This note derives a linear quadratic regulator which is robust to real parametric uncertainty, by using the overbounding method of Petersen and Hollot. The resulting controller is determined from the solution of a single modified Riccati equation. This controller has the same guaranteed robustness properties as standard linear quadratic designs for known systems. It is proven that when applied to a structural system, the controller achieves its robustness by minimizing the potential energy of uncertain stiffness elements, and minimizing the rate of dissipation of energy by the uncertain damping elements.