On the stability of a certain class of nonlinear sampled-data systems

Abstract

A sufficient condition for stability of a class of sampled-data feedback systems containing a memory-less, nonlinear gain element is obtained. The new stability theorem for the class of systems discussed requires that the following relationship be satisfied on the unit circle: <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\Re G^{\ast}(z)[1 + q(z - 1)] + \frac{1}{K} - \frac{K'|q|}{2} | (z - 1)G^{\ast}(z)|^{2} \leq 0</tex> . In this papers the stability criterion embodied in this theorem can be readily obtained from the frequency response of the linear plant. This method is essentially similar to Popov's method applied to the study of nonlinear continuous systems. Furthermore, Tsypkin's resuits for the discrete case are obtained as a special case when <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q=0</tex> . Several examples are discussed, and the results are compared with Lyapunov's quadratic and quadratic plus integral forms as well as with other methods. For these examples, the results obtained from the new theorem yield less conservative values of gain than Lyapunov's method. Furthermore, for certain linear plants the new theorem also yields the necessary and sufficient conditions.