Design of State Feedback Adaptive Fuzzy Controllers for Second-Order Systems Using a Frequency Stability Criterion

This paper proposes a design method for second-order systems that guarantees the stability of an adaptive fuzzy controller with state feedback. The system consists of a linear unstable plant, the Takagi-Sugeno controller, a nonlinear reference model, and an adaptation mechanism. In this paper, gradient-based adaptation is used to change the consequents of the controller rules so that the closed-loop system behaves like the reference model. The proposed method utilizes a frequency-domain criterion in the form of the modified circle theorem. The controller function is assumed to be a nonlinearity described by a sector condition, which means that the function lies between two planes. During the process of adaptation, this function is verified so it stays in the sector guaranteeing stability. The method described here is illustrated by an example of a control system containing an unstable plant with an unknown pole in the right half-plane. The motivation of this paper is to propose a frequency-domain method for the design of state feedback adaptive fuzzy controllers. Comparing with time-domain methods that require advanced software tools, the proposed method offers simple graphical interpretation on the Nyquist plane.