Relative stability of closed-loop systems

Frequency response plots have long been used to predict transient as well as steady-state responses of closed loop linear systems, avoiding the exact and laborious solution of differential equations. This paper considers the relations between specific properties of frequency response plots and the corresponding properties of the early portions of the time responses. It is shown that the common output/input transfer function plots are in reality plots of the Laplace transform of the impulse (transient) response. The Laplace transform of a time function is a real integral in which an exponential kernal is multiplied by the time function, and the product summed from time zero to infinity. This process amounts to a mechanism by which the kernal scans the time function as the frequency is varied. This paper gives a method for interpreting Laplace transform plots and extending their use, and presents a method for estimating the size, frequency, damping, and phase of oscillations in the corresponding time response. ¿Standard¿ plots are included to aid in estimating particular properties. The method is relatively quick in application, and makes use of functions and techniques already familiar in Laplace transform analysis. In addition, it is not limited to the responses of systems having rational transforms, or to responses to impulses or step inputs, but can be used to estimate the response to almost any desired input, and is applicable to systems having time-varying elements.