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

The instability of adaptive schemes for small 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. A sharp boundary between stability and instability is signal dependent and much less demanding than the usual strict positive realness property. The new positivity condition has a simple signal energy interpretation.