On the absolute stability of automatic control systems

Two methods prevail in the literature for establishing the frequency-stability criterion of systems containing a single nonlinearity. The purpose of the present investigation is to bridge this gap by establishing the relationship between the Popov approach and Lyapunov approach. The result, which is an extension of Lyapunov´s direct method, is general in application, and it is shown that the problem on the existence of a Lyapunov function is transformed to the existence of a positive-definite function satisfying a certain condition. The method is particularly suitable for converting stability conditions from the time domain to the frequency domain, and applications to continuous and sampled-data systems are given.