Abstract
This paper considers the L 2 stability of n-input/n-output linear, time-invariant feedback systems. It is demonstrated that the ring of all linear, bounded, time-invariant (LBTI) operators of L 2 into itself is isomorphic with a commutative ring K(0) of bounded and holomorphic complex functions with domain the open right-half plane (ORHP). Necessary and sufficient conditions for n-input/n-output stability are derived from the conditions of invertibility of matrices over the ring K(0). Furthermore, a comprehensive analysis is given of the geometric interpretation of the stability conditions leading to a generalized Nyquist criterion. It is shown that the properties of a system matrix A∈K(0)n×n associated with its invertibility in K(0)n×ncan be deduced from simple encirclement conditions in the complex plane involving the loci of the eigenvalues of A.
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