DB Strukov

F Amato

FM Bayat

HB Bao

HG Wu

JD Cao

JJ Huang

Junjian Huang

LO Chua

M Itoh

PB Sanjay

QK Song

R Pakkiyappan

SP Wen

Wei Xiong

X Li

X Wang

YG Liu

Advances in Difference Equations

English

Crossref

Xiong and Huang Advances in Diﬀerence Equations  (2016) 2016:101 DOI 10.1186/s13662-016-0789-3RESEARCH Open AccessFinite-time control and synchronizationfor memristor-based chaotic system viaimpulsive adaptive strategyWei Xiong1 and Junjian Huang2**Correspondence:jjhuangcqe@163.com2Department of Computer Science,Chongqing University of Education,Chongqing, 400067, P.R. ChinaFull list of author information isavailable at the end of the articleAbstractThis paper investigates the stabilizing and synchronization problems of amemristor-based Chua chaotic system in a ﬁnite time. A lemma concerning theﬁnite-time stability for an impulsive system is proposed by extending the ﬁnite-timestability theory. Then some ﬁnite-time stabilizing and synchronization criterion arepresented which guarantee the ﬁnite-time stabilization and synchronization for themodel considered. Finally, the eﬃciency of the control scheme is furtherdemonstrated by the simulation examples.Keywords: ﬁnite time; control and synchronization; memristor; impulsive adaptivestrategy1 IntroductionIn , the memristor which is considered to be the missing fourth passive circuit ele-ment was postulated []. However, this important postulation has not caused attention inalmost  years. Until in , Hewlett-Packard Labs announced the development of amemristor based on nanotechnology []. As we know, the memristor takes its place alongwith the other three existing elements: the resistor, the capacitor, and the inductor. In-creasing focus was put on the memristor for its potential applications in programmablelogic, signal processing, neural networks, and so on [].Moreover, as the novel element, the circuit based on the memristor shares many in-teresting phenomenon. Recently, the research memristor chaotic circuits have becomea focal topic [–]. In [], the author presented a novel fourth-order memristor-basedChua oscillator by replacing Chua’s diode with an active two-terminal circuit. The stabi-lization problemof amemristor-based chaotic systemwas investigated in []. As themostimportant phenomenon, the synchronization was also discussed []. In [], the adaptivesynchronization problem of memristor-based Chua circuits was investigated.As time goes on, more and more researchers began to realize the important role ofthe synchronization time. To attain a high convergence speed, many eﬀective methodshave been introduced and ﬁnite-time control is one of them. Finite-time synchroniza-tion means the optimality in convergence time. Much research work has been done onchaos synchronization based on ﬁnite time (see for instance [–] and the references© 2016 Xiong and Huang. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in anymedium, pro-vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, andindicate if changes were made.Xiong and Huang Advances in Diﬀerence Equations  (2016) 2016:101 Page 2 of 9Figure 1 Chua’s memristor-based chaotic circuitswith a ﬂux-controlled memristor.therein). However, the ﬁnite-time synchronization problem has not been fully investi-gated in the literature, and it still remains open. Motivated by the above discussion, weinvestigate the ﬁnite-time synchronization problem for a memristor-based Chua circuit.Based on the ﬁnite-time stability theory, a novel lemma which guarantees the impulsivesystem is ﬁnite-time stable is presented. Then the impulsive [–] adaptive control lawis proposed to realize ﬁnite-time synchronization of the model considered. Numericalsimulations demonstrate the eﬀectiveness and correctness of this results.The paper is organized as follows. Some preliminaries are presented in the next section.Section  proposes the main results of this paper. In Section , the numerical simulationsare presented, which is followed by the conclusion in Section .2 PreliminariesIn [], the author proposed a novel nonlinear circuits, with a ﬂux-controlled memristorwhich replaces the Chua diode. Figure  shows a memristor-based Chua oscillator with aﬂux-controlled memristor.Applying a Kirchhoﬀ voltage, the current law, and the volt-ampère relationship of thecomponents, the state equation of a Chua memristor-based chaotic system can be de-scribed as follows:⎧⎪⎪⎪⎨⎪⎪⎪⎩u˙ = C (u – u +Gu –W (φ)u),u˙ = C (u – u) + Ri,i˙ = – Lu +rL i,φ˙ = u.()For convenience, letting x = u, x = u, x = i, x = φ, α = /C, β = /L, γ = r/L, ξ =G,C = , and R = , then the model can be rewritten as the following equation:⎧⎪⎪⎪⎨⎪⎪⎪⎩x˙ = α(x – x + ξx –W (x)x),x˙ = x – x + x,x˙ = –βx + γ x,x˙ = x.()If we set α = , β = /, γ = ., ξ = /, a = / and b = /, and the initial values are(–, , ,–.), then system () is a chaotic system and the chaotic attractor is shownin Figure .In order to compute simply, letting x = [x,x,x,x]T , then system () can be describedas follows:x˙ = Ax + φ(x), ()Xiong and Huang Advances in Diﬀerence Equations  (2016) 2016:101 Page 3 of 9Figure 2 The chaotic attractor of thememristor-based Chua circuit.whereA =⎡⎢⎢⎢⎣α(ξ – ) α   –   –β γ    ⎤⎥⎥⎥⎦, φ(x) =⎡⎢⎢⎢⎣–αW (x)x⎤⎥⎥⎥⎦,andW (x) = a + bx, a, b, γ , ξ , α, and β are positive constants.Similar to [], the nonlinear functions q(φ),W (φ) are given byq(φ) = aφ + bφ, ()W (φ) = dq(φ)dφ = a + bφ. ()Throughout this paper, the following assumption and lemma are necessary for our mainresults.Assumption  System () is a chaotic system, namely, the state is bound, we assume thatthe following assumptions hold:|x| ≤M, |x| ≤M,whereM,M are real constants.Lemma  Suppose the function is continuous and non-negative when t ∈ [,∞) and sat-isﬁes the following conditions:{V˙ (t)≤ –ρV η(t), t = tk ,V (t+k )≤ δV (tk), k = , , . . . ,()where ρ > ,  < η < ,  < δ < , k = {, , . . . ,m}, is a ﬁnite natural number set and m is apositive integer, then the following inequality holds:V –η(t)≤ V –η(t) – ρ( – η)(t – t), t ≤ t ≤ T , ()where T is a constant which represents the setting time.Xiong and Huang Advances in Diﬀerence Equations  (2016) 2016:101 Page 4 of 9Proof Without loss of generality, let t = . In order to prove () holds, the following func-tion H(t) is constructed:H(t) = V –η(t) –V –η() + ρ( – η)t. ()Clearly, if the function H(t) satisﬁes H(t)≤ , then the equality () holds.One can easily observe thatH() = V –η() –V –η()≤ . ()Next, we will prove that H(t)≤  holds for t ∈ [t, t]. Otherwise, there exists t∗ such thatH(t) > , t ∈ (t∗, t), H(t) < , t ∈ (t, t∗), ()H(t∗)= , H˙(t∗)> . ()Combing () with (), one hasH˙(t∗)= ( – η)V–η(t∗)V˙(t∗)+ ρ( – η)= ( – η)V–η(t∗)(–ρV–η(t∗))+ ρ( – η)=  ()which contradicts (). Namely, H(t)≤  holds for t ∈ [t, t).When t = t, we getH(t) –H(t–)=H(t+)–H(t–)= V –η(t+)–V –η(t–)=(β–η – )V –η(t–)≤ . ()It yieldsH(t)≤H(t–) ≤ . ()Then we suppose that H(t)≤  holds for t ∈ [tk–, tk]. For t ∈ [tk , tk+], we haveH(tk+) = V –η(tk+) –V –η() + α( – η)tk+= β–ηV –η(t–k+)–V –η()α( – η)t–k+=[β–η – ]V –η(t–k+)–H(t–k+)≤ [β–η – ]V –η(t–k+)–H(t+k)≤ . ()From ()-(), we know that H(t)≤  holds. This completes the proof. Xiong and Huang Advances in Diﬀerence Equations  (2016) 2016:101 Page 5 of 93 Main resultsIn this section, the ﬁnite-time control and synchronization problems via an impulsiveadaptive strategy are investigated, respectively. Taking the impulsive adaptive strategy intoaccount in (), one obtains{x˙(t) = Ax(t) + φ(x(t)) + u(t), t = tk ,x(tk) = Ik(x(tk)) = –Bx(tk), t = tk ,k ∈ , ()where x(tk) = x(t+k ) – x(t–k ), x(t+k ) = limt→t+k x(t), x(t–k ) = limt→t–k x(t),  = {, , . . . ,n,n, . . . ,nk}, is a ﬁnite natural number set. For simplicity, it is assumed that x(t–k ) = x(tk), whichmeans that x(tk) is left continuous. Letting u(t) = –kx(t) – k sign(x(t))|x(t)|γ , we have thefollowing theorem.Theorem  Suppose Assumption  holds. There exists a positive constant γ satisfying  <γ <  such that the memristor-based chaotic system is ﬁnite-time stable if the followingconditions hold:(i) q = λmax[AT +A – (k + )I] < ;(ii) d = λmax(I + B)T (I + B) < .Proof Construct the following Lyapunov candidate function:V (t,x) = xTx. ()Calculating the derivative along the trajectory of () we haveD+V (t,x) = x˙Tx + xT x˙=(Ax + φ(x) + u(t))Tx + xT(Ax + φ(x) + u(t))= xT(A +AT – kI)x + xTφ(t,x) – kxT sign(x(t))∣∣x(t)∣∣γ . ()From Assumption , one hasxTφ(x)≤ xTx + φT (x)φ(x)≤ xTx + [(a + bx)x]≤ xTx + x[a + bM] ≤ (I + J)xTx, ()where J = [[a + bM], , , ]T .From () and (), one obtainsD+V (t,x)≤ xT[AT +A – (k + )I]x – kxT sign(x(t))∣∣x(t)∣∣γ≤ qV (t) – k∑i=|xi|γ+, ()where q = λmax[AT +A – (k + )I]. From the fact that  < γ < , one obtains( ∑i=|xi|γ+) γ+≥( ∑i=|xi|) . ()Xiong and Huang Advances in Diﬀerence Equations  (2016) 2016:101 Page 6 of 9Alson∑i=|xi|γ+ ≥( n∑i=|xi|) γ+=(xTx) γ+ . ()Then, from condition () of Theorem , we have for t = tkD+V (t,x)≤ –kV (x) γ+ . ()When t = tk , one obtainsV(t+k)= (x + Bx)T (x + Bx)= x(I + BT)(I + B)x≤ dV (tk). ()Through condition in Theorem  and Lemma , we can easily see that the system () isﬁnite-time stable, which complete the proof. Next, we investigate the problem of ﬁnite-synchronization for a memristor-basedchaotic system. Based on the drive-response synchronization concept, letting system ()be the drive system, the response system with control input u is as follows:y˙ = Ay + φ(y) + u. ()Taking the impulsive adaptive eﬀects into account, the response system () is as follows:⎧⎪⎨⎪⎩y˙ = Ay + φ(y) + u(t), t = tk ,y = Ik(y) = –Be, t = tk ,y(t+ ) = y, t ≥ ,()where e = [x – y]T = [x – y,x – y,x – y,x – y]T , tk are the impulsive instants whichsatisfy t < t < · · · < tk– < tk and limk→∞ tk =∞. Lettingu = –ke(t) – k sign(e(t))∣∣e(t)∣∣γ , ()where the constants k, k are the control strength coeﬃcients to be designed, the realnumber γ satisﬁes ≤ γ < .Combing () with (), the error system can be described as follows:{e˙ = Ae + φ(e) + u(t), t = tk ,e = Be, t = tk ,k = , , . . . ,()where φ(e) = [αW (y)y – αW (x)x, , , ]T . Hence, we have the following theorem.Xiong and Huang Advances in Diﬀerence Equations  (2016) 2016:101 Page 7 of 9Theorem  Suppose Assumption  holds. There exists a positive constant γ satisfying  <γ <  such that the memristor-based Chua systems () and () can be synchronized underthe impulsive adaptive strategy if the following conditions hold:(i) AT +A – kI – abMMI < ;(ii) d = λmax(I + B)T (I + B) < .Proof Construct the following Lyapunov candidate functionV (e) = eTe. ()For t ∈ [tk , tk+), the derivative of V (t) along the trajectory of () isD+V (e) = eT e˙ = eT (A – kI)e(t) + eTφ(e) – keT sign(e(t))∣∣e(t)∣∣α . ()From Assumption , we haveeTφ(e) = α[(a + by)y –(a + bx)x]e= –αe + ab(yy – yx + yx – xx)≤ abMMeTe. ()Combing () with (), we haveD+V (e)≤ eT(AT +A – kI – abMMI)e – k∑i=|ei|γ+. ()From the fact that  < γ < , one obtains( ∑i=|ei|γ+) γ+≥( ∑i=|ei|) . ()Alson∑i=|ei|γ+ ≥( n∑i=|ei|) γ+=(eTe) γ+ . ()Thus, based on the condition (i) in Theorem , we have for t = tkD+V (e)≤ –kV (e) γ+ . ()When t = tk , one obtainsV(t+k)= (e + Be)T (e + Be)= e(I + BT)(I + B)e≤ dV (tk). ()Xiong and Huang Advances in Diﬀerence Equations  (2016) 2016:101 Page 8 of 9Figure 3 Memristor-based chaotic system underan adaptive impulsive strategy.Through Theorem  and Lemma , we can easily see that the system () is ﬁnite-timestable. This implies that the response system () is synchronized with the drive system() in a ﬁnite time. 4 Simulation resultsThe numerical simulations are carried out using the fourth-order Runge-Kutta method.The initial states of the drive and response systems are (–, , , ) and (, , , ). Theparameters of the drive systems are α = , β = /, γ = ., ξ = /, a = /, and b =/. Solving the inequality in Theorem , and choosing B = diag(–.,–.,–.,–.),k = ., k = ., γ = ., the response system synchronizes with the drive system asshown in Figure . It is easily shown that the state response curve of the error system isstable.5 ConclusionIn this paper, the ﬁnite-time control and synchronization problems of memristor-basedchaotic systems have been investigated. Some novel impulsive adaptive control lawswhichguarantee the memristor-based Chua circuits is stabilized and synchronized in ﬁnite timehave been proposed. Moreover, simulation results were given to verify the eﬀectivenessand feasibility of the method. Our future research topics mainly consider the time delayeﬀects on the ﬁnite-time stability of the memristor-based nonlinear system.Competing interestsThe authors declare that they have no competing interests.Authors’ contributionsAll authors drafted the manuscript, and they read and approved the submitted version.Author details1College of Computer, Chongqing College of Electronic Engineering, Chongqing, 401331, P.R. China. 2Department ofComputer Science, Chongqing University of Education, Chongqing, 400067, P.R. China.AcknowledgementsThe work described in this paper was partially supported by the National Natural Science Foundation of China (GrantNo. 61403050) and the Scientiﬁc and Technological Research Program of Chongqing Municipal Education Commission(KJ1501412, KJ1501409, KJ1501301), and the Foundation of CQUE (KY201519B, KY201520B).Received: 8 November 2015 Accepted: 22 February 2016References1. Chua, LO: Memristor - the missing circuit element. IEEE Trans. Circuit Theory 18(5), 507-511 (1971)2. Strukov, DB, Snider, GS, Stewart, DR, Williams, RS: The Missing Memristor Found. Nature 453, 80-83 (2008)Xiong and Huang Advances in Diﬀerence Equations  (2016) 2016:101 Page 9 of 93. Bayat, FM, Shouraki, SB: Programming of memristor crossbars by using genetic algorithm. Proc. Comput. Sci. 3,232-237 (2011)4. Itoh, M, Chua, LO: Memristor oscillators. Int. J. Bifurc. Chaos 18, 3183-3206 (2008)5. 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Finite-time control and synchronization for memristor-based chaotic system via impulsive adaptive strategy

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Finite-time control and synchronization for memristor-based chaotic system via impulsive adaptive strategy

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