Frequency domain analysis of HOSM systems

The frequency domain analysis of dynamically perturbed Higher Order Sliding Mode (HOSM) systems is tackled using Describing Function (DF) and Harmonic Balance (HB) techniques. The goal of this analysis is to study possible limit cycles in such systems. DFs of Nested 3rd and 4th order algorithms are obtained for the first time. Then, HB equation is used to analyze the real sliding motion in the HOSM system, where the sliding set converges to a limit cycle. Chattering (limit cycle) in the HOSM systems is studied, and the chattering parameters (amplitude and frequency) are computed. A definition of Tolerance Limits, which characterizes the acceptable performance in the real HOSM system, is applied to verify if the chattering parameters fit the amplitude and frequency limits. Next, Performance Phase Margin and Performance Gain Margin definitions, which give the metrics for robustness of real HOSM to unmodeled dynamics, are applied to assess the robustness of the limit cycle emerged in the real HOSM system. Examples and simulations that validate the obtained results are presented.