# Nonlinearly Parametrized Modeling and Adaptive Control for a Generic Hypersonic Vehicle

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## Abstract

**:**

## 1. Introduction

- (i)
- Nonlinearly parametrized models are built by a curve-fitting technique according to aerodynamic data, which improve the accuracy of models and extends the scope of a nonlinear controlled object.
- (ii)
- An effective coordinate transformation and a new parameter separation technique are introduced to make the unknown parameters separate from the nonlinear dynamics.
- (iii)
- The adaptive backstepping control method improves the adaptability and robustness of the control algorithm, only requiring the structure of the upper-bound function bounding knowledge.

## 2. Modeling of Hypersonic Vehicle Model

**Remark**

**1.**

- (i)
- Because the nonlinearly parametrized form of ${C}_{L}$ is sinusoidal, the coefficient ${C}_{L}$ can be approximate to a linear relation of $\alpha $ in a small range value of $\alpha $.
- (ii)
- Considering the uncertainties of aerodynamic coefficients and developing the accuracy of curve fitting, we define the aerodynamic coefficients in (10)–(12) as the unknown aerodynamic coefficients, which can be expressed as

**Assumption**

**1.**

**Assumption**

**2.**

**Remark**

**2.**

**Lemma**

**1.**

**Lemma**

**2.**

## 3. Control Design and Stability Analysis

#### 3.1. The Attitude Subsystem Design

**Theorem**

**1.**

**Proof.**

**Initial Step**: Define ${\xi}_{1}={y}_{1}-{y}_{0}^{*}$ with ${y}_{0}^{*}=0$ and $\tilde{\Psi}\left(t\right)=\Psi -\widehat{\Psi}\left(t\right)$ where $\tilde{\Psi}\left(t\right)$ is the estimate error. Choose the Lyapunov function

**Second Step**: Choose the Lyapunov function

**Third Step**: We construct the positive and proper Lyapunov function

#### 3.2. The Velocity Subsystem Design

**Theorem**

**2.**

**Proof.**

**Remark**

**3.**

## 4. Simulation Results

#### 4.1. Simulation Systems

#### 4.2. Simulation Results

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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V | vehicle velocity | g | acceleration of gravity |

$\gamma $ | flight-path angle | m | vehicle mass |

$\alpha $ | angle of attack | ${I}_{yy}$ | moment of inertia |

q | pitch rate | $\rho $ | air density |

$\Phi $ | engine throttle | S | reference area of the wing |

$\delta $ | elevator angular deflection | $\overline{c}$ | mean aerodynamic chord |

h | height |

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**MDPI and ACS Style**

Yang, S.; Li, X.
Nonlinearly Parametrized Modeling and Adaptive Control for a Generic Hypersonic Vehicle. *Processes* **2023**, *11*, 263.
https://doi.org/10.3390/pr11010263

**AMA Style**

Yang S, Li X.
Nonlinearly Parametrized Modeling and Adaptive Control for a Generic Hypersonic Vehicle. *Processes*. 2023; 11(1):263.
https://doi.org/10.3390/pr11010263

**Chicago/Turabian Style**

Yang, Shaohua, and Xia Li.
2023. "Nonlinearly Parametrized Modeling and Adaptive Control for a Generic Hypersonic Vehicle" *Processes* 11, no. 1: 263.
https://doi.org/10.3390/pr11010263