Linear Quadratic Gaussian Balancing for Discrete-Time Infinite-Dimensional Linear Systems

In this paper, we study the existence of linear quadratic Gaussian (LQG)–balanced realizations for discrete-time infinite-dimensional systems. LQG-balanced realizations are those for which the smallest nonnegative self-adjoint solutions of the control and filter Riccati equations are equal. We show that the control (filter) Riccati equation has a nonnegative self-adjoint solution if and only if the system is output (input) stabilizable. Our main result is that the transfer function of a discrete-time linear system has an approximately controllable and observable LQG-balanced realization iff it has an input and output stabilizable realization. The corresponding control and filter Riccati equations have unique nonnegative self-adjoint solutions. Moreover, 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 smallest nonnegative self-adjoint solutions of the control and filter Riccati equations is independent of the particular realization.