  This work explores the trade-off between the number of samples required to
accurately build models of dynamical systems and the degradation of performance
in various control objectives due to a coarse approximation. In particular, we
show that simple models can be easily fit from input/output data and are
sufficient for achieving various control objectives. We derive bounds on the
number of noisy input/output samples from a stable linear time-invariant system
that are sufficient to guarantee that the corresponding finite impulse response
approximation is close to the true system in the $\mathcal{H}_\infty$-norm. We
demonstrate that these demands are lower than those derived in prior art which
aimed to accurately identify dynamical models. We also explore how different
physical input constraints, such as power constraints, affect the sample
complexity. Finally, we show how our analysis fits within the established
framework of robust control, by demonstrating how a controller designed for an
approximate system provably meets performance objectives on the true system.
