Parameter Convergence in the Recursive Identification of Multivariable Systems

A multivariable 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 n + v distinct frequency components be present in each input of the plant, where v is the maximum of the observability indexes, and n the order of the plant. On the other hand, the necessary condition requires n + mv distinct frequency components distributed over the inputs, where m is the number of inputs, and at least v frequency components in each separate input. Through examples, it is shown that convergence may or may not occur for intermediate number of frequency components, depending on their location and on the plant parameters.