  The aim in model order reduction is to approximate an input-output map
described by a large-scale dynamical system with a low-dimensional and
cheaper-to-evaluate reduced order model. While high fidelity can be achieved by
a variety of methods, only a few of them allow for rigorous error control. In
this paper, we propose a rigorous error bound for the reduction of linear
systems with balancing-related reduction methods. More specifically, we
consider the simulation over a finite time interval and provide an a posteriori
adaption of the standard a priori bound for Balanced Truncation and Balanced
Singular Perturbation Approximation in that setting, which improves the error
estimation while still yielding a rigorous bound. Our result is based on an
error splitting induced by a Fourier series approximation of the input and a
subsequent refined error analysis. We make use of system-theoretic concepts,
such as the notion of signal generator driven systems, steady-states and
observability. Our bound is also applicable in the presence of nonzero initial
conditions. Numerical evidence for the sharpness of the bound is given.
