Information pattern for linear discrete-time models with stochastic coefficients

A linear discrete-time system with constant coefficients has an information pattern that does not grow in complexity with time, an information state. This paper shows that a particular canonical form for such systems, the phase-variable model, retains this property when coefficients vary as Gaussian time series with rational spectra. The distribution of the output conditional on past data is normal, and its parameters, being functions of the information state, can be calculated in real time in a simple way. The property is fundamental for effective optimal control. A priori characteristics for the Gaussian time series must be specified, but a maximum likelihood method is proposed for estimating any unknown characteristics from a long sample of input-output data. Also, a parameter-free statistic is found for testing the validity of the phase-variable model in actual cases.