Stabilization of Recurrent Fuzzy Systems via Sum of Squares-based Hybrid Control

This paper presents an approach for stabilization of equilibria in recurrent fuzzy systems. This type of dynamic fuzzy systems being defined via linguistic rules can be interpreted as interpolation between constant gradients, and therefore as hybrid dynamical system. It is shown that the latter viewpoint allows for a precise description of the system dynamics, but on the other hand lacks transparency. In order to render a given equilibrium of the recurrent fuzzy system globally asymptotically stable, local polynomial controllers are computed via sum of squares optimization to allow only for deterministic mode transitions on a micro level. In addition, the controlled recurrent fuzzy system can then be interpreted as finite deterministic automaton, thus allowing for analysis of system properties on a more abstract macro level. Relaxations are proposed in cases where recurrent fuzzy systems may not be rendered deterministic and the method is applied to two examples.