A Note on the Optimality of Balanced Truncation for a Class of Infinite Dimensional Systems

Balanced truncation is a widespread model reduction method that uses balanced realizations for finite dimensional systems. The latter are state space realizations where the controllability and observability gramians are equal to the same diagonal positive matrix. In this paper, a generalization of balanced realization for a class of infinite dimensional LTI systems is employed to perform balanced truncation. It is shown that balanced truncation is optimal in the Hilbert-Schmidt sense if a particular time-varying balanced realization is used for the original LTI system. It appears that the use of time-varying balanced realizations to study the optimality and perform balanced model reduction for LTI systems is novel.