Synthesis of Limit Cycles and Reference Tracking via Switching Affine Systems

This paper from arXiv (2605.06181) tackles the challenge of approximating limit cycles in nonlinear ODEs with switching affine systems. Previous work was largely confined to simple two-region or low-dimensional (often planar) partitions. The authors break this limitation by using more general partitions in higher-dimensional state spaces, augmented with external signals. They formulate the synthesis task as a constrained numerical optimization problem: starting from sampled data of the nonlinear dynamics, the method minimizes the error between the data and the limit cycle generated by the switching affine model, while enforcing stability constraints to guarantee global stability. This makes the approach suitable for data-driven modeling of complex oscillatory systems.

Building on the approximation scheme, the paper addresses reference tracking for switching affine systems with periodic behavior. While the approximation uses a common Lyapunov function for global stability, the tracking approach switches to multiple Lyapunov functions to achieve less conservative convergence results. The principle and effectiveness are demonstrated through a set of examples. This work opens up practical applications in control systems where periodic behaviors need to be modeled or tracked, such as in robotics, power electronics, and biological systems, offering a systematic way to design stable limit cycles from data.