# An Intelligent Control and a Model Predictive Control for a Single Landing Gear Equipped with a Magnetorheological Damper

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model of the MR Damper

#### 2.1. Structure of the Landing Gear Equipped with an MR Damper

#### 2.2. Mathematical Model

_{1}and z

_{2}are the displacements of the aircraft and the wheel, respectively. For a closed system, the polytropic gas law is determined by the pneumatic force F

_{a}:

_{1}− z

_{2}is the stroke. The viscous force F

_{v}represents the viscous-induced stress or the Newtonian behavior of fluid as follows:

_{1}= 245 kg, and sink speed v = 3 m/s are given in Figure 4. It can be seen that there is a small error of under 5% between the results of the mathematical model and the experiments.

## 3. Control Design

#### 3.1. Control Target

_{d}= F

_{v}+ F

_{a}+ F

_{MR}. During touchdown, the landing gear must absorb all the potential and kinetic energy of the aircraft, as can be seen in the following example:

#### 3.2. Model Predict Controller

_{target}) as long as possible during the first stroke, as can be seen in Figure 8 [30]. From Equation (9), under a landing condition (m

_{1}, v), the right-hand side is constant. Moreover, the tire absorbs a small amount of energy, so the total energy that is absorbed by the damper is constant under a landing condition whether the MR damper is used or not. Thus, the target damping force is calculated based on the performance of the passive damper:

_{target}and a length ${s}_{ideal}\approx 0.9{s}_{max}$. The values of F

_{target}under various landing conditions are given in Figure 9.

_{target}, so the cost function of MPC can be given as follows:

_{target}.

#### 3.3. Bandit Neural Network Controller

**Initialization**:

**While**looping until $Q\left(ki\right)\ge 0.9$ or $ki\ge 200$

- R ≤
**bandit**(W): - If $J\left(W\right)>J\left({W}_{max}\right):$
- ${W}_{max}=W$
- R = 0
- else:
- R = 0.9

_{max}, which has the maximum value of action Q($ki$). In the beginning, this weight matrix of the neural network is zero. The greedy algorithm is used to select exploiting or exploring actions. After exploiting actions, the agent will receive the maximum reward R = 0.9, which represents the maximum expected value of shock absorber efficiency. While exploring actions, the agent will be punished with R = 0 if the cost function $J\left(W\right)$ of the current weight matrix is larger than the cost function of $J\left({W}_{max}\right)$. The value of action Q(ki) is then updated based on that reward, as can be seen in Equation (20). The algorithm converges when this value of action Q(ki) reaches the maximum reward. The numerical simulation result of the greedy bandit algorithm is shown in Figure 12. It can be seen that the algorithm converges after more than 100 loops. The maximum value of the cost function is nearly 90%.

## 4. Result and Discussion

_{sky}is the skyhook gain that is turned depending on the landing condition, as can be seen in Figure 13. In order to use that controller efficiently, the aircraft mass and sink speed are assumed to be known before touchdown.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Landing gear’s components [17].

**Figure 3.**Drops test experiment [28].

**Figure 4.**The comparison between experimental data and simulation results of landing gear in case of m

_{1}= 245 kg, v = 2.5 m/s.

**Figure 8.**The concept of ideal force control [30].

**Figure 14.**The performance of landing gear using the passive damper, skyhook control, MPC, and intelligent control in case of m

_{1}= 245 kg, v = 2.5 m/s.

Symbol | Quantity | Value | Unit |
---|---|---|---|

A_{p} | cross-area of the head piston | 2.1 × 10^{−3} | m^{2} |

b | tire force index used to assess the nonlinearity of the tires | 1.13 | |

C | viscous damping coefficient | 9.77 | kNs/m |

g | gravitational acceleration | 9.81 | m/s^{2} |

m_{1} | sprung mass (aircraft mass) | 200~245 | kg |

m_{2} | un-sprung mass | 18 | kg |

n | polytropic process index | 1.3 | |

p_{0} | initial air chamber charging pressure | 100 | kPa |

k_{T} | tire force constant | 163 | kN/m |

v | initial sink speed of aircraft at touchdown | 1.5–2.5 | m/s |

V_{0} | initial air chamber volume | 4.26 × 10^{−4} | m^{3} |

u | control input (electrical current) | 0~1 | A |

Aircraft Mass (kg) | ||||
---|---|---|---|---|

200 | 225 | 245 | ||

Sink speed(m/s) | 1.5 | η_{1} | η_{2} | η_{3} |

2 | η_{4} | η_{5} | η_{6} | |

2.5 | η_{7} | η_{8} | η_{9} |

**Table 3.**Damper performance using passive damping, skyhook control, MPC, and intelligent control in differing landing cases.

Passive Damper | Skyhook Control | MPC | Intelligent Control | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{F}}_{\mathit{T}}^{\mathit{m}\mathit{a}\mathit{x}}$ (kN) | ${\mathit{s}}^{\mathit{m}\mathit{a}\mathit{x}}$ (m) | η (%) | ${\mathit{F}}_{\mathit{T}}^{\mathit{m}\mathit{a}\mathit{x}}$ (kN) | ${\mathit{s}}^{\mathit{m}\mathit{a}\mathit{x}}$ (m) | η (%) | ${\mathit{F}}_{\mathit{T}}^{\mathit{m}\mathit{a}\mathit{x}}$ (kN) | ${\mathit{s}}^{\mathit{m}\mathit{a}\mathit{x}}$ (m) | η (%) | ${\mathit{F}}_{\mathit{T}}^{\mathit{m}\mathit{a}\mathit{x}}$ (kN) | ${\mathit{s}}^{\mathit{m}\mathit{a}\mathit{x}}$ (m) | η (%) | |

m_{1} = 200 kg | ||||||||||||

v = 1.5 m/s | 4.72 | 0.172 | 79.5 | 4.72 | 0.172 | 79.5 | 4.72 | 0.166 | 81.0 | 4.72 | 0.162 | 82.3 |

v = 2 m/s | 5.98 | 0.177 | 80.5 | 5.98 | 0.177 | 80.5 | 5.98 | 0.173 | 82.3 | 6.01 | 0.159 | 86.8 |

v = 2.5 m/s | 7.26 | 0.183 | 82.8 | 7.26 | 0.183 | 82.8 | 7.26 | 0.180 | 84.9 | 7.45 | 0.166 | 89.0 |

m_{1} = 225 kg | ||||||||||||

v = 1.5 m/s | 4.77 | 0.179 | 84.8 | 4.77 | 0.179 | 84.8 | 4.79 | 0.175 | 86.5 | 4.77 | 0.170 | 88.5 |

v = 2 m/s | 6.85 | 0.183 | 85.0 | 6.85 | 0.183 | 85.0 | 6.04 | 0.179 | 87.9 | 6.05 | 0.172 | 91.2 |

v = 2.5 m/s | 8.68 | 0.184 | 71.9 | 7.50 | 0.186 | 87.4 | 7.3 | 0.186 | 88.5 | 7.75 | 0.175 | 91.0 |

m_{1} = 245 kg | ||||||||||||

v = 1.5 m/s | 5.82 | 0.183 | 73.2 | 5.13 | 0.180 | 84.8 | 5.53 | 0.169 | 87.5 | 4.81 | 0.176 | 92.9 |

v = 2 m/s | 7.37 | 0.186 | 72.4 | 6.5 | 0.183 | 86.4 | 6.35 | 0.179 | 90.1 | 6.08 | 0.179 | 94.4 |

v = 2.5 m/s | 9.55 | 0.189 | 68.3 | 8.05 | 0.186 | 87.2 | 8.04 | 0.180 | 92.6 | 7.54 | 0.184 | 95.2 |

m_{1} = 200 kg (Random Case 1) | ||||||||||||

v = 2.25 m/s | 6.62 | 0.180 | 81.6 | 6.62 | 0.180 | 81.6 | 6.62 | 0.178 | 82.7 | 6.72 | 0.162 | 88.2 |

m_{2} = 225 kg (Random Case 2) | ||||||||||||

v = 2.75 m/s | 9.18 | 0.189 | 75.6 | 8.32 | 0.187 | 86.2 | 8.65 | 0.183 | 87.9 | 8.20 | 0.182 | 93.8 |

m_{3} = 245 kg (Random Case 3) | ||||||||||||

v = 1.75 m/s | 6.50 | 0.184 | 73.4 | 5.82 | 0.180 | 85.4 | 5.80 | 0.177 | 86.7 | 5.44 | 0.177 | 93.5 |

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## Share and Cite

**MDPI and ACS Style**

Le, Q.-N.; Park, H.-M.; Kim, Y.; Pham, H.-H.; Hwang, J.-H.; Luong, Q.-V.
An Intelligent Control and a Model Predictive Control for a Single Landing Gear Equipped with a Magnetorheological Damper. *Aerospace* **2023**, *10*, 951.
https://doi.org/10.3390/aerospace10110951

**AMA Style**

Le Q-N, Park H-M, Kim Y, Pham H-H, Hwang J-H, Luong Q-V.
An Intelligent Control and a Model Predictive Control for a Single Landing Gear Equipped with a Magnetorheological Damper. *Aerospace*. 2023; 10(11):951.
https://doi.org/10.3390/aerospace10110951

**Chicago/Turabian Style**

Le, Quang-Ngoc, Hyeong-Mo Park, Yeongjin Kim, Huy-Hoang Pham, Jai-Hyuk Hwang, and Quoc-Viet Luong.
2023. "An Intelligent Control and a Model Predictive Control for a Single Landing Gear Equipped with a Magnetorheological Damper" *Aerospace* 10, no. 11: 951.
https://doi.org/10.3390/aerospace10110951