A refinement approach to the multivariable tracking error problem. (15th May 2022) - Pretorius, Arnold; Boje, Edward
Abstract: This article presents a new refinement design routine aimed at solving the robust multivariable tracking error problem with reduced conservatism. The prevailing multivariable quantitative feedback theory (QFT) approach is to implicitly overbound the model‐error tracking set frequency response in magnitude using the triangle inequality, with the intention of arriving at a set of univariate design constraints that independently describe the feedback controller solution space. While this method is effective in the low‐frequency range (where the loop gain tends to be large), the mid‐ to high‐frequency design regions (where diagonal dominance is not possible in general) suffers from arbitrarily large design conservatism. This inhibits minimum gain‐phase solutions and necessitates undesirable over‐design in the frequency band that can contribute to large, expensive control action. The proposed method follows a refinement approach that makes use of information from an a priori feedback control design. In this way, the tracking error problem is reposed as a differential design, and gain‐phase information from the previous iteration can be captured to reduce the conservatism imposed when applying the triangle inequality. Additionally, a constrained optimization routine is used to select a prototype feedforward filter that can relax the constraint set, thereby increasing the accessible solution space of the diagonal feedback controller. Finally, a nondiagonal, multivariable feedforward filter bound generation routine is defined that relies on existence conditions and is free of induced design conservatism. The viability of this design methodology is demonstrated on two benchmark problems of varying complexity, in order to demonstrate the widespread efficacy.