Adaptive type‐2 fuzzy system for synchronisation and stabilisation of chaotic non‐linear fractional order systems

This study proposes an adaptive interval type‐2 Takagi–Sugeno–Kang (IT2 TSK) fuzzy system with a supervisory mode to control and stabilise a certain class of non‐linear fractional order systems. In this study, a fractional order adaptation law is derived which adjusts the free parameters and bounds them by utilising a projection algorithm. The global Mittag–Leffler stability of the closed‐loop system is proved in the sense that all the involved signals are uniformly bounded. Moreover, if the non‐linear system tends to be unstable, a supervisory controller starts cooperating with the adaptive IT2 TSK fuzzy controller to guarantee the stability of the closed‐loop system. In addition, a new inference mechanism for the adaptive IT2 TSK fuzzy system is introduced for which the antecedent part is chosen as a type‐2 fuzzy set and the consequent parameters are represented as interval sets. According to the practical nature of the proposed inference equation, it would be applicable in online and real‐time applications. Numerical simulations show the validity and effectiveness of the introduced control strategy for stabilisation and control of a general class of non‐linear fractional order systems perturbed by disturbance and uncertainty.