New zero-input overflow stability proofs based on Lyapunov theory

Abstract

In this paper we demonstrate some new proofs of suppressing zero-input overflow oscillations in resive digital filters. These proofs are based on the second method of Lyapunov For second-order digital filters with complex conjugated poles the state describes a trajectory in the phase plane, spiralling towards the origin, as long as no overflow correction is applied. Following this state signal an energy function can be defined,which is a natural candidate for a Lyapunov function. For the second-order direct form digital filter with a saturation characteristic this energy function is a Lyapunov function. However, this function is not the only possible Lyapunov function of this filter. All energy functions with anenergy matrix that is diagonally dominant, guarantee zero-input stability, if a saturation characteristic is used for overflow correction. In this paper we determine the condition a general second-order digital filters has to fulfil so that there exists at least one energy function with a matrix, which is diagonally dominant.