Linear Optimal Regulation to Zero Dynamics

A general problem encompassing output regulation and pattern generation can be formulated as the design of controllers to achieve convergence to a persistent trajectory within the zero dynamics on which an output vanishes. We develop an optimal control theory for such design by adding the requirement to minimize the
$H_{2}$
norm of a closed-loop transfer function. Within the framework of eigenstructure assignment, the optimal control is proven identical to the standard
$H_{2}$
control in form. However, the solution to the Riccati equation for the linear quadratic regulator is not stabilizing. Instead, it partially stabilizes the closed-loop dynamics excluding the zero dynamics. The optimal control architecture is shown to have the feedback of the deviation from the subspace of the zero dynamics and the feedforward of the control input to remain in the subspace.