# RBFNN-Based Anti-Input Saturation Control for Hypersonic Vehicles

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Control Model and Theoretical Basis

#### 2.1.1. Control Model

#### 2.1.2. RBFNN Model

#### 2.1.3. Problem Formulation

#### 2.2. Controller Design

#### 2.2.1. Design of Speed Subsystem Controller

#### 2.2.2. Design of Height Subsystem Controller

**Step 1:**By defining the altitude tracking error ${e}_{h}=h-{h}_{d}$, ${h}_{d}$ as the reference flight altitude, performing first-order differentiation, and combining the first equation of Equation (3), it can be obtained that

**Step 2:**We defined the trajectory inclination angle tracking error ${e}_{\gamma}=\gamma -{\gamma}_{d}$, where ${\gamma}_{d}$ was the reference command, performed first-order differentiation, and combined the second equation of Equation (3) to obtain

**Step 3:**By defining the angle of attack tracking error ${e}_{\alpha}=\alpha -{\alpha}_{d}$, and performing first-order differentiation on it, it can be obtained that

**Step 4:**The normal number to be designed defines the pitch angular velocity tracking error.

#### 2.3. Stability Proof

## 3. Results

**Scenario 1:**due to systematic and random errors in speed sensors and height sensors, the initial error ${e}_{V}=0.5\mathrm{ft}/\mathrm{s}$, ${e}_{h}=20\mathrm{f}\mathrm{t}$ is considered, and 20% aerodynamic parameter perturbation and unknown external disturbances are added on this basis, i.e., $C={C}_{0}\left(1+0.2\mathrm{sin}\left(0.1\pi t\right)\right)$, ${C}_{0}$ is the nominal aerodynamic parameter.

**Scenario 2:**considering the same initial error in Scenario 1, consider the more severe input saturation constraint of the actuator, $\mathrm{\Phi}\in [0.05,0.5]$, ${\delta}_{e}\in [-{15}^{\circ},{15}^{\circ}]$.

## 4. Conclusions

- (1)
- Implementing the RBFNN adaptive controller has yielded significant quantifiable results. Notably, it has led to a remarkable reduction in the maximum error fluctuation of speed and height, with a decrease of over 50%. This outcome underscores the controller’s effectiveness in improving tracking accuracy and stability within the control system, contributing to its robustness in dealing with parameter uncertainties and disturbances.
- (2)
- The anti-saturation auxiliary system has shown noteworthy quantitative results when operating under severe input saturation constraints. Although such limitations may lead to a speed error of 1.5%, this auxiliary system has demonstrated an impressive 20% improvement in convergence speed. These results indicate the system’s resilience in ensuring precise tracking, even in scenarios characterized by severe input saturation, highlighting its potential to mitigate the effects of saturation constraints and enhance control system performance.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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Geometrical Parameters | Parameter Values |
---|---|

Quality (m) | 4378 kg |

Moment of inertia (I_{yy}) | 6.770 × 10^{5} kg.m^{2} |

Parameters | Lower Limit | Upper Limit |
---|---|---|

V | 7500 ft/s | 11,000 ft/s |

h | 85,000 ft | 88,000 ft |

γ | −1° | 1° |

$\alpha $ | −5° | 5° |

Q | −10 deg | 10 deg |

$\varnothing $ | 0.05 | 1.5 |

${\delta}_{e}$ | −20 deg | 20 deg |

q | 500 psf | 2000 psf |

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## Share and Cite

**MDPI and ACS Style**

Zhang, B.; Liang, Y.; Rao, S.; Kuang, Y.; Zhu, W.
RBFNN-Based Anti-Input Saturation Control for Hypersonic Vehicles. *Aerospace* **2024**, *11*, 108.
https://doi.org/10.3390/aerospace11020108

**AMA Style**

Zhang B, Liang Y, Rao S, Kuang Y, Zhu W.
RBFNN-Based Anti-Input Saturation Control for Hypersonic Vehicles. *Aerospace*. 2024; 11(2):108.
https://doi.org/10.3390/aerospace11020108

**Chicago/Turabian Style**

Zhang, Bangchu, Yiyong Liang, Shuitao Rao, Yu Kuang, and Weiyu Zhu.
2024. "RBFNN-Based Anti-Input Saturation Control for Hypersonic Vehicles" *Aerospace* 11, no. 2: 108.
https://doi.org/10.3390/aerospace11020108