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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Wco</tex> to include symmetric MIMO systems where the matrix <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Wco</tex> is defined as an integral. In this correspondence, we define this matrix as a solution of a Lyapunov equation and prove the fundamental identity <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Wco² = WcWₒ</tex> where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Wc</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Wₒ</tex> are the controllability and observability Gramians defined via Lyapunov equations. This definition of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Wco</tex> complements the integral definition and has some advantages from theoretical and numerical points of view.