= Scalar delayed-observation LQG regulator
{c}
{title2=$K=H=(\sqrt5-1)/2$}
For $X_{n+1}=X_n+U_n+\varepsilon_{n+1}$ and $Y_{n+1}=X_n+\eta_{n+1}$ with independent unit-variance normal noises, the equilibrium prediction-error variance is $P=(1+\sqrt5)/2$. 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 $R+(1+R)K^2P=1+\sqrt5$. The one-step observation delay is essential to the value of $P$.
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