Stability of adaptive delta modulators with 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. It is known that a standard accumulator-based ADM has the following highly undesirable characteristic: virtually all combinations of the algorithm parameters result in 4-cycles, and the avoidance of 4-cycles requires a nongeneric initialization. Further, the 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, we study the use of a forgetting factor in the ADM loop, and provide a detailed stability analysis and design guidelines. Intuitively, adding a forgetting factor to the classical ADM algorithm prevents 4-periodic cycles from occurring by damping them. In particular, we show that for suitably chosen design parameters, the ADM with forgetting factor can track a constant signal arbitrarily closely under mild assumptions. We provide simulations to demonstrate how much better the modified algorithm performs relative to the original ADM algorithm in a remote control setting.