research on robust control and exponential stabilization for large scale impulsive hybrid network...

J SupercomputDOI 10.1007/s11227-013-1064-y

Research on robust control and exponentialstabilization for large scale impulsive hybrid networksystems with time-delay

Lanping Chen · Zhengzhi Han · Zhenghua Ma

© Springer Science+Business Media New York 2013

Abstract This paper investigates the stability of network control systems, regarded asa class of large-scale hybrid systems, for ubiquitous computing environment. The prob-lem of robust exponential stabilization for the hybrid systems is addressed, which arecomposed of impulsive subsystems with time-delay and parameter uncertainties. Usingthe Lyapunov–Krasovskii functional approach and linear matrix inequality method,an adaptive robust controller is designed to stabilize the uncertain continuous subsys-tems. Then the delay-dependent exponential stability conditions for the whole hybridsystem are derived by analyzing the stability of the subsystems. An example is givento show the effectiveness of the proposed design method.

Keywords Network · Ubiquitous computing · Time-delay · Adaptive control ·Impulsive · Robust · Exponential stability

1 Introduction

Ubiquitous computing is one of the most profound technologies of the twenty-firstcentury. Ubiquitous computing enhances computer use by making many comput-ers available throughout the network environment. With the use of computers, manypractical systems involve a mixture construction of continuous and discrete dynam-ics. These systems are usually called hybrid dynamic systems (HDSs) in which the

L. Chen (B) · Z. HanSchool of Electronic, Information and Electrical Engineering,Shanghai Jiaotong University, Shanghai 200240, Chinae-mail: [email protected]

L. Chen · Z. MaCollege of Information and Engineering Science, Changzhou University, Jiangsu 213164, China

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two kinds of dynamics coexist and interact. HDSs cover many useful and importantpractical systems such as networked control systems, robotics, automotive electronics,logic-based control systems, sampled-data control systems, etc. In recent years, thereis an increasing interest in studying the stability and stabilization of HDSs [1–4].

Digital communication networks can be prescribed as large-scale impulsive hybridsystems. In addition, many practical systems consist of a large number of intercon-nected subsystems that have complex structures such as power systems, urban trafficnetworks, etc. Robust control for such large-scale systems has been one of the mainconcerns of researchers in the past years, and many studies have been conducted withmany interesting results (see [5–7]). However, the systems investigated in the litera-ture do not contain the time delays. However, the time-delay phenomena often arisefrom many industrial, economic and social fields. It is commonly believed that thetime delay is one of the major sources for the instability of systems. So the problemof stability analysis of time delay systems has been extremely attractive and manyresearch studies have focused on this problem during the past two decades [8–12].

In this paper, we study a class of large-scale uncertain impulsive hybrid networksystems with time-delay. Compared with existing results in the literature, the noveltyof our results lies in the consideration of the time delay in the large-scale impulsivehybrid systems. In such systems, the uncertainties are usually defined as state andtime-delay state matrices that contain time-invariant polyopic uncertainties (normallyunbounded) and the nonlinear unknown perturbation. Based on Lyapunov–Krasovskiiapproach and linear matrix inequality,we first design a robust adaptive controller tostabilize the continuous subsystems. Then we propose the delay-dependent sufficientconditions which ensure exponential stability for the whole hybrid system.

The rest of the paper is organized as follows: Section 2 gives some mathematicalpreliminaries. Section 3 presents our main results of theorems on the exponentialstability for impulsive hybrid system with time delay. Finally, an example is given anddiscussed in Sect. 4.

2 Preliminaries

In this section, for easy presentation and to facilitate the discussion for the remainingpart of the paper, we first present some mathematical preliminaries.

Consider the following interconnected system that consists of N subsystems. Here,each subsystem is a uncertain nonlinear impulsive hybrid dynamical system withtime-delay:

⎧⎨

⎩

xi = Ai xi (t) + Di xi (t − h(t)) + Bi uic(t) + fi (X (t), X (t − h(t))), t �= tki = 1, 2, . . . , N

�x(t+k ) = dk x(t−k ), t = tk, k = 1, 2, . . . ,

(1)

where i ∈ {1, 2, . . . , N }, N is the total number of subsystems in the large-scale systemunder consideration. xi (t) ∈ Rn is state vector. uic(t) is the control input. The timedelay h(t) satisfies 0 ≤ h(t) ≤ h. where 0 < t1 < t2 < · · · < tk < · · · is a series of

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Research on robust control and exponential stabilization

time. The state x(tk) satisfies that x(t−k ) = limt→t−kx(t) and x(t+k ) = x(tk). dk is the

impulsive gain at t = tk .The following assumptions further characterize the class of systems that are under

consideration.

Assumption 1 Matrices Ai and Di are not precisely known, but belong to the convexhull of Ai j and Di j , respectively, that are given by

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

Ai ∈ Rn×n : Ai =M∑

j=1

ξ j Ai j

M∑

j=1

ξ j = 1, ξ j > 0

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

(2)

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

Di ∈ Rn×n : Di =M∑

j=1

ξ j Di j

M∑

j=1

ξ j = 1, ξ j > 0

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

(3)

Assumption 2 For i ∈ I (1, N ), the functions fi (x(t), x(t − h(t)) denote nonlinearperturbations with respect to the current state x(t) and delayed state x(t − h(t)).It is assumed that there exist unknown parameters ai , bi that satisfy the followingcondition.

‖ fi (X (t), X (t − h(t))‖ ≤ ai‖X (t)‖ + bi‖X (t − h(t))‖

≤N∑

i=1

ai‖xi (t)‖ +N∑

i=1

bi‖xi (t − h(t))‖,∀xi (t), xi (t − h(t)) (4)

To this end, we employ the adaptive parameters ai and bi to estimate the unknownparameters ai and bi , respectively. The estimation error of the adaptive parametersare denoted as ai = ai − ai and bi = bi − bi .

Definition 1 Any solution of a continuous system is said to be globally exponentiallystable if there exist some constants α > 0 and M > 1 such that for any initial datax(t0) = φ

‖x(t, t0, φ)‖ = M‖φ‖e−α(t−t0).

Definition 2 Let the positive-definite function V be C1 in x and t . If there is a constantr > 0 such that

V (tk+1) ≤ e−r V (tk), t = tk and V (t) ≤ −r V (t), t �= tk

Then we call V (t) a Lyapunov function with exponential decay for an impulsive hybridsystem as defined in (1).

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3 Main results

In the section, the stabilization criteria of the nonlinear impulsive hybrid system withuncertainties and time-delay effects are derived based on stability analysis of theuncertain continuous differential equations and discrete difference equations.

Lemma 1 Let matrices X and Y with appropriate dimensions, we can have

XTY + Y T X ≤ σ XT X + σ−1Y TY

for one positive scalar σ > 0.

Lemma 2 Using Newton–Leibniz formula

x(t − h(t)) = x(t) −t∫

t−h(t)

x(s)ds

Then the continuous i th subsystem can be transformed into

xi (t) =M∑

j=1

ξ j

⎧⎪⎨

⎪⎩(Ai j + Di j )xi (t)−Di j

⎡

⎢⎣Ai j

t∫

t−h(t)

xi (s)ds+Di j

t∫

t−h(t)

xi (s−h(s))ds

⎤

⎥⎦

+Di j

t∫

t−h(t)

Bi u(s)ds + Di j

t∫

t−h(t)

fi (X (s), X (s − h(s)))ds

⎫⎪⎬

⎪⎭+ Bi uic(t)

+ fi (X (t), X (t − h(t))) (5)

In what follows, for the sake of convenience, we we first consider the stability ofthe continuous subsystems of impulsive hybrid system (1).

Theorem 3.1 For the uncertain continuous subsystem of hybrid system (1), if thefollowing LMI hold:

(� �

�T L

)

≤ 0,

(M NN T R

)

≤ 0

where

� =N∑

i=1

[(Ai j + Di j )

T P + P(Ai j + Di j ) + 2(2Pi Di j Bi BTi DT

i j Pi

+eδi h Pi Di j Di j DTi j DT

i j P + eδi h Pi Di j Ai j ATi j DT

i j Pi + 2Pi Bi + 4Pi Bi BTi Pi )

+(2ai1 + 2bi21 + Qi + εPi )]

L = e−δi h(2bi I − Qi ), �= Pi Bi kdi, M =‖[e−δi h(ai +1)+k2i ]I‖, N =‖kikdi I‖

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R = ∥∥[e−δi h(bi + 1) + k2

di

]I∥∥, 1 = −eδi h

∥∥PT

i DTi j Di j Pi

∥∥ + ∥

∥PTi Pi

∥∥ + I

21 = −eδi h∥∥PT

i DTi j Di j Pi

∥∥ + eδi h

∥∥PT

i Pi∥∥

Pi and Qi are positive-definite matrices, I is identity matrix, ε > 0 is the exponentialdecay rate, δi is a positive constant. Then the adaptive tracking feedback controller

uic(t) = kixi (t) + kdixi (t − h(t)) +√

ε

4γ1ai +

√ε

4γ2bi

that renders the state trajectories of the subsystem (1) is robust exponential stable,where ki, kdi denote feedback gains of state and delay-state, respectively. ai , bi arethe adaptive parameters designed as the following tuned law with scalars γ1, γ2 > 0

˙ai = 2γ1( − eδi h

∥∥xT

i (t)PTi DT

i j Di j Pi xi (t)∥∥ + ∥

∥xTi (t)PT

i Pi xi (t)∥∥

+∥∥xT

i (t)xi (t)∥∥) + εai

˙bi = 2γ2( − eδi h

∥∥xT

i (t)PTi DT

i j Di j Pi xi (t)∥∥ + ∥

∥xTi (t)PT

i Pi xi (t)∥∥

+∥∥xT

i (t − h(t))xi (t − h(t))∥∥) + εbi

Proof Suppose that there exists a Lyapunov–Krasovskii function candidate as

Vi (t) = Vi1(t) + Vi2(t) + Wi

= xTi (t)Pi xi (x) +

T∫

t−h(t)

eδi (s−t)xTi (s)Qi xi (s)ds + 1

2γ1ai

2 + 1

2γ2bi

2(6)

where Pi and Qi are positive matrices, δi , γi1 and γi2 are positive scalars.Then calculating the time derivatives of Vi1(t) along the trajectory of continuous

subsystem (1), and using Lemma 2, we can get

Vi1(t) =M∑

j=1

ξ j xTi (t)

[(Ai j + Di j )

T Pi + Pi (Ai j + Di j )]xi (t)

−M∑

j=1

ξ j

t∫

t−h(t)

2Pi Di j Ai j xi (s)xi (t)ds

−M∑

j=1

ξ j

t∫

t−h(t)

2Pi Di j Di j xi (s − h(s))xi (t)ds

−M∑

j=1

ξ j

t∫

t−h(t)

2Pi Di j Bi ui (s)xi (t)ds

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−M∑

j=1

ξ j

t∫

t−h(t)

2Pi Di j fi(X (s), X (s − h(s))

)xi (t)ds

+ 2Pi Bi uic(t)xi (t) + 2Pi fi(X (t), X (t − h(t))

)xi (t) (7)

Based on Lemma 1, the above formula can be equivalently written as

Vi1(t) ≤M∑

j=1

ξ j xTi (t)

[(Ai j +Di j )

T Pi +Pi (Ai j +Di j )]xi (t) − 2

M∑

j=1

ξ j

t∫

t−h(t)

xTi (s)AT

i j

× DTi j Pi xi (t)ds − 2

M∑

j=1

ξ j

t∫

t−h(t)

xTi (s − h(s))DT

i j Di j Pi x(t)ds − 2M∑

j=1

ξ j

×t∫

t−h(t)

uTi (s)BT

i DTi j Pi xi (t)ds+2

M∑

j=1

ξ j

⎡

⎢⎣

t∫

t−h(t)

(N∑

i=1

ai‖xi (s)‖)∥∥DT

i j Pi xi (t)∥∥ds

+t∫

t−h(t)

(N∑

i=1

bi‖xi (s − h(s))‖)∥∥DT

i j Pi xi (t)∥∥ds

⎤

⎥⎦ + 2uT

i (t)BTi Pi xi (t)

+ 2

[(N∑

i=1

ai‖xi (t)‖)∥∥Pi xi (t)

∥∥ +

(N∑

i=1

bi∥∥xi (t − h(t))

∥∥

)∥∥Pi xi (t)

∥∥

]

(8)

Then we construct the adaptive controller as mentioned in Theorem 1. Here, wecan get:

−2M∑

j=1

ξ j

t∫

t−h(t)

uTi (s)BT

i DTi j Pi xi (t)ds

= −2M∑

j=1

ξ j

⎡

⎢⎣

t∫

t−h(t)

[kix

Ti (s)+kdix

Ti (s−h(s))

]ds+

√ε

4γ1ai +

√ε

4γ2bi

⎤

⎥⎦ BT

i DTi j Pi xi (t)ds

≤ 2M∑

j=1

ξ j

⎡

⎢⎣

t∫

t−h(t)

∥∥kix

Ti (s)+kdix

Ti (s−h(s))

∥∥2ds+4

∥∥BT

i DTi j Pi xi (t)

∥∥2+ ε

8γ1a2

i + ε

8γ2b2

i

⎤

⎥⎦

(9)

Substitute formula (9) into (8), there is

123


Vi1(t) ≤M∑

j=1

ξ j xTi (t)

[

(Ai j + Di j )T Pi + Pi (Ai j + Di j )

]

xi (t) + 2M∑

j=1

ξ j

∏

i

+2M∑

j=1

ξ j

∏

i i

+2M∑

j=1

ξ j

t∫

t−h(t)

∥∥kix

Ti (s) + kdix

Ti (s − h(s))

∥∥2ds + 2

M∑

j=1

ξ j

(ε

4γ1a2

i + ε

4γ2b2

i

)

+2M∑

j=1

ξ j

⎡

⎢⎣eδi h(ai + bi )xT

i (t)Pi DTi j Di j Pi xi (t) +

t∫

t−h(t)

N∑

i=1

e−δi hai∥∥xT

i (s)xi (s)∥∥ds

+t∫

t−h(t)

N∑

i=1

e−δi hbi∥∥xT

i (s − h(s))xi (s − h(s))∥∥ds

⎤

⎥⎦ + 2xT

i (t)Pi Bi kdixi (t − h(t))

+ 2

[

(ai + bi eδi h)xT

i (t)PTi Pi xi (t) +

N∑

i=1

ai∥∥xT

i (t)xi (t)∥∥

+N∑

i=1

bi e−δi h

∥∥xT

i (t − h(t))xi (t − h(t))∥∥

]

(10)

where∏

i

= eδi h xTi (t)Pi Di j Ai j AT

i j DTi j Pi xi (t) + eδi h xT

i (t)Pi Di j Di j DTi j DT

i j Pxi (t)

+4xTi (t)Pi Di j Bi BT

i DTi j Pi xi (t)+2xT

i (t)ki BTi Pi xi (t)+4xT

i (t)Pi Bi BTi Pi xi (t)

∏

i i

= e−δi h

t∫

t−h(t)

xTi (s)xi (s)ds + e−δi h

T∫

t−h(t)

xTi (s − h(s))xi (s − h(s))

Similarly, the derivative of Vi2(t) and Wi are given as

Vi2(t) = −δi Vi2+eδi (t−t)xTi (t)Qi xi (t)−e−δi h(t)xT

i (t−h(t))Qi xi (t−h(t)) (11)

Wi = − 1

γ1(ai − ai ) ˙ai − 1

γ2(bi − bi )

˙bi (12)

Considering appropriate estimations of the parameter, and using the inequality−ε(a − a)a ≤ ε

2 (a − a)2 − ε2 (a)2, clearly, it is easy to verify that the following

equality holds

− 1

γ1(ai − ai ) ˙ai − 1

γ2(bi − bi )

˙bi

≤ 2xTi (t)(ai1 − ai1 + bi21 − bi21)xi (t) − 2bi xT

i (t − h(t))22xi (t − h(t))

+2bi xTi (t −h(t))22xi (t −h(t))+ ε

2γ1(ai − ai )

2− ε

2γ1a2

i + ε

2γ2(bi − bi )

2− ε

2γ2b2

i

(13)

where 22 = e−δi h .

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Putting together the derivatives of Vi1, Vi2 and Wi , we can get Vi (t). For thecomplete system (1), we choose the Lyapunov function as the following form

V =N∑

i=1

Vi (14)

In order to ensure the exponential stability of the continuous subsystems of the systemdefined in (1), we need to have V (t) ≤ −εV (t) should be hold obviously. Takingζ1(t) and ζ2(t) as the following form ζT

1 (t) = [xT

i (t), xTi (t −h(t))

], ζT

2 (t) = [xT

i (s),xT

i (s − h(s))], then we get

V ≤ −N∑

i=1

(δi + ε)Vi2 +N∑

i=1

ζT1 (t)

(cc� �

�T L

)

ζ1(t)

+2N∑

i=1

e−δi h

t∫

t−h(t)

ζT2 (t)

(M NN T R

)

ζ2(t)ds + ε (15)

where � = ∑Mj=1 ξ j� and ε = ∑N

i=1

(εγ1

ai2 + ε

γ2bi

2). It is clear that

(� �

�T L

)≤ 0

implies that(

� �

�T L

)≤ 0 holds. Furthermore, applying the Schur complement,

the hypotheses LMI is satisfied which can be directly derived from inequality (15).Since ε can be rendered arbitrarily small by choosing an appropriate small value forthe parameter � = max{ ε

γ1, ε

γ2}, the converging region can be rendered arbitrarily

small. Thus, if this condition hold, then V ≤ 0 implies that the solution x(t) ofuncertain continuous subsystem (1) with time-delay is robust exponential stable, suchthat V (t) ≤ V (t0)e−ε(t−t0). Thus, the desired result is proved. �

Next, we study the impulsive discrete subsystem stability of hybrid system, thenanalyze the synthetical stability for the whole hybrid system.

Theorem 3.2 If there exists a constant ε such that the following inequality

max0≤k≤∞

{ln(1 + dk)

2

tk+1 − tk

}

≤ ε (16)

holds, then the uncertain impulsive hybrid system (1) is global exponential stable at

origin and its convergence rate is −(ε − λ), where λ = max 0≤k≤∞{ ln(1+dk)

2

tk+1−tk

}.

Proof Considering discrete subsystem of hybrid system (1), from the above Lyapunovfunction V (t), we have:

V (tk) =N∑

i

⎡

⎢⎣xT

i (tk)Pi xi (tk) +tk∫

tk−h(t)

exp[δi (s − t)]xTi (s)Qi xi (s)ds

⎤

⎥⎦ (17)

123


Let βk = max{1, (1 + dk)2}, then the following inequality holds

V (tk) =N∑

i

⎡

⎢⎢⎣(1 + dik)

2xTi (t−k )Pxi (t

−k ) +

t−k∫

t−k −h(t)

exp[δi (s − t)]xTi (s)Qi xi (s)ds

⎤

⎥⎥⎦

≤ βk V (t−k ) (18)

Combining the result of theorem 1 with inequality (18), for any t ∈ (tk, tk+1], we have

V (t) ≤ V (tk) exp [−ε(t − tk)]

≤ βk V (t−k ) exp [−ε(t − tk)] ≤ βkβk−1V (t−k−1) exp[−ε(t − tk−1)

](19)

By induction, and from the condition (16), we have

V (t) ≤ β1β2 . . . βk exp

⎡

⎣−ε

k∑

j=0

(t j+1 − t j )

⎤

⎦ V (t0)

≤ V (t0) exp

[

lnk∑

i=1

βi − ε(t − t0)

]

≤ V (t0) exp [λ(tk − t0) − ε(t − t0)]

≤ V (t0) exp [−(ε − λ)(t − t0)] (20)

It is easy to see that V (t) → 0 as t → ∞, which implies that the solution of HDSs(1) is globally exponential stable. The proof of the theorem is completed. �Remark 3.1 In the design procedure, we consider uncertainties, state matrices A andB as the convex combination, not the usual bounded function which has been widelystudied such as literature [13–15].

If there is no perturbation, that is, f (x(t), t) = 0, the stability problem of system(1) is reduced to analyze the stability of impulsive system. This problem has beenwidely studied in recent literature (see e.g., [16,17]).

Remark 3.2 Theorem 3.1 enables to prove that the continuous subsystem of impulsivehybrid system (1), under the conditions, is exponential stable. Theorem 3.2 verify toensure the whole impulsive hybrid system (1) exponential stable, when the discreteimpulsive subsystem does not have to be exponential stable, if only the ratio of theimpulsive gains dk and the impulsive interval (tk − tk−1) satisfy the condition (16).

Remark 3.3 We can construct a adaptive controller for hybrid impulsive system (1)such as u = u1 + u2. According to Theorem 3.1, design of adaptive controller ucof continuous subsystem is easy to obtain. u2 are defined as impulsive signal. Andthe two controllers should satisfy: u1 = uc�(k), u2 = dk x(tk)δ(k), where δ(k) is theDirac impulse, �(k) = 1 as tk−1 < t < tk , and otherwise �(k) = 0.

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4 Examples

In this section, we give two examples to illustrate the results of previous sections.

Example 4.1 Consider the following uncertain impulsive time-delay system:

{xi = Ai xi (t) + Di xi (t − h(t)) + Bi uic(t) + fi (xi (t), xi (t − h(t))), t �= tk�x(t+k ) = dk x(t−k ), t = tk, k = 1, 2, . . . ,

(21)

where the continuous subsystems of the system (23) is equivalent to the followingsystem:

xi = (ξ j Ai1 + (1 − ξ j )Ai2

)xi (t) + (

ξ j Di1 + (1 − ξ j )Di2)xi (t − h(t))

+(ξ j Bi1 + (1 − ξ j )Bi2

)uic + fi (xi (t), xi (t − h(t))), i = 1, 2

where the state vector x1(t) = [x11(t) x12(t)]T, x2(t) = [x21(t) x22(t)]T, state-delayvector x1(t − h(t)) = [x11(t − h(t)) x12(t − h(t))]T, x2(t − h(t)) = [x21(t − h(t))x22(t − h(t))]T.

The parameters of the system are specified as follows:

A11 =(−1.5 0.1

−0.1 −1

)

, A12 =(−1.2 0.1

−0.1 −0.6

)

,

A21 =(

2 00 −0.9

)

, A22 =(−1 0

0 0.4

)

,

D11 =(−0.6 0.5

−1 −0.8

)

, D12 =(−0.5 0.4

−0.9 −0.7

)

,

D21 =(−1 0

−1 −1

)

, D22 =(

1 01 1.5

)

,

B11 =(

01

)

, B12 =(

10

)

, B21 =(

10

)

, B22 =(

01

)

Choose ξ j = 0.5,�tk chosen as 2, dk = 0.2, δ1 = δ2 = 0.5, ε = 0.2. Assume thatthe nonlinear perturbations f (x(t), x(t − h(t)) satisfy (4), choose γ1 = γ2 = 2.

Furthermore, we also obtain the following adaptive time-variant controller

uic = kixi (t) + kdixi (t − h) +√

ε

4γ1ai +

√ε

4γ2bi

by taking ki = 2.8, kdi = 2.3, and the adaptive law ai , bi satisfy the condition ofTheorem 3.1.

Figures 1 and 2 show the state trajectories of the impulsive delayed system with aconstant interval time �tk = 2. The results also show that a longer time delay willresult in a slower convergence rate.

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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

time t

x1(t

)

x11x12x11’x12’

Fig. 1 State trajectories of x11, x12, the solid and dashed-dotted curves are for cases with h = 0.2 and0.7, respectively

0 1 2 3 4 5 6 7 8−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

time t

x2(t

)

x21x22x21’x22’


Example 4.2 Consider the following uncertain linear impulsive system with time-delay:

{xi = Ai xi (t) + Di xi (t − h(t)) + Bi uic(t), t �= tk�x(t+k ) = dk x(t−k ), t = tk, k = 1, 2, . . . ,

(22)

where Ai = −1.2I , Bi = −0.5I , where I is 4 × 4 identity matrix.

Other parameters chosen are similar as Example 4.1. According to Theorem 3.1,taking ki = 3.5, kdi = 2.8, the state trajectories of the impulsive delayed system with

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L. Chen et al.

1 1.5 2 2.5 3 3.5 4 4.5 5−1

−0.5

0

0.5

1

time t

x1(t

)

x11x12x11x12


0 1 2 3 4 5−1

−0.5

0

0.5

1

time t

x2(t

)

x21x22x21x22


a constant interval time �tk = 2 are shown in Figs. 3 and 4. The results show that thelinear hybrid system with time delay is exponentially stable.

Remark 4.1 For any hybrid network systems with impulses as system (1), it is expo-nential asymptotically stable if and only if the system satisfying with Theorem 3.2.Hence, for nonlinear hybrid system Example 4.1 shows that the system is exponentialstability and has robust exponential stability property with respect to a certain extentof nonlinear disturbances. On the other hand, in Example 4.2, adaptive controller andimpulses help the whole system to achieve the exponential stability property.

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5 Conclusion

In this paper we have considered the exponential stability problem of a class of uncer-tain large-scale impulsive hybrid network systems with time-delay. First, by construct-ing appropriate Lyapunov–Krasovskii functions and designing adaptive controller asshown in Theorem 3.1, we prove robust exponential stability of continuous subsystemof (1), where the uncertainties include parameter uncertainty and unknown perturba-tion. Next, we show the sufficient conditions of the exponential stability for impulsivehybrid systems. Finally, to illustrate the effectiveness of the proposed design method,two numerical examples are presented using the results obtained in the paper.

Acknowledgments The authors are grateful for the support of the National Natural Science Foundationof China (Grant No. 61074003).

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