The Sequential Construction of Minimal Partial Realizations from Finite Input–Output Data

Abstract

Any strictly proper transfer function matrix of a continuous or discrete, linear, constant, multivariable system can be written as the product of a numerator polynomial matrix with the inverse of another polynomial matrix, the denominator. Since a realization is easily constructed from the polynomial matrix representation, the minimal partial realization problem is translated to that of extracting -a minimal order partial denominator polynomial matrix from a finite length matrix sequence. It is shown that minimal partial denominator matrices evolve recursively that is, a minimal partial denominator matrix for any finite length sequence is a combination of the minimal partial denominator matrices of its proper subsequences. A computationally efficient algorithm that sequentially constructs a minimal partial denominator matrix for a given finite length sequence is presented. A theorem by Anderson and Brasch leads to a definition of uniqueness for the resulting denominator matrix based upon its invariant factors. Parameters used during execution of the algorithm are shown to be sufficient for enumerating all invariant factor sets in the equivalence class of minimal partial realizations. The results apply to continuous and discrete linear systems including finite state machines.