An Algebraic Interpretation to the Operator-Theoretic Approach to Stabilizability. Part I: SISO Systems

The purpose of this paper is to show that a duality exists between the fractional ideal approach [23, 26] and the operator-theoretic approach [4, 6, 8, 9, 33, 34] to stabilization problems. In particular, this duality helps us to understand how the algebraic properties of systems are reflected by the operator-theoretic approach and conversely. In terms of modules, we characterize the domain and the graph of an internally stabilizable plant or that of a plant which admits a (weakly) coprime factorization. Moreover, we prove that internal stabilizability implies that the graph of the plant and the graph of a stabilizing controller are direct summands of the global signal space. These results generalize those obtained in [6, 8, 9, 33, 34]. Finally, we exhibit a class of signal spaces over which internal stabilizability is equivalent to the existence of a bounded inverse for the linear operator mapping the errors e1 and e2 of the closed-loop system to the inputs u1 and u2. Mathematics Subject Classifications (2000): 93C05, 93D25, 93B25, 93B28, 16D40, 30D55, 47A05.