Comments on the Distributed-Lag Operator

The response of closed-loop systems having plants with distributed lag is an important and complicated problem which appears amenable only to approximate methods of solution. The response of such systems has been treated, for example, by using sampled-data-systems concepts-a digital computer, however, is required. Root-locus methods have been suggested, but not implemented, which eliminate the requirement for a computer; consequently, system responses, root loci, and pole locations for specific gains (characteristics of general interest) have not been given in the literature. As a first step in remedying this lack, in this paper a root-locus-type method is applied to a specific plant (of wide interest) with distributed lag. As a result, plots of exact and approximate low-order solutions are given, data on root locations and coefficients for higher order approximate solutions are tabulated, and typical root loci are sketched. It is found that low-order rational approximation results in reasonably exact system responses. Further, it is shown that the domain of the variables for valid approximations is not (essentially) the entire complex plane but rather excludes the fourth quadrant. Equivalently, the first quadrant is excluded for complex-conjugate values.