This paper proposes a robust adaptive fuzzy PID control scheme augmented with a supervisory controller for unknown systems. In this scheme, a generalized fuzzy model is used to describe a class of unknown systems. The control strategy allows each part of the control law, i.e., a supervisory controller, a compensator, and an adaptive fuzzy PID controller, to be designed incrementally according to different guidelines. The supervisory controller in the outer loop aims at enhancing system robustness in the face of extra disturbances, variation in system parameters, and parameter drift in the adaptation law. Furthermore, an H∞ control design method using the fuzzy Lyapunov function is presented for the design of the initial control gains that guarantees transient performance at the start of closed-loop control, which is generally overlooked in many adaptive control systems. This design of the initial control gains is a compound search strategy called conditional linear matrix inequality (CLMI) approach with IROA (Improved random optimal algorithm), it leads to less complex designs than a standard LMI method by fuzzy Lyapunov function. Numerical studies of the tracking control of an uncertain inverted pendulum system demonstrate the effectiveness of the control strategy. From results of this simulation, the generalized fuzzy model reduces the rule number of T-S fuzzy model indeed.

This study introduces the fuzzy Lyapunov function to the fuzzy PID control systems, modified fuzzy systems, with an optimized robust tracking performance. We propose a compound search strategy called conditional linear matrix inequality (CLMI) approach which was composed of the proposed improved random optimal algorithm (IROA) concatenated with the simplex method to solve the linear matrix inequality (LMI) problem. If solutions of a specific system exist, the scheme finds more than one solutions at a time, and these fixed potential solutions and variable PID gains are ready for tracking performance optimization. The effectiveness of the proposed control scheme is demonstrated by the numerical example of a cart-pole system.