 Scalar delayed-observation LQG regulator

For $X_{n+1}=X_n+U_n+{\epsilon }_{n+1}$ and $Y_{n+1}=X_n+{\eta }_{n+1}$ with independent unit-variance normal noises, the equilibrium prediction-error variance is $P=(1+\sqrt{5})/2$. Orthogonality to the next innovation gives the filtering gain $H=P/(1+P)$. Minimizing the quadratic average cost gives a control Riccati coefficient $R$ satisfying $R^2=R+1$, so $R=P$ and $K=R/(1+R)=H$. Completing the control square yields minimal cost $R+(1+R)K^{2}P=1+\sqrt{5}$. The one-step observation delay is essential to the value of $P$.