Frequency domain conditions for parameter convergence of a multivariable adaptive identifier

A multivariate adaptive identifier is designed on the basis of a canonical form leading to a unique set of parameters to identify. Frequency-domain conditions for parameter convergence are derived. The sufficient condition to guarantee convergence in all cases is that p +v distinct frequency components be present in each input of the plant, where v is the maximum of the observability indexes and p the order of the plant. On the other hand, a necessary condition requires p+2v distinct frequency components distributed over the inputs and at least v frequency components in each separate input. Through examples, it can be shown that convergence may or may not occur for an intermediate number of frequency components, depending on their location and on the plant parameters