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
W_{co}
to include symmetric MIMO systems where the matrix
W_{co}
is defined as an integral. In this correspondence, we define this matrix as a solution of a Lyapunov equation and prove the fundamental identity
W_{co}^{2} = W_{c}W_{o}
where
W_{c}
and
W_{o}
are the controllability and observability Gramians defined via Lyapunov equations. This definition of
W_{co}
complements the integral definition and has some advantages from theoretical and numerical points of view.