Nonlinear Dyn DOI 10.1007/s11071-014-1586-8
ORIGINAL PAPER
Robust synchronization of uncertain chaotic neural networks with randomly occurring uncertainties and non-fragile output coupling delayed feedback controllers V. Vembarasan · P. Balasubramaniam · Chee Seng Chan
Received: 4 June 2013 / Accepted: 6 July 2014 © Springer Science+Business Media Dordrecht 2014
Abstract This paper deals with the synchronization control problem for the uncertain chaotic neural networks with randomly occurring uncertainties and randomly occurring control gain fluctuations. By introducing an improved Lyapunov–Krasovskii functional and employing reciprocally convex approach, a delaydependent non-fragile output feedback controller is designed to achieve synchronization with the help of a drive–response system and the linear matrix inequality approach. Finally, numerical results and its simulations are given to show the effectiveness of the derived results. Keywords Chaos synchronization · Linear matrix inequalities · Lyapunov–Krasovskii functional · Neural networks · Non-fragile controllers · Randomly occurring uncertainties
V. Vembarasan · P. Balasubramaniam (B) Department of Mathematics, Gandhigram Rural Institute - Deemed University, Gandhigram 624 302, Tamilnadu, India e-mail:
[email protected] V. Vembarasan · C. S. Chan Faculty of Computer Science and Information Technology, Centre of Image and Signal Processing, University of Malaya, Kuala Lumpur 50603, Malaysia e-mail:
[email protected] C. S. Chan e-mail:
[email protected]
1 Introduction Since Pecora and Carroll [1] proposed the drive– response concept to achieve the synchronization of chaotic systems, researchers have presented a variety of synchronization schemes of such systems and developed many potential applications in physical, chemical, ecological systems [2], engineering applications such as secure communication [3], image encryption [4], image segmentation [5] and two-dimensional motion control [6]. The phenomenon of synchronization of two chaotic systems is fundamental in science and has enormous applications in technology. There are many control techniques for synchronizing chaotic systems, such as linear error feedback control [7,8], impulsive control [9–11], back stepping control [12,13] and sliding mode control [14,15]. Meanwhile, chaotic neural networks (CNNs) as special complex networks can exhibit some complicated dynamics and even chaotic behaviors [16]. A typical characteristic of chaotic systems is their sensitive dependence on initial conditions. It means that it is generally difficult to achieve synchronization between chaotic systems. In a practical CNNs, time delays occur inevitably due to the finite speed of signal transmission, which may degrade the synchronization performance of the network [17–19]. Therefore, the study of synchronization in CNNs with time delays has become an active area of research in the past decade. A lot of results have been provided on time-delayed CNNs that have been widely used in different areas, such as cryptography, secure communication, geography, associate
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memory, and combinatorial optimization. Thus, much attention has been drawn to the study of CNNs with time delay (for example, see [20–23] and references therein). On the other hand, parameter uncertainties are unavoidable due to the modeling inaccuracies, variations of the operating point, aging of the devices, etc. Therefore, the issue of robustness analysis has been taken into account in all sorts of systems by many researchers [24,25]. Ma et al. [26] have proposed a new type of uncertainties named as randomly occurring uncertainties (ROUs) due to the fact that the uncertainties may be subject to random changes in environment circumstances, for instance, repairs of components and sudden environmental disturbances and thus uncertainties may occur in a probabilistic way with certain types and intensity. Hu et al. [27] studied robust sliding mode control for discrete stochastic systems with mixed time delays, randomly occurring uncertainties and randomly occurring nonlinearities. In [28], authors investigated the robust synchronization problem for uncertain nonlinear chaotic systems with ROUs that adjust certain Bernoulli-distributed white noise sequences. However, there are only a very few works for synchronization of chaotic systems with ROUs [29–32]. In many applications, the interesting problem is to design a memoryless state-feedback controller u(t) = K 1 e(t), see Refs. [7,8]. Involvement of time delay in the feedback loop eliminates the need for explicitly determining any information about the underlying dynamics other than the period of the desired orbit. But in practice, time delays always influence the dynamic properties of delayed CNNs, which may cause periodic oscillations, bifurcation and chaotic attractors and so on. Thus if the information on the size of the time-varying delay τ (t) is available, a delayed feedback controller of the form: u(t) = K 1 e(t) + K 2 e(t − τ (t)) has been considered (for details, see [33,34]). The more general form of a delayed feedt back controller is u(t) = K 1 e(t) + K 2 t−τ (t) e(s)ds, see [35,36]. But in many real networks, only output signals can be measured (see [37,38]), and it takes the form: u(t) = K 1 ( f (y(t)) − f (x(t))) + K 2 ( f (y(t − τ (t))) − f (x(t − τ (t)))). Further, the distributed delay in the delayed output feedback controller of the form u(t) = K 1 ( f (y(t)) − f (x(t))) + K 2 ( f (y(t − τ (t))) − t f (x(t −τ (t))))+ K 3 t−ρ(t) ( f (y(s))− f (x(s)))ds has been considered in [39]. Meanwhile, it is noted that in practical the designed controller should be able to
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tolerate some uncertainty in its coefficients due to the fact that the uncertainty is not avoided by many reasons, such as finite word length in digital systems, the imprecision inherent in analog systems and the need for additional tuning of parameters in the final controller implementation [40,41]. The problem of nonfragile synchronization control for complex networks has been discussed in [42], with time-varying coupling delay and missing data. Further an exponential synchronization condition has been obtained for ensuring the exponential mean square stability of the error system. Tang et. al [43] have studied the distributed synchronization and pinning distributed synchronization of stochastic coupled neural networks via randomly occurring control. Very recently, Fang et al. [44] studied a non-fragile procedure for the problem of synchronization of CNNs with ROUs in controllers, which obeys certain Bernoulli-distributed white noise sequences. In all the above-mentioned works [42,44], the non-fragile controller of the form u(t) = (K 1 + ΔK 1 (t))e(t) has been considered. Motivated by the ROUs [27], and delayed output feedback controller designs [37,38], in this paper a novel non-fragile controller is designed in the response system of the form, u(t) = (K 1 +α(t)ΔK 1 (t))( f (y(t))− f (x(t)))+(K 2 + α(t)ΔK 2 (t))( f (y(t − τ (t))) − f (x(t − τ (t)))), which is not considered in any of the previous literature. Inspired by the above works, in this paper, we derive the criteria to synchronize two identical CNNs with ROUs in system parameters and in novel output feedback non-fragile controller designs using the Lyapunov stability theory. Moreover, the control gain matrices of the non-fragile controllers can be determined based on linear matrix inequality (LMI) which can be easily solved by any LMI solvers [45]. The main contribution of this paper is that the proposed results guarantee the mean square asymptotic stability of error system with respect to the newly designed delayed output feedback non-fragile controller in the response system with ROUs and in the system parameters. Finally, numerical simulations are illustrated to show the effectiveness and design procedure of the derived results. The rest of the paper is organized as follows. In Sect. 2, the problem formulation of CNNs with ROUs and mixed time-varying delays, and some preliminaries are given. The main results and the design methods of the CNNs with ROUs in system parameters and in nonfragile controllers are provided in Sect. 3. In Sect. 4, two examples are demonstrated to show the effective-
Synchronization of chaotic neural networks with ROUs
ness of the derived results. The conclusion is given in Sect. 5. Notations Throughout this paper, Rn and Rm×n denote the n-dimensional Euclidean space and the set of all m × n real matrices, respectively. I denotes the identity matrix. 0m×n denotes an m × n zero matrix. The superscript T denotes the transposition and the notation X ≥ Y (similarly, X > Y ), where X and Y are symmetric matrices, means that X −Y is positive semidefinite (similarly, positive definite). The notation ∗ always denotes the symmetric block in one symmetric matrix. E{x} and E{x|y}, respectively, mean the expectation of the stochastic variable x and the expectation of the stochastic variable x conditional on the stochastic variable y. Pr {δ} means the occurrence probability of the event δ.
2 Problem description and preliminaries Consider the CNNs described by the following equation x(t) ˙ = −(A + δ(t)ΔA(t))x(t) + (B + δ(t)ΔB(t)) f (x(t)) + (C + δ(t)ΔC(t)) f (x(t − τ (t))) t + (D + δ(t)ΔD(t)) f (x(s))ds + J (t), t−ρ(t)
(1) where x(t) = (x1 (t), x2 (t), . . . , xn (t))T is the neuron state vector of the CNN; A > 0 is a diagonal matrix; B, C and D are, respectively, denote the connection weight matrix, the delayed weight matrix and the distributively delayed connection weight matrix; J (t) = (J1 (t), J2 (t), . . . , Jn (t))T is an external input vector; τ (t) > 0 and ρ(t) > 0 denote the discrete time-varying delay and the distributed time-varying delay, respectively, and are assumed to satisfy 0 ≤ τm ≤ τ (t) ≤ τ M , τ˙ (t) ≤ μ < ∞, and 0 ≤ ρ(t) ≤ ρ M , where τm , τ M , μ and ρ M are constants; f (x) = ( f 1 (x1 ), f 2 (x2 ), . . . , f n (xn ))T represents the neuron activation function; The initial condition associated with model (1) is given by x(s) = φ(s), s ∈ [−d, 0], where φ(s) is bounded and continuously differential on [−d, 0], d = max{τ M , ρ M }. Throughout this paper, we always assume that activation function f is generalized activation function rather assuming Lipschitz activation function.
Assumption (H) For any i ∈ {1, 2, . . . , n}, there exist some real constants si− and si+ which may be posif i (ι2 ) tive, zero or negative, such that si− ≤ fi (ι1ι1)− ≤ −ι2 + si , ∀ ι1 , ι2 ∈ R, ι1 = ι2 . Remark 1 Activation functions for the hidden units are needed to introduce nonlinearity into the networks. The reason is that a composition of linear functions is again a linear function. However, it is the nonlinearity (i.e., the capability to represent nonlinear functions) that makes multi-layer networks so powerful. Almost any nonlinear function does the job, although for backpropagation learning it must be differentiable and it helps if the function is bounded. Most commonly used activation function is sigmoid function. This function is especially advantageous for use in neural networks trained by back-propagation, because it is easy to differentiate and thus can dramatically reduce the computation burden for training. In particular, if the activation function is chosen to be sigmoid function, then system (1) describes the dynamics of continuous-time Hopfield neural networks. Similarly, if we choose activation function f (x(t)) = 21 |x(t) + 1| − |x(t) − 1| , then system (1) describes the dynamics of cellular neural networks. It is worth noting that generalized activation function allows that f to be monotone, bounded, or differentiable. Since f is bounded, it can be easily proved that the neural network (1) has at least one equilibrium point by the Schauder fixed point theorem. We consider system (1) as the drive system. From the unidirectional linear coupling approach, a response system for (1) can be described by the following equation y˙ (t) = −(A + δ(t)ΔA(t))y(t) + (B + δ(t)ΔB(t)) f (y(t)) + (C + δ(t)ΔC(t)) f (y(t − τ (t))) t f (y(s))ds + (D + δ(t)ΔD(t)) t−ρ(t)
+ J (t) + u(t),
(2)
where A, B, C, D are matrices which are the same as in the model (1); u(t) is the controller. The initial condition associated with model (2) is given by y(s) = ϕ(s), s ∈ [−d, 0], where ϕ(s) is bounded and continuously differential function on [−d, 0]. In practical situation, the output signals of the drive system (1) can be received by the response system (2).
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Define the synchronization error as e(t) := y(t) − x(t), subtracting system (1) from system (2), yields the synchronization error dynamical system as follows e(t) ˙ = −(A + δ(t)ΔA(t))e(t) + (B + δ(t)ΔB(t))g(e(t)) + (C + δ(t)ΔC(t))g(e(t − τ (t))) t + (D + δ(t)ΔD(t)) g(e(s))ds +u(t), t−ρ(t)
e(s) = ϕ(s) − φ(s), s ∈ [−d, 0], t ≥ 0,
(3)
where g(e(t)) = f (e(t) + x(t)) − f (x(t)). One can check that the functions gi satisfy the following condition (H1) For any i ∈ {1, 2, . . . , n}, there exist some real constants si− and si+ which may be positive, zero or negative, such that si− ≤ gi ι(ι) ≤ si+ , ∀ ι = 0. The real-valued system matrices ΔA(t), ΔB(t), ΔC(t), and ΔD(t) of (1), (2), and (3) are of the form [ΔA(t) ΔB(t) ΔC(t) ΔD(t)] = E F(t)[G A G B G C G D ],
(4)
in which E, G A , G B , G C , and G D are known real constant matrices of appropriate dimensions with an unknown time-varying matrix satisfying F T (t)F(t) ≤ I.
(5)
The aim of this paper is to design a controller input u(t) such that the response system (2) synchronizes with the drive system (1). In this paper, we consider the controller u(t) = (K 1 + α(t)ΔK 1 (t))( f (y(t)) − f (x(t))) +(K 2 + α(t)ΔK 2 (t))( f (y(t − τ (t))) − f (x(t − τ (t)))),
(6)
in the response system, where K 1 , K 2 are the controller gain matrices to be determined, and the real-valued matrices ΔK 1 (t), ΔK 2 (t) represent possible controller gain fluctuations. It is assumed that ΔK 1 (t), ΔK 2 (t) has the following structure [ΔK 1 (t) ΔK 2 (t)] = E F(t)[G K 1 G K 2 ], where G K 1 , G K 2 are known constant matrices and other parameters are defined same as in (4) and (5). Now substitute (6) in (3), we obtain
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e(t) ˙ = −(A + δ(t)ΔA(t))e(t) + (B + δ(t)ΔB(t) + K 1 + α(t)ΔK 1 (t))g(e(t)) + (C + δ(t)ΔC(t) + K 2 + α(t)ΔK 2 (t))g(e(t − τ (t))) t + (D + δ(t)ΔD(t)) g(e(s))ds, t−ρ(t)
e(s) = ϕ(s) − φ(s), s ∈ [−d, 0], t ≥ 0.
(7)
The stochastic variables δ(t) and α(t) are introduced to describe the phenomena of randomly occurring uncertainties and randomly occurring controllers, respectively, which are Bernoulli-distributed white noise sequences taking on values of zero or one as given in [26] Pr {δ(t) = 1} = δ,
Pr {δ(t) = 0} = 1 − δ,
Pr {α(t) = 1} = α,
Pr {α(t) = 0} = 1 − α,
(8)
where δ and α in [0, 1] are known constants. Definition 1 (Wu et al. [28]) The error system (7) is said to be mean square stable if for any > 0, there is a γ ( ) > 0 such that E{ e(t) 2 }
0, when E{ e(0) 2 } < γ ( ). In addition, if limt→∞ E{ e(t) 2 } = 0, for any initial conditions, then the error system (7) is said to be globally mean square asymptotically stable. 3 Main results In this section, we will establish a criterion to implement the synchronization of CNNs with ROUs in system parameters and in non-fragile controller designs. Before stating our main results, we recall the following well-known lemma which will be used in the sequel. Lemma 1 (Balasubramaniam et al. [24]) Let Ω = Ω T , E and G be real matrices of appropriate dimensions, satisfying F T (t)F(t) ≤ I then Ω + E F(t)G + G T F T (t)E T < 0, if and only if there exists a positive scalar > 0 such that M + −1 E E T + G T G < 0. Further, for presentation convenience, in the following, we denote
1 = diag(s1− s1+ , s2− s2+ , . . . , sn− sn+ ), s1− + s1+ s2− + s2+ sn− + sn+ , ,...,
2 = diag . 2 2 2
Synchronization of chaotic neural networks with ROUs
Theorem 1 Assume that the condition (H1) holds. For given positive scalars δ, α and μ, if there exist symmetric positive definite matrices P, Q 1 , Q 2 , Q 3 , R1 , R2 , T with compatiR3 , R4 , S, further for any matrices R2 T ≥ 0, positive diagonal ble dimensions, and T T R2 matrices T1 , T2 , and the positive scalars 1 , 2 , such that the following LMI holds ⎡ ⎤ Π Ψ1 1 Ψ2 Ψ3 2 Ψ4 ⎢ ⎥ 0 0 0 ⎥ ⎢ ∗ − 1 I ⎢ ⎥ ⎢∗ (9) ∗ − 1 I 0 0 ⎥ ⎢ ⎥ < 0, ⎢ ⎥ ∗ ∗ − 2 I 0 ⎦ ⎣∗ ∗ ∗ ∗ ∗ − 2 I where, ⎡
Ω
τm T
⎢ ⎢ ∗ −2P + R 1 ⎢ ⎢ Π =⎢ ∗ ∗ ⎢ ⎢ ⎣∗ ∗ ∗
2 −τ 2 τM m 2
(τ M − τm )T
τM T √ 2
0
0
0
−2P + R2
0
0
∗
−2P + R3
0
∗
∗
−2P + R4
∗
and Ω = (Ωi, j )10×10 with Ω1,1 = −P A − A T P + Q 1 + Q 2 + Q 3 − R1 − R3 − R3T τ M − τm − R4 − T1 1 , τ M + τm Ω1,2 = R1 , Ω1,5 = PB + Y1 + T1 2 , Ω1,6 = PC + Y2 , 1 Ω1,7 = (R3 + R3T ), τM 1 Ω1,8 = (R4 + R4T ), τ M + τm 1 1 Ω1,9 = (R3 + R3T ) + (R4 + R4T ), τM τ M + τm Ω1,10 = P D, Ω2,2 = −Q 1 − R1 − R2 , Ω2,3 = R2 − T, Ω2,4 = T,
T
⎤ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎦
⎡ τM Ψ1 = ⎣ P E 0 · · · 0 τm P E (τ M −τm )P E √ P E 2 9 ⎡
Ψ2 = ⎣−δG TA 0 · · · 0 δG TB δG CT 0 ·· · 0 δG TD 0 ·· · 0⎦ , 3
⎡
· · 0 PB + Y1 PC + Y2 0 ·· · 0 P D ⎦ , = ⎣−P A 0 · 3
3
τM − τm )E √ E 2 ⎤
9
⎡
4
2 τM
⎤ − τm2 E⎦ , 2
Ψ4 = ⎣0 ·· · 0 αG TK 1 αG TK 2 0 ·· · 0⎦ , 4
8
and the other parameters are defined to be zero, then the response system (2) can globally mean square asymptotically synchronize the drive system (1), and the gain matrices of control law (6) is K 1 = P −1 Y1 and K 2 = P −1 Y2 . Proof Consider the Lyapunov–Krasovskii functionals 5 (LKFs) with terms described by V (e(t)) = v=1 Vv (e(t)) for system (7) V1 (e(t)) = e(t)Pe(t), t V2 (e(t)) = e T (s)Q 1 e(s)ds t−τm t
e T (s)Q 2 e(s)ds
+
+
t−τ M t
e T (s)Q 3 e(s)ds,
t−τ (t) 0 t
V3 (e(t)) = τm
−τm
e˙ T (s)R1 e(s)dsdθ ˙
t+θ
+ (τ M −τm )
Ω3,4 = −T + R2 , Ω6,6 = −T2 , 1 1 Ω7,7 = − 2 (R3 + R3T ), Ω7,9 = − 2 (R3 + R3T ), τM τM 1 (R4 + R4T ), Ω8,8 = − 2 τ M − τm2 1 (R4 + R4T ), Ω8,9 = − 2 τ M − τm2 1 1 (R4 + R4T ), Ω9,9 = − 2 (R3 + R3T ) − 2 τM τ M − τm2 Ω10,10 = −S, ⎤ ⎡
3
Ψ3 = ⎣ E 0 ·· · 0 τm E (τ M
Ω3,3 = −(1 − μ)Q 3 − R2 − R2T + T + T T − T2 1 , 2 S − T1 , Ω3,6 = T2 2 , Ω4,4 = −Q 2 − R2 , Ω5,5 = ρ M
⎤ 2 −τ 2 τM m P E ⎦, 2 ⎤
V4 (e(t)) =
0
0 t
−τ M θ −τm
+
V5 (e(t)) = ρ M
−τ M 0 −ρ M
−τm −τ M
t
e˙ T (s)R2 e(s)dsdθ, ˙
t+θ
e˙ T (s)R3 e(s)dsdλdθ ˙
t+λ 0 t
θ
e˙ T (s)R4 e(s)dsdλdθ, ˙
t+λ t
g T (e(s))Sg(e(s))dsdθ.
t+θ
Define the infinitesimal operator L of V (e(t)) as follows LV (e(t)) = lim
h→0+
1 {E{V (e(t + h))|e(t)} − V (e(t))}. h
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Along the trajectory of system (7), it can be calculated that E{LV1 (e(t))} = E{2e T (t)P e(t)}, ˙ T E{LV2 (e(t))} ≤ E e (t)Q 1 e(t) − e T (t − τm )
(10)
× Q 1 e(t − τm ) +e T (t)Q 2 e(t) − e T (t − τ M ) × Q 2 e(t − τ M ) +e T (t)Q 3 e(t) − e T (t − τ (t)) × (1 − μ)Q 3 e(t − τ (t)) , E{LV3 (e(t))} = E τm2 e˙ T (t)R1 e(t) ˙ t e˙ T (s)R1 e(s)ds ˙ −τm
(11)
t−τm
˙ +(τ M − τm )2 e˙ T (t)R2 e(t) t−τm e˙ T (s)R2 e(s)ds ˙ , −(τ M −τm ) t−τ M
E{LV4 (e(t))} ≤ E
τ2
e˙ (t)R3 e(t) ˙ t e˙ T (s)R3 e(s)dsdθ ˙
M T
− +
2 0
2
−τm −τ M
+e T (t)Q 2 e(t) − e T (t − τ M )Q 2 e(t − τ M )
˙ e˙ (t)R4 e(t) t
e˙ T (s)R4 e(s)dsdθ ˙ ,
t+θ
(13)
g(e(s))ds t−ρ(t) t
g(e(s))ds
S .
(14)
t−ρ(t)
Now by applying Jensen’s Inequality Lemma [46] and Theorem 1 in [47], one can get t e˙ T (s)R1 e(s)ds ˙ −τm t−τm
≤ −[e(t) − e(t − τm )]T R1 [e(t) − e(t − τm )], −(τ M − τm ) −(τ M − τm )
(15) t−τm
t−τ M
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e˙ T (s)R2 e(s)ds ˙
t−τ (t) t−τ (t)
e˙ T (s)R2 e(s)ds ˙
+e T (t)Q 3 e(t) −e T (t − τ (t))(1 − μ)Q 3 e(t − τ (t)) 2 ˙ +e˙ T (t) τm2 R1 + (τ M − τm R2 )e(t) −[e(t) − e(t − τm )]T R1
T
t
(17)
−e T (t − τm )Q 1 e(t − τm )
2 T ρM g (e(t))Sg(e(t))
− ×
On the other hand by condition (H1), we have T −T1 1 T1 2 e(t) e(t) ∗ −T1 g(e(t)) g(e(t)) T −T2 1 T2 2 e(t − τ (t)) + ∗ −T2 g(e(t − τ (t))) e(t − τ (t)) × ≥ 0. g(e(t − τ (t))) Now adding (10)–(17), we get ˙ + e T (t)Q 1 e(t) E{LV (e(t))} ≤ E 2e T (t)P e(t)
−τ M t+θ 2 τ M − τm2 T
−
E{LV5 (e(t))} ≤ E
(12)
(τ M − τm ) [e(t − τm ) − e(t − τ (t))]T τ (t) − τm ×R2 [e(t − τm ) − e(t − τ (t))] (τ M − τm ) [e(t − τ (t)) − e(t − τ M )]T − τ M − τ (t) ×R2 [e(t − τ (t)) − e(t − τ M )], T e(t − τm ) − e(t − τ (t)) R2 T ≤− e(t − τ (t)) − e(t − τ M ) T T R2 e(t − τm ) − e(t − τ (t)) × . (16) e(t − τ (t)) − e(t − τ M )
≤−
×[e(t) − e(t − τm )] T e(t − τm ) − e(t − τ (t)) − e(t − τ (t)) − e(t − τ M ) e(t − τm ) − e(t − τ (t)) R2 T × T T R2 e(t − τ (t)) − e(t − τ M ) 2 2 − τ2 τ τ m +e˙ T (t) M R3 + M R4 e(t) ˙ 2 2 0 t − e˙ T (s)R3 e(s)dsdθ ˙
−τ M t+θ −τm t
e˙ T (s)R4 e(s)dsdθ ˙ t+θ 2 Sg(e(t)) +g (e(t))ρ M t T t −
−τ M T
−
g(e(s))ds t−ρ(t)
S
g(e(s))ds
t−ρ(t)
Synchronization of chaotic neural networks with ROUs
−T1 1 T1 2 e(t) + ∗ −T1 g(e(t)) T −T2 1 T2 2 e(t − τ (t)) + ∗ −T2 g(e(t − τ (t))) e(t − τ (t)) × , g(e(t − τ (t))) ! + ΔΩ(t) ξ(t) , ≤ E ξ T (t) Ω (18)
T
e(t) g(e(t))
where
By noting K 1 = P −1 Y1 , K 2 = P −1 Y2 , pre- and postmultiplying (21), by ⎧ ⎨
⎡
diag
ξ T (t) = ⎣e T (t) e T (t − τm ) e T (t − τ (t)) e T (t − τ M ) g T (e(t)) t g T (e(t − τ (t)))
known Schur Complement Lemma [45], the inequality (20) is equivalent to the following matrix inequality ⎤ ⎡ ! Ψ1 1 Ψ2 Ψ3 2 Ψ4 Ω ⎥ ⎢ 0 0 0 ⎥ ⎢ ∗ − 1 I ⎥ ⎢ ⎢∗ (21) ∗ − 1 I 0 0 ⎥ ⎥ < 0. ⎢ ⎥ ⎢ ∗ ∗ − 2 I 0 ⎦ ⎣∗ ∗ ∗ ∗ ∗ − 2 I
T t−τm e(s)ds
T t
t−τ (t)
e(s)ds t−τ M
10
and
t−τ (t)
t−τ (t)
T
g(e(s))ds t−ρ(t)
e(s)ds
P R4−1 , I, . . . ,
⎫ ⎬
I ⎭
⎧ ⎨
4
respectively, and from view of the inequality −P Rk−1 P ≥ 2P − Rk , k = 1, 2, 3, 4, it is easy to see that the LMI condition (9) can guarantee that (21) holds. Hence, (20) is implied by the LMI condition (9). In other words, the LMI condition (9) and by Definition 1 implies that the response system (2) is globally mean square asymptotically synchronize the drive system (1). The proof is completed.
⎦,
+Ψ3 F(t)Ψ4 + Ψ4T F T (t)Ψ3T , ˆ 10×10 = (Ωˆ i, j ) = (Ωi, j ) = (Ω)10×10 , (Ω)
⎤
ˆ = ⎣−P A 0 · · · 0 PB + PK 1 PC + PK 2 0 · · · 0 P D ⎦ . 3
By Lemma 1, we have ΔΩ(t) ≤ 1−1 Ψ1 Ψ1T + 1 Ψ2 Ψ2T + 2−1 Ψ3 Ψ3T + 2 Ψ4 Ψ4T .
P R3−1 ,
4
10
⎤
i, j varies between 1 and 10; except that Ωˆ 1,5 = PB + PK 1 + T1 2 , Ωˆ 1,6 = PC + PK 2 , ⎡
P R2−1 ,
⎫ ⎬ diag I, . . . , I , R1−1 P, R2−1 P, R3−1 P, R4−1 P, I, . . . , I , ⎭ ⎩
T
ˆ T R1 ˆ + (τ M − τm )2 ˆ T R2 ˆ ! = Ωˆ + τm2 Ω 2 2 2 τ τ − τm T ˆ T R3 ˆ R4 , ˆ + M ˆ + M 2 2 ΔΩ(t) = Ψ1 F(t)Ψ2 + Ψ2T F T (t)Ψ1T
3
I, . . . , I , ⎩
P R1−1 ,
(19)
Substitute the inequality (19) in (18), one can get the following inequality
Remark 2 As discussed in the introduction, the designed novel non-fragile controller (6) is not considered in any of the previous literature. Hence the derived results are more general than those results discussed in the literature [42,44]. Moreover, it is worth pointing out that the non-fragile controller with ROUs is interesting but has not attracted enough attention. Thus, the theoretical results proposed in this paper enrich interest to study on synchronization of CNNs with ROUs and time-varying delay. To the best of authors knowledge, this is the first time to deal with the non-fragile synchronization control for CNNs with ROUs and time-varying delay. Consider the following drive and response system without ROUs
(20)
x(t) ˙ = −Ax(t) + B f (x(t)) + C f (x(t − τ (t))) t f (x(s))ds + J (t), (22) +D
In order to get the expected LMI condition of the errorstate system (7), we will need to make some standard manipulations on the relation (20). By using well-
y˙ (t) = −Ay(t) + B f (y(t)) + C f (y(t − τ (t))) t f (y(s))ds + J (t) + u(t), (23) +D
! + −1 Ψ1 Ψ1T + 1 Ψ2 Ψ2T + −1 Ψ3 Ψ3T Ω 1 2 + 2 Ψ4 Ψ4T < 0.
t−ρ(t)
t−ρ(t)
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x (t) 1
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0 −2 −4
0
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5
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(c)
e1(t)
e(t)
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250
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Fig. 1 State trajectories (a, b) and error trajectories (c) of drive system (1) and response system (2) in Example 1
⎡
where u(t) = K 1 ( f (y(t)) − f (x(t))) +K 2 ( f (y(t −τ (t))) − f (x(t −τ (t)))).
(24)
Defining the error as e(t) = y(t) − x(t), then one can get the following synchronization error system
τM T √ Ω τm T (τ M − τm )T 2 ⎢ ⎢ ∗ −2P + R1 0 0 ⎢ ⎢∗ ∗ −2P + R2 0 ⎢ ⎣∗ ∗ ∗ −2P + R3 ∗ ∗ ∗ ∗
< 0,
2 −τ 2 τM m 2
T
0 0 0 −2P + R4
⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
(26)
(25)
and the other parameters are defined in Theorem 1, then the response system (23) can globally asymptotically synchronize the drive system (22), and the gain matrices of control law (24) are K 1 = P −1 Y1 and K 2 = P −1 Y2 .
Corollary 1 Assume that the condition (H1) holds. For given positive scalars δ, α and μ, if there exist symmetric positive definite matrices P, Q 1 , Q 2 , Q 3 , R1 , R2 , T with compatiR3 , R4 , S, further for any matrices R2 T ble dimensions, and ≥ 0, positive diagonal T T R2 matrices T1 , T2 , such that the following LMI holds
Remark 3 The robust non-fragile control design problem is solved in Theorem 1 for the addressed synchronization for uncertain CNNs with ROUs and time delays. We have derived an LMI-based sufficient conditions for the existence of delayed output feedback controllers that ensure the mean square asymptotic stability of the uncertain CNNs. The feasibility of the controller design problem can be readily checked by the solvability of LMIs, which are dependent on the lower
e(t) ˙ = −Ae(t) + (B + K 1 )g(e(t)) +(C + K 2 )g(e(t − τ (t))) t +D g(e(s))ds. t−ρ(t)
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State Trajectories of Master System
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10
8
8
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0
0
−2
−2
−4
−4
−6
−6
−8
−8
−10 −3
−2
−1
0
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2
3
−10 −3
−2
−1
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2
3
y1(t)
Fig. 2 Behavior of a drive system (1) and b response system (2) in phase space in Example 1
and upper bounds of the time-varying delays. It means that the derived results are delay dependent, that is the LMI conditions (9) and (26) involve the terms τm and τ M . In general, these LMI conditions are not feasible for all values of the lower and upper bound of the time delay. So the feasibility depends on the values of τm and τ M . The solvability of such delay-dependent LMIs can be readily checked by resorting to the MATLAB LMI Solvers. Remark 4 In many cases, the information of the derivative of time delays is unknown because it is a difficult task to obtain the precise values (even their boundedness or the boundedness of their derivatives) of time delay systems. Regarding this circumstance, rateindependent criteria for delays τ (t) and ρ(t) satisfying the conditions 0 ≤ τm ≤ τ (t) ≤ τ M , 0 ≤ ρ(t) ≤ ρ M can be derived by choosing Q 3 in Theorem 1 and Corollary 1.
Remark 5 Recently, the research of neural networks with Markovian chain has gained much attention. Furthermore, a neural network sometimes has finite modes that jump from one to another at different times, and such a jumping can be governed by a Markovian chain. Therefore, it is very important to study the synchronization of CNNs with Markovian jumping parameters (see [23]) and ROUs (see [27]) from the view of theory and practice. The results will appear in the near future. 4 Numerical examples Example 1 Consider the CNNs (1) and (2) with the following parameters 1.8 −0.15 A = I, B = , −5.2 3.5 −1.7 −0.12 C= , −0.26 −2.5
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(a)
2 x (t) 1
y1(t)
0 −2 −4
0
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time (sec)
(b)
10 x (t) 2
5
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0 −5 −10
0
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300
350
400
450
500
time (sec)
(c)
20 e (t) 1
10
e (t)
e(t)
2
0 −10 −20
0
50
100
150
200
250
300
350
400
450
500
time (sec)
Fig. 3 State trajectories (a, b) and error trajectories (c) of drive system (1) and response system (2) without control input in Example 1
0.6 0.15 D= , −2 −0.12 and the activation functions are chosen as f (x(t)) = tan h(x(t)) and f (y(t)) = tan h(y(t)), which mean that 1 = 0I, 2 = 0.5I . The parameters associated with system uncertainties are given as follows 0.5 0 0.4 0 E= , GA = , 0 0.5 0 0.4 0 −0.3 GB = , 0.3 0 0.4 0 GC = , 0 −0.4 0 −0.3 GD = , F(t) = sin(t)I, 0.3 0 also δ = 0.3, α = 0.3. t
The discrete time-varying delay τ (t) = ete+1 , and the distributed time-varying delay ρ(t) = 0.32sin(t). Noticing that τ (t) is a monotone increasing function with respect to t, we can easily obtain that τm = 0.5
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and τ M = 1. In addition, ρ M = 0.32. Moreover, it et can be calculated that τ˙ (t) = (et +1) 2 ≤ 0.25, which implies μ = 0.25. Using the MATLAB LMI Solvers to solve the LMIs in Theorem 1, we obtain the following feasible solution matrices. For the purpose of space limitations, we list positive definite matrices alone as follows 1.9044 −0.1755 , P = 104 × −0.1755 0.1704 1.6512 −0.0110 Q 1 = 103 × , −0.0110 0.2326 1.3754 −0.0151 3 Q 2 = 10 × , −0.0151 0.2201 1.2550 −0.0120 Q 3 = 103 × , −0.0120 0.1900 7.6011 0.1457 R1 = 103 × , 0.1457 1.0876 8.9123 0.0381 R2 = 103 × , 0.0381 1.3526
Synchronization of chaotic neural networks with ROUs 1.5
(a)
δ(t)
1
0.5
0
−0.5
0
1
2
3
4
5
6
7
8
9
10
6
7
8
9
10
time (sec) 1.5
(b)
α(t)
1
0.5
0
−0.5
0
1
2
3
4
5
time (sec)
Fig. 4 Time evolutions of a δ(t) and b α(t); δ(t) and α(t) switch from values of 0 and 1 according to their expectations in Example 1
288.4075 −72.4142 , −72.4142 159.6004 1.4598 0.0064 3 R4 = 10 × , 0.0064 0.3310 1.5224 0.2149 S = 105 × , 0.2149 0.1528 R3 =
1 = 104 × 3.3618, 2 = 104 × 1.7820. Consider the following non-fragile control u(t) = (K 1 + α(t)ΔK 1 (t))( f (y(t)) − f (x(t))) +(K 2 + α(t)ΔK 2 (t))( f (y(t − τ (t))) − f (x(t − τ (t)))), where K 1 and K 2 will be designed. ΔK 1 (t) and ΔK 2 (t) represent variations in the control gains. From the structures of ΔK 1 (t) and ΔK 2 (t), we have 0.5 0 0.1 0 E= , GK1 = , 0 0.5 0 0.1 0 0.1 GK2 = , F(t) = sin(t)I, 0.1 0
By Theorem 1, we know that the response system (2) can globally mean square asymptotically synchronize the drive system (1), and the gain matrices K 1 , K 2 can be obtained as −2.7286 0.1772 , K1 = 2.7234 −5.0467 1.6503 0.0408 K2 = . 0.5191 2.1141 The Euler numerical scheme is applied in the simulations by choosing the time step size h = 0.1 and time segment T = 500. Here we can consider two cases (A) and (B). Case (A): when an external input vector J (t) = [0 0]T , the trajectories of drive, response and error system with initial conditions x(s) = [0.2 0.3]T , y(s) = [−0.1 0.6]T , s ∈ [−1, 0] are shown in Fig. 1, while Fig. 2 depicts its phase trajectories of drive and response system. When u(t) = 0 in response system (2), the state trajectories of the CNN are shown in Fig. 3. It should be noted that the drive system (1) and the response system (2) without
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an external input vector J (t) = [2sin(t) 2sin(t)]T , the trajectories of drive, response, and error system with initial conditions x(s) = [0.2 0.3]T , y(s) = [−0.1 0.6]T , s ∈ [−1, 0] are shown in Fig. 6. In both the cases, and from simulations, one can find that synchronization of system (1) is realized via the feedback gain matrices K 1 , K 2 and those simulations match the obtained results perfectly.
0.5 u (t) 1
u (t) 2
0
u(t)
−0.5
−1
−1.5
Example 2 Consider the CNNs (22) and (23) with the following parameters 1.8 −0.15 A = I, B = , −5.2 3.5 −1.7 −0.12 C= , −0.26 −2.5 0.6 0.15 0 D= ,J = , −2 −0.12 0
−2 0
5
10
15
20
25
time (sec)
Fig. 5 The graph of the non-fragile control actions that are being applied to the response system in Example 1
control input cannot be synchronized. The time evolutions of δ(t) and α(t) are given in Fig. 4, which show that variables δ(t) and α(t) switch from values of 0 and 1 according to their expectations. The graph of the non-fragile control actions that are being applied to the response system is shown in Fig. 5. Case (B): when
(a)
and = the activation functions are chosen as f (x(t)) 1 1 2 |x(t) + 1| − |x(t) − 1| , and f (y(t)) = 2 |y(t) + 1| − |y(t) − 1| , which mean that 1 = 0I, 2 =
10
x1(t) y1(t)
0
−10
−20
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(b)
10
x2(t) y (t) 2
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−10
−20
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250
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450
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time (sec)
(c)
0.4 e1(t)
0.2
e (t)
e(t)
2
0 −0.2 −0.4
0
50
100
150
200
250
300
350
400
450
500
time (sec)
Fig. 6 State trajectories (a, b) and error trajectories (c) of drive system (1) and response system (2) in Example 1 with an external input vector J (t)
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Synchronization of chaotic neural networks with ROUs
(a)
1 x (t) 1
0.5
y1(t)
0 −0.5 −1
0
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(b)
10 x (t) 2
5
y2(t)
0 −5 −10
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350
400
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(c)
0.4 e1(t)
e(t)
0.2
e2(t)
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50
100
150
200
250
300
350
400
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500
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Fig. 7 State trajectories (a, b) and error trajectories (c) of drive system (22) and response system (23) in Example 2
t
0.5I . The discrete time-varying delay τ (t) = ete+1 , and the distributed time-varying delay ρ(t) = 0.5sin(t). Noticing that τ (t) is a monotone increasing function with respect to t, we can easily obtain that τm = 0.5 and τ M = 1. In addition, ρ M = 0.5. Moreover, it can be et calculated that τ˙ (t) = (et +1) 2 ≤ 0.25, which implies μ = 0.25. Using the MATLAB LMI Solvers to solve the LMIs in Corollary 1, we obtain the following feasible solution matrices. For the purpose of space limitations, we list positive definite matrices alone as follows
1.2450 −0.0930 , P = 10 × −0.0930 0.1113 178.3643 −1.4742 Q1 = , −1.4742 20.7292 159.3471 −2.6182 Q2 = , −2.6182 18.3717 155.0812 −2.2640 Q3 = , −2.2640 16.5267 3
R1 R2 R3 R4
570.2880 20.3077 = , 20.3077 82.5321 590.3105 10.7587 = , 10.7587 101.7865 88.7456 6.3514 = , 6.3514 42.1449 110.3928 −0.6617 = . −0.6617 27.9775
By Corollary 1, we know that the response system (23) can globally asymptotically synchronize the drive system (22), and the gain matrices K 1 , K 2 can be obtained as −2.4103 0.2873 , K1 = 3.3714 −4.9356 1.6444 0.0601 K2 = . 0.3470 2.2409 We use the Euler numerical scheme in the simulations with the time step size h = 0.1 and time segment T = 500. The trajectories of drive, response
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(b)
6
4
4
2
2
y (t)
0
2
2
x (t)
(a)
State Trajectories of Slave System
6
0
−2
−2
−4
−4
−6 −1.5
−1
−0.5
0
0.5
1
1.5
−6 −1.5
−1
−0.5
x (t)
0
0.5
1
1.5
y (t)
1
1
Fig. 8 Behavior of a drive system (22) and b response system (23) in phase space in Example 2
and error system with initial conditions x(s) = [0.2 0.3]T , y(s) = [−0.1 0.6]T , s ∈ [−1, 0] are shown in Fig. 7. Figure 8 depicts its phase trajectories of drive and response system. The state trajectories of the CNN without controller in the response system (23) is shown in Fig. 9. It should be noted that the drive system (22) and the response system (23) without control input cannot be synchronized. The graph of the control actions that are being applied to the response system is shown in Fig. 10. From the simulations, we can find that synchronization of system (22) is realized via the feedback gain matrices K 1 , K 2 and those simulations match the obtained results perfectly. Remark 6 In Example 1, the activation function is chosen to be f (x(t)) = tan h(x(t)), f (y(t)) = tan h(y(t)), then system (7) describes the dynamics of Hopfield-type CNNs. Similarly, if we choose acti-
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vation function f (x(t)) = 21 |x(t) + 1| − |x(t) − 1| , f (y(t)) = 21 |y(t) + 1| − |y(t) − 1| as in Example 2, then system (7) describes the dynamics of Cellular-type CNNs, which prove the derived results can be applied for class of neural networks, such as Hopfield neural networks and Cellular neural networks. 5 Conclusions In this paper, the synchronization problem has been investigated for CNNs with ROUs in system parameters and in non-fragile controller designs. By constructing proper LKFs involving triple integral terms and employing recently developed reciprocally convex optimization techniques, we have derived LMI-based criteria to guarantee synchronization of the CNNs with mixed time delays. Also, the new non-fragile control
Synchronization of chaotic neural networks with ROUs 1
(a)
x (t) 1
0.5
y (t) 1
0 −0.5 −1
0
50
100
150
200
250
300
350
400
450
500
time (sec)
(b)
10 x (t) 2
5
y (t) 2
0 −5 −10
0
50
100
150
200
250
300
350
400
450
500
time (sec)
(c)
20 e1(t)
e(t)
10
e2(t)
0 −10 −20
0
50
100
150
200
250
300
350
400
450
500
time (sec)
Fig. 9 State trajectories (a, b) and error trajectories (c) of drive system (22) and response system (23) without control input in Example 2
0.5 u (t) 1
u2(t)
0
u(t)
−0.5
−1
−1.5
−2 0
5
10
15
20
25
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Fig. 10 The graph of the control actions that are being applied to the response system in Example 2
law has been used to obtain its estimation gains which can be easily obtained by solving the LMIs. Two numerical examples have been illustrated to show the performance of the proposed controller design. The future
research topics would include the extension of the main results developed in this paper to more general CNNs such as systems with Markovian jumping parameters, mode-dependent time delays and stochastic perturbations. Acknowledgments The authors would like to express their sincere gratitude to the Editor-in-Chief, Associate Editor, and Anonymous Reviewers for their valuable comments and suggestions to improve the quality of the manuscript. The research work of Mr. V. Vembarasan is supported by DST INSPIRE Fellowship Grant DST/INSPIRE Fellowship/2011/278 dated 21.12.2011 and 03.10.2012, from Ministry of Science and Technology, Government of India; Also this research work is supported by the High Impact Research MoE Grant UM.C/625/1/HIR/ MoHE/FCSIT/08, H-22001-00-B0008 from the Ministry of Higher Education Malaysia.
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