Control of Chaotic Resonance Phenomena using Prototypes in Manifold Forms

Abstract

This paper presents a modeling strategy using prototypes in manifold forms for unknown chaotic behaviors, observed possibly in power electronics, robot manipulators, lasers, etc.; it models from only a finite number of one dimensional time series observables in order to simplify both the modeling and the controller design. With the intention of dividing the operating space into a set of smaller regions, the Self-Organizing Map (SOM) is employed as a modeling groundwork and prototypes in manifold forms attached to the SOM are created in the least square sense for each region. Once a set of prototypes representing the operating space is established, the regional controllers associated with the prototypes are designed with a traditional PID control law. Switching of the controllers is done synchronously by the SOM, which chooses the regional operating space, linked with an active regional prototype valid in a certain operating regime. Simulations on the chaotic oscillator regulating the unknown chaos to a fixed point or a stable periodic orbit illustrate the efficiency of the proposed modeling and control architecture.