For a standard Kalman filter
\begin{align} x_{t+1}&= Ax_t + w_t\\ y_t &= Cx_t + v_t, \end{align}
I am interested in designing the matrix $C$ such that the error covariance is minimized and $\operatorname{tr}(CC^T)\le p$. Can it be formulated as a semidefinite program (SDP)?
My attempt
The optimization can be written using the Riccati equation as:
$$ \begin{array}{ll} \underset {P \succeq0,C, K} {\text{minimize}} & \operatorname{tr} (P) \\ \text{subject to} & (A - K C) P (A - K C)^T + K R K^T + Q = P, \operatorname{tr}(CC^T)\le p \end{array} $$
Using my previous question in Kalman filter solution via LMI: summary of methods, we can write:
\begin{align} \begin{array}{ll} \underset {X,Y,Z,C} {\text{minimize}} & \operatorname{tr} (Z) \\ \text{such that} & \begin{bmatrix} X & XA - YC & Y & X\\ (XA - YC)^T & X & 0& 0\\ Y^T & 0 & R^{-1}&0\\ X& 0 & 0 & Q^{-1} \end{bmatrix} \succeq 0, \\ & \begin{bmatrix} Z & I\\ I & X \end{bmatrix} \succeq 0, \\ & \operatorname{tr}(CC^T)\le p \end{array} \end{align}
but there are still bilinear terms.
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