On positive real transfer functions and the convergence of some recursive schemes

The convergence with probability one of a recently suggested recursive identification method by Landau is investigated. The positive realness of a certain transfer function is shown to play a crucial role, both for the proof of convergence and for convergence itself. A completely analogous analysis can be performed also for the extended least squares method and for the self-tuning regulator of Åström and Wittenmark. Explicit conditions for convergence of all these schemes are given. A more general structure is also discussed, as well as relations to other recursive algorithms.