LQG Balancing for Continuous-Time Infinite-Dimensional Systems

In this paper we study the existence of linear quadratic Gaussian (LQG)-balanced realizations for continuous-time infinite-dimensional systems. LQG-balanced realizations are those for which the optimal cost operator for the system and its dual system are equal (and diagonal). The class of systems we consider is that of distributional resolvent linear systems which includes well-posed linear systems as a subclass. We prove the existence of LQG-balanced realizations under a finite cost condition for both the system and its dual system. We also show that an LQG-balanced realization of a well-posed transfer function is well-posed. We further show that approximately controllable and observable LQG-balanced realizations are unique up to a unitary state-space transformation. Finally, we show that the spectrum of the product of the optimal cost operator of a system and its dual system is independent of the particular realization. Our method of proof shows the connections with coprime factorizations, Lyapunov-balanced realizations, and discrete-time systems. The main reason for studying LQG-balanced realizations is that truncated LQG-balanced realizations provide a good approximation of the original system. We show that, under certain conditions, this is also true in the infinite-dimensional case by proving an error bound in the gap-metric.