Limit cycles in high-order nonlinear systems

This paper investigates the stability of limit cycles in nonlinear feedback systems with several describing function (d.f.) predicted limit cycle solutions. The investigations showed that for a single-frequency limit cycle the necessary conditions for stability of Loeb and Gelb could be satisfied yet the limit cycle was unstable. Also the simple and often used graphical criterion was found to be incorrect. On the other hand, the incremental-describing function (i.d.f.) method used for both asynchronous and synchronous perturbations gave sufficient conditions, within the expected d.f. accuracy, for all the situations. It is further shown that the synchronous i.d.f. and the Loeb criterion are normally identical. The combined mode limit cycle oscillations were also satisfactorily predicted by a twoinput d.f. and i.d.f.