# 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

- Zhang, B.-L.; Han, Q.-L.; Zhang, X.-M. Recent advances in vibration control of offshore platforms. Nonlinear Dyn.
**2017**, 89, 755–771. [Google Scholar] [CrossRef] - Yang, J.S. Hybrid active and passive control of a very large floating beam structure. Nonlinear Dyn.
**2017**, 87, 1835–1845. [Google Scholar] [CrossRef] - Han, S.-Y.; Tang, G.-Y.; Chen, Y.-H.; Tang, X.-X.; Yang, X. Optimal vibration control for vehicle active suspension discrete-time systems with actuator time delay. Asian J. Control
**2013**, 15, 1579–1588. [Google Scholar] [CrossRef] - Zhang, B.-L.; Han, Q.-L.; Zhang, X.-M. Robust non-fragile sampled-data control for offshore steel jacket platforms. Nonlinear Dyn.
**2016**, 83, 1939–1954. [Google Scholar] [CrossRef] - Zhang, B.-L.; Ma, L.; Han, Q.-L. Sliding mode H∞ control for offshore steel jacket platforms subject to nonlinear self-excited wave force and external disturbance. Nonlinear Anal. Real World Appl.
**2013**, 14, 163–178. [Google Scholar] [CrossRef] - Zhang, B.-L.; Huang, Z.-W.; Han, Q.-L. Delayed non-fragile H∞ control for offshore steel jacket platforms. J. Vib. Control
**2015**, 21, 959–974. [Google Scholar] [CrossRef] - Wang, L.; Yang, B.; Chen, Y.; Zhang, X.; Orchard, J. Improving neural-network classifiers using nearest neighbor partitioning. IEEE Trans. Neural Netw. Learn. Syst.
**2017**, 28, 2255–2267. [Google Scholar] [CrossRef] [PubMed] - Yu, Z.; Liu, Y.; Yu, X.; Pu, K.Q. Scalable distributed processing of K nearest neighbor queries over moving objects. IEEE Trans. Knowl. Data Eng.
**2015**, 27, 1383–1396. [Google Scholar] [CrossRef] - Kandasamy, R.; Cui, F.; Townsend, N. A review of vibration control methods for marine offshore structures. Ocean Eng.
**2016**, 127, 279–297. [Google Scholar] [CrossRef] - Faltinsen, O.M. Sea Loads on Ships and Offshore Structures; Cambridge University Press: Cambridge, UK, 1990. [Google Scholar]
- Kandasamy, R.; Cui, F.; Townsend, N. A systematic study of the lowest order small slope approximation for a Pierson–Moskowitz spectrum. IEEE Geosci. Remote Sens. Lett.
**2011**, 8, 158–162. [Google Scholar] - Annalisa, C.; Constance, M.S. Characterizing JONSWAP rogue waves and their statistics via inverse spectral data. Wave Motion
**2017**, 71, 5–17. [Google Scholar] - Cho, H.-Y.; Kweon, H.-M.; Jeong, W.-M.; Kim, S.-I. A study on the optimal equation of the continuous wave spectrum. Int. J. Nav. Archit. Ocean Eng.
**2015**, 7, 1056–1063. [Google Scholar] [CrossRef] - Zhang, B.-L.; Liu, Y.-J.; Han, Q.-L.; Tang, G.-Y. Optimal tracking control with feedforward compensation for offshore jacket platforms with active mass damper mechanisms. J. Vib. Control
**2016**, 22, 695–709. [Google Scholar] [CrossRef] - Zhang, B.-L.; Han, Q.-L. Network-based modelling and active control for offshore steel jacket platform with TMD mechanisms. J. Sound Vib.
**2014**, 333, 6796–6814. [Google Scholar] [CrossRef] - Zhang, B.-L.; Han, Q.-L.; Zhang, X.-M. Event-triggered H∞ reliable control for offshore structures in network environments. J. Sound Vib.
**2016**, 368, 1–21. [Google Scholar] [CrossRef] - Yang, J.S. Robust mixed H
_{2}/H ∞ active control for offshore steel jacket platform. Nonlinear Dyn.**2014**, 2, 1503–1514. [Google Scholar] [CrossRef] - Zhang, B.-L.; Han, Q.-L.; Zhang, X.-M.; Yu, X. Sliding mode control with mixed current and delayed states for offshore steel jacket platforms. IEEE Trans. Control Syst. Technol.
**2014**, 22, 1769–1783. [Google Scholar] [CrossRef] - Nourisola, H.; Ahmadi, B. Robust adaptive sliding mode control based on wavelet kernel principal component for offshore steel jacket platforms subject to nonlinear wave-induced force. J. Vib. Control
**2016**, 22, 3299–3311. [Google Scholar] [CrossRef] - Han, S.-Y.; Chen, Y.-H.; Tang, G.-Y. Sensor fault and delay tolerant control for networked control systems subject to external disturbances. Sensors
**2017**, 17, 700. [Google Scholar] [CrossRef] [PubMed] - Han, S.Y.; Zhang, C.-H.; Tang, G.-Y. Approximation optimal vibration for networked nonlinear vehicle active suspension with actuator time delay. Asian J. Control
**2017**, 19, 983–995. [Google Scholar] [CrossRef] - Han, S.-Y.; Chen, Y.-H.; Tang, G.-Y. Fault diagnosis and fault-tolerant tracking control for discrete-time systems with faults and delays in actuator and measurement. J. Frankl. Inst.
**2017**, 354, 4719–4738. [Google Scholar] [CrossRef] - Han, S.Y.; Wang, D.; Chen, Y.H.; Tang, G.Y.; Yang, X.X. Optimal tracking control for discrete-time systems with multiple input delays under sinusoidal disturbances. Int. J. Control Autom. Syst.
**2015**, 13, 292–301. [Google Scholar] [CrossRef] - Goebel, M.; Raitums, U. Constrained control of a nonlinear two point boundary value problem, I. J. Glob. Optim.
**1994**, 4, 367–395. [Google Scholar] [CrossRef] - Chanane, B. Optimal control of nonlinear systems: A recursive approach. Comput. Math. Appl.
**1998**, 35, 29–33. [Google Scholar] [CrossRef] - Beard, R.W.; Saridis, G.N.; Wen, J.T. Galerkin approximation of the generalized Hamilton-Jacobi-Bellman equation. Automatica
**1997**, 33, 2159–2177. [Google Scholar] [CrossRef] - Tang, G.-Y.; Gao, D.-X. Approximation design of optimal controllers for nonlienar systems with sinusoidal disturbances. Nonlinear Anal. Theory Methods Appl.
**2007**, 66, 403–414. [Google Scholar] [CrossRef] - Ma, H.; Tang, G.-Y.; Zhao, Y.-D. Feedforward and feedback optimal control for offshore structures subjected to irregular wave forces. Ocean Eng.
**2006**, 33, 1105–1117. [Google Scholar] [CrossRef]

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