control//controller design//LQG//separation principle

The separation principle is a theorem about linear systems with an observer in the loop which says that the state-feedback gain and the estimator gain can be designed independently, and it is what lets an engineering team give the estimator to one person and the controller to another. Design \(K\) as if every state were measured, design the estimator gain \(L\) as if there were no control at all, connect them with \(u=-K\hat x\), and the closed loop behaves as the union of the two designs.

The separation principle is a theorem about linear systems with an observer in the loop which says that the state-feedback gain and the estimator gain can be designed independently, and it is what lets an engineering team give the estimator to one person and the controller to another. Design KKK as if every state were measured, design the estimator gain LLL as if there were no control at all, connect them with u=−Kx^u=-K\hat xu=−Kx^, and the closed loop behaves as the union of the two designs.

The reason is visible in the equations. Write the estimation error e=x−x^e=x-\hat xe=x−x^. For a Luenberger observer or a Kalman filter on x˙=Ax+Bu\dot x=Ax+Bux˙=Ax+Bu, y=Cxy=Cxy=Cx, the pair (x,e)(x,e)(x,e) evolves as

[x˙e˙]=[A−BKBK0A−LC][xe].\begin{bmatrix}\dot x\\ \dot e\end{bmatrix}=\begin{bmatrix}A-BK & BK\\ 0 & A-LC\end{bmatrix}\begin{bmatrix}x\\ e\end{bmatrix}.[x˙e˙​]=[A−BK0​BKA−LC​][xe​].

The matrix is block triangular, so its eigenvalues are those of A−BKA-BKA−BK together with those of A−LCA-LCA−LC: the poles of the controller and the poles of the estimator, each untouched by the other. The error dynamics do not depend on uuu at all, which is why the estimator can ignore what the controller wants.

For the linear-Gaussian problem the statement is stronger: the combination is the optimal output-feedback controller (LQG). Deciding as if the estimate were the truth is called certainty equivalence, and in decision problems under partial observation (POMDP) it is exact only in this linear-quadratic-Gaussian case; elsewhere it is a good heuristic while uncertainty is small compared with the consequences.

It guarantees poles and optimality, nothing about robustness. The upper-right block BKBKBK couples the estimation error into the state, so a model mismatch the estimator does not know about can push the combined loop close to instability even when both pieces are excellent. The poles of the nominal design say nothing about the margins of the real plant.

It also says nothing about speed. The two sets of poles coexist, so a slow estimator drags the closed loop; the usual rule puts the estimator poles several times faster than the controller poles, limited in turn by sensor noise.

Outside linear systems separation does not hold in general. A nonlinear controller fed by an extended Kalman filter works in practice when the estimate is good, and an action that also improves the estimate (looking around a corner before moving) can be worth more than one that only regulates (dual control).