Optimal Hankel Norm Model Reduction by Truncation of Trajectories

We show how optimal Hankel-norm approximations of dynamical systems allow for a straightforward interpretation in terms of system trajectories. It is shown that for discrete time single-input systems optimal reductions are obtained by cutting 'balanced trajectories', i.e., by disconnecting the past and future in the input-output pairs relating to left- and right singular vectors of the system. A self-contained proof of optimality is given, and formulas are derived in terms of Faddeev sequences. Some parallels with the literature are briefly indicated.