On the cross-Gramian for symmetric MIMO systems

Recently, Laub et al. [1] and Fernando and Nicholson [2] extended the definition of the cross-Gramian matrix to include symmetric MIMO systems where the matrix is defined as an integral. In this correspondence, we define this matrix as a solution of a Lyapunov equation and prove the fundamental identity where and are the controllability and observability Gramians defined via Lyapunov equations. This definition of complements the integral definition and has some advantages from theoretical and numerical points of view.