Chaos synchronization and parameters estimation of chaotic Lur’e systems with full unknown parameters via sampled-data control

Chaos synchronization and parameters estimation of chaotic Lur’e systems with full unknown... Abstract In this article, we investigate the parameters estimation and chaos synchronization problem for chaotic Lur’e systems based on the sampled-data control method, adaptive control and Lyapunov stability theory. An adaptive parameters update rule and a sampled-data controller are constructed by using sampled-data drive signals to realize the parameters estimation and chaos synchronization. Different from existing sampled-data control strategies, all parameters of the Lur’e systems are unknown in our article, which is more reasonable in applications. The corresponding theoretical proof is given to guarantee the effectiveness of the proposed strategy. Finally, numerical examples are given to show the effectiveness of the proposed method. 1. Introduction Since the pioneering work of Pecora & Carroll (1990), the control and synchronization of chaotic systems has been significantly studied for its potential applications in practical systems, for instance, biology, engineering, chaos secure communication (Herceg et al., 2016; Martínez-Guerra et al., 2016) and some other nonlinear fields (Toopchi & Wang, 2014; Chandrasekar & Rakkiyappan, 2016; Zheng, 2016). Due to the unpredictability and complicated dynamic behaviour of chaotic systems, parameters of chaotic systems are unknown in many practical situations. Thus, to solve this problem, some methods have been proposed in the literature (Zhao et al., 2013; Wang et al., 2014; Chen et al., 2015; Hamel et al., 2015; Zhao et al., 2015; Almatroud Othman et al., 2016; Han et al., 2016; Wang et al., 2016). For example, the problem of the finite-time chaos synchronization and parameters estimation of time-delay Lur’e systems is discussed in Wang et al. (2014). In Zhao et al. (2013), the authors investigate the synchronization and parameters estimation of chaotic systems based on a special matrix structure. These controllers for chaos synchronizing and parameters estimation by using continuous-time chaotic signal can be implemented by analogue circuits. Recently, the study on the digital redesign techniques for chaos control in discontinuous-time chaotic systems has noticeably become an active area of research, due to the advantage and the rapid development of the modern high-speed computers, microelectronics, and communication networks. For the case that the full parameters of chaotic systems are unknown, some articles investigate the synchronization and parameters estimation of chaotic systems via impulsive control (Chen et al., 2014). For example, the impulsive synchronization of chaotic systems is studied in Liu et al. (2014) and Zhang et al. (2014). In these articles, parameters adaptive rules are designed by using the split-second feedback signal in real time from the master systems. Recently, authors in Gao & Hu (2016) propose a new method in which both the adaptive synchronization controller and the parameters adaptive rule are modelled by impulsive differential equations to realize the adaptive impulsive synchronization of chaotic system with full unknown parameters. Also, the last decade has seen a wealth of research on sampled-data control technology, because sampled-data control is more reasonable in applications (Zhang et al., 2013; Xiao et al., 2014; Shi et al., 2016) than impulsive control. A number of sampled-data controllers have been introduced for chaos synchronization. The authors study the sampled-data control for synchronization of chaotic Lur’e systems using the free-weighting matrix approach and linear matrix inequalities (LMIs) in Lu & Hill (2008). Authors in Ge et al. (2014) and Chen et al. (2012) study the sampled-data chaos synchronization of Lur’e systems based on piecewise Lyapunov functionals. Recently, references in Ge et al. (2015) introduce a new method which makes full use of the information in the nonlinear part of the system for the chaos synchronization of chaotic Lur’e systems using sampled-data controller. Much more literatures can be found in Hua et al. (2015); Zeng et al. (2015) and Zhang et al. (2016). In addition, Lur’e systems can represent a general class of chaotic systems, such as Chua’s chaotic system (Liu, 2015), Hopfield neural networks (Zhao et al., 2015) and Chen chaotic system. Up to now, the chaos synchronization of chaotic Lur’e systems has been one of the focal points in many research and application fields. There are a large number of control methods have been proposed for the chaos synchronization of chaotic Lur’e systems. For example, Chen et al. (2004) consider the T-S fuzzy model and the delayed feedback controller. The authors investigate the effect of the time delay on chaos synchronization of Lur’e systems (Ge et al., 2004). However, there exist few works for the chaos synchronization or parameters estimation problem of Lur’e systems with full unknown parameters via sampled-data control. Therefore, in this brief we intend to investigate the chaos synchronization and parameters estimation of Lur’e systems with full unknown parameters via sampled-data control. We theoretically prove that synchronization of such chaotic Lur’e systems can be obtained using our method. Moreover, sufficient conditions are derived for synchronization of such chaotic Lur’e systems. Finally, we give one example to illustrate the effectiveness of proposed method. Notation: Throughout this article, $${\rm R}^{n \times m}$$ is the set of all $$n \times m$$ real matrices; $$P > 0\, (P \geq 0)$$ means that $$P$$ is a real symmetric and positive-definite (semi-positive-definite) matrix; diag($$\ldots$$) denotes a block-diagonal matrix; and symmetric term in a symmetric matrix is denoted by *. 2. Problem statement and preliminaries Consider the following master system: x˙(t)=Ax(t)+Bf(Cx(t)) (1) and the slave system {y˙(t)=A~y(t)+B~f(Cy(t))+u(t),t∈[tk,tk+1)u(t)=K(y(tk)−x(tk)) (2) with the parameters update rule {A~˙(t)=−βa(y(tk)−x(tk))yT(t)B~˙(t)=−βb(y(tk)−x(tk))fT(Cy(t)),t∈[tk,tk+1) (3) where $$x, y\,{\in}\,R^{n}$$ are the state vectors. $$u(t)\,{\in}\,R^{n}$$ is the control input of slave system. Let $${\it \Delta}$$ denotes the updating period of the control input $$u(t)$$, then the discrete $$y(t_{k})-x(t_{k})$$ can be transformed to a continuous time signal $$u(t)=K(y(t_{k})-x(t_{k}))$$ for $$t\,{\in}\,$$[$$t_{k}t_{k +1}) C\,{\in}\,R^{n \times n}$$ is known constant real matrices, $$A = (a_{ij})_{n \times n}, B = (b_{ij})_{n \times n}$$ are unknown real parameters matrices of system (2), and $$\tilde{{A}}(t) = (\tilde{{a}}_{ij} (t))_{n \times n}$$, $$\tilde{{B}}(t) = (\tilde{{b}}_{ij} (t))_{n \times n}$$ are their estimations. $$f = (f_{\mathrm{1}}(t), f_{\mathrm{2}}(t), \ldots, f_{n}(t))^{T}\!: R^{n}\to R^{n}$$ is a nonlinear function. $$\beta _{a}\,{\in}\,R$$, $$\beta_{b}\,{\in}\,R$$ and $$K\,{\in}\,R^{n \times n}$$ are the sampled-data controller gain to be designed. $$t_{k}$$ denotes the updating instant time, and satisfies $$0 = t_0 < t_{\mathrm{1}} < \ldots < t_{k} = +\infty $$. It is assume that Δ=tk+1−tk⩽h, (4) where $$h > 0$$ represents the upper bound of the sampling period. We assume that $$f(\bullet)$$: $$R^{n}\to R^{n}$$ satisfies a sector condition with $$f_{i}(\bullet )$$, $$i =$$ 1,2,$$\ldots$$, $$n$$, belonging to sector [0, $$\lambda $$]: fi(ζ)(fi(ζ)−λζ)⩽0 ∀ζ∈Rn;i=1,2,...,n which means that there exists a scalar $$\lambda $$ such that 0⩽fi(σ1)−fi(σ2)σ1−σ2⩽λ (5) for any $$\sigma_{\mathrm{1}}$$, $$\sigma_{\mathrm{2}}\,{\in}\, R^{n}$$. Define the synchronization error as $$e(t) = y(t)-x(t)$$, then we can get the error system from systems (1) and (2): e˙(t)=A~y(t)−Ax(t)+B~f(Cy(t))−Bf(Cx(t))+u(t),t∈[tk,tk+1) (6) Definition 1 The master system (1) and the slave system (2) are said to be asymptotically synchronized if and only if the error system (6) is globally asymptotically stable for e(t) $$\equiv $$ 0. That is $$e(t)\to $$0 as $$t\to \infty$$. Our major objective is to design the sampling period $${\it \Delta} $$, sampled-data controller gain matrix $$K\,{\in}\,R^{n \times n}$$, $$\beta_{a}\,{\in}\,R$$ and $$\beta_{b}\,{\in}\,R$$ to stabilize the error system. 3. Adaptive synchronization control and parameters estimation Here, we will focus on the adaptive synchronization and parameters estimation problem described above by utilizing the adaptive control, sampled-data feedback control method and the input delay approach (He et al., 2004; Lu & Hill, 2008). First, we represent the sampling instant $$t_{k}$$ as tk=t−(t−tk)=t−τ(t), (7) where $$\tau (t) = t- tk$$. It is clear that $$0 \leqslant \tau (t)\leqslant h$$ and $$\dot{{\tau }}(t)=1$$ for any $$t_{k}\leqslant t\leqslant t_{k}+$$1. Let p^(t)=A^(t)y(t)q^(t)=B^(t)f(Cy(t)), (8) where $$\hat{{A}}=\tilde{{A}}-A=(\hat{{a}}_{ij} )_{n \times n} $$ and $$\hat{{B}}=\tilde{{B}}-B=(\hat{{b}}_{ij} )_{n \times n} $$ denote the parameters error. Then the master system (2), error system (6) and the parameters update rule (3) can be rewritten as y˙(t)=A~y(t)+B~f(Cy(t))+Keτ(t) (9) e˙(t)=Ae(t)+Bℏ(Ce;x)+p^(t)+q^(t)+Keτ(t) (10) and {A~˙(t)=−βaeτ(t)yT(t)B~˙(t)=−βbeτ(t)fT(Cy(t)) (11) where $$l_{\tau}(t)$$ denotes $$l(t-\tau(t))$$ for any function $$l(t)$$, and $$\hbar (Ce; x) = f(C(e(t)+x(t)))-f(Cx(t))$$. Theorem 1 For given $$h > 0$$, the master system (1) and the slave system (9) are synchronous if there exist matrices $$Q_{\mathrm{1}}>$$ 0, $$Q_{\mathrm{2 }}>$$ 0, $${\it \Upsilon} =$$ diag($${\it \gamma}_{\mathrm{1}}$$, $${\it \gamma} _{\mathrm{2}}, \ldots, {\it \gamma}_{n})> 0$$, $$G_{a}=$$ diag($$\beta_{a}$$, $$\beta_{a}, \ldots, \beta_{a}) \,{\in}\,R^{n \times n}$$, $$G_{b}=$$ diag($$\beta_{b}$$, $$\beta_{b}, \ldots, \beta_{b}) \,{\in}\,R^{n \times n}$$, $$K\,{\in}\,R^{n \times n}$$, a matrix: S=(S11S12⋯S16S12TS22⋯S26⋮⋮⋮S16TS26T⋯S66)⩾0 (12) and any appropriate dimensional matrices $$L_{i}$$, $$M_{i}$$, $$i = 1,2,3,\ldots 6$$, such that the following LMIs hold: Ψ=(Ψ11Ψ12⋯Ψ16Ψ12TΨ22⋯Ψ26⋮⋮⋮Ψ16TΨ26T⋯Ψ66)<0 (13) and Γ=(S11⋯S16L1⋱⋮⋮S66L6∗Q2)⩾0, (14) where Ψ11=L1+L1T−M1A−ATM1TΨ12=M1−ATM2T+L2T+Q1Ψ13=−ATM3T−M1K+L3T−L1Ψ14=−ATM4T−M1B+L4T+λCTΥTΨ15=−ATM5T−M1+L5TΨ16=−ATM6T−M1+L6TΨ22=M2+M2T+hQ2Ψ23=M3T−M2K−L2Ψ24=M4T−M2BΨ25=M5T−M2Ψ26=M6T−M2Ψ33=−M3K−(M3K)T−L3−L3TΨ34=−M3B−(M4K)T−L4TΨ35=−(M5K)T−M3−L5T+GaΨ36=−(M6K)T−M3−L6T+GbΨ44=−M4B−(M4B)T−2ΥΨ45=−(M5B)T−M4Ψ46=−(M6B)T−M4Ψ55=−M5−M5TΨ56=−M6T−M5Ψ66=−M6−M6T Proof. Choose the Lyapunov–Krasovskii function as V(t)=∑i=13Vi(t), (15) where $$V_{1} (t)=e^{T}Q_{1} e+\int_{-h}^0 {\int_{t+\omega }^t {\dot{{e}}(s)Q_{2} \dot{{e}}(s)dsd\omega}}, V_{2} (t)=\sum\limits_{i=1}^n {\sum\limits_{i=1}^n {\hat{{a}}_{ij}^{2} } }$$, $$V_{3} (t)=\sum\limits_{i=1}^n {\sum\limits_{i=1}^n {\hat{{b}}_{ij}^{2}}}$$. From the assumption, we know that $$V_{i}(t)$$ is positive for $$I = 1$$, 2, 3. Calculating the derivative of $$V(t)$$ along the trajectories of (6), we can get V˙1(t)=2eTQ1e˙+he˙T(t)Q2e˙(t)−∫t−hte˙(s)Q2e˙(s)ds (16) From the parameters update rule (11), we have A~˙(t)=(a~˙ij(t))n×n=(−βae1τy1−βae1τy2⋯−βae1τyn−βae2τy1−βae2τy1⋯−βae2τy1⋮⋮⋮−βaenτy1−βaenτy2⋯−βaenτyn) (17) where $$e_{i\tau } = e_{i} (t-\tau (t)) = e_{i} (t_{k})$$. Then, we can get the following results: V˙2(t)=2∑i=1n∑i=1na~˙ija^ij=2eτTGaA^y=2eτTGap^ (18) Similarly, we can get V˙3(t)=2∑i=1n∑i=1nb~˙ijb^ij=2eτTGbq^, (19) where $$G_{a}={\rm diag}(-\beta_{a}, -\beta_{a},\ldots, -\beta_{a})\,{\in}\, R^{n \times n}$$, and $$G_{b}= {\rm diag}(-\beta_{b}, -\beta _{b},\ldots, -\beta_{b}) \,{\in}\,R^{n \times n}$$. According to (10), for any appropriate dimensional matrices $$M_{j} (j = 1,2,3,4,5,6)$$, we can get the following equation: 0=(2eTM1+2e˙TM2+2eτTM3+2ℏTM4+2p^TM5+2q^T(t)M6)×[e˙(t)−Ae(t)−Bℏ(Ce;x)−KLeτ(t)−p^(t)−q^(t)] (20) From Leibniz–Newton formula, it is easy to find that: e(t)−eτ(t)=e(t)−e(t−τ(t))−∫t−τ(t)te˙(s)ds=0 (21) So, for appropriately dimensional matrices $$L_{i}(I = 1, 2, 3, 4, 5)$$, we can have the following equation: 0=[2eTL1+2e˙TL2+2eτTL3+2ℏTL4+2p^TL5+2q^T(t)L6] ×[e(t)−eτ(t)−∫t−τ(t)te˙(s)ds] (22) On the other hand, for any semi-positive-definite matrix S=(S11S12⋯S16S12TS22⋯S26⋮⋮⋮S16TS26T⋯S66)⩾0 (23) the following inequality holds: 0⩽hςTSς−∫t−τ(t)tςTSςds, (24) where $$\varsigma^{T}(t)=(e^{T}(t),\dot{{e}}^{T}(t),\hslash ^{T}(Ce(t);x),e_{\tau }^{T} (t),\hat{{p}}^{T}(t),\hat{{q}}^{T}(t))$$ From (5), we can find the inequality: 0⩽ℏi(cie;x)cie=fi(ci(e(t)+x(t)))−fi(cix(t))cie⩽λ (25) The following inequality then holds: 0⩽−∑i=1n2γiℏi(ℏi−λcie)=−2ℏTΥ(ℏ−λCe) (26) for any $${\it \Upsilon} =$$ diag($${\it \gamma}_{\mathrm{1}}$$, $${\it \gamma}_{\mathrm{2}}$$, $$\ldots$$, $${\it \gamma}_{n})$$. Adding the right-hand side of (20), (22) and (26) to $$\dot{{V}}(t)$$, we can get V˙(t) ⩽2eTQ1e˙+he˙T(t)Q2e˙(t)−∫t−hte˙(s)Q2e˙(s)ds+2eτTGap^+2eτTGbq^  +(2eTM1+2e˙TM2+2eτTM3+2ℏTM4+2p^TM5+2q^T(t)M6)  ×[e˙(t)−Ae(t)−Bℏ(Ce;x)−KLeτ(t)−p^(t)−q^(t)]  +[2eTL1+2e˙TL2+2eτTL3+2ℏTL4+2p^TL5+2q^T(t)L6],  ×[e(t)−eτ(t)−∫t−τ(t)te˙(s)ds]  +hςT(t)Sς(t)−∫t−τ(t)tς(t)TSς(t)ds ⩽ςT(t)Ψς(t)−∫t−τ(t)tξ(t,ω)TΓξ(t,ω)dω (27) where $$\xi (t,\omega )^{T}=(\varsigma^{T}(t),\dot{{e}}^{T}(s))$$. If $${\it \Psi} < 0$$ and $$\Gamma \geqslant $$ 0, then $$\dot{{V}}(t)<-\sigma \left\| {e(t)} \right\|^{2}$$ for a sufficiently small $$\sigma $$, which means that the master system (1) and the slave system (9) are synchronous. We set $$M_{\mathrm{2}}K = V$$. $${\it \Psi} < 0 $$ can ensure that $$N + N^{T}$$ is negative definite and $$N$$ is nonsingular, then we can get the sampled-date controller gain matrix $$K = N^{\mathrm{-1}}V$$. Moreover, the sampling period h, the feedback control gain $$\beta_{a}$$ and $$\beta _{b}$$ can be derived from LMIs (13) and (14). □ 4. Numerical examples Consider the following Chua’s chaotic system as the master system (Wang et al., 2014): {x˙1(t)=a(x2(t)−m1x1(t)+f(x1(t)))x˙2(t)=x1(t)−x2(t)+x3(t)x˙3(t)=−bx2(t) (28) with the nonlinear characteristics $$f(x_{1} (t))=\frac{1}{2}(m_{1} -m_{0} )(\left| {x_{1} (t)+1} \right|-\left| {x_{1} (t)-1} \right|)$$, which belongs to sector [0,1]. The system (28) can be rewritten as the Lur’e form with system parameters: A=(−am1a1−11−b),B=(a(m1−m0)00),C=[100]. Parameters $$a = 9, m_{\mathrm{1}} = 2/7, m = -1/7$$, and $$b = 14.286$$. The initial conditions of the master system and the slave system are $$x(t) = [0.2, 0.3, 0.2]T$$ and $$y(t) = [1.5, 1.2, 0.9]T$$. The Chua’s system (28) can have the double scroll attractors as shown in Fig. 1. Fig. 1. View largeDownload slide Chaotic behaviors of Chua’s chaotic system (28) with parameters $$a = 9$$, $$m_{\mathrm{1}}=2/7, m_{\mathrm{0}}=-1/7$$, and $$b = 14.286$$. Fig. 1. View largeDownload slide Chaotic behaviors of Chua’s chaotic system (28) with parameters $$a = 9$$, $$m_{\mathrm{1}}=2/7, m_{\mathrm{0}}=-1/7$$, and $$b = 14.286$$. From Theorem 1, we can get the maximum value of the sampling period $$h = 0.2$$, and the sampled-date controller gain $$\beta _{a} = 3.51, \beta_{b} = 1.28$$ and $$K = diag(-2.5, -1.89, -3.31)$$. Let $${\it \Delta} = 0.2$$. The slave system is constructed as: {y˙1(t)=a~y2(t)+a~1y1(t)+a~2(|y1(t)+1|−|y1(t)−1|)−2.5e1(tk)y˙2(t)=y1(t)−y2(t)+y3(t)−1.89e2(tk)y˙3(t)=a~3y2(t)−3.31e3(tk) (29) with the parameters update rule: {a~(t)=−3.51e1(tk)y2(t)a~1(t)=−3.51e1(tk)y1(t)a~2(t)=−1.28e1(tk)(|y1(t)+1|−|y1(t)−1|)a~3(t)=−3.51e3(tk)y2(t), (30) where $$\widetilde{a}_{1} =-\widetilde{a}\widetilde{m}_{1}$$, $$\widetilde{a}_{2}\,=(\widetilde{a}_{2}\,/2)(\widetilde{m}_{1}-\widetilde{m}_{0}),\,\,{{\widetilde{a}}_{3}}\,=\,-\widetilde{b}.$$, $$\widetilde{a}_{3} =-\widetilde{{b}}$$. The initial values of the parameters estimation can be given arbitrarily. Here, we set them as $$\tilde{{a}}(0)=8$$, $$\tilde{{a}}_{1} (0)=-2$$, $$\tilde{{a}}_{2} (0)=-1$$, $$\tilde{{a}}_{3} (0)=-14$$. The control inputs $$u(t)$$ is represented in Fig. 2. Fig. 2. View largeDownload slide State trajectories of the slave control input $$u(t), u_{\mathrm{1}}(t) = -2.5e_{\mathrm{1}}(t_{k})$$, $$u_{\mathrm{2}}(t) = -1.89e_{\mathrm{2}}(t_{k})$$, $$u_{\mathrm{3}}(t) = -3.31e_{\mathrm{3}}(t_{k})$$, $$t\,{\in}\,$$[$$t_{k}$$, $$t_{k +1})$$, $$t_{k +1}-t_{k}= 0.2, k = 0,1,2,\ldots$$. Fig. 2. View largeDownload slide State trajectories of the slave control input $$u(t), u_{\mathrm{1}}(t) = -2.5e_{\mathrm{1}}(t_{k})$$, $$u_{\mathrm{2}}(t) = -1.89e_{\mathrm{2}}(t_{k})$$, $$u_{\mathrm{3}}(t) = -3.31e_{\mathrm{3}}(t_{k})$$, $$t\,{\in}\,$$[$$t_{k}$$, $$t_{k +1})$$, $$t_{k +1}-t_{k}= 0.2, k = 0,1,2,\ldots$$. The trajectory of error system (10) is shown in Fig. 3, the master and the slave systems achieve synchronization by the sampled-data controller, which means that the error system (6) is globally asymptotically stable for $$e(t) \equiv 0, e(t)\to 0$$ as $$t\to \infty $$. Fig. 3. View largeDownload slide The state trajectories of error system, $$e_{i}(t) = y_{i} (t)-x_{i}(t)$$, $$i = 1, 2, 3.\ e_{i}(t)\to 0$$ as $$t\to \infty $$. Fig. 3. View largeDownload slide The state trajectories of error system, $$e_{i}(t) = y_{i} (t)-x_{i}(t)$$, $$i = 1, 2, 3.\ e_{i}(t)\to 0$$ as $$t\to \infty $$. Moreover, in Fig. 4, the curves of the parameters estimation $$\tilde{{a}}(t), \tilde{{m}}_{1} (t), \tilde{{m}}_{0} (t)$$, and $$\tilde{{b}}(t)$$ converge to their true values 9, 2/7, $$-$$1/7 and 14.286, respectively. Fig. 4. View largeDownload slide Curves of parameters estimation $$\tilde{{a}}(t)$$, $$m_{1} (t)$$, $$m_{0} (t)$$, and $$\tilde{{b}}(t)$$. $$\tilde{{a}}(t)\to 9, \tilde{{m}}_{1} (t)\to 2/7, \tilde{{m}}_{0} (t)\to -1/7$$, and $$\tilde{{b}}(t)\to 14.286$$ as $$t\to \infty $$. Fig. 4. View largeDownload slide Curves of parameters estimation $$\tilde{{a}}(t)$$, $$m_{1} (t)$$, $$m_{0} (t)$$, and $$\tilde{{b}}(t)$$. $$\tilde{{a}}(t)\to 9, \tilde{{m}}_{1} (t)\to 2/7, \tilde{{m}}_{0} (t)\to -1/7$$, and $$\tilde{{b}}(t)\to 14.286$$ as $$t\to \infty $$. 5. Conclusions We discuss the adaptive sampled-data control for synchronization and parameters estimation of a class of Lur’e systems with full unknown parameters. An adaptive controller and update laws have been constructed via sampled-data signals from master system. And the sufficient conditions are derived to theoretically ensure the effectiveness of the method. Furthermore, the simulations on chaotic Chua’s system show the effectiveness of our method. Funding This work was supported by the Open Research Fund of State Key Laboratory of Cryptology [MMKFKT201613]; and the Independent Innovation Fund of Huazhong University of Science & Technology [2016YXMS067]. References Almatroud Othman, A. , Noorani, M. S. 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Chaos synchronization and parameters estimation of chaotic Lur’e systems with full unknown parameters via sampled-data control

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Oxford University Press
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© The authors 2017. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
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0265-0754
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1471-6887
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Abstract

Abstract In this article, we investigate the parameters estimation and chaos synchronization problem for chaotic Lur’e systems based on the sampled-data control method, adaptive control and Lyapunov stability theory. An adaptive parameters update rule and a sampled-data controller are constructed by using sampled-data drive signals to realize the parameters estimation and chaos synchronization. Different from existing sampled-data control strategies, all parameters of the Lur’e systems are unknown in our article, which is more reasonable in applications. The corresponding theoretical proof is given to guarantee the effectiveness of the proposed strategy. Finally, numerical examples are given to show the effectiveness of the proposed method. 1. Introduction Since the pioneering work of Pecora & Carroll (1990), the control and synchronization of chaotic systems has been significantly studied for its potential applications in practical systems, for instance, biology, engineering, chaos secure communication (Herceg et al., 2016; Martínez-Guerra et al., 2016) and some other nonlinear fields (Toopchi & Wang, 2014; Chandrasekar & Rakkiyappan, 2016; Zheng, 2016). Due to the unpredictability and complicated dynamic behaviour of chaotic systems, parameters of chaotic systems are unknown in many practical situations. Thus, to solve this problem, some methods have been proposed in the literature (Zhao et al., 2013; Wang et al., 2014; Chen et al., 2015; Hamel et al., 2015; Zhao et al., 2015; Almatroud Othman et al., 2016; Han et al., 2016; Wang et al., 2016). For example, the problem of the finite-time chaos synchronization and parameters estimation of time-delay Lur’e systems is discussed in Wang et al. (2014). In Zhao et al. (2013), the authors investigate the synchronization and parameters estimation of chaotic systems based on a special matrix structure. These controllers for chaos synchronizing and parameters estimation by using continuous-time chaotic signal can be implemented by analogue circuits. Recently, the study on the digital redesign techniques for chaos control in discontinuous-time chaotic systems has noticeably become an active area of research, due to the advantage and the rapid development of the modern high-speed computers, microelectronics, and communication networks. For the case that the full parameters of chaotic systems are unknown, some articles investigate the synchronization and parameters estimation of chaotic systems via impulsive control (Chen et al., 2014). For example, the impulsive synchronization of chaotic systems is studied in Liu et al. (2014) and Zhang et al. (2014). In these articles, parameters adaptive rules are designed by using the split-second feedback signal in real time from the master systems. Recently, authors in Gao & Hu (2016) propose a new method in which both the adaptive synchronization controller and the parameters adaptive rule are modelled by impulsive differential equations to realize the adaptive impulsive synchronization of chaotic system with full unknown parameters. Also, the last decade has seen a wealth of research on sampled-data control technology, because sampled-data control is more reasonable in applications (Zhang et al., 2013; Xiao et al., 2014; Shi et al., 2016) than impulsive control. A number of sampled-data controllers have been introduced for chaos synchronization. The authors study the sampled-data control for synchronization of chaotic Lur’e systems using the free-weighting matrix approach and linear matrix inequalities (LMIs) in Lu & Hill (2008). Authors in Ge et al. (2014) and Chen et al. (2012) study the sampled-data chaos synchronization of Lur’e systems based on piecewise Lyapunov functionals. Recently, references in Ge et al. (2015) introduce a new method which makes full use of the information in the nonlinear part of the system for the chaos synchronization of chaotic Lur’e systems using sampled-data controller. Much more literatures can be found in Hua et al. (2015); Zeng et al. (2015) and Zhang et al. (2016). In addition, Lur’e systems can represent a general class of chaotic systems, such as Chua’s chaotic system (Liu, 2015), Hopfield neural networks (Zhao et al., 2015) and Chen chaotic system. Up to now, the chaos synchronization of chaotic Lur’e systems has been one of the focal points in many research and application fields. There are a large number of control methods have been proposed for the chaos synchronization of chaotic Lur’e systems. For example, Chen et al. (2004) consider the T-S fuzzy model and the delayed feedback controller. The authors investigate the effect of the time delay on chaos synchronization of Lur’e systems (Ge et al., 2004). However, there exist few works for the chaos synchronization or parameters estimation problem of Lur’e systems with full unknown parameters via sampled-data control. Therefore, in this brief we intend to investigate the chaos synchronization and parameters estimation of Lur’e systems with full unknown parameters via sampled-data control. We theoretically prove that synchronization of such chaotic Lur’e systems can be obtained using our method. Moreover, sufficient conditions are derived for synchronization of such chaotic Lur’e systems. Finally, we give one example to illustrate the effectiveness of proposed method. Notation: Throughout this article, $${\rm R}^{n \times m}$$ is the set of all $$n \times m$$ real matrices; $$P > 0\, (P \geq 0)$$ means that $$P$$ is a real symmetric and positive-definite (semi-positive-definite) matrix; diag($$\ldots$$) denotes a block-diagonal matrix; and symmetric term in a symmetric matrix is denoted by *. 2. Problem statement and preliminaries Consider the following master system: x˙(t)=Ax(t)+Bf(Cx(t)) (1) and the slave system {y˙(t)=A~y(t)+B~f(Cy(t))+u(t),t∈[tk,tk+1)u(t)=K(y(tk)−x(tk)) (2) with the parameters update rule {A~˙(t)=−βa(y(tk)−x(tk))yT(t)B~˙(t)=−βb(y(tk)−x(tk))fT(Cy(t)),t∈[tk,tk+1) (3) where $$x, y\,{\in}\,R^{n}$$ are the state vectors. $$u(t)\,{\in}\,R^{n}$$ is the control input of slave system. Let $${\it \Delta}$$ denotes the updating period of the control input $$u(t)$$, then the discrete $$y(t_{k})-x(t_{k})$$ can be transformed to a continuous time signal $$u(t)=K(y(t_{k})-x(t_{k}))$$ for $$t\,{\in}\,$$[$$t_{k}t_{k +1}) C\,{\in}\,R^{n \times n}$$ is known constant real matrices, $$A = (a_{ij})_{n \times n}, B = (b_{ij})_{n \times n}$$ are unknown real parameters matrices of system (2), and $$\tilde{{A}}(t) = (\tilde{{a}}_{ij} (t))_{n \times n}$$, $$\tilde{{B}}(t) = (\tilde{{b}}_{ij} (t))_{n \times n}$$ are their estimations. $$f = (f_{\mathrm{1}}(t), f_{\mathrm{2}}(t), \ldots, f_{n}(t))^{T}\!: R^{n}\to R^{n}$$ is a nonlinear function. $$\beta _{a}\,{\in}\,R$$, $$\beta_{b}\,{\in}\,R$$ and $$K\,{\in}\,R^{n \times n}$$ are the sampled-data controller gain to be designed. $$t_{k}$$ denotes the updating instant time, and satisfies $$0 = t_0 < t_{\mathrm{1}} < \ldots < t_{k} = +\infty $$. It is assume that Δ=tk+1−tk⩽h, (4) where $$h > 0$$ represents the upper bound of the sampling period. We assume that $$f(\bullet)$$: $$R^{n}\to R^{n}$$ satisfies a sector condition with $$f_{i}(\bullet )$$, $$i =$$ 1,2,$$\ldots$$, $$n$$, belonging to sector [0, $$\lambda $$]: fi(ζ)(fi(ζ)−λζ)⩽0 ∀ζ∈Rn;i=1,2,...,n which means that there exists a scalar $$\lambda $$ such that 0⩽fi(σ1)−fi(σ2)σ1−σ2⩽λ (5) for any $$\sigma_{\mathrm{1}}$$, $$\sigma_{\mathrm{2}}\,{\in}\, R^{n}$$. Define the synchronization error as $$e(t) = y(t)-x(t)$$, then we can get the error system from systems (1) and (2): e˙(t)=A~y(t)−Ax(t)+B~f(Cy(t))−Bf(Cx(t))+u(t),t∈[tk,tk+1) (6) Definition 1 The master system (1) and the slave system (2) are said to be asymptotically synchronized if and only if the error system (6) is globally asymptotically stable for e(t) $$\equiv $$ 0. That is $$e(t)\to $$0 as $$t\to \infty$$. Our major objective is to design the sampling period $${\it \Delta} $$, sampled-data controller gain matrix $$K\,{\in}\,R^{n \times n}$$, $$\beta_{a}\,{\in}\,R$$ and $$\beta_{b}\,{\in}\,R$$ to stabilize the error system. 3. Adaptive synchronization control and parameters estimation Here, we will focus on the adaptive synchronization and parameters estimation problem described above by utilizing the adaptive control, sampled-data feedback control method and the input delay approach (He et al., 2004; Lu & Hill, 2008). First, we represent the sampling instant $$t_{k}$$ as tk=t−(t−tk)=t−τ(t), (7) where $$\tau (t) = t- tk$$. It is clear that $$0 \leqslant \tau (t)\leqslant h$$ and $$\dot{{\tau }}(t)=1$$ for any $$t_{k}\leqslant t\leqslant t_{k}+$$1. Let p^(t)=A^(t)y(t)q^(t)=B^(t)f(Cy(t)), (8) where $$\hat{{A}}=\tilde{{A}}-A=(\hat{{a}}_{ij} )_{n \times n} $$ and $$\hat{{B}}=\tilde{{B}}-B=(\hat{{b}}_{ij} )_{n \times n} $$ denote the parameters error. Then the master system (2), error system (6) and the parameters update rule (3) can be rewritten as y˙(t)=A~y(t)+B~f(Cy(t))+Keτ(t) (9) e˙(t)=Ae(t)+Bℏ(Ce;x)+p^(t)+q^(t)+Keτ(t) (10) and {A~˙(t)=−βaeτ(t)yT(t)B~˙(t)=−βbeτ(t)fT(Cy(t)) (11) where $$l_{\tau}(t)$$ denotes $$l(t-\tau(t))$$ for any function $$l(t)$$, and $$\hbar (Ce; x) = f(C(e(t)+x(t)))-f(Cx(t))$$. Theorem 1 For given $$h > 0$$, the master system (1) and the slave system (9) are synchronous if there exist matrices $$Q_{\mathrm{1}}>$$ 0, $$Q_{\mathrm{2 }}>$$ 0, $${\it \Upsilon} =$$ diag($${\it \gamma}_{\mathrm{1}}$$, $${\it \gamma} _{\mathrm{2}}, \ldots, {\it \gamma}_{n})> 0$$, $$G_{a}=$$ diag($$\beta_{a}$$, $$\beta_{a}, \ldots, \beta_{a}) \,{\in}\,R^{n \times n}$$, $$G_{b}=$$ diag($$\beta_{b}$$, $$\beta_{b}, \ldots, \beta_{b}) \,{\in}\,R^{n \times n}$$, $$K\,{\in}\,R^{n \times n}$$, a matrix: S=(S11S12⋯S16S12TS22⋯S26⋮⋮⋮S16TS26T⋯S66)⩾0 (12) and any appropriate dimensional matrices $$L_{i}$$, $$M_{i}$$, $$i = 1,2,3,\ldots 6$$, such that the following LMIs hold: Ψ=(Ψ11Ψ12⋯Ψ16Ψ12TΨ22⋯Ψ26⋮⋮⋮Ψ16TΨ26T⋯Ψ66)<0 (13) and Γ=(S11⋯S16L1⋱⋮⋮S66L6∗Q2)⩾0, (14) where Ψ11=L1+L1T−M1A−ATM1TΨ12=M1−ATM2T+L2T+Q1Ψ13=−ATM3T−M1K+L3T−L1Ψ14=−ATM4T−M1B+L4T+λCTΥTΨ15=−ATM5T−M1+L5TΨ16=−ATM6T−M1+L6TΨ22=M2+M2T+hQ2Ψ23=M3T−M2K−L2Ψ24=M4T−M2BΨ25=M5T−M2Ψ26=M6T−M2Ψ33=−M3K−(M3K)T−L3−L3TΨ34=−M3B−(M4K)T−L4TΨ35=−(M5K)T−M3−L5T+GaΨ36=−(M6K)T−M3−L6T+GbΨ44=−M4B−(M4B)T−2ΥΨ45=−(M5B)T−M4Ψ46=−(M6B)T−M4Ψ55=−M5−M5TΨ56=−M6T−M5Ψ66=−M6−M6T Proof. Choose the Lyapunov–Krasovskii function as V(t)=∑i=13Vi(t), (15) where $$V_{1} (t)=e^{T}Q_{1} e+\int_{-h}^0 {\int_{t+\omega }^t {\dot{{e}}(s)Q_{2} \dot{{e}}(s)dsd\omega}}, V_{2} (t)=\sum\limits_{i=1}^n {\sum\limits_{i=1}^n {\hat{{a}}_{ij}^{2} } }$$, $$V_{3} (t)=\sum\limits_{i=1}^n {\sum\limits_{i=1}^n {\hat{{b}}_{ij}^{2}}}$$. From the assumption, we know that $$V_{i}(t)$$ is positive for $$I = 1$$, 2, 3. Calculating the derivative of $$V(t)$$ along the trajectories of (6), we can get V˙1(t)=2eTQ1e˙+he˙T(t)Q2e˙(t)−∫t−hte˙(s)Q2e˙(s)ds (16) From the parameters update rule (11), we have A~˙(t)=(a~˙ij(t))n×n=(−βae1τy1−βae1τy2⋯−βae1τyn−βae2τy1−βae2τy1⋯−βae2τy1⋮⋮⋮−βaenτy1−βaenτy2⋯−βaenτyn) (17) where $$e_{i\tau } = e_{i} (t-\tau (t)) = e_{i} (t_{k})$$. Then, we can get the following results: V˙2(t)=2∑i=1n∑i=1na~˙ija^ij=2eτTGaA^y=2eτTGap^ (18) Similarly, we can get V˙3(t)=2∑i=1n∑i=1nb~˙ijb^ij=2eτTGbq^, (19) where $$G_{a}={\rm diag}(-\beta_{a}, -\beta_{a},\ldots, -\beta_{a})\,{\in}\, R^{n \times n}$$, and $$G_{b}= {\rm diag}(-\beta_{b}, -\beta _{b},\ldots, -\beta_{b}) \,{\in}\,R^{n \times n}$$. According to (10), for any appropriate dimensional matrices $$M_{j} (j = 1,2,3,4,5,6)$$, we can get the following equation: 0=(2eTM1+2e˙TM2+2eτTM3+2ℏTM4+2p^TM5+2q^T(t)M6)×[e˙(t)−Ae(t)−Bℏ(Ce;x)−KLeτ(t)−p^(t)−q^(t)] (20) From Leibniz–Newton formula, it is easy to find that: e(t)−eτ(t)=e(t)−e(t−τ(t))−∫t−τ(t)te˙(s)ds=0 (21) So, for appropriately dimensional matrices $$L_{i}(I = 1, 2, 3, 4, 5)$$, we can have the following equation: 0=[2eTL1+2e˙TL2+2eτTL3+2ℏTL4+2p^TL5+2q^T(t)L6] ×[e(t)−eτ(t)−∫t−τ(t)te˙(s)ds] (22) On the other hand, for any semi-positive-definite matrix S=(S11S12⋯S16S12TS22⋯S26⋮⋮⋮S16TS26T⋯S66)⩾0 (23) the following inequality holds: 0⩽hςTSς−∫t−τ(t)tςTSςds, (24) where $$\varsigma^{T}(t)=(e^{T}(t),\dot{{e}}^{T}(t),\hslash ^{T}(Ce(t);x),e_{\tau }^{T} (t),\hat{{p}}^{T}(t),\hat{{q}}^{T}(t))$$ From (5), we can find the inequality: 0⩽ℏi(cie;x)cie=fi(ci(e(t)+x(t)))−fi(cix(t))cie⩽λ (25) The following inequality then holds: 0⩽−∑i=1n2γiℏi(ℏi−λcie)=−2ℏTΥ(ℏ−λCe) (26) for any $${\it \Upsilon} =$$ diag($${\it \gamma}_{\mathrm{1}}$$, $${\it \gamma}_{\mathrm{2}}$$, $$\ldots$$, $${\it \gamma}_{n})$$. Adding the right-hand side of (20), (22) and (26) to $$\dot{{V}}(t)$$, we can get V˙(t) ⩽2eTQ1e˙+he˙T(t)Q2e˙(t)−∫t−hte˙(s)Q2e˙(s)ds+2eτTGap^+2eτTGbq^  +(2eTM1+2e˙TM2+2eτTM3+2ℏTM4+2p^TM5+2q^T(t)M6)  ×[e˙(t)−Ae(t)−Bℏ(Ce;x)−KLeτ(t)−p^(t)−q^(t)]  +[2eTL1+2e˙TL2+2eτTL3+2ℏTL4+2p^TL5+2q^T(t)L6],  ×[e(t)−eτ(t)−∫t−τ(t)te˙(s)ds]  +hςT(t)Sς(t)−∫t−τ(t)tς(t)TSς(t)ds ⩽ςT(t)Ψς(t)−∫t−τ(t)tξ(t,ω)TΓξ(t,ω)dω (27) where $$\xi (t,\omega )^{T}=(\varsigma^{T}(t),\dot{{e}}^{T}(s))$$. If $${\it \Psi} < 0$$ and $$\Gamma \geqslant $$ 0, then $$\dot{{V}}(t)<-\sigma \left\| {e(t)} \right\|^{2}$$ for a sufficiently small $$\sigma $$, which means that the master system (1) and the slave system (9) are synchronous. We set $$M_{\mathrm{2}}K = V$$. $${\it \Psi} < 0 $$ can ensure that $$N + N^{T}$$ is negative definite and $$N$$ is nonsingular, then we can get the sampled-date controller gain matrix $$K = N^{\mathrm{-1}}V$$. Moreover, the sampling period h, the feedback control gain $$\beta_{a}$$ and $$\beta _{b}$$ can be derived from LMIs (13) and (14). □ 4. Numerical examples Consider the following Chua’s chaotic system as the master system (Wang et al., 2014): {x˙1(t)=a(x2(t)−m1x1(t)+f(x1(t)))x˙2(t)=x1(t)−x2(t)+x3(t)x˙3(t)=−bx2(t) (28) with the nonlinear characteristics $$f(x_{1} (t))=\frac{1}{2}(m_{1} -m_{0} )(\left| {x_{1} (t)+1} \right|-\left| {x_{1} (t)-1} \right|)$$, which belongs to sector [0,1]. The system (28) can be rewritten as the Lur’e form with system parameters: A=(−am1a1−11−b),B=(a(m1−m0)00),C=[100]. Parameters $$a = 9, m_{\mathrm{1}} = 2/7, m = -1/7$$, and $$b = 14.286$$. The initial conditions of the master system and the slave system are $$x(t) = [0.2, 0.3, 0.2]T$$ and $$y(t) = [1.5, 1.2, 0.9]T$$. The Chua’s system (28) can have the double scroll attractors as shown in Fig. 1. Fig. 1. View largeDownload slide Chaotic behaviors of Chua’s chaotic system (28) with parameters $$a = 9$$, $$m_{\mathrm{1}}=2/7, m_{\mathrm{0}}=-1/7$$, and $$b = 14.286$$. Fig. 1. View largeDownload slide Chaotic behaviors of Chua’s chaotic system (28) with parameters $$a = 9$$, $$m_{\mathrm{1}}=2/7, m_{\mathrm{0}}=-1/7$$, and $$b = 14.286$$. From Theorem 1, we can get the maximum value of the sampling period $$h = 0.2$$, and the sampled-date controller gain $$\beta _{a} = 3.51, \beta_{b} = 1.28$$ and $$K = diag(-2.5, -1.89, -3.31)$$. Let $${\it \Delta} = 0.2$$. The slave system is constructed as: {y˙1(t)=a~y2(t)+a~1y1(t)+a~2(|y1(t)+1|−|y1(t)−1|)−2.5e1(tk)y˙2(t)=y1(t)−y2(t)+y3(t)−1.89e2(tk)y˙3(t)=a~3y2(t)−3.31e3(tk) (29) with the parameters update rule: {a~(t)=−3.51e1(tk)y2(t)a~1(t)=−3.51e1(tk)y1(t)a~2(t)=−1.28e1(tk)(|y1(t)+1|−|y1(t)−1|)a~3(t)=−3.51e3(tk)y2(t), (30) where $$\widetilde{a}_{1} =-\widetilde{a}\widetilde{m}_{1}$$, $$\widetilde{a}_{2}\,=(\widetilde{a}_{2}\,/2)(\widetilde{m}_{1}-\widetilde{m}_{0}),\,\,{{\widetilde{a}}_{3}}\,=\,-\widetilde{b}.$$, $$\widetilde{a}_{3} =-\widetilde{{b}}$$. The initial values of the parameters estimation can be given arbitrarily. Here, we set them as $$\tilde{{a}}(0)=8$$, $$\tilde{{a}}_{1} (0)=-2$$, $$\tilde{{a}}_{2} (0)=-1$$, $$\tilde{{a}}_{3} (0)=-14$$. The control inputs $$u(t)$$ is represented in Fig. 2. Fig. 2. View largeDownload slide State trajectories of the slave control input $$u(t), u_{\mathrm{1}}(t) = -2.5e_{\mathrm{1}}(t_{k})$$, $$u_{\mathrm{2}}(t) = -1.89e_{\mathrm{2}}(t_{k})$$, $$u_{\mathrm{3}}(t) = -3.31e_{\mathrm{3}}(t_{k})$$, $$t\,{\in}\,$$[$$t_{k}$$, $$t_{k +1})$$, $$t_{k +1}-t_{k}= 0.2, k = 0,1,2,\ldots$$. Fig. 2. View largeDownload slide State trajectories of the slave control input $$u(t), u_{\mathrm{1}}(t) = -2.5e_{\mathrm{1}}(t_{k})$$, $$u_{\mathrm{2}}(t) = -1.89e_{\mathrm{2}}(t_{k})$$, $$u_{\mathrm{3}}(t) = -3.31e_{\mathrm{3}}(t_{k})$$, $$t\,{\in}\,$$[$$t_{k}$$, $$t_{k +1})$$, $$t_{k +1}-t_{k}= 0.2, k = 0,1,2,\ldots$$. The trajectory of error system (10) is shown in Fig. 3, the master and the slave systems achieve synchronization by the sampled-data controller, which means that the error system (6) is globally asymptotically stable for $$e(t) \equiv 0, e(t)\to 0$$ as $$t\to \infty $$. Fig. 3. View largeDownload slide The state trajectories of error system, $$e_{i}(t) = y_{i} (t)-x_{i}(t)$$, $$i = 1, 2, 3.\ e_{i}(t)\to 0$$ as $$t\to \infty $$. Fig. 3. View largeDownload slide The state trajectories of error system, $$e_{i}(t) = y_{i} (t)-x_{i}(t)$$, $$i = 1, 2, 3.\ e_{i}(t)\to 0$$ as $$t\to \infty $$. Moreover, in Fig. 4, the curves of the parameters estimation $$\tilde{{a}}(t), \tilde{{m}}_{1} (t), \tilde{{m}}_{0} (t)$$, and $$\tilde{{b}}(t)$$ converge to their true values 9, 2/7, $$-$$1/7 and 14.286, respectively. Fig. 4. View largeDownload slide Curves of parameters estimation $$\tilde{{a}}(t)$$, $$m_{1} (t)$$, $$m_{0} (t)$$, and $$\tilde{{b}}(t)$$. $$\tilde{{a}}(t)\to 9, \tilde{{m}}_{1} (t)\to 2/7, \tilde{{m}}_{0} (t)\to -1/7$$, and $$\tilde{{b}}(t)\to 14.286$$ as $$t\to \infty $$. Fig. 4. View largeDownload slide Curves of parameters estimation $$\tilde{{a}}(t)$$, $$m_{1} (t)$$, $$m_{0} (t)$$, and $$\tilde{{b}}(t)$$. $$\tilde{{a}}(t)\to 9, \tilde{{m}}_{1} (t)\to 2/7, \tilde{{m}}_{0} (t)\to -1/7$$, and $$\tilde{{b}}(t)\to 14.286$$ as $$t\to \infty $$. 5. Conclusions We discuss the adaptive sampled-data control for synchronization and parameters estimation of a class of Lur’e systems with full unknown parameters. An adaptive controller and update laws have been constructed via sampled-data signals from master system. And the sufficient conditions are derived to theoretically ensure the effectiveness of the method. Furthermore, the simulations on chaotic Chua’s system show the effectiveness of our method. Funding This work was supported by the Open Research Fund of State Key Laboratory of Cryptology [MMKFKT201613]; and the Independent Innovation Fund of Huazhong University of Science & Technology [2016YXMS067]. References Almatroud Othman, A. , Noorani, M. S. 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IMA Journal of Mathematical Control and InformationOxford University Press

Published: May 13, 2017

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