Nonlinear Dyn DOI 10.1007/s11071-017-3709-5 ORIGINAL PAPER Synchronization and stabilization of fractional order nonlinear systems with adaptive fuzzy controller and compensation signal Pouria Jafari · Mohammad Teshnehlab · Mahsan Tavakoli-Kakhki Received: 16 November 2016 / Accepted: 31 July 2017 © Springer Science+Business Media B.V. 2017 Abstract In this paper, a direct adaptive fuzzy con- troller with compensation signal is presented to control and stabilize a class of fractional order systems with unknown nonlinearities. Based on a Lyapunov function candidate the global Mittag–Leffler stability is proved and a new fractional order adaptation law is derived. The adaptation law adjusts free parameters of the fuzzy controller and bounds them by utilizing a novel frac- tional order projection algorithm. Furthermore, due to the use of compensation term, the proposed approach does not demand suitable membership functions in the fuzzy system. In addition, the stability of the closed- loop system is guaranteed by utilizing a supervisory controller. Numerical simulations show the validity and effectiveness of the introduced scheme for various frac- tional order nonlinear models that perturbed by distur- bance and uncertainty. Keywords Direct adaptive fuzzy control · Compen- sation control · Fractional order nonlinear system · Fractional order Lyapunov theory · Supervisory control P. Jafari · M. Teshnehlab (B ) Intelligent Systems Laboratory (ISLab.), Faculty of Electrical Engineering, K.N. Toosi University of Technology, Tehran, Iran e-mail: teshnehlab@eetd.kntu.ac.ir M. Teshnehlab · M. Tavakoli-Kakhki Industrial Control Center of Excellence, Faculty of Electrical Engineering, K.N. Toosi University of Technology, Tehran, Iran 1 Introduction The fractional calculus that was started by the ideas of Leibniz in 1695 represents the generalization of stan- dard differential calculus up to non-integer orders [1, 2]. This field remained ignored by applied sciences dur- ing several centuries, due to the lack of application backgrounds, insufficient geometrical interpretation, and many conflicting definitions of fractional means. However, in the last years, because of their increased flexibility (with respect to integer order equations) which allows a more precise modeling of complex sys- tems, the fractional order equations have progressively attracted attention and play a considerable role in var- ious areas, such as: mathematics, physics, electronics, chemistry, mechanics, and control theory [3–7]. Furthermore, fractional order controllers (FOCs) have so far been implemented to modify the robust- ness and the performance of the control applications. In 1996, Oustaloup proposed the idea of fractional calculus in control of dynamic systems [8] where he demonstrated the superior performance of the CRONE (French abbreviation for Commande Robuste d’Ordre NonEntier) method over the classical PID controller. Later, Podlubny introduced a fractional order PID con- troller and demonstrated the better efficiency of this type of controllers, in comparison with the integer PID controllers, when used to control of fractional order systems (FOSs) [9]. The control of FOSs by FOCs has begun to receive a lot of consideration recently [10–14] and different fractional order stability theo- 123