Abstract
A laboratory model exhibiting the property of distributed lag has been built. The model consists of elements made of teledeltos paper and a silver paint surface by laminating film. Treated as a distributed line it was terminated in its characteristic impedance and used in a closed-loop system to obtain measured results for comparison with theoretical calculations. A Riemann surface is used to investigate the locus of the singularities of the closed-loop system containing distributed lag alone and with a simple time lag. The transform w=s is introduced to facilitate this investigation. On the w plane the constant gain in decibels and constant argument curves are shown to be linearly spaced vertical and horizontal lines respectively. Each s plane pole or zero appears on the w plane as a pair of poles or zeros. This allows determination of the locus of the singularities of the closed-loop system by locating only a few points. The w plane representation of distributed lag corresponds to the s plane representation of dead time. On transformation from the w plane back to the s plane it is observed that the singularities of the closed-loop system are of the form (square root)s= a plus or minus jb. Minimum loop gain and the frequency of oscillation are determined from the loci of the roots on the w plane for several systems and these are compared to those measured for the corresponding system. A time domain solution is presented. This solution is obtained by evaluating the complex inversion integral for the Laplace transformation using a digital computer. The solution is carried out for a step input to a closed-loop system containing distributed lag alone and with simple time lag. The Fortran listing of the computer program for this and comparisons of computed with measured results are included.
Ball, Billie Joe (1962). Distributed lag in feedback control systems. Doctoral dissertation, Texas A&M University. Texas A&M University. Libraries. Available electronically from
https : / /hdl .handle .net /1969 .1 /DISSERTATIONS -171716.