Model reference fuzzy adaptive control for uncertain dynamical systems with time delays

This paper investigates a model reference fuzzy adaptive control (MRFAC) scheme for uncertain dynamical systems with known structures but unknown parameters which are dependent on known variables, multiple delayed state uncertainties, and disturbances. Each delayed uncertainty is assumed to be bounded by an unknown gain. A fuzzy basis function expansion (FBFE) is used to represent the unknown parameters of the controlled system from the strategic manipulation of the model following tracking errors. The proposed MRFAC scheme uses two on-line estimations, which allows for the inclusion of identifying the gains of the delayed state uncertainties and training the weights of the FBFE simultaneously. Stability and robustness of the MRFAC scheme is analyzed in the sense of Lyapunov. It is shown that the proposed control scheme can guarantee parameter estimation convergence and stability robustness of the closed-loop system with the model following tracking errors uniformly ultimately bounded in the presence of plant parameter uncertainties, delayed state uncertainties, and external disturbances. The theoretical results are evaluated through a gyroscopic system with a single actuating input.

1 Introduction

Fuzzy control with adaptation can provide an effective solution to the control of plants that are complex, unknown parameters, or unknown variations in plant parameters, and have available quantitative knowledge from repetitive adjustment of the system with better performance than those of fuzzy controls with constant rule bases, especially for systems with nonlinearities [1], [3], [5], [6], [9], [10], [12]. Recently, several stable adaptive fuzzy control schemes have been introduced to provide good results to the trajectory tracking [5], [7], [10]. Their scheme requires the assumptions that the dynamics of the system is exactly known and is feedback linearizable with well-defined vector relative degree. However, in most cases nonlinearities existing in the dynamical system are not known a priori and feedback linearization is less suited for systems with significant nonminimum phase effects [1], [3]. To facilitate the tracking control with fast convergence of nonlinear dynamical systems, Golea et al. [5] proposed a fuzzy model reference adaptive controller using Takagi-Sugeno (T-S) fuzzy controller and PI type adaptation law with the inclusion of a priori analytic information and Vishnupad and Shin [6] presented an adaptive fuzzy tuner for the optimization of non-linear, multi-variable systems while the gradient-descent method is used to adaptively tune the bases of the membership functions used in the fuzzy logic optimization.