Canonical Matrix Factorisation and Polynomial Riccati Equations

The polynomial matrix factorisation of an arbitrary Hermitian matrix with additional constraints on the column degrees of a factor is considered. It is proved that the successive factor extraction algorithm always finds the solution if it exists. A necessary and sufficient condition for the existence ofa factorisation is found in terms of poles of the inverse matrix. Polynomial Riccati equations are introduced. They have a similar structure to algebraic Riccati equations but coefficients and the unknown matrix are polynomial. The equivalence of polynomial matrix factorisation, polynomial Riccati equations and algebraic Riccati equations is established. Singular cases of these problems are considered which correspond to singularity of matrices in the corresponding H ∞ control problem. A numerical example illustrates the proposed algorithms.