Separation principle

Remarks

The proof has two parts. First, for every admissible control, the error $x - \hat{x}$ is a Gaussian process whose law does not depend on $u$, with covariance $\Sigma(t)$. Second, writing $x = \hat{x} + (x - \hat{x})$ splits the cost into a term that does not depend on $u$ and the cost of a fully observed linear-quadratic problem for $\hat{x}$. The linear-quadratic regulator then gives the optimal feedback for $\hat{x}$. The gain $R^{-1} B^{T} P$ is the same as without noise, and the filter gain $\Sigma C^{T} \Theta^{-1}$ does not depend on the cost: estimation and control separate. A deterministic version holds for observer-based feedback of $x' = A x + B u$, $y = C x$. With the observer $\hat{x}' = A \hat{x} + B u + L(y - C \hat{x})$ and $u = -K \hat{x}$, the error $e = x - \hat{x}$ solves $e' = (A - L C) e$. In the coordinates $(x, e)$ the closed-loop matrix is block triangular, so its eigenvalues are those of $A - B K$ together with those of $A - L C$. The principle depends on linear dynamics, Gaussian noise and quadratic cost; for nonlinear problems the best control may need to act so as to improve future estimates.

Going deeper

For a student

The estimator is the Kalman–Bucy filter and the controller is the linear-quadratic regulator gain, and each comes from its own Riccati equation. The two Riccati equations are dual: the filter equation is the control equation for $(A^{T}, C^{T})$ run forward in time. In practice this lets engineers design the estimator and the controller separately. The deterministic version is the reason why pole placement for the controller and for the observer can be done independently.