  Models that balance accuracy against computational costs are advantageous
when designing dynamic systems with optimization studies, as several hundred
predictive function evaluations might be necessary to identify the optimal
solution. The efficacy and use of derivative function surrogate models (DFSMs),
or approximate models of the state derivative function, have been
well-established in the literature. However, previous studies have assumed an a
priori state dynamic model is available that can be directly evaluated to
construct the DFSM. In this article, we propose an approach to extract the
state derivative information from system simulations using piecewise polynomial
approximations. Once the required information is available, we propose a
multi-fidelity DFSM approach as a predictive model for the system's dynamic
response. This multi-fidelity model consists of summation between a linear-fit
lower-fidelity model and an additional nonlinear error corrective function that
compensates for the error between the high-fidelity simulations and
low-fidelity models. We validate the model by comparing the simulation results
from the DFSM to the high-fidelity tools. The DFSM model is, on average, five
times faster than the high-fidelity tools while capturing the key time domain
and power spectral density~(PSD) trends. Then, an optimal control study using
the DFSM is conducted with outcomes showing that the DFSM approach can be used
for complex systems like floating offshore wind turbines~(FOWTs) and help
identify control trends and trade-offs.
