Limit cycle analysis of nonlinear sampled-data systems by gain-phase margin approach

This work analyzes the limit cycle phenomena of nonlinear sampled-data systems by applying the methods of gain-phase margin testing, the M-locus and the parameter plane. First, a sampled-data control system with nonlinear elements is linearized by the classical method of describing functions. The stability of the equivalent linearized system is then analyzed using the stability equations and the parameter plane method, with adjustable parameters. After the gain-phase margin tester has been added to the forward open-loop system, exactly how the gain-phase margin and the characteristics of the limit cycle are related can be elicited by determining the intersections of the M-locus and the constant gain and phase boundaries. A concise method is presented to solve this problem. The minimum gain-phase margin of the nonlinear sampled-data system at which a limit cycle can occur is investigated. This work indicates that the procedure can be easily extended to analyze the limit cycles of a sampled-data system from a continuous-data system cases considered in the literature. Finally, a sampled-data system with multiple nonlinearities is illustrated to verify the validity of the procedure.