Identiﬁcation and the information matrix: how to get just sufﬁciently rich?
In prediction error identiﬁcation, the information matrix plays a central role. Speciﬁcally, when the system is in the model set, the covariance matrix of the parameter estimates converges asymptotically, up to a scaling factor, to the inverse of the information matrix. The existence of a ﬁnite covariance matrix thus depends on the positive deﬁniteness of the information matrix, and the rate of convergence of the parameter estimate depends on its “size”. When the system is not in the model set, the nonsingularity of the information matrix at all identiﬁable values of the parameter vector is a necessary condition for the asymptotic convergence of the identiﬁcation algorithm. The information matrix is also the key tool in the solution of optimal experiment design procedures, which have become a focus of recent attention. Introducing a geometric framework, we provide a complete analysis, for arbitrary model structures, of the minimum degree of richness required to guarantee the nonsingularity of the information matrix. We then particularize these results to all commonly used model structures, both in open loop and in closed loop. In a closed-loop setup, our results provide an unexpected and precisely quantiﬁable trade-off between controller degree and required degree of external excitation.