# Model Predictive Control Tuning by Inverse Matching for a Wave Energy Converter

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## Abstract

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## 1. Introduction

## 2. Methods

#### 2.1. Controller Matching Problem

#### 2.2. An Analytical Solution When $N=2$

## 3. Results: Application to a WEC

`tfest`over the frequency range of interest $f\in [0.1\phantom{\rule{0.277778em}{0ex}}1.0]$ (Hz) assuming that the energy in ocean waves lies within a relatively narrow frequency band (periods of $1\le T\le 10$ s). Note that, as previously stated, the WEC considered here is a model scale device, which is tested in a wave basin. As such, the frequency range of interest is higher than that of an open ocean device. From the second-order model (37), the natural frequency and the damping ratio are found by ${\omega}_{n}=\sqrt{16.2}$ = 4.025 (rad/s) = 0.641 (Hz) and $\zeta =1.38/\left(2{\omega}_{n}\right)=0.171$, respectively. Hence, the resonant frequency is located at ${\omega}_{n}\sqrt{1-2{\zeta}^{2}}$ = 3.905 (rad/s) = 0.622 (Hz), which is an approximate value because of the existence of the zero in the transfer function. At the resonant frequency, the output is in phase with the input, and for small input frequencies, 90 degrees phase lead occurs while 90 degrees phase lag occurs for large input frequencies. Figure 2 shows the Bode diagram of the original model (36) and the simplified model (37), which shows a good agreement between the two models. Hence, we proceed to MPC design based on the simplified model (37).

#### 3.1. Numerical Simulations

- The system has no constraints.
- The system has a nearly hard constraint on its position.
- The system has a hard constraint on its control input and a soft constraint on its position.

`OV.minECR`and

`OV.maxECR`values (

`OV`denotes the output variable) for the position used in the MPC Toolbox are set as $2\times {10}^{-4}$, where an equal concern for the relaxation (ECR) value is zero for a completely hard constraint and a larger ECR value means a softer constraint. When constraints are imposed, the value function Equation (1) is rewritten as

`OV.minECR`,

`OV.maxECR`,

`MV.minECR`, and

`MV.maxECR`, respectively in the MPC Toolbox. It is easy to see that when an ECR value is zero, the constraint is a hard one that cannot be violated. The larger ECR value is, the softer the constraint is.

`MV.minECR`and

`MV.maxECR`values (

`MV`denotes the manipulated variable) are set as 0 for the control input and

`OV.minECR`=

`OV.maxECR`= 0.01 are selected for the position. For all scenarios, the control horizon is equal to the prediction horizon N. It should be mentioned that the parameters are scaled to improve numerical accuracy.

#### 3.1.1. Scenario 1: System with No Constraint

#### 3.1.2. Scenario 2: System with Nearly Hard Constraint on the Position

`OV.minECR`=

`OV.maxECR`= $2\times {10}^{-4}$) is imposed on the position of the WEC device and the prediction horizon $N=50$ is assumed to anticipate constraint violation early enough to allow corrective action. A ‘nearly’ hard constraint means that the constraint is quite well satisfied on the whole but a small constraint violation is allowed. In practice, a completely hard constraint should not be imposed on the output because the plant is always subject to disturbance and QP infeasibility is inevitable. We apply the numerical solver ‘YALMIP’ to yield the following weight matrices:

#### 3.1.3. Scenario 3: System with Hard Constraint on the Control Force and Soft Constraint on the Position

`MV.minECR`=

`MV.maxECR`= 0) on the control force and a soft constraint (

`OV.minECR`=

`OV.maxECR`= 0.01) on the position. In general, control force and position constraints cannot always be simultaneously satisfied, and so the position constraint is usually softened for QP feasibility [23]. Bacelli and Ringwood [28] extended the discussion of the force and position constraints for WECs and proposed sufficient conditions for the satisfaction of both constraints for a given hydrodynamic model and set of sea conditions.

#### 3.2. Experimental Tests

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**Comparison of the control forces, positions, power absorptions, and cumulative average power obtained by the proportional-integral (PI) controller and the model predictive control (MPC) (scenario 1).

**Figure 4.**Comparison of the control forces, positions, power absorptions, and cumulative average power obtained by the PI controller and the MPC (scenario 2).

**Figure 5.**Comparison of the control forces, positions, power absorptions, and cumulative average power obtained by the PI controller and the MPC (scenario 3).

**Figure 6.**Time history of control forces, velocities, and power captures obtained by PI controller and MPC.

**Figure 7.**Wave elevation in frequency domain and in time domain (inner box) when PI controller and MPC were tested.

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

Rigid-body mass (float & slider), M (kg) | 858 |

Displaced volume, ∀ (m${}^{3}$) | 0.858 |

Float radius, r (m) | 0.88 |

Float draft, T (m) | 0.53 |

Water density, $\rho $ (kg/m${}^{3}$) | 1000 |

Water depth, h (m) | 6.1 |

Linear hydrostatic stiffness, S (kN/m) | 23.9 |

Infinite-frequency added mass, ${m}_{\infty}$ (kg) | 782 |

Max vertical travel, $\left(\right)$ (m) | 0.6 |

Scenario | Constraint | ECR Values | N |
---|---|---|---|

1 | No constraint | N/A | 2 |

2 | Nearly hard constraint on position $\in [-0.1\phantom{\rule{0.222222em}{0ex}}0.1]$ | OV.minECR = OV.maxECR = $2\times {10}^{-4}$ | 50 |

3 | Hard constraint on control input $\in [-1500\phantom{\rule{0.222222em}{0ex}}1500]$ Soft constraint on position $\in [-0.1\phantom{\rule{0.222222em}{0ex}}0.1]$ | MV.minECR = MV.maxECR = 0OV.minECR = OV.maxECR = 0.01 | 50 |

Name | Prediction Steps | Slew Rate Limit | Power Absorbed [W] |
---|---|---|---|

PI | N/A | N/A | −42.7 |

MPC ($N=50$) | 50 | (None) | −37.6 |

MPC ($N=50$, $\left|\mathrm{MV}\right|<50$) | 50 | $\left|\mathrm{MV}\right|<50$ | −22.8 |

MPC ($N=200$) | 200 | (None) | −41.2 |

**Table 4.**Power captured (W and % change from unconstrained) by using unconstrained and constrained MPCs.

Controller | Power Capture [W] | % Reduction |
---|---|---|

Unconstrained | −34.1 | 0 |

${U}_{max}=900$ N (75%) | −33.8 | 0.7 |

${U}_{max}=600$ N (50%) | −31.9 | 6.3 |

${U}_{max}=300$ N (25%) | −23.5 | 31.0 |

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**MDPI and ACS Style**

Cho, H.; Bacelli, G.; Coe, R.G.
Model Predictive Control Tuning by Inverse Matching for a Wave Energy Converter. *Energies* **2019**, *12*, 4158.
https://doi.org/10.3390/en12214158

**AMA Style**

Cho H, Bacelli G, Coe RG.
Model Predictive Control Tuning by Inverse Matching for a Wave Energy Converter. *Energies*. 2019; 12(21):4158.
https://doi.org/10.3390/en12214158

**Chicago/Turabian Style**

Cho, Hancheol, Giorgio Bacelli, and Ryan G. Coe.
2019. "Model Predictive Control Tuning by Inverse Matching for a Wave Energy Converter" *Energies* 12, no. 21: 4158.
https://doi.org/10.3390/en12214158