In this work an adaptive sliding mode controller in the presence of uncertainty, as well as the external disturbance is considered. A concise introduction and investigation of the dynamic behavior of a novel class of chaotic systems with fractional order derivatives for synchronization is presented. It is supposed that the high bounds of uncertainty and external disturbance are unknown. The proposed controller is designed based on error dynamics and acceptable adaptive laws. The sliding mode dynamic stability and the condition to start sliding are proved by Lyapunov stability theory. With this new proposed approach, Chen and Lorenz system with fractional order derivatives are synchronized. Finally, simulation results with MATLAB software showed that the designed comparative sliding mode controller was able to synchronize chaotic systems with fractional order derivatives in the presence of the mentioned adverse factors. The main characteristic of the proposed method compared to other methods is providing acceptable adaptive laws for satisfactory functioning against uncertainty and external disturbance and eliminate the chattering phenomenon for synchronization of non-identical chaotic systems with fractional order derivatives.

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This paper presents a novel adaptive control for a class of nonlinear switched systems by introducing a sufficient condition for stabilization. Based on the possible instability of all sub-systems, a variable structure (VS) switching rule with an adaptive approach and sliding sector was offered. Moreover, the stability condition of the system can be determined by solving linear matrix inequalities (LMIs) to ensure asymptotic stability. The application of H∞ analysis of nonlinear switched systems was also investigated through the design of the mentioned adaptive control system and defining a VS switching rule. Finally, simulation results were presented to validate the novelty of the proposed method.

INTRODUCTIONThe broad field of chaos theory has been among the most interesting issues researchers have studied in recent decades. Concepts related to chaos theory and its related disciplines are employed in different fields, including medicine (Sarbaz et al., 2012;Aghababa and Borjkhani 2014;Provata et al., 2012), economics (Pan et al., 2012;Airaudo and Zanna, 2012), functioning of laser diodes (Banerjee et al., 2012;Gao, 2012), and mathematics (Hosseinalipour, 2013;Kupka, 2014). The control of chaos-related phenomena has attracted wide attention from many different kinds of researchers.Although the idea of fractional-order operators has a history as long as that of integer-order operators, interest in this field is expanding due to an increasing amount of attention from scientists and mathematicians. In recent decades, fractional-order operators have been the driving force behind an increasing number of investigations. With the development of studies in this area, practical and theoretical investigations into the application of fractional-order operators in engineering sciences have now become widespread in the academic community (Padula and Visioli, 2014;Pakzad et al., 2013;Tripathy et al., 2015a;Tripathy et al., 2015b). For example, fractional-order calculations have been applied in mechanical and electrical engineering, biology, economics, and mathematics, among other fields (Hernandez et al., 2014;Wang and Li, 2014;Cortes and Elejabarrieta, 2007).A chaotic system is a highly complex, dynamic nonlinear system. The important and salient characteristics of chaotic systems include their extreme sensitivity to initial conditions, making the synchronisation of chaotic systems vital. The rapid increase in interest in fractional-order chaotic systems has manifested in investigations into the chaotic behavior of fractional-order horizontal platform systems (Aghababa, 2014) and in the proliferation of other published articles on these systems (Yin et al., 2013;Li and Chen, 2014;Li and Tong, 2013).During the past two decades, the control and synchronisation of fractional-order and integer-order chaotic systems have largely attracted scientists and researchers due to their potential applications in secure communications, biological systems, medicine, and other fields. For example, an active-control technique has been provided for the identical and non-identical synchronisation of fractional-order chaotic systems (Srivastava et al., 2014). In a different application, function-projective synchronisations (FPS) of identical and non-identical modified finance systems (MFS) and the Shimizu-Morioka system (S-MS) have been studied via active control technique (Kareem et al., 2012). Fast projective synchronisation of fractional-order chaotic and reverse chaotic systems with its application to an affine cipher using date of birth (DOB) have also been reviewed (Muthukumar PROCEEDINGS OF THE LATVIAN ACADEMY OF SCIENCES. Section B, Vol. 71 (2017)