Here we consider the problem of providing near optimal performance (in this context, "near optimal performance" means performance as close to optimality as desired) for a large set of possible models. We adopt the linear quadratic regulator (LQR) framework in the single-input-single-output (SISO) setting, and prove that given a compact set of controllable and observable plant models of a fixed order, we can construct a single linear periodic controller (LPC) which provides near optimal LQR performance. Since the controller is linear, it automatically has the nice feature that there is some degree of tolerance to unmodeled dynamics. The approach is also shown to work if the goal is the more modest one of pole placement, and it can be simplified if there is additional structure to the plant model.