On multivariable partial realization†

In this paper some aspects of the partial realization problem for multivariable formal power series are considered. The dynamic structure of the systems underlying such series is analysed on the basis of the results of Kalman, and a canonical realization algorithm is described based on this concept of structure. Special attention is paid to the aspect of uniqueness of the realization; necessary and sufficient conditions are given for the uniqueness of the extension sequence. A specific class of power series is defined for which a very simple uniqueness criterion is derived. Generally, noise-disturbed data sequences belong to this specific class; it is shown that the unique partial realization for such a noise-disturbed power series may serve as an intermediate step to arrive at an approximate realization of limited order.