General analysis and stability study of finite pulsed feedback systems

Abstract

A finite pulsed system is defined as a system containing one or more samplers whose outputs are pulses of finite width and amplitude. In this report a method is presented for the exact analysis of finite pulsed feedback systems. The actual pulse width and the pulse shape are considered in its analytical treatment. The technique is based on the superposition principle, as well as some operational properties of the Laplace transform. It is applicable to a general linear system. The application of this method to systems with periodic samplers leads to a set of simultaneous first-order difference equations with constant coefficients. These equations can be solved for C(z), the z-transform of the output at t = nT. The stability of the system can be determined from the roots of C(z). It is found that the stability conditions are independent of the input but are functions of the pole-zero configurations of both the open- and closed-loop transfer functions, as well as the sampler characteristics. The technique is demonstrated by means of an example in the Appendix. Tables are also included to reduce the algebraic burden. These will be found useful in solution of second- and third-order systems. They can be extended whenever the need arises. The responses of an error pulsed second-order system to a step, ramp, and a sinusoidal input function are also plotted to show the effect of different pulse widths. These indicate that, in each case, as the pulse width h approaches the sampler period the output approaches the equivalent continuous system response. In a similar way it can be shown that as h-0 and the gain of the system is increased inversely as h, the output corresponds to the equivalent sampled-data system.