A new optimal control approach to uncertain Euler-Lagrange equations: $ H_\infty $ disturbance estimator and generalized $ H_2 $ tracking controller

Taewan Kim; Jung Hoon Kim; Taewan Kim; Jung Hoon Kim

doi:10.3934/math.20241642

AIMS Mathematics

2024, Volume 9, Issue 12: 34466-34487. doi: 10.3934/math.20241642

Previous Article Next Article

Research article Special Issues

A new optimal control approach to uncertain Euler-Lagrange equations: $ H_\infty $ disturbance estimator and generalized $ H_2 $ tracking controller

Taewan Kim ¹,
Jung Hoon Kim ^{1,2
,
,}

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

Received: 03 October 2024 Revised: 20 November 2024 Accepted: 03 December 2024 Published: 09 December 2024
MSC : 93B18, 93B36, 93B50, 93B53

This paper proposed a new optimal control method for uncertain Euler-Lagrange systems, focusing on estimating model uncertainties and improving tracking performance. More precisely, a linearization of the nonlinear equation was achieved through the inverse dynamic control (IDC) and an $ H_\infty $ optimal estimator was designed to address model uncertainties arising in this process. Subsequently, a generalized $ H_2 $ optimal tracking controller was obtained to minimize the effect of the estimation error on the tracking error in terms of the induced norm from $ L_2 $ to $ L_\infty $. Necessary and sufficient conditions for the existences of these two optimal estimator and controller were characterized through the linear matrix inequality (LMI) approach, and their synthesis procedures can be operated in an independent fashion. To put it another way, this developed approach allowed us to minimize not only the modeling error between the real Euler-Lagrange equations and their nominal models occurring from the IDC approach but also the maximum magnitude of the tracking error by solving some LMIs. Finally, the effectiveness of both the $ H_\infty $ optimal disturbance estimator and the generalized $ H_2 $ tracking controller were demonstrated through some comparative simulation and experiment results of a robot manipulator, which was one of the most representative examples of Euler-Lagrange equations.
- Euler-Lagrange equations,
- inverse dynamic control,
- $ H_\infty $ optimal estimator,
- generalized $ H_2 $ optimal controller,
- linear matrix inequalities
Citation: Taewan Kim, Jung Hoon Kim. A new optimal control approach to uncertain Euler-Lagrange equations: $ H_\infty $ disturbance estimator and generalized $ H_2 $ tracking controller[J]. AIMS Mathematics, 2024, 9(12): 34466-34487. doi: 10.3934/math.20241642

Related Papers:

Abstract

This paper proposed a new optimal control method for uncertain Euler-Lagrange systems, focusing on estimating model uncertainties and improving tracking performance. More precisely, a linearization of the nonlinear equation was achieved through the inverse dynamic control (IDC) and an $ H_\infty $ optimal estimator was designed to address model uncertainties arising in this process. Subsequently, a generalized $ H_2 $ optimal tracking controller was obtained to minimize the effect of the estimation error on the tracking error in terms of the induced norm from $ L_2 $ to $ L_\infty $. Necessary and sufficient conditions for the existences of these two optimal estimator and controller were characterized through the linear matrix inequality (LMI) approach, and their synthesis procedures can be operated in an independent fashion. To put it another way, this developed approach allowed us to minimize not only the modeling error between the real Euler-Lagrange equations and their nominal models occurring from the IDC approach but also the maximum magnitude of the tracking error by solving some LMIs. Finally, the effectiveness of both the $ H_\infty $ optimal disturbance estimator and the generalized $ H_2 $ tracking controller were demonstrated through some comparative simulation and experiment results of a robot manipulator, which was one of the most representative examples of Euler-Lagrange equations.

References

[1]	K. Zhou, J. C. Doyle, K. Glover, Robust and optimal control, Prentice hall, 1995.
[2]	K. Zhou, J. C. Doyle, Essentials of robust control, Prentice Hall, 1997.
[3]	F. Lin, Robust control design: an optimal control approach, John Wiley & Sons, Inc., 2007. https://doi.org/10.1002/9780470059579
[4]	F. L. Lewis, D. M. Dawson, C. T. Abdallah, Robot manipulator control: theory and practice, 2 Eds., CRC Press, 2003. https://doi.org/10.1201/9780203026953
[5]	A. A. Siqueira, M. H. Terra, M. Bergerman, Robust control of robots: fault tolerant approaches, Springer Science & Business Media, 2011. https://doi.org/10.1007/978-0-85729-898-0
[6]	M. W. Spong, S. Hutchinson, M. Vidyasagar, Robot modeling and control, 2 Eds., John Wiley & Sons, Inc., 2020.
[7]	M. L. Corradini, V. Fossi, A. Giantomassi, G. Ippoliti, S. Longhi, G. Orlando, Minimal resource allocating networks for discrete time sliding mode control of robotic manipulators, IEEE Trans. Ind. Inf., 8 (2012), 733–745. https://doi.org/10.1109/TII.2012.2205395 doi: 10.1109/TII.2012.2205395
[8]	X. Jia, J. Yang, T. Shi, W. Wang, Y. Pan, H. Yu, Robust precision motion control based on enhanced unknown system dynamics estimator for high-DoF robot manipulators, IEEE-ASME Trans. Mechatron., 2024. https://doi.org/10.1109/TMECH.2024.3385785 doi: 10.1109/TMECH.2024.3385785
[9]	G. I. Song, J. H. Kim, Time-delay compensation-based robust control of mechanical manipulators: operator-theoretic analysis and experiment validation, Math. Methods Appl. Sci., 47 (2024), 318–335. https://doi.org/10.1002/mma.9656 doi: 10.1002/mma.9656
[10]	H. Zhang, Y. Zhao, Y. Wang, L. Liu, Adaptive neural network control of robotic manipulators with input constraints and without velocity measurements, IET Control Theory Appl., 18 (2024), 1232–1247. https://doi.org/10.1049/cth2.12660 doi: 10.1049/cth2.12660
[11]	H. Y. Park, J. H. Kim, Model-free control approach to uncertain Euler-Lagrange equations with a Lyapunov-based $L_{\infty}$-gain analysis, AIMS Math., 8 (2023), 17666–17686. https://doi.org/10.3934/math.2023902 doi: 10.3934/math.2023902
[12]	O. R. Kang, J. H. Kim, Robust sliding mode control for robot manipulators with analysis on trade-off between reaching time and $L_{\infty}$ gain, Math. Methods Appl. Sci., 47 (2024), 7270–7287. https://doi.org/10.1002/MMA.9972 doi: 10.1002/MMA.9972
[13]	H. Y. Park, J. H. Kim, K. Yamamoto, A new stability framework for trajectory tracking control of biped walking robots, IEEE Trans. Ind. Inf., 18 (2022), 6767–6777. https://doi.org/10.1109/TII.2021.3139909 doi: 10.1109/TII.2021.3139909
[14]	D. Kwak, J. H. Kim, T. Hagiwara, Robust stability analysis of sampled-data systems with uncertainties characterized by the $L_{\infty}$-induced norm: gridding treatment with convergence rate analysis, IEEE Trans. Autom. Control, 68 (2023), 8119–8125. https://doi.org/10.1109/TAC.2023.3288631 doi: 10.1109/TAC.2023.3288631
[15]	W. Tai, X. Li, J. Zhou, S. Arik, Asynchronous dissipative stabilization for stochastic Markov-switching neural networks with completely-and incompletely-known transition rates, Neural Networks, 161 (2023), 55–64. https://doi.org/10.1016/j.neunet.2023.01.039 doi: 10.1016/j.neunet.2023.01.039
[16]	H. T. Choi, J. H. Kim, Set-invariance-based interpretations for the $L_1$ performance of nonlinear systems with non-unique solutions, Int. J. Robust Nonlinear Control, 33 (2023), 1858–1875. https://doi.org/10.1002/rnc.6469 doi: 10.1002/rnc.6469
[17]	O. R. Kang, J. H. Kim, The $l_{\infty}$-induced norm of multivariable discrete-time linear systems: Upper and lower bounds with convergence rate analysis, AIMS Math., 8 (2023), 29140–29157. https://doi.org/10.3934/math.20231492 doi: 10.3934/math.20231492
[18]	J. Zhou, D. Xu, W. Tai, C. K. Ahn, Switched event-triggered $\mathcal{H}_{\infty}$ security control for networked systems vulnerable to aperiodic DoS attacks, IEEE Trans. Network Sci. Eng., 10 (2023), 2109–2123. https://doi.org/10.1109/TNSE.2023.3243095 doi: 10.1109/TNSE.2023.3243095
[19]	L. Wang, J. Su, G. Xiang, Robust motion control system design with scheduled disturbance observer, IEEE Trans. Ind. Electron., 63 (2016), 6519–6529. https://doi.org/10.1109/TIE.2016.2578840 doi: 10.1109/TIE.2016.2578840
[20]	W. H. Chen, J. Yang, L. Guo, S. Li, Disturbance-observer-based control and related methods-an overview, IEEE Trans. Ind. Electron., 63 (2015), 1083–1095. https://doi.org/10.1109/TIE.2015.2478397 doi: 10.1109/TIE.2015.2478397
[21]	T. Li, H. Xing, E. Hashemi, H. D. Taghirad, M. Tavakoli, A brief survey of observers for disturbance estimation and compensation, Robotica, 41 (2023), 3818–3845. https://doi.org/10.1017/S0263574723001091 doi: 10.1017/S0263574723001091
[22]	K. Kim, Y. Hori, Experimental evaluation of adaptive and robust schemes for robot manipulator control, IEEE Trans. Ind. Electron., 42 (1995), 653–662. https://doi.org/10.1109/41.475506 doi: 10.1109/41.475506
[23]	E. Schrijver, J. van Dijk, Disturbance observers for rigid mechanical systems: equivalence, stability, and design, J. Dyn. Sys. Meas. Control, 124 (2002), 539–548. https://doi.org/10.1115/1.1513570 doi: 10.1115/1.1513570
[24]	J. N. Yun, J. B. Su, Design of a disturbance observer for a two-link manipulator with flexible joints, IEEE Trans. Control Syst. Technol., 22 (2013), 809–815. https://doi.org/10.1109/TCST.2013.2248733 doi: 10.1109/TCST.2013.2248733
[25]	Z. Ruan, J. Hu, J. Mei, Robust optimal triple event-triggered intermittent control for uncertain input-constrained nonlinear systems, Commun. Nonlinear Sci. Numer. Simul., 129 (2024), 107718. https://doi.org/10.1016/j.cnsns.2023.107718 doi: 10.1016/j.cnsns.2023.107718
[26]	I. Karafyllis, D. Theodosis, M. Papageorgiou, Lyapunov-based two-dimensional cruise control of autonomous vehicles on lane-free roads, Automatica, 145 (2022), 110517. https://doi.org/10.1016/j.automatica.2022.110517 doi: 10.1016/j.automatica.2022.110517
[27]	H. T. Choi, J. H. Kim, The $\mathscr{L}_{1}$ controller synthesis for piecewise continuous nonlinear systems via set invariance principles, Int. J. Robust Nonlinear Control, 33 (2023), 8670–8692. https://doi.org/10.1002/rnc.6843 doi: 10.1002/rnc.6843
[28]	D. Kwak, T. Hagiwara, J. H. Kim, A new quasi-finite-rank approximation of compression operators on $\mathscr{L}{\infty}[0, H)$ with applications to sampled-data and time-delay systems: piecewise linear kernel approximation approach, J. Frankl. Inst., 361 (2024), 107271. https://doi.org/10.1016/j.jfranklin.2024.107271 doi: 10.1016/j.jfranklin.2024.107271
[29]	M. A. Llama, R. Kelly, V. Santibanez, Stable computed-torque control of robot manipulators via fuzzy self-tuning, IEEE Trans. Syst. Man Cybern., 30 (2000), 143–150. https://doi.org/10.1109/3477.826954 doi: 10.1109/3477.826954
[30]	H. Wang, Y. Xie, Adaptive inverse dynamics control of robots with uncertain kinematics and dynamics, Automatica, 45 (2009), 2114–2119. https://doi.org/10.1016/j.automatica.2009.05.011 doi: 10.1016/j.automatica.2009.05.011
[31]	Y. Chen, G. Ma, S. Lin, J. Gao, Adaptive fuzzy computed-torque control for robot manipulator with uncertain dynamics, Int. J. Adv. Robot. Syst., 9 (2012), 237. https://doi.org/10.5772/54643 doi: 10.5772/54643
[32]	D. V. Balandin, R. S. Biryukov, M. M. Kogan, Finite-horizon multi-objective generalized $H_2$ control with transients, Automatica, 106 (2019), 27–34. https://doi.org/10.1016/j.automatica.2019.04.023 doi: 10.1016/j.automatica.2019.04.023
[33]	M. A. Rotea, The generalized $H_2$ control problem, Automatica, 29 (1993), 373–385. https://doi.org/10.1016/0005-1098(93)90130-L doi: 10.1016/0005-1098(93)90130-L
[34]	C. Scherer, S. Weiland, Linear matrix inequalities in control, Dutch Institute for Systems and Control, 1994.
[35]	L. Villani, C. C. de Wit, B. Brogliato, An exponentially stable adaptive control for force and position tracking of robot manipulators, IEEE Trans. Autom. Control, 44 (1999), 798–802. https://doi.org/10.1109/9.754821 doi: 10.1109/9.754821
[36]	H. Liu, T. Zhang, Adaptive neural network finite-time control for uncertain robotic manipulators, J. Intell. Robot. Syst., 75 (2014), 363–377. https://doi.org/10.1007/s10846-013-9888-5 doi: 10.1007/s10846-013-9888-5
[37]	M. Golestani, R. Chhabra, M. Esmaeilzadeh, Finite-time nonlinear $H_\infty$ control of robot manipulators with prescribed performance, IEEE Control Syst. Lett., 7 (2023), 1363–1368. https://doi.org/10.1109/LCSYS.2023.3241137 doi: 10.1109/LCSYS.2023.3241137

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)