Necessary and sufficient conditions stability for -input, -output convolution feedback systems with a finite number of unstable poles

This paper considers -input, -output convolution feedback systems characterized by and , where the open-loop transfer function contains a finite number of unstable multiple poles and is a constant nonsingular matrix. Theorem 1 gives necessary and sufficient conditions for stability. A basic device is the following: the principal part of the Laurent expansion of at the unstable poles is factored as a ratio of two right-coprime polynomial matrices. There are two necessary and sufficient conditions: the first is the usual infimum one, and the second is required to prevent the closed-loop transfer function from being unbounded in some small neighborhood of each open-loop unstable pole. The latter condition is given an interpretation in concepts of McMillan degree theory. The modification of the theorem for the discrete-time case is immediate.