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Nonlinear Swing-Up and Stabilization of Inverted Pendulum on a Cart Using Energy Control Method Based on Multiple Lyapunov Functions, Sliding Mode Control, and Pole Placement
Version 1
: Received: 14 May 2024 / Approved: 15 May 2024 / Online: 15 May 2024 (07:32:34 CEST)
How to cite:
Rible, G. P. Nonlinear Swing-Up and Stabilization of Inverted Pendulum on a Cart Using Energy Control Method Based on Multiple Lyapunov Functions, Sliding Mode Control, and Pole Placement. Preprints2024, 2024051003. https://doi.org/10.20944/preprints202405.1003.v1
Rible, G. P. Nonlinear Swing-Up and Stabilization of Inverted Pendulum on a Cart Using Energy Control Method Based on Multiple Lyapunov Functions, Sliding Mode Control, and Pole Placement. Preprints 2024, 2024051003. https://doi.org/10.20944/preprints202405.1003.v1
Rible, G. P. Nonlinear Swing-Up and Stabilization of Inverted Pendulum on a Cart Using Energy Control Method Based on Multiple Lyapunov Functions, Sliding Mode Control, and Pole Placement. Preprints2024, 2024051003. https://doi.org/10.20944/preprints202405.1003.v1
APA Style
Rible, G. P. (2024). Nonlinear Swing-Up and Stabilization of Inverted Pendulum on a Cart Using Energy Control Method Based on Multiple Lyapunov Functions, Sliding Mode Control, and Pole Placement. Preprints. https://doi.org/10.20944/preprints202405.1003.v1
Chicago/Turabian Style
Rible, G. P. 2024 "Nonlinear Swing-Up and Stabilization of Inverted Pendulum on a Cart Using Energy Control Method Based on Multiple Lyapunov Functions, Sliding Mode Control, and Pole Placement" Preprints. https://doi.org/10.20944/preprints202405.1003.v1
Abstract
This work explores state-of-the-art approaches for stabilizing an inverted pendulum on a cart system, which has been used for over a century to demonstrate ideas in nonlinear and linear control theory, owing to its simple practical implementation yet complex system dynamics. We introduce energy-based control methods and propose a simple linear controller to quickly force the system to the downward equilibrium before swing-up. By constraining the system to start from a stable equilibrium configuration, the problem of global stability is reduced to ensuring that the system can follow all heteroclinic orbits that start very close to the downward equilibrium. Multiple Lyapunov functions are employed to ensure non-zero tracking, increasing the system's domain of attraction than is achieved by traditional controllers based on a single Lyapunov function. We stabilize our inverted pendulum system in the upright configuration through pole placement using a linear quadratic regulator known for fast settling times and smooth system response and through sliding mode controllers known for their robustness to disturbances and model uncertainties. While sliding mode controllers exhibit faster stabilization of the pendulum angle, they fail to stabilize the cart position in a reasonable amount of time. The paper concludes by suggesting various controller techniques that can enhance the swing-up and stabilization controller architectures, leaving room for future exploration. To facilitate further research and education, an open-source MATLAB simulation code is provided, encapsulating the system dynamics and controller designs. This resource offers flexibility for customization, making it a valuable tool for both researchers and educators working with inverted pendulum systems.
Keywords
nonlinear; linear; pendulum; underactuated; Lyapunov; sliding mode; LQR; pole placement; PID; swing-up; stability; energy control
Subject
Computer Science and Mathematics, Robotics
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
