Nonlinear Dynamics

, Volume 70, Issue 3, pp 1977–1987

Synchronization of chaotic systems under sampled-data control

Authors

  • S. Jeeva Sathya Theesar
    • Department of MathematicsGandhigram Rural Institute—Deemed University
    • Institute for Mathematical ResearchUniversiti Putra Malaysia
    • Institute for Mathematical ResearchUniversiti Putra Malaysia
  • P. Balasubramaniam
    • Department of MathematicsGandhigram Rural Institute—Deemed University
Original Paper

DOI: 10.1007/s11071-012-0590-0

Cite this article as:
Theesar, S.J.S., Banerjee, S. & Balasubramaniam, P. Nonlinear Dyn (2012) 70: 1977. doi:10.1007/s11071-012-0590-0

Abstract

The problem of global asymptotical synchronization of chaotic Lur’e systems using sampled-data controller is considered in this paper. Sufficient conditions are obtained in terms of effective synchronization linear matrix inequality using a piecewise sawtooth structure of the sampling in time by constructing the new discontinuous Lyapunov functionals. The sampled-data feedback control gain is obtained from the derived condition. The Chua system and horizontal platform system are taken for numerical demonstration to show the effectiveness of the proposed condition.

Keywords

Discontinuous Lyapunov functionalsLur’e systemsSampled-data controlSynchronization

1 Introduction

Yang and Chua [1, 2] have introduced the sampling control technique and implemented analog circuits for controlling chaos based on the sampling sequence of output of the chaotic system as a feedback control input. The main motivation for the sampled-data controller is to exploit the digital control techniques in real-time. In the digital controller, the output of the chaotic system is sampled and the sampled data is used to construct the appropriate control signal. Since then, sampled-data systems have become key research areas [3]. Synchronization is an important phenomena in dynamical systems and in the last two decades, it has become a major research area in nonlinear science. From the pioneering work of Pecora and Carroll [4] on the master-slave (drive-response) concept for achieving the synchronization of chaotic systems, a variety of alternative schemes for ensuring the control and synchronization of such systems have been proposed (for, e.g., [515]) due to their potential applications in various fields, including chaos generator design, secure communication, chemical reaction, biological systems, and information science. On the other hand, Lur’e representation of nonlinear feedback systems can form a general class of nonlinear systems including chaotic systems such as Chua’s circuit, n-scroll attractors, and hyperchaotic attractors [7] by imposing sector bounding nonlinear criteria. A number of master-slave synchronization schemes for chaotic Lur’e systems have been proposed [1620] along with controllers such as observer-based control [8, 11], adaptive control [6], and state delayed feedback control [7, 2123]. It is noted that the above techniques utilized continuous time controllers to synchronize chaos. In many real-world applications, it is difficult to guarantee that the state variables transmitted to controllers are continuous. As the rapid development of computer hardware, the sampled-data control technology has shown superiority over other control approaches. Thus, the use of the sampled-data controller to synchronize the chaotic systems comes into play, and subsequently authors in [13, 14] have developed sampled-data techniques for synchronization of master-slave systems at discrete time instants as well as the input delay approach. The operation of synchronization under sampled-data feedback control can be described as follows: Firstly, the system states that both the master and slave systems form an error signal, which is fed to the sampler with a sampling interval Δt. Then the sampled system states information is processed by the sampled-data controller to produce an appropriate control signal. Finally, the control signal is kept constant during the sampling interval by the zero-order holder (ZOH) and fed to the slave system to realize the synchronization. The synchronization scheme has been illustrated in Fig. 1.
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-012-0590-0/MediaObjects/11071_2012_590_Fig1_HTML.gif
Fig. 1

Master-Slave synchronization scheme under sampled-data feedback controller

For synchronizing continuous-time chaotic systems via sampled-data, mostly digital redesign techniques were applied and numerical results have been demonstrated without adequate theoretical development. On the other hand, recently sufficient conditions for global asymptotical synchronization of such chaotic Lur’e systems have been obtained in [13] as Linear Matrix Inequality (LMI) conditions with time-varying delay transformed from the sampled-data error system. Recently, the criterion for global synchronization has been further improved in [14] by constructing augmented Lyapunov functionals and free weighting matrices in terms of LMI conditions.

It must be noted that without utilizing the sawtooth structure of the sampling completely the LMI conditions have been obtained in [13, 14] by constructing the continuous time Lyapunov–Krasovskii functional (LKF), which is inadequate and very conservative. Though the sampled-data control is being transformed to a continuous time controller under ZOH, the sawtooth structure (discontinuous in time) of sampling was not utilized appropriately in aforementioned literature. Thus, there is a quest to obtain conditions utilizing the complete sawtooth structure of the sampling period which is a piecewise continuous function. Motivated by the above discussion, in this paper, we aim to present new criteria for global synchronization of Lur’e systems under the sampled-data controller. The design of sampled-data control gain is dependent of obtained LMI conditions derived by constructing discontinuous Lyapunov functionals. The motivation for constructing discontinuous LKF is prevailed from recently developed sampled-data control problem for linear systems [24, 25]. Moreover, in order to reduce the conservatism in sampling-time-dependent criterion, the decomposition approach of delay interval is employed. The derived LMI conditions can be solved by most of the existing numerical LMI solvers (for example, LMI lab in Matlab Robust Control toolbox). To justify the proposed result of this paper, LMI criteria are numerically solved using Matlab LMI Lab. Numerical simulation on the Chua system [7] and horizontal platform system [26] are presented to exhibit the effectiveness of the proposed method.

Notations

The notations in this paper is quite standard. The superscript T denotes the transposition and the notation XY (respectively, X>Y), where X and Y are symmetric matrices, means that XY is a positive semidefinite (respectively, positive definite). ℜn and ℜn×n denote n-dimensional Euclidean space and the set of all n×n real matrices, respectively. I is the identity matrix. The notation ⋆ always denotes the symmetric block in one symmetric matrix. Matrices, if not explicitly stated, are assumed to have compatible dimensions. Let τM>0 and \(\mathcal{C}([-\tau_{M},0];\Re^{n})\) denote the family of continuous functions φ from [−τM,0] to ℜn with norm \(\|\varphi\|=\sup_{-\tau_{M}\le \theta \le 0}\|\varphi\|\). \(\mathfrak{L}_{2}[0,\,\infty)\) stands for the space of square integrable functions on [0, ∞).

2 Synchronization problem description

Consider the general form of the Master-Slave type of coupled Lur’e systems under the sampled-data controller as follows:
$$ \mathit{Master} : \begin{cases} \dot{x}(t) =Ax(t)+Bh(D^{T}x(t)),&\\ \noalign {\vspace {4pt}} p(t) =C\, x(t),&\end{cases} $$
(1)
$$ \mathit{Slave} : \begin{cases} \dot{y}(t) =Ay(t)+Bh(D^{T}y(t))+u(t),&\\ \noalign {\vspace {4pt}} q(t) =C\, y(t),&\end{cases} $$
(2)
$$ \begin{array}[b]{@{}l} \mathit{Control} : u(t)= K\, (p(t_{k})-q(t_{k}) ),\\ \noalign {\vspace {4pt}} \quad t_{k}\le t<t_{k+1}, \end{array} $$
(3)
where Master and Slave systems are Lur’e chaotic systems with state vectors x(t), y(t)∈ℜn, and the output vectors of subsystems p(t), q(t)∈ℜp, respectively. Here, the system matrices, A∈ℜn×n,  B∈ℜn×l, D∈ℜn×l, and C∈ℜp×n are known real constant matrices. h(DTx(t)), h(DTy(t)) denote the vector-valued nonlinear functions satisfying h(0)=0.

(H1)

The nonlinear function h(⋅)=[h1,h2,…,hl]T:ℜl→ℜl satisfy the following inequality
$$ \bigl[h_{i}(\xi)-\sigma_{i}^{+}\xi \bigr] \bigl[h_{i}(\xi)-\sigma_{i}^{-}\xi \bigr] \le 0,\quad\forall\xi, $$
(4)
for all i=1,2,…,l. Thus, the nonlinear function satisfies sector bounding condition and hi(⋅) is said to belong in the sector \([ \sigma_{i}^{-}, \sigma_{i}^{+} ]\). Denote the updating instant time of the ZOH by tk. The control input to the slave system is denoted by u(t)∈ℜn under sample time tk. K∈ℜn×p is the sampled-data feedback controller gain matrix to be determined. Moreover, p(tk), and q(tk) are discrete measurements of p(t) and q(t) at the sampling instant tk, respectively. Suppose that the updating signal (successfully transmitted signal from the sampler to the controller and to the ZOH) at the instant tk has experienced a constant signal transmission delay η. It is assumed that the sampling intervals are bounded and satisfy
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-012-0590-0/MediaObjects/11071_2012_590_Equa_HTML.gif
where h is a positive scalar and represents the largest sampling interval. Thus, we have
$$ t_{k+1}-t_{k} +\eta \le h+\eta \le \tau_{M}. $$
(5)
Here, τM denotes the maximum time span between the time tkη at which the state is sampled and the time tk+1 at which the next update arrives at the destination. The main aim of this study is to design the sampled-data controller (3) for the synchronization between Master (1) and Slave (2). That is the error between two dynamics must be equal to zero asymptotically. Let the error between Master and Slave systems be e(t)=x(t)−y(t). Moreover, based on the above sampled-data controller design formulation, the feedback controller takes the form u(tk)=Ke(tkη). Thus, considering the behavior of ZOH, we have
$$ u(t) = K\, e(t_{k}-\eta),\quad t_{k}\le t<t_{k+1},\ k=0,1,2,\ldots $$
(6)
with tk+1 being the next updating instant time of the ZOH after tk. Defining τ(t)=ttk+η, tkt<tk+1, then the following closed loop error dynamical system is obtained:
$$ \dot{e}(t) = Ae(t)+Bf\bigl(D^{T}e(t)\bigr)-KCe\bigl(t-\tau(t)\bigr), $$
(7)
where f(DTe(t))=h(DTx(t))−h(DTy(t)). Thus, we have ητ(t)<tk+1tk+ητM, and \(\dot{\tau}(t)=1\), for ttk. It can be easily checked that fi(0)=0 and the nonlinearity fi(⋅) belongs to the sector \([\sigma_{i}^{-}, \sigma_{i}^{+} ]\) for all i=1,2,…,l and ∀ξ∈ℝ,
$$ \bigl[f_{i}(\xi)-\sigma_{i}^{+}\xi \bigr] \bigl[f_{i}(\xi)-\sigma_{i}^{-}\xi \bigr] \le 0 ,\quad i=1,2,\ldots,l. $$
(8)
The general sector conditions are rewritten as
$$ \sigma_{i}^{-} \le \frac{f_{i}(\xi)}{\xi} \le \sigma_{i}^{+},\quad i=1,2,\ldots,l. $$
(9)
Also, consider \(\varSigma_{1}= \operatorname {diag}\{\sigma_{1}^{-}\sigma_{1}^{+},\sigma_{2}^{-}\sigma_{2}^{+},\ldots, \sigma_{l}^{-}\sigma_{l}^{+}\}\), and \(\varSigma_{2}= \operatorname {diag}\{\frac{\sigma_{1}^{-}+\sigma_{1}^{+}}{2},\frac{\sigma_{2}^{-}+\sigma_{2}^{+}}{2},\ldots, \frac{\sigma_{l}^{-}+\sigma_{l}^{+}}{2}\}\). It is implied from the above formulation that the synchronization problem between (1) and (2) is converted into an equivalent absolute stability problem of the error dynamical systems (7). In this paper, we aim at establishing easily computable yet less conservative synchronization criteria by finding maximum sampling time hmax.

Now we state the following definitions and lemmas which will be used in the sequel.

Definition 1

The Master (1) and Slave (2) systems are said to be asymptotically synchronized if and only if the error dynamical systems (7) is globally asymptotically stable for the equilibrium point e(t)≡0. That is, e(t)→0 as t→∞.

Lemma 1

[9]

For any constant matrixX∈ℝn×n, X=XT>0, there exists positive scalarτsuch that 0≤τ(t)≤τ, and a vector-valued function\(\dot{x}:[-\tau,0]\rightarrow\mathbb{R}^{n}\), the integration\(-\tau\int_{t-\tau}^{t}\dot{x}^{T}(s)X\dot{x}(s)\,ds\)is well defined,
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-012-0590-0/MediaObjects/11071_2012_590_Equb_HTML.gif

Lemma 2

([24] Extended Wirtinger Inequality)

Letz(t)∈W[a,b) andz(a)=0. Then for any matrixR>0, the following inequality holds:
$$ \int_{a}^{b}z^{T}(s)Rz(s)\,ds \leq \frac{4(b-a)^{2}}{\pi^{2}} \int_{a}^{b} \dot{z}^{T}(s)R\dot{z}(s)\,ds. $$

Lemma 3

[25]

Let there exists positive numbersα, β, and a functionalV:ℝ×W[−τM,0]×L2[−τM,0]→ℝ such that
$$ \alpha\bigl|\phi(0)\bigr|^{2}\le V(t,\phi,\dot{\phi}) \le \beta\| \phi\|_{W}^{2}. $$
Let the function\(\bar{V}(t)=V(t,\, x_{t},\,\dot{x}_{t})\), wherext(θ)=x(t+θ), and\(\dot{x}_{t}(\theta)=\dot{x}(t+\theta)\)withθ∈[−τM,0] is continuous from the right forx(t) satisfying the system (7), absolutely continuous forttkand satisfies
$$ \lim_{t\rightarrow t_{k}^{-}}\bar{V}(t) \ge \bar{V}(t_{k}). $$
Through (7), \(\dot{\bar{V}}(t)\le -\epsilon|e(t)|^{2}\)forttkand for some scalarϵ>0, hence (7) is asymptotically stable.
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-012-0590-0/MediaObjects/11071_2012_590_Equf_HTML.gif

3 Main results

In this section, LMI criterions for global asymptotic synchronization is given in Theorem 1. The sampled-data feedback control gain K can be obtained from Theorem 1 in order to achieve the synchronization. Before proceeding to the main results, let us divide the sampling interval as [−τM,−η], and [−η,0]. Let N be any positive integer. There exists a real number α such that \(\alpha=\frac{\tau_{M}-\eta}{N}\). Thus, decomposition of the first interval yields [−(η+),−(η+(j−1)α)], where j=1,2,…,N.

Theorem 1

Consider the sampled-data controller (6). With prescribed nonlinear functions satisfying (8), the closed-loop error system (7) is said to be globally asymptotically stable, if there exist symmetric positive definite matricesP, S1, S2, Qj, Rj, W, diagonal matrixΛ, andUwith compatible dimensions such that the following LMI holds forγ>0 andu∈{1,2,…,l}:
$$ \tilde{\varPhi}_{u} <0, $$
(10)
where
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-012-0590-0/MediaObjects/11071_2012_590_Equg_HTML.gif
$$\tilde{\omega}_{i}= \begin{cases} S_{1}-S_{2}, & i=1,\\ \noalign {\vspace {1pt}} -S_{1}-S_{2}+Q_{1}-R_{1}, & i=2,\\ \noalign {\vspace {1pt}} Q_{i-1}-Q_{i-2}&\\ \noalign {\vspace {1pt}} -R_{i-1}-R_{i-2}, & i=3,\ldots,N+1,\\ \noalign {\vspace {1pt}} -Q_{N}-R_{N}, & i=N+2, \end{cases} $$
and
$$\mathcal{R}=\sum_{i=1}^{N}R_{i}. $$
Then theMaster (1) andSlave (2) systems are synchronized by the sampled-data controller (3). The sampled-data feedback control gain is given byK=U−1L.

Proof

Consider the following discontinuous Lyapunov–Krasovskii functional for the error system:
$$ \everymath{\displaystyle }\begin{array}{@{}rcl} V(t,\, e_{t},\,\dot{e}_{t}) &=& V_{1}(t)+V_{2}(t) + \sum_{j=1}^{N}V_{3j}(t)+V_{4}(t), \\ \noalign {\vspace {-10pt}} \end{array} $$
(11)
where
$$ V_{1}(t) = e^{T}(t)Pe(t), $$
(12)
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-012-0590-0/MediaObjects/11071_2012_590_Equ13_HTML.gif
(13)
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-012-0590-0/MediaObjects/11071_2012_590_Equ14_HTML.gif
(14)
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-012-0590-0/MediaObjects/11071_2012_590_Equ15_HTML.gif
(15)
Note that V4(t) can be rewritten as
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-012-0590-0/MediaObjects/11071_2012_590_Equ16_HTML.gif
(16)
where tkt<tk+1, for positive integer k≥0. Since \([e(s)-e(t_{k}-\eta) ]_{s=t_{k}-\eta}=0\), by the extended Wirtinger’s inequality, V4(t)≥0. Also V4(t) vanishes at t=tk. Hence, the condition
$$ \lim_{t\rightarrow t_{k}^{-}} V(t,\, e_{t},\,\dot{e}_{t}) \ge V(t_{k}) $$
holds. Taking the time derivative of (11) along the trajectories of (7), we have
$$ \dot{V}_{1}(t) = 2e^{T}(t)P\dot{e}(t), $$
(17)
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-012-0590-0/MediaObjects/11071_2012_590_Equ18_HTML.gif
(18)
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-012-0590-0/MediaObjects/11071_2012_590_Equ19_HTML.gif
(19)
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-012-0590-0/MediaObjects/11071_2012_590_Equ20_HTML.gif
(20)
Also, one can exhibit u such that u∈{1,2,…,N} for τ(t)∈[−(η+),−(η+(u−1)α)]. Then by Lemma 1, we have
$$ -\alpha\int_{t-(\eta+u\alpha)}^{t-(\eta+(u-1)\alpha)}\dot{e}^{T}(\theta)R_{u} \dot{e}(\theta)\,d\theta \le \tilde{\zeta}^{T}(t)\varOmega_{1}\tilde{\zeta}(t), $$
(21)
where
$$\tilde{\zeta}(t)=\left [\begin{array}{c} e(t-(\eta+(u-1)\alpha))\\ e(t-\tau(t))\\ e(t-(\eta+u\alpha))\end{array} \right ], $$
and
$$\varOmega_{1} = \left [\begin{array}{c@{\quad }c@{\quad }c} -R_{u} & R_{u} & 0\\ \star & -2R_{u} & R_{u}\\ \star & \star & -R_{u}\end{array} \right ]. $$
When ju again by using the Lemma 1, it is easy to get
$$ \everymath{\displaystyle }\begin{array}[b]{@{}l} -\alpha\int_{t-(\eta+j\alpha)}^{t-(\eta+(j-1)\alpha)} \dot{e}^{T}(\xi)R_{j}\dot{e}(\xi)\,d\xi \le \hat{\zeta}^{T}(t) \varOmega_{2}\hat{\zeta}(t), \\ \noalign {\vspace {-11pt}} \end{array} $$
(22)
where https://static-content.springer.com/image/art%3A10.1007%2Fs11071-012-0590-0/MediaObjects/11071_2012_590_IEq16_HTML.gif, and https://static-content.springer.com/image/art%3A10.1007%2Fs11071-012-0590-0/MediaObjects/11071_2012_590_IEq17_HTML.gif. Similarly, the last term in (11) can be written as
$$ \everymath{\displaystyle }\begin{array}[b]{@{}l} -\eta\int_{t-\eta}^{t} \dot{e}^{T}(\theta)S_{2}\dot{e}(\xi)\,d\xi \\ \quad \leq \left [\begin{array}{c} e(t)\\ e(t-\eta)\end{array} \right ]^{T} \left [\begin{array}{c@{\quad }c} -S_{2} & S_{2}\\ \star & -S_{2}\end{array} \right ] \left [\begin{array}{c} e(t)\\ e(t-\eta)\end{array} \right ]. \end{array} $$
(23)
Also, from the sector bounding conditions (4), and (8), for appropriately dimensioned diagonal matrix Λ>0 it can be obtained that
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-012-0590-0/MediaObjects/11071_2012_590_Equ24_HTML.gif
(24)
On the other hand, one can have the following equation for appropriately dimensioned matrix U and a scalar γ>0,
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-012-0590-0/MediaObjects/11071_2012_590_Equ25_HTML.gif
(25)
Now accumulating (7), (17)–(25), and letting L=UK, and
$$\zeta(t)=\left [\begin{array}{c} e(t-\tau(t)) \\ e(t) \\ e(t-\eta) \\ e(t-(\eta+\alpha)) \\ \vdots \\ e(t-(\eta+N\alpha)) \\ f(D^{T}e(t)) \\ \dot{e}(t)\end{array} \right ], $$
we get,
$$ \dot{V}(t,\, e_{t},\,\dot{e}_{t})\leq\zeta^{T}(t) \tilde{\varPhi}_{u}\zeta(t). $$
(26)
By Schur complements on (26), (10) guarantees that \(\dot{V}(t,\, e_{t},\,\dot{e}_{t})\le -\epsilon|e(t)|^{2}\), for some ϵ>0, and hence the closed-loop error system (7) is stable. Thus, the master and slave systems are synchronized asymptotically. The sampled data feedback control gain is given by K=U−1L. This completes the proof. □

Remark 1

Note that in proving above Theorem 1, the Lyapunov candidate \(\int_{t-\tau(t)}^{t}e^{T}(\xi) \tilde{R}\, e(\xi)\,d\xi\) is not used. That is, the information \(\dot{\tau}(t)=1\) has not been utilized due to discontinuous nature of sampled-data controller and the constructed Lyapunov functional. By this fact, it is important to mention that the obtained sufficient conditions for synchronization can be less conservative. Without utilizing the discontinuous nature of sampled-data controller, the following theorem can be deduced for the case of continuous Lyapunov functions for sampled-data synchronization.

Theorem 2

Consider the sampled-data controller (6). With prescribed nonlinear functions satisfying (8), the closed-loop error system (7) is said to be globally asymptotically stable, if there exist symmetric positive definite matricesP, S1, S2, S3, Qj, Rj, diagonal matrixΛ, andUwith compatible dimensions such that the following LMI holds forγ>0 andu∈{1,2,…,l}:
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-012-0590-0/MediaObjects/11071_2012_590_Equ27_HTML.gif
(27)
where\(\bar{\varPhi}_{u}\)contains the same entries of LMI (10), \(\tilde{\varPhi}_{u}<0\)other than the following entries:
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-012-0590-0/MediaObjects/11071_2012_590_Equn_HTML.gif
The feedback control gain isK=U−1L.

Proof

For obtaining the LMI condition for synchronization under sampled-data controller without utilizing the discontinuous nature, consider the Lyapunov functional as \(\tilde{V}(t)=V_{1}(t)+V_{2}(t)+\sum_{j=1}^{N}V_{3j}(t)+V_{5}(t)\), where \(V_{5}(e_{t},t)=\int_{t-\tau(t)}^{t}e^{T}(\xi)\, S_{3}\, e(\xi)\,d\xi\), and the other elements V1, V2, and V3j are given in (12), (13), and (14), respectively. Following the procedure given in the proof of the Theorem 1, the LMI condition (27) can be obtained. This completes the proof. □

Remark 2

Continuous time Lyapunov functionals for sampled-data synchronization are considered in [13, 14]. It must be noted that those obtained LMI conditions did not utilize the sawtooth structure of the sampling completely. On the other hand, Theorem 1 presents new LMI conditions considering the sawtooth structure (discontinuous in time) of sampling. Thus, the results are less conservative than [13, 14].

Moreover, Theorems 1 and 2 give the synchronization conditions for the master-slave system via the discontinuous Lyapunov functional and the continuous functional, respectively. The major difference between the above two Lyapunov functional lies in V4(t) by making use of the sawtooth structure of the sampling time and delay induced in ZOH, which has been adopted from [24, 25]. Thus, Theorem 1 can be less conservative than Theorem 2 and provides appropriate conditions for sampled-data control gain. Also, the derived criterion are verified through numerical simulations in Matlab which are presented in the next section.

4 Illustrative examples

In this section, in order to show the effectiveness of the proposed result, the Chua circuit system and horizontal platform system have been solved numerically using Matlab.

Chua system

Consider the master-slave synchronization of two identical Chua’s circuits system under sampled-data feedback control. Let us take the following representation of Chua’s circuit system:
$$ \begin{array}{@{}l} \dot{x}_{1}(t) = a\bigl(x_{2}(t)-\phi\bigl(x_{1}(t)\bigr)\bigr), \\ \noalign {\vspace {4pt}} \dot{x}_{2}(t) = x_{1}(t)-x_{2}(t)+x_{3}(t), \\ \noalign {\vspace {4pt}} \dot{x}_{1}(t) = -bx_{2}(t) \end{array} $$
(28)
with a nonlinear characteristic ϕ(x(t)) of Chua’s diode given by
$$ m_{1}x_{1}(t)+\frac{1}{2}(m_{0}-m_{1}) \bigl(\bigl|x_{1}(t)+c\bigr|-\bigl|x_{1}(t)-c\bigr|\bigr), $$
and parameters a=9, b=14.28, c=1, m0=−(1/7), m1=2/7. With these parameters, the Chua system produces the double scroll attractor which is given in Fig. 2. Based on [7, 14], the system can be represented in the Lur’e form with
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-012-0590-0/MediaObjects/11071_2012_590_Equp_HTML.gif
and h(ξ)=(14/2)(|ξ+1|−|ξ−1|) belonging to the sector [0 1].
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-012-0590-0/MediaObjects/11071_2012_590_Fig2_HTML.gif
Fig. 2

The chaotic attractor formed by Chua system (28)

Consider the controller of the form u(t)=K(p(tk)−q(tk)) with p(t)=x1(t),q(t)=y1(t). Solving the LMI (10) given in Theorem 1 by setting N=2, γ=0.6, and η=0.1 using Matlab LMI Toolbox, we obtain τM=0.3196. The corresponding solution and the control gain K are:
$$\begin{array}{@{}l} P=10e^{-006}\left [\begin{array}{c@{\quad }c@{\quad }c} 0.0657 & -0.1148 & 0.0124\\ -0.1148 & 0.5398 & -0.1330\\ 0.0124 & -0.1330 & 0.0389\end{array} \right ] , \\ \noalign {\vspace {5pt}} K=\left [\begin{array}{c@{\quad }c@{\quad }c} 1.8392 & 0.5215 & 0.3204\end{array} \right ]^{T} . \end{array} $$
For different values of η and γ, the obtained τM results are listed in Table 1 along with the corresponding control gain matrices. Also, while solving LMI (10), the LMI is found feasible when η≤0.24. That is the delay in ZOH that cannot be more than 0.24 in order to use the sampled-data control technique by the above proposed method.
Table 1

Sampling times and sampled-data feedback gain matrices for Chua system example

η

γ

τM

hmax

Control gain K

0

0.63

0.3764

0.3764

\(\left [\begin{array}{c} 2.0388 \\ 0.4100 \\ -0.0498\end{array} \right ]\)

0.1

1

0.2654

0.1654

\(\left [\begin{array}{c} 2.1064 \\ 1.1050 \\ 0.7069\end{array} \right ]\)

0.1

0.1

0.1044

0.0044

\(\left [\begin{array}{c} 4.0721 \\ 1.5508 \\ -1.1517\end{array} \right ]\)

0.1

0.5

0.3056

0.2056

\(\left [\begin{array}{c} 1.6064 \\ 0.7761 \\ 0.8481\end{array} \right ]\)

0.1

0.63

0.3200

0.2200

\(\left [\begin{array}{c} 1.9768 \\ 0.4413 \\ 0.0634\end{array} \right ]\)

0.2

0.63

0.2640

0.0640

\(\left [\begin{array}{c} 1.9707 \\ 0.4754 \\ 0.1457\end{array} \right ]\)

The maximum sampling time obtained by the LMI conditions in [13, 14] are 0.22 and 0.32, respectively, using continuous Lyapunov functionals. In this paper, we obtained 0.3764 as the maximum sampling time by using discontinuous Lyapunov functionals, which models the sawtooth structure of the samplings completely. Hence, the criterions presented in the Theorem 1 are less conservative than the existing results. Moreover, it is clear from Fig. 3 that the response of closed-loop error system converges to zero state asymptotic with control gain [2.03880.4100 −0.0498]T implying the occurrence of the synchronization. The sampled-data controller used in the simulation is illustrated in Fig. 4.
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-012-0590-0/MediaObjects/11071_2012_590_Fig3_HTML.gif
Fig. 3

The error state trajectories showing the stableness

https://static-content.springer.com/image/art%3A10.1007%2Fs11071-012-0590-0/MediaObjects/11071_2012_590_Fig4_HTML.gif
Fig. 4

State trajectories of sampled-data control signal fed into slave dynamics for Chua system example

Horizontal platform system

In offshore engineering and earthquake engineering, horizontal platform systems (HPS) are widely used [27]. The mathematical model of HPS can be described by
$$ \begin{array}{@{}l} a\ddot{x}+d\dot{x}+rg\sin x - \frac{3g}{R_0}(b-c)\cos x\sin x =m\cos \omega t, \\ \noalign {\vspace {-5pt}} \end{array} $$
(29)
where x is the rotation of the platform relative to the earth, a, b, c are the inertia moments of the platform for the three axes: d is the damping coefficient, r is the proportional constant of the accelerometer, g is the acceleration constant of gravity, R0 is the radius of the earth, and mcosωt is the harmonic torque. In [26], master–slave synchronization for HPS using a time delayed feedback control has been studied. The general master–slave synchronization scheme for HPS is given as
$$ \mathit{Master} : \left\{\begin{array}{@{}l} \dot{x}(t) =Ax(t)+Bh(Dx(t))+m(t),\\ \noalign {\vspace {3pt}} p(t) =x(t),\end{array} \right. $$
(30)
$$ \mathit{Slave} : \left\{\begin{array}{@{}ll} \dot{y}(t) &=Ay(t)+Bh(Dy(t))\\ \noalign {\vspace {3pt}} &\quad {}+m(t)+u(t),\\ \noalign {\vspace {3pt}} q(t) &=y(t),\end{array}\right. $$
(31)
where https://static-content.springer.com/image/art%3A10.1007%2Fs11071-012-0590-0/MediaObjects/11071_2012_590_IEq31_HTML.gif, https://static-content.springer.com/image/art%3A10.1007%2Fs11071-012-0590-0/MediaObjects/11071_2012_590_IEq32_HTML.gif, h(ξ)=sinξ−(la/2rg)sin2ξ, https://static-content.springer.com/image/art%3A10.1007%2Fs11071-012-0590-0/MediaObjects/11071_2012_590_IEq33_HTML.gif, \(D=(1\;0)\), and \(l=\frac{3g(b-c)}{aR_{0}}\) is a constant. It is an important point to note that the existing studies on synchronization for HPS have been focused on the linearized error dynamics and linear feedback control. To overcome this, constant time delayed feedback control [26] has been proposed and Lur’e form is used to represent the nonlinear dynamics of HPS. On the other hand, due to finite speed of transmission and spreading, the control signal feed into slave dynamics must be sampled. To accommodate this necessary requirement, different from the existing result [26], in this paper, the following sampled-data control is considered:
$$ \mathit{Control} : u(t)=K\, \bigl(x(t_{k})-y(t_{k}) \bigr), \quad t_{k}\le t<t_{k+1}. $$
(32)
Following the proposed scheme in Sect. 2, the error dynamics are formed with f(ξ) belonging to the sector [L,L+], where L=−1−|la/rg|, and L+=1+|la/rg|, ∀ξ∈ℝ. For this example, choose the parameters as a=0.3 kg mm, b=0.5 kg mm, c=0.2 kg mm, d=0.4 kg mm s−1, r=0.11559633 kg m, R0=6378000 m, g=9.8 ms−2, m=3.4 N, and ω=1.8 rad ms−1.
Solving the LMI (10) given in Theorem 1 for the parameters N=3, γ=0.5, and η=0.1 using the Matlab LMI Toolbox, we obtain τM=0.5338. The corresponding solution and the control gain K are
$$\everymath{\displaystyle }\begin{array}[b]{@{}l} P=10e^{-006}\left [\begin{array}{c@{\quad }c@{\quad }c} 0.6224 & -0.0055\\ \noalign {\vspace {2pt}} -0.0044 & 0.0425\end{array} \right ] , \\ \noalign {\vspace {5pt}} K=\left [\begin{array}{c@{\ }c} 1.0340 & -4.9237 \end{array} \right ]^{T} . \end{array} $$
Table 2 is presented to exhibit the obtained numerical results for HPS example under different values of η and γ and the obtained τM along with the corresponding control gain matrices. Sampled-data controller fed into slave HPS dynamics is plotted in Fig. 5. It is evident that the HPS are essentially a class of nonlinear systems which are widely used in earthquake engineering. The above obtained results for synchronization of HPS under the sampled-data controller is a new and different from the existing results. Representing the HPS in Lur’e form gives a freedom to approach the synchronization problem in a control theory point of view and the proposed results are more effective than the existing results in literature.
https://static-content.springer.com/image/art%3A10.1007%2Fs11071-012-0590-0/MediaObjects/11071_2012_590_Fig5_HTML.gif
Fig. 5

State trajectories of sampled-data control signal fed into slave dynamics for HPS example

Table 2

Sampling times and sampled-data feedback gain matrices for HPS example

η

γ

τM

hmax

Control gain K

0

1

0.5979

0.5979

\(\left [\begin{array}{c} 1.1019 \\ -6.9177 \end{array} \right ]\)

0

0.1

0.1605

0.1605

\(\left [\begin{array}{c} 4.2234 \\ -6.4887 \end{array} \right ]\)

0.1

0.5

0.5338

0.4338

\(\left [\begin{array}{c} 1.0340 \\ -4.9237 \end{array} \right ]\)

0.1

1

0.5410

04410

\(\left [\begin{array}{c} 1.1026 \\ -6.9127 \end{array} \right ]\)

5 Conclusion

In this paper, the global asymptotical synchronization problem of chaotic Lur’e systems using a sampled-data controller is addressed. A new discontinuous Lyapunov functional is constructed by considering the sawtooth structure of the sampling and effective synchronization linear matrix inequality sufficient conditions were obtained. Numerical simulations on the Chua system and horizontal platform system were clearly established. The application of the sampled-data system are vast and the study on synchronization of the sampled-data systems is our future work.

Copyright information

© Springer Science+Business Media B.V. 2012