Spectral factorizations with unmixed pole and zero sets
Automatica (Journal of IFAC)
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In this paper a new characterization of the class of all minimal square spectral factors of a given rational spectral density is presented. This characterization, which is established without assumptions on poles and zeroes of the spectral density, extends a result presented in [A. Ferrante, IEEE Trans. Automat. Control, 39 (1994), pp. 2122{2126]. The characterization consists of two bijective maps which relate the set of minimal square spectral factors to a set of invariant subspaces of a certain matrix and to a set of symmetric solutions of an algebraic Riccati equation (ARE). In the second part of the paper it is proven that these two maps are homeomorphisms. This result extends and applies to spectral factorization theory recent results of H. Wimmer [Integral Equations Operator Theory, 21 (1995), pp. 362{375], where it is proven that the well-known relation between solutions of ARE and invariant subspaces of a certain matrix is, in fact, a homeomorphism.