I am trying to solve a problem similar to a H2 feedforward control synthesis problem with an additional constraint. I was wondering if it is possible to reformulate the problem to be a set of linear matrix inequalities (LMI) through some combination of transformations and slack variables. Constraints 1-3 and 4-5 can be written separately as LMIs, but I am having difficulty combining the constraints. Dealing with the relationship of $Q A_{aug}$ in constraint (3) and $A_fR$ and $PA_f$ in constraints (4) and (5) is what is giving me the most trouble. The problem is as follows:
Given matrices $A \in \mathbb{R}^n, B \in \mathbb{R}^{n \times m}, C \in \mathbb{R}^{p \times n}$, and $\delta$ which is an arbitrarily small number.
Find matrices $A_f \in \mathbb{R}^{n_c}, B_f \in \mathbb{R}^{n_c \times m}, C_f \in \mathbb{R}^{p \times n_c},Q \in \mathbb{R}^{n+n_c},W \in \mathbb{R}^p, P \in \mathbb{R}^{n_c}$, and $R \in \mathbb{R}^{n_c}$ such that the following optimization problem is satisfied.
$$\begin{align} \text{minmize} \quad &\mu \\ \text{s.t.} \quad &trace(W)<\mu \tag{1}\\ &\begin{bmatrix}W&C_{aug}\\ C_{aug}^T&Q\end{bmatrix} \succ 0 \tag{2}\\ &\begin{bmatrix}Q&QA_{aug}&QB_{aug}\\ A_{aug}^TQ&Q&0\\ B_{aug}^TQ&0&I\end{bmatrix} \succ 0 \tag{3}\\ &Q \succ 0\\ &W\succ 0\\ &\begin{bmatrix}\delta R&\frac{1}{2}(A_f R- RA_f^T)\\ -\frac{1}{2}(A_f R- RA_f^T)&\delta R\end{bmatrix} \succ 0 \tag{4}\\ &\begin{bmatrix}P& PA_f\\ A_f^T P & P\end{bmatrix} \succ 0 \tag{5}\\ &P \succ 0\\ &R\succ 0\\ \end{align}$$
where $A_{aug} = \begin{bmatrix}A&0\\0&A_f\end{bmatrix}, B_{aug} = \begin{bmatrix}B\\B_f\end{bmatrix}, C_{aug} = \begin{bmatrix}C&-C_f\end{bmatrix}$.
If it helps, (5) corresponds to the eigenvalues of $A_f$ being inside the unit circle, (4) corresponds to the eigenvalues of $A_f$ being restricted to the real axis, and (1-3) correspond to minimizing the H2 norm of a parallel connection of the discrete statespace systems ss($A$,$B$,$C$,$0$) and ss($A_f$,$B_f$,$C_f$,$0$). Please let me know if there is anything that is unclear.
Thank You