The absolute stability of high-order discrete-time systems utilizing the saturation nonlinearity

Previously, we investigated the stability of a class of discrete-time filters involving a restricted class of linear systems, the all-pole systems, and a single, specific, memoryless nonlinearity-the saturation nonlinearity. A rather effective criterion was derived for determining if a given system is free of all periodic oscillations except the trivial null solution. The derivation relied on the observation that certain rather exclusive properties, called passivity properties, are associated with the saturation nonlinearity. Here, we generalize the results in two directions. First, the linear part of the system is allowed to be quite general, i.e., both poles and zeros may occur in the transfer function. Thus in the context of digital filters, the restriction to the direct-form realization is lifted. Secondly, the criterion obtained here guarantees the absolute stability of the system and not just the absence of nontrivial periodic oscillations. Thus the possibility of aperiodic or "chaotic" behavior is eliminated. The broadened formulation enhances the utility of the criterion, making it applicable to the large class of recursive digital processing systems, including digital filters, which use the saturation arithmetic to cope with overflow in the course of forming the sum of the results of many multiplicative operations. Also, the methods developed indicate an approach for the stability investigation of large digital transmission networks containing nonlinearities. The technique used for deriving the new result is entirely different. The new element is the demonstration that the hypothesis which is in the form of a condition in the frequency domain implies the existence of a Lyapunov function.