# A Model Free Adaptive Scheme for Integrated Control of Civil Aircraft Trajectory and Attitude

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## Abstract

**:**

## 1. Introduction

## 2. Data Generation

## 3. MIMO MFAC Method with Saturation Constraint

## 4. MIMO MFAC Method with Hard Constraints

## 5. Control Scheme with Mixed Constraints

- (1)
- Estimate the value of PPJM ${\mathit{\Phi}}_{f,{L}_{y}+{L}_{u}}\left(k\right)$ by Formula (23).
- (2)
- Reset the value of ${\widehat{\mathit{\Phi}}}_{{L}_{y}+1}\left(k\right)$ when the following conditions are satisfied:if $|{\widehat{\varphi}}_{ii({L}_{y}+1)}\left(k\right)|<{b}_{2}$ or $|{\widehat{\varphi}}_{ii({L}_{y}+1)}\left(k\right)|>\alpha {b}_{2}$ or $\mathrm{sign}\left({\widehat{\varphi}}_{ii({L}_{y}+1)}\left(k\right)\right)\ne \mathrm{sign}\left({\widehat{\varphi}}_{ii({L}_{y}+1)}\left(1\right)\right)$, $\phantom{\rule{4pt}{0ex}}i=1,\cdots ,m$, then$${\widehat{\varphi}}_{ii({L}_{y}+1)}\left(k\right)={\widehat{\varphi}}_{ii({L}_{y}+1)}\left(1\right)$$$${\widehat{\varphi}}_{ij({L}_{y}+1)}\left(k\right)={\widehat{\varphi}}_{ij({L}_{y}+1)}\left(1\right)$$
- (3)
- Calculate the current control input using the estimated value of PPJM ${\mathit{\Phi}}_{f,{L}_{y}+{L}_{u}}\left(k\right)$.$$\begin{array}{cc}\hfill \mathit{u}\left(k\right)& =\mathit{u}\left(k-1\right)\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& +\frac{{\widehat{\mathit{\Phi}}}_{{L}_{y}+1}^{\mathrm{T}}\left(k\right)\left({\rho}_{{L}_{y}+1}\left({\mathit{y}}^{*}\left(k+1\right)-\mathit{y}\left(k\right)\right)\right)}{\lambda +\parallel {\widehat{\mathit{\Phi}}}_{{L}_{y}+1}\left(k\right){\parallel}^{2}}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& -\frac{{\widehat{\mathit{\Phi}}}_{{L}_{y}+1}^{\mathrm{T}}\left(k\right){\displaystyle \sum _{i=1}^{{L}_{y}}}{\rho}_{i}{\widehat{\mathit{\Phi}}}_{i}\left(k\right)\Delta \mathit{y}\left(k-i+1\right)}{\lambda +\parallel {\widehat{\mathit{\Phi}}}_{{L}_{y}+1}\left(k\right){\parallel}^{2}}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& -\frac{{\widehat{\mathit{\Phi}}}_{{L}_{y}+1}^{\mathrm{T}}\left(k\right){\displaystyle \sum _{i={L}_{y}+2}^{{L}_{y}+{L}_{u}}}{\rho}_{i}{\widehat{\mathit{\Phi}}}_{i}\left(k\right)\Delta \mathit{u}\left(k-i+Ly+1\right)}{\lambda +\parallel {\widehat{\mathit{\Phi}}}_{{L}_{y}+1}\left(k\right){\parallel}^{2}}\hfill \end{array}$$$i,j=1,\cdots ,m$; ${\rho}_{1},\cdots ,{\rho}_{{L}_{y}+{L}_{u}}\in (0,1]$; $\eta \in (0,2]$; $\lambda >0$, $\mu >0$. According to Equation (21), impose the saturation constraint on the control input.

## 6. Simulation Results

## 7. Discussion

## 8. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

MIMO | Multi-input multi-output |

MFAC | Model free adaptive control |

NextGen | Next generation air traffic transportation system |

SESAR | Single european sky air traffic management research |

UAV | Unmanned aerial vehicle |

PID | Proportion integration differentiation |

NLI | Non-linear inversion |

I/O | Input / Output |

SPSA | Simultaneous perturbation stochastic approximation |

UC | Unfalsified control |

IFT | Iterative feedback tuning |

VRFT | Virtual reference feedback tuning |

FFDL | Full-format dynamic linearization |

PPJM | Pseudo partitioned Jacobian matrix |

MBC | Model-based control |

DDC | Data-driven control |

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Aircraft Parameters | Controller Parameters |
---|---|

m = 48,000 kg | ${L}_{u}=1$ |

$\rho =0.41\phantom{\rule{0.277778em}{0ex}}\mathrm{kg}/{\mathrm{m}}^{3}$ | ${L}_{y}=2$ |

$S=123.55\phantom{\rule{0.277778em}{0ex}}{\mathrm{m}}^{2}$ | $\eta =0.001$ |

$b=28.34$ m | $\mu =1000$ |

$\overline{c}=4.35$ m | $\lambda =0.1$ |

A = 1,278,369.56 $\mathrm{kg}\xb7{\mathrm{m}}^{2}$ | ${\rho}_{2}={\rho}_{3}=1$ |

B = 3,781,267.79 $\mathrm{kg}\xb7{\mathrm{m}}^{2}$ | ${\rho}_{1}=\left[\begin{array}{ccccc}1.48& 0& 0& 0& 0\\ 0& 1.85& 0& 0& 0\\ 0& 0& 6.17& 0& 0\\ 0& 0& 0& 3.15& 0\\ 0& 0& 0& 0& 740\end{array}\right]\times {10}^{-3}$ |

C = 4,877,649.98 $\mathrm{kg}\xb7{\mathrm{m}}^{2}$ | |

E = 135,588.17 $\mathrm{kg}\xb7{\mathrm{m}}^{2}$ | |

${C}_{{l}_{{\delta}_{ail}}}=-0.02\phantom{\rule{0.277778em}{0ex}}{\mathrm{rad}}^{-1}$ | |

${C}_{{l}_{{\delta}_{rud}}}=0\phantom{\rule{0.277778em}{0ex}}{\mathrm{rad}}^{-1}$ | |

${C}_{{m}_{{\delta}_{ele}}}=-0.03\phantom{\rule{0.277778em}{0ex}}{\mathrm{rad}}^{-1}$ | |

${C}_{{n}_{{\delta}_{ail}}}=0.002\phantom{\rule{0.277778em}{0ex}}{\mathrm{rad}}^{-1}$ | |

${C}_{{n}_{{\delta}_{rud}}}=-0.07\phantom{\rule{0.277778em}{0ex}}{\mathrm{rad}}^{-1}$ |

Control Input Increment | Control Input |
---|---|

$-0.05\le \Delta {\delta}_{ail}\le 0.05$ | $-15\le {\delta}_{ail}\le 15$ |

$-0.05\le \Delta {\delta}_{ele}\le 0.05$ | $-30\le {\delta}_{ele}\le 30$ |

$-0.05\le \Delta {\delta}_{rud}\le 0.05$ | $-1\le {\delta}_{rud}\le 1$ |

$-5\le \Delta {F}_{thr}\le 5$ | $0\le {F}_{thr}\le 6\times {10}^{4}$ |

$-0.5\le \Delta \gamma \le 0.5$ | $-10\le \gamma \le 10$ |

$\mathit{\varphi}$ | $\mathit{\theta}$ | $\mathit{\psi}$ | ${\mathit{x}}_{\mathit{E}}$ | ${\mathit{z}}_{\mathit{E}}$ | |
---|---|---|---|---|---|

NLI | 4.82 | 0.00 | 0.23 | 120.28 | 24.65 |

FFDL-MFAC | 8.36 | 1.27 | 0.09 | 881.43 | 2.22 |

${\mathit{\delta}}_{\mathit{ail}}$ | ${\mathit{\delta}}_{\mathit{ele}}$ | ${\mathit{\delta}}_{\mathit{rud}}$ | ${\mathit{F}}_{\mathit{thr}}$ | $\mathit{\gamma}$ | |
---|---|---|---|---|---|

NLI | 0.35 | 14.21 | 0.00 | 153,239.35 | 0.00 |

FFDL-MFAC | 2.34 | 0.37 | 0.03 | 64,522.67 | 0.67 |

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**MDPI and ACS Style**

Jiang, G.; Liu, G.; Yu, H.
A Model Free Adaptive Scheme for Integrated Control of Civil Aircraft Trajectory and Attitude. *Symmetry* **2021**, *13*, 347.
https://doi.org/10.3390/sym13020347

**AMA Style**

Jiang G, Liu G, Yu H.
A Model Free Adaptive Scheme for Integrated Control of Civil Aircraft Trajectory and Attitude. *Symmetry*. 2021; 13(2):347.
https://doi.org/10.3390/sym13020347

**Chicago/Turabian Style**

Jiang, Gaoyang, Genfeng Liu, and Hansong Yu.
2021. "A Model Free Adaptive Scheme for Integrated Control of Civil Aircraft Trajectory and Attitude" *Symmetry* 13, no. 2: 347.
https://doi.org/10.3390/sym13020347