are given to demonstrate the effectiveness of our proposed criteria. Acknowledgments This work was supported in part by the National Key Scientific Research Project (61233003), the National Natural Science Foundation of China (60935001, 61174061, 61074033, and 60934006), the Doctoral Fund of Ministry of Education of China (20093402110019) and Anhui Provincial Natural Science Foundation (11040606M143), and the Fundamental Research Funds for the Central Universities and the Program for New Century Excellent Talents in University.

304

Xinghua Liu and Hongsheng Xi

References 1. Bellen A, Guglielmi N, Ruehli AE. Methods for linear systems of circuit delay differential equations of neutral type. IEEE Trans Circ Syst. 1999;46:212–215. 2. Chen B, Li H, Shi P, Lin C, Zhou Q. Delay-dependent stability analysis and controller synthesis for Markovian jump systems with state and input delays. Inf Sci. 2009;179:2851–2860. 3. Shen CC, Zhong SM. New delay-dependent robust stability criterion for uncertain neutral systems with time-varying delay and nonlinear uncertainties. Chaos, Solitons, Fractals. 2009;40:2277–2285. 4. Lien CH, Chen JD. Discrete-delay-independent, discrete-delay-dependent criteria for a class of neutral systems. J Dyn Syst Meas Control. 2003;125:33–41. 5. Sworder DD, Rogers RO. An LQG solution to a control problem with solar thermal receiver. IEEE Trans Autom Control. 1983;28:971–978. 6. Yue D, Han QL. A delay-dependent stability criterion of neutral systems and its application to a partial element equivalent circuit model, Proceedings of the 2004 American Control Conference. Piscataway: IEEE Press; 2004, pp. 5438–5442. 7. Qiu F, Cui BT, Ji Y. Vol. 11. Further results on robust stability of neutral system with mixed timevarying delays and nonlinear perturbations; 2010, pp. 895–906. 8. Ji HB. Algebra foundation of control theory, Hefei: University of Science and Technology of China Press; 2008. 9. Karimi HR, Zapateiro M, Luo N. Stability analysis and control synthesis of neutral systems with time-varying delays and nonlinear uncertainties. Chaos, Solitons, Fractals. 2009;42:595–603. 10. Shao H. New delay-dependent stability criteria for systems with interval delay. Automatica. 2009;45:744–749. 11. Gu K. An improved stability criterion for systems with distributed delays. Int J Robust Nonlinear Control. 2003;13:819–831. 12. Park JH. Novel robust stability criterion for a class of neutral systems with mixed delays and nonlinear perturbations. Appl Math Comput. 2005;161:413–421. 13. Zhang JH, Peng S, Qiu JQ. Robust stability criteria for uncertain neutral system with time delay and nonlinear uncertainties. Chaos, Solitons and Fractals. 2008;38(1):160–167. 14. Hale JK, Verduyn Lunel SM. Introduction to functional differential equation. New York: Springer; 1993. 15. Sun J, Liu GP, Chen J. Delay-dependent stability and stabilization of neutral time-delay systems. Int J Robust and Nonlinear Control. 2009;19:1364–1375. 16. Wu J, Chen T, Wang L. Delay-dependent robust stability and control for jump linear systems with delays. Syst Control Lett. 2006;55(11):939–949. 17. Xie L. Output feedback control of systems with parameter uncertainty. Int J Control. 1996;63:741–750. 18. Parlakci MNA. Delay-dependent robust stability criteria for uncertain neutral systems with mixed timevarying discrete and neutral delays. Asian J Control. 2008;9:411–421. 19. Xiao N, Jia YM. Delay distribution dependent stability criteria for interval time-varying delay systems. J Frankl Inst. 2012;349:3142–3158. 20. Kwon OM, Park JH. Delay-range-dependent stabilization of uncertain dynamic systems with interval time-varying delays. Appl Math Comput. 2009;208:58–68. 21. Park PG, Ko JW, Jeong C. Reciprocally convex approach to stability of systems with time-varying delays. Automatica. 2011;47:235–238. 22. Shi P, Xia Y, Liu G, Ress D. On designing of sliding-mode control for stochastic jump systems. IEEE Trans Autom Control. 2006;51(1):97–103. 23. Han QL. A descriptor system approach to robust stability of uncertain neutral systems with discrete and distributed delays, proceedings of american control conference, Denver, Jun. 2003, pp. 5098–5103. 24. Han QL. On stability of linear neutral systems with mixed time delays: a discretized lyapunov functional approach. Automatica. 2005;41:1209–1218. 25. Han QL, Yu L. Robust stability of linear neutral systems with nonlinear parameter perturbations. IEE Proceedings-Control Theory and Appl. 2004;151(5):539–546. 26. Brayton RK. Bifurcation of periodic solutions in a nonlinear difference-differential equation of neutral type. Q Appl Math. 1966;24:215–224. 27. He S, Liu F. Exponential stability for uncertain neutral systems with Markov jumps. J Control Theory Appl. 2009;7:35–40. 28. Niculescu SI. Delay effects on stability: a robust control approach. Berlin: Springer; 2001. 29. Xu SY, Lam J. A survey of linear matrix inequality techniques in stability analysis of delay systems. Int J Syst Sci. 2008;39:1095–1113.

Stochastic Stability Analysis for Neutral Markovian Jump

305

30. Xu SY, Lam J, Mao X. Delay-dependent H∞ control and filtering for uncertain Markovian jump systems with time-varying delays. IEEE Trans Circ Syst I Regular Papers. 2007;54:2070–2077. 31. Xu SY, Lam J, Zou Y. Further results on delay-dependent robust stability conditions of uncertain neutral systems. Int J Robust Nonlinear Control. 2005;15:233–246. 32. Zhang WA, Yu L. Delay-dependent robust stability of neutral systems with mixed delays and nonlinear perturbations. Acta Autom Sin. 2007;33(8):863–866. 33. Chen WH, Guan ZH, Lu X. Delay-dependent output feedback stabilisation of Markovian jump system with Time-delay. IEE Proceedings-Control Theory Appl. 2004;151:561–566. 34. Blair WP, Sworder DD. Feedback control of a class of linear discrete systems with jump parameters and quadratic cost criteria. Int J Control. 1975;21:833–841. 35. Li XG, Zhu XJ, Cela A, Reama A. Stability analysis of neutral systems with mixed delays. Automatica. 2008;44:2968–2972. 36. Mao X. Stability of stochastic differential equations with Markovian switching. Stoch Process Appl. 1999;79(1):45–67. 37. Chen Y, Xue AK, Lu RQ, Zhou SS. On robustly exponential stability of uncertain neutral systems with time-varying delays and nonlinear perturbations. Nonlinear Anal. 2008;68:2464–2470. 38. He Y, Wu M, She JH, Liu GP. Delay-dependent robust stability criteria for uncertain neutral systems with mixed delays. Syst Control Lett. 2004;51:57–65. 39. Kuang Y. Delay differential equations with applications in population dynamics. Boston: Academic Press; 1993. 40. Cao YY, Lam J. Computation of robust stability bounds for time-delay systems with nonlinear timevarying perturbations. Int J Syst Sci. 2000;31:359–365. 41. Cao YY, Yan W, Xue A. Improved delay-dependent stability conditons and h∞ control for jump timedelay systems, In Proceedings of 43rd IEEE Conference on Decision and Control, Atlantis, Bahamas, Dec.2004, pp. 4527–4532. 42. Zhang Y, He Y, Wu M, Zhang J. Stabilization for Markovian jump systems with partial information on transition probability based on free-connection weighting matrices. Automatica. 2011;47(1):79–84. 43. Zou Z, Wang Y. New stability criterion for a class of linear systems with time-varying delay and nonlinear perturbations. IEE Proceedings-Control Theory Appl. 2006;153(5):623–626.