Thus the problem can be broken into two separate parts, which facilitates the design.
The separation principle does not hold in general for nonlinear systems.
Another instance of the separation principle arises in the setting of linear stochastic systems, namely that state estimation (possibly nonlinear) together with an optimal state feedback controller designed to minimize a quadratic cost, is optimal for the stochastic control problem with output measurements.
When process and observation noise are Gaussian, the optimal solution separates into a Kalman filter and a linear-quadratic regulator.
Consider a deterministic LTI system: where We can design an observer of the form and state feedback Define the error e: Then Now we can write the closed-loop dynamics as Since this is a triangular matrix, the eigenvalues are just those of A − BK together with those of A − LC.