Model-free control approach to uncertain Euler-Lagrange equations with a Lyapunov-based $ L_\infty $-gain analysis

Hae Yeon Park; Jung Hoon Kim; Hae Yeon Park; Jung Hoon Kim

doi:10.3934/math.2023902

AIMS Mathematics

2023, Volume 8, Issue 8: 17666-17686. doi: 10.3934/math.2023902

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Model-free control approach to uncertain Euler-Lagrange equations with a Lyapunov-based $ L_\infty $-gain analysis

Hae Yeon Park ¹,
Jung Hoon Kim ^{1,2
,
,}

1.
Department of Electrical Engineering, Pohang University of Science and Technology (POSTECH), Pohang 37673, Republic of Korea
2.
The Institute for Convergence Research and Education in Advanced Technology, Yonsei University, Incheon 21983, Republic of Korea

Received: 15 March 2023 Revised: 08 May 2023 Accepted: 16 May 2023 Published: 23 May 2023
MSC : 70Q05, 93B52, 93C10, 93D05, 93D09

This paper considers a model-free control approach to Euler-Lagrange equations and proposes a new quantitative performance measure with its Lyapunov-based computation method. More precisely, this paper aims to solve a trajectory tracking problem for uncertain Euler-Lagrange equations by using a model-free controller with a proportional-integral-derivative (PID) control form. The $ L_\infty $-gain is evaluated for the closed-loop systems obtained through the feedback connection between the Euler-Lagrange equation and the model-free controller. To this end, the input-to-state stability (ISS) for the closed-loop systems is first established by deriving an appropriate Lyapunov function. The study further extends these arguments to develop a computational approach to determine the $ L_\infty $-gain. Finally, the theoretical validity and effectiveness of the proposed quantitative performance measure are demonstrated through a simulation of a $ 2 $-degree-of-freedom ($ 2 $-DOF) robot manipulator, which is one of the most representative examples of Euler-Lagrange equations.
- Euler-Lagrange equations,
- input-to-state stability (ISS),
- model-free control,
- optimal tracking control,
- $ L_\infty $-gain,
- Lyapunov function
Citation: Hae Yeon Park, Jung Hoon Kim. Model-free control approach to uncertain Euler-Lagrange equations with a Lyapunov-based $ L_\infty $-gain analysis[J]. AIMS Mathematics, 2023, 8(8): 17666-17686. doi: 10.3934/math.2023902

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Abstract

This paper considers a model-free control approach to Euler-Lagrange equations and proposes a new quantitative performance measure with its Lyapunov-based computation method. More precisely, this paper aims to solve a trajectory tracking problem for uncertain Euler-Lagrange equations by using a model-free controller with a proportional-integral-derivative (PID) control form. The $ L_\infty $-gain is evaluated for the closed-loop systems obtained through the feedback connection between the Euler-Lagrange equation and the model-free controller. To this end, the input-to-state stability (ISS) for the closed-loop systems is first established by deriving an appropriate Lyapunov function. The study further extends these arguments to develop a computational approach to determine the $ L_\infty $-gain. Finally, the theoretical validity and effectiveness of the proposed quantitative performance measure are demonstrated through a simulation of a $ 2 $-degree-of-freedom ($ 2 $-DOF) robot manipulator, which is one of the most representative examples of Euler-Lagrange equations.

References

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