On the Asymtotic Behaviour of an Adaptive Pole-Placement Algorithm

Abstract

In this paper convergence of an adaptive pole-placement algorithm based on the minimization of a predefined criterion is analysed. It is supposed that an external random excitation with possibly decreasing variance is injected into the system. The main result is the proof of the convergence w.p. 1 of the regulator parameter estimates to the optimal ones, i.e. convergence w.p.1 of the closed loop poles to the desired ones, derived in two methodologically different ways. Convergence is ensured under mild conditions concerning the number of estimated parameters and requiring irreducibility of the optimal regulator transfer function. It is demonstrated that the algorithm converges even in the case of an external random excitation whose variance tends to zero, despite the fact that the persistence of excitation is not satisfied. Consistency proof for the parameter estimates in minimum variance self-tuning control algorithms can be derived as a special case.