Computer based methods are developed for the analysis, synthesis and design of linear compensators for a wide class of single-loop non-linear feedback systems characterized by a separable non-linear element of any general multivalued form. Preliminary analysis is performed in the complex frequency domain applying approximate analytical methods to predict and quantify possible modes of limit cycle operation. Graphical displays are used to determine initial compensator structure which will ensure the avoidance of the critical region in this domain. Subsequently, a numerical synthesis procedure is applied to adjusting the compensator parameters to meet a set of additional time domain functional subject to specified system constraints and, if possible, the absence of autonomous limit-cycle operation.