The dynamics of bursting in simple adaptive feedback systems with leakage

Three low-order simple examples of adaptive systems representative of rarely related applications are considered. These are model reference adaptive control, adaptive echo cancellation, and autoregressive moving average predictors. Bursting can be observed in the three applications when the input signal are not persistently exciting. It is shown that bursting is the time-domain manifestation of oscillatory behavior and that when the adaptive algorithms use leakage, it is possible to give bounds to the signals during bursts. In these operating conditions, bursting is transient and it is possible to estimate the length of time it will take to disappear. The conclusion to be drawn from the bifurcation analysis is that an appropriate level of leakage is necessary to provide protection against bursting due to a specific level of disturbances. Addition of leakage is associated with artificially forcing the parameter estimates towards a value which is determined a priori and is not associated with data-driven adaptation. In operation under more reasonable, i.e., nonbursting conditions, this leakage has a performance loss implication.>