Non-linear effects of action of magnetic fields on models of the tumour growth

Abstract

Applications of adaptive systems require a certain robustness of the interconnected estimator/controller structure. By restriction to identification based parameter adaptive schemes we prove asymptotic (Ljapunov-) stability of the Least-Squares parameter estimator under persistent excitation and examine the stability of the adaptive loop. However, Ljapunov stability proofs do not allow to evaluate the control performance of the system relative to a certain performance index. In the main part of the contribution we therefore establish time-varying stability margins of parameter estimators by extension of results in sector stability theory of constant gain optimal systems. We will furthermore introduce optimality margins for Riccati-observers and extend the results to time-varying optimality margins of LS-estimators. We analyse the results referring to the estimator’s long time behaviour and robustness. The results are promising with regard to on-line evaluation of local stability and optimality of parameter adaptive systems and contribute to a deeper understanding of the relation between persistent excitation, robustness and adaptation. The results may also be applied for new adaptation strategies in adaptive control of time varying processes.