State feedback control for tracking limit cycles of planar periodic systems

Abstract This paper introduces a novel method for tracking sampled planar limit cycles using a particular class of switching affine closed-loop systems, while simultaneously preserving the stability properties of the underlying oscillation. In contrast to prior approaches that either rely on periodic input signals to enforce stability or are restricted to tracking polygonal curves, the proposed framework synthesizes affine control laws for reference tracking. This enables formal guarantees for the existence of limit cycles as well as for stability, while ensuring accurate reference tracking. The synthesis of control parameters is formulated as a constrained numerical optimization problem. Initialized from reference data, the optimization minimizes the deviation between the target trajectory and the resulting limit cycle. Stability and tracking performance are explicitly enforced through constraints derived from analytic conditions. The fundamental principle and practical effectiveness of the method are demonstrated by means of a numerical example.