# Neural Network-Based Adaptive Fractional-Order Backstepping Control of Uncertain Quadrotors with Unknown Input Delays

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

## 3. Dynamic Model of Quadrotor UAV

Parameter | Nominal Value | Parameter | Nominal Value |
---|---|---|---|

$m$ | 0.53 kg | ${I}_{x}$ | $6.228\times {10}^{-3}$ kg·m^{2} |

${I}_{y}$ | $6.228\times {10}^{-3}$ kg·m^{2} | ${I}_{z}$ | $1.121\times {10}^{-2}$ kg·m^{2} |

$l$ | 0.232 m | $b$ | $3.130\times {10}^{-5}$ |

$d$ | $7.500\times {10}^{-7}$ | $g$ | 9.8 m/s^{2} |

## 4. Controller Design

**Figure 2.**Schematic diagram of the closed-loop neural network-based adaptive quadrotor control system.

#### 4.1. Position Control

_{x}(t − τ

_{1}) ≈ y

_{2}− u

_{x}(t) and an additional differential equation can be deduced from Equation (17), i.e.,

_{1}= x and ${x}_{2}=\dot{x}$, the second-order differential equation in the x direction can be represented as:

#### 4.2. Attitude Control

#### 4.3. Stability Analysis

**Theorem**

**1.**

**Proof of Theorem**

**1.**

_{1}< 0, $\left({D}^{{\beta}_{1}}{z}_{1}-{z}_{1}\right){e}^{-{r}_{1}t}$ is evaluated by:

## 5. Numerical Simulation and Discussions

#### 5.1. Closed-Loop Simulation with Modeling Uncertainties

^{2}, the RMSE can be reduced by 23.7% if $\beta $ is switched from −0.9 to 0.5. Moreover, it can be observed from Figure 3 that the settling time of the closed-loop quadrotor system decreases as $\beta $ increases from −0.9 to 0.5, i.e., the trajectory with a larger $\beta $ will converge faster to the desired path than those with a smaller $\beta $. These findings suggest that the adaptive fractional-order backstepping controller with $0<\beta \le 0.5$ can demonstrate a better tracking performance (in terms of settling time and tracking error) and a higher robustness to modeling uncertainties than a conventional adaptive (integer-order) backstepping controller ($\beta =0$). However, a surprising result emerging from Figure 3 and Table 3 is that both the tracking error and the settling time expand at $\beta =0.9$. A possible explanation for this finding is that the stability of the proposed fractional-order controller may not be guaranteed at large $\beta $ values, as suggested by Theorem 1. Additionally, in these sets of simulations under various payloads, the system stability at $\beta =0.9$ is degraded compared to the case when $\beta =0.5$.

#### 5.2. Closed-Loop Simulation with Unknown Input Delays

_{1}over a wider time domain, which results in a larger virtual control ${\alpha}_{1}$ at larger time delays.

#### 5.3. Closed-Loop Simulation with Modleing Uncertainties and Unknown Input Delays

^{2}and $\tau =0.1$ s is considered. This observation is consistent with the implication of Theorem 1 that the stability of fractional-order controllers with large positive β values may not be guaranteed in this work.

^{2}and an input delay of 0.10 s, the control signal at $\beta =0.9$ experiences an explosion, leading to the destabilization of the closed-loop system.

## 6. Comparison to Other Controllers

^{T}P + PA − PBR

^{−1}B

^{T}P + Q = 0, with A being the coefficient matrix of the linearized quadrotor model and Q being a positive-definite weight matrix. In [40], to mitigate the impact of perturbations and address the issue of steady-state error, the LQR controller was augmented with an integrator. Thus, the difference between the system output and the reference signal is expressed as the time derivative of an augmented state-space variable, resulting from the inclusion of a referred integrator, denoted by the symbol ξ. Then, the augmented state-space model of the position subsystem in the z direction is:

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

## Appendix B

**Figure A1.**Simulink diagram of the closed-loop quadrotor UAV system with the proposed neural network-based adaptive fractional-order backstepping controller.

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**Figure 3.**Simulated results that demonstrate the robustness of the varying fractional orders in the proposed controller to uncertainty in the quadrotor dynamics with various payloads: (

**a**) ${d}_{3}=0$ m/s

^{2}, (

**b**) ${d}_{3}=2$ m/s

^{2}, (

**c**) ${d}_{3}=4$ m/s

^{2}, (

**d**) ${d}_{3}=6$ m/s

^{2}. (Assume that ${d}_{1}={d}_{2}={d}_{4}={d}_{5}={d}_{6}=0$ and no input time delay.)

**Figure 4.**Simulated control inputs of the varying fractional orders in the proposed controller to uncertainty in the quadrotor dynamics with various payloads: (

**a**) ${d}_{3}=0$ m/s

^{2}, (

**b**) ${d}_{3}=2$ m/s

^{2}, (

**c**) ${d}_{3}=4$ m/s

^{2}, (

**d**) ${d}_{3}=6$ m/s

^{2}. (Assume that ${d}_{1}={d}_{2}={d}_{4}={d}_{5}={d}_{6}=0$ and no input time delay.)

**Figure 5.**Simulated results that demonstrate the robustness of the varying fractional orders in the proposed controller to various input delays: (

**a**) $\tau =0.05$ s, (

**b**) $\tau =0.1$ s. (Assume that $\tau ={\tau}_{1}={\tau}_{2}={\tau}_{3}={\tau}_{4}$ and no payload is applied.)

**Figure 6.**Simulated control inputs of the varying fractional orders in the proposed controller to various input delays: (

**a**) $\tau =0.05$ s, (

**b**) $\tau =0.1$ s. (Assume that $\tau ={\tau}_{1}={\tau}_{2}={\tau}_{3}={\tau}_{4}$ and no payload is applied.)

**Figure 7.**Simulated results that demonstrate the robustness of the varying fractional orders in the proposed controller to uncertainty in the quadrotor dynamics with various payloads and uncertain input time delays: (

**a**) ${d}_{3}=2$ m/s

^{2}and $\tau =0.05$ s, (

**b**) ${d}_{3}=2$ m/s

^{2}and $\tau =0.10$ s, (

**c**) ${d}_{3}=4$ m/s

^{2}and $\tau =0.05$ s, (

**d**) ${d}_{3}=4$ m/s

^{2}and $\tau =0.10$ s. (Assume that ${d}_{1}={d}_{2}={d}_{4}={d}_{5}={d}_{6}=0$ and $\tau ={\tau}_{1}={\tau}_{2}={\tau}_{3}={\tau}_{4}$.)

**Figure 8.**Simulated control inputs of the varying fractional orders in the proposed controller to uncertainty in the quadrotor dynamics with various payloads and uncertain input time delays: (

**a**) ${d}_{3}=2$ m/s

^{2}and $\tau =0.05$ s, (

**b**) ${d}_{3}=2$ m/s

^{2}and $\tau =0.10$ s, (

**c**) ${d}_{3}=4$ m/s

^{2}and $\tau =0.05$ s, (

**d**) ${d}_{3}=4$ m/s

^{2}and $\tau =0.10$ s. (Assume that ${d}_{1}={d}_{2}={d}_{4}={d}_{5}={d}_{6}=0$ and $\tau ={\tau}_{1}={\tau}_{2}={\tau}_{3}={\tau}_{4}$.)

**Figure 9.**Comparison of the simulated trajectories with three controllers ($\beta =0.5$ in RBF-ADFOBC) in response to various payloads and uncertain input time delays: (

**a**) ${d}_{3}=2$ m/s

^{2}and $\tau =0.05$ s, (

**b**) ${d}_{3}=2$ m/s

^{2}and $\tau =0.10$ s, (

**c**) ${d}_{3}=4$ m/s

^{2}and $\tau =0.05$ s, (

**d**) ${d}_{3}=4$ m/s

^{2}and $\tau =0.10$ s. (Assume that ${d}_{1}={d}_{2}={d}_{4}={d}_{5}={d}_{6}=0$ and $\tau ={\tau}_{1}={\tau}_{2}={\tau}_{3}={\tau}_{4}$.)

**Figure 10.**Comparison of the simulated control signals with three controllers ($\beta =0.5$ in RBF-ADFOBC) in response to various payloads and uncertain input time delays: (

**a**) ${d}_{3}=2$ m/s

^{2}and $\tau =0.05$ s, (

**b**) ${d}_{3}=2$ m/s

^{2}and $\tau =0.10$ s, (

**c**) ${d}_{3}=4$ m/s

^{2}and $\tau =0.05$ s, (

**d**) ${d}_{3}=4$ m/s

^{2}and $\tau =0.10$ s. (Assume that ${d}_{1}={d}_{2}={d}_{4}={d}_{5}={d}_{6}=0$ and $\tau ={\tau}_{1}={\tau}_{2}={\tau}_{3}={\tau}_{4}$.)

Param | Value | Param | Value | Param | Value | Param | Value | Param | Value |
---|---|---|---|---|---|---|---|---|---|

${a}_{1}$ | 5 | ${a}_{2}$ | 5 | ${b}_{1}$ | 1 | ${b}_{2}$ | 1 | ${c}_{11}$ | 5 |

${c}_{12}$ | 5 | ${\gamma}_{11}$ | 10 | ${\gamma}_{12}$ | 10 | ${\gamma}_{13}$ | 10 | ${\nu}_{11}$ | 10 |

${\nu}_{12}$ | 10 | ${\nu}_{13}$ | 10 | ${h}_{1}$ | 2 | ${r}_{1}$ | 0.1 | ${\lambda}_{1}$ | 2 |

${a}_{3}$ | 5 | ${a}_{4}$ | 5 | ${b}_{3}$ | 1 | ${b}_{4}$ | 1 | ${c}_{31}$ | 5 |

${c}_{32}$ | 5 | ${\gamma}_{31}$ | 10 | ${\gamma}_{32}$ | 10 | ${\gamma}_{33}$ | 10 | ${\nu}_{31}$ | 10 |

${\nu}_{32}$ | 10 | ${\nu}_{33}$ | 10 | ${h}_{3}$ | 2 | ${r}_{3}$ | 0.1 | ${\lambda}_{3}$ | 2 |

${a}_{5}$ | 5 | ${a}_{6}$ | 5 | ${b}_{5}$ | 1 | ${b}_{6}$ | 1 | ${c}_{51}$ | 5 |

${c}_{52}$ | 5 | ${\gamma}_{51}$ | 10 | ${\gamma}_{52}$ | 10 | ${\gamma}_{53}$ | 10 | ${\nu}_{51}$ | 10 |

${\nu}_{52}$ | 10 | ${\nu}_{53}$ | 10 | ${h}_{5}$ | 2 | ${r}_{5}$ | 0.1 | ${\lambda}_{5}$ | 2 |

${a}_{7}$ | 5 | ${a}_{8}$ | 5 | ${b}_{7}$ | 1 | ${b}_{8}$ | 1 | ${c}_{71}$ | 5 |

${c}_{72}$ | 5 | ${\gamma}_{71}$ | 10 | ${\gamma}_{72}$ | 10 | ${\gamma}_{73}$ | 10 | ${\nu}_{71}$ | 10 |

${\nu}_{72}$ | 10 | ${\nu}_{73}$ | 10 | ${h}_{7}$ | 2 | ${r}_{7}$ | 0.1 | ${\lambda}_{7}$ | 2 |

${a}_{9}$ | 5 | ${a}_{10}$ | 5 | ${b}_{9}$ | 1 | ${b}_{10}$ | 1 | ${c}_{91}$ | 5 |

${c}_{92}$ | 5 | ${\gamma}_{91}$ | 10 | ${\gamma}_{92}$ | 10 | ${\gamma}_{93}$ | 10 | ${\nu}_{91}$ | 10 |

${\nu}_{92}$ | 10 | ${\nu}_{93}$ | 10 | ${h}_{9}$ | 2 | ${r}_{9}$ | 0.1 | ${\lambda}_{9}$ | 2 |

${a}_{11}$ | 5 | ${a}_{12}$ | 5 | ${b}_{11}$ | 1 | ${b}_{12}$ | 1 | ${c}_{111}$ | 5 |

${c}_{112}$ | 5 | ${\gamma}_{111}$ | 10 | ${\gamma}_{112}$ | 10 | ${\gamma}_{113}$ | 10 | ${\nu}_{111}$ | 10 |

${\nu}_{112}$ | 10 | ${\nu}_{113}$ | 10 | ${h}_{11}$ | 2 | ${r}_{11}$ | 0.1 | ${\lambda}_{11}$ | 2 |

**Table 3.**Root mean squared error (RMSE) of the simulated tracks under different payloads and different fractional orders in the proposed controller.

Payloads (m/s^{2}) | $\mathit{\beta}=-0.9$ | $\mathit{\beta}=-0.5$ | $\mathit{\beta}=-0.1$ | $\mathit{\beta}=0$ | $\mathit{\beta}=0.1$ | $\mathit{\beta}=0.5$ | $\mathit{\beta}=0.9$ |
---|---|---|---|---|---|---|---|

0 | 0.6493 | 0.5963 | 0.5127 | 0.4953 | 0.4830 | 0.4786 | 0.7251 |

2 | 0.6459 | 0.5943 | 0.5132 | 0.4962 | 0.4841 | 0.4800 | 0.7241 |

4 | 0.6387 | 0.5900 | 0.5129 | 0.4964 | 0.4846 | 0.4806 | 0.7208 |

6 | 0.6295 | 0.5842 | 0.5119 | 0.4961 | 0.4847 | 0.4802 | 0.7139 |

**Table 4.**Root mean squared value of the control signal ${u}_{1}$ in the first 30 s under different payloads and different fractional orders in the proposed controller.

Payloads (m/s^{2}) | $\mathit{\beta}=-0.9$ | $\mathit{\beta}=-0.5$ | $\mathit{\beta}=-0.1$ | $\mathit{\beta}=0$ | $\mathit{\beta}=0.1$ | $\mathit{\beta}=0.5$ | $\mathit{\beta}=0.9$ |
---|---|---|---|---|---|---|---|

0 | 5.4361 | 5.3713 | 5.3344 | 5.3429 | 5.3530 | 5.3857 | 5.9943 |

2 | 4.4824 | 4.4071 | 4.3638 | 4.3733 | 4.3848 | 4.4214 | 5.1261 |

4 | 3.5554 | 3.4652 | 3.4122 | 3.4234 | 3.4369 | 3.4796 | 4.3215 |

6 | 2.6900 | 2.5764 | 2.5065 | 2.5196 | 2.5354 | 2.5849 | 3.6169 |

**Table 5.**Root mean squared value of the control signal ${u}_{2}$ in the first 30 s under different payloads and different fractional orders in the proposed controller.

Payloads (m/s^{2}) | $\mathit{\beta}=-0.9$ | $\mathit{\beta}=-0.5$ | $\mathit{\beta}=-0.1$ | $\mathit{\beta}=0$ | $\mathit{\beta}=0.1$ | $\mathit{\beta}=0.5$ | $\mathit{\beta}=0.9$ |
---|---|---|---|---|---|---|---|

0 | 2.0106 | 1.5832 | 1.9039 | 2.3651 | 2.4281 | 2.5267 | 5.6396 |

2 | 2.0257 | 1.6009 | 1.9399 | 2.3753 | 2.4359 | 2.5359 | 5.6718 |

4 | 2.0623 | 1.6288 | 1.9738 | 2.3862 | 2.4447 | 2.5460 | 5.7331 |

6 | 2.2329 | 1.7023 | 2.0095 | 2.3990 | 2.4547 | 2.5592 | 5.8290 |

**Table 6.**Root mean squared value of the control signal ${u}_{3}$ in the first 30 s under different payloads and different fractional orders in the proposed controller.

Payloads (m/s^{2}) | $\mathit{\beta}=-0.9$ | $\mathit{\beta}=-0.5$ | $\mathit{\beta}=-0.1$ | $\mathit{\beta}=0$ | $\mathit{\beta}=0.1$ | $\mathit{\beta}=0.5$ | $\mathit{\beta}=0.9$ |
---|---|---|---|---|---|---|---|

0 | 4.5057 | 4.0187 | 4.7854 | 4.7979 | 4.8203 | 4.8216 | 6.1584 |

2 | 4.5158 | 4.0317 | 4.8005 | 4.8029 | 4.8229 | 4.8225 | 6.1636 |

4 | 4.5264 | 4.0436 | 4.8138 | 4.8068 | 4.8242 | 4.8224 | 6.1761 |

6 | 4.5476 | 4.0609 | 4.8264 | 4.8102 | 4.8249 | 4.8224 | 6.4069 |

**Table 7.**Root mean squared value of the control signal ${u}_{4}$ in the first 30 s under different payloads and different fractional orders in the proposed controller.

Payloads (m/s^{2}) | $\mathit{\beta}=-0.9$ | $\mathit{\beta}=-0.5$ | $\mathit{\beta}=-0.1$ | $\mathit{\beta}=0$ | $\mathit{\beta}=0.1$ | $\mathit{\beta}=0.5$ | $\mathit{\beta}=0.9$ |
---|---|---|---|---|---|---|---|

0 | 0.3847 | 0.3851 | 0.3871 | 0.3884 | 0.3903 | 0.4207 | 2.3505 |

2 | 0.3847 | 0.3851 | 0.3871 | 0.3884 | 0.3903 | 0.4207 | 2.3505 |

4 | 0.3847 | 0.3851 | 0.3871 | 0.3884 | 0.3903 | 0.4207 | 2.3505 |

6 | 0.3847 | 0.3851 | 0.3871 | 0.3884 | 0.3903 | 0.4207 | 2.3505 |

**Table 8.**Root mean squared error (RMSE) of the simulated tracks under different input time delays and different fractional orders in the proposed controller.

$\mathit{\tau}$ (s) | $\mathit{\beta}=-0.9$ | $\mathit{\beta}=-0.5$ | $\mathit{\beta}=-0.1$ | $\mathit{\beta}=0$ | $\mathit{\beta}=0.1$ | $\mathit{\beta}=0.5$ | $\mathit{\beta}=0.9$ |
---|---|---|---|---|---|---|---|

0.05 | 0.6484 | 0.5926 | 0.5187 | 0.5022 | 0.4898 | 0.4836 | 0.7429 |

0.10 | 0.6380 | 0.5855 | 0.5292 | 0.5150 | 0.5040 | 0.5075 | 0.7923 |

**Table 9.**Root mean squared value of the control signal ${u}_{1}$ in the first 30 s under different input time delays and different fractional orders in the proposed controller.

$\mathit{\tau}$ (s) | $\mathit{\beta}=-0.9$ | $\mathit{\beta}=-0.5$ | $\mathit{\beta}=-0.1$ | $\mathit{\beta}=0$ | $\mathit{\beta}=0.1$ | $\mathit{\beta}=0.5$ | $\mathit{\beta}=0.9$ |
---|---|---|---|---|---|---|---|

0.05 | 5.4411 | 5.3787 | 5.3541 | 5.3685 | 5.3829 | 5.4450 | 6.4784 |

0.10 | 5.4721 | 5.3984 | 5.3765 | 5.3830 | 5.3975 | 5.4649 | 7.4987 |

**Table 10.**Root mean squared value of the control signal ${u}_{2}$ in the first 30 s under different input time delays and different fractional orders in the proposed controller.

$\mathit{\tau}$ (s) | $\mathit{\beta}=-0.9$ | $\mathit{\beta}=-0.5$ | $\mathit{\beta}=-0.1$ | $\mathit{\beta}=0$ | $\mathit{\beta}=0.1$ | $\mathit{\beta}=0.5$ | $\mathit{\beta}=0.9$ |
---|---|---|---|---|---|---|---|

0.05 | 1.9331 | 1.5050 | 1.6891 | 2.3141 | 2.3788 | 2.5456 | 5.6385 |

0.10 | 1.9289 | 1.4889 | 1.6140 | 2.3026 | 2.3620 | 2.4665 | 13.3469 |

**Table 11.**Root mean squared value of the control signal ${u}_{3}$ in the first 30 s under different input time delays and different fractional orders in the proposed controller.

$\mathit{\tau}$ (s) | $\mathit{\beta}=-0.9$ | $\mathit{\beta}=-0.5$ | $\mathit{\beta}=-0.1$ | $\mathit{\beta}=0$ | $\mathit{\beta}=0.1$ | $\mathit{\beta}=0.5$ | $\mathit{\beta}=0.9$ |
---|---|---|---|---|---|---|---|

0.05 | 4.4377 | 3.9316 | 4.6976 | 4.7950 | 4.8502 | 4.9601 | 6.1635 |

0.10 | 4.4185 | 3.8899 | 4.6636 | 4.7569 | 4.8149 | 4.8853 | 15.0158 |

**Table 12.**Root mean squared value of the control signal ${u}_{4}$ in the first 30 s under different input time delays and different fractional orders in the proposed controller.

$\mathit{\tau}$ (s) | $\mathit{\beta}=-0.9$ | $\mathit{\beta}=-0.5$ | $\mathit{\beta}=-0.1$ | $\mathit{\beta}=0$ | $\mathit{\beta}=0.1$ | $\mathit{\beta}=0.5$ | $\mathit{\beta}=0.9$ |
---|---|---|---|---|---|---|---|

0.05 | 0.3842 | 0.3846 | 0.3866 | 0.3879 | 0.3885 | 0.4227 | 2.3533 |

0.10 | 0.3839 | 0.3841 | 0.3862 | 0.3873 | 0.3882 | 0.4223 | 2.3530 |

**Table 13.**Root mean squared error (RMSE) of the simulated tracks under different payloads, different input time delays and different fractional orders in the proposed controller.

Payloads (m/s^{2}) | $\mathit{\tau}$ (s) | $\mathit{\beta}=-0.9$ | $\mathit{\beta}=-0.5$ | $\mathit{\beta}=-0.1$ | $\mathit{\beta}=0$ | $\mathit{\beta}=0.1$ | $\mathit{\beta}=0.5$ | $\mathit{\beta}=0.9$ |
---|---|---|---|---|---|---|---|---|

2 | 0.05 | 0.6493 | 0.5944 | 0.5192 | 0.5027 | 0.4903 | 0.4843 | 0.7422 |

0.10 | 0.6436 | 0.5892 | 0.5298 | 0.5151 | 0.5040 | 0.5064 | 0.8158 | |

4 | 0.05 | 0.6474 | 0.5942 | 0.5190 | 0.5025 | 0.4902 | 0.4844 | 0.7410 |

0.10 | 0.6447 | 0.5902 | 0.5296 | 0.5147 | 0.5037 | 0.5048 | 116.7240 |

**Table 14.**Root mean squared value of the control signal ${u}_{1}$ in the first 30 s under different payloads, different input time delays and different fractional orders in the proposed controller.

Payloads (m/s^{2}) | $\mathit{\tau}$ (s) | $\mathit{\beta}=-0.9$ | $\mathit{\beta}=-0.5$ | $\mathit{\beta}=-0.1$ | $\mathit{\beta}=0$ | $\mathit{\beta}=0.1$ | $\mathit{\beta}=0.5$ | $\mathit{\beta}=0.9$ |
---|---|---|---|---|---|---|---|---|

2 | 0.05 | 4.4959 | 4.4187 | 4.3881 | 4.4058 | 4.4231 | 4.4969 | 5.7538 |

0.10 | 4.5443 | 4.4461 | 4.4162 | 4.4248 | 4.4422 | 4.5285 | 17.8763 | |

4 | 0.05 | 4.0347 | 3.9476 | 3.9126 | 3.9323 | 3.9515 | 4.0328 | 5.4364 |

0.10 | 4.0941 | 3.9813 | 3.9454 | 3.9547 | 3.9736 | 4.0731 | 9850.6 |

**Table 15.**Root mean squared value of the control signal ${u}_{2}$ in the first 30 s under different payloads, different input time delays and different fractional orders in the proposed controller.

Payloads (m/s^{2}) | $\mathit{\tau}$ (s) | $\mathit{\beta}=-0.9$ | $\mathit{\beta}=-0.5$ | $\mathit{\beta}=-0.1$ | $\mathit{\beta}=0$ | $\mathit{\beta}=0.1$ | $\mathit{\beta}=0.5$ | $\mathit{\beta}=0.9$ |
---|---|---|---|---|---|---|---|---|

2 | 0.05 | 1.9505 | 1.5200 | 1.7307 | 2.3233 | 2.3912 | 2.5839 | 5.6791 |

0.10 | 1.9681 | 1.5059 | 1.7089 | 2.3123 | 2.3691 | 2.5970 | 26.4734 | |

4 | 0.05 | 1.9692 | 1.5315 | 1.7540 | 2.3293 | 2.4005 | 2.6100 | 5.7090 |

0.10 | 2.2984 | 1.5267 | 1.7438 | 2.3197 | 2.3781 | 2.6325 | 60.9009 |

**Table 16.**Root mean squared value of the control signal ${u}_{3}$ in the first 30 s under different payloads, different input time delays and different fractional orders in the proposed controller.

Payloads (m/s^{2}) | $\mathit{\tau}$ (s) | $\mathit{\beta}=-0.9$ | $\mathit{\beta}=-0.5$ | $\mathit{\beta}=-0.1$ | $\mathit{\beta}=0$ | $\mathit{\beta}=0.1$ | $\mathit{\beta}=0.5$ | $\mathit{\beta}=0.9$ |
---|---|---|---|---|---|---|---|---|

2 | 0.05 | 4.4485 | 3.9436 | 4.7216 | 4.8083 | 4.8637 | 4.9636 | 6.1716 |

0.10 | 4.4291 | 3.9022 | 4.6844 | 4.7717 | 4.8261 | 4.9236 | 27.0198 | |

4 | 0.05 | 4.4540 | 3.9507 | 4.7341 | 4.8152 | 4.8723 | 4.9649 | 6.1762 |

0.10 | 4.4402 | 3.9089 | 4.6960 | 4.7793 | 4.8321 | 4.9332 | 67.0480 |

**Table 17.**Root mean squared value of the control signal ${u}_{4}$ in the first 30 s under different payloads, different input time delays and different fractional orders in the proposed controller.

Payloads (m/s^{2}) | $\mathit{\tau}$ (s) | $\mathit{\beta}=-0.9$ | $\mathit{\beta}=-0.5$ | $\mathit{\beta}=-0.1$ | $\mathit{\beta}=0$ | $\mathit{\beta}=0.1$ | $\mathit{\beta}=0.5$ | $\mathit{\beta}=0.9$ |
---|---|---|---|---|---|---|---|---|

2 | 0.05 | 0.3835 | 0.3840 | 0.3855 | 0.3875 | 0.3892 | 0.4207 | 2.3505 |

0.10 | 0.3845 | 0.3849 | 0.3866 | 0.3883 | 0.3900 | 0.4211 | 2.3509 | |

4 | 0.05 | 0.3850 | 0.3855 | 0.3875 | 0.3891 | 0.3905 | 0.4213 | 2.3515 |

0.10 | 0.3852 | 0.3866 | 0.3877 | 0.3899 | 0.3931 | 0.4222 | 2.3520 |

**Table 18.**$Q$ and $R$ matrices used in the LQR optimal gain calculation and the resulting optimal gains for each subsystem.

Subsystem | $\mathit{Q}$ | $\mathit{R}$ | $\mathit{K}$ |
---|---|---|---|

X Position (${x}_{1}$) | diag(1000, 20, 100) | 3 | [19.8524, 5.2640, −5.7735] |

Y Position (${x}_{3}$) | diag(1000, 20, 100) | 3 | [19.8524, 5.2640, −5.7735] |

Z Position (${x}_{5}$) | diag(1000, 20, 100) | 3 | [19.8524, 5.2640, −5.7735] |

Roll Angle (${x}_{7}$) | diag(1000, 20, 100) | 3 | [19.0702, 2.6276, −5.7735] |

Pitch Angle (${x}_{9}$) | diag(1000, 20, 100) | 3 | [19.0702, 2.6276, −5.7735] |

Yaw Angle (${x}_{11}$) | diag(1000, 20, 100) | 3 | [19.0811, 2.6635, −5.7735] |

**Table 19.**Root mean squared error (RMSE) of the simulated tracks under different payloads and different input time delays for the three different controllers.

Payloads (m/s^{2}) | $\mathit{\tau}$ (s) | FOPID | LQR | $\mathbf{RBF}-\mathbf{ADFOBC}(\mathit{\beta}=0.5)$ |
---|---|---|---|---|

2 | 0.05 | 0.7580 | 0.5155 | 0.4843 |

0.10 | 0.7480 | 0.5988 | 0.5064 | |

4 | 0.05 | 0.7623 | 0.5199 | 0.4844 |

0.10 | 0.7614 | 1.9011 | 0.5048 |

**Table 20.**Comparison of the root mean squared value of the control signals in the three controllers under different payloads, different input time delays and different fractional orders.

Payloads (m/s^{2}) | $\mathit{\tau}$ (s) | FOPID | LQR | $\mathbf{RBF}-\mathbf{ADFOBC}(\mathit{\beta}=0.5)$ | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{u}}_{1}$ | ${\mathit{u}}_{2}$ | ${\mathit{u}}_{3}$ | ${\mathit{u}}_{4}$ | ${\mathit{u}}_{1}$ | ${\mathit{u}}_{2}$ | ${\mathit{u}}_{3}$ | ${\mathit{u}}_{4}$ | ${\mathit{u}}_{1}$ | ${\mathit{u}}_{2}$ | ${\mathit{u}}_{3}$ | ${\mathit{u}}_{4}$ | ||

2 | 0.05 | 4.4721 | 2.0972 | 0.6381 | 2.0977 | 34.9813 | 18.0742 | 17.0831 | 12.1074 | 4.4969 | 2.5839 | 4.9636 | 0.4207 |

0.10 | 4.4824 | 2.0707 | 0.8075 | 2.0977 | 55.8172 | 31.7920 | 23.8891 | 12.1074 | 4.5285 | 2.5970 | 4.9236 | 0.4211 | |

4 | 0.05 | 3.9968 | 2.0992 | 0.6452 | 2.0977 | 29.2486 | 18.3960 | 17.1631 | 12.1074 | 4.0328 | 2.6100 | 4.9649 | 0.4213 |

0.10 | 4.0391 | 3.2332 | 1.2922 | 2.0977 | 1073.9 | 33.2098 | 32.3988 | 12.1074 | 4.0731 | 2.6325 | 4.9332 | 0.4222 |

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## Share and Cite

**MDPI and ACS Style**

Yang, Y.; Zhang, H.H.
Neural Network-Based Adaptive Fractional-Order Backstepping Control of Uncertain Quadrotors with Unknown Input Delays. *Fractal Fract.* **2023**, *7*, 232.
https://doi.org/10.3390/fractalfract7030232

**AMA Style**

Yang Y, Zhang HH.
Neural Network-Based Adaptive Fractional-Order Backstepping Control of Uncertain Quadrotors with Unknown Input Delays. *Fractal and Fractional*. 2023; 7(3):232.
https://doi.org/10.3390/fractalfract7030232

**Chicago/Turabian Style**

Yang, Yi, and Haiyan H. Zhang.
2023. "Neural Network-Based Adaptive Fractional-Order Backstepping Control of Uncertain Quadrotors with Unknown Input Delays" *Fractal and Fractional* 7, no. 3: 232.
https://doi.org/10.3390/fractalfract7030232