Proving stability of delta-sigma modulators using invariant sets

An invariant set, S, is a set of points in state space having the property that all trajectories emanating from points in S remain in S. Such sets are useful in the context of delta-sigma modulators since an invariant set yields rigorous theoretical bounds on the state variables and so establishes the stability of the modulator. This paper extends previously reported work for the second-order modulator with a constant input to higher-order modulators and time-varying inputs. An invariant set for a 3^rd-order delta-sigma modulator is given which definitively proves that this modulator is stable