  This paper presents a generalizable methodology for data-driven
identification of nonlinear dynamics that bounds the model error in terms of
the prediction horizon and the magnitude of the derivatives of the system
states. Using higher-order derivatives of general nonlinear dynamics that need
not be known, we construct a Koopman operator-based linear representation and
utilize Taylor series accuracy analysis to derive an error bound. The resulting
error formula is used to choose the order of derivatives in the basis functions
and obtain a data-driven Koopman model using a closed-form expression that can
be computed in real time. Using the inverted pendulum system, we illustrate the
robustness of the error bounds given noisy measurements of unknown dynamics,
where the derivatives are estimated numerically. When combined with control,
the Koopman representation of the nonlinear system has marginally better
performance than competing nonlinear modeling methods, such as SINDy and NARX.
In addition, as a linear model, the Koopman approach lends itself readily to
efficient control design tools, such as LQR, whereas the other modeling
approaches require nonlinear control methods. The efficacy of the approach is
further demonstrated with simulation and experimental results on the control of
a tail-actuated robotic fish. Experimental results show that the proposed
data-driven control approach outperforms a tuned PID (Proportional Integral
Derivative) controller and that updating the data-driven model online
significantly improves performance in the presence of unmodeled fluid
disturbance. This paper is complemented with a video:
https://youtu.be/9_wx0tdDta0.
