ABSTRACT
This chapter reviews the interaction between the unknown system and the adaptive filter that can be nicely arranged as a feedback loop consisting of a linear part closed by a nonlinear part. It gives an overview of the hyperstability theorem and positive real transfer functions. The chapter shows how the system identification problem can be structured in a manner amenable to the application of hyperstability theory, leading to a direct form algorithm which is provably convergent in the sufficient order, noise-free case. It examines that the undermodelled case have shown the absence of divergence in the undermodelled case, under many difficult technical assumptions, but convergence properties have been difficult to ascertain. It addresses a discrete-time proof of the hyperstability theorem. The chapter shows how hyperstability theory can be applied to the noise-free, sufficient-order identification problem. It concludes that the strict positive real condition on ℱ(z) is sufficient for closed-loop stability.
