A Hamiltonian system based approach for the computation of the maximal rank-minimizing solution of the LMI arising from a singular LQR problem

In this paper we deal with the linear matrix inequality (LMI) arising from a singular linear quadratic regulator (LQR) problem. The maximal rank-minimizing solution
$K_{\max}$
of the LMI plays a central role in obtaining a proportional-derivative feedback law for the optimal input. The optimal cost of the LQR, too, depends on this solution
$K_{\max}$
. In this paper, we provide a method to compute this maximal rank-minimizing solution
$K_{\max}$
of the singular LQR LMI. We compute this solution using the notions of the weakly unobservable or the slow space and the strongly reachable or the fast space of the Hamiltonian system arising from the singular LQR problem. In this process, we also provide a novel characterization of the fast space in terms of the system matrices.