Synchronization of chaotic systems under sampled-data control
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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 controlSynchronization1 Introduction
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 X≥Y (respectively, X>Y), where X and Y are symmetric matrices, means that X−Y is a positive semidefinite (respectively, positive definite). ℜ^{n} and ℜ^{n×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 τ_{M}>0 and \(\mathcal{C}([-\tau_{M},0];\Re^{n})\) denote the family of continuous functions φ from [−τ_{M},0] to ℜ^{n} 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
(H1)
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]
Lemma 2
([24] Extended Wirtinger Inequality)
Lemma 3
[25]
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 [−τ_{M},−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+jα),−(η+(j−1)α)], where j=1,2,…,N.
Theorem 1
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
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 V_{1}, V_{2}, and V_{3j} 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 V_{4}(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
Sampling times and sampled-data feedback gain matrices for Chua system example
η | γ | τ_{M} | h_{max} | 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 | \(\left [\begin{array}{c} 1.9707 \\ 0.4754 \\ 0.1457\end{array} \right ]\) |
Horizontal platform system
Sampling times and sampled-data feedback gain matrices for HPS example
η | γ | τ_{M} | h_{max} | 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.