# Position Control with ADRC for a Hydrostatic Double-Cylinder Actuator

^{*}

## Abstract

**:**

## 1. Introduction

## 2. System Principle and Mathematical Model

#### 2.1. System Principle

#### 2.2. System Mathematical Model

_{h}is the equivalent coefficient of the oil compression converted into elastic deformation of the hose, k

_{1}and k

_{2}are the elastic coefficients of the hose and B

_{e}is the damping coefficient of the hose wall. a is the average radius of the hose.

_{P1}and A

_{P2}are the effective area of the power cylinder, A

_{A1}and A

_{A2}are the effective area of the target cylinder respectively. X

_{pi}and X

_{po}are the displacements of the power cylinder and the target cylinder, respectively. V

_{h1}is the pressurized volume enclosed by the hose connecting two rodless chambers and V

_{h2}is the one enclosed by the hose connecting two rod chambers.

_{t}is the force as the external load, m is the load mass, P

_{1}and P

_{2}are the pressure of the rod chamber and the rodless chamber of the target cylinder respectively, and f

_{v}is the friction in the target cylinder as given by

_{s}is the maximum static friction and μ

_{visc}is the viscous friction coefficient.

#### 2.3. Controller Design

#### 2.3.1. Tracking Differentiator (TD)

_{1}are the input and the output of the tracking differentiator, respectively, v

_{2}is the differential output of v

_{1}, r is the adjustable tracking speed, and h and h

_{0}are the simulation intervals for the TD and fhan function, respectively. When h = h

_{0}, the high-frequency oscillation in v

_{2}can be eliminated, and when h > h

_{0}, the noise can be removed from v

_{2}.

#### 2.3.2. Extended State Observer (ESO)

_{1},x

_{2},ω(t),t) is added to the original system to make it expand into a new state variable.

_{1}, z

_{2}and b

_{0}are the estimations of x

_{1}, x

_{2}and b, respectively. z

_{3}is the total disturbance estimation of the system. β

_{01}, β

_{02}and β

_{03}are three constants in the ESO. fal(x,α,δ) is the nonlinear function of the ESO.

#### 2.3.3. Nonlinear State Error Feedback (NLSEF)

_{1}and β

_{2}are the gain factors. α

_{1}and α

_{2}generally satisfy 0 < α

_{1}< 1 < α

_{2}.

_{3}to estimate the total disturbance in real time.

_{3}(k)/b

_{0}is the compensation disturbance component and u

_{0}(k) is the integrator series component controlled by the nonlinear feedback.

## 3. Results

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Schematic diagram of the whole system. 1. Force transducer; 2. Load mass; 3. LVDT; 4. Target cylinder; 5. Hose; 6. Accumulator; 7. On/off valve; 8. Power cylinder; 9. Ball screw; 10. Stepping motor; 11. Loading system.

**Figure 6.**Experimental system.1. Stepping motor; 2. Ball screw; 3. Power cylinder; 4. Target cylinder; 5. Force transducer; 6. LVDT; 7. Pressure transducer; 8. Accumulator (oil compensation tank); 9. Hose.

**Figure 7.**(

**a**). Displacement of two cylinders. (

**b**). Displacement of the target cylinder and pressure in the rodless chamber.

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

m | 50 kg | r | 0.003 m | c_{1} | 140 |

A_{P}_{1} | 3.77 × 10^{−4} m^{2} | l | 10 m | γ_{1} | 0.5 |

A_{P}_{2} | 4.91 × 10^{−4} m^{2} | k_{11} | 6 × 10^{5} | c_{2} | 120 |

A_{A}_{1} | 3.14 × 10^{−4} m^{2} | k_{12} | 5000 | γ_{2} | 0.1 |

A_{A}_{2} | 2.01 × 10^{−4} | k_{2} | 1 × 10^{6} | c_{j} | [−1.0 −0.5 0 0.5 1] |

f_{s} | 50 N | μ_{visc} | 200 N·s/m | b_{j} | 5 |

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

P | 7 | β_{02} | 100 | α_{2} | 0.4 |

I | 0.01 | β_{03} | 30 | b_{0} | 0.02 |

h | 0.002 | β_{1} | 50 | b | 5 |

h_{0} | 0.002 | β_{2} | 0.1 | r | 0.005 |

β_{01} | 50 | α_{1} | 0.5 | δ | 0.001 |

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**MDPI and ACS Style**

Wang, B.; Ji, H.; Chang, R. Position Control with ADRC for a Hydrostatic Double-Cylinder Actuator. *Actuators* **2020**, *9*, 112.
https://doi.org/10.3390/act9040112

**AMA Style**

Wang B, Ji H, Chang R. Position Control with ADRC for a Hydrostatic Double-Cylinder Actuator. *Actuators*. 2020; 9(4):112.
https://doi.org/10.3390/act9040112

**Chicago/Turabian Style**

Wang, Bin, Hengyu Ji, and Rui Chang. 2020. "Position Control with ADRC for a Hydrostatic Double-Cylinder Actuator" *Actuators* 9, no. 4: 112.
https://doi.org/10.3390/act9040112