This paper presents a data-driven methodology for linear embedding of nonlinear systems. Utilizing structural knowledge of general nonlinear dynamics, the authors exploit the Koopman operator to develop a systematic, data-driven approach for constructing a linear representation in terms of higher order derivatives of the underlying nonlinear dynamics. With the linear representation, the nonlinear system is then controlled with an LQR feedback policy, the gains of which need to be calculated only once. As a result, the approach enables fast control synthesis. We demonstrate the efficacy of the approach with simulations and experimental results on the modeling and control of a tail-actuated robotic fish and show that the proposed policy is comparable to backstepping control. To the best of our knowledge, this is the first experimental validation of Koopman-based LQR control.
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Derivative-Based Koopman Operators for Real-Time Control of Robotic Systems
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.
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- NSF-PAR ID:
- 10302016
- Date Published:
- Journal Name:
- IEEE Transactions on Robotics
- ISSN:
- 1552-3098
- Page Range / eLocation ID:
- 1 to 10
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1109/TRO.2021.3076581
