In this section, we provide a numerical example to illustrate our low-rank controller design. We assume that \(n=15,\,m=p=10\). The \(Q,\,R,\,W,\,U\) matrices are all chosen to be identity matrix. The matrix \(A,\,B,\,C\) are randomly generated, where each entry is independently drawn from a uniform distribution on \([0,1]\).
We consider the LQG performance of a low-rank controller versus the performance of an optimal controller with no rank constraint. We define the relative performance loss as
\begin{align} \frac{J\text{ of the low rank controller}}{J\text{ of the optimal controller}}-1.\notag \\ \end{align}We will choose \(q\) from \(5\) to \(9\) and for each \(q\), we will perform \(1000\) random experiments. Fig \ref{fig:lqgloss} is the box and whisker diagram of the relative LQG performance loss generated by the random experiments. One can see that even if \(q=5\), meaning that we only use half degrees of freedom to design the controller, the LQG loss is still small, with median loss at \(6\%\).