Deep Koopman Operator With Control for Nonlinear Systems

Recently Koopman operator has become a promising data-driven tool to facilitate real-time control for unknown nonlinear systems. It maps nonlinear systems into equivalent linear systems in embedding space, ready for real-time linear control methods. However, designing an appropriate Koopman embeddin...

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Bibliographic Details
Published in:IEEE robotics and automation letters Vol. 7; no. 3; pp. 7700 - 7707
Main Authors: Shi, Haojie, Meng, Max Q.-H.
Format: Journal Article
Language:English
Published: Piscataway IEEE 01.07.2022
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:
Aerospace electronics
Algorithms
Cart-pole problem
Control methods
Control systems
Damping
Deep learning
Deep learning methods
Dynamical systems
Embedding
Linear control
Linear quadratic regulator
Linear systems
Linearity
Machine learning
machine learning for robot control
model learning for control
Neural networks
Nonlinear control
Nonlinear dynamical systems
Nonlinear dynamics
Nonlinear systems
Nonlinearity
Optimal control
Real time
Real-time systems
Robot arms
ISSN:2377-3766, 2377-3766
Online Access:Get full text
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Summary:Recently Koopman operator has become a promising data-driven tool to facilitate real-time control for unknown nonlinear systems. It maps nonlinear systems into equivalent linear systems in embedding space, ready for real-time linear control methods. However, designing an appropriate Koopman embedding function remains a challenging task. Furthermore, most Koopman-based algorithms only consider nonlinear systems with linear control input, resulting in lousy prediction and control performance when the system is fully nonlinear with the control input. In this work, we propose an end-to-end deep learning framework to learn the Koopman embedding function and Koopman Operator together to alleviate such difficulties. We first parameterize the embedding function and Koopman Operator with the neural network and train them end-to-end with the K-steps loss function. Then, an auxiliary control network is augmented to encode the nonlinear state-dependent control term to model the nonlinearity in the control input. This encoded term is considered the new control variable instead to ensure linearity of the modeled system in the embedding system. We next deploy Linear Quadratic Regulator (LQR) on the linear embedding space to derive the optimal control policy and decode the actual control input from the control net. Experimental results demonstrate that our approach outperforms other existing methods, reducing the prediction error by order of magnitude and achieving superior control performance in several nonlinear dynamic systems like damping pendulum, CartPole, and the seven DOF robotic manipulator.
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ISSN:2377-3766
2377-3766
DOI:10.1109/LRA.2022.3184036