# Nonlinear Position Control Using Differential Flatness Concept with Load Torque Observer for Electro Hydraulic Actuators with Sinusoidal Load Torque

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

- The proposed method improves the position tracking performance under the sinusoidal load torque in EHAs.
- The proposed position controller is designed with the consideration of the EHS dynamics.
- The proposed load torque observer is designed to estimate the load torque without considering its maximum frequency.

## 2. Modeling

## 3. Controller Design

#### 3.1. Position Controller

**Remark**

**1.**

**Proposition**

**1.**

**Proof.**

**Remark**

**2.**

**Theorem**

**1.**

**Proof.**

#### 3.2. Load Torque Observer

**Theorem**

**2.**

**Proof.**

#### 3.3. Stability Analysis of the Closed-Loop System

**Theorem**

**3.**

**Proof.**

## 4. Simulations

#### 4.1. Case 1

#### 4.2. Case 2

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Merrit, H.E. Hydraulic Control System; Wiley and Sons: New York, NY, USA, 1967. [Google Scholar]
- Chen, T.; Wu, Y. An optimal variable structure control with integral compensation for electrohydraulic position servo control systems. IEEE Trans. Ind. Elec.
**1992**, 39, 460–463. [Google Scholar] [CrossRef] - Jerouane, M.; Lamnabhi-Lagarrigue, F. A new sliding mode controller for a hydraulic actuators. In Proceedings of the Conference on Decision and Control, Orlando, FL, USA, 4–7 December 2001; pp. 908–913. [Google Scholar]
- Bonchis, A.; Corke, P.I.; Rye, D.C.; Ha, Q.P. Variable structure methods in hydraulic servo systems control. Automatica
**2001**, 37, 589–595. [Google Scholar] [CrossRef] - Li, Y.; He, L. Counterbalancing speed control for hydrostatic drive heavy vehicle under longdown-slope. IEEE/ASME Trans. Mechatron.
**2015**, 20, 1533–1542. [Google Scholar] [CrossRef] - Hahn, H.; Piepenbrink, A.; Leimbach, K.-D. Input/output linearization control of an electro servo-hydraulic actuator. In Proceedings of the IEEE 1994 Conference on Control Applications, Glasgow, UK, 24–26 August 1994; pp. 995–1000. [Google Scholar]
- Vossoughi, G.; Donath, M. Dynamic feedback linearization for electrohydraulically actuated control systems. J. Dyn. Syst. Meas. Control
**1995**, 117, 468–477. [Google Scholar] [CrossRef] - Eryilmaz, B.; Wilson, B.H. Improved Tracking Control of Hydraulic Systems. J. Dyn. Syst. Meas. Control
**2001**, 123, 457–462. [Google Scholar] [CrossRef] - Ayalew, B.; Kulakowski, B.T. Cascade tuning for nonlinear position control of an electro-hydraulic actuator. In Proceedings of the IEEE 2006 American Control Conference, Minneapolis, MN, USA, 14–16 June 2006; pp. 4627–4632. [Google Scholar]
- Alleyne, A.; Liu, R. Systematic control of a class of nonlinear systems with application to electrohydraulic cylinder pressure control. IEEE Trans. Control Syst. Technol.
**2000**, 8, 623–634. [Google Scholar] [CrossRef] - Yao, B.; Bu, F.; Reedy, J.; Chiu, G.T.-C. Adaptive robust motion control of single-rod hydraulic actuators: Theory and experiments. IEEE/ASME Trans. Mechatron.
**2000**, 5, 79–91. [Google Scholar] - Kaddissi, C.; Kenné, J.; Saad, M. Identification and real-time control of an electrohydraulic servo system based on nonlinear backstepping. IEEE/ASME Trans. Mechatron.
**2007**, 12, 12–22. [Google Scholar] [CrossRef] - Zeng, H.; Sepehri, N. Tracking control of hydraulic actuators using a LuGre friction model compensation. ASME J. Dyn. Syst. Meas. Control
**2008**, 120, 014502. [Google Scholar] [CrossRef] - Guan, C.; Pan, S. Nonlinear adaptive robust control of single-rod electro-hydraulic actuator with unknown nonlinear parameters. IEEE Trans. Control Syst. Technol.
**2008**, 16, 434–445. [Google Scholar] [CrossRef] - Kim, W.; Won, D.; Shin, D.; Chung, C.C. Output feedback nonlinear control for electro-hydraulic systems. Mechatronics
**2012**, 22, 766–777. [Google Scholar] [CrossRef] - Won, D.; Kim, W.; Tomizuka, M. High gain observer based integral sliding mode control for position tracking of electro-hydraulic systems. IEEE/ASME Trans. Mechatron.
**2017**, 22, 2695–2704. [Google Scholar] [CrossRef] - Wrat, G.; Ranjan, P.; Bhola, M.; Mishra, S.K.; Das, J. Position control and performance analysis of hydraulic system using two pump-controlling strategies. Proc. Inst. Mech. Eng. Part I J. Syst. Control Eng.
**2019**, 233, 1093–1105. [Google Scholar] [CrossRef] - Imani, M.; Ghoreishi, S.F.; Braga-Neto, U.M. Bayesian control of large MDPs with unknown dynamics in data-poor environments. In Proceedings of the Advances in Neural Information Processing Systems, Montreal, QC, Canada, 3–8 December 2018; pp. 8146–8156. [Google Scholar]
- Banerjee, S.; Samynathan, B.; Abraham, J.; Chatterjee, A. Real-time error detection in nonlinear control systems using machine learning assisted state-space encoding. IEEE Trans. Depend. Secur. Comput.
**2019**. [Google Scholar] [CrossRef] - Won, D.; Kim, W.; Shin, D.; Chung, C.C. High gain disturbance observer based backstepping control with output tracking error constraint for electro-hydraulic systems. IEEE Trans. Control Syst. Technol.
**2015**, 23, 787–795. [Google Scholar] [CrossRef] - Guo, Q.; Zhang, M.Y.; Celler, B.G.; Su, S.W. Backstepping control of electro-hydraulic system based on extended-state-observer with plant dynamics largely unknown. IEEE Trans. Ind. Electron.
**2016**, 63, 6909–6920. [Google Scholar] [CrossRef] - Wang, C.; Quan, L.; Zhang, S.; Meng, H.; Lan, Y. Reduced-order model based active disturbance rejection control of hydraulic servo system with singular value perturbation theory. ISA Trans.
**2017**, 67, 455–465. [Google Scholar] [CrossRef] [PubMed] - Won, D.; Kim, W.; Tomizuka, M. Nonlinear control with high gain extended state observer for position tracking of electro-hydraulic systems. IEEE/ASME Trans. Mechatron.
**2020**. [Google Scholar] [CrossRef] - Fliess, M.; Lévine, J.; Martin, P.; Rouchon, P. Flatness and defect of non-linear systems: Introductory theory and examples. Int. J. Control
**1995**, 61, 1327–1361. [Google Scholar] [CrossRef] [Green Version]

**Figure 4.**Simulation results for Case 1: position tracking performance (${x}_{1}$ and ${x}_{1}^{d}$) (

**a**), position tracking error (${e}_{1}$) (

**b**), current (u) (

**c**), magnitude of the load torque (${m}_{L}$) (

**d**) and Load torque (${\tau}_{L}$) (

**e**).

**Figure 5.**Simulation results for Case 2: position tracking performance (${x}_{1}$ and ${x}_{1}^{d}$) (

**a**), position tracking error (${e}_{1}$) (

**b**), current (u) (

**c**), magnitude of the load torque (${m}_{L}$) (

**d**) and Load torque (${\tau}_{L}$) (

**e**).

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Suh, S.; Kim, W.
Nonlinear Position Control Using Differential Flatness Concept with Load Torque Observer for Electro Hydraulic Actuators with Sinusoidal Load Torque. *Mathematics* **2020**, *8*, 1484.
https://doi.org/10.3390/math8091484

**AMA Style**

Suh S, Kim W.
Nonlinear Position Control Using Differential Flatness Concept with Load Torque Observer for Electro Hydraulic Actuators with Sinusoidal Load Torque. *Mathematics*. 2020; 8(9):1484.
https://doi.org/10.3390/math8091484

**Chicago/Turabian Style**

Suh, Sangmin, and Wonhee Kim.
2020. "Nonlinear Position Control Using Differential Flatness Concept with Load Torque Observer for Electro Hydraulic Actuators with Sinusoidal Load Torque" *Mathematics* 8, no. 9: 1484.
https://doi.org/10.3390/math8091484