# A Numerical Implementation of Fractional-Order PID Controllers for Autonomous Vehicles

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## Abstract

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^{ρ}-controller; particle swarm optimization; Laplacian operator; Oustaloup’s approach; continued fractional expansion approach

## 1. Introduction

## 2. The Fractional-Order PID Controller

#### 2.1. The CFE Approximation

#### 2.2. Oustaloup’s Approximation

## 3. The Design of the Fractional-Order PID Controller for Autonomous Cars

#### 3.1. Tuning Fractional-Order PID of Linear Transfer Motion

- The $P{I}^{\gamma}{D}^{\rho}$-PSO-controller via the CFE approach:

- The $P{I}^{\gamma}{D}^{\rho}$-PSO-controller via Oustaloup’s approach:

- The PID controller via the bacteria-foraging-algorithm (BFA) approach: herein, we implement the BFA to obtain the PID controller. The output form is as follows:

- The PID controller via the Ziegler–Nichols (ZN) approach: herein, we implement the ZN algorithm to obtain the PID controller. The output is of the following form:

- The PID controller via the Cohen-Coon (CC) approach: here, we applied the CC algorithm to obtain the PID controller. This controller has the following form:

#### 3.2. Tuning the Fractional-Order PID Controller for Angular Transfer Motion

- The $P{I}^{\gamma}{D}^{\rho}$-PSO-controller via the CFE approach:

- The $P{I}^{\gamma}{D}^{\rho}$-PSO-controller via Oustaloup’s approach:

- The PID controller via the bacteria foraging algorithm (BFA): in this part, we obtain the following result:

- The PID controller via the Ziegler–Nichols (ZN) approach: in this part, we have the following:

- The PID controller via the Cohen–Coon (CC) approach: herein, we have

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Block diagram of the PSO algorithm running to tune the $P{I}^{\gamma}{D}^{\rho}$-controller.

**Figure 2.**Step responses of ${H}_{1}\left(s\right)$, ${H}_{2}\left(s\right)$, ${H}_{3}\left(s\right)$, ${H}_{4}\left(s\right)$ and ${H}_{5}\left(s\right)$.

**Figure 3.**Step responses of ${H}_{6}\left(s\right)$, ${H}_{7}\left(s\right)$, ${H}_{8}\left(s\right)$, ${H}_{9}\left(s\right)$ and ${H}_{10}\left(s\right)$.

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

Population size. | 20 |

Max. number of iterations. | 100 |

Range of ${K}_{p}$. | (0, 60] |

Range of ${K}_{i}$. | (0, 66] |

Range of ${K}_{d}$. | (0, 61] |

Range of $\gamma $. | (0, 1) |

Range of $\rho $. | (0, 1) |

Gains/Methods | ZN | CC | BFA | CFE | Oustaloup |
---|---|---|---|---|---|

Proportional gain $\left({K}_{p}\right)$. | 2.5 | 3.02 | 12.73 | 48 | 0.17 |

Integral gain $\left({K}_{i}\right)$. | 0.582 | 0.472 | 14.08 | 24.2411 | 9.8834 |

Differential gain $\left({K}_{d}\right)$. | 4.271 | 2.81 | 22.50 | 51 | 61 |

$\gamma $ | 1 | 1 | 1 | 0.9110 | 0.2823 |

$\rho $ | 1 | 1 | 1 | 0.7119 | 0.976 |

**Table 3.**Step responses of ${H}_{1}\left(s\right)$, ${H}_{2}\left(s\right)$, ${H}_{3}\left(s\right)$, ${H}_{4}\left(s\right)$ and ${H}_{5}\left(s\right)$.

Step Response | ${\mathit{H}}_{1}\left(\mathit{s}\right)$ | ${\mathit{H}}_{2}\left(\mathit{s}\right)$ | ${\mathit{H}}_{3}\left(\mathit{s}\right)$ | ${\mathit{H}}_{4}\left(\mathit{s}\right)$ | ${\mathit{H}}_{5}\left(\mathit{s}\right)$ |
---|---|---|---|---|---|

Rise time. | 0.2789 | 0.2443 | 0.7812 | 4.0109 | 3.6089 |

Settling time. | 5.4935 | 5.3397 | 13.7308 | 31.669 | 20.479 |

Settling minimum. | 0.9029 | 0.9000 | 0.9014 | 0.9001 | 0.9008 |

Settling maximum. | 1.0004 | 1.0068 | 1.1513 | 1.2346 | 1.1892 |

Overshoot. | 0.0479 | 0.7059 | 15.1346 | 23.4691 | 18.9203 |

Undershoot. | 0 | 0 | 0 | 0 | 0 |

Peak. | 1.0004 | 1.0068 | 1.1513 | 1.2346 | 1.1892 |

Peak time. | 14.385 | 10.107 | 3.3052 | 10.4523 | 9.5893 |

Gains/Methods | ZN | CC | BFA | CFE | Oustaloup |
---|---|---|---|---|---|

Proportional gain $\left({K}_{p}\right)$. | 1.94 | 2.22 | 8.46 | 48 | 59 |

Integral gain $\left({K}_{i}\right)$. | 1.02 | 1.01 | 10.57 | 24.2411 | 9.8834 |

Differential gain $\left({K}_{d}\right)$. | 0.922 | 0.745 | 13.10 | 51 | 61 |

$\gamma $ | 1 | 1 | 1 | 0.9110 | 0.821 |

$\rho $ | 1 | 1 | 1 | 0.7119 | 0.7167 |

**Table 5.**Step responses of ${H}_{6}\left(s\right)$, ${H}_{7}\left(s\right)$, ${H}_{8}\left(s\right)$, ${H}_{9}\left(s\right)$ and ${H}_{10}\left(s\right)$.

Step Response | ${\mathit{H}}_{5}\left(\mathit{s}\right)$ | ${\mathit{H}}_{6}\left(\mathit{s}\right)$ | ${\mathit{H}}_{8}\left(\mathit{s}\right)$ | ${\mathit{H}}_{7}\left(\mathit{s}\right)$ | ${\mathit{H}}_{8}\left(\mathit{s}\right)$ |
---|---|---|---|---|---|

Rise time. | 0.2831 | 0.2316 | 1.0025 | 2.8371 | 2.6644 |

Settling time. | 1.3539 | 1.2385 | 19.6450 | 26.0728 | 20.0318 |

Settling minimum. | 0.9056 | 0.9053 | 0.8478 | 0.8669 | 0.8747 |

Settling maximum. | 1.1989 | 1.2447 | 1.2566 | 1.2999 | 1.2888 |

Overshoot. | 19.921 | 24.5356 | 25.6579 | 29.9886 | 28.8789 |

Undershoot. | 0 | 0 | 0 | 0 | 0 |

Peak. | 1.1989 | 1.2447 | 1.2566 | 1.2999 | 1.2888 |

Peak time. | 0.6748 | 0.5863 | 2.8777 | 6.2479 | 6.0252 |

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**MDPI and ACS Style**

Batiha, I.M.; Ababneh, O.Y.; Al-Nana, A.A.; Alshanti, W.G.; Alshorm, S.; Momani, S.
A Numerical Implementation of Fractional-Order PID Controllers for Autonomous Vehicles. *Axioms* **2023**, *12*, 306.
https://doi.org/10.3390/axioms12030306

**AMA Style**

Batiha IM, Ababneh OY, Al-Nana AA, Alshanti WG, Alshorm S, Momani S.
A Numerical Implementation of Fractional-Order PID Controllers for Autonomous Vehicles. *Axioms*. 2023; 12(3):306.
https://doi.org/10.3390/axioms12030306

**Chicago/Turabian Style**

Batiha, Iqbal M., Osama Y. Ababneh, Abeer A. Al-Nana, Waseem G. Alshanti, Shameseddin Alshorm, and Shaher Momani.
2023. "A Numerical Implementation of Fractional-Order PID Controllers for Autonomous Vehicles" *Axioms* 12, no. 3: 306.
https://doi.org/10.3390/axioms12030306