Robust H∞ control for impulsive switched complex delayed networks

Mathematical and Computer Modelling 56 (2012) 257–267

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Robust H∞ control for impulsive switched complex delayed networks Shukai Li, Jianxiong Zhang ∗ , Wansheng Tang Institute of Systems Engineering, Tianjin University, 300072, China

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Article history: Received 6 July 2010 Received in revised form 15 October 2011 Accepted 27 December 2011 Keywords: Complex networks Impulsive switched systems H∞ control Time delays Linear matrix inequalities

abstract In this paper, the robust H∞ control problem of impulsive switched complex delayed dynamical networks with input constraints is investigated. By the multiple Lyapunov functions theory, sufficient condition is developed to ensure robustly global stability and a prescribed H∞ disturbance attenuation level for the resulting closed-loop networks in terms of linear matrix inequalities (LMIs). A convex optimization problem is formulated to determine the optimal state feedback H∞ controller which guarantees the optimal H∞ disturbance attenuation level. Numerical examples are given to illustrate the effectiveness of the proposed method. © 2011 Elsevier Ltd. All rights reserved.

1. Introduction Complex networks have received an increasing attention of researchers from different fields in recent years. Many largescale systems can be modeled by complex dynamical networks with the nodes representing individuals in the system and the edges representing the special connections among them. Among these are physical systems, communication networks, electricity distribution networks, genetic networks, food web, social networks, etc. [1–4]. In [5,6], the stability analysis and control of a class of complex dynamical networks are investigated. In particular, the synchronization behavior of complex networks has received much of the focus [7–11]. Time delays are ubiquitous in many dynamical systems and may modify drastically dynamic behavior of the system. In [12–14], the synchronization stability criteria for general complex networks with time varying delays are studied. Furthermore, the real-world complex networks often present some uncertainties. The adaptive control problems of uncertain complex dynamical networks have been investigated in [15,16]. Many practical systems in physics, biology, engineering, and economics usually exhibit impulsive dynamical behaviors due to abrupt changes at the switching points. We call these switched systems with impulse effect as impulsive switched systems [17,18]. In real-world complex networks, the nodes may be some switched impulsive systems, such as biological neural networks, optimal control models in economics, frequency-modulated signal processing systems, etc. Moreover, the connection topology of a network may change by jumps or switches due to the link failures or new creation in a network. Therefore, it is very interesting to study the complex delayed dynamical networks with switched behaviors on both its nodes and connection topology, and impulsive dynamical behaviors at the switching points. In [19,20], the robust stabilization of complex switched networks is investigated. The synchronization of complex dynamical networks with switching topology is studied in [21]. However, little literature can be found to deal with the impulsive switched complex delayed dynamical networks. In this paper, we present an uncertain impulsive switched complex dynamical network consisting of N nonidentical nodes with diffusive and delayed coupling, and consider the problem of the robust H∞ state feedback control for the complex dynamical network with input constraints. As the network states will change abruptly at the switching points, we construct multiple Lyapunov functions to analyze the stability property of the network. Based on multiple Lyapunov functions and robust H∞ control technique, sufficient condition is developed to ensure robustly global stability and a

∗

Corresponding author. Tel.: +86 02227404446; fax: +86 02227401810. E-mail addresses: [email protected], [email protected] (J. Zhang).

0895-7177/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2011.12.045

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S. Li et al. / Mathematical and Computer Modelling 56 (2012) 257–267

prescribed H∞ disturbance attenuation level for the resulting closed-loop network in terms of linear matrix inequalities (LMIs). Furthermore, the existence of the optimal state feedback H∞ controller can be solved by a convex optimization problem, which ensures an optimal H∞ disturbance attenuation level for the resulting closed-loop network with input constraints. Numerical examples show the effectiveness of the proposed method. The rest of this paper is organized as follows. In Section 2, an uncertain impulsive switched complex delayed dynamical network model and some preliminaries are presented. In Sections 3, the robust H∞ control for the impulsive switched complex delayed dynamical network is considered. In Section 4, numerical examples show the effectiveness of the proposed method. Conclusions are finally drawn in Section 5. 2. Problem formulation and preliminaries Consider an uncertain impulsive switched complex dynamical network consisting of N non-identical nodes with diffusive and delayed coupling, in which each node of the network is an m-dimensional dynamical system. The state equations of the whole network are described by

 N   σ   ˙ x ( t ) = ( A + ∆ A ) x ( t ) + f ( x ( t )) + cij k Γ xj (t − d) i σ i σ i i σ i i  k k k    j =1 +(Bσk i + ∆Bσk i )ui (t ) + Dσk i vi (t ), t ̸= tk ,   ∆xi (t ) = Eki xi (t ), t = tk ,     zi (t ) = Gσk i xi (t ), xi (t ) = φi (t ), t ∈ [−d, 0], i = 1, 2, . . . , N ,

(1)

where xi (t ) = (xi1 (t ), xi2 (t ), . . . , xim (t ))T ∈ Rm and ui (t ) ∈ Rq are the state and control of the i-th node, respectively, d is the coupling delay, φi (t ) is a continuously differential function, the switching signal is denoted by σk ∈ {1, 2, . . . , n} with constant n being the total number of the switching modes, tk is an impulsive switched time point and t0 < t1 < t2 < · · · < t∞ , ∆xi (tk ) = xi (tk+ ) − xi (tk− ), xi (tk− ) = xi (tk ) = limh→0+ xi (tk − h), xi (tk+ ) = limh→0+ xi (tk + h), Aσk i , Bσk i , Dσk i , Gσk i are constant matrices with appropriate dimensions, fσk i : Rm → Rm is a continuously differentiable vector-valued function satisfying that

  fσ i (x) − fσ i ( y) ≤ µσ i ∥x − y∥ , k k k where µσk i is a positive constant, C σ

σk

∀ x, y ∈ R m ,

(2)

σk

= [cij ]N ×N represents the outer switched coupling configuration of the complex σ

network, in which cij k is defined as follows: if there is a connection from the node j to the node i (i ̸= j), cij k > 0, otherwise σ

σ

σ

cij k = 0, and the entries of matrix C σk satisfy the diffusive condition cii k = − j=1,j̸=i cij k , Γ ∈ Rm×m is the inner coupling matrix in each node, vi (t ) ∈ Rp is the disturbance input which is square integrable on [0, ∞), i.e., v(·) ∈ L2 [0, ∞), zi (t ) ∈ Rs is the controlled output, ∆Aσk i , ∆Bσk i are unknown real norm-bounded matrix valued functions, representing time-varying parameter uncertainties of the form

N

[∆Aσk i , ∆Bσk i ] = Mσk i F (t )[Hσk i , Nσk i ]

(3)

where Mσk i , Hσk i , Nσk i are known real constant matrices, F (t ) is an unknown real time-varying matrix satisfying F (t )F (t ) ≤ I, where I is an identity matrix with appropriate dimensions. The switching law of network (1) is that, at the time point tk , the complex dynamical network switches to the σk subsystem from the σk−1 subsystem. Reformulate network (1) in virtue of the Kronecker product as T

 σk  x˙ (t ) = Aσk x(t ) + fσk (x(t )) + (C ⊗ Γ )x(t − d) + Bσk u(t ) + Dσk v(t ), ∆x(t ) = Ek x(t ), t = tk ,  z (t ) = Gσk x(t ), x(t ) = φ(t ), t ∈ [−d, 0],

t ̸= tk , (4)

where x(t ) = (xT1 (t ), xT2 (t ), . . . , xTN (t ))T , u(t ) = (uT1 (t ), uT2 (t ), . . . , uTN (t ))T , v(t ) = (v1T (t ), v2T (t ), . . . , vNT (t ))T , Aσk = diag{Aσk 1 + ∆Aσk 1 , Aσk 2 + ∆Aσk 2 , . . . , Aσk N + ∆Aσk N }, Bσk = diag{Bσk 1 + ∆Bσk 1 , Bσk 2 + ∆Bσk 2 , . . . , Bσk N + ∆Bσk N }, Ek = diag{Ek1 , Ek2 , . . . , EkN }, Gσk = diag{Gσk 1 , Gσk 2 , . . . , Gσk N }. For network (4), we consider a state feedback controller with the form u(t ) = diag{Lσk 1 , Lσk 2 , . . . , Lσk N }x(t ) = Lσk x(t )

(5)

for the σk subsystem, where Lσk is the control gain to be determined. In practice, all the physical control systems have to operate under constraints on the magnitude of the control input due to the physical limitations of actuators [22]. For this purpose, the control input u(t ) is assumed to satisfy the following constraints

− ul ≤ ul (t ) ≤ ul , where ul (t ) is the l-th element in the control input, ul is a known constant.

(6)

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259

Applying controller (5) with constraints (6) to network (4), one can obtain the following closed-loop impulsive switched network

 σk  x˙ (t ) = Aσk x(t ) + fσk (x(t )) + (C ⊗ Γ )x(t − d) + Bσk Lσk x(t ) + Dσk v(t ), ∆x(t ) = Ek x(t ), t = tk ,  z (t ) = Gσk x(t ), x(t ) = φ(t ), t ∈ [−d, 0].

t ̸= tk , (7)

The robust H∞ control problem of the uncertain impulsive switched complex network with coupling delay in this paper can be formulated as follows: for the closed-loop impulsive switched network (7), given a scalar γ > 0, obtain the state feedback controller (5) with input constraints (6) such that the following conditions are satisfied (1) the closed-loop impulsive switched network (7) with v(t ) = 0 is robustly globally stable; (2) under the zero initial condition, the controlled output z (t ) satisfies

∥z (t )∥2 ≤ γ ∥v(t )∥2 ,

(8)

for any nonzero v(·) ∈ L2 [0, ∞) and all the admissible uncertainties satisfying (3). In this case, the closed-loop impulsive switched network (7) is said to be globally stable with the disturbance attenuation γ . 3. Robust H∞ control design In this section, based on the multiple Lyapunov functions theory, we will study the robust H∞ control for the uncertain impulsive switched complex delayed dynamical network. To remove the dependence of the network performance on the initial state, similar to the approach adopted in [23], we consider that the initial condition is subjected to a bounded region D , which is defined by D = {x(t ) ∈ RmN | x(t ) = Uh(t ), h(t )T h(t ) ≤ 1, t ∈ [−d, 0]},

(9)

where U is a known matrix. The following theorem presents a sufficient condition for the robust H∞ control for network (7) with input constraints (6). Theorem 3.1. Suppose that there exist positive scalars εσk , ησk , positive definite matrices P σk , Q σk , Zσk , and any matrices Lσk with appropriate dimensions such that the following LMIs hold

Ωσk Q σ ((C σk )T ⊗ Γ T )  k  DTσk   µσk P σk   ΥσTk   G P



(C σk ⊗ Γ )Q σk −Q σk

σk σk

P σk



P σk−1

(I + Ek )P σk−1  T I U U



U P σ0 0

Zσk

Lσk

Lσk

P σk

T

Dσk

µσk P σk

Υσk

P σk GTσk

P σk

0

0 0

0 0

0 0

0 0

0 0

0

−γ 2 I

0 0

0 0

−εσk I

0

0

−ησk I

0 0

0 0

0 0

0 0

P σk−1 (I + Ek ) P σk

T



−I 0



    0   < 0, 0   0 

(10)

−Q σk

> 0,

(11)

UT 0  > 0, −1 d Q σ0

(12)

> 0,

(13)





(Zσk )ll ≤ u2l ,

(14) T

T

where Ωσk = Aσk P σk + P σk ATσk + Bσk Lσk + Lσk BTσk + εσk I + ησk Mσk MσTk , Υσk = P σk HσTk + Lσk NσTk , P σk = diag{P σk 1 , P σk 2 , . . . , P σk N }, Q σk = diag{Q σk 1 , Q σk 2 , . . . , Q σk N }, µσk = max{µσk i , i = 1, 2, . . . , N }, (Zσk )ll denotes the l-th diagonal element of the matrix Zσk . Then the closed-loop network (7) with input constraints (6) and initial condition (9) is robustly globally stable with −1

the disturbance attenuation γ . Moreover, the state feedback H∞ controller is given by u(t ) = Lσk P σk x(t ) for the σk subsystem.

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Proof. At first, based on the multiple Lyapunov functions theory, we show the stability result for the closed-loop network (7) with input constraints (6) and v(t ) = 0. For any t ∈ (tk , tk+1 ], construct a Lyapunov–Krasovskii function candidate for network (7) as V (t ) = xT (t )Pσk x(t ) +

t



xT (s)Qσk x(s)ds,

(15)

t −d

where Pσk = diag{Pσk 1 , Pσk 2 , . . . , Pσk N }, Qσk = diag{Qσk 1 , Qσk 2 , . . . , Qσk N }. Calculating the left and upper Dini’s derivative of V (t ) along the trajectory of network (7) with v(t ) = 0 yields D− V (t ) = 2xT (t )Pσk Aσk x(t ) + fσk (x(t )) + (C σk ⊗ Γ )x(t − d)



 + Bσk Lσk x(t ) + xT (t )Qσk x(t ) − xT (t − d)Qσk x(t − d).

(16)

By condition (2), together with the fact that AT B + BT A ≤ α AT A + α −1 BT B, ∀ α > 0, and matrices A and B with appropriate dimensions, one can get 2xT (t )Pσk fσk (x(t )) ≤ εσk xT (t )Pσk PσTk x(t ) + εσ−k1 µ2σk xT (t )x(t ),

(17)

2x (t )Pσk (∆Aσk + ∆Bσk Lσk )x(t ) ≤ ησk x (t )Pσk Mσk Mσk Pσk x(t ) + ησ−k1 xT (t )(Hσk + Nσk Lσk )T (Hσk + Nσk Lσk )x(t ).

(18)

T

T

T

T

Combining (16)–(18), one has D− V (t ) ≤

T 

Φσk ((C σk )T ⊗ Γ T )Pσk

x(t ) x(t − d)



Pσk (C σk ⊗ Γ ) −Qσk

x( t ) , x( t − d )





(19)

where Φσk = Pσk Aσk + ATσk Pσk + Pσk Bσk Lσk + LTσk BTσk PσTk + εσk Pσk PσTk + ησk Pσk Mσk MσTk PσTk + Qσk + εσ−k1 µ2σk I + ησ−k1 (Hσk + Nσk Lσk )T (Hσk + Nσk Lσk ). −1

−1

−1

−1

−1

In addition, let Pσk = P σk , Qσk = Q σk , Lσk = Lσk P σk . Pre and post-multiplying both sides of (10)–(11) by diag{P σk , Q σk , −1

−1

I , I , I , I , I } and diag{P σk−1 , P σk }, respectively, it is shown that inequalities (10)–(11) are equivalent to the following inequalities.

Ψσk ((C ) ⊗ Γ T )Pσk  DTσk Pσk   µσk I   ΞσTk  



σk T

Gσk I



Pσk−1 Pσk (I + Ek )

Pσk (C σk ⊗ Γ ) −Qσk 0 0 0 0 0

(I + Ek )T Pσk Pσk



Pσk Dσk 0 −γ 2 I 0 0 0 0

µσk I

Ξσk

0 0

0 0 0

−εσk I 0 0 0

−ησk I 0 0

GTσk 0 0 0 0 −I 0

I 0   0   0  < 0,  0   0 −1 −Qσk



> 0,

(20)

(21)

where Ψσk = Pσk Aσk + ATσk Pσk + Pσk Bσk Lσk + LTσk BTσk PσTk + εσk Pσk PσTk + ησk Pσk Mσk MσTk PσTk , Ξσk = HσTk + LTσk NσTk . By the Schur complement formula [24], it follows from condition (20) that

Φσk ((C ) ⊗ Γ T )Pσk



σk T

Pσk (C σk ⊗ Γ ) < 0, −Qσk



(22)

which shows from (19) that D− V (t ) < 0. On the other hand, for the impulsive switched time point tk , it has V (tk+ ) − V (tk− ) = xT (tk )[(I + Ek )T Pσk (I + Ek )]x(tk ) − xT (tk )Pσk−1 x(tk )

= xT (tk )[(I + Ek )T Pσk (I + Ek ) − Pσk−1 ]x(tk ).

(23)

It follows from the Schur complement formula that (21) is equivalent to

(I + Ek )T Pσk (I + Ek ) − Pσk−1 < 0, which, together with (23), imply that V (t ) is decreasing at the impulsive switched time, i.e., V (tk+ ) − V (tk− ) < 0.

(24)

S. Li et al. / Mathematical and Computer Modelling 56 (2012) 257–267

261

Additionally, it follows from (9) that xT (0)Pσ0 x(0) +



0

xT (s)Qσ0 x(s)ds ≤ λmax (U T Pσ0 U ) + dλmax (U T Qσ0 U ).

(25)

−d

By the Schur complement formula, (12) implies that

λmax (U T Pσ0 U ) + dλmax (U T Qσ0 U ) < 1.

(26)

0

Combining (25) and (26), one can obtain xT (0)Pσ0 x(0) + −d xT (s)Qσ0 x(s)ds < 1, i.e., V (0) < 1. Note that 1

−1

∥ul (t )∥2 = ∥Lσk l x(t )∥2 = ∥Lσk l Pσk 2 Pσ2k x(t )∥2 ≤ Lσk l Pσ−k1 LTσk l xT (t )Pσk x(t ) ≤ Lσk l Pσ−k1 LTσk l V (0) < Lσk l Pσ−k1 LTσk l , where Lσk l is the l-th row of the matrix Lσk , and LTσk is a column vector with mN dimensions. l It is shown that (13) and (14) are equivalent to Lσk Pσ−k1 LTσk < Zσk ,

(Zσk )ll ≤ u2l ,

(27)

which implies −ul ≤ ul (t ) ≤ ul , i.e., constraints (6) are satisfied. Therefore, the closed-loop network (7) with constraints (6) and v(t ) = 0 is robustly globally stable. Next, we shall show that the impulsive switched complex network (7) satisfies (8). By conditions (17)–(18), calculating the Dini’s derivative of V (t ) along the trajectory of the closed-loop network (7) yields D− V (t ) = 2xT (t )Pσk Aσk x(t ) + fσk (x(t )) + (C σk ⊗ Γ )x(t − d) + Bσk Lσk x(t ) + Dσk v(t )



+ xT (t )Qσk x(t ) − xT (t − d)Qσk x(t − d)  T  Φσk Pσk (C σk ⊗ Γ ) x( t ) σ T T k −Qσk ≤ x(t − d) ((C ) ⊗ Γ )Pσk v(t ) DTσk Pσk 0



 Pσk Dσk x(t ) 0  x(t − d) . v(t ) 0 

(28)

By the Schur complement, condition (20) implies

Φσk + GTσk Gσk ((C σk )T ⊗ Γ T )Pσ k



DTσk Pσk

Pσk (C σk ⊗ Γ ) −Qσk 0



Pσk Dσk 0  < 0. −γ 2 I

(29)

It follows from (28) and (29) that D− V (t ) + z (t )T z (t ) − γ 2 v(t )T v(t ) < 0,

t ∈ (tk , tk+1 ].

(30)

Then, under the zero initial condition, integrating both sides of (30) from 0 to +∞, one can obtain k  

+∞



(z (t )T z (t ) − γ 2 v(t )T v(t ))dt ≤ − lim

k→∞

0

 = lim

k→∞

j =0

tj+1

D− V (t )dt

tj

 k  V (0) + (V (tj+ ) − V (tj− )) − V (tk−+1 ) j =1

< lim (V (0) − V (tk−+1 )) k→∞

= V (0) − lim V (tk+1 ) k→∞

= 0.

(31)

Thus,

∥z (t )∥2 < γ ∥v(t )∥2 . The network (7) satisfies (8) for all nonzero v(·) ∈ L2 [0, ∞).



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In practice, we often hope that the system has a smaller H∞ disturbance attenuation γ . Then the optimal H∞ control of impulsive switched complex delayed networks with input constraints (6) is introduced as follows. Consider the closed-loop network (7) with input constraints (6). If the following optimization problem min

(εσk ,ησk ,P σk ,Q σk ,Zσk ,Lσk )

γ2

(32)

subject to (10)–(14), −1

exists a set of solutions (εσk , ησk , P σk , Q σk , Zσk , Lσk ), then u(t ) = Lσk P σk x(t ) is the optimal H∞ controller which ensures the optimal H∞ disturbance attenuation level for the resulting closed-loop network. Remark 3.1. Note that the above optimal H∞ control problem is a convex optimization problem. It can be solved by using the Matlab LMI toolbox, which implements an interior-point algorithm. If there exists a set of feasible solutions, then the optimal state feedback H∞ controller can be obtained. In addition, when ∆Aσk i , ∆Bσk i in network (1) are equal to 0, i.e., the uncertainties disappear, one can obtain a deterministic impulsive switched complex delayed dynamical network, which is described as

 x˙ (t ) = Aσk x(t ) + fσk (x(t )) + (C σk ⊗ Γ )x(t − d) + Bσk u(t ) + Dσk v(t ),   ∆x(t ) = Ek x(t ), t = tk , z (t ) = Gσk x(t ),  x(t ) = φ(t ), t ∈ [−d, 0].

t ̸= tk , (33)

The closed-loop system for the deterministic impulsive switched complex dynamical network with the state feedback controller (5) is obtained as

 x˙ (t ) = Aσk x(t ) + fσk (x(t )) + (C σk ⊗ Γ )x(t − d) + Bσk Lσk x(t ) + Dσk v(t ),   ∆x(t ) = Ek x(t ), t = tk , z (t ) = Gσk x(t ),   x(t ) = φ(t ), t ∈ [−d, 0].

t ̸= tk , (34)

By virtue of the proof technique in Theorem 3.1, the robust H∞ control for the deterministic impulsive switched complex dynamical network (33) with controller (5) and input constraints (6) is present as follows. Corollary 3.1. Suppose that there exist positive scalars εσk , positive definite matrices P σk , Q σk , Zσk , and any matrices Lσk with appropriate dimensions such that the following LMIs hold

Θσk Q σ ((C σk )T ⊗ Γ T )  k  DTσk   µσk P σk   G P



(C σk ⊗ Γ )Q σk −Q σk

Dσk

µσk P σk

P σk GTσk

P σk

0

0

−γ I

0 0

0 0

0 0

0 0 0

0 0 0

−εσk I

0 −I 0

−Q σk

σk σk

P σk



P σk−1

(I + Ek )P σk−1  T I U U



U P σ0 0

Zσk

Lσk

Lσk

P σk

T

P σk−1 (I + Ek ) P σk

T



> 0,

2

0 0



    < 0, 0   0 

(35)

(36)

UT 0  > 0, d−1 Q σ0

(37)

> 0,

(38)





(Zσk )ll ≤ u2l ,

(39) T

where Θσk = Aσk P σk + P σk ATσk + Bσk Lσk + Lσk BTσk + εσk I. Then the closed-loop network (34) with input constraints (6) and initial condition (9) is robustly globally stable with the disturbance attenuation γ . Moreover, the state feedback H∞ controller is given −1

by u(t ) = Lσk P σk x(t ) for the σk subsystem.

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263

Correspondingly, the optimal H∞ control of the deterministic impulsive switched complex delayed network with controller (5) and input constraints (6) is introduced as follows. If the following optimization problem min

(εσk ,P σk ,Q σk ,Zσk ,Lσk )

γ2

(40)

subject to (35)–(39), −1

exists a set of solutions (εσk , P σk , Q σk , Zσk , Lσk ), then u(t ) = Lσk P σk x(t ) is the optimal H∞ control which ensures the optimal H∞ disturbance attenuation level for the closed-loop network (34). Remark 3.2. It should be pointed that, for the proposed impulsive switched network, the switched behaviors on the nodes and connection topology are occurring simultaneously. In a similar way, the proposed methods can be extended to handle the case that the switched behaviors on the nodes and connection topology are not occurring simultaneously. 4. Numerical examples In this section, two numerical examples are given to illustrate the effectiveness of the proposed methods. Example 4.1. Consider an impulsive switched complex network (1) with 3 nodes. The total number of the switching modes n = 2, and σk ∈ {1, 2}. For σk = 1, each node in the network is a nonlinear system in [25], and for σk = 2, each node is a Lorenz system in [26]. The parameter matrices in the network are given as follows. 0 0 −0.5

 A1i =

 −10  28 A2i =  0

1 0 −0.5 10 −1 0





 ,   f1i (xi (t )) =  5 ln xi1 (t ) + x2i1 (t ) + 1 



0 0  , 8

−

0 0



0 1 , −0.5



0 −xi1 xi3 , xi1 xi2

 f2i (xi (t )) =

3



Eki = diag{0.2, 0.2, 0.2}, k = 1, 3, 5, . . . , Eki = diag{−0.5, −0.5, −0.5}, k = 2, 4, 6, . . . , B1i = B2i = diag{1, 1, 1}, D1i = D2i = diag{0.5, 0.5, 0.5}, G1i = G2i = diag{0.1, 0.1, 0.1}, i = 1, 2, 3. Assume the diffusive outer coupling matrices are



−2

1

1 1

C =

1 −3 2

1 2 , −3



 2

C =

−3 1 2

1 −2 1

2 1 , −3



and the inner coupling matrix is given by Γ = diag{1, 1, 1}, the coupling delay d = 0.1. The uncertainties of the complex network (1) are given as follows M1i = M2i = H1i = H2i = N1i = N2i = diag{0.1, 0.1, 0.1}, F (t ) = diag{sin(t ), sin(t ), sin(t )}. The control input u(t ) is subjected to the constraints −60 ≤ ul (t ) ≤ 60, and U = diag{1, 1, 1, 1, 1, 1, 1, 1, 1}. Suppose that the switching signal alternates as 1 → 2 → 1 → 2 → 1 → 2 → · · ·. We apply the optimal H∞ control method to the above system with controller (5). By solving the optimization problem (32), the optimal H∞ disturbance attenuation is obtained as γ = 0.11, and the corresponding controller gains can be obtained as

−14.7753 = −6.2202 0.1361  −21.2028 = −9.6405 0.1318  −12.3465 = −13.1072 0.0000 

L11

L13

L22

−6.1029 −13.648 −0.1374

0.2250 −0.2389 , −12.5648



−9.5420 −19.1747 −0.1496

0.2136 −0.2427 , −17.6577

−11.1881 −12.0175 0.0002

 −0.0008 0.0005 , −8.6495



−20.6437 L12 = −9.0157 0.1175  −13.5552 L21 = −14.5132 0.0000  −13.477 L23 = −14.5158 0.0001 

−8.8593 −18.8291 −0.1391 −12.3348 −13.4810 0.0001 −12.3225 −13.5295 0.0001

0.2118 −0.2469 , −16.3141



 −0.0004 0.0002 , −11.2330  0.0001 −0.0003 . −11.1708

Let initial conditions x(t ) = (0.4, −0.2, 0.4, −0.3, 0.3, −0.4, 0.3, 0.4, 0.5)T , t ∈ [−d, 0]. The simulation results of the state responses for the closed-loop system (7) with v(t ) = 0 are given in Fig. 1. Under the state feedback H∞ control, the impulsive switched complex network is robustly globally stable and the optimal disturbance attenuation γ is satisfied.

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Fig. 1. The state responses of the impulsive switched complex dynamical network (1) with 3 nodes under the H∞ controller (5).

Example 4.2. Consider an impulsive switched complex network (1) with 30 nodes. The total number of the switching modes and the switching law are the same as those in Example 4.1. For σk = 1, each node in the network is a chaotic system in [25], and for σk = 2, each node is a chaotic Hopfield neural network system in [27]. The parameter matrices in the network are given as follows 0 0 −0.5

 A1i =

1 0 −0.5

0 1 , −0.5



0 0   f1i =  5 − 5 exp(−x )  , i1



 A2i =



1 + exp(−xi1 )

 f2i =

−1 0 0

0 −1 0

0 0 , −1



2 tanh(xi1 ) + 1.2 tanh(xi2 ) − 7 tanh(xi3 ) 1.1 tanh(xi1 ) − 0.1 tanh(xi2 ) + 2.8 tanh(xi3 ) . 0.8 tanh(xi1 ) − 2 tanh(xi2 ) + 4 tanh(xi3 )



The outer coupling matrices C 1 and C 2 of the network are generated by the BA scale-free model [28], respectively. Fig. 2 shows the connection topology of C 1 and Fig. 3 shows the connection topology of C 2 . The inner coupling matrix and the uncertainties of the complex network (1) are the same to those in Example 4.1. The coupling delay d = 0.1. The control input u(t ) is subjected to the constraints −80 ≤ ul (t ) ≤ 80, and U = 0.2I. We apply the optimal H∞ control method to the above system with controller (5). By solving the optimization problem (32), the optimal H∞ disturbance attenuation is obtained as γ = 0.12, and the corresponding controller gains can be obtained. To keep the paper concise, the controller gains Lσk ∈ R90×90 , σk ∈ {1, 2} are not presented here for the dimensions are very large. Let initial conditions xi (t ) = (−0.2 + 0.004(3i − 2), −0.2 + 0.004(3i − 1), −0.2 + 0.004(3i))T , t ∈ [−d, 0], i = 1, 2, . . . , 30. The simulation results of the state responses for the closed-loop system (7) with v(t ) = 0 are given in Fig. 4. Under the state feedback H∞ control, the impulsive switched complex network is robustly globally stable and the optimal disturbance attenuation γ is satisfied. 5. Conclusion In this paper, the robust H∞ control problem for the impulsive switched uncertain complex dynamical networks with coupling delay is investigated. Sufficient conditions are developed to ensure robustly global stability and a prescribed H∞ disturbance attenuation level for the resulting closed-loop network in terms of LMIs. The existence of the optimal state feedback H∞ control with input constraints can be solved by a convex optimization problem. Numerical examples show the effectiveness of the proposed method.

S. Li et al. / Mathematical and Computer Modelling 56 (2012) 257–267

Fig. 2. The connection topology of C 1 .

Fig. 3. The connection topology of C 2 .

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Fig. 4. The state responses of the impulsive switched complex dynamical network (1) with 30 nodes under the H∞ controller (5).

Acknowledgments The authors thank the anonymous referees for their valuable comments and suggestions. This work was supported by the National Natural Science Foundation of China (No. 61004015), the Research Fund for the Doctoral Programme of Higher Education of China (No. 200900321200340), the Program for New Century Excellent Talents in Universities of China, and the Program for Changjiang Scholars and Innovative Research Team in University of China. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]

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