Stochastic LQR Design With Disturbance Preview

This paper considers the discrete-time, stochastic LQR problem with $p$ steps of disturbance preview information where $p$ is finite. We first derive the solution for this problem on a finite horizon with linear, time-varying dynamics and time-varying costs. Next, we derive the solution on the infinite horizon with linear, time-invariant dynamics and time-invariant costs. Our proofs rely on the well-known principle of optimality. We provide an independent proof for the principle of optimality that relies only on nested information structure. Finally, we show that the finite preview controller converges to the optimal noncausal controller as the preview horizon $p$ tends to infinity. We also provide a simple example to illustrate both the finite and infinite horizon results.