$mathcal{H}_{infty}$ -Optimal Strictly Positive Real Parallel Feedforward Control

This paper presents static and dynamic parallel feedforward controller synthesis methods that render a linear time-invariant system strictly positive real (SPR) in an $\mathcal{H}_{\infty}$ -optimal fashion. The parallel feedforward controller is designed in such a manner that when the output of the system is added to the output of the parallel feedforward controller, the transfer matrix from the system input to the new output is SPR. In order to ensure that the difference between the new output and the original system output is small, the maximum singular value of a static parallel feedforward controller or the weighted $\mathcal{H}_{\infty}$ norm of a dynamic parallel feedforward controller is minimized. The proposed synthesis methods are convex optimization problems that make use of linear matrix inequality and equality constraints. The controllers are implemented numerically on a flexible-joint robotic manipulator and compared to a parallel feedforward controller from the literature. It is shown in closed-loop simulation that a significant improvement in tracking error is achieved with one of the proposed dynamic parallel feedforward controller synthesis methods.