Synthesis of MIMO controllers for extending the stability range of periodic solutions in forced nonlinear systems

This paper deals with a central issue in bifurcations and chaos control applications, i.e., the stabilization of periodic motions in sinusoidally forced nonlinear systems. Specifically, the problem of designing multi-input-multi-output (MIMO) finite-dimensional linear time-invariant controllers maximizing the amplitude of the sinusoidal input for which the corresponding periodic solutions are guaranteed to be stable, is considered. By exploiting linearization techniques and the multi-variable circle criterion, a synthesis algorithm is developed to determine the controller which maximizes a suitable lower bound of the amplitude of the input. The algorithm requires the solution of a sequence of linear matrix inequalities (LMIs) of increasing size. The Brusselator oscillator is employed as a case study to show that the synthesized controllers, though optimizing a lower bound, provide quite satisfactory control performance.