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Finitetime control and synchronization for memristorbased chaotic system via impulsive adaptive strategy
Advances in Difference Equations volume 2016, Article number: 101 (2016)
Abstract
This paper investigates the stabilizing and synchronization problems of a memristorbased Chua chaotic system in a finite time. A lemma concerning the finitetime stability for an impulsive system is proposed by extending the finitetime stability theory. Then some finitetime stabilizing and synchronization criterion are presented which guarantee the finitetime stabilization and synchronization for the model considered. Finally, the efficiency of the control scheme is further demonstrated by the simulation examples.
Introduction
In 1971, the memristor which is considered to be the missing fourth passive circuit element was postulated [1]. However, this important postulation has not caused attention in almost 40 years. Until in 2008, HewlettPackard Labs announced the development of a memristor based on nanotechnology [2]. As we know, the memristor takes its place along with the other three existing elements: the resistor, the capacitor, and the inductor. Increasing focus was put on the memristor for its potential applications in programmable logic, signal processing, neural networks, and so on [3].
Moreover, as the novel element, the circuit based on the memristor shares many interesting phenomenon. Recently, the research memristor chaotic circuits have become a focal topic [4–10]. In [4], the author presented a novel fourthorder memristorbased Chua oscillator by replacing Chua’s diode with an active twoterminal circuit. The stabilization problem of a memristorbased chaotic system was investigated in [10]. As the most important phenomenon, the synchronization was also discussed [9]. In [9], the adaptive synchronization problem of memristorbased Chua circuits was investigated.
As time goes on, more and more researchers began to realize the important role of the synchronization time. To attain a high convergence speed, many effective methods have been introduced and finitetime control is one of them. Finitetime synchronization means the optimality in convergence time. Much research work has been done on chaos synchronization based on finite time (see for instance [11–22] and the references therein). However, the finitetime synchronization problem has not been fully investigated in the literature, and it still remains open. Motivated by the above discussion, we investigate the finitetime synchronization problem for a memristorbased Chua circuit. Based on the finitetime stability theory, a novel lemma which guarantees the impulsive system is finitetime stable is presented. Then the impulsive [16–18] adaptive control law is proposed to realize finitetime synchronization of the model considered. Numerical simulations demonstrate the effectiveness and correctness of this results.
The paper is organized as follows. Some preliminaries are presented in the next section. Section 3 proposes the main results of this paper. In Section 4, the numerical simulations are presented, which is followed by the conclusion in Section 5.
Preliminaries
In [6], the author proposed a novel nonlinear circuits, with a fluxcontrolled memristor which replaces the Chua diode. Figure 1 shows a memristorbased Chua oscillator with a fluxcontrolled memristor.
Applying a Kirchhoff voltage, the current law, and the voltampère relationship of the components, the state equation of a Chua memristorbased chaotic system can be described as follows:
For convenience, letting \(x_{1} = u_{1}\), \(x_{2} = u_{2}\), \(x_{3} = i_{3}\), \(x_{4} = \phi\), \(\alpha = 1/C_{1}\), \(\beta = 1/L_{1}\), \(\gamma = r/L\), \(\xi = G\), \(C_{2} = 1\), and \(R = 1\), then the model can be rewritten as the following equation:
If we set \(\alpha = 10\), \(\beta = 100 / 7\), \(\gamma = 0.1\), \(\xi = 9 / 7\), \(a = 1 / 7\) and \(b = 2 / 7\), and the initial values are \(( 10^{  10},0,0,  0.515 )\), then system (2) is a chaotic system and the chaotic attractor is shown in Figure 2.
In order to compute simply, letting \(x = [ x_{1},x_{2},x_{3},x_{4} ]^{T}\), then system (2) can be described as follows:
where
and \(W ( x_{4} ) = a + 3bx_{4}^{2}\), a, b, γ, ξ, α, and β are positive constants.
Similar to [7], the nonlinear functions \(q ( \phi )\), \(W ( \phi )\) are given by
Throughout this paper, the following assumption and lemma are necessary for our main results.
Assumption 1
System (2) is a chaotic system, namely, the state is bound, we assume that the following assumptions hold:
where \(M_{1}\), \(M_{2}\) are real constants.
Lemma 1
Suppose the function is continuous and nonnegative when \(t \in [ 0,\infty )\) and satisfies the following conditions:
where \(\rho > 0\), \(0 < \eta < 1\), \(0 < \delta < 1\), \(k = \{ 1,2,\ldots,m\}\), is a finite natural number set and m is a positive integer, then the following inequality holds:
where T is a constant which represents the setting time.
Proof
Without loss of generality, let \(t_{0} = 0\). In order to prove (7) holds, the following function \(H ( t )\) is constructed:
Clearly, if the function \(H ( t )\) satisfies \(H ( t ) \le 0\), then the equality (7) holds.
One can easily observe that
Next, we will prove that \(H ( t ) \le 0\) holds for \(t \in [ t_{0},t_{1} ]\). Otherwise, there exists \(t^{ *}\) such that
Combing (10) with (11), one has
which contradicts (11). Namely, \(H ( t ) \le 0\) holds for \(t \in [ t_{0},t_{1} )\).
When \(t = t_{1}\), we get
It yields
Then we suppose that \(H ( t ) \le 0\) holds for \(t \in [ t_{k  1},t_{k} ]\). For \(t \in [ t_{k},t_{k + 1} ]\), we have
From (8)(15), we know that \(H ( t ) \le 0\) holds. This completes the proof. □
Main results
In this section, the finitetime control and synchronization problems via an impulsive adaptive strategy are investigated, respectively. Taking the impulsive adaptive strategy into account in (3), one obtains
where \(\Delta x ( t_{k} ) = x ( t_{k}^{ +} )  x ( t_{k}^{ } )\), \(x ( t_{k}^{ +} ) = \lim_{t \to t_{k}^{ +}} x ( t )\), \(x ( t_{k}^{ } ) = \lim_{t \to t_{k}^{ }} x ( t )\), \(\ell = \{ 1,2, \ldots, n,n_{1}, \ldots, n_{k} \}\), is a finite natural number set. For simplicity, it is assumed that \(x ( t_{k}^{ } ) = x ( t_{k} )\), which means that \(x ( t_{k} )\) is left continuous. Letting \(u ( t ) =  k_{1}x ( t )  k_{2}\operatorname{sign} ( x ( t ) )\vert x ( t ) \vert ^{\gamma}\), we have the following theorem.
Theorem 1
Suppose Assumption 1 holds. There exists a positive constant γ satisfying \(0 < \gamma < 1\) such that the memristorbased chaotic system is finitetime stable if the following conditions hold:

(i)
\(q = \lambda_{\max} [ A^{T} + A  ( 2k_{1} + 1 )I ] < 0\);

(ii)
\(d = \lambda_{\max} ( I + B )^{T} ( I + B ) < 1\).
Proof
Construct the following Lyapunov candidate function:
Calculating the derivative along the trajectory of (16) we have
From Assumption 1, one has
where \(J = [ [ a + bM_{4}^{2} ]^{2},0,0,0 ]^{T}\).
From (18) and (19), one obtains
where \(q = \lambda_{\max} [ A^{T} + A  ( 2k_{1} + 1 )I ]\). From the fact that \(0 < \gamma < 1\), one obtains
Also
Then, from condition (1) of Theorem 1, we have for \(t \ne t_{k}\)
When \(t = t_{k}\), one obtains
Through condition in Theorem 1 and Lemma 1, we can easily see that the system (16) is finitetime stable, which complete the proof. □
Next, we investigate the problem of finitesynchronization for a memristorbased chaotic system. Based on the driveresponse synchronization concept, letting system (3) be the drive system, the response system with control input u is as follows:
Taking the impulsive adaptive effects into account, the response system (25) is as follows:
where \(e = [ x  y ]^{T} = [ x_{1}  y_{1},x_{2}  y_{2},x_{3}  y_{3},x_{4}  y_{4} ]^{T}\), \(t_{k}\) are the impulsive instants which satisfy \(t_{1} < t_{2} < \cdots < t_{k  1} < t_{k}\) and \(\lim_{k \to \infty} t_{k} = \infty\). Letting
where the constants \(k_{1}\), \(k_{2}\) are the control strength coefficients to be designed, the real number γ satisfies \(0 \le \gamma < 1\).
Combing (3) with (26), the error system can be described as follows:
where \(\phi ( e ) = [\alpha W ( y_{4} )y_{1}  \alpha W ( x_{4} )x_{1},0,0,0]^{T}\). Hence, we have the following theorem.
Theorem 2
Suppose Assumption 1 holds. There exists a positive constant γ satisfying \(0 < \gamma < 1\) such that the memristorbased Chua systems (3) and (26) can be synchronized under the impulsive adaptive strategy if the following conditions hold:

(i)
\(A^{T} + A  2k_{1}I  2abM_{1}M_{2}I < 0\);

(ii)
\(d = \lambda_{\max} ( I + B )^{T} ( I + B ) < 1\).
Proof
Construct the following Lyapunov candidate function
For \(t \in [ t_{k},t_{k + 1} )\), the derivative of \(V ( t )\) along the trajectory of (28) is
From Assumption 1, we have
Combing (30) with (31), we have
From the fact that \(0 < \gamma < 1\), one obtains
Also
Thus, based on the condition (i) in Theorem 2, we have for \(t \ne t_{k}\)
When \(t = t_{k}\), one obtains
Through Theorem 2 and Lemma 1, we can easily see that the system (28) is finitetime stable. This implies that the response system (26) is synchronized with the drive system (3) in a finite time. □
Simulation results
The numerical simulations are carried out using the fourthorder RungeKutta method. The initial states of the drive and response systems are \(( 10^{  10},0,0,0 )\) and \(( 0,0,0,0 )\). The parameters of the drive systems are \(\alpha = 10\), \(\beta = 100 / 7\), \(\gamma = 0.1\), \(\xi = 9 / 7\), \(a = 1 / 7\), and \(b = 2 / 7\). Solving the inequality in Theorem 2, and choosing \(B = \operatorname{diag}(  0.9,  0.9,  0.9,  0.9)\), \(k_{1} = 0.02\), \(k_{2} = 0.01\), \(\gamma = 0.3\), the response system synchronizes with the drive system as shown in Figure 3. It is easily shown that the state response curve of the error system is stable.
Conclusion
In this paper, the finitetime control and synchronization problems of memristorbased chaotic systems have been investigated. Some novel impulsive adaptive control laws which guarantee the memristorbased Chua circuits is stabilized and synchronized in finite time have been proposed. Moreover, simulation results were given to verify the effectiveness and feasibility of the method. Our future research topics mainly consider the time delay effects on the finitetime stability of the memristorbased nonlinear system.
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Acknowledgements
The work described in this paper was partially supported by the National Natural Science Foundation of China (Grant No. 61403050) and the Scientific and Technological Research Program of Chongqing Municipal Education Commission (KJ1501412, KJ1501409, KJ1501301), and the Foundation of CQUE (KY201519B, KY201520B).
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Keywords
 finite time
 control and synchronization
 memristor
 impulsive adaptive strategy