A New Algorithm to Search for a Limit Cycle in Nonlinear Control Systems

One normally solves for a limit cycle by solving a set of coupled nonlinear algebraic equations with the aid of a suitable optimization technique. In this paper a new approach is presented to search for a limit cycle locally in nonlinear control systems with multiple nonlinearities. Describing functions are used to derive a quasilinear model for the control system. Eigenvalue and eigenvector sensitivities are then employed to drive the least damped eigenvalue of the quasilinear model to the imaginary axis. It is shown that with the aid of the minimum norm solution one can vary the state variable amplitudes and the system parameters simultaneously. This results in an algorithm with accelerated convergence and improved numerical stability. A simple case study is included and the development of a simple criterion to test the stability of the limit cycle is also reported.