# Adaptive Damping Variable Sliding Mode Control for an Electrohydrostatic Actuator

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model of the EHA

#### 2.1. Modeling of the Brushless DC Motor (BLDCM)

#### 2.2. Modeling of Pump and Cylinder

## 3. Adaptive Damping Variable Sliding Mode Control Strategy

#### 3.1. Problem Formulation

#### 3.2. Nonlinear Projection Mapping

#### 3.3. Design of the Extended State Observer (ESO)

_{1}is a Hurwitz matrix, a positive definite matrix P satisfying ${B}_{1}^{T}P+P{B}_{1}=-I$ must exist and I is an identity matrix.

#### 3.4. Design of the Adaptive Damping Variable Sliding Mode Controller (ADV-SMController)

_{d}stands for the targeted tracking position of the system.

_{r}is the time interval to reach the sliding surface.

_{1}, together with the parametric adaption, is devised as:

#### 3.5. Stability Analysis

#### 3.6. Control Method Design of the EHA

## 4. Experimental Verification

#### 4.1. Experimental Setup

_{SMC}= 210.7, which are determined via online tuning to facilitate the implementation.

_{n}= 50, ξ

_{max}= 1, ξ

_{min}= 0.1, η

_{DV-SMC}= 210.7.

^{3}, 25}, which is also determined via online tuning. The variable θ is bounded by θ

_{max}= [10, 5000, 25] and θ

_{min}= [0, 1000, 0]. In addition, the initial estimate is set as $\widehat{\mathsf{\theta}}=[1,2000,5]$. Other parameters are the same as those for DV-SMC.

#### 4.2. Results

^{−3}m for PID, 3.0 × 10

^{−4}m for SMC, 3.6 × 10

^{−4}m for DV-SMC, and 1.2 × 10

^{−4}m for ADV-SMC. These performance gaps range from 2.3 × 10

^{−3}m (PID) to 1.7 × 10

^{−4}m (SMC), which are significant (Table 2). With the same robust gain parameters and within the last period, the maximum position tracking errors for ADV-SMC, DV-SMC, and SMC are 0.2, 0.4, and 0.3 mm, respectively. It is worth noting that the tracking accuracy of DV-SMC does not exceed that of SMC. A possible explanation is that the first and second derivatives of x

_{d}are set as 0 to suppress the overshoot in DV-SMC, which weakens the capability in steady-state response. In this case, removing the feedforward compensation term from basic SMC has an impact on the sinusoidal tracking accuracy and leads to the accumulating of tracking errors.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Maré, J.-C.; Fu, J. Review on signal-by-wire and power-by-wire actuation for more electric aircraft. Chin. J. Aeronaut.
**2017**, 30, 857–870. [Google Scholar] [CrossRef] - Alle, N.; Hiremath, S.S.; Makaram, S.; Subramaniam, K.; Talukdar, A. Review on electro hydrostatic actuator for flight control. Int. J. Fluid Power
**2016**, 17, 125–145. [Google Scholar] [CrossRef] - Shang, Y.; Li, X.; Qian, H.; Wu, S.; Pan, Q.; Huang, L.; Jiao, Z. A Novel Electro Hydrostatic Actuator System With Energy Recovery Module for More Electric Aircraft. IEEE Trans. Ind. Electron.
**2020**, 67, 2991–2999. [Google Scholar] [CrossRef] - Marco, V.; Marco, T.; Giuseppe, F.; Luca, C. Electromechanical Actuator for Helicopter Rotor Damper Application. IEEE Trans. Ind. Appl.
**2014**, 50, 1007–1014. [Google Scholar] - Qiao, G.; Liu, G.; Shi, Z.; Wang, Y.; Ma, S.; Lim, T.C. A review of electromechanical actuators for More/All Electric aircraft systems. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci. Nov.
**2017**, 232, 4128–4151. [Google Scholar] [CrossRef] [Green Version] - Bossche, D. The A380 flight control electro-hydrostatic actuators, achievements and lessons learnt. In Proceedings of the 25th International Congress of the Aeronautical Sciences, Hamburg, Germany, 3–8 September 2006; pp. 1–8. [Google Scholar]
- Qi, H.T.; Fu, Y.L.; Qi, X.Y.; Lang, Y. Architecture Optimization of More Electric Aircraft Actuation System. Chin. J. Aeronaut.
**2011**, 24, 506–513. [Google Scholar] [CrossRef] [Green Version] - MOOG Inc. Electro Hydrostatic Actuators; MOOG Inc.: Salt Lake City, UT, USA, 2014. [Google Scholar]
- Zhao, J.A.; Fu, Y.L.; Ma, J.M.; Fu, J.; Chao, Q.; Wang, Y. Review of cylinder block/valve plate interface in axial piston pumps: Theoretical models, experimental investigations, and optimal design. Chin. J. Aeronaut.
**2021**, 34, 111–134. [Google Scholar] [CrossRef] - Shea, A.; Jahns, T.M. Hardware integration for an integrated modular motor drive including distributed control. In Proceedings of the 2014 IEEE Energy Conversion Congress and Exposition (ECCE), Pittsburgh, PA, USA, 14–18 September 2014; pp. 4881–4887. [Google Scholar]
- Wang, J.; Li, Y.; Han, Y. Integrated modular motor drive design with GaN power FETs. IEEE Trans. Ind. Appl.
**2015**, 51, 3198–3207. [Google Scholar] [CrossRef] - Yang, Y.S.; Niu, W.Q.; Zhao, J. Design and Numerical Study of Electro-Hydrostatic Actuator. J. Phys. Conf. Ser.
**2019**, 012088, 1–9. [Google Scholar] [CrossRef] - Ren, G.; Esfandiari, M.; Song, J.; Sepehri, N. Position Control of an Electro-hydrostatic Actuator with Tolerance to Internal Leakage. IEEE Trans. Control Syst. Technol.
**2016**, 24, 2224–2232. [Google Scholar] [CrossRef] - Zhang, H.; Liu, X.; Wang, J.; Karimi, H.R. Robust H∞ Sliding mode control with pole placement for a fluid power electrohydraulic actuator (EHA) system. Int. J. Adv. Manuf. Technol.
**2014**, 73, 1095–1104. [Google Scholar] [CrossRef] [Green Version] - Wang, C.; Quan, L.; Jiao, Z.; Zhang, S. Nonlinear Adaptive Control of Hydraulic System With Observing and Compensating Mismatching Uncertainties. IEEE Trans. Control Syst. Technol.
**2018**, 26, 927–938. [Google Scholar] [CrossRef] - Yao, Z.K.; Yao, J.Y.; Yao, F.Y. Model reference adaptive tracking control for hydraulic servo systems with nonlinear neural-networks. ISA Trans.
**2019**, 100, 396–404. [Google Scholar] [CrossRef] [PubMed] - Wang, M.K.; Fu, Y.L.; Zhao, J.A. A Novel Cascade Control Based on Damp Variable Sliding Mode Control for an Electro-hydrostatic Actuator. J. Beijing Univ. Aeronaut. Astronaut.
**2020**. (In Chinese) [Google Scholar] [CrossRef] - Yang, R.R.; Fu, Y.L.; Zhang, G.L. A Novel Sliding Mode Control Framework for Electrohydrostatic Actuator. Math. Probl. Eng.
**2018**, 7159891. [Google Scholar] [CrossRef] - Ataklti, E.A.; Fu, Y.L. Sliding mode control of electro-hydrostatic actuator based on extended state observer. In Proceedings of the 29th Chinese Control And Decision Conference, Chongqing, China, 28–30 May 2017; pp. 758–763. [Google Scholar]
- Wang, M.; Wang, Y.; Yang, R.; Fu, Y.; Zhu, D. A Sliding Mode Control Strategy for an ElectroHydrostatic Actuator with Damping Variable Sliding Surface. Actuators
**2020**, 10, 3. [Google Scholar] [CrossRef] - Ali, S.A.; Christen, A.; Begg, S.; Langlois, N. Continuous–discrete time-observer design for state and disturbance estimation of electrohydraulic actuator systems. IEEE Trans. Ind. Electron.
**2016**, 63, 4314–4324. [Google Scholar] [CrossRef] [Green Version] - Throne, D.; Martinez, F.; Marguire, R.; Arens, D. Integrated Motor/Drive Technology with Rockwell Connectivity. Rexroth, Bosch Group, Lohr am Main, Germany, Tech. Rep. Available online: http://www.cmafh.com/enewsletter/PDFs/IntegratedMotorDrives.pdf (accessed on 17 April 2021).
- Kim, H.-J.; Park, H.-S.; Kim, J.-M. Expansion of Operating Speed Range of High-Speed BLDC Motor Using Hybrid PWM Switching Method Considering Dead Time. Energies
**2020**, 13, 5212. [Google Scholar] [CrossRef] - Berri, P.C.; Vedova, M.D.L.D.; Maggiore, P. A Simplified Monitor Model for EMA Prognostics. Matec Web Conf.
**2018**, 233, 00016. [Google Scholar] [CrossRef] - Yao, B.; Bu, F.; Reedy, J.; Chiu, G.C. Adaptive Robust Motion Control of Single-Rod Hydraulic Actuators: Theory and Experiments. IEEE/ASME Trans. Mechatron.
**2000**, 5, 79–91. [Google Scholar]

**Figure 7.**Experimental responses to 1 Hz sinusoidal signal: (

**a**) tracking error; (

**b**) parametric estimation; (

**c**) control output.

Parameter | Value |
---|---|

Piston effective area (m^{2}) | 1.134 × 10^{−4} |

Effective stroke (m) | 0.1 |

Fluid elastic modulus (N/m^{2}) | 6.86 × 10^{8} |

Hydraulic cylinder volume (m^{3}) | 4 × 10^{−4} |

Mass of cylinder and load (kg) | 243 |

Pump displacement (m^{3}/rad) | 3.98 × 10^{−7} |

Phase resistance (Ω) | 0.2 |

Phase inductance (mH) | 1.33 |

Motor spindle moment of inertia (kg⸱m^{2}) | 4 × 10^{−4} |

Torque coefficient (N⸱m/A) | 0.351 |

Back EMF coefficient (V/(rad/s)) | 0.234 |

Bus voltage (V) | 270 |

5 mm | 50 mm | |||
---|---|---|---|---|

ST/s | OS | ST/s | OS | |

PID | 1+ | 0.024 | 1.8+ | 0.029 |

SMC | 0.925 | 0.022 | 1.8+ | 0.029 |

DV-SMC | 0.285 | 0 | 1.63 | 0 |

ADV-SMC | 0.205 | 0 | 1.03 | 0 |

Error | Average | Maximum | Variance |
---|---|---|---|

PID | 0.001589 | 0.002421 | 5.17 × 10^{−7} |

DV-SMC | 0.000252 | 0.000362 | 1.02 × 10^{−8} |

SMC | 0.000205 | 0.000297 | 6.64 × 10^{−9} |

ADV-SMC | 4.42 × 10^{−5} | 0.000123 | 1.74 × 10^{−9} |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Li, L.; Wang, M.; Yang, R.; Fu, Y.; Zhu, D.
Adaptive Damping Variable Sliding Mode Control for an Electrohydrostatic Actuator. *Actuators* **2021**, *10*, 83.
https://doi.org/10.3390/act10040083

**AMA Style**

Li L, Wang M, Yang R, Fu Y, Zhu D.
Adaptive Damping Variable Sliding Mode Control for an Electrohydrostatic Actuator. *Actuators*. 2021; 10(4):83.
https://doi.org/10.3390/act10040083

**Chicago/Turabian Style**

Li, Linjie, Mingkang Wang, Rongrong Yang, Yongling Fu, and Deming Zhu.
2021. "Adaptive Damping Variable Sliding Mode Control for an Electrohydrostatic Actuator" *Actuators* 10, no. 4: 83.
https://doi.org/10.3390/act10040083