Design of a 3×3 multivariable feedback system with large plant uncertainty†

Abstract

A recent paper presented a quantitative synthesis theory based on Schauder's fixed point theorem, for uncertain multiple input-output feedback systems. This paper is devoted to practical design execution, by means of a detailed 3×3 problem with large plant uncertainty, including open-loop instability and some non-minimum-phase plant elements. The outstanding features of the theory are : 1. The quantitative nature of the assigned tolerances on the nine closed-loop system response functions. In this problem four are specified basically non-interacting and the other five interacting. 2, The design problem is translated into separate quantitative single-loop problems, each with plant uncertainty, external disturbances and closed-loop tolerances derived from the original problem. The solutions for these single-loop problems are guaranteed to solve the original multivariable problem, whose system characteristic equation need never be examined. 3. There is developed a systematic means of (a) optimization of the loops, wherein loops may be improved at no expense to any other loops, until equilibrium is achieved and this is no longer possible, (b) if desired, trade-offs between the loops where some may be sacrificed for the improvement of others. Analytical trade-off relations are established for the higher frequency range. The problem was simulated with excellent design verification.