Adaptive Control of Heterogeneous Platoons with Guaranteed Collision Avoidance

TL;DR

This paper introduces a novel adaptive control framework for heterogeneous vehicle platoons that guarantees collision avoidance for actual dynamics by employing a virtual platoon concept and set-theoretic adaptive laws.

Contribution

It presents a new adaptive control approach that ensures collision avoidance in heterogeneous platoons, extending beyond reference dynamics to actual vehicle behavior.

Findings

Ensures collision avoidance for actual platoon dynamics.

Uses virtual platoon concept for analysis without interaction.

Provides Lyapunov-based stability and convergence proofs.

Abstract

This work proposes a framework for Cooperative Adaptive Cruise Control of a vehicular platoon characterized by unidirectional communication and heterogeneous parameters. In the proposed framework, the actual (heterogeneous) platoon is made to converge to a reference (homogeneous) platoon via adaptive laws designed using of set-theoretic model reference adaptive control. Yet, in contrast to the state-of-art that is based on ensuring collision avoidance on the reference platoon dynamics only, the approach we propose can ensure collision avoidance on the actual platoon dynamics. This result is possible thanks to the introduction of a novel concept of virtual platoon, only used for analysis, but that does not interact with the actual platoon. The stability and convergence properties of the proposed framework are established using Lyapunov-based analysis in conjunction with the aforementioned virtual platoon concept.

Equations

$${\Omega }_c=\left\{{\Vert \eta \Vert }_M\in {\mathbb{R}}_{+}|{\Vert \eta \Vert }_M<c\right\},$$

$${\psi }^{{}^{\prime }}({\Vert \eta \Vert }_M)=\frac{d\psi ({\Vert \eta \Vert }_M)}{d{\Vert \eta \Vert }_M^2}.$$

$$2{\psi }^{{}^{\prime }}({\Vert \eta \Vert }_M){\Vert \eta \Vert }_M^2-\psi ({\Vert \eta \Vert }_M)>0.$$

$$\psi ({\Vert \eta \Vert }_M)=\frac{{\Vert \eta \Vert }_M^2}{c-{\Vert \eta \Vert }_M}$$

$$\begin{array}{rl}& \dot{\chi }(t)=\mathcal{A}\chi (t)+\mathcal{B}\mu (t)\\ & {\rm Y}(t)=\mathcal{C}\chi (t)+\mathcal{D}\mu (t)\end{array}$$

$$g(t)={\mathfrak{L}}^{-1}\left\{G(s)\right\}\ge 0,\forall t\ge 0.$$

$$\begin{array}{rl}& {\dot{\chi }}_j(t)={\mathcal{A}}_j{\chi }_j(t)+{\mathcal{B}}_j{{\rm Y}}_{j-1}(t)\\ & {{\rm Y}}_j(t)={\mathcal{C}}_j{\chi }_j(t)\end{array}$$

$$\underset{\omega }{\sup }|G_j(ȷ\omega )|=\underset{\omega }{\sup }\left|\frac{{{\rm Y}}_j(ȷ\omega )}{{{\rm Y}}_{j-1}(ȷ\omega )}\right|\le 1,\forall \omega ,\forall j.$$

$$s_{d,j}(t)=r_j+h{v}_j(t),j\in {\alpha }_N,$$

$$s_{j-1,j}(t)=y_{i-1}(t)-y_j(t)-L_j,j\in {\alpha }_N,$$

$$\begin{array}{rl}{e}_j(t)& =s_{j-1,j}(t)-s_{d,j}(t)\\ & =(y_{j-1}(t)-y_j(t)-L_j)-(r_j+h{v}_j(t)).\end{array}$$

$$\begin{array}{cc}\left(\begin{array}{c}{\dot{e}}_j\\ {\dot{v}}_j\\ {\dot{a}}_j\end{array}\right)=\left(\begin{array}{c}{v}_{j-1}-v_j-h{a}_j\\ a_j\\ -\frac{1}{{\tau }_j}{a}_j+\frac{1}{{\tau }_j}{\Lambda }_j{u}_j\end{array}\right),& j\in {\alpha }_N,\end{array}$$

$$u_j(t)=u_{bl,j}(t)+u_{ad,j}(t),\forall j\in {\alpha }_N\cup \left\{0\right\}$$

$$h{\dot{u}}_{bl,j}=-u_{bl,j}+k_p{e}_j+k_d{\dot{e}}_j+u_{bl,j-1},$$

$$h{\dot{u}}_{bl,0}=-u_{bl,0}+u_{in},$$

$$\begin{array}{rl}\underset{{\dot{x}}_{v,j}}{\underbrace{\left[\begin{array}{c}{\dot{e}}_{v,j}\\ {\dot{v}}_{v,j}\\ {\dot{a}}_{v,j}\\ {\dot{u}}_{bl,v,j}\end{array}\right]}}& =\underset{A_m}{\underbrace{\left[\begin{array}{cccc}0& -1& -h& 0\\ 0& 0& 1& 0\\ 0& 0& -\frac{1}{\bar{\tau }}& \frac{1}{\bar{\tau }}\\ \frac{k_p}{h}& -\frac{k_d}{h}& -k_d& -\frac{1}{h}\end{array}\right]}}\underset{x_{v,j}}{\underbrace{\left[\begin{array}{c}{e}_{v,j}\\ v_{v,j}\\ a_{v,j}\\ u_{bl,v,j}\end{array}\right]}}\\ & +\underset{B_w}{\underbrace{\left[\begin{array}{cc}1& 0\\ 0& 0\\ 0& 0\\ \frac{k_d}{h}& \frac{1}{h}\end{array}\right]}}\underset{w_{v,j-1}}{\underbrace{\left[\begin{array}{c}{v}_{v,j-1}\\ u_{bl,v,j-1}\end{array}\right]}},\forall j\in {\alpha }_N,\end{array}$$

$${\dot{x}}_{v,j}=A_m{x}_{v,j}+B_w{w}_{v,j-1},\forall j\in {\alpha }_N,$$

$$\underset{{\dot{x}}_{v,0}}{\underbrace{\left[\begin{array}{c}{\dot{e}}_{v,0}\\ {\dot{v}}_{v,0}\\ {\dot{a}}_{v,0}\\ {\dot{u}}_{bl,v,0}\end{array}\right]}}=\underset{A_{m,0}}{\underbrace{\left[\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 0& -\frac{1}{\bar{\tau }}& \frac{1}{\bar{\tau }}\\ 0& 0& 0& -\frac{1}{h}\end{array}\right]}}\underset{x_{v,0}}{\underbrace{\left[\begin{array}{c}{e}_{v,0}\\ v_{v,0}\\ a_{v,0}\\ u_{bl,v,0}\end{array}\right]}}+\underset{B_r}{\underbrace{\left[\begin{array}{c}0\\ 0\\ 0\\ \frac{1}{h}\end{array}\right]}}{u}_{in}.$$

$$\begin{array}{rl}\underset{{\dot{\bar{x}}}_v}{\underbrace{\left[\begin{array}{c}{\dot{x}}_{v,0}\\ {\dot{x}}_{v,1}\\ ⋮\\ {\dot{x}}_{v,N}\end{array}\right]}}& =\underset{{\bar{A}}_m}{\underbrace{\left[\begin{array}{cccc}{A}_{m,0}& & & 0_{4\times 4}\\ {\bar{B}}_w& A_m& & 0_{4\times 4}\\ & \ddots & \ddots & \\ 0_{4\times 4}& & {\bar{B}}_w& A_m\end{array}\right]}}\underset{{\bar{x}}_v}{\underbrace{\left[\begin{array}{c}{x}_{v,0}\\ x_{v,1}\\ ⋮\\ x_{v,N}\end{array}\right]}}\\ & +\underset{\bar{B}}{\underbrace{\left[\begin{array}{c}{B}_r\\ 0_{4\times 1}\\ ⋮\\ 0_{4\times 1}\end{array}\right]}}{u}_{in},\end{array}$$

$$\begin{array}{rl}& {\bar{x}}_{eq,v}=1_{N+1}\otimes \underset{\triangleq x_{eq,v}}{\underbrace{(0v_{c}00{)}^T}},\\ & \text{for}{x}_{v,0}(0)=x_{eq,v}\text{and}{u}_{in}=0.\end{array}$$

$$h>0,k_p>0,k_d>0,k_d>\bar{\tau }{k}_p.$$

$$v_{c,new}=\underset{t\to \infty }{\lim }{v}_{v,0}(t)$$

$$\begin{array}{rl}{\bar{s}}_{v,j-1,j}(t)& =y_{v,j-1}(t)-y_{v,j}(t)-L_j-r_j\\ & =s_{v,j-1,j}(t)-r_j.\end{array}$$

$$\begin{array}{rl}& {\bar{s}}_{v,j-1,j}(0)\ge 0,i.e.,s_{v,j-1,j}(0)\ge r_j\\ & v_{v,j}(0)\ge 0,\forall j\in {\alpha }_N.\end{array}$$

$$G_{v,j}(s)=\frac{\bar{\tau }{s}^3+s^2+k_{d}s+k_p}{(hs+1)(\bar{\tau }{s}^3+s^2+k_{d}s+k_p)}=\frac{1}{hs+1}.$$

$${\dot{\bar{s}}}_{v,j-1,j}(t)=v_{v,j-1}(t)-v_{v,j}(t).$$

$$F_{v,j}(s)=\frac{h}{hs+1}.$$

$$|G_{v,j}(ȷ\omega )|=\frac{1}{\sqrt{h^2{\omega }^2+1}}$$

$$\begin{array}{rl}\underset{{\dot{x}}_{r,j}}{\underbrace{\left[\begin{array}{c}{\dot{e}}_{r,j}\\ {\dot{v}}_{r,j}\\ {\dot{a}}_{r,j}\\ {\dot{u}}_{bl,r,j}\end{array}\right]}}& =\underset{A_m}{\underbrace{\left[\begin{array}{cccc}0& -1& -h& 0\\ 0& 0& 1& 0\\ 0& 0& -\frac{1}{\bar{\tau }}& \frac{1}{\bar{\tau }}\\ \frac{k_p}{h}& -\frac{k_d}{h}& -k_d& -\frac{1}{h}\end{array}\right]}}\underset{x_{r,j}}{\underbrace{\left[\begin{array}{c}{e}_{r,j}\\ v_{r,j}\\ a_{r,j}\\ u_{bl,r,j}\end{array}\right]}}\\ & +\underset{B_w}{\underbrace{\left[\begin{array}{cc}1& 0\\ 0& 0\\ 0& 0\\ \frac{k_d}{h}& \frac{1}{h}\end{array}\right]}}\underset{w_{j-1}}{\underbrace{\left[\begin{array}{c}{v}_{i-1}\\ u_{bl,i-1}\end{array}\right]}},\forall j\in {\alpha }_N,\end{array}$$

$$x_{r,j}(0)=x_{v,j}(0),\forall j\in {\alpha }_N$$

Full text

I Introduction

In recent years, the continuous advances in vehicle-to-vehicle communication have been opening the field of connected automated vehicles, paving the way to cooperative automated driving. Several studies suggest that cooperative automated vehicles can lead to substantial increase in traffic capacity due to the deployment of vehicular platoons [1, 2]. Nowadays, many vehicles are already equipped with adaptive cruise control (ACC) technology, enabling to maintain a desired inter-vehicle distance [3, 4, 5] with the help of on-board sensors like lasers and radars. Cooperative Adaptive Cruise Control (CACC) extends the ACC technology by adding wireless vehicle-to-vehicle communication, thus enhancing the connectivity of vehicles beyond on-board sensing [6, 7]. As compared to ACC, CACC ensures a smaller inter-vehicle gap [8], and provides better performance in terms of string stability [9] and disturbance decoupling [10] criteria. String stability describes the attenuation of disturbances [8, 11, 12] (e.g., induced by sudden brake or acceleration), as they propagate downstream along the string of vehicles composing the platoon.

The most standard CACC protocols consider predecessor-follower unidirectional [9] and sometimes bidirectional [13, 14] communication among adjacent vehicles. Typical desirable properties of the platoon are asymptotic stability [15, 16] and string stability [17, 8, 18]. However, it has been shown that asymptotic stability and string stability cannot automatically ensure collision avoidance within adjacent vehicles in the platoon. External positivity has been regarded as a promising property to ensure collision avoidance within the platoon [19, 20]. External positivity guarantees a positive output (i.e., inter-vehicle distance) for a positive input (i.e., the velocity of the predecessor vehicle). Using external positivity, collision avoidance is studied in [19, 20] for ACC, and later extended in [21] for CACC.

One of the pioneering works in CACC is [8] which assumes a known homogeneous actual platoon (AP). If the AP is heterogeneous, then homogeneous dynamics can be used to design a homogeneous reference platoon (RP) in the framework of model reference adaptive control (MRAC) [9, 14], so as to deal with the situation that not only the AP is heterogeneous, but its parameters are possibly unknown. The main idea behind the MRAC strategy is that desirable properties like asymptotic stability, string stability and external positivity can be imposed on the RP. Then, the AP can asymptotically attain the desirable properties by converging to the dynamics of the RP. Although effective, this strategy has the drawback that the desirable properties can only be attained asymptotically, which implies that external positivity (and thus collision avoidance) for the RP does not necessarily imply collision avoidance on the AP at all times. This represents a major open problem in the literature that is addressed in this work by introducing a new platoon concept and new analysis tools.

A first significant contribution of this work is the introduction of a virtual platoon (VP) concept that, differently from the RP, is not part of the control design, but is only used for analysis. The advantage of the VP is that, because it does not interact with the AP as it happens with the RP (refer to Fig. 2), it can be used to ensure properties of string stability and collision avoidance for the entire platoon, differently from the standard analysis in the literature based on a predecessor-follower pair [9, 14]. Our analysis shows that both RP and AP converge to VP asymptotically; hence, external positivity and string stability properties are established for the entire AP asymptotically. Then, a second significant contribution of this work is to prove that not only the AP asymptotically achieves external positivity, but that it actually attains collision avoidance at all times. This marks a difference with ensuring collision avoidance for the RP only. This result is possible thanks to the introduction of a set-theoretic MRAC architecture [22], in place of a standard MRAC architecture, that allows to use a generalized barrier Lyapunov function [23]. A novel Lyapunov analysis is presented to guarantee that not only the tracking errors between the AP and RP converge asymptotically to zero, but also that a worst-case user-defined bound can be guaranteed for the tracking error. This bound in turn can be used to ensure collision avoidance at all times not only for the RP but also for the AP. In this strategy, the VP model acts as a robust analysis tool and facilitates the appropriate use of Barbalat’s Lemma [24]. With the aforementioned tracking error bound and the VP’s external positivity condition, we derive a lower bound on the standstill distance of the AP ensuring collision avoidance at all times.

The paper is organized as follows. Preliminary information is provided in Section II. Section III provides the dynamics for the CACC platooning and the baseline control law. Section IV presents the dynamics of the VP and analysis of asymptotic stability, external positivity, and string stability. In Section V, RP dynamics, adaptive augmentation, and platoon convergence are given. Section VI provides the condition for collision avoidance.

Notation**.**

$\mathbb{R}$, $\mathbb{R}^{n}$ and $\mathbb{R}^{n\times m}$ represent the sets of real numbers, real vectors of dimension $n$ and real matrices of dimension ${n\times m}$, respectively. $\mathbb{R}_{+}$ denotes set of positive real numbers. $\mathbb{N}$ is set of natural numbers. We use $\left|\cdot\right|$ to denote the absolute value of a constant and $\left\|\cdot\right\|$ for the $2$ norm of a vector and the Frobenius norm of a matrix. The notation $\sup\left|\cdot\right|$ is the least upper bound of a function. $\mathfrak{L}^{-1}$ denotes the inverse Laplace transform. $\mathcal{L}_{2}$ and $\mathcal{L}_{\infty}$ denote the space of square integrable signals and the space of bounded signals, respectively. $I_{n\times n}$ is the identity matrix of dimension ${n\times n}$. The zero and the one vector of dimension $n$ are indicated respectively by $0_{n}=[0\;0\;\cdots\;(n\;\text{times})\;\cdots\;0]^{T}$ and ${1}_{n}=[1\;1\;\cdots\;(n\;\text{times})\;\cdots\;1]^{T}$, while the zero matrix of dimension $n\times m$ is represented by $0_{n\times m}=[0_{n}\;0_{n}\;\cdots\;(m\;\text{times})\;\cdots\;0_{n}]$. The symbol “$\otimes$” is used for Kronecker product.

II Preliminaries

In this section, some key concepts of generalized barrier Lyapunov function [22], string stability [8, 9], and external positivity [19, 21] are recalled.

Definition 1**.**

(Generalized barrier Lyapunov function [22]) Let $\left\|\eta\right\|_{M}=\sqrt{\eta^{T}M\eta}$ be the weighted Euclidean norm, with $\eta\in\mathbb{R}^{p}$, and $M\in\mathbb{R}^{p\times p}$ being symmetric positive definite. Consider a function $\psi(\cdot):\Omega_{c}\rightarrow\mathbb{R}$, where

[TABLE]

with $c\in\mathbb{R}_{+}$ a user-defined constant bound. The function $\psi(\cdot)$ is called a generalized barrier Lyapunov function if the following conditions are satisfied:

1. i)

If $\left\|\eta\right\|_{M}=0$, then $\psi(\left\|\eta\right\|_{M})=0$. 2. ii)

$\psi(\left\|\eta\right\|_{M})>0$* inside $\Omega_{c}$ excluding $\left\|\eta\right\|_{M}=0$.* 3. iii)

$\lim_{\left\|\eta\right\|_{M}\rightarrow c}\psi(\left\|\eta\right\|_{M})=\infty$. 4. iv)

$\psi(\left\|\eta\right\|_{M})$* is continuously differentiable on $\Omega_{c}$.* 5. v)

$\psi^{{}^{\prime}}(\left\|\eta\right\|_{M})>0$* inside $\Omega_{c}$, where*

[TABLE] 6. vi)

Inside $\Omega_{c}$, the following holds

[TABLE]

The term generalized barrier Lyapunov function comes from the fact that the conditions in Definition 1 generalize the notion of barrier Lyapunov function [23]. A possible generalized barrier Lyapunov function, also used in this work, is

[TABLE]

with $c$ being user-defined. The above function satisfies all the properties in Definition 1.

Definition 2**.**

(External Positivity [19, 21]) Consider the linear system

[TABLE]

with zero initial conditions. Let $G(s)=\mathcal{C}(sI-\mathcal{A})^{-1}\mathcal{B}+\mathcal{D}$ be the transfer function of the system. The system is said to be externally positive (i.e., $\mu(t)\geq 0,\forall t\geq 0\Rightarrow\Upsilon(t)\geq 0,\forall t\geq 0$), if and only if its impulse response is non-negative, i.e.,

[TABLE]

Definition 3**.**

(String Stability [8, 9]) Consider the cascaded interconnected system

[TABLE]

with zero initial conditions, $1\leq j\leq N$, and $\Upsilon_{0}(t)$ an external input. Let $G_{j}(s)=\mathcal{C}_{j}(sI-\mathcal{A}_{j})^{-1}\mathcal{B}_{j}$ be the transfer function from $\Upsilon_{j-1}(\jmath\omega)$ to $\Upsilon_{j}(\jmath\omega)$. The cascaded interconnected system is said to be string stable if

[TABLE]

where $\jmath$ is the complex unit.

III Platoon Model

Consider a heterogeneous platoon consisting of $N+1$ vehicles as shown in Fig. 1, where $v_{j}(t)$ is the velocity (m/s) of vehicle $j$ and $s_{j-1,j}(t)$ is the distance (m) between vehicle $j$ and its predecessor vehicle $j-1$. Consider the set $\alpha_{N}=\left\{j\in\mathbb{N}\mid 1\leq j\leq N\right\}$ with $j=0$ used for the leading vehicle. The objective of the platoon is to maintain a desired distance $s_{d,j}(t)$ between vehicle $j$ and its predecessor. A constant time headway (CTH) spacing policy [25] is used to define the desired distance (m)

[TABLE]

where $h>0$ is the time gap (s) and $r_{j}>0$ is the standstill distance (m) for vehicle $j$. The distance between vehicle $j$ and vehicle $j-1$ is calculated as

[TABLE]

where $y_{j}(t)\in\mathbb{R}$ is the rear bumper position (m) and $L_{j}>0$ is length (m) of vehicle $j$ as in Fig. 1. Then, the spacing error for vehicle $j$ can be defined as

[TABLE]

In the following, we may omit the time dependence whenever obvious. The dynamics of vehicle $j$ is defined as [9]

[TABLE]

where $u_{j}(t)\in\mathbb{R}$ and $a_{j}(t)\in\mathbb{R}$ are the input (${\text{m}}/{\text{s}^{2}}$) and the acceleration (${\text{m}}/{\text{s}^{2}}$), $\Lambda_{j}>0$ is the engine efficiency and $\tau_{j}>0$ is the engine time constant (s) for vehicle $j$ [9]. In this work, we take $\Lambda_{j}$ and $\tau_{j}$ to be unknown, thus requiring an adaptive design.

The interconnection of the dynamics (12) will be referred to as AP. The control objective is to design $u_{j}$ in (12) such that

• •

All the spacing errors in the AP converges to zero, i.e., $\lim_{t\rightarrow\infty}e_{j}(t)=0$, $\forall j\in\alpha_{N}$,

• •

collision avoidance is achieved in the AP, i.e., $s_{j-1,j}(t)>0,\ \forall t\geq 0,\ \forall j\in\alpha_{N}$.

Other properties guaranteed by the proposed design (e.g., string stability and external positivity) will be presented later by making use of the VP concept.

The control input $u_{j}(t)$ is taken in the following form [26]

[TABLE]

where $u_{bl,j}(t)$ and $u_{ad,j}(t)$ denote the baseline controller and the adaptive controller, respectively. Hereafter, we introduce the baseline controller, while the adaptive controller is designed in Section V.

III-A Baseline Controller

A baseline controller has been proposed in the literature [8] under the assumption that all vehicles are homogeneous (i.e., $\Lambda_{j}=1$ and $\tau_{j}$ is identical $\forall j\in\alpha_{N}$ in (12)). Such a controller takes the form

[TABLE]

where $k_{p}>0$ and $k_{d}>0$ are the controller design parameters. The feedback from $u_{bl,j-1}$ is obtained via wireless communication, in line with the standard unidirectional communication in a CACC protocol (cf. Fig. 1). The leader vehicle control input is as follows

[TABLE]

where $u_{in}(t)\in\mathcal{L}_{\infty}$ is the external input representing the desired acceleration $(\text{m}/\text{s}^{2})$ of the leader vehicle. Since the vehicles dynamic in (12) are taken heterogeneous, the assumption used in [8] fails, so that the baseline controller must be augmented by an adaptive term as in (13).

Remark 1**.**

To facilitate external positivity and string stability analysis, a homogeneous VP is now introduced. It is worth stressing that the VP is only used for analysis and does not interact with the RP or the AP as shown in Fig. 2. This marks a difference with existing literature where the RP is used for analysis as well as for interacting with the AP [9, 21].

IV Virtual Platoon Dynamics and Its Properties

A homogeneous VP is defined by using nominal homogeneous parameters in the dynamics (12)- $\tau_{j}=\bar{\tau}>0$ and $\Lambda_{j}=1,\;\forall j\in\alpha_{N}$ and using the baseline control dynamics in (14). The resulting dynamics is

[TABLE]

which can be compactly represented as follows

[TABLE]

where subscript $v$ denotes the VP, $x_{v,j}(t)\in\mathbb{R}^{4}$ is the state vector of vehicle $j$ in the VP, and $w_{v,j-1}(t)\in\mathbb{R}^{2}$ contains signals from the preceding vehicle.

Using (15), the dynamics of the leader of the VP can be written as follows

[TABLE]

Remark 2**.**

Since the leader has no predecessor, $e_{0}$ in (18) is a fictitious zero state. It is only introduced to ensure consistency with the other vehicle dynamics. Without loss of generality, a reduced-dimensional state can be defined as $\underline{x}_{v,0}=(v_{v,0}\quad a_{v,0}\quad u_{bl,v,0})$ which will be used subsequently to perform stability analysis for the VP, and will also be defined for the RP and the AP.

IV-A Stability Analysis of the Virtual Platoon

According to Definition 3, string stability can be used to describe the ability to attenuate external disturbances as they propagate through the platoon due to the interconnected vehicle dynamics. To analyze the string stability of the VP, let us write the interconnected system dynamics in a compact form, resulting from (17)-(18)

[TABLE]

where $\bar{B}_{w}=[0_{4\times 1}\quad B_{w}(:,1)\quad 0_{4\times 1}\quad B_{w}(:,2)]\in\mathbb{R}^{4\times 4}$, $\bar{x}_{v}\in\mathbb{R}^{4(N+1)\times 1}$, $\bar{A}_{m}\in\mathbb{R}^{4(N+1)\times 4(N+1)}$ and $\bar{B}\in\mathbb{R}^{4(N+1)\times 1}$. The equilibrium point

[TABLE]

of (19) has been proven to be asymptotically stable in [8], where $v_{c}$ is any constant velocity, provided that the following Routh–Hurwitz conditions are satisfied

[TABLE]

Let us define the constant equilibrium velocity in such a way that when $\lim_{t\to\infty}u_{in}(t)=0$, which implies $\lim_{t\to\infty}a_{v,0}(t)=0$, then

[TABLE]

exists and finite.

Theorem 1**.**

The vector $\bar{x}_{v}(t)$ of the VP defined in (19) converges to the equilibrium point $\bar{x}_{eq,v,new}=1_{N+1}\otimes(0\quad v_{c,new}\quad 0\quad 0)^{T}$, where $v_{c,new}$ is defined in (22).

Proof.

Since $u_{in}(t)$ is such that $x_{v,0}(t)$ converges to $(0\quad v_{c,new}\quad 0\quad 0)^{T}$, following the arguments of [8], it can be shown that the origin of the dynamics of $(\bar{x}_{v}(t)-\bar{x}_{eq,v})$ is asymptotically converging to zero since $\lim_{t\to\infty}u_{in}(t)=0$. ∎

IV-B External Positivity and String Stability Analysis of the Virtual Platoon

Let us define the pseudo spacing between vehicle $(j)$ and vehicle $(j-1)$ in the VP

[TABLE]

Let us introduce the following assumption:

Assumption 1**.**

The initial conditions satisfy

[TABLE]

This assumption guarantees that the initial condition of the platoon is physically well posed, e.g., there is no collision at $t=0$ and velocities are non-negative.

The external positivity and the string stability for the VP can be proven along similar lines as the literature [19, 21], via the following two lemmas.

Lemma 1**.**

(External positivity of inter-vehicle spacing) Consider the VP dynamics (17) and (18). The linear system defined by the transfer function $G_{v,j}(s)=\frac{v_{v,j}(s)}{v_{v,j-1}(s)}$ is externally positive. Furthermore, the linear system defined by the transfer function $F_{v,j}(s)=\frac{\bar{s}_{v,j-1,j}(s)}{v_{v,j-1}(s)}$ is also externally positive.

Proof.

From the VP dynamics (17), $G_{v,j}(s)$ is calculated as [9]

[TABLE]

The time derivative of $\bar{s}_{v,j-1,j}(t)$ in (23) gives

[TABLE]

From (25) and (26), the transfer function $F_{v,j}(s)$ is calculated as

[TABLE]

The impulse response of (25) is $g_{v,j}(t)=h^{-1}e^{-h^{-1}t},\forall t\geq 0$. We have $g_{v,j}(t)\geq 0,\forall t\geq 0$, which satisfies Definition 2. Furthermore, the impulse response of (27) is $f_{v,j}(t)=e^{-h^{-1}t},\forall t\geq 0$. We have $f_{v,j}(t)\geq 0,\forall t\geq 0$, which also satisfies Definition 2.

∎

Remark 3**.**

From Definition 2 and Lemma 1, it is concluded that a positive input $v_{v,j-1}(t)$ would ensure that both the outputs $v_{v,j}(t)$ and $\bar{s}_{v,j-1,j}(t)$, $\forall t\geq 0$ remain positive, so that the collision avoidance of VP is ensured under Assumption 1.

Remark 4**.**

The goal of this work is to guarantee collision avoidance in the AP, which is different from the state-of-the-art approach of guaranteeing collision avoidance in the RP [21]. Such a goal requires a different analysis and the introduction of $\bar{s}_{v,j-1,j}(t)$ instead of $s_{v,j-1,j}(t)$, as it will become evident in Section VI.

Lemma 2**.**

The VP in (17) is string stable.

Proof.

String stability of the linear system defined by the transfer function $G_{v,j}(s)=\frac{a_{v,j}(s)}{a_{v,j-1}(s)}$ (equivalently, $G_{v,j}(s)=\frac{v_{v,j}(s)}{v_{v,j-1}(s)}$) can be analyzed from (8) and (25), and the following is concluded

[TABLE]

implying, $|G_{v,j}(\jmath\omega)|\leq 1,\;\forall\omega,\;\forall j\;\in\>\alpha_{N}\>\setminus\>\left\{0\right\}$. Therefore, from Definition 3, the VP in (17) is string stable. ∎

The above properties of VP are going to be useful in the analysis of subsequently designed MRAC architecture.

V MRAC Augmented CACC Platooning

In order to address heterogeneity in the platoon, the adaptive controller in (13) is now designed. First, we design homogeneous RP dynamics. Differently from the VP dynamics that do not interact with the AP, the RP dynamics are part of the adaptive control design and thus interact with the AP.

V-A Reference Dynamics

The homogeneous RP dynamics is designed as

[TABLE]

subject to the initial condition

[TABLE]

where subscript $r$ denotes the RP, $x_{r,j}(t)\in\mathbb{R}^{4}$ and $w_{j-1}(t)\in\mathbb{R}^{2}$ are the RP vector and the AP input vector, respectively. The dynamics of the leader of the RP can be written as follows

[TABLE]

Remark 5**.**

Due to the coupling between the AP and the RP, it is not straightforward to derive string stability and external positivity for the whole RP in the sense of Definitions 2 and 3. On the other hand, such properties can be directly shown for VP (cf. Lemma 1 and Lemma 2). In subsequent analysis, it is shown that the RP and the AP ultimately converge to the VP. Unlike existing analysis tools that rely on properties defined for pairs of adjacent predecessor-follower vehicles [21], the analysis we present does not rely on a unidirectional predecessor-follower interconnection and is potentially applicable to a broader class of platoon interconnections [27].

In order to write the interconnected form for the RP dynamics in (29), after some manipulation, the following dynamics is achieved

[TABLE]

where $\tilde{w}_{j-1}(t)=w_{j-1}(t)-w_{r,j-1}(t)$. Therefore, the interconnected form for the dynamics in (29) is as follows

[TABLE]

where $x_{r,0}(t)=x_{v,0}(t)\;\forall t\geq 0$, $\bar{x}_{r}(t)\in\mathbb{R}^{4(N+1)\times 1}$, $\tilde{\bar{w}}(t)\in\mathbb{R}^{2N\times 1}$ and $B_{\tilde{\bar{w}}}\in\mathbb{R}^{4(N+1)\times 2N}$.

V-B MRAC Design

Substituting (13) into (12) results in

[TABLE]

which can be compactly written as

[TABLE]

where $x_{j}(t)\in\mathbb{R}^{4}$, $A_{j}\in\mathbb{R}^{4\times 4}$, $B_{u}\in\mathbb{R}^{4}$ and $\theta_{j}\in\mathbb{R}$.

The following ideal controller gain would make the AP dynamics in (35) match the RP dynamics in (29).

Lemma 3**.**

There exist a constant vector $k_{x,j}\in\mathbb{R}^{4}$ such that

[TABLE]

Proof.

It can be shown that the following expression of $k_{x,j}^{T}$ satisfies (36)

[TABLE]

∎

It should be noted that the ideal controller gain $k_{x,j}$ in Lemma 3 requires knowledge of $\Lambda_{j}$ and $\tau_{j}$, which are unknown and therefore, $u_{ad,j}(t)$ is defined as follows

[TABLE]

where $\hat{k}_{x,j}(t)\in\mathbb{R}^{4}$ is the online estimate of $k_{x,j}$. Define the tracking error as

[TABLE]

Define the parameter estimation error as

[TABLE]

Using (35), (36), (37), (38) and (39), the tracking error dynamics is

[TABLE]

Let the parameter update law for the gain in (37) be designed as

[TABLE]

where $P_{m}$ is a positive definite matrix satisfying

[TABLE]

for any design matrix $Q_{m}=Q_{m}^{T}>0$. From (41), the parameter estimation error dynamics is as follows

[TABLE]

Theorem 2**.**

For the dynamics of AP in (35) and the dynamics of RP in (29), the control input (37) and the adaptive update law (41) ensure the following:

The origin of the error $(\tilde{x}_{j}(t),\tilde{k}_{x,j}(t))$ dynamics in (40) and (43), is Lyapunov stable $\forall j\in\alpha_{N}\cup\left\{0\right\}$. 2. 2.

The state tracking error is guaranteed to satisfy

[TABLE]

where $c$ is defined in (4).

Proof.

$1)$ Let us define the following radially unbounded Lyapunov candidate

[TABLE]

Taking the time derivative of (45), substituting (40) in it, and using (42), the following is obtained

[TABLE]

Considering the adaptive law as in (41), $\dot{V}_{j}(\tilde{x}_{j}(t),\tilde{k}_{x,j}(t))$ in (46) reduces to

[TABLE]

In (47), $\dot{V}_{j}(\tilde{x}_{j}(t),\tilde{k}_{x,j}(t))$ is negative semi definite, therefore the origin of $(\tilde{x}_{j}(t)(t),\tilde{k}_{x,j}(t))$ is Lyapunov stable. $2)$ From the fact that $\dot{V}_{j}(\tilde{x}_{j}(t),\tilde{k}_{x,j}(t))\leq 0$, we have $V_{j}(\tilde{x}_{j}(t),\tilde{k}_{x,j}(t))\leq V_{j}(\tilde{x}_{j}(0),\tilde{k}_{x,j}(0))<\infty$. Hence, $(\psi(\left\|\tilde{x}_{j}(t)\right\|_{P_{m}}),\tilde{k}_{x,j}(t))\in\mathcal{L}_{\infty}$ from (45), and consequently, $\left\|\tilde{x}_{j}\right\|_{P_{m}}<c,\forall t\geq 0$ from the Definition 1, which implies $\tilde{x}_{j}(t)\in\mathcal{L}_{\infty}$. ∎

V-C MRAC Design for Leader

From (13) and (15), the dynamics of the leader of the AP can be written as follows

[TABLE]

Let us assume a matrix $\underline{A}_{c}$ as

[TABLE]

where $\underline{A}_{c}$ is Hurwitz for the following Routh-Hurwitz condition:

[TABLE]

The following ideal controller gain would make the AP dynamics in (48) match the RP dynamics in (31).

Lemma 4**.**

There exist constant vectors $k_{\underline{x}_{0}}\in\mathbb{R}^{3}$ and $k_{\tilde{\underline{x}}_{0}}\in\mathbb{R}^{3}$ such that

[TABLE]

Proof.

It can be shown that the following $k_{\underline{x}_{0}}$ and $k_{\tilde{\underline{x}}_{0}}$ satisfy (51)

[TABLE]

[TABLE]

∎

It should be noted that the ideal controller gains $k_{\underline{x}_{0}}$ and $k_{\tilde{\underline{x}}_{0}}$ require knowledge of $\Lambda_{0}$ and $\tau_{0}$, which are unknown and therefore, $u_{ad,0}(t)$ for (48) is defined as follows

[TABLE]

where $\hat{k}_{\underline{x}_{0}}(t)\in\mathbb{R}^{3}$ is the online estimate of $k_{\underline{x}_{0}}$ and $\hat{k}_{\tilde{\underline{x}}_{0}}(t)\in\mathbb{R}^{3}$ is the online estimate of ${k}_{\tilde{\underline{x}}_{0}}$, whereas $\tilde{\underline{x}}_{0}(t)$ is the tracking error defined as

[TABLE]

Define parameter estimation error as

[TABLE]

[TABLE]

Using (48), (51), (52), (53), (54) and (55), the tracking error dynamics is

[TABLE]

Let the parameter update law for the gain in (52) be designed as

[TABLE]

[TABLE]

where $P_{m,0}$ is a positive definite matrix satisfying

[TABLE]

for any design matrix $Q_{c}=Q_{c}^{T}>0$. From (57) and (58), the parameter estimation error dynamics are as follows

[TABLE]

[TABLE]

Theorem 3**.**

For the leader dynamics of AP in (48) and the leader dynamics of RP in (31), the control input (52) and the adaptive update laws (57), (58) ensure the following:

The origin of the error $(\tilde{\underline{x}}_{0}(t),\tilde{k}_{\underline{x}_{0}}(t),\tilde{k}_{\tilde{\underline{x}}_{0}}(t))$ dynamics in (56),(60) and (61), is Lyapunov stable. 2. 2.

The state tracking error is guaranteed to satisfy

[TABLE]

where $c$ is defined in (4).

Proof.

$1)$ Let us define the following radially unbounded Lyapunov candidate

[TABLE]

Taking the time derivative of (63), substituting (56) in it, and using (59), the following can be obtained

[TABLE]

Considering the adaptive law as in (57) and (58), $\dot{V}_{0}(\tilde{\underline{x}}_{0}(t),\tilde{k}_{\underline{x}_{0}}(t),\tilde{k}_{\tilde{\underline{x}}_{0}}(t))$ in (64) reduces to

[TABLE]

In (65), $\dot{V}_{0}(\tilde{\underline{x}}_{0}(t),\tilde{k}_{\underline{x}_{0}}(t),\tilde{k}_{\tilde{\underline{x}}_{0}}(t))$ is negative semi definite, therefore the origin of $(\tilde{\underline{x}}_{0}(t),\tilde{k}_{\underline{x}_{0}}(t),\tilde{k}_{\tilde{\underline{x}}_{0}}(t))$ is Lyapunov stable.

$2)$ From the fact that $\dot{V}_{0}(\tilde{\underline{x}}_{0}(t),\tilde{k}_{\underline{x}_{0}}(t),\tilde{k}_{\tilde{\underline{x}}_{0}}(t))\leq 0$, we have $V_{0}(\tilde{\underline{x}}_{0}(t),\tilde{k}_{\underline{x}_{0}}(t),\tilde{k}_{\tilde{\underline{x}}_{0}}(t))\leq V_{0}(\tilde{\underline{x}}_{0}(0),\tilde{k}_{\underline{x}_{0}}(0),\tilde{k}_{\tilde{\underline{x}}_{0}}(0))<\infty$. Hence, $(\psi(\left\|\tilde{\underline{x}}_{0}(t)\right\|_{P_{m,0}}),\tilde{k}_{\underline{x}_{0}}(t)\tilde{k}_{\tilde{\underline{x}}_{0}}(t))\in\mathcal{L}_{\infty}$ from (63), and consequently, $\left\|\tilde{\underline{x}}_{0}(t)\right\|_{P_{m,0}}<c,\forall t\geq 0$ from the Definition 1, which also implies $\tilde{\underline{x}}_{0}(t)\in\mathcal{L}_{\infty}$. ∎

Remark 6**.**

Since $\tilde{e}_{0}(t)=e_{0}(t)-e_{0,r}(t)=0\;\forall t\geq 0$, the state tracking error between the leader of the AP and the leader of the RP satisfies $\tilde{x}_{0}(t)=(\tilde{e}_{0}(t)\quad\tilde{v}_{0}(t)\quad\tilde{a}_{0}(t)\quad\tilde{u}_{bl,0}(t))\in\mathcal{L}_{\infty}$.

V-D Platoon Convergence

In this part of the work, we provide a proof of convergence of the AP dynamics in (35) to the external positive and string stable VP dynamics as $t\rightarrow\infty$. Define the error between the RP dynamics in (33) and the VP dynamics in (19) as

[TABLE]

Calculating the time derivative of (66) and using the dynamics in (19) and (33), the following error dynamics is derived

[TABLE]

Theorem 4**.**

The solution of (67) is uniformly ultimately bounded (UUB).

Proof.

From the initial condition in (30) and the error in (66) the following is achieved

[TABLE]

We can further infer that, for $\tilde{\bar{w}}(t)=0$, the equilibrium point of (67) is $\bar{z}_{eq}=1_{N+1}\otimes 0_{4}$. Furthermore, from the Routh–Hurwitz condition given in (21), the system $\dot{\bar{z}}(t)=\bar{A}_{m}\bar{z}(t)$ is asymptotically stable around the equilibrium points.

Based on Theorem 2, $\tilde{x}_{j}(t)\in\mathcal{L}_{\infty}$, therefore, $\tilde{w}_{j-1}(t)\in\mathcal{L}_{\infty}$ from the RP dynamics in (29). Furthermore, $\tilde{\bar{w}}(t)\in\mathcal{L}_{\infty}$ from the interconnected RP dynamics in (33). According to (67), $\bar{z}$ is the state of an asymptotically stable linear system with bounded external input $\tilde{\bar{w}}(t)$. Therefore, any solution to (67) is UUB. ∎

Theorem 5**.**

The state tracking error in (38) asymptotically converges to zero, i.e., $\lim_{t\rightarrow\infty}\left(x_{j}(t)-x_{r,j}(t)\right)=0,\quad\forall j\in\alpha_{N}$.

Proof.

Since $\tilde{k}_{x,j}(t)\in\mathcal{L}_{\infty}$ from Theorem 2 and $k_{x,j}$ is constant, thus $\hat{k}_{x,j}(t)\in\mathcal{L}_{\infty}$ from (39). From (67), $\bar{z}(t)\in\mathcal{L}_{\infty}$, and $\bar{x}_{v}(t)\in\mathcal{L}_{\infty}$ by design. Therefore, $\bar{x}_{r}(t)\in\mathcal{L}_{\infty}$ from the error dynamics (66). Since $\bar{x}_{r}(t)$ is the vector of $x_{r,j}(t)$, then $x_{r,j}(t)\in\mathcal{L}_{\infty}$. Consequently, ${x}_{j}(t)\in\mathcal{L}_{\infty}$ and $u_{bl,j}(t)\in\mathcal{L}_{\infty}$, from the error dynamics in (38). From (37), $u_{ad,j}(t)\in\mathcal{L}_{\infty}$, hence $u_{j}(t)\in\mathcal{L}_{\infty}$ from (13). In (40), $A_{m}$ is Hurwitz; $B_{u},\>\theta_{j}$ are constant; therefore, $\dot{\tilde{x}}_{j}(t)\in\mathcal{L}_{\infty}$. This concludes that $\tilde{x}(t)$ is uniformly continuous.

As from Definition 1 [22], $\psi^{{}^{\prime}}(\left\|\tilde{x}_{j}(t)\right\|_{M})$ is strictly increasing with respect to $(\left\|\tilde{x}_{j}(t)\right\|_{M})$, from (2) and (4) the following holds

[TABLE]

Therefore, using (69) in (47), the following is achieved

[TABLE]

where $\lambda_{min}(Q_{m})$ is the minimum eigenvalue of the positive definite matrix $Q_{m}$. Applying integration on both sides of (70) w.r.t. time $t$ gives

[TABLE]

After rearranging (71), the following is concluded

[TABLE]

From (72), we have $\tilde{x}_{j}(t)\in\mathcal{L}_{2}$. Therefore, from Theorem 2 and Remark 6, we have $\tilde{x}_{j}(t)\in\mathcal{L}_{2}\cap\mathcal{L}_{\infty}\quad\forall j\in\alpha_{N}\cup\left\{0\right\}$. Using Barbalat’s Lemma (Lemma $8.2$ of [24]), asymptotic convergence of $\tilde{x}_{j}(t)$ is ensured, i.e., $\lim_{t\rightarrow\infty}\tilde{x}_{j}(t)=0\quad\forall j\in\alpha_{N}\cup\left\{0\right\}$. ∎

Proposition 1**.**

The RP dynamics in (29) asymptotically converges to the VP dynamics in (17), i.e., $\lim_{t\rightarrow\infty}\bar{z}(t)=0$. Hence, the AP dynamics in (35) also asymptotically converges to the VP dynamics, i.e., $\lim_{t\rightarrow\infty}\left(x_{j}(t)-x_{v,j}(t)\right)=0$.

Proof.

Note that $\tilde{\bar{w}}(t)$ is a vector comprising $\tilde{w}_{j}(t)$’s and the elements $\tilde{w}_{j}(t)$ are included in the elements of $\tilde{x}_{j}(t),\quad\forall j\in\alpha_{N}\cup\left\{0\right\}$. From Theorem 5, $\lim_{t\rightarrow\infty}\tilde{x}_{j}(t)=0\quad\forall j\in\alpha_{N}\cup\left\{0\right\}$. Thus, $\lim_{t\rightarrow\infty}\tilde{\bar{w}}(t)=0\quad\forall j\in\alpha_{N}\cup\left\{0\right\}$. Therefore, from (67), the following is concluded

[TABLE]

It has been proven $\forall j\in\alpha_{N}$ that $\lim_{t\rightarrow\infty}\tilde{x}_{j}(t)=0$ in Theorem 5 and $\lim_{t\rightarrow\infty}z_{j}(t)=0$. Therefore, the AP dynamics $x_{j}(t)$ is asymptotically tracks the VP dynamics $x_{v,j}(t)$, i.e., $\lim_{t\rightarrow\infty}\left(x_{j}(t)-x_{v,j}(t)\right)=0,\quad\forall j\in\alpha_{N}\cup\left\{0\right\}$. ∎

From Theorem 1 and Proposition 1, it can be shown that $\lim_{t\rightarrow\infty}v_{0}(t)=v_{c,new}$.

Corollary 1**.**

The spacing error of the AP converges to zero, i.e., $\lim_{t\rightarrow\infty}e_{j}(t)=0$ $\forall j\in\alpha_{N}\cup\{0\}$.

Proof.

Since, from Theorem 1, $\lim_{t\rightarrow\infty}e_{v,j}(t)=0,\;\forall j\in\alpha_{N}\cup\{0\}$ and from Proposition 1 $\lim_{t\rightarrow\infty}\left(x_{j}(t)-x_{v,j}(t)\right)=0$, we have $\lim_{t\rightarrow\infty}\left(e_{j}(t)-e_{v,j}(t)\right)=0$. It is concluded that $\lim_{t\rightarrow\infty}e_{j}(t)=0$ $\forall j\in\alpha_{N}\cup\{0\}$. ∎

Remark 7**.**

The importance of the Corollary 1 is to show that not only $\lim_{t\rightarrow\infty}{e}_{v,j}(t)=0$, but also $\lim_{t\rightarrow\infty}{e}_{j}(t)=0$, $\forall j\in\alpha_{N}\cup\{0\}$.

VI Collision Avoidance in the Actual Platoon

Lemma 1 only guarantees collision avoidance of the VP, but it does not guarantee collision avoidance of the AP. Therefore, in this section, we derive the conditions for collision avoidance of the AP.

Lemma 5**.**

The state tracking error $\tilde{x}_{j}^{v}(t)=x_{j}(t)-x_{v,j}(t)$ satisfies

[TABLE]

where $\Omega=\max_{\forall t\in[0,\infty)}\int_{0}^{t}\left\|e^{\bar{A}_{m}^{{}^{\prime}}(t-\tau)}\right\|\left\|B_{\tilde{\bar{w}}}^{{}^{\prime}}\right\|d\tau$111Since $\bar{A}_{m}^{{}^{\prime}}$ is Hurwitz, $\Omega$ is a finite constant. with $\bar{A}_{m}^{{}^{\prime}}=\bar{A}_{m}(5:4(N+1),5:4(N+1))\in\mathbb{R}^{4N\times 4N}$, $B_{\tilde{\bar{w}}}^{{}^{\prime}}=B_{\tilde{\bar{w}}}(5:4(N+1),:)\in\mathbb{R}^{4N\times 2N}$ and $\bar{\mathcal{Z}}=\max(\frac{c}{\lambda_{min}(P_{m,0})},\frac{c}{\lambda_{min}(P_{m})})$, $\lambda_{min}(P_{m,0})$ and $\lambda_{min}(P_{m})$ being the minimum eigenvalues of the matrices $P_{m,0}$ and $P_{m}$, respectively.

Proof.

Using the initial condition in (68), the dynamics in (67) is integrated on both sides with respect to $t$

[TABLE]

Using the bound on $\tilde{x}_{j}(t)$ in (44), a bound for $\tilde{w}_{j}(t)$ is as follows

[TABLE]

Using (76), bound for the $\tilde{\bar{w}}(t)$ is derived as follow

[TABLE]

Using (75) and (77), bound for $\bar{z}(t)$ is given as follow

[TABLE]

The following state tracking error bound can also be concluded from the bound in (78)

[TABLE]

Using the bound for the $\left\|\tilde{x}_{j}(t)\right\|$ in (44) and $z_{j}(t)$ in (79), the following inequality holds

[TABLE]

This completes the proof. ∎

The following main result guarantees collision avoidance of the entire AP.

Theorem 6**.**

If

[TABLE]

the actual spacing between the vehicles in the AP is always positive, i.e., $s_{j-1,j}(t)>0$.

Proof.

Let us define the spacing error of the VP for the spacing in (23)

[TABLE]

where $s_{d,v,j}(t)=r_{j}+hv_{v,j}(t)$. Further defining the error between (11) and (82), the following holds

[TABLE]

where $\tilde{v}_{j}^{v}(t)=v_{j}(t)-v_{v,j}(t)$ and $\bar{s}_{j-1,j}(t)=s_{j-1,j}(t)-r_{j}$.

After rearranging (83), the spacing of RP dynamics is written as follows

[TABLE]

Since from Lemma 1 and Remark 3, $\bar{s}_{v,j-1,j}(t)\geq 0,\forall t\geq 0$, the following inequality is achieved from (84)

[TABLE]

The bound in (74) implies

[TABLE]

Using the bound in (86) in inequality (85), the following bound is achieved

[TABLE]

From (85) and (87)

[TABLE]

Therefore, using this bound in (10), the following inequality for the actual spacing is concluded

[TABLE]

From the inequality in (81) and (89), the following is concluded for the actual spacing

[TABLE]

∎

VII Conclusion

In contrast to analyzing predecessor-follower pairs as the state-of-the-art platooning methods, this work has proposed a virtual-platoon-based (VP-based) analysis, which offers a generalized framework ensuring desirable properties (asymptotic stability, string stability, external positivity, etc.) for the entire platoon. A set-theoretic MRAC-based CACC architecture was designed for the actual platoon, subject to uncertain and possibly heterogeneous engine efficiency. Rigorous mathematical analysis, utilizing the Lyapunov technique in conjunction with the VP concept, demonstrated stability and the platoon’s convergence of the state of the heterogeneous actual platoon to the state of a homogeneous reference platoon. Furthermore, the set-theoretic MRAC-based CACC architecture allows to satisfy a user-defined performance bound for the state tracking error. A critical limiting condition for the standstill distance is derived from the bound on the tracking error, ensuring that the actual platoon can avoid inter-vehicle collisions for all times including the initial transient interval. Exploring the proposed VP-base analysis in more general multi-hop communication protocols would be an exciting avenue for future research.