A nyquist type criterion for the stability of multivariable linear systems

This paper considers the L2 - stability on n-input/output linear time-invariant feedback systems. It proves that the ring of all linear, continuous operators mapping L2 into itself is isomorphic to a commutative ring K(0) of holomorphic and bounded complex functions in the open right half plane. Necessary and sufficient conditions for stability of n-input/output systems are then derived from the conditions of invertivility 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. An essential aspect of the stability criterion is the conditions for the invertibility of a matrix A over K(0) which is related to the geometric properties of an appropriate region ?? of the complex plane. It is shown that these geometric properties can be deduced from simple encirclement conditions in the plane involving the loci of the eigenvalues of A.