On a General Structure of the Stabilizing Controllers Based on Stable Range

In this paper, we prove that some stabilizing controllers of a plant, which admits a left/right-coprime factorization, have a special form where their stable and unstable parts are separated. The dimension of the unstable part depends on the algebraic concept of stable range of the ring A of SISO stable plants. Moreover, we prove that, if the stable range of A is equal to 1, then every plant—defined by a transfer matrix with entries in the quotient field of A and admitting a left/right-coprime factorization—can be stabilized by a stable controller (strong stabilization). In particular, using a result of Treil proving that the stable range of H∞(D) is equal to 1, we show that every stabilizable plant—defined by a transfer matrix with entries in the quotient field of H∞(D) or H∞(C+)—is strongly stabilizable and, equivalently, every couple of stabilizable plants can be simultaneously stabilized by a controller (simultaneous stabilization). Finally, using the fact that the topological stable range of H∞(D) is equal to 2, a result due to Suárez, we show that every unstabilizable SISO plant—defined by a transfer function with entries in the quotient field of H∞(D)—is as close as we want to a stabilizable plant in the product topology.