A stability-instability boundary for disturbance-free slow adaptation and unmodeled dynamics

The instability of adaptive schemes at very low or even infinitesimal values of parameter adjustment gain has the form of a slow drift of adjustable parameters. With an averaging analysis we derive not only a sufficient condition for stability of this drift, but also a sufficient condition for it to be unstable. The result is a sharp boundary between stability and instability, which has a simple interpretation of signal energy.