Chaos control of new Mathieu-van der Pol systems by fuzzy logic constant controllers

In this paper, a new fuzzy logic controller—fuzzy logic constant controller (FLCC) is introduced to chaotic signals controlling. The main ideas of the FLCC are described as follows: (1) proving the two chaotic systems are going to achieve asymptotically stable via Lyapunov direct method; (2) via detecting the sign of the errors, the appropriate fuzzy logic control scheme is operated; (3) choosing the upper bound and the lower bound of the error derivatives of the chaotic signals to be the consequent parts (corresponding controllers). Due to controllers in traditional method – derived by Lyapunov direct method, are always complicated, nonlinear form or the functions of errors, a new simplest controller—FLCC is presented in this paper to synchronize two chaotic signals. Through the FLCC, there are three main contributions can be obtained: (1) the mathematical models of the nonlinear chaotic systems can be unknown, all we have to do is eywords: haos control uzzy logic constant controller LCC ew Mathieu–van der Pol systems capturing the signals of the unknown systems; (2) through the fuzzy logic rules, the strength of controllers can be adjusted via the corresponding membership functions (which are decided by the values of error derivatives); (3) by the FLCC, the chaotic system can be much more exactly and efficiently controlled to the trajectory of our goal than traditional ones. Three cases, original point, regular function and chaotic Qi system (with large values of initial conditions), are given to illustrate the effectiveness of our new FLC. i system . Introduction Since Ott et al. [1] gave the famous OGY control method in 990, the applications of the various methods to control a chaotic ehavior in natural sciences and engineering are well known. For xample, the adaptive control [2–5], the method of chaos control ased on sampled data [6], the method of pulse feedback of systemtic variable [7], the active control [8,9] and linear error feedback ontrol [10,11]. However, when Lyapunov stability of zero soluion of states is studied, the stability of solutions on the whole eighborhood region of the origin is demanded. In recent years, some chaos control based on fuzzy systems has een proposed since the fuzzy set theory was initiated by Zadeh 12], such as fuzzy sliding mode controlling technique [13,14], MI-based synchronization [15] and extended backstepping slidng mode controlling technique [16]. The fuzzy logic control (FLC) cheme has been widely developed for almost 40 years and as been successfully applied to many applications [17]. Many esearchers have worked to improve the performance of the FLCs nd ensure their stability. Li and Gatland in Refs. [18,19] proposed ∗ Tel.: +886 3 5712121x55179; fax: +886 3 5720634. E-mail address: [email protected] 568-4946/$ – see front matter © 2011 Elsevier B.V. All rights reserved. oi:10.1016/j.asoc.2011.08.024 © 2011 Elsevier B.V. All rights reserved. a more systematic design method for PDand PI-type FLCs [20]. Choi et al. [21] presents a single input FLC ensuring stability. Ying [22] presents a practical design method for nonlinear fuzzy controllers, and many other researchers have results on the matter of the stability of FLCs, in Castillo et al. [23] and Cázarez et al. [24] was presented an extension of the Margaliot work [25] to built stable type-2 fuzzy logic controllers in Lyapunov sense. Recently, Yau and Shieh [26] proposed an amazing new idea to design fuzzy logic controllers—constructing fuzzy rules subject to a common Lyapunov function such that the master-slave chaos systems satisfy stability in the Lyapunov sense. In Ref. [26], there are two main controllers in their slave system. One is used in elimination of nonlinear terms and the other is built by fuzzy rules subject to a common Lyapunov function. Therefore, the resulting controllers are nonlinear form. Otherwise, in Ref. [26], the regular form is necessary. In order to carry out the new method, the original system must to be transformed into their regular form. In this paper, we propose a new strategy which is also constructing fuzzy rules subject to a Lyapunov direct method. Error dynamics are used to be upper bound and lower bound. Through this new approach, a simplest controller, constant controller, can be obtained and the difficulty in realization of complicated controllers in chaos synchronization by Lyapunov direct method can be also coped. Unlike conventional approaches, the resulting S.-Y. Li / Applied Soft Computing 11 (2011) 4474–4487 4475