Automatic synthesis of feedforward elements in quantitative feedback theory

Abstract

Recent developments in quantitative feedback theory (QFT) lead to feedforward design problems with both magnitude and phase constraints. In these cases, manual feedforward tuning becomes much more challenging and time consuming than the traditional prefilter shaping taking place on the Bode plot. This article presents a general procedure for the automatic synthesis of such elements. Feedforward bounds in the complex plane are expressed as constraints of a linear programming problem in which the Bode real-complex relation is implicitly considered, ensuring a stable rational solution. The methodology is successfully tested in a well-known benchmark problem. K E Y W O R D S2DOF, feedforward synthesis, QFT INTRODUCTIONQuantitative feedback theory (QFT) 1-4 owes its name to the fact that feedback is used in the minimum amount required to fight the effects of plant uncertainty and unknown disturbances. Deeply rooted in classical control theory, QFT aims to provide low-order, low-bandwidth feedback controllers fulfilling a set of frequency-domain specifications. By doing so, QFT leads to designs with minimum cost of feedback, understood as the effect of sensor noise in the system's actuator and plant. This philosophy defines the way in which QFT confronts two-degree-of-freedom (2DOF) problems, that is, problems in which both feedforward and feedback actions are involved. Unlike inversion-based techniques, in QFT the feedback element is designed first, and its purpose is to reduce the effect of uncertainty to a level in which a single feedforward element can handle the whole set of plants, making all of its members meet the specification. 1,5 The paradox is that the closer the feedback controller gets to its theoretical optimum, 6 the harder it is to find a solution to the feedforward synthesis problem. However, given that most designers accept certain overdesign in exchange for low order controllers, the feedforward design usually becomes a trivial task. This is particularly the case in the classical QFT tracking problem, in which specifications are defined as upper and lower tolerances on the magnitude of the tracking frequency response. With this kind of specification, the feedforward controller-a prefilter preceding the feedback loop-is easily tuned on the Bode plot. All this could explain why only a few articles deal with automated methods for prefilter shaping. [7][8][9] This contrasts with the considerable effort that has been put into the automatic tuning of QFT feedback controllers. [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28] Abbreviations: 2DOF, two-degree-of-freedom; QFT, quantitative feedback theory.This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made.

A refinement approach to the multivariable tracking error problem

“…The designer is therefore required to constantly allocate additional headroom to all interim controller designs, in order to prevent subsequent loop shaping issues as a result of the Bode gain‐phase relationship. An appealing option could be to incorporate filter synthesis from Reference 19, which would theoretically allow for lower gain‐phase feedback controller solutions, but this is not encapsulated in the scope of this article.…”