On Certain Convergence Questions in System Identification

This paper examines the asymptotic properties of the maximum likelihood estimates of the unknown parameters and the unknown initial state of linear, stable, constant coefficient, discrete time dynamic systems where plant noise and observation noise are present. Necessary and sufficient conditions are obtained for the system parameter estimates to converge with probability one, to be asymptotically normal and to converge in mean square. These conditions require that the system representation be unique and impose a simple constraint on the input sequence. Under these conditions, the initial state estimate is shown to be asymptotically unbiased and have finite covariance.