# Exploring the Role of Sampling Time in String Stabilization for Platooning: An Experimental Case Study

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## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

#### 2.1. Preliminaries and Notation

#### 2.2. Platooning Setup

#### 2.3. Controller Design for String Stability

**Definition 1.**

**Lemma 1.**

**Lemma 2.**

#### 2.4. Problem of Interest

## 3. Materials and Methods

#### 3.1. PL-TOON Experimental Platform

#### 3.2. Continuous-Time Controller Design

#### 3.3. Discrete-Time Implementation

**Proposition 1.**

**Proof.**

## 4. Simulation and Experimental Results

#### 4.1. Experiment Description

#### 4.2. Simulation Results

#### 4.3. Experimental Results

## 5. Conclusions and Future Work

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Feedback control loop for the position ${y}_{i}$ in each vehicle. The time headway spacing policy is included in H.

**Figure 4.**Tracking errors for the string-stable continuous-time platoon of size 15. The first agent is represented by darker blue, while the last agent corresponds to the dark red curve.

**Figure 5.**Feedback control loop implementation in discrete time. Both the controller $\overline{C}$ and the time headway block $\overline{H}$ are implemented digitally, whereas the input and output of $G\left(s\right)$ are converted to the appropriate time domain using a digital-to-analog converter (D/A) and an analog-to-digital converter (A/D), respectively.

**Figure 6.**Pole locations of the closed-loop sampled-time transfer function $\overline{T}\left(z\right)$ for varying h with 4 values of the sample time $\mathsf{\Delta}$. (

**Top left**) $\mathsf{\Delta}=0.05$ (s); (

**top right**) $\mathsf{\Delta}=0.1$ (s); (

**bottom left**) $\mathsf{\Delta}=0.15$ (s); (

**bottom right**) $\mathsf{\Delta}=0.2$ (s).

**Figure 7.**Magnitude plots for the transfer function $\overline{T}\left(z\right)$ for varying h with 4 values of the sample time $\mathsf{\Delta}$. (

**Top left**) $\mathsf{\Delta}=0.05$ (s); (

**top right**) $\mathsf{\Delta}=0.1$ (s); (

**bottom left**) $\mathsf{\Delta}=0.15$ (s); (

**bottom right**) $\mathsf{\Delta}=0.2$ (s).

**Figure 8.**Setup of the experiment to perform showing 3 PL-TOON agents. The platoon starts in formation with the desired distance ${\u03f5}_{i}$ between vehicles and then the setpoint ${\u03f5}_{1}$ changes its value periodically, which provokes a tracking error in the first vehicle that then propagates to the rest of the platoon.

**Figure 9.**Simulation results for the tracking error for the vehicle platoon with a sampling time $\mathsf{\Delta}=0.02$ s.

**Figure 10.**Simulation results for the tracking error for the vehicle platoon with a sampling time $\mathsf{\Delta}=0.125$ s.

**Figure 11.**Simulation results for the tracking error for the vehicle platoon with a sampling time $\mathsf{\Delta}=0.17$ s.

**Figure 12.**Tracking error for the experimental vehicle platoon with different sample times $\mathsf{\Delta}$ in the range $0.02$–$0.20$ s.

**Figure 13.**Tracking error for the experimental vehicle platoon with a sampling time $\mathsf{\Delta}=0.02$ (s).

**Figure 14.**Tracking error for the experimental vehicle platoon with a sampling time $\mathsf{\Delta}=0.12$ (s).

**Figure 15.**Tracking error for the experimental vehicle platoon with a sampling time $\mathsf{\Delta}=0.18$ (s).

$\mathbf{\Delta}=0.02$ (s) | $\mathbf{\Delta}=0.12$ (s) | $\mathbf{\Delta}=0.18$ (s) | |
---|---|---|---|

${\overline{e}}_{1}$ | 5.2655 | 4.9257 | 3.7994 |

${\overline{e}}_{2}$ | 2.1130 | 1.9896 | 1.9605 |

${\overline{e}}_{3}$ | 2.6374 | 2.2446 | 2.6644 |

${\overline{e}}_{4}$ | 1.8563 | 2.9717 | 2.9070 |

${\overline{e}}_{5}$ | 1.6580 | 2.6531 | 3.1332 |

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**MDPI and ACS Style**

Villenas, F.I.; Vargas, F.J.; Peters, A.A.
Exploring the Role of Sampling Time in String Stabilization for Platooning: An Experimental Case Study. *Mathematics* **2023**, *11*, 2923.
https://doi.org/10.3390/math11132923

**AMA Style**

Villenas FI, Vargas FJ, Peters AA.
Exploring the Role of Sampling Time in String Stabilization for Platooning: An Experimental Case Study. *Mathematics*. 2023; 11(13):2923.
https://doi.org/10.3390/math11132923

**Chicago/Turabian Style**

Villenas, Felipe I., Francisco J. Vargas, and Andrés A. Peters.
2023. "Exploring the Role of Sampling Time in String Stabilization for Platooning: An Experimental Case Study" *Mathematics* 11, no. 13: 2923.
https://doi.org/10.3390/math11132923