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Date of publication xxxx 00, 0000, date of current version xxxx 00, 0000. Digital Object Identifier 10.1109/ACCESS.2017.DOI
Fixed-time synchronization of memristive fuzzy BAM cellular neural networks with time-varying delays based on feedback controllers MINGWEN ZHENG1 ,2 , LIXIANG LI3 , HAIPENG PENG3 ,JINGHUA XIAO2 ,YIXIAN YANG3 ,YANPING ZHANG2 1
School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China School of Mathematics and Statistics, Shandong University of Technology, Zibo 255000, China (e-mail:
[email protected]) Information Security Center, State Key Laboratory of Networking and Switching Technology, Beijing University of Posts and Telecommunications,Beijing 100876, China 2 3
Corresponding author: Lixiang Li (e-mail:
[email protected]). This paper is supported by the National Key Research and Development Program (Grant Nos.2016YFB0800602), the National Natural Science Foundation of China (Grant Nos.61472045, 61573067, 61771071).
ABSTRACT This paper mainly investigates the fixed-time synchronization problem of memristive fuzzy bidirectional associative memory cellular neural networks (MFBAMCNNs) with time-varying delays. MFBAMCNNs are formulated by virtue of differential inclusion and set-valued map theories. By utilizing the definition of fixed-time synchronization and some inequality techniques, some novel criteria for easy verification are derived to ensure the fixed-time synchronization of the drive-response MFBAMCNNs based on the Lyapunov stability theory and nonlinear feedback controllers. The results of the main theorem can be easily extended to the fuzzy BAM cellular neural networks without memristor and the memristive fuzzy BAM cellular neural networks without fuzzy logic. In addition, the settling time of fixed-time synchronization, which does not depend on the initial values, can be simply calculated. At last, two numerical examples are presented to verify the effectiveness of main results. INDEX TERMS Enter key words or phrases in alphabetical Fixed-time synchronization, Memristor, MFBAMCNNs, Lyapunov function, Nonlinear feedback controller
I. INTRODUCTION
N 1987, B. Kosok firstly proposed the bidirectional associative memory neural networks (BAMNNs), which are made up of two neuron layers, i.e. X-layer and Y-layer, and have the features of heteroassociative, content-addressable memory [1]. These two layers of neurons are completely interconnected, and there is no signal transmission between neurons on the same layer. In real life, BAMNNs have powerful information processing abilities and some good application fields, such as information associative memory, image processing, artificial intelligence, and so on. Thus, many scholars have studied various forms of BAMNNs models [2]– [6]. The memristor (memory resistor), which was proposed by Leon Chua for the first time [7], is well-known as the forth basic passive nonlinear circuit element (please see Fig. 1). Until the HP labs developed the real memristor in 2008
I
VOLUME 4, 2016
[8] (please see Fig. 2), it was widely recognized [2], [9]– [12]. The memristor has the characteristics of nanometer size, nonvolatile and hysteresis loops, which make it work like neuronal synapses. Based on the advantages of the memristor, it can replace the traditional resistors to simulate the biological synapses, and the synaptic density in the memristive neural network is further moved closer to the biological neural network. Naturally, the researchers combine the memristor with BAMNNs to form a class of neural networks named memristive bidirectional associative memory neural networks (MBAMNNs), which have been extensively investigated in the aspect of its dynamic behaviors [4], [5], [13]–[18]. For example, Ali et al. investigated a class of memristor-based neutral-type stochastic bidirectional associative memory neural networks; with the aid of a new Lyapunov-Krasovskii function, they obtained some delay1
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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2018.2805183, IEEE Access Author et al.: Preparation of Papers for IEEE TRANSACTIONS and JOURNALS
FIGURE 1. The position of memristor in four basic circuit elements.
dependent passivity LMI inequality conditions [13]. Wang et al. discussed the global asymptotic stability of drive-response MBAMNNs with different delays based on a novel sampleddata feedback controller [5]. Since uncertainties or vagueness are unavoidable in the real world, Yang et al. combined fuzzy theory with cellular neural networks as a suitable method to deal them, thus forming the fuzzy cellular neural networks (FCNNs) [19], [20]. The stability of FCNNs plays an important role in image processing and pattern recognition applications. Many scholars have done extensive researches on FCNNs [21]– [24]. As far as we know, few researchers have considered the combination of FCNNs and MBAMNNs to form a new class of memristive fuzzy BAM cellular neural networks (MFBAMCNNs). In [25], P.Balasubramaniam et al. studied the global asymptotic stability of a class of bidirectional associative memory fuzzy cellular neural networks with multiple types of delays by using the free-weighting matrix method. Xu et al. investigated the exponential stability of FBAMCNNs with delays and impulses. However, the above research objects are not MFBAMCNNs [26]. Compared to existing literature, MFBAMCNNs have more complex network structures and dynamic behaviors due to the addition of memristor and fuzzy logic. There are two main difficul2
ties in the study of MFBAMCNNs. First, the addition of the memristor model, which makes MFBAMCNNs become a right-hand discontinuous switching differential equations, and traditional research methods do not suitable for such systems. Secondly, the addition of fuzzy logic makes MFBAMCNNs more complex, and how to deal with the fuzzy item effectively is a problem worth considering. This is one of the motivations of this paper. On the other hand, since Pecora and Carrol first introduced the drive-response concept to synchronize two chaotic systems [27], the synchronization control of chaotic systems has been widely used in secure communication, image encryption, motion control, and so on. As a result, many synchronization control strategies and schemas are proposed for chaotic systems [4], [12], [13], [15], [23], [28]–[30]. In these synchronization studies of chaotic neural networks, the convergence rate (time) is a key consideration of synchronization performance indicator. According to the convergence time, the synchronization control mainly include the asymptotic (exponential) synchronization, finite-time synchronization and fixed-time synchronization. Among them, the asymptotic synchronization is based on the enough large convergence time, and the finite-time and fixed-time synchronization can synchronize the drive-response chaotic systems VOLUME 4, 2016
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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2018.2805183, IEEE Access Author et al.: Preparation of Papers for IEEE TRANSACTIONS and JOURNALS
FIGURE 2. The schematic diagram of memristor developed by HP Laboratory [8].
in a finite time, which is called the settling time. In practical applications, we sometimes expect the drive-response systems to achieve synchronization as quickly as possible. The critical difference between the finite-time and fixedtime synchronization is whether the settling time depends on the initial conditions of the drive-response systems. The finite-time synchronization depends on the initial conditions, while the fixed-time synchronization does not depend on it. In many practical cases, it is not easy to obtain the initial conditions of the chaotic systems, which make the settling time difficult to be determined. The fixed-time synchronization is derived from the definition of fixed-time stability, which was proposed by Polyakov et al. for the first time [31], [32], can overcome this difficulty. In [28], Wang et al. investigated the fixed-time master-slave synchronization of Cohen-Grossberg neural networks with time-varying delays and parameter uncertainties by virtue of Filippov discontinuous theory and Lyapunov stability theory, and obtained some sufficient conditions to ensure fixed-time synchronization of master-slave systems. In [33], Liu and Chen discussed the finite-time and fixed-time cluster synchronization of complex networks under pinning control. However, there are few results on the fixed-time synchronization of MFBAMCNNs. In addition, how to design a suitable fixed-time synchronization controller for drive-response MFBAMCNNs systems is a VOLUME 4, 2016
challenging issue. This is another research motivation of this paper. Motivated by the above discusses, we combine fuzzy cellular neural networks with MBAMNNs to form a new class of neural network model and study its fixed-time synchronization problem of the drive-response systems for the first time. With the help of the definition of fixed-time synchronization, differential inclusion theory, set-valued map, some inequalities techniques and Lyapunov stability theory, some easily verifiable sufficient conditions are obtained. In addition, it is worth noting that the settling time that does not depend on the initial conditions can be easily calculated directly. The main contributions of this paper are summarized as follows. (1) Compared with the existing results on fixed-time synchronization of other neural network models [28], [34], the fixed-time synchronization problem of MFBAMCNNs with time-varying delays is studied for the first time. The existing BAMNNs models without the memristor or without the fuzzy logic can be regarded as a special case of the MFBAMCNNs model. (2) By the constructions of nonlinear feedback controllers and a simple Lyapunov function, some novel easily verifiable algebraic inequality conditions are obtained to ensure the fixed-time synchronization of the proposed model. (3) The main results of this paper are more general. It can 3
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be easily extended to the fuzzy BAM cellular neural networks without memristor and the memristive fuzzy BAM cellular neural networks without fuzzy logic. The remainder of this paper is organized as follows. In Section II, the MFBAMCNNs mathematical model, the definition of the fixed-time synchronization, some assumptions and lemmas are presented as preliminaries. Through strict proof, some novel criteria to guarantee the drive-response MFBAMCNNs achieve the fixed-time synchronization are obtained in Section III. After two numerical simulations illustrate the validity of our results in Section IV, we sum up the whole paper in Section V. Notations: In the whole paper, R, Rm , Rn denote real number set, m-dimensional Euclidean space and ndimensional Euclidean space, respectively. R+ represents positive real number set. C([−τ, 0], Rn ), C([−σ, 0], Rn ) denotes a set of continuous function from interval [−τ, 0] and [−σ, 0] to space Rn . Let I , {1, 2, · · · , n}, J , {1, 2, · · · , m} be a natural number set, respectively.
J . fj (·), gi (·) represent the W V activation function of the jth and ith neurons. and represent fuzzy OR and fuzzy (1) (2) (1) (2) AND operations. αij (αji ) and βij (βji ) are respectively the connection weights of the fuzzy feedback MIN template and the fuzzy feedforward MAX template with delays. (1) (2) (1) (2) (1) (2) cij (cji ), Sij (Sji ) and Tij (Tji ) denote the connection weights of the fuzzy feedforward template, the elements of the fuzzy feedforward MIN template and fuzzy feedforward MAX template without delays. τ (t), σ(t) are the timevarying delays, which meet 0 ≤ τ (t) ≤ τ , 0 ≤ σ(t) ≤ σ. Ii1 and Ij2 denote the bias values of the ith and jth neurons. (1)
(2)
(1)
di (xi (t)) =
II. NETWORK MODEL AND PRELIMINARIES
Inspired by Refs. [14], [35], we consider a class of the MFBAMCNNs model described by the following differential equations m X (1) (1) aij (xi (t))fj (yj (t)) x˙ i (t) = −di (xi (t))xi (t) + j=1
+ + + +
m X j=1 m X j=1 m _ j=1 m ^
(1) bij (xi (t
(1)
(1)
(1)
cij wj +
m ^
(2) aji (yj (t))
(1)
(1)
βij fj (yj (t − τ (t))) +
m _
(1)
(2) bji (yj (t))
(1)
Sij wj
(1)
(1)
(1)
(1) n X
(2)
aji (yj (t))gi (xi (t))
(2)
bji (yj (t − τ (t)))gi (xi (t − σ(t)))
i=1
+
(2)
(2)
cji wi
+
n ^
(2)
αji gi (xi (t − σ(t)))
i=1 (2)
βji gi (xi (t − σ(t))) +
n _
(2)
(2)
Sji wi
i=1 (2)
(2)
Tji wi
j=1 n X
1
(2) (Wji + (2) Cj i=1 (1) Wij × sgnij , (1) Ci (1) Mij × sgnij , (1)
(2)
Mji ) × sgnji +
(1)
Ri
1 (2)
Rj
, ,
Ci
Wji
(2)
Cj
=
Mji
(2)
Cj
× sgnji , × sgnji ,
where sgnij is the symbolic function with a value of 1 (1) (2) when i = j and 0 otherwise. Ci and Cj are capaci(1) (1) (2) (2) tors. Wij , Mij , Wji , Mji denote the memductances of (11) (12) (21) (22) (11) the resistors Rij , Rij , Rji , Rji , respectively. Rij is the resistor between xi (t) and the activation function (12) fj (yj (t)); Rij is the resistor between xi (t) and the activa(21) tion function fj (yj (t − τj (t))); Rji is the resistor between (22) yj (t) and the activation function fi (xi (t)); Rji is the resistor between yj (t) and the activation function gi (yi (t−σ(t))). Furthermore, according to the feature of memristor, we use the memristor simplified model, which is shown in Fig.3. Its memristive weights is presented as follows
(2)
+ Ij , t ≥ 0,
i=1
where x(t) = (x1 (t), x2 (t), · · · , xn (t))T , y(t) = (y1 (t), y2 (t), · · · , ym (t))T denote the states of neurons in X-layer and Y-layer (the voltage of the capacitors (1) (2) Ci and Cj in the actual circuits), respectively. i ∈ I, j ∈ 4
Ci
1
(2)
j=1
Tij wj + Ii ,
n X
i=1 n ^
=
αij fj (yj (t − τ (t)))
i=1
+
m 1 X (1) (1) (Wij + Mij ) × sgnij + (1)
j=1
n X
i=1 n _
(1)
(2)
(2)
+
(1)
aij (xi (t)) =
− σ(t)))fj (yj (t − τ (t)))
= −dj (yj (t))yj (t) + +
(2)
dj (xj (t)) =
bij (xi (t)) =
j=1
y˙ j (t)
(1)
di (xi (t)), dj (xj (t)), aij (xi (t)), bij (xi (t)), (2) (2) aji (xj (t)), bji (xj (t)) denote the memristive weights. In the memristor-based neural network real circuits, they are given as follows
(1) di (xi (t))
(
(1) (1) dˆi , |xi (t)| ≤ Γi , (1) (1) dˇ , |xi (t)| > Γ ,
(
(2) (2) dˆj , |yj (t)| ≤ Γj , (2) (2) dˇ , |yj (t)| > Γ ,
=
i
(2)
dj (yj (t)) =
j
i
j
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2
× 10-4
Orignal memristor model Simplified memristor model
I(t)
1
0
-1
-2 -1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
V(t)
FIGURE 3. The hysteresis loop (V (t) − I(t)) of the original memristor model and the simplified memristor model (two-state).
(1)
(
(1) a ˆij , |xi (t)| (1) a ˇij , |xi (t)|
(
(2) a ˆji , |yj (t)| (1) a ˇji , |yj (t)|
(1)
aij (xi (t)) = (2) aji (yj (t))
(1) bij (xi (t
=
≤ >
(2)
ˆb(1) , |xi (t − σ(t))| ≤ Γ(1) , ij i ˇb(1) , |xi (t − σ(t))| > Γ(1) ,
(
ˆb(2) , |yj (t − τ (t))| ≤ Γ(2) , ji j ˇb(2) , |yj (t − τ (t))| > Γ(2) ,
ij
(2)
>
( − σ(t))) =
bji (yj (t − τ (t))) =
≤
(1) Γi , (1) Γi , (2) Γj , (2) Γj ,
i
ji
x˙ i (t)
j
(1) (2) (1) (1) where the switching jumps Γi > 0, Γj > 0, dˆi , dˇi , (2) (2) (1) (1) (2) (2) ˆ(1) ˇ(1) ˆ(2) ˇ(2) dˆi , dˇi , a ˆij , a ˇij , a ˆji , a ˇji , bij , bij , bji , bji are known
constants, i ∈ I, j ∈ J . The initial values of MFBAMCNNs (1) are give by ( (1) xi (s) = ϕi (s), −τ ≤ s ≤ 0, (3) (1) yj (s) = ψj (s), −σ ≤ s ≤ 0, where
(1) ϕi (s)
VOLUME 4, 2016
n
∈ C([−τ, 0], R
(1) ), ψj (s)
m
∈ C([−σ, 0], R ),
(1)
i.e. ϕi (s) and ψj (s) are continuous function on interval [−τ, 0] and [−σ, 0]. Obviously, the memristive fuzzy BAM cellular neural networks (1) is a differential equation with discontinuous rightsides according the definitions of the memristive connection weights. It shows that the traditional processing method of dealing with differential equations does not apply to the model (1). By virtue of differential inclusion and set-valued map theories [36], [37], the MFBAMCNNs (1) can be regarded as a state-dependent switching system as shown below (1)
(1)
∈ −co[di , di ]xi (t) m X (1) (1) + co[aij , aij ]fj (yj (t)) j=1
+
m X
(1)
(1)
co[bij , bij ]fj (yj (t − τ (t)))
j=1
+ +
m X j=1 m _ j=1
(1)
m ^
(1)
cij wj +
(1)
αij fj (yj (t − τ (t)))
j=1 (1) βij fj (yj (t
− τ (t))) +
m _
(1)
(1)
Sij wj
j=1 5
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+
m ^
(1)
(1)
(1)
Tij wj + Ii ,
+
(4)
(2) (2) ∈ −co[dj , dj ]yj (t) n X (2) (2) + co[aji , aji ]gi (xi (t)) i=1 n X (2) (2) + co[bji , bji ]gi (xi (t − σ(t))) i=1 n n X ^ (2) (2) (2) + cji wi + αji gi (xi (t − σ(t))) i=1 i=1 n n _ _ (2) (2) (2) + βji gi (xi (t − σ(t))) + Sji wi i=1 i=1 n ^ (2) (2) (2) + Tji wi + Ij , t ≥ 0, i=1
(1)
= −dxi (t)xi (t) +
m X
+ + + + v˙ j (t)
=
+ + +
j=1 m ^ j=1 m _
+ + + +
j=1 m ^
(1)
(1)
(1)
(1)
Tij wj + Ii ,
= −dyj (t)yj (t) +
n X
j=1 (1)
αij fj (vj (t − τ (t))) (1)
βij fj (vj (t − τ (t))) + (1)
(1)
(1)
Tij wj + Ii
(2) −dj (vj (t))vj (t) n X
+ + +
i=1 n ^ i=1 n _ i=1
6
byji (t)gi (xi (t
ayji (t)gi (xi (t))
− σ(t))) +
n X
i=1 n ^ i=1 n _ i=1 n ^
(5)
(2) (2) cji wi
(2)
− σ(t)))
βji gi (xi (t − σ(t))) +
i=1
(2)
(6)
+
n X
(2)
aji (vj (t))gi (ui (t))
− σ(t))) +
n X
(2)
(2)
cji wi
i=1 (2) αji gi (ui (t
− σ(t)))
(2)
βji gi (ui (t − σ(t))) +
n _
(2)
(2)
Sji wi
i=1 (2) (2) Tji wi
+
(2) Ij
+ rjv (t), t ≥ 0,
(2)
u˙ i (t)
= −dui (t)ui (t) +
(2)
Sji wi
m X
auij (t)fj (vj (t))
j=1
+ n _
(1)
where ϕi (s) ∈ C([−τ, 0], Rn ), ψj (s) ∈ C([−σ, 0], Rm ), (2) (2) i.e. ϕi (s) and ψj (s) are continuous functions on interval [−τ, 0] and [−σ, 0], respectively. Using the same processing method as the model (1) by applying the set-valued map and differential inclusion theories, we have
i=1 (2) αji gi (xi (t
(1)
Sij wj
+ riu (t),
(2) bji (vj (t))gi (ui (t
i=1 n X
m _ j=1
(2)
j=1
y˙ j (t)
(1)
(1)
Sij wj
j=1 (1)
j=1 m ^
(1)
cij wj
where and rjv (t) are the fixed-time synchronization controllers to be designed. The initial values of the MFBAMCNNs (6) are defined as ( (2) ui (s) = ϕi (s), −τ ≤ s ≤ 0, (7) (2) vj (s) = ψj (s), −σ ≤ s ≤ 0,
(1)
cij wj
(1)
m _
j=1 m _
m X
i=1
αij fj (yj (t − τ (t))) (1)
j=1 m ^
(1)
bij (ui (t))fj (vj (t − τ (t))) +
riu (t)
j=1
βij fj (yj (t − τ (t))) +
m X
i=1
j=1
bxij (t)fj (yj (t − τ (t))) +
(1)
aij (ui (t))fj (vj (t))
j=1
axij (t)fj (yj (t)) m X
m X j=1
CNNs (4) can be equivalently to the following form
+
(2)
+ Ij , t ≥ 0.
= −di (ui (t))ui (t) +
u˙ i (t)
(1) (1) (1) (1) (1) = min{dˆi , dˇi }, di = max{dˆi , dˇi }, where (2) (2) (2) (2) (1) (2) dj = min{dˆj , dˇj }, dj = max{dˆj , dˇj }, (1) (1) (1) (1) (1) (1) aij = min{ˆ aij , a ˇij }, aij = max{ˆ aij , a ˇij }, (2) (2) (2) (2) (2) (2) aji = min{ˆ aji , a ˇji }, aji = max{ˆ aji , a ˇji }, (1) (1) ˇ(1) (1) (1) ˇ(1) (2) ˆ ˆ bij = min{bij , bij }, bij = max{bij , bij }, bji = (2) (2) (2) (2) (2) min{ˆbji , ˇbji }, bji = max{ˆbji , ˇbji }, i ∈ I, j ∈ J . (1) (1) Let measurable functions dxi (t) ∈ co[di , di ], dyj (t) ∈ (2) (2) (1) (1) (2) (2) co[dj , dj ], axij (t) ∈ co[aij , aij ], ayji (t) ∈ co[aji , aji ], (1) (1) (2) (2) bxij (t) ∈ co[bij , bij ], byji (t) ∈ co[bji , bji ], then MFBAM-
m X
(2)
if the MFBAMCNNs (1) are assumed to be the drive (master) system, then the response (slave) MFBAMCNNs are designed as follows
(1) di
x˙ i (t)
(2)
Tji wi
i=1
j=1
y˙ j (t)
n ^
+
m X j=1 m ^
buij (t)fj (vj (t − τ (t))) +
m X
(1)
(1)
cij wj
j=1 (1)
αij fj (vj (t − τ (t)))
j=1 VOLUME 4, 2016
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+ +
m _
(1)
βij fj (vj (t − τ (t))) +
j=1 m ^
m _
(1)
(1)
−
Sij wj
j=1 (1)
(1)
(1)
Tij wj + Ii
Fjy (t)
+ riu (t),
=
= −dvj (t)vj (t) +
n X
avji (t)gi (ui (t))
+
(8)
+ + +
n X i=1 n ^ i=1 n _ i=1 n ^
bvji (t)gi (ui (t − σ(t))) +
n X
(2)
(2)
−
cji wi
i=1
Gyj (t)
(2)
αji gi (ui (t − σ(t)))
=
(2)
βji gi (ui (t − σ(t))) +
(2)
(2)
−
Sji wi
i=1 (2)
(2)
Tji wi
(2)
+ Ij
+ rjv (t), t ≥ 0,
+
(1)
(1)
(2)
(2)
where dui (t) ∈ co[di , di ], dvj (t) ∈ co[dj , dj ], auij (t) ∈ (1)
(2)
(2)
(1)
(1)
co[aij , aij ], avji (t) ∈ co[aji , aji ], buij (t) ∈ co[bij , bij ], (2)
(2)
bvji (t) ∈ co[bji , bji ]. A. THE DEFINITION OF FIXED-TIME SYNCHRONIZATION
In this subsection, we give the definition of fixed-time synchronization between drive-response (master-slave) systems (1) and (6). We define the error systems between (1) and (6) as exi (t) = ui (t)−xi (t), eyj (t) = vj (t)−yj (t), i ∈ I, j ∈ J . Then, we have ( e˙ xi (t) = Dix (t) + Fix (t) + Gxi (t) + riu (t), (9) e˙ yj (t) = Djx (t) + Fjy (t) + Gyj (t) + rjv (t), where
Fix (t)
Dix (t) = −(dui (t)ui (t) − dxi (t)xi (t)), Djy (t) = −(dvj (t)vj (t) − dyj (t)yj (t)), m m X X = auij (t)fj (vj (t)) − axij (t)fj (yj (t)) j=1
+
m X
j=1
buij (t)fj (vj (t − τ (t)))
j=1
− Gxi (t)
= − +
m X j=1 m ^ j=1 m ^ j=1 m _ j=1
VOLUME 4, 2016
avji (t)gi (ui (t)) −
n X
n X
ayji (t)gi (xi (t))
i=1
bvji (t)gi (ui (t − σ(t)))
n X i=1 n ^
byji (t)gi (xi (t − σ(t))), (2)
αji gi (ui (t − σ(t)))
i=1 n _
i=1
(1)
j=1 n X
i=1
i=1
+
(1)
βij fj (yj (t − τ (t))),
i=1
j=1
v˙ j (t)
m _
bxij (t)fj (yj (t − τ (t))), (1)
αij fj (vj (t − τ (t))) (1)
αij fj (yj (t − τ (t))) (1)
βij fj (vj (t − τ (t)))
−
n ^ i=1 n _ i=1 n _
(2)
αji gi (xi (t − σ(t))) (2)
βji gi (ui (t − σ(t))) (2)
βji gi (xi (t − σ(t))).
i=1
The initial condition of the error system (9) can be expressed as ( (2) (1) exi (s) = ϕi (s) − ϕi (s), (10) (2) (1) eyj (s) = ψj (s) − ψj (s). Definition 1: The systems (1) and (6) are said to achieve the finite-time synchronization, for i ∈ I, j ∈ J , if there exists T (exi (0), eyj (0)) : S a locally bounded function n x R →R+ {0} that depends on ei (0), eyj (0), such that exi (t, exi (0)) = 0, eyj (t, eyj (0)) = 0 for all t ≥ T (exi (0), eyj (0)), where exi (t, exi (0)), eyj (t, eyj (0)) are the solutions of the Cauchy problem (9). The function T (exi (0), eyj (0)) is called the settling-time function. Definition 2: The systems (1) and (6) are said to obtain the fixed-time synchronization, if they can achieve the finite-time synchronization and the settling-time function T (exi (0), eyj (0)) is globally bounded, i.e. there exists a fixed constant Tmax ∈ R+ such that T (exi (0), eyj (0)) ≤ Tmax , for any exi (0), eyj (0) ∈ R. Remark 1: According to Definition 2, let e(t) , (ex1 (t), ex2 (t), · · · , exn (t), ey1 (t), · · · , eym (t))T and T (e(0)) , T (exi (0), eyj (0)) , we have the following equivalent form lim ke(t)k = 0, t→T (e(0)) e(t) ≡ 0, ∀t ≥ T (e(0))), T (e(0)) ≤ Tmax , ∀e(0) ∈ C([−τ, 0], Rn ). where k · k denotes the Euclidean norm. Remark 2: Compared the Definition 1 and Definition 2, the main difference between the finite-time synchronization and the fixed-time synchronization is whether the settlingtime function T and Tmax depend on the initial conditions. 7
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B. SOME ASSUMPTIONS AND LEMMAS
III. MAIN RESULTS
In this subsection, we will give some assumptions and lemmas in order to obtain our main results. Assumption 1. For i ∈ I, j ∈ J , the activation functions fj (·), gi (·) meet the Lipschitz conditions, i.e. for any x1 , x2 ∈ R, x1 6= x2 , there exist positive constants ki , lj satisfying the following conditions
Here, we will design the fixed-time synchronization controllers and derive the sufficient conditions to guarantee the synchronization between the drive MFBAMCNN (1) and the response MFBAMCNN (6). To obtain the fixed-time synchronization, we design the following controllers riu (t), rjv (t), which are added to the response MFBAMCNN (6)
|fj (x1 ) − fj (x2 )| ≤ kj |x1 − x2 |, |gi (x1 ) − gi (x2 )| ≤ li |x1 − x2 |.
riu (t) = −λxi exi (t) − sgn(exi (t))(ιxi + ωix |exi (t − σ(t))| + µxi |exi (t)|γ1 + νix |exi (t)|γ2 ), v rj (t) = −λyj eyj (t) − sgn(eyj (t))(ιyj + ωjy |eyj (t − τ (t))|
Lemma 1: [38] Suppose x1 , x2 are any two states of model (1), then we have the following inequalities
+ µyj |eyj (t)|γ1 + νjy |eyj (t)|γ2 ),
m m m ^ ^ (1) X (1) (1) αij fj (x2 ) − αij fj (x1 ) ≤ |αij ||fj (x2 ) − fj (x1 )|, j=1
j=1
where λxi , λyj , ωix , ωjy , µxi , νix , µyj , νjy are nonnegative real numbers to be determined, γ1 > 1, 0 < γ2 < 1 are constants. For simplicity, we will transform the Eq. (9) and Eq. (10) to the following inequalities according to Lemma 1 and Lemma 2. Obviously, we have
j=1
m m m _ _ (1) X (1) (1) ≤ β f (x ) − β f (x ) |βij ||fj (x2 ) − fj (x1 )|, j 2 j 1 ij ij j=1
j=1
j=1
X ^ n n ^ n (2) (2) (2) ≤ α g (x ) |αji ||gi (x2 ) − gi (x1 )|, α g (x ) − i 1 i 2 ji ji i=1 i=1 i=1 m m m _ X _ (2) (2) (2) ≤ |βji ||gi (x2 ) − gi (x1 )|. g (x ) β g (x ) − β i 1 i 2 ji ji
Fix (t)
≤
(2) If Assumption 1 and fj (±Γj )
Lemma 2: 0 hold, we have the following inequalities
=
(1) 0, gi (±Γi )
=
Gxi (t)
≤ +
|buij (t)fj (vj (t)) − bxij fj (yj (t))| ≤ b∗ij kj |vj (t) − yj (t)|, |avji (t)gi (ui (t)) − ayji (t)gi (xi (t))| ≤ a∗∗ ji li |ui (t) − xi (t)|, −
byji (t)gi (xi (t))|
≤
b∗∗ ji li |ui (t)
− xi (t)|,
i=1
zip ≥ N 1−p
N N N X X p X q zi , ziq ≥ zi . i=1
i=1
Fjy (t)
≤
Gyj (t)
m X
b∗ij kj eyj (t − τ (t)),
j=1 (1)
|αij |kj |eyj (t − τ (t))
m X j=1 n X
(1)
|βij |kj |eyj (t − τ (t)); x a∗∗ ji li ei (t) +
≤
n X
n X
(15)
x b∗∗ ji li ei (t − σ(t))
i=1 (2)
|αji |li |exi (t − σ(t))
i=1
+
n X
(2)
|βji |li |exi (t − τ (t)).
i=1
According to Eq. (14) and Eq. (15), the error system (9) can be rewritten as
e˙ xi (t)
≤ Dix (t) +
m X
kj a∗ij eyj (t)
j=1
(12) +
i=1
Lemma 5: [32] Suppose there exists a continuous radiS cally unbounded function V : Rn → R+ {0} such that 1) V (χ) = 0 ⇒ χ = 0; 2) any solution e(t) of system (9) satisfies V˙ (e(t)) ≤ −aV s (e(t)) − bV r (e(t)) for a, b > 0, s > 1, 0 < r < 1. Then the system (9) can be fixed-time stable, and the settling time can be calculated as follows 1 1 T (e0 (t)) ≤ Tmax = + . (13) a(s − 1) b(1 − r) 8
m X
i=1
(1) (1) (1) (1) where a∗ij = max{|ˆ aij |, |ˇ aij |}, b∗ij = max{|ˆbij |, |ˇbij |}, (2) (2) ˆ(2) ˇ(2) a∗∗ aji |, |ˇ aji |}, b∗∗ ji = max{|ˆ ji = max{|bji |, |bji |}, mj , ni are the same as those in Assumption 1. Proof: Please see Ref. [9] for its proof process. Lemma 3: For Eq.(9), the following inequalities hold ( (1) (1) (1) (1) sgn(exi (t))Dix (t) ≤ −di |exi (t)| + Γi |dˆi − dˇi |, (2) (2) (2) (2) sgn(eyj (t))Djy (t) ≤ −dj |eyj (t)| + Γj |dˆj − dˇj |. (11) Lemma 4: [39] Let the real numbers z1 , z2 , · · · , zN ≥ 0, p > 1, 0 < q ≤ 1, then the following inequalities hold N X
a∗ij kj eyj (t) +
j=1
|auij (t)fj (vj (t)) − axij fj (yj (t))| ≤ a∗ij kj |vj (t) − yj (t)|,
|bvji (t)gi (ui (t))
m X j=1
j=1
i=1
i=1
(14)
m X
(1)
(1)
kj (b∗ij + |αij | + |βij |)eyj (t − τ )
j=1
e˙ yj (t)
−λxi exi (t) − sgn(exi (t))(ιxi + ωix |exi (t − σ(t))| +µxi |exi (t)|γ1 + νix |exi (t)|γ2 ), (16) n X x ≤ Djy (t) + li a∗∗ ji ei (t) i=1
+
n X
(2)
(2)
x li (b∗∗ ji + |αji | + |βji |)ei (t − σ)
i=1
−λyj eyj (t) − sgn(eyj (t))(ιyj + ωjy |eyj (t − τ (t))| +µyj |eyj (t)|γ1 + νjy |eyj (t)|γ2 ). VOLUME 4, 2016
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Theorem 1: Under Assumption 1 and the controllers (14), if the following algebraic conditions hold m X (1) x a∗∗ ji li − (di + λi ) < 0, j=1 n X ∗ (2) aij kj − (dj + λyj ) < 0, i=1 n X (1) (1) (1) (Γi |dˆi − dˇi | − ιxi ) i=1 (17) m X (2) ˆ(2) (2) y ˇ + (Γ | d − d | − ι ) < 0, j j j j j=1 n X (1) (1) kj (b∗ij + |αij | + |βij |) − ωjy < 0, i=1 m X (2) (2) x li (b∗∗ ji + |αji | + |βji |) − ωi < 0.
+
V (t) = V1 (t) + V2 (t),
+
V1 (t) =
n X
|exi (t)|, V2 (t)
=
i=1
m X
− − = + +
D+ V1 (t)
=
n X
sgn(exi (t))e˙ xi (t)
i=1
≤ +
n X i=1 m X
m X Dix (t) + kj a∗ij eyj (t)
sgn(exi (t))
+
+
(1) |αij |
+
(1) |βij |)eyj (t
−
VOLUME 4, 2016
i=1 n X
n X
µxi |exi (t)|γ1 −
i=1 n X i=1 n X
n X
ωix |exi (t − σ(t))|
i=1
νix |exi (t)|γ2
i=1
(1) (−di
(1)
− λxi )|exi (t)| (1)
(Γi |dˆi
n X m X
n X m X
n X
(21)
(1) − dˇi | − ιxi )
a∗ij kj |eyj (t)| −
n X
ωix |exi (t − σ(t))|
i=1 (1)
(1)
kj (b∗ij + |αij | + βij )|eyj (t − τ (t))|
µxi |exi (t)|γ1 −
i=1
D+ V2 (t)
=
n X
n X
νix |exi (t)|γ2 .
i=1
sgn(eyj (t))e˙ yj (t)
j=1 m X
n X x Djy (t) + lj a∗∗ ji ei (t)
sgn(eyj (t))
j=1 n X
i=1 (2)
(2)
x li (b∗∗ ji + |αji | + |βji |)ei (t − σ)
i=1
−λyj eyj (t) − sgn(eyj (t))(ιyj + ωjy |eyj (t − τ (t))| +µyj |eyj (t)|γ1 + νjy |eyj (t)|γ2 ) m X (2) y (2) ˆ(2) (2) ˇ ≤ − dj |ej (t)| + Γj |dj − dj | j=1
+
m X n X
− τ)
+
m X n X
x a∗∗ ji li |ei (t)|
x b∗∗ ji li |ei (t − σ(t))|
j=1 i=1
+
m X n X
(2)
(2)
li (|αji | + βji )|exi (t − σ(t))|
j=1 i=1
−
m X
i=1
j=1
n X m X
m X
i=1 j=1
ιxi −
j=1 i=1
− sgn(exi (t))(ιxi + ωix |exi (t − σ(t))| +µxi |exi (t)|γ1 + νix |exi (t)|γ2 ) n X (1) x (1) ˆ(1) (1) ˇ ≤ − di |ei (t)| + Γi |di − di | a∗ij kj |eyj (t)|
n X
Similarly, the upper right-hand Dini derivative of V2 (t) can be calculated as follows
j=1 −λxi exi (t)
+
λxi |exi (t)| −
i=1 j=1
j=1
kj (b∗ij
n X
i=1 j=1
+
Obviously, V (t) ≥ 0. Calculating the upper right-hand Dini derivative of V1 (t) with respect to t along the solution of (10), we have
(1)
i=1
(20)
j=1
(1)
kj (|αij | + βij )|eyj (t − τ (t))|
i=1
≤ |eyj (t)|.
n X m X i=1 j=1
(19)
where
b∗ij kj |eyj (t − τ (t))|
i=1 j=1
j=1
then the drive MFBAMCNN (1) and the response MFBAMCNN (6) can achieve the fixed-time synchronization. Furthermore, the settling time can be calculated by 1 Tmax = x 1−γ 1−γ 1 mini,j {n µi , m 1 µyj } · 21−γ1 (γ1 − 1) 1 + . x mini,j {νi , νjy }(1 − γ2 ) (18) Proof: Define the Lyapunov function as follows
n X m X
−
λxj |eyj (t)| −
m X
ιyj
j=1
ωjy |eyj (t − τ (t))|
j=1 9
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− =
m X
µyj |eyj (t)|γ1 −
m X
νjy |eyj (t)|γ2
≤ − min{µxi } i
j=1
j=1 m X
(2)
(−dj − λyj )|eyj (t)|
i
+ +
j=1 m X n X j=1 i=1 m X n X
m X
x a∗∗ ji li |ei (t)| −
j
li (b∗∗ ji
+
(2) |αji |
−
m X
+
(2) βji )|eyj (t
|eyj (t)|γ1
j=1 m X
− min{νjy } j
|eyj (t)|γ2
j=1
j=1
j=1 i=1 m X − µyj |eyj (t)|γ1 j=1
1−γ1
≤ − min{n
− τ (t))|
i
µxi }
X n i=1
X γ2 n x x − min{νi } |ei (t)|
νjy |eyj (t)|γ2 .
i
i=1
j=1
1−γ1
X m
µyj }
D+ V (t)
γ2 X m − min{νjy } |exj (t)| .
+ +
j=1 i=1 n m X X
a∗ij kj
j=1 n X
(1)
(1)
i=1 m X
(2)
(2)
−
(2) (dj
+
|eyj (t)|
λyj )
i=1
(Γi |dˆi
(1)
− dˇi | − ιxi ) (2)
(Γj |dˆj − dˇj | − ιyj )
j=1
+
n m X X
(1)
(1)
kj (b∗ij + |αij | + |βij |)
i=1
j=1
y −ωj |ej (t − τ (t))| n X m X (2) (2) + li (b∗∗ ji + |αji | + |βji |) i=1
j=1
x −ωi |ej (t − τ (t))| − −
n X
µxi |exi (t)|γ1 −
n X
i=1
i=1
m X
m X
µyj |eyj (t)|γ1 −
j=1
νix |exi (t)|γ2 νjy |eyj (t)|γ2 .
j=1
According to the condition (16) and Lemma 4, we get
+
D V (t)
≤−
n X
µxi |exi (t)|γ1
i=1
−
m X j=1
µyj |eyj (t)|γ1
−
n X
νix |exi (t)|γ2
i=1
−
m X j=1
νjy |eyj (t)|γ2
j
j
γ1
|eyj (t)|
− min{m
= D+ V1 (t) + D+ V2 (t) m n X X (1) x ∗∗ aji li − (di + λi ) |exi (t)| ≤
γ1
|exi (t)|
Merging Eq.(21) and Eq.(22) into Eq.(18), we have
+
10
m X
− min{µyj }
ωjy |eyj (t − τ (t))|
|exi (t)|γ2
i=1
j=1 m X (2) (2) (2) + (Γj |dˆj − dˇj | − ιyj )
|exi (t)|γ1
i=1 n X
− min{νix }
(22)
n X
j=1
j=1
Let a = mini,j {n1−γ1 µxi , m1−γ1 µyj }, b = mini,j {νix , νjy }, according to the Lemma 4, we have D+ V (t)
≤ −a(V1γ1 (t) + V2γ1 (t)) − b(V1γ2 (t) + V2γ2 (t)) ≤ −a · 21−γ1 V γ1 (t) − bV γ2 (t).
By Lemma 5, the origin of error system (9) is fixe-time stable. Equivalently, the drive MFBAMCNNs (1) and the response system MFBAMACNNs (6) can achieve the fixedtime synchronization. Furthermore, the settling time can be calculated by 1 1 + Tmax = 1−γ 1 a·2 (γ1 − 1) b(1 − γ2 ) 1 = mini,j {n1−γ1 µxi , m1−γ1 µyj } · 21−γ1 (γ1 − 1) 1 . + x mini,j {νi , νjy }(1 − γ2 ) The proof is finished. Remark 3: From the conclusions of Theorem 1, we can see that the controller parameters µxi , µyj , νix , νjy , γ1 , γ2 do not appear in the condition (17), but directly determine the result of the settling time Tmax . That is to say, we do not need to consider them when we select the control parameters that meet the condition (17). Remark 4: According to the settling time formula (18), we know that Tmax is inversely proportional to µxi , µyj , νix , νjy when γ1 and γ2 are fixed. The conclusions of Theorem 1 can be easily extended to common fuzzy BAM cellular neural networks without the memristor. Considering the following drive-response FBAMCNNs m X (1) (1) x˙ i (t) = −di xi (t) + aij fj (yj (t)) j=1 VOLUME 4, 2016
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+ + + +
m X j=1 m ^ j=1 m _
(1)
bij fj (yj (t − τ (t))) +
m X
where sui (t), svj (t) are designed as follows
(1)
sui (t) = −λxi exi (t) − sgn(exi (t))(ωix |exi (t − σ(t))| + µxi |exi (t)|γ1 + νix |exi (t)|γ2 ), svj (t) = −λyj eyj (t) − sgn(eyj (t))(ωjy |eyj (t − τ (t))|
j=1 (1)
αij fj (yj (t − τ (t))) m _
(1)
βij fj (yj (t − τ (t))) +
j=1 m ^
(1)
cij wj
(1)
j=1 (1)
(1)
(1)
Tij wj + Ii ,
(23)
j=1 n X
(2)
y˙ j (t) = −dj yj (t) +
(2)
aji gi (xi (t))
i=1
+ + + +
n X i=1 n ^
(2) bji gi (xi (t
− σ(t))) +
(2) (2) cji wi
+
i=1 (2) αji gi (xi (t
i=1 n _
e˙ yj
(2)
i=1 n ^
n _
(2)
+
m X j=1 m ^ j=1 m _ j=1 m ^
+
m X
+
≤ 0.
(1)
(1)
bij fj (vj (t − τ (t))) +
m X
(1)
(1)
cij wj
j=1 (1)
αij fj (vj (t − τ (t))) m _
(1)
(1)
(1)
Sij wj
j=1 (1)
(1)
(1)
Tij wj + Ii
+ sui (t),
(24)
n X
(2)
aji gi (ui (t))
i=1
+ +
n X
i=1 n _ i=1 n ^ i=1
VOLUME 4, 2016
(26) (2) aji li exi (t)
+
svj (t)
n X
(2) (2) (2) li bji + |αji | + |βji | exi (t − σ(t)).
j=1
βij fj (vj (t − τ (t))) +
(2)
+
+
n X
For Eq. (23) and Eq. (24), we have the following corollary. Corollary 1: Under the controllers (14), if the following algebraic conditions hold m X (2) (1) aji li − di − λxi ) < 0, j=1 n X (1) (2) aij kj − dj − λyj ) < 0, i=1 (27) n X (1) (1) (1) y kj (bij + |αij | + |βij |) − ωj < 0, i=1 m X (2) (2) (2) li (bji + |αji | + |βji |) − ωix < 0.
aij fj (vj (t))
v˙ j (t) = −dj vj (t) +
i=1 n ^
(2) −dj eyj (t)
i=1 (2) Ij , t
j=1
+
≤
(2)
Sji wi
j=1
+
(1) (1) (1) kj bij + |αij | + |βij | eyj (t − τ (t)),
i=1
i=1 (2) (2) Tji wi
(1)
+
m X j=1
− σ(t)))
βji gi (xi (t − σ(t))) +
= −di ui (t) + +
Define the error systems exi (t) = ui (t) − vi (t), eyj (t) = vj (t) − yj (t), i ∈ I, j ∈ J between the drive system (23) and the response system (24), and according to Assumption 1 and Lemma 1, we have m X (1) (1) e˙ xi ≤ −di exi (t) + aij kj eyj (t) + sui (t) j=1
n X
i=1
u˙ i (t)
+ µyj |eyj (t)|γ1 + νjy |eyj (t)|γ2 ),
(1)
Sij wj
(25)
(2)
bji gi (ui (t − σ(t))) +
n X
(2)
(2)
cji wi
then the drive-response FBAMCNNs (23) and (24) can achieve the fixed-time synchronization. Furthermore, the settling time can be calculated by Eq. (18). Proof: The result can be obtained directly by the proof of Theorem 1. We can also easily extend the results to the case without fuzzy logic. Consider the following drive-response memristive BAM cellular neural networks m X (1) (1) x˙ i (t) = −di (xi (t))xi (t) + aij (xi (t))fj (yj (t)) j=1
i=1 (2)
+
αji gi (ui (t − σ(t)))
m X
(1)
bij (xi (t))fj (yj (t − τ (t)))
j=1 (2)
βji gi (ui (t − σ(t))) +
n _
(2)
(2)
+
Sji wi
(2)
(2)
(2)
+ Ij
+ svj (t), t ≤ 0,
(1)
(1)
(1)
cij wj + Ii ,
(28)
j=1
i=1
Tji wi
m X
y˙ j (t)
(2)
= −dj (yj (t))yj (t) +
n X
(2)
aji (yj (t))gi (xi (t))
i=1 11
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+
n X
(2)
IV. NUMERICAL EXAMPLES
(1)
Now, we will validate the effectiveness of Theorem 1 and Corollary 2 by two numerical simulations. Example 1. Consider the following MFBAMCNNs with time-varying delays as the drive system
bji (yj (t))gi (xi (t − σ(t)))
i=1
+
n X
(2)
cji wi
(2)
+ Ij .
i=1
u˙ i (t)
(1)
= −di (ui (t))ui (t) +
m X
(1)
aij (ui (t))fj (vj (t))
j=1
+ +
m X j=1 m X
x˙ i (t)
(1)
(1)
(1)
cij wj + Ii
+ (29) +
(2)
n X
+
i=1 n X
+
+ +
(2) Ij
+
rjv (t), t
(1)
(1)
cij wj +
For Eq.(28) and Eq.(29), we can obtain the following corollary. Corollary 2: Under the controllers (14), if the following algebraic inequalities hold m X (1) x a∗∗ ji li − (di + λi ) < 0, j=1 n X (2) a∗ij kj − (dj + λyj ) < 0, i=1 n X (1) (1) (1) (Γi |dˆi − dˇi | − ιxi ) i=1 (30) m X (2) ˆ(2) (2) y ˇ + (Γ | d − d | − ι ) < 0, j j j j j=1 n X kj b∗ij − ωjy < 0, i=1 m X x li b∗∗ ji − ωi < 0,
2 _
2 ^
(1)
αij fj (yj (t − τ (t)))
j=1 (1)
βij fj (yj (t − τ (t)))
2 _
(1)
(1)
Sij wj +
j=1
≥ 0.
i=1
y˙ j (t)
2 ^
(1)
(1)
(1)
Tij wj + Ii ,
(31)
j=1 (2)
= −dj (yj (t))yj (t) +
2 X
(2)
aji (xi (t))gi (xi (t))
i=1
+
2 X
(2)
bji (yj (t))gi (xi (t − σ(t)))
i=1
+ + +
2 X
(2)
(2)
cji wi
+
n ^
(2)
αji gi (xi (t − σ(t)))
i=1
i=1 2 _
(2)
βji gi (xi (t − σ(t))) +
i=1 2 ^
2 _
(2)
(2)
Sji wi
i=1 (2)
(2)
Tji wi
(2)
+ Ij , t ≥ 0.
i=1
The corresponding response MFBAMCNNs is defined as follows u˙ i (t)
2 X
(1)
= −di (ui (t))ui (t) +
(1)
aij (ui (t))fj (vj (t))
j=1
j=1
then the drive MBAMCNNs (28) and response MBAMCNNs(29) can achieve the fixed-time synchronization. Meanwhile, the settling time can be calculated by Eq. (18). Proof: The result can be obtained directly by the proof of Theorem 1. Remark 5: The research on fixed-time synchronization of neural networks is still in the initial stage, and the related research results are still relatively few. We first studied the fixed-time synchronization problem of drive-response MFBAMCNNs. The nonlinear feedback controller (14) we designed is simple in form and contains more freely adjustable parameters, which can provide convenience for the design of v˙ j (t) the actual controller. 12
2 X
j=1
(2)
bji (vj (t))gi (ui (t − σ(t))) (1) (2) cji wi
(1)
bij (xi (t))fj (yj (t − τ (t)))
j=1
(2)
aji (vj (t))gi (ui (t))
i=1
+
2 X j=1
+ riu (t),
= −dj (vj (t))vj (t) + n X
(1)
aij (xi (t))fj (yj (t))
j=1 (1)
bij (ui (t))fj (vj (t − τ (t)))
j=1
v˙ j (t)
2 X
(1)
= −di (xi (t))xi (t) +
+
2 X
(1)
bij (ui (t))fj (vj (t − τ (t)))
j=1
+
2 X
(1)
(1)
cij wj +
j=1
+
2 _
2 ^
(1)
αij fj (vj (t − τ (t)))
j=1 (1)
βij fj (vj (t − τ (t))) +
j=1
+
2 ^
2 _
(1)
(1)
Sij wj
j=1 (1)
(1)
(1)
Tij wj + Ii
+ riu (t),
(32)
j=1 (2)
= −dj (vj (t))vj (t) +
2 X
(2)
aji (ui (t))gi (ui (t))
i=1 VOLUME 4, 2016
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+ + + +
2 X i=1 2 X i=1 2 _ i=1 2 ^
(2)
(2)
cji wi
+
2 ^
(2)
(2)
(2)
(1)
Γi
i=1 (2)
βji gi (ui (t − σ(t))) +
2 _
(2)
(2)
Sji wi
i=1 (2)
(2)
Tji wi
(2)
+ Ij
(1)
+ rjv (t), t ≥ 0. (2)
1.2, |x1 | ≤ 1, (1) 1.5, |x2 | ≤ 1, d2 (x2 ) = 1.0, |x1 | > 1, 1.2, |x2 | > 1, 1.0, |y1 | ≤ 2, (2) 1.5, |y2 | ≤ 2, (2) d (y2 ) = d1 (y1 ) = 1.2, |y1 | > 2, 2 1.1, |y2 | > 2,
d1 (x1 ) =
−1.5, |x1 | ≤ 1, (1) −0.8, |x1 | ≤ 1, a12 (x1 ) = −1.8, |x1 | > 1, −0.5, |x1 | > 1, −0.4, |x2 | ≤ 1, (1) −1.8, |x2 | ≤ 1, (1) a21 (x2 ) = a (x2 ) = 0.4, |x2 | > 1, 22 −1.5, |x2 | > 1, −0.5, |y | ≤ 2, −0.5, |y1 | ≤ 2, 1 (2) (2) a11 (y1 ) = a (y1 ) = 0.5, |y1 | > 2, 12 0.5, |y1 | > 2, −0.6, |y2 | ≤ 2, (2) −0.8, |y2 | ≤ 2, (2) a21 (y2 ) = a22 (y2 ) = 0.6, |y2 | > 2, 0.8, |y2 | > 2,
a11 (x1 ) =
−1.2, |x1 | ≤ 1, (1) −1.2, |x1 | ≤ 1, b12 (x1 ) = −1.5, |x1 | > 1, −0.8, |x1 | > 1, −0.1, |x2 | ≤ 1, (1) −1.6, |x2 | ≤ 1, (1) b21 (x2 ) = b (x2 ) = 0.2, |x2 | > 1, 22 −1.2, |x2 | > 1, 1.5, |y | ≤ 2, −1.2, |y1 | ≤ 2, 1 (2) (2) b11 (y1 ) = b (y1 ) = 1.2, |y1 | > 2, 12 1.5, |y1 | > 2, 1.1, |y2 | ≤ 2, (2) 2.8, |y2 | ≤ 2, (2) b21 (y2 ) = b22 (y2 ) = 1.5, |y2 | > 2, 2.2, |y2 | > 2, (1)
(2)
ϕ2 (s) = ψ2 (s) = 0.6 − 0.2cos(t);
(2)
αji gi (ui (t − σ(t)))
Let the switching jumps Γi = 1, Γj = 2, then the memristor weights parameters of the drive-response MFBAMCNNs (31) and (32) are selected as follows
(1)
(1)
ϕ1 (s) = ψ1 (s) = −0.4 + 0.1sin(t);
i=1
(1)
(1)
ϕ2 (s) = ψ2 (s) = −0.2 + 0.1cos(t);
(2)
bji (vj (t))gi (ui (t − σ(t)))
= 1; Γ2j = 2;
In the controller (14), to ensure the conditions (17) hold, we choose the control parameters λx = (2, 2)T , ιx = (2, 2)T , ω x = (2.5, 2.5)T , µx = (6, 6)T , ν x = (2, 2)T , λy = (2, 2)T , ιy = (2, 2)T , ω y = (2.5, 2.5)T , µy = (6, 6)T , ν y = (2, 2)T , γ1 = 1.2, γ2 = 0.6, li = kj = 0.5. Next, we will verify the conditions (17) of Theorem 1. According to the above parametervalues, we 1.8 0.8 0.5 0.5 ∗ ∗∗ have (aji )2×2 = , (aij )2×2 = , 0.4 1.8 0.6 0.8 1.5 1.2 1.5 1.5 (b∗ji )2×2 = , (b∗∗ ij )2×2 = 1.5 2.8 . By simple 0.2 1.6 calculation, these parameters make the conditions (17) of Theorem 1 hold. With the help of Matlab, we can solve the numerical solutions of the drive-response MFBAMCNNs (31) and (32). Fig. 4 shows the phase diagrams of the drive MFBAMCNNs (31). Fig. 5 presents the state trajectories between the drive MFBAMCNNs (31) and the response MFBAMCNNs (32). Fig. 6 illustrates the error curves of the drive MFBAMCNNs (31) and the response MFBAMCNNs (32). We calculate the settling time Tmax = 3.7154 based on Eq. (18) in Theorem 1. Example 2. Consider the following common driveresponse MBAMCNNs without fuzzy logic
b11 (x1 ) =
(1)
et ; τ (t) = σ(t) = 1 + et fj (x) = tanh(|x| − 1); gi (x) = tanh(|x| − 2); 0.4 0.2 −0.2 −0.02 (1) (2) (1) (2) C =C = ;α = α = ; 0.2 0.4 −0.01 −0.1 −0.1 −0.01 0.1 0.2 β (1) = β (2) = ; S (1) = S (2) = ; −0.1 −0.1 0.2 0.1 0.2 0.1 T (1) = T (2) = ; I (1) = I (2) = (0, 0)T ; 0.1 0.2 w
(1) ϕ1 (s) VOLUME 4, 2016
=
=w
(2)
(1) ψ1 (s)
(1)
= −di (xi (t))xi (t) + +
= 0.1 + 0.2sin(t);
(1)
aij (xi (t))fj (yj (t))
2 X
(1)
(1)
bij (xi (t))fj (yj (t − τ (t))) + Ii ,
(33)
j=1
y˙ j (t)
(2)
= −dj (yj (t))yj (t) +
2 X
(2)
aji (yj (t))gi (xi (t))
i=1
+
2 X
(2)
(2)
bji (yj (t))gi (xi (t − σ(t))) + Ij .
i=1
The corresponding response system is shown below u˙ i (t)
(1)
= −di (ui (t))ui (t) +
2 X
(1)
aij (ui (t))fj (vj (t))
j=1
+
2 X
(1)
(2)
bij (ui (t))fj (vj (t − τ (t))) + Ii
j=1 +riu (t),
v˙ j (t)
(2)
= −dj (vj (t))vj (t) +
(34) 2 X
(2)
aji (vj (t))gi (ui (t))
i=1
T
= (1.2, 1.2) ;
2 X j=1
(2)
Note: In the definitions of bji (yj ) and bji (xi ), xi and yj are denoted as xi (t − σ(t)) and yj (t − τ (t)) in order to simplify the expressions. Other parameters are shown below.
(1)
x˙ i (t)
+
2 X
(2)
(2)
bji (vj (t))gi (ui (t − σ(t))) + Ij
i=1 13
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2.5
2
2
1.5
1.5 1
1
y 2 (t)
x 2 (t)
0.5 0.5
0 0
-0.5 -0.5
-1
-1
-1.5 -4
-3
-2
-1
0
1
-1.5 -1.5
2
-1
-0.5
x 1 (t)
0
0.5
1
1.5
2
2.5
y 1 (t)
FIGURE 4. The phase diagrams of the drive MFBAMCNNs (31).
2.5
2 x1
x2
2
u1
1
u2
1.5 1
x2 & u 2
x1 & u 1
0
-1
0.5 0
-2
-0.5 -3
-1
-4
-1.5 0
5
10
15
20
0
5
t
10
15
20
t
2.5
2
2
1.5
y2
1.5
v2
1
y2 & v2
y1 & v1
1 0.5
0.5 0
0 -0.5
-0.5
-1
y1
-1
v1
-1.5
-1.5 0
5
10
t
15
20
0
5
10
15
20
t
FIGURE 5. The state trajectories between the drive MFBAMCNNs (31) and the response MFBAMCNNs (32) under the controllers (14).
14
VOLUME 4, 2016
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0.5
0.2
0.4
0.1 0
0.3
e x2
e x1
-0.1 0.2
-0.2 0.1 -0.3 0
-0.4
-0.1
-0.5 0
5
10
15
20
0
5
10
15
20
15
20
t
0.5
0.1
0.4
0
0.3
-0.1
e y2
e y1
t
0.2
-0.2
0.1
-0.3
0
-0.4
-0.1
-0.5 0
5
10
15
20
0
5
10
t
t
FIGURE 6. The state error curves of the drive MFBAMCNNs (31) and the response MFBAMCNNs (32) under the controllers (14).
+rjv (t), (1)
(2)
where the switching jumps Γi = 1; Γj memristor weights are given as follows
= 2 and the
1.2, |x1 | ≤ 1, (1) 1.5, |x2 | ≤ 1, d2 (x2 ) = 1.5, |x1 | > 1, 2.0, |x2 | > 1, 1.1, |y1 | ≤ 2, (2) 1.05, |y2 | ≤ 2, (2) d1 (y1 ) = d (y2 ) = 1.3, |y1 | > 2, 2 1.25, |y2 | > 2,
(1)
d1 (x1 ) =
−1.5, |x1 | ≤ 1, (1) −0.6, |x1 | ≤ 1, a12 (x1 ) = −1.6, |x1 | > 1, 0.4, |x1 | > 1, 0.4, |x2 | ≤ 1, (1) 0.4, |x2 | ≤ 1, (1) a21 (x2 ) = a (x2 ) = 0.35, |x2 | > 1, 22 0.42, |x2 | > 1, 0.2, |y1 | ≤ 2, (2) 0.56, |y1 | ≤ 2, (2) a11 (y1 ) = a12 (y1 ) = 0.25, |y1 | > 2, 0.61, |y1 | > 2, 0.2, |y2 | ≤ 2, (2) 0.5, |y2 | ≤ 2, (2) a21 (y2 ) = a (y2 ) = 0.25, |y2 | > 2, 22 0.56, |y2 | > 2, (1)
a11 (x1 ) =
−1.1, |x1 | ≤ 1, (1) −1.2, |x1 | ≤ 1, b12 (x1 ) = −1.3, |x1 | > 1, −1.05, |x1 | > 1, −0.8, |x2 | ≤ 1, −1.5, |x2 | ≤ 1, (1) (1) b21 (x2 ) = b22 (x2 ) = −1.3, |x2 | > 1, −0.9, |x2 | > 1, 0.45, |y1 | ≤ 2, (2) 0.85, |y1 | ≤ 2, (2) b11 (y1 ) = b (y1 ) = 0.38, |y1 | > 2, 12 1.0, |y1 | > 2, −0.60, |y | ≤ 2, 1.6, |y2 | ≤ 2, 2 (2) (2) b21 (y2 ) = b (y2 ) = 0.65, |y2 | > 2, 22 1.4, |y2 | > 2, (1)
b11 (x1 ) =
VOLUME 4, 2016
We assume that the memristor weights of the response system (34) is the same as that of the drive system (33). Other parameters are shown below τ (t) = 0.4 + 0.1cos(t); σ(t) = 0.6 + 0.1sin(t); fj (x) = tanh(|x| − 1); gi (x) = tanh(|x| − 2); (1)
(1)
(2)
(2)
(1)
(1)
ϕ1 (s) = ψ1 (s) = −0.2; ϕ2 (s) = ψ2 (s) = 0.6; (2)
(2)
ϕ1 (s) = ψ1 (s) = 0.4; ϕ2 (s) = ψ2 (s) = −0.8; Similarly, we choose the control parameters λx = (2, 2)T , ιx = (2, 2)T , ω x = (1, 1)T , µx = (4, 4)T , ν x = (2, 2)T , λy = (2, 2)T , ιy = (2, 2)T , ω y = (1, 1)T , µy = (4, 4)T , ν y = (2, 2)T , γ1 = 1.5, γ2 = 0.8. Next, we will verify the conditions (30) of Corollary 2. According to the 1.6 0.6 ∗ above parameter values, we have (aji )2×2 = , 0.4 0.42 0.45 1.0 1.3 1.2 (a∗∗ = , (b∗ji )2×2 = , ij )2×2 0.65 1.6 1.5 0.9 0.45 1.0 (b∗∗ ij )2×2 = 0.65 1.6 . The above parameters make the conditions (30) of Corollary 2 hold. Here is our simulation results. Fig.8 presents the state trajectories of the drive MBAMCNNs (33) and the response MBAMCNNs (34) under the controllers (14). Fig. 9 illustrates the state error curves between the driveresponse MBAMCNNs (33) and (34) under the controllers 15
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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2018.2805183, IEEE Access Author et al.: Preparation of Papers for IEEE TRANSACTIONS and JOURNALS
2.5
2
2
1.5
1.5 1
1
y 2 (t)
x 2 (t)
0.5 0.5
0 0
-0.5 -0.5
-1
-1
-1.5 -4
-3
-2
-1
0
1
-1.5 -1.5
2
-1
-0.5
0
x 1 (t)
0.5
1
1.5
2
2.5
y 1 (t)
FIGURE 7. The phase diagrams of the drive system (33).
1.5
0.8 x1
x2
u1
1
u2
0.6 0.4
x2 & u 2
x1 & u 1
0.5
0
0.2 0
-0.5 -0.2 -1
-0.4
-1.5
-0.6 0
5
10
15
20
0
5
t
10
0.8
20
0.8 y1
0.6
y2
0.6
v1
0.4
0.4
0.2
0.2
y2 & v2
y1 & v1
15
t
0
0
-0.2
-0.2
-0.4
-0.4
-0.6
-0.6
-0.8
v2
-0.8 0
5
10
t
15
20
0
5
10
15
20
t
FIGURE 8. The state trajectories of the drive-response MBAMCNNs (33) and (34) under the controllers (14).
16
VOLUME 4, 2016
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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2018.2805183, IEEE Access Author et al.: Preparation of Papers for IEEE TRANSACTIONS and JOURNALS
0.4
0.5
0.2 0
e x2
e x1
0 -0.5
-0.2 -1 -0.4
-0.6
-1.5 0
5
10
15
20
0
5
t
10
15
20
15
20
t
0.2
0.1
0
0
-0.2
-0.1
-0.4
e y2
e y1
-0.2 -0.6
-0.3 -0.8 -0.4
-1
-0.5
-1.2
-0.6
-1.4 0
5
10
15
20
0
5
t
10
t
FIGURE 9. The state error curves between the drive-response MBAMCNNs(31) and (32) under the controllers (14).
(14). We can also calculate the settling time Tmax = 3.2500 based on Eq. (18). Remark 6: In the existing literature on fixed-time synchronization of neural networks [28], [33], [34], the model and the fixed-time synchronization controllers proposed in this paper are more general. The proof process of this paper is easier to understand. Meanwhile, it can be seen from the two numerical examples that the settling-time can be easily calculated.
time stability or synchronization of the MFBAMCNN model. For example, some uncertainty, stochastic phenomena or various impulsive is inevitable in practical applications [40]– [44]. Therefore, it may be an interesting problem to study the MFBAMCNN system with uncertainty or stochastic phenomena. Moreover, how to design a continuous fixed-time synchronization controller, or an adaptive controller similar to that in Refs. [45], [46] is also a problem worth studying in the future. REFERENCES
V. CONCLUSION
In this paper, we address the fixed-time synchronization for the drive-response MFBAMCNNs with time-varying delays. MFBAMCNNs are a class of neural networks that combine FCNN with MBAMNN, and have more complex structure and dynamic behavior. Most of previous work did not involved such neural networks. Since the discontinuous rightsides of MFBAMCNNs, we deal with the memristive connection weights by using differential inclusion and set-valued map theory. By the definition of fixed-time synchronization, inequality techniques, and Lyapunov stability theory, some novel criteria are derived to ensure the fixed-time synchronization of the drive-response MFBAMCNNs based on nonlinear feedback controllers. In addition, the settling time can be obtained by simple calculation. At last, the effectiveness of our results is validated by two numerical examples. So far, there are still many worthwhile studies on the fixedVOLUME 4, 2016
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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2018.2805183, IEEE Access Author et al.: Preparation of Papers for IEEE TRANSACTIONS and JOURNALS
MINGWEN ZHENG received the B.S. degree in School of Science from Shan Dong University of Technology, ZiBo, China in 2002, and the M.S. degree in School of Computer and Communication Engineering from China University of Petroleum, Dong Ying, China, in 2009. Since 2014, he is going on study as a Ph.D student Beijing University of Posts and Telecommunications, Beijing, China. His current research interests include complex dynamical network, memristor-based neural network and fractional-order systems.
YIXIAN YANG received the M.S. degree in applied mathematics in 1986 and the Ph.D. degree in electronics and communication systems in 1988 from Beijing University of Posts and Telecommunications, Beijing, China. He is the Managing Director of information security center, Beijing University of Posts and Telecommunications, Beijing, China. The Yangtze River scholar Program professor, National Outstanding Youth Fund winners, the National Teaching Masters. Major in coding and cryptography, information and network security, signal and information processing; having authored more than 40 national and provincial key scientific research project, published more than 300 high-level papers and 20 monographs.
LIXIANG LI received the M.S. degree in circuit and system from Yanshan University, Qinhuangdao, China, in 2003, and the Ph.D. degree in signal and information processing from Beijing University of Posts and Telecommunications, Beijing, China, in 2006. The National Excellent Doctoral theses winner, the New Century Excellent Talents in university, Henry Fok education foundation, Hong Kong Scholar Award winner, Beijing higher education program for young talents winner. Visiting Potsdam Germany in 2011. Engaged in research of swarm intelligence and network security; having more than 80 papers and a monograph published and being supported by 10 national foundations in recent five years. She is currently a professor at the School of Computer Science and Technology, Beijing University of Posts and Telecommunications, China. Her research interests include swarm intelligence, information security and network security. Dr. L.Li is the co-author of 70 scientific papers and 10 Chinese patents.
PENGHAI PENG received the M.S. degree in system engineering from Shenyang University of Technology, Shenyang, China, in 2006, and the Ph.D. degree in signal and information processing from Beijing University of Posts and Telecommunications, Beijing, China, in 2010. He is currently an associate professor at the School of Computer Science and Technology, Beijing University of Posts and Telecommunications, China. His research interests include information security, network security, complex networks and control of dynamical systems. Dr. H. Peng is the co-author of 50 scientific papers and over 10 Chinese patents.
YANPING ZHANG received the B.S. degree in College of mathematics and science from QuFu Normal University, Shandong, China in 2003, and the M.S. degree in School of mathematics from Southeast University, Nanjing, China, in 2006. Since 2006, she is a lecture in Shandong University of Technology, Zibo, China. Her current research interests include fuzzy mathematics, fuzzy sets.
JINGHUA XIAO received the B.Sc, M.Sc and Ph.D.degrees in Physics from Beijing Normal University, Beijing, China in 1987, 1990 and 1999. He is currently a professor at Beijing University of Posts and Telecommunications, Beijing, China. His research interests include nonlinear dynamics, complex networks. He has authored over 70 refereed papers in international journals and five of them were published in PRL.
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