Stability analysis of an adaptive system with unmodelled dynamics!

Stability bounds for fast and slow adaptation are derived for a simple adaptive system with one unmodelled (‘parasitic’) pole which is approximated by a right half-plane zero. In fast adaptation the product of the adaptive gain and power of the reference input must be smaller than the parasitic pole. In slow adaptation most of the input signal energy must be in the frequency range lower than the square root of the parasitic pole. It is shown that a plant bypass ensures a global stability property by making the linear time-invariant part of the adaptive loop strictly positive real (SPR). The price paid is an increase of the tracking error at very low frequencies and with slow adaptation when the equilibrium of the system without bypass is exponentially stable.