How to derive continuous-time Linear Quadratic Gaussian state estimator?

Question

In a Linear Quadratic Gaussian problem, one can prove a separation principle, thereby separating the objective of control design from state estimation, when the linear system has process and measurement noise. The continuous-time solution involves solving for two dual Riccati differential equations for controller and state estimator, respectively.

Most of the papers and lectures I've read spend a lot of time proving the separation principle, as well as the optimality of the linear feedback. However, maybe it's a limitation in my understanding, I don't see any proof/derivation of how the continuous-time state estimator relates to the process/measurement noise covariances and the Riccati equation. The discrete-time variant, specifically Kalman Filter, proof is aplenty, but I don't like to rely on any discussion in discrete-time.

Even more to the basic, I would like to understand precisely even the problem formulation of the continuous-time LQG state estimator. What is it trying to minimize, and what are the "manipulated variables"? (i.e. what is the cost function?)

Answer 1

Thanks to @Pait's comment, I've found my answer in the Kalman-Bucy filter. To answer my latter questions, an optimal linear state estimator has two objectives:

• Ensure that the state estimates are unbiased ($E[e(t)-\hat{e}(t)]=0$)
• Minimize the variance of the state estimation error

By restricting the state estimation to linear in nature, objective #1 automatically confers the Luenberger observer structure (all time-varying):

$$\dot{x}(t)=Ax+Bu+L(y-C\hat{x})$$

Then, given a symmetric positive-definite matrix, $M(t)$, we can construct a cost function as:

$$J=E[(x-\hat{x})M(t)(x-\hat{x})^T]=tr[M(t)P(t)]$$

where $P(t)=E[(x-\hat{x})(x-\hat{x})^T]$ is the state estimate error covariance matrix.

Objective #2 is then re-stated to minimize the cost function, $J$, by selecting the Kalman gain, $L(t)$. It is then possible to show that the optimal solution involves solving for the Riccati equation for $P(t)$, then use it to set the optimal Kalman gain.

The full derivation can be found in Athens, Tse, A direct derivation of the optimal linear filter using the maximum principle, 1967., which is well-written and easy to follow.