Generalized Functions and a Delay-Integrator with Negative Feedback

摘要

An analysis, using the techniques of system theory, of a difference-differential equation in an application involving generalized functions. The tools of Laplace transforms, block diagrams, and timing diagrams are used to obtain the Green's function or impulse response, and the response to impulses of higher order. The physical unit that gives rise to difference-differential equations is the delay-integrator with negative feedback. It is connected by an electrical transmission line to a direct coupled (operational, analog computer) amplifier. The critical value of the delay parameter is determined, and stable and unstable responses are exhibited at different parameters. The main result is that the impulse response of the system with time-lag is the Green's function only if the dynamics are expressed in terms of the output; without delay, the impulse response is the Green's function of the state equations themselves. (Author)