Stability of Adaptive Delta Modulators with a Forgetting Factor and Constant Inputs

Motivated by applications to feedback control over communication networks where the actuation and feedback signals are transmitted over communication channels, we study the stability of Adaptive Delta Modulators (ADM) when the coded signal is a constant. The importance of such a setting arises because a common control task is to track a dc input. In an earlier paper we had shown that a standard accumulator based adaptive delta modulator has the following characteristic: that virtually all combinations of the algorithm parameters result in 4-cycles, that the avoidance of 4-cycles requires a nongeneric initialization, and that steady state oscillations that generically arise in the course of these cycles can have amplitudes that can be arbitrarily close to the initial error. Consequently, in this paper we study the use of a forgetting factor in the ADM loop, and provide a detailed stability analysis and design guidelines.