Limit Cycle Conditions in Optimalizing Controllers

Abstract

A method is developed for determining search limit cycle frequency and magnitude in single dimensional optimalizing (peak holding) systems. The performance criterion measurement process is modeled by a nonlinear function (often adequately approximated by a parabola) in cascade with a linear filter, denoted G(jω), representing measurement delay. The analysis is based upon a single describing function (DF) of the logic circuitry, the parameter adjustment integrator, and the performance criterion nonlinearity. The DF is frequency dependent, adding some complexity to the standard, although seldom applied, graphical solution method, but the procedure is otherwise straightforward. Limit cycle amplitude and frequency are determined directly from DF and G locus intersection values in the usual way. Prediction accuracy averages approximately 5 percent for the examples considered, and is often even better. As should be expected, greatest predictive accuracy is obtained when considerable smoothing (measurement filtering) is present in the feedback loop. When measurement filtering is negligible, limit cycle conditions are readily obtained by more direct methods.¹