Stability of the equilibria of adaptive systems with leakage estimator

A study is made of the stability of the equilibria of the differential equations that describe an adaptive controller in a closed loop with a linear time-invariant (LTI) undermodeled plant when the parameter update law is a leaky gradient, i.e. a σ-modified estimator. It has been shown by L. Hsu and R. Costa (1987) for the full-order case that under certain limiting conditions the resulting dynamic system has three, possibly unstable, equilibrius points. This result is extended by further characterizing the class of undermodeled LTI plants for which the equilibria exist and are (un)stable. It is shown that the equilibria are stable if a given compensator stabilizes the plant. This compensator is, up to the plant steady-state gain, known to the designer