Controller synthesis for stabilizing periodic orbits in forced nonlinear systems

Delayed feedback controllers are an appealing tool for the stabilization of periodic orbits in nonlinear systems. Unfortunately, their inherent infinite dimensional structure prevents the definition of reliable design procedures. This paper considers the use of finite dimensional linear time invariant controllers for the stabilization of periodic solutions in a general class of sinusoidally forced nonlinear systems. Such controllers-which can be interpreted as rational approximations of the delayed ones-provide an LMI-based synthesis technique, by mixing results concerning absolute stability of nonlinear systems and robustness of uncertain linear systems. In particular, the synthesis algorithm is able to select the controller maximizing a lower bound of the maximum amplitude of the forcing input, for which the corresponding periodic solutions are guaranteed to be stable. A single-mode CO₂ laser is employed to illustrate the main features of the developed synthesis technique