# Force Loading Tracking Control of an Electro-Hydraulic Actuator Based on a Nonlinear Adaptive Fuzzy Backstepping Control Scheme

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## Abstract

**:**

## 1. Introduction

## 2. Experimental Set Up and Dynamic Model

#### 2.1. Experimental Setup of the EHFLS

#### 2.2. Dynamic Model of the EHFLS

## 3. Controller Design

#### 3.1. Adaptive Backstepping Controller

**Step 1**Define the force loading tracking error ${e}_{1}$ as:

**Remark**

**1.**

**Step 2**Referring to Equation (14), the time derivative of ${\alpha}_{1}$ is given as:

**Remark**

**2.**

**Step 3**Because the final controlled variable ${x}_{2}$ is visualized, it is not necessary to imagine another virtual control variable at this step.

#### 3.2. Adaptive Fuzzy Controller

**Remark**

**3.**

**Step****1**- Define $m$ fuzzy set ${A}_{j}^{}$ to state variables ${e}_{j}$ (in this work, the order of the state space form of ${e}_{j}$ is 1, so $n=1$).
**Step****2**- The control output of fuzzy system can be expressed as by employing the strategy of product inference engine, singleton fuzzifier, and center average defuzzifier:$${\widehat{u}}_{j}\left({e}_{j}|{\widehat{\mathsf{\xi}}}_{\mathit{fj}}\right)=\frac{{\displaystyle \sum _{l=1}^{{m}_{}}{\overline{y}}_{\mathrm{u}}^{l}\left({\mu}_{{A}_{j}^{}}\left({e}_{j}\right)\right)}}{{\displaystyle \sum _{l=1}^{{m}_{}}\left({\mu}_{{A}_{j}^{}}\left({e}_{j}\right)\right)}},$$

**Proof.**

## 4. Experimental Results and Analysis

## 5. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 6.**Force loading tracking performance with the PID + VF subjected to the random position disturbances. (

**a**) force loading tracking performance; (

**b**) force loading tracking error.

**Figure 7.**Force loading tracking performance with the backstepping controller subjected to the random position disturbances. (

**a**) force loading tracking performance; (

**b**) force loading tracking error.

**Figure 8.**Force loading tracking performance with the ABC subjected to the random position disturbances. (

**a**) force loading tracking performance; (

**b**) force loading tracking error.

**Figure 9.**Force loading tracking performance with the NAFBC subjected to the random position disturbances. (

**a**) force loading tracking performance; (

**b**) force loading tracking error.

**Figure 10.**Force loading tracking performance with the PID + VF subjected to the step position disturbances. (

**a**) force loading tracking performance; (

**b**) force loading tracking error.

**Figure 11.**Force loading tracking performance with the backstepping controller subjected to the step position disturbances. (

**a**) force loading tracking performance; (

**b**) force loading tracking error.

**Figure 12.**Force loading tracking performance with the ABC subjected to the step position disturbances. (

**a**) force loading tracking performance; (

**b**) force loading tracking error.

**Figure 13.**Force loading tracking performance with the NAFBC subjected to the step position disturbances. (

**a**) force loading tracking performance; (

**b**) force loading tracking error.

Description | Parameters | Values | Units |
---|---|---|---|

Oil effective bulk modulus | ${\beta}_{e}$ | $1\times {10}^{9}$ | ${\mathrm{N}/\mathrm{m}}^{2}$ |

Total chamber volume | ${V}_{ft}$ | $3.8\times {10}^{-4}$ | ${\mathrm{m}}^{3}$ |

Cylinder effective area | ${A}_{fp}$ | $1.88\times {10}^{-3}$ | ${\mathrm{m}}^{2}$ |

Total mass | ${m}_{fp}$ | 530 | kg |

Viscous damping coefficient | ${B}_{fp}$ | 7000 | ${\mathrm{N}/\mathrm{ms}}^{-1}$ |

Stiffness of force sensor | ${k}_{f}$ | $1.9\times {10}^{7}$ | $\mathrm{N}/\mathrm{m}$ |

Saturated input of servo-valve | ${u}_{\mathrm{max}}$ | 10 | V |

Rated flow of servo-valve | ${Q}_{r}$ | 38 | L/min |

Total leakage coefficient | ${C}_{tp}$ | $6.9\times {10}^{-13}$ | ${\mathrm{m}}^{3}/\mathrm{s}\cdot \mathrm{Pa}$ |

System supply pressure | ${P}_{fs}$ | 9 | Mpa |

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

${k}_{p}$ | 0.0007 | ${F}_{2}$ | 300 |

${k}_{i}$ | 0.002 | ${F}_{3}$ | 200 |

${k}_{d}$ | 0 | ${\gamma}_{1}$ | $2\times {10}^{-11}$ |

${k}_{vf}$ | 0.0035 | ${\gamma}_{2}$ | $3.5\times {10}^{4}$ |

${k}_{1}$ | $100$ | ${\gamma}_{3}$ | $6\times {10}^{-12}$ |

${k}_{2}$ | $2.8\times {10}^{8}$ | ${\phi}_{1}$ | 0.01 |

${k}_{3}$ | $2\times {10}^{5}$ | ${\phi}_{2}$ | 50 |

${F}_{1}$ | 500 | ${\phi}_{3}$ | 150 |

**Table 3.**Peak errors and the RMSE of four control algorithms with random position external disturbances.

Control Algorithm | Peak Error (N) | RMSE |
---|---|---|

PID + VF | 2435.7 | 858.675 |

Backstepping controller | 2215.9 | 654.325 |

ABC | 1995.6 | 603.215 |

NAFBC | 1405.2 | 445.389 |

**Table 4.**Peak errors and the RMSE of four control algorithms with step position external disturbances.

Control Algorithm | Peak Error (N) | RMSE |
---|---|---|

PID + VF | 3952.3 | 1160.837 |

Backstepping controller | 3743.8 | 1070.723 |

ABC | 2914.6 | 957.904 |

NAFBC | 2610.5 | 909.292 |

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**MDPI and ACS Style**

Li, X.; Zhu, Z.-C.; Rui, G.-C.; Cheng, D.; Shen, G.; Tang, Y.
Force Loading Tracking Control of an Electro-Hydraulic Actuator Based on a Nonlinear Adaptive Fuzzy Backstepping Control Scheme. *Symmetry* **2018**, *10*, 155.
https://doi.org/10.3390/sym10050155

**AMA Style**

Li X, Zhu Z-C, Rui G-C, Cheng D, Shen G, Tang Y.
Force Loading Tracking Control of an Electro-Hydraulic Actuator Based on a Nonlinear Adaptive Fuzzy Backstepping Control Scheme. *Symmetry*. 2018; 10(5):155.
https://doi.org/10.3390/sym10050155

**Chicago/Turabian Style**

Li, Xiang, Zhen-Cai Zhu, Guang-Chao Rui, Dong Cheng, Gang Shen, and Yu Tang.
2018. "Force Loading Tracking Control of an Electro-Hydraulic Actuator Based on a Nonlinear Adaptive Fuzzy Backstepping Control Scheme" *Symmetry* 10, no. 5: 155.
https://doi.org/10.3390/sym10050155