  This paper studies the relationship between state feedback policies and
disturbance response policies for the standard Linear Quadratic Regulator
(LQR). For open-loop stable plants, we establish a simple relationship between
the optimal state feedback controller $u_t=K_\star x_t$ and the optimal
disturbance response controller
$u_t=L^{(H)}_{\star;1}w_{t-1}+\cdots+L^{(H)}_{\star;H}w_{t-H}$ with $H$-order.
Here $x_t, w_t, u_t$ stands for the state, disturbance, control action of the
system, respectively. Our result shows that $L_{\star,1}^{(H)}$ is a good
approximation of $K_\star$ and the approximation error $\|K_\star -
L_{\star,1}^{(H)}\|$ decays exponentially with $H$. We further extend this
result to LQR for open-loop unstable systems, when a pre-stabilizing controller
$K_0$ is available.
