On the cross-Gramian for symmetric MIMO systems

Abstract

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.