Disclaimer: I am from another field, the language in the following problem therefore may be different from the usual optimal control lingo.
Consider the system
$$x_{t+1} = Ax_t + Bu_t + C\varepsilon_{t+1}\\ y_t = Gx_t + D v_t $$
where $\varepsilon$ and $v$ are Gaussian white noise. Minimizing the usual quadratic cost function, the Linear-Quadratic-Gaussian regulator reads
$$u_t = -F\hat{x}_t$$
where $\hat{x}_t$ is computed using the Kalman filter.
Using the equation for the optimal control, I am looking for a way to cast the state/transition equation in the following form:
$$x_{t+1} = \tilde{A}x_t + \tilde{C} \varepsilon_{t+1} + \tilde{D} v_t \ $$
Any help will be greatly appreciated. Thanks.
Update:
Substitute the optimal control $u_t = -F\hat{x}_t$ in the state dynamics $x_{t+1}$ to get:
$$x_{t+1} = Ax_t - BF\hat{x}_t + C \varepsilon_{t+1}$$
Define the error $e_t = x_t - \hat{x}_t$. Substiutte for $\hat{x}_t$ and rearrange to arrive at:
$$x_{t+1} = (A-BF)x_t + BFe_t + C \varepsilon_{t+1}$$
Now what's left to do is to determine the distribution and/or dynamics of $e_t$. Given $e_t$, I can describe the dynamics of the state if the LQG-regulator is followed.
At the end of the day, I want to solve for the dynamics of the optimal control with observable state given that the optimal control with hidden state was followed. That is, I am looking for the dynamics of $\tilde{u}_t = -F x_t$, where $x_{t+1} = Ax_t - BF\hat{x}_t + C \varepsilon_{t+1}$.
Thanks in advance for your suggestions or comments.