Adaptive Control for Modified Projective Synchronization-Based Approach for Estimating All Parameters of a Class of Uncertain Systems: Case of Modified Colpitts Oscillators
A method of estimation of all parameters of a class of nonlinear uncertain dynamical systems is considered, based on the modified projective synchronization (MPS). The case of modified Colpitts oscillators is investigated. Through a suitable transformation of the dynamical system, sufficient conditions for achieving synchronization are derived based on Lyapunov stability theory. Global stability and asymptotic robust synchronization of the considered system are investigated. The proposed approach offers a systematic design procedure for robust adaptive synchronization of a large class of chaotic systems. The combined effect of both an additive white Gaussian noise (AWGN) and an artificial perturbation is numerically investigated. Results of numerical simulations confirm the effectiveness of the proposed control strategy. 1. Introduction Synchronization of chaotic systems and their potential applications in wide areas of physics and engineering sciences is currently a field of great interest ([1, 2] and references cited therein). The first idea of synchronizing two identical chaotic systems with different initial conditions is introduced by Pecora and Carroll [3] and the method is realized in electronic circuits. Synchronization techniques have been improved in recent years, and many different methods are applied theoretically and experimentally to synchronize the chaotic systems which include back stepping design technique [4], projective synchronization (PS) [5], modified projective synchronization (MPS) [6, 7], generalized synchronization [8], adaptive modified projective synchronization [9], lag synchronization [10], anticipating synchronization [11], phase synchronization [12], and their combinations [13]. Synchronization may involve several systems without a prescribed hierarchy (bidirectional) as it is the case in synchronization of networks of systems [14, 15], often happening naturally, for instance, in certain biological systems. Another intensive area of research to emphasize within bidirectional synchronization is the study of the consensus paradigm (see an excellent text in [16]). Amongst all kinds of chaos synchronization, MPS is the state-of-the-art of synchronization schemes. MPS means that the master and slave systems could be synchronized up to a constant scaling matrix. Recently, various control methods which include adaptive control [17, 18] and active control [7, 19, 20] have been introduced. Most of the works done on MPS of chaotic systems have used active control method since it is easy to design a control input and to deal with
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