By Theorem \ref{thm:nonidentifiable}, one way to prevent the adversary from identifying \(\mathcal{G}(z)\) is to enforce the factorization (\ref{eq:factorization}) on the controller transfer function \(\mathcal{K}(z)\). Let us define the following “virtual” control input:
\begin{equation} \label{eq:tildeudef} \label{eq:tildeudef}\tilde{u}(k)\triangleq\tilde{\mathcal{K}}(z)y(k).\\ \end{equation}Hence, \(u(k)=\mathcal{K}(z)y(k)=F\tilde{u}(k)\). The factorization on \(\mathcal{K}(z)\) implies the CPS diagram illustrated in Fig \ref{fig:factorization}.
Since we are restricting ourselves to use a low-rank controller, the performance of the system may not be optimal. In this section, we consider the problem of optimizing the following infinite horizon LQG performance:
\begin{equation} \label{eq:lqgcost} \label{eq:lqgcost}J=\limsup_{T\rightarrow\infty}\frac{1}{T}\min_{u(k)}\mathbb{E}\left[\sum_{k=0}^{T-1}x(k)^{T}Wx(k)+u(k)^{T}Uu(k)\right],\\ \end{equation}under the constraint that \(F\in\mathbb{R}^{p\times q}\) where \(q\) is given. The \(W,\,U\) matrices are assumed to be positive semidefinite. We shall first consider how to design \(\tilde{\mathcal{K}}(z)\) when \(F\) is given. We then provide a heuristic algorithm to compute the optimal \(F\) based on convex relaxation.