Feedback Stabilization over Commutative Rings: The Matrix Case

Abstract

This paper provides a solution of the feedback stabilization problem over commutative rings for matrix transfer functions. Stabilizability of a transfer matrix is realised as local stabilizability over the entire spectrum of the ring. For stabilizable plants, certain modules generated by its fractions and that of the stabilizing controller are shown to be projective compliments of each other. In the case of rings with irreducible spectrum, this geometric relationship shows that the plant is stabilizable if and only if the above modules of the plant are projective of ranks equal to the number of inputs and the outputs. If the maxspectrum of the ring is Noetherian and of zero (Krull) dimension, then this result shows that the stabilizable plants have doubly coprime fractions. Over unique factorization domains the above stabilizability condition is interpreted in terms of matrices formed by the fractions of the plant. Certain invariants of these matrices known as elementary factors, are defined and it is shown that the plant is stabilizable if and only if these elementary factors generate the whole ring. This condition thus provides a generalization of the coprime factorizability as a condition for stabilizabilty. A formula for the class of all stabilizing controllers is then developed that generalizes the previous well-known formula in factorization theory. For multidimensional transfer functions these results provide concrete conditions for stabilizabilty. Finally, it is shown that the class of polynomial rings over principal ideal domains is an additional class of rings over which stabilizable plants always have doubly coprime fractions.