Delayed feedback controllers are an appealing tool for stabilization of periodic orbits in chaotic systems. Despite their conceptual simplicity, specific and reliable design procedures are difficult to obtain, partly also because of their inherent infinite-dimensional structure. This chapter considers the use of finite dimensional linear time invariant controllers for stabilization of periodic solutions in a general class of sinusoidally forced nonlinear systems. For such controllers which can be interpreted as rational approximations of the delayed ones we provide a computationally attractive synthesis technique based on Linear Matrix Inequalities (LMIs), by mixing results concerning absolute stability of nonlinear systems and robustness of uncertain linear systems. The resulting controllers prove to be effective for chaos suppression in electronic circuits and systems, as shown by two different application examples.

20.1 Introduction

Periodic motion is a classical and thoroughly investigated topic in nonlinear systems science. Recently, a strong renewed interest has been observed in the subject of stability of periodic solutions due to its significant relations with chaos control problems (see ^[1] and references therein), where stabilization of periodic dynamics is often a satisfactory target. More precisely, the underlying key idea for low-energy feedback control of chaos is to stabilize one among the infinite unstable periodic orbits embedded in the chaotic attractor.

Among other approaches, delayed feedback controllers represent a simple and appealing tool for stabilizing...