Minimal Degree Coprime Factorization of Rational Matrices

Given a rational matrix G with complex coefficients and a domain Γ in the closed complex plane, both arbitrary, we develop a complete theory of coprime factorizations of G over Γ, with denominators of McMillan degree as small as possible. The main tool is a general pole displacement theorem which gives conditions for an invertible rational matrix to dislocate by multiplication a part of the poles of G. We apply this result to obtain the parametrized class of all coprime factorizations over Γ with denominators of minimal McMillan degree nb—the number of poles of G outside Γ. Specific choices of the parameters and of Γ allow us to determine coprime factorizations, as for instance, with polynomial, proper, or stable factors. Further, we consider the case in which the denominator has a certain symmetry, namely it is J all-pass with respect either to the imaginary axis or to the unit circle. We give necessary and sufficient solvability conditions for the problem of coprime factorization with J all-pass denominator of McMillan degree nb and, when a solution exists, we give a construction of the class of coprime factors. When no such solution exists, we discuss the existence of, and give solutions to, coprime factorizations with J all-pass denominators of minimal McMillan degree (> nb). All the developments are carried out in terms of descriptor realizations associated with rational matrices, leading to explicit and computationally efficient formulas.