Uncertain Fractional Order Systems Control by Fractional Order Sliding Mode Indirect Adaptive Fuzzy Controller

Nowadays, the noticeable growth of fractional order calculus in engineering science has transformed this research field into one of the most popular fields of research in engineering and in particular in control engineering. Accordingly, several controllers have been designed based on fractional order mathematics. Adaptive fuzzy controllers are among the designed fractional order controllers. In the literatures, it has been shown that the uncertain fractional order systems can be well compensated by fractional order adaptive fuzzy controllers in the presence of disturbance. For this reason, in this paper, an indirect adaptive TSK fuzzy controller with fractional order sliding mode control is proposed to control a certain class of nonlinear fractional order systems.The stability of the closed-loop system is investigated and using a fractional order Lyapunov function candidate, the adaptation laws are extracted. A fractional order adaptation law is used to adjust the free parameters in the consequence part of the adaptive TSK system and a robust adaptive law is used to reduce the influence of the approximation error between the output of the system nonlinear model and the output of the fuzzy system. Hence, using the fractional order Lyapunov theorem, the Mittag-Leffler stability of the closed-loop system is guaranteed. Finally, the numerical simulation shows the efficiency of the proposed control strategy for a gyroscope uncertain nonlinear fractional order system in the presence of uncertainty.