# Digital-Control-Based Approximation of Optimal Wave Disturbances Attenuation for Nonlinear Offshore Platforms

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## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

#### 2.1. Jacket-Type Offshore Platform

#### 2.2. Irregular Wave Disturbances

#### 2.3. Discrete Optimal Irregular Wave Disturbances Attenuation Problem

**Lemma**

**1.**

**Lemma**

**2.**

## 3. Main Results

#### 3.1. Design of AOWDAC

**Theorem**

**1.**

**Proof**

**of**

**Theorem**

**1.**

#### 3.2. The Feasibility of AOWDAC

## 4. Simulation Results

#### 4.1. Parameters of Wave Disturbance Attenuation Problem

#### 4.2. Analysis of Wave Disturbance Attenuation Abilities of Proposed AOWDAC

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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Name of Parameter | Variable | Value | Unit |
---|---|---|---|

Mass of AMD | ${m}_{2}$ | 11,855 | $kg$ |

Nature frequency of AMD | ${\omega}_{2}$ | $2.33$ | rad/s |

Structure damping ratio of AMD | ${\xi}_{2}$ | $9.32$ | % |

First modal mass of offshore structure | ${m}_{1}$ | 2,371,100 | kg |

Nature frequency | ${\omega}_{1}$ | $2.20$ | rad/s |

Structure damping ratio | ${\xi}_{1}$ | 4 | % |

Shape function of first mode | $\phi \left(z\right)$ | $1-cos(\pi z/2L)$ | / |

Equivalent characteristic diameter of legs | ${d}_{3}$ | $1.7$ | m |

Nonlinear parameter | ${\alpha}_{1}$ | 1,500,534 | / |

Nonlinear parameter | ${\alpha}_{2}$ | 2,371,100 | / |

Nonlinear parameter | ${\alpha}_{3}$ | 2,382,955 | / |

Name of Parameter | Variable | Value | Unit |
---|---|---|---|

Water depth | d | $13.2$ | m |

Drag coefficient | ${C}_{d}$ | $1.2$ | / |

Inertial coefficient | ${C}_{m}$ | $2.0$ | / |

Density of ocean fluid | $\rho $ | $0.09$ | / |

Iteration Number | 1 | 2 | 3 | 4 |
---|---|---|---|---|

Performance index J | $4.7354$ | $1.3662$ | $0.9884$ | $0.9792$ |

**Table 4.**Peak values of the displacement, velocity and energy consumption, of offshore platforms under difference control schemes.

Control Scheme | Displacement (m) | Velocity (m/s) | $\mathit{u}\left({10}^{5}\right)$N |
---|---|---|---|

Open-Loop | $0.3035$ | $0.4468$ | |

Optimal feedback and feedfoward vibration controller (36) | $0.1914$ | $0.2352$ | $2.9925$ |

Approximation of optimal wave disturbances attenuation controller (33) | 0.1127 | 0.0910 | 1.6182 |

**Table 5.**The root-mean-square values of the displacement, velocity and energy consumption, of offshore platforms under difference control schemes.

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**MDPI and ACS Style**

Zhong, X.-F.; Sun, Y.-H.; Han, S.-Y.; Zhou, J.; Wang, D.
Digital-Control-Based Approximation of Optimal Wave Disturbances Attenuation for Nonlinear Offshore Platforms. *Energies* **2017**, *10*, 1997.
https://doi.org/10.3390/en10121997

**AMA Style**

Zhong X-F, Sun Y-H, Han S-Y, Zhou J, Wang D.
Digital-Control-Based Approximation of Optimal Wave Disturbances Attenuation for Nonlinear Offshore Platforms. *Energies*. 2017; 10(12):1997.
https://doi.org/10.3390/en10121997

**Chicago/Turabian Style**

Zhong, Xiao-Fang, Yu-Hong Sun, Shi-Yuan Han, Jin Zhou, and Dong Wang.
2017. "Digital-Control-Based Approximation of Optimal Wave Disturbances Attenuation for Nonlinear Offshore Platforms" *Energies* 10, no. 12: 1997.
https://doi.org/10.3390/en10121997