# Sampled-Data Cooperative Adaptive Cruise Control for String-Stable Vehicle Platooning with Communication Delays: A Linear Matrix Inequality Approach

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## Abstract

**:**

## 1. Introduction

- This study introduces a practical CACC technique for vehicle platooning, incorporating the MSC technique to enhance the reliability of V2V communication.
- The proposed controller design conditions, applicable to variable-sampling intervals, are established in the time domain as LMIs, simultaneously ensuring both individual stability and string stability.
- An improved LKF, designed with partitioned sampling intervals and considering essential states for the CACC system configuration, is proposed. This improved LKF reduces conservatism in the design conditions of the sampled-data controller and optimizes computational complexity.

## 2. Problem Statement

#### 2.1. Vehicle Longitudinal Dynamics

#### 2.2. Platooning Error Dynamics

**Definition**

**1.**

**Remark**

**1.**

**Remark**

**2.**

**Problem**

**1.**

- 1 .
- The equilibrium of ${x}_{i1}\left(t\right)$ is asymptotically stable when ${u}_{i-1}\left(t\right)=0$, ensuring individual stability;
- 2 .
- The following inequality is guaranteed, which ensures string stability [29]:$$\begin{array}{c}\hfill \parallel {\mathcal{S}}_{i}{\left(t\right)\parallel}_{{\mathcal{L}}_{2}}\le {\parallel {\mathcal{S}}_{i-1}\left(t\right)\parallel}_{{\mathcal{L}}_{2}},\phantom{\rule{3.33333pt}{0ex}}2\le i\le N,\end{array}$$$$\begin{array}{c}\hfill {\int}_{0}^{{t}_{f}}{\mathcal{S}}_{i}^{T}\left(s\right){\mathcal{S}}_{i}\left(s\right)ds\le V\left(0\right)+{\int}_{0}^{{t}_{f}}{\mathcal{S}}_{i-1}^{T}\left(s\right){\mathcal{S}}_{i-1}\left(s\right)ds,\end{array}$$

**Remark**

**3.**

#### 2.3. Required Lemmas

**Lemma**

**1**

**Lemma**

**2**

## 3. LMI-Based Sampled-Data Controller Design

**Remark**

**4.**

**Remark**

**5.**

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

## 4. Simulation

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The time responses of the control inputs achieved through string-stable vehicle platooning.

**Figure 2.**The state trajectories of the position, velocity, and acceleration achieved through string-stable vehicle platooning.

**Figure 3.**The time responses of the spacing error and velocity error achieved through string-stable vehicle platooning.

**Figure 4.**The time responses of the control inputs achieved through string-unstable vehicle platooning.

**Figure 5.**The state trajectories of the position, velocity, and acceleration achieved through string-unstable vehicle platooning.

**Figure 6.**The time responses of the spacing error and velocity error achieved through string-unstable vehicle platooning.

**Figure 9.**The time responses of the spacing error with $({h}_{1},{h}_{2})=(0.1,0.3)$, using Theorem 2.

**Figure 10.**The time responses of the spacing error with $({h}_{1},{h}_{2})=(0.1,0.3)$, using the controller designed in [36].

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**MDPI and ACS Style**

Jang, Y.H.; Kim, H.S.
Sampled-Data Cooperative Adaptive Cruise Control for String-Stable Vehicle Platooning with Communication Delays: A Linear Matrix Inequality Approach. *Machines* **2024**, *12*, 165.
https://doi.org/10.3390/machines12030165

**AMA Style**

Jang YH, Kim HS.
Sampled-Data Cooperative Adaptive Cruise Control for String-Stable Vehicle Platooning with Communication Delays: A Linear Matrix Inequality Approach. *Machines*. 2024; 12(3):165.
https://doi.org/10.3390/machines12030165

**Chicago/Turabian Style**

Jang, Yong Hoon, and Han Sol Kim.
2024. "Sampled-Data Cooperative Adaptive Cruise Control for String-Stable Vehicle Platooning with Communication Delays: A Linear Matrix Inequality Approach" *Machines* 12, no. 3: 165.
https://doi.org/10.3390/machines12030165