On Stability of Linear Varying-Parameter Systems

Abstract

A linear varying-parameter system is defined to be stable if and only if every bounded input produces a bounded output. It is shown that a necessary and sufficient condition for stability is that the impulsive response of the system W(t, τ) should be integrable (considered as a function τ) for all t. From this result and the fact that the system function H(s; t) is the laplace transform of W(t, τ), it is deduced that a necessary condition for stability of a linear varying-parameter system is that the system function H(s; t) should be analytic and bounded in the right half and on the imaginary axis of the s-plane for all t. This result represents a generalization of the familiar frequency domain criterion which is commonly used in connection with fixed systems. The generalized criterion is applied to the investigation of stability of a variable feedback system.