Robust stabilization of recurrent fuzzy systems via switching control

Abstract

A method for stabilization of known equilibria in recurrent fuzzy systems is presented, which particularly accounts for model uncertainties. Since the dynamics of recurrent fuzzy systems are defined over a rectangular grid, it is first observed that for stability analysis, only gradient conditions at grid points have to be considered given that the inputs are piecewise constant. Therefore, a robust structure variable controller is proposed, switching between constant inputs. In order to prevent the system from deadlock phenomena due to the switching of the system, the structure variable control is augmented by a piecewise polynomial controller, guaranteeing asymptotic stability. The proposed method is applied to the example of an inverted pendulum. I. INTRODUCTIONAmong the many concepts of utilizing fuzzy logic for modeling of dynamic processes, two generally different approaches can be identified: The first is driven by the need for high precision of the system model, which mostly leads to dynamic fuzzy models of high complexity. The second approach tries to incorporate the basic idea of fuzzy logic to model a system in a transparent and linguistically interpretable way. Recurrent fuzzy systems [1], piecewise bilinear systems [2] and more generally fuzzy systems with singleton consequences [3] can be subsumed in the latter class. While offering a high degree of linguistic interpretability, their drawback is the inherit model uncertainty stemming from the approximate nature of these dynamic fuzzy systems.Nevertheless, given an approximate model, it is desirable to control the process despite model uncertainties. The question on how to stabilize equilibria in recurrent fuzzy systems by means of fuzzy controllers was already addressed (see, e.g., [4]), neglecting robustness issues. For the closely related class of piecewise bilinear systems, [5] presented an approach based on feedback linearization, taking uncertainties into account.This paper now presents a control strategy for recurrent fuzzy systems based on structure variable control, taking model uncertainties explicitly into consideration. Because the dynamics of a recurrent fuzzy system can be interpreted as being defined piecewise over polytopes, a key observation concerning stability analysis is that for constant control inputs, dynamic properties only have to be considered at the vertices of the polytopes, which reduces the computational complexity significantly. This observation is akin to a line