Stability theorems for the relaxation of the strictly positive real condition in hyperstable adaptive schemes

The hyperstability theorems of Popov have played an important role in establishing the convergence of adaptive schemes, notably adaptive output error identification and adaptive control. The error system of these schemes has the form of a feedback loop with a time-invariant forward path and a passive time-varying feedback path. The strict positive realness of the forward path suffices to establish asymptotic stability of the feedback loop and therefore establishes convergence of the adaptive scheme. In this paper we study conditions which preserve the asymptotic stability but permit relaxation of the strict positive real condition at high frequencies, subject to restrictions on algorithm gain parameters and frequency content of the input signal. These theorems are important for the design of robust adaptive methods.