A study on new right/left inverses of nonsquare polynomial matrices
International Journal of Applied Mathematics and Computer Science - SPECIAL SECTION: Efficient Resource Management for Grid-Enabled Applications
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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 $H_{\infty}(\mathbb{D})$ is equal to 1, we show that every stabilizable plant---defined by a transfer matrix with entries in the quotient field of $H_{\infty}(\mathbb{D})$ or $H_{\infty}({\mathbb{C}}_+)$---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 $H_{\infty}(\mathbb{D})$ 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 $H_{\infty}(\mathbb{D})$---is as close as we want to a stabilizable plant in the product topology.