A method For the numerical synthesis of compensators for non-linear control systems is proposed. The non-linear systems investigated are the type in which the system can be represented by two separate parts: a linear element and a non-linear element, as shown in Fig, 1. The approach adopted is called the method of inequalities, in which the problem is formulated as a set of inequalities, which represent the specified closed loop performance of the system as well as constraints of an engineering and financial kind. The non-linear part of the system is linearized using an exponential input describing function. An inequality is then formulated concerning the non-linear element to ensure that every point in the admissible space represents a closed loop system which does not exhibit autonomous limit cycle oscillation. This inequality is defined by using an enhanced sinusoidal describing function which takes account of the effects of higher harmonic signal components as well as of a fundamental sinusoid in the search for regions of possible limit cycle operation.