A Computer Loopshaping Algorithm for Controllers for Minimum-Phase, Uncertain Systems Starting from QFT Type Bounds

An automated method for loopshaping a controller for linear, minimum-phase, uncertain systems in the frequency domain is presented. The method assumes the existence of lower bounds on the open loop transfer function as well as a stability robustness bound. For the example presented, the bounds on the open loop transfer function were generated using Quantitative Feedback Theory (QFT) techniques. The first part of the algorithm describes the search for points on the nominal open loop transfer function based on Bode's Theorems [2] and assumes that a rational transfer function can be made to pass through (or close to) the lower right corner of the stability robustness bound. After points for the nominal transfer function are found, points for the controller transfer function can be extracted from these. The second part of the algorithm, based again on Bode's Theorems, describes an iterative procedure to curve fit the controller data into a rational transfer function. Maintaining a low order for the compensator structure and assuring the rationality of the transfer function were the main concerns of the authors when developing the algorithm. Furthermore, because the decisions confronting the designer at any given time are of a very simple nature, the procedure can be extended into an automated computer algorithm. The computer package is developed in programming language "c" and its use is best illustrated with an example.