Adaptive T2 Fuzzy System for Control of Nonlinear Fractional Order Systems

What is it about?

This paper proposes an adaptive Interval Type-2 Takagi-Sugeno-Kang (IT2 TSK) fuzzy system with a supervisory mode to control and stabilize a certain class of nonlinear fractional order systems. In this study, a fractional order adaptation law is derived which adjusts the free parameters and bounds them by utilizing 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 nonlinear 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 stabilization and control of a general class of nonlinear fractional order systems perturbed by disturbance and uncertainty.

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Why is it important?

In this paper, a new adaptive IT2 TSK fuzzy controller is designed to control the uncertain chaotic fractional order systems. For the first time, a Fractional Order direct Adaptive IT2 TSK (FOAIT2 TSK) fuzzy controller is applied to estimate the ideal control signal for an unknown fractional order nonlinear system. The fractional order adaptation law is proposed to adjust the free interval parameters in the consequence part of the adaptive IT2 TSK controller. In addition, in the field of fractional order adaptive fuzzy controllers, the free parameters are uniformly bounded in the adaptation law for the first time by applying a fractional order projection algorithm. In a new application, a supervisory controller is also designed to be implemented parallel to FOAIT2 TSK fuzzy controller. By using the fractional order projection algorithm and the supervisory controller, the necessary and sufficient conditions are provided for applying fractional order Barbalat lemma. Hence, the global Mittag-Leffler stability of the closed-loop system with the uniformly bounded signals is guaranteed. Moreover, since the uncertainty and disturbance are inseparable parts of the fractional order systems, to compensate their effects, a new inference mechanism has been presented for FOAIT2 TSK fuzzy controller. In this new type-2 fuzzy system, antecedent parameters have type-2 fuzzy membership functions and variables of the consequent part include interval sets. Since the proposed inference method has a closed mathematical form, it is useful for control system design and stability analyse of the considered IT2 FLC. In addition, due to the simple nature of the introduced inference equation, it is easy to be implemented in the online and real-time applications.