On the Relationship of Optimal State Feedback and Disturbance Response Controllers

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 $u_t=L_{\star ;1}^{(H)}{w}_{t-1}+\cdots +L_{\star ;H}^{(H)}{w}_{t-H}$ 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 $\mathrm{\parallel }{K}_{\star }-L_{\star ,1}^{(H)}\mathrm{\parallel }$ 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.