# Research on Three-Closed-Loop ADRC Position Compensation Strategy Based on Winch-Type Heave Compensation System with a Secondary Component

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## Abstract

**:**

## 1. Introduction

## 2. Dynamic Model of Heave Compensation System

#### 2.1. Mathematical Modeling

#### 2.1.1. The Model of the Electro-Hydraulic Servo Valve

_{sf}represents the output flow, I is the coil input current, K

_{v}is the flow gain, s is the Laplace operator, ω

_{n}is the natural frequency, and ξ

_{n}is the damping ratio. Due to its inherent frequency generally being much higher than the system bandwidth, it can be approximated as a proportional link:

#### 2.1.2. Variable Control Oil Cylinder

_{g}is the effective piston area inside the cylinder, x

_{g}is the piston displacement, C

_{t}is the total leakage coefficient, p is the pressure difference between the high-pressure and low-pressure chambers, V

_{t}is the total volume of the two chambers, β

_{e}is the volumetric modulus of oil, C

_{i}is the internal leakage coefficient, and C

_{e}is the external leakage coefficient.

_{g}represents the total mass of the moving components, B

_{g}is the viscous damping coefficient, k

_{g}is the spring stiffness, and F

_{fg}is the resistance force acting on the piston.

#### 2.1.3. Secondary Component Displacement

_{er}represents the displacement, D

_{ermax}is the maximum displacement, and x

_{gmax}is the maximum displacement of the piston.

_{s}represents the constant pressure of the oil supply, J

_{er}is the rotational inertia converted to the output shaft, θ

_{er}is the main shaft rotation angle of the secondary component, B

_{er}is the viscous damping coefficient converted to the output shaft, and M

_{L}is the external load torque.

#### 2.1.4. Cable-Winding System

_{l}represents the displacement of the load in the vertical direction, x

_{l}

_{0}is the initial position of the load, x

_{h}is the displacement due to the heave of the ship, i

_{jt}is the reduction ratio of the gearbox, R is the radius of the winch drum, Δl is the elongation of the cable, Δl

_{d}is the dynamic elongation of the cable, and Δl

_{s}is the static elongation of the cable.

_{eq}represents the equivalent mass of the load and cable, k

_{l}is the cable’s elastic coefficient, and C

_{l}is the cable’s damping coefficient.

#### 2.2. Simulation Model and Parameter Settings

- The system operated in an ideal constant-temperature and constant-pressure hydraulic environment. The hydraulic components and pipelines were sufficiently rigid, and there was no pressure loss along the pipeline.
- The hydraulic oil was considered to be incompressible, with a constant density and viscosity.
- Friction forces at various locations, such as the cable and valve spool, were assumed to be constant and did not vary with operating conditions and temperature changes.

## 3. Three-Closed-Loop ADRC Position Control Strategy Design

#### 3.1. ADRC Controller Design and Composition

#### 3.1.1. The Tracking Differentiator (TD)

_{1}(t), its derivative signal r

_{2}(t) can be computed. In the discrete form, it can be expressed as follows:

_{0}is the filtering factor; and h is the integration step size, where typically h

_{0}can be equal to h, but to reduce overshoot and oscillations, h

_{0}is usually taken to be greater than h.

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

_{01}, β

_{02}, and β

_{03}are observer gain parameters. The setting of these parameters can be determined based on the bandwidth method given by Gao Zhiqiang [30] as follows:

_{1}is the estimated state signal, z

_{2}is the estimated state velocity signal, and z

_{3}is the estimated state acceleration signal.

#### 3.1.3. Nonlinear States Error Feedback (NLSEF)

_{0}, which represents the error feedback control signal.

#### 3.1.4. Disturbance Estimation Compensation

_{0}is the compensation factor. Increasing b

_{0}can reduce the oscillations, but it will simultaneously decrease the compensation for disturbances, affecting the disturbance suppression effectiveness.

#### 3.1.5. Stability Analysis

#### 3.2. Controller Design and Parameter Setting

_{1}(t) and r

_{2}(t). The second-order extended state observer (ESO), which is established for each system, observes the feedback signals. Z

_{1}represents the estimated state signal, Z

_{2}is the estimated state velocity signal, and Z

_{3}is the estimated state acceleration signal. The observer gain parameters β

_{0}

_{1}, β

_{0}

_{2}, and β

_{0}

_{3}are adjusted through the gain-scheduling method, and the tuning of these parameters can be determined using the bandwidth method provided by Gao Zhiqiang [30]. The specific control parameters are listed in Table 3. Finally, the signals r

_{1}(t) and r

_{2}(t) from the tracking differentiator (TD) and the signals Z from the extended state observer (ESO) are input into the nonlinear state error feedback (NLSEF) controller. By adjusting the compensation factor b

_{0}, the error and disturbance are estimated and compensated for.

## 4. Simulation Experiment Research and Analysis

#### 4.1. Step Disturbance Response Analysis

#### 4.2. Sine Disturbance Response Analysis

#### 4.3. A Certain Sea Area in China Ship Heave Motion Disturbance Response Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

Acronyms | Definition |
---|---|

ADRC | Active disturbance rejection control |

PID | Proportional integral derivative |

AMESim | Software name |

A4VSO, A4VSG | Rexroth product model |

HNC | Rexroth controller |

AHC | Active heave compensation system |

SMC | Sliding mode control |

MPC | Model predictive control |

IDA-PBC | Interconnection and damping assignment productivity based control |

ABFTSMC | Adaptive barrier fast terminal sliding mode control |

UAVs | Unmanned aerial vehicles |

FTSMC | Fast terminal sliding mode control |

NMPC | Nonlinear model predictive control |

TD | Tracking differentiator |

ESO | Extended state observer |

NLSEF | Nonlinear states error feedback |

MRU | Motion reference unit |

Symbol | Definition | Symbol | Definition |
---|---|---|---|

Q_{sf} | Output flow | θ_{er} | Main shaft rotation angle of the secondary component |

I | Coil input current | B_{er} | Viscous damping coefficient |

K_{v} | Flow gain | M_{L} | External load torque |

s | Laplace operator | x_{l} | Displacement of the load |

ω_{n} | Natural frequency | x_{l}_{0} | Initial position of the load |

ξ_{n} | Damping ratio | x_{h} | Displacement due to the heave of the ship |

q | Flow entering the high-pressure chamber | i_{jt} | Reduction ratio of the gearbox |

A_{g} | Effective piston area inside | R | Radius of the winch drum |

x_{g} | Piston displacement | Δl | Elongation of the cable |

C_{t} | Total leakage coefficient | Δl_{d} | Dynamic elongation of the cable |

p | Pressure difference | Δl_{s} | Static elongation of the cable |

V_{t} | Total volume | m_{eq} | Equivalent mass of the load and cable |

β_{e} | Volumetric modulus of oil | k_{l} | Cable’s elastic coefficient |

C_{i} | Internal leakage coefficient | C_{l} | Cable’s damping coefficient |

C_{e} | External leakage coefficient | r | Velocity factor |

m_{g} | Total mass of the moving components | h_{0} | Filtering factor |

B_{g} | Viscous damping coefficient | h | Integration step size |

k_{g} | Spring stiffness | fhan | Maximum speed control function |

F_{fg} | Resistance force acting | β_{01}, β_{02}, β_{03} | Observer gain parameters |

D_{er} | Secondary component displacement | b_{0} | Compensation factor |

D_{ermax} | Secondary component maximum displacement | ζ | Heave compensation root-mean-square error compensation rate |

x_{gmax} | Maximum displacement of the piston | Δ | Load position error in heave compensation |

p_{s} | Constant pressure of the oil supply | T | Statistical duration |

J_{er} | Rotational inertia converted to the output shaft | x | Amplitude of the disturbance |

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**Figure 8.**The sine disturbance response curve of three-closed-loop position control system with amplitude 0.5 m and period 10 s.

**Figure 13.**Local magnification of the simulation curve for the heave motion of the ship under the three-closed-loop control system.

Main Components | Sub-Model | Functional Description |
---|---|---|

Motor | PM000 | Standard electric motor |

Pump | PP01 | Constant-pressure variable pump |

Servo valve | SV00 | Three-position four-way directional valve |

Relief valve | RV012 | Safety valve |

Accumulator | HA000 | Diaphragm-type accumulator |

Variable cylinder spring chamber | BAP016 | Variable cylinder reset function chamber |

Variable cylinder piston chamber | BAF01 | Variable cylinder control function chamber |

Mass block | MECMAS21/MAS001 | Variable cylinder mass property simulation |

Hydraulic motor | HYDVPM01 | Bidirectional variable hydraulic motor |

Winch | WINCH01 | Ideal winch |

Cable or wire rope | MECROPE0/REND001 | Rigid rope |

Parameter Name | Value | Unit |
---|---|---|

Hydraulic motor displacement Der | 40 | mL/r |

Maximum motor speed n_{mdmax} | 664 | r/min |

Gearbox reduction ratio i_{jt} | 26.4 | — |

Winch drum radius R | 190 | mm |

System pressure p_{s} | 18 | MPa |

Accumulator volume V | 20 | L |

Parameters | h | r_{TD} | r_{NLSEF} | β_{01} | β_{02} | β_{03} | b_{0} |
---|---|---|---|---|---|---|---|

Inner loop ADRC | 0.001 | $1\times {10}^{7}$ | $5\times {10}^{7}$ | 10000 | $1\times {10}^{7}$ | $1\times {10}^{8/3}$ | $1\times {10}^{6}$ |

Middle loop ADRC | 0.001 | 900 | $1\times {10}^{5}$ | 10000 | $3.75\times {10}^{7}$ | $6.25\times {10}^{10}$ | 780 |

Outermost loop ADRC | 0.001 | 900 | $4.5\times {10}^{5}$ | 10000 | $3.75\times {10}^{7}$ | $6.25\times {10}^{10}$ | 7500 |

Parameters | Rise Time (s) | Peak Time (s) | Maximum Overshoot | Settling Time (s) |
---|---|---|---|---|

No-overshoot PID | 1.3445 | 2.3400 | 0 | 1.7077 |

Overshoot PID | 1.3347 | 1.8900 | 1.18% | 1.6563 |

ADRC | 1.0257 | 1.5700 | 0 | 1.3532 |

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## Share and Cite

**MDPI and ACS Style**

Li, S.; Wu, Q.; Liu, Y.; Qiao, L.; Guo, Z.; Yan, F.
Research on Three-Closed-Loop ADRC Position Compensation Strategy Based on Winch-Type Heave Compensation System with a Secondary Component. *J. Mar. Sci. Eng.* **2024**, *12*, 346.
https://doi.org/10.3390/jmse12020346

**AMA Style**

Li S, Wu Q, Liu Y, Qiao L, Guo Z, Yan F.
Research on Three-Closed-Loop ADRC Position Compensation Strategy Based on Winch-Type Heave Compensation System with a Secondary Component. *Journal of Marine Science and Engineering*. 2024; 12(2):346.
https://doi.org/10.3390/jmse12020346

**Chicago/Turabian Style**

Li, Shizhen, Qinfeng Wu, Yufeng Liu, Longfei Qiao, Zimeng Guo, and Fei Yan.
2024. "Research on Three-Closed-Loop ADRC Position Compensation Strategy Based on Winch-Type Heave Compensation System with a Secondary Component" *Journal of Marine Science and Engineering* 12, no. 2: 346.
https://doi.org/10.3390/jmse12020346