#
${\mathcal{L}}_{1}$ Adaptive Control Based on Dynamic Inversion for Morphing Aircraft

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## Abstract

**:**

## 1. Introduction

- Many other researchers focus on the 3DOF vehicle dynamics model, which is simply based on the longitudinal dynamics that generates pitching and forward motion. Different from them, we establish a 6DOF dynamics model which can fully describe the dynamic characteristics of the vehicle. This expands the application scope of our method.
- Compared to the traditional LPV-based morphing aircraft controller, we adopt a nonlinear control method to realize the design of our flight controller, which improves the control precision. The combination of dynamic inverse and ${\mathcal{L}}_{1}$ adaptive control provides a balance between control performance and robustness.
- Different from many other works that only adopt NDI to design the basic controller, we use NDI and INDI, respectively, to design it. In addition, the ${\mathcal{L}}_{1}$ adaptive controller we proposed aims at the error dynamics system instead of the vehicle system itself, which is distinct from nearly all other works adopting this method and can ensure the desired command tracking to the maximum extent.

## 2. Model Description

#### 2.1. Equations of Force and Moment

#### 2.2. Dynamics Model of Morphing Aircraft

#### 2.2.1. Dynamics Equations of V, $\gamma $, and $\chi $

#### 2.2.2. Dynamic Equations of $\alpha $, $\beta $, and $\mu $

#### 2.2.3. Dynamic Equations of p, q, and r

#### 2.3. Data Acquisition in Symmetry Morphing Process

## 3. Dynamic Inversion (DI) Design for Decoupling

#### 3.1. Dynamic Inversion Design for the Attitude Control Loop

#### 3.2. Dynamic Inversion Design for the Angular Rate Control Loop

#### 3.3. Analysis of Dynamic Inversion Controller

**Assumption 1.**

**Assumption 2.**

## 4. Adaptive Flight Controller Design

#### 4.1. Error Dynamic System

#### 4.2. Linear Quadratic Regulator Controller Design

#### 4.3. ${\mathcal{L}}_{1}$ Adaptive Controller Design

**Assumption 3.**

**Assumption 4.**

**Assumption 5.**

**State predictor**

**Adaptation laws**

**Control Law**

## 5. Stability Analysis

**Lemma 1.**

**Proof.**

**Lemma 2.**

**Proof.**

**Theorem 1.**

**Proof.**

## 6. Simulation

#### 6.1. Comparison

- The NDI method is based on a cascade design with an angular rate control loop and an attitude control loop, the same as that in Figure 5. But the NDI method only uses NDI to realize decoupling control and adds an LQR controller to maintain the desired tracking performance.
- SMC is a classical nonlinear control methodology which is widely used for flight controller design. It introduces a discontinuous switching term associated with a sliding variable, which can not only ensure the system reaches the sliding manifold in finite time, but also approximates and compensates the unknown interference effectively. Referring to [1,38], and based on the multi-loop cascade control structure similar to the aforementioned NDI, sliding surfaces employing first and second-order dynamic sliding mode technologies have been constructed in both an angular rate loop and an attitude loop, which are then used to derive SMC control law.
- MLP is a kind of artificial neural network with forward architecture with a simple connection, which can deal with nonlinear separable problems. In this section, the MLP control method is also NDI-based, in which multi-layer neural networks are added in the control channels of $\alpha $, $\beta $, and $\mu $, respectively, to compensate for tracking errors caused by NDI (details can be found in [29]). The addition of MLP can improve control performance in the presence of morphing.

#### 6.1.1. Scenario 1

#### 6.1.2. Scenario 2

#### 6.2. Sensitivity Analysis

- In Scenario 3, the uncertainties were introduced, which included sensor measurement errors and control surface disturbances, shown in Table 5.
- In Scenario 4, aiming at the model uncertainties, aerodynamic coefficient perturbation was considered, and aerodynamic forces and moments coefficients were increased by 30% to check if the controller can still achieve a satisfactory performance.
- In Scenario 5, the morphing rates were changed and the aircraft was allowed to change from the loiter configuration ($\Lambda ={15.97}^{\circ}$) to the dash configuration in 8 s, 15 s, and 20 s to observe whether the control performance was affected.

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

NDI | Nonlinear dynamic inversion |

INDI | Incremental nonlinear dynamic inversion |

LQR | Linear quadratic regulator |

LPV | Linear parameter varying |

MPDLF | Multiple parameter-dependent Lyapunov function |

RBF | Radial basis function |

6DOF | Six degrees of freedom |

3DOF | Three degrees of freedom |

MRAC | Model reference adaptive control |

CG | Center of gravity |

DI | Dynamic inversion |

SMC | Sliding mode control |

MLP | Multilayer perceptron |

RMSE | Root mean square errors |

## Appendix A

## Appendix B

#### Appendix B.1

#### Appendix B.2

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**Figure 1.**Reference frames and transformation [2].

**Figure 4.**${C}_{{Y}_{\beta}}$, ${C}_{{l}_{\beta}}$, and ${C}_{{n}_{\beta}}$ at $Ma$ = 0.5, attitude = 5000 m.

**Figure 6.**The structure of the adaptive controller based on LQR and ${\mathcal{L}}_{1}$ adaptive control.

**Figure 11.**The estimate of parameters ${\widehat{\omega}}_{\alpha}$, ${\widehat{\mathit{\theta}}}_{\alpha}$, and ${\widehat{\sigma}}_{\alpha}$.

Parameters | Loitor Configuration ($\mathbf{\Lambda}$ = ${15.97}^{\circ}$) | Dash Configuration ($\mathbf{\Lambda}$ = ${60}^{\circ}$) |
---|---|---|

Gross weight, m | 907 kg | |

Wing weight, ${m}_{w}$ | 60 kg | |

Length | 6.68 m | |

Wing span, b | 6.802 m | 3.842 m |

Wing area, ${S}_{ref}$ | 4.5 m | 5.765 m |

Mean aerodynamic chord, $\overline{c}$ | 0.688 m | 1.935 m |

Aspect ratio | 10.281 | 2.561 |

sweep angle $\Lambda $ | ${15.97}^{\circ}$ | ${60}^{\circ}$ |

Software | Solver | Type | Step Size | CPU | RAM |
---|---|---|---|---|---|

MATLAB | ode4 (Runge-Kutta) | fixed step | 0.001s | i7-10875H | 16G |

Time (s) | V (m/s) | $\mathit{\gamma}$ (deg) | $\mathit{\chi}$ (deg) | ||||||
---|---|---|---|---|---|---|---|---|---|

min | max | RMSE | min | max | RMSE | min | max | RMSE | |

${\mathcal{L}}_{1}$-DI | 0 | 0.0993 | 0.0157 | 0 | 0.0844 | 0.0122 | 0 | 4.2945 | 0.7734 |

SMC | $1.2754\times {10}^{-5}$ | 0.1710 | 0.0565 | 0 | 0.1536 | 0.0650 | 0 | 5.2775 | 1.4959 |

MLP | 0 | 0.3247 | 0.0686 | 0 | 0.0986 | 0.0168 | 0 | 9.3999 | 1.9038 |

NDI | 0 | 0.6070 | 0.2602 | 0 | 0.1415 | 0.0536 | 0 | 18.8475 | 6.6398 |

Times (s) | V (m/s) | $\mathit{\gamma}$ (deg) | $\mathit{\chi}$ (deg) | Sweep Angle (deg) |
---|---|---|---|---|

0∼15 | 150 | 0 | 0 | 15.97∼60 |

15∼30 | 160 | 3 | 0 | 60 |

30∼45 | 150 | 0 | 0 | 60∼15.97 |

45∼60 | 150 | 0 | 0∼90 | 15.97 |

60∼75 | 150 | 0 | 90 | 15.97∼60 |

75∼90 | 140 | −3 | 90 | 60 |

90∼105 | 150 | 0 | 90 | 60∼15.97 |

States | Measurement Error Range |
---|---|

V | $[-0.5\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}/\mathrm{s},0.5\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}/\mathrm{s}]$ |

$\alpha ,\beta $ | $\left[-{0.2}^{\circ},{0.2}^{\circ}\right]$ |

$p,q,r$ | $\left[-{0.15}^{\circ}/\mathrm{s},{0.15}^{\circ}/\mathrm{s}\right]$ |

$\varphi ,\mathit{\theta},\psi $ | $\left[-{1.5}^{\circ},{1.5}^{\circ}\right]$ |

${\delta}_{a},{\delta}_{e},{\delta}_{r}$ | $[-10\%,10\%]$ |

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**MDPI and ACS Style**

Cheng, L.; Li, Y.; Yuan, J.; Ai, J.; Dong, Y.
*Aerospace* **2023**, *10*, 786.
https://doi.org/10.3390/aerospace10090786

**AMA Style**

Cheng L, Li Y, Yuan J, Ai J, Dong Y.
*Aerospace*. 2023; 10(9):786.
https://doi.org/10.3390/aerospace10090786

**Chicago/Turabian Style**

Cheng, Lingquan, Yiyang Li, Jiayi Yuan, Jianliang Ai, and Yiqun Dong.
2023. "*Aerospace* 10, no. 9: 786.
https://doi.org/10.3390/aerospace10090786