“Stabilizing” the stabilizing controllers

The main purpose of this paper is to revisit the internal/simultaneous/robust stabilization problems without assuming the existence of doubly coprime factorizations for the transfer matrices. Indeed, it has been recently shown in the literature that an internally stabilizable does not generally admit doubly coprime factorizations. Firstly, we give new necessary and sufficient conditions for internal stabilizability by means of matrix equalities. On these characterizations, the fact that internal stabilizability does not imply the existence of coprime factorizations becomes obvious. Secondly, we recall that a necessary condition for strong stabilizability is the existence of a doubly coprime factorization for the transfer matrix. Then, using the concept of stable range sr(A) of a ring A, introduced in (algebraic, topological) K-theory, we prove that sr(A) = 1 implies that every transfer matrix defined over the quotient field of A and which admits a leftor a rightcoprime factorization is strongly stabilizable. In particular, this result holds for A = H∞(D), H∞(C+), W+ and A(D), solving a question asked by A. Feintuch in [8]. Thirdly, we point out that the simultaneous stabilization problem is not equivalent to the strong stabilization problem if the plants do not admit doubly coprime factorizations. Using the fractional ideal approach to stabilization problems, we give a necessary and sufficient condition for a pair of SISO plants to be simultaneously stabilizable without assuming the existence of coprime factorizations. Finally, using the parametrization of all stabilizing controllers of an internally stabilizable SISO plant (which does not necessarily admit coprime factorizations), we show how to transform the non-linear sensitivity minimization problem into an affine, and thus, convex minimization problem.