  Inspired by competitive policy designs approaches in online learning, new
control paradigms such as competitive-ratio and regret-optimal control have
been recently proposed as alternatives to the classical $\mathcal{H}_2$ and
$\mathcal{H}_\infty$ approaches. These competitive metrics compare the control
cost of the designed controller against the cost of a clairvoyant controller,
which has access to past, present, and future disturbances in terms of ratio
and difference, respectively. While prior work provided the optimal solution
for the regret-optimal control problem, in competitive-ratio control, the
solution is only provided for the sub-optimal problem. In this work, we derive
the optimal solution to the competitive-ratio control problem. We show that the
optimal competitive ratio formula can be computed as the maximal eigenvalue of
a simple matrix, and provide a state-space controller that achieves the optimal
competitive ratio. We conduct an extensive numerical study to verify this
analytical solution, and demonstrate that the optimal competitive-ratio
controller outperforms other controllers on several large scale practical
systems. The key techniques that underpin our explicit solution is a reduction
of the control problem to a Nehari problem, along with a novel factorization of
the clairvoyant controller's cost. We reveal an interesting relation between
the explicit solutions that now exist for both competitive control paradigms by
formulating a regret-optimal control framework with weight functions that can
also be utilized for practical purposes.
