Minimum Sensitivity Analysis for Accurate Open-loop Controllers in Linear Systems using Weighted Gramians

The objective is to find accurate open-loop controls for linear systems, which bring the state to the exact desired target in the absence of the disturbance (accurate), and minimize the worst case error at the final state (minimally sensitive) when disturbed. In particular, a suboptimal open-loop control is proposed for the multiplicative disturbance case. The proposed suboptimal control has a structural analogy to the Gramian based minimum energy controller but differs by using a weighted Gramian. Further analysis on this unconventional weighted Gramian shows that its inverse relates to the upper bound of the worst perturbation at the final state. A two dimensional heat exchanger example shows the effectiveness of the proposed control.