Dynamics and Improved Robust Adaptive Control Strategy for the Finite Time Synchronization of Uncertain Nonlinear Systems

Dynamics and Improved Robust Adaptive Control Strategy for the Finite Time Synchronization of Uncertain Nonlinear Systems

Kammogne Soup Tewa Alain (Department of Physics, Faculty of Sciences, University of Dschang, Dschang, Cameroon), Kengne Romanic (Department of Physics, Faculty of Sciences, University of Dschang, Dschang, Cameroon) and Fotsin Hilaire Bertrand (Department of Physics, Faculty of Sciences, University of Dschang, Dschang, Cameroon)
Copyright: © 2017 |Pages: 29
DOI: 10.4018/IJSDA.2017100103
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This letter addresses a robust adaptive control for the synchronization method based on a modified polynomial observer (slave system) which tends to follow exponentially the chaotic Colpitts circuits brought back to a topology of the Chua oscillator (master system) with perturbations. The authors derive some less stringent conditions for the exponential and asymptotic stability of adaptive robust control systems at finite time. They provide a proof of stability and convergence (hence, that synchronization takes place) via Lyapunov stability method. That is, the observer (slave system) must synchronize albeit noisy measurements and reject the effect of perturbations on the system dynamics. To highlight their contribution, the authors also present some simulation results with the purpose to compare the proposed method to the classical polynomial observer. Finally, numerical results are used to show the robustness and effectiveness of the proposed control strategy.
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1. Introduction

The various models associated with the Chua’s circuit have been recently found helpful in demonstrating and explaining the various facets of chaos (Chua and Lin, 1900). Attempts are made in the literature to discover new members of the large family of Chua's circuits and to find their close relatives (Chua, 1993). Recently, many new three and four dimension (3D and 4D respectively) chaotic attractors have been proposed and the new derived chaotic system can be achieved by adding or changing the linear/nonlinear term of existing chaotic system (Vaidyanathan and Azar, 2015a,b,c,d; Vaidyanathan et al., 2015a,b,c; Moysis & Azar, 2017). The present communication deals with the well-known Colpitts oscillator, which was shown to be topologically similar to Chua’s circuit (Sarafian and Kaplan, 1995). The nonlinearity of the active device in the Colpitts oscillator is modified to be purely odd, then the circuit exhibits chaotic phenomena closely related to those exhibited by the classical Chua’s circuit. The impact of nonlinearity on circuit analysis for the design of new electronic devices have generated much interest in nonlinear circuit theory (Vaidyanathan et al., 2017a, 2017b; Volos et al., 2016; Azar et al., 2017a; Pham et al., 2017; Vaidyanathan and Azar, 2016a, 2016b, 2016c, 2016d). A key to the understanding the behaviour of dynamical system is essentially based on the topology of nonlinear term (Wang et al., 2017; Vaidyanathan, 2016e). There exists a robust relationship between the Colpitts oscillator and the Chua’s oscillator (Kennedy, 1995), one can show that both systems are not simply close relatives, but they are even strongly related and can be regarded as being conjugate one to another. The dynamics behavior of the Colpitts oscillator that results from this kind of modeling is reported in this paper and is related to the well-known chaotic behavior of Chua’s circuit. In the past, these works were essentially theoretical and the technological aspect was limited (Kennedy, 1995). It ought to be mentioned that the relationship between Colpitts and Chua oscillators with a symmetric nonlinearity leads to a singular phenomenon with only very few cases reported. The research on this model is motivated by the discovery of a simple third-order ordinary differential equations of the form whose solution are chaotic. The nonlinear function J is called a Jerk, because it describes the third-time derivative of acceleration in a mechanical system (Schot, 1978, Sundarapandian et al., 2017). We should note that, the representation of the Colpitts oscillator as jerky dynamics has another useful and important advantage: based on their jerky dynamics, a classification of different dynamical systems is possible, because the transformation of certain functionally different three-dimensional dynamical systems can lead to the same jerky dynamics (Ralf et al., 2002).

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