New zero-input overflow stability proofs based on Lyapunov theory

The authors demonstrate some proofs of zero-input overflow-oscillation suppression in recursive digital filters. The 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, spiraling toward the origin, as long as no overflow correction is applied. Following this state signal, an energy function that is a natural candidate for a Lyapunov function can be defined. For the second-order direct-form digital filter with a saturation characteristic, this energy function is a Lyapunov function. However, it is not the only possible Lyapunov function of this filter. All energy functions with an energy matrix that is diagonally dominant guarantee zero-input stability if a saturation characteristic is used for overflow correction. The authors determine the condition that a general second-order digital filter has to fulfil so that there exists at least one energy function with a matrix that is diagonally dominant