# An Improved Parameter Identification Algorithm for the Friction Model of Electro-Hydraulic Servo Systems

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## Abstract

**:**

## 1. Introduction

## 2. System Model

#### 2.1. LuGre Friction Model

_{0}is the bristle stiffness coefficient; σ

_{1}denotes the microscopic damping coefficient; σ

_{2}represents the viscous friction coefficient; F

_{c}denotes the Coulomb friction force; F

_{s}is the maximum static friction force; v

_{s}is the Stribeck velocity; and z denotes the average bristle deformation. Accurate determination of the six parameters σ

_{0}, σ

_{1}, σ

_{2}, F

_{c}, F

_{s}, and v

_{s}is the key to the study of friction compensation using the LuGre friction model.

_{0}is the sampling period and k is the number of sampling points.

#### 2.2. Identification Model

_{n}, which drives the hydraulic cylinder to produce displacement x. The friction force of the electro-hydraulic servo systems is F

_{f}.

_{n}(k) of the observable measured hydraulic cylinder into Equations (2)–(7).

_{n}(k) (k = 0, 1, …, n) are substituted into the parameter identification model. An appropriate identification algorithm is used to calculate the friction parameters of LuGre friction model.

## 3. Improved Adaptive Genetic Identification Algorithm

#### 3.1. Adaptive Evolution Module Design

_{m}to make the algorithm able to avoid local optimums and obtain the global optimum. Then, the crossover is performed to reduce the crossover probability P

_{c}. When the population concentration is lower than 0.5, differences within the population are large. Crossover can increase the crossover probability P

_{c}to ensure the full evolution of individuals to generate better individuals. The follow-up mutation can reduce the mutation probability P

_{m}, i.e., the probability of destroying high-quality individuals.

_{i}is the fitness value of individuals in the population.

_{1}is the crossover coefficient.

_{2}is the mutation coefficient.

#### 3.2. Optimal Solution Accuracy Optimization Module Design

_{j,i}is the fitness value of the ith individual in the jth evolution; k

_{0}is the number of evolutions of the population; m denotes the total number of evolutions; n is the total number of individuals in the population; δ, γ, and α are the starting coefficients of the optimal solution accuracy optimization module.

_{i}the gene position released at the ith time.

## 4. Simulation Results

_{n}(k). The parameter settings of the LuGre friction model are provided in Table 2.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Baghestan, K.; Rezaei, S.M.; Talebi, H.A.; Zareinejad, M. An energy-saving nonlinear position control strategy for electro-hydraulic servo systems. ISA Trans.
**2015**, 59, 268–279. [Google Scholar] [CrossRef] [PubMed] - Feng, L.; Yan, H. Nonlinear adaptive robust control of the electro-hydraulic servo system. Appl. Sci.
**2020**, 10, 4494. [Google Scholar] [CrossRef] - Tivay, A.; Zareinejad, M.; Rezaei, S.M.; Baghestan, K. A switched energy saving position controller for variable-pressure electro-hydraulic servo systems. ISA Trans.
**2014**, 53, 1297–1306. [Google Scholar] [CrossRef] [PubMed] - Sang, Y.; Gao, H.; Xiang, F. Practical friction models and friction compensation in high-precision electro-hydraulic servo force control systems. Instrum. Sci. Technol.
**2014**, 42, 184–199. [Google Scholar] [CrossRef] - Gao, B.; Shen, W.; Zheng, L.; Zhang, W.; Zhao, H. A Review of Key Technologies for Friction Nonlinearity in an Electro-Hydraulic Servo System. Machines
**2022**, 10, 568. [Google Scholar] [CrossRef] - Wang, Y.; Wang, Z. Research on friction disturbance compensation method in low-speed region of permanent magnet synchronous motor. In Proceedings of the 2022 IEEE 6th Information Technology and Mechatronics Engineering Conference (ITOEC), Chongqing, China, 4–6 March 2022; pp. 2056–2060. [Google Scholar]
- Xue, L. Parameters Identification and Friction Compensation Based on the LuGre Model in Servo System. Master’s Thesis, University of Jinan, Jinan, China, 2013. [Google Scholar]
- Liu, B.; Yao, H.; Nie, S. Parameter Identification of LuGre Friction Model Based on Interval Analysis. China Mech. Eng.
**2013**, 24, 2647. [Google Scholar] - He, Y.; Wang, J.; Hao, R. Adaptive robust dead-zone compensation control of electro-hydraulic servo systems with load disturbance rejection. J. Syst. Sci. Complex.
**2015**, 28, 341–359. [Google Scholar] [CrossRef] - Goforth, F.J.; Gao, Z. An active disturbance rejection control solution for hysteresis compensation. In Proceedings of the 2008 American Control Conference, Seattle, DC, USA, 11–13 June 2008; pp. 2202–2208. [Google Scholar]
- Le Tien, L.; Albu-Schaffer, A.; De Luca, A.; Hirzinger, G. Friction observer and compensation for control of robots with joint torque measurement. In Proceedings of the 2008 IEEE/RSJ International Conference on Intelligent Robots and Systems, Nice, France, 22–26 September 2008; pp. 3789–3795. [Google Scholar]
- Freidovich, L.; Robertsson, A.; Shiriaev, A.; Johansson, R. LuGre-model-based friction compensation. IEEE Trans. Control Syst. Technol.
**2009**, 18, 194–200. [Google Scholar] [CrossRef] - Wang, X.; Wang, S. High performance adaptive control of mechanical servo system with LuGre friction model: Identification and compensation. J. Dyn. Syst. Meas. Control
**2012**, 134, 011021. [Google Scholar] [CrossRef] - Jiang, S.; Zhang, K.; Wang, H.; Zhong, D.; Su, J.; Liu, Z. Research on adaptive friction compensation of digital hydraulic cylinder based on LuGre friction model. Shock Vib.
**2021**, 2021, 8854424. [Google Scholar] [CrossRef] - Zhang, W.; Li, P.; Xu, H. An active disturbance rejection friction compensation in permanent magnet synchronous motor servo system. In Proceedings of the 30th Chinese Control Conference, Shenyang, China, 9–11 June 2018; pp. 3715–3719. [Google Scholar]
- Xu, L. Application of the Newton iteration algorithm to the parameter estimation for dynamical systems. J. Comput. Appl. Math.
**2015**, 288, 33–43. [Google Scholar] [CrossRef] - De Wit, C.C.; Lischinsky, P. Adaptive friction compensation with partially known dynamic friction model. Int. J. Adapt. Control Signal Process.
**1997**, 11, 65–80. [Google Scholar] [CrossRef] - Rutenbar, R.A. Simulated annealing algorithms: An overview. IEEE Circuits Devices Mag.
**1989**, 5, 19–26. [Google Scholar] [CrossRef] - Li, J.; Cheng, J.-H.; Shi, J.-Y.; Huang, F. Brief introduction of back propagation (BP) neural network algorithm and its improvement. In Advances in Computer Science and Information Engineering: Volume 2; Springer: Berlin/Heidelberg, Germany, 2012; pp. 553–558. [Google Scholar]
- Poli, R.; Kennedy, J.; Blackwell, T. Particle swarm optimization. Swarm Intell.
**2007**, 1, 33–57. [Google Scholar] [CrossRef] - Mirjalili, S.; Mirjalili, S. Genetic algorithm. In Evolutionary Algorithms and Neural Networks. Studies in Computational Intelligence; Springer: Cham, Switzerland, 2019; pp. 43–55. [Google Scholar]
- Irakoze, R.; Yakoub, K.; Kaddouri, A. Identification of piezoelectric LuGre model based on particle swarm optimization and real-coded genetic algorithm. In Proceedings of the 2015 IEEE 28th Canadian Conference on Electrical and Computer Engineering (CCECE), Halifax, NS, Canada, 3–6 May 2015; pp. 1451–1457. [Google Scholar]
- Chen, Q.; Bao, J.; Wang, M.; Ye, H.; Peng, F. Simulated Annealing Algorithm for Friction Parameters Identification. TELKOMNIKA Indones. J. Electr. Eng.
**2013**, 11, 245–252. [Google Scholar] [CrossRef] - Ding, S.; Su, C.; Yu, J. An optimizing BP neural network algorithm based on genetic algorithm. Artif. Intell. Rev.
**2011**, 36, 153–162. [Google Scholar] [CrossRef] - Katoch, S.; Chauhan, S.S.; Kumar, V. A review on genetic algorithm: Past, present, and future. Multimed. Tools Appl.
**2021**, 80, 8091–8126. [Google Scholar] [CrossRef] [PubMed] - Lambora, A.; Gupta, K.; Chopra, K. Genetic algorithm-A literature review. In Proceedings of the 2019 international conference on Machine Learning, Big Data, Cloud and Parallel Computing (COMITCon), Faridabad, India, 14–16 February 2019; pp. 380–384. [Google Scholar]
- Johanastrom, K.; Canudas-De-Wit, C. Revisiting the LuGre friction model. IEEE Control Syst. Mag.
**2008**, 28, 101–114. [Google Scholar] [CrossRef] - Yue, F.; Li, X. Robust adaptive integral backstepping control for opto-electronic tracking system based on modified LuGre friction model. ISA Trans.
**2018**, 80, 312–321. [Google Scholar] [CrossRef] [PubMed] - Fang, Y.-C.; Tsai, C.-M.; Chung, C.-L. A study of optical design and optimization of zoom optics with liquid lenses through modified genetic algorithm. Opt. Express
**2011**, 19, 16291–16302. [Google Scholar] [CrossRef] [PubMed]

**Figure 6.**Identification results of difference between maximum static friction ${F}_{s}$ and coulomb friction ${F}_{c}$.

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

Search range of the bristle stiffness coefficient σ_{0} | [0, 5 × 10^{9}] |

Search range of the microscopic damping coefficient σ_{1} | [0, 1000] |

Search range of the viscous friction coefficient σ_{2} | [0, 50,000] |

Search range of the Coulomb friction force F_{c} | [0, 1000] |

Search range of the maximum static friction force F_{s} | [0, 1000] |

Search range of the Stribeck velocity v_{s} | [0.001, 0.1] |

Sampling period t_{0} | 0.001 s |

Crossover coefficient k_{1} | 0.8 |

Mutation coefficient k_{2} | 0.7 |

Starting coefficient γ | 0.1 |

Starting coefficient α | 0.6 |

Constant β | 0.4 |

Number of iterations m | 500 |

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

Bristle stiffness coefficient σ0 | 3.9 × 109 N/m |

Microscopic damping coefficient σ1 | 200 N/(m/s) |

Viscous friction coefficient σ2 | 48,900 N/(m/s) |

Coulomb friction force Fc | 506 N |

Maximum static friction force Fs | 632 N |

Stribeck velocity vs | 0.005 m/s |

Piston rod mass M | 200 kg |

Parameter Type | Different Genetic Algorithm | Identification Value | Error (%) |
---|---|---|---|

Bristle stiffness coefficient σ_{0} | Traditional | 3.9003 × 10^{9} | 0.0077 |

Adaptive | 3.8996 × 10^{9} | 0.010 | |

Improved adaptive | 3.8999 × 10^{9} | 0.0026 | |

Microscopic damping coefficient σ_{1} | Same as above | 187.19 | 6.4 |

205.23 | 2.6 | ||

201.37 | 0.69 | ||

Viscous friction coefficient σ_{2} | 4.32 × 10^{4} | 57 | |

4.84 × 10^{4} | 1.0 | ||

4.88 × 10^{4} | 0.20 | ||

Coulomb friction force F_{c} | 567.94 | 12 | |

512.22 | 1.2 | ||

507.01 | 0.20 | ||

Difference between the maximum static friction force F_{s} and the Coulomb friction force F_{c} | 61.58 | 51 | |

123.20 | 2.2 | ||

125.10 | 0.71 | ||

Stribeck velocity v_{s} | 0.00400 | 20 | |

0.00480 | 4.0 | ||

0.00498 | 0.4 |

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**MDPI and ACS Style**

Liao, J.; Zhou, F.; Zheng, J.
An Improved Parameter Identification Algorithm for the Friction Model of Electro-Hydraulic Servo Systems. *Sensors* **2023**, *23*, 2076.
https://doi.org/10.3390/s23042076

**AMA Style**

Liao J, Zhou F, Zheng J.
An Improved Parameter Identification Algorithm for the Friction Model of Electro-Hydraulic Servo Systems. *Sensors*. 2023; 23(4):2076.
https://doi.org/10.3390/s23042076

**Chicago/Turabian Style**

Liao, Jian, Fuming Zhou, and Jianbo Zheng.
2023. "An Improved Parameter Identification Algorithm for the Friction Model of Electro-Hydraulic Servo Systems" *Sensors* 23, no. 4: 2076.
https://doi.org/10.3390/s23042076