Since \(u(k)=F\tilde{u}(k)\), we can rewrite the system equation as
\begin{align} x(k+1)=Ax(k)+\tilde{B}\tilde{u}(k)+w(k),\notag \\ \end{align}where \(\tilde{B}\triangleq BF\). Furthermore, the objective function of LQG can be rewritten as
\begin{align} J=\limsup_{T\rightarrow\infty}\frac{1}{T}\min_{\tilde{u}(k)}\mathbb{E}\left[\sum_{k=0}^{T-1}x(k)^{T}Wx(k)+\tilde{u}(k)^{T}\tilde{U}\tilde{u}(k)\right],\notag \\ \end{align}where \(\tilde{U}\triangleq F^{T}UF\in\mathbb{R}^{q\times q}\). Therefore, the optimal control is given by a Kalman filter and a LQR controller \cite{Schenato_2007}:
The state estimation of the Kalman filter (with a fixed gain) is given by:
\begin{align} \hat{x}(k) & =\hat{x}(k|k-1)+K(y(k)-C\hat{x}(k|k-1)),\notag \\ \hat{x}(k+1|k) & =A\hat{x}(k)+Bu(k).\notag \\ \end{align}where \(K=PC^{T}(CPC^{T}+R)^{-1},\) and \(P\) is the fixed point of the following Riccati equation:
\begin{align} P=APA^{T}+Q-APC^{T}(CPC^{T}+R)^{-1}CPA^{T}.\notag \\ \end{align}The optimal control can then be derived as a linear function of the state estimate:
\begin{align} \tilde{u}(k)=\tilde{L}\hat{x}(k),\\ \end{align}where
\begin{align} \tilde{L}=-(\tilde{B}^{T}\tilde{S}\tilde{B}+\tilde{U})^{-1}\tilde{B}^{T}\tilde{S}A,\notag \\ \end{align}and \(\tilde{S}\) is the solution of the following Riccati equation
\begin{align} \label{eq:tildeS} \label{eq:tildeS}\tilde{S}=A^{T}\tilde{S}A+W-A^{T}\tilde{S}\tilde{B}(\tilde{B}^{T}\tilde{S}\tilde{B}+\tilde{U})^{-1}\tilde{B}^{T}\tilde{S}A.\\ \end{align}The corresponding \(\tilde{K}(z)\) is given by
\begin{align} \tilde{K}(z)=z\tilde{L}\left[zI-(I-KC)(A+B\tilde{L})\right]^{-1}K.\notag \\ \end{align}The corresponding LQG cost is given by
\begin{align} \label{eq:optlqgcost} J^{*} & =\operatorname*{tr}(\tilde{S}Q)+\operatorname*{tr}[(W+A^{T}\tilde{S}A-\tilde{S})(P-KCP)]\notag \\ & \label{eq:optlqgcost}=\operatorname*{tr}(\tilde{S}Y)+\operatorname*{tr}[W(P-KCP)],\\ \end{align}where
\begin{equation} \begin{split}\displaystyle Y&\displaystyle\triangleq Q+A(P-KCP)A^{T}-(P-KCP)\\ &\displaystyle=PC^{T}(CPC^{T}+R)^{-1}CP\succeq 0.\end{split}\\ \end{equation}