On the utilization of Macroscopic Information for String Stability of a Vehicular Platoon
Marco Mirabilio Alessio Iovine Elena De Santis Maria Domenica Di Benedetto Giordano Pola
eess.SY cs.SY







Incommensurate two-dimensional checkerboard charge density wave in the low dimensional superconductor 
    S. Haravifard
    December 30, 2023
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    The use of macroscopic information for the control of a vehicular platoon composed of autonomous vehicles is investigated. A mesoscopic control law is provided, and String Stability is proved by Lyapunov functions and Input-to-State Stability (ISS) concepts. Simulations are implemented in order to validate the controller and to show the efficacy of the proposed approach for mitigating traffic oscillations. 
    
    Keywords: String Stability, Input-to-State Stability, platoon control, mesoscopic modeling, Cooperative Adaptive Cruise Control.












§ INTRODUCTION


Nowadays, Vehicle-to-Infrastucture (V2I) and Vehicle-to-Vehicle (V2V) communication technologies are a reality in the smart transportation domain (see <cit.>), and their utilization in Cooperative Adaptive Cruise Control (CACC) is widely expected  to improve traffic conditions (see <cit.>, <cit.>, <cit.>). Indeed, traffic jamming transition has been shown to strongly depend on the amplitude of fluctuations of the leading vehicle (see <cit.>), and interconnected autonomous vehicles are sensed to reduce stop-and-go waves propagation and traffic oscillations via the concept of String Stability  <cit.>, <cit.>,  <cit.>, <cit.>.  

String Stability relies on the idea that disturbances acting on an agent of the cluster should not amplify backwards in the string. 

In the case of vehicular platooning, disturbances may be due to reference speed variation, external inputs acting on each vehicle, wrong modeling, etc. Several cases of information sharing have been considered for each leader-follower interaction, but a common characteristic is that some microscopic variables are always shared among the whole platoon, e.g. the acceleration of the platoon's leading vehicle (see <cit.>) or its desired speed profile (see <cit.>). It needs a V2V communication among the whole platoon, or a V2I bidirectional exchange of information. 


This paper analyses the benefits of the information propagation in a String Stability framework using both microscopic and macroscopic information for control purposes.

Each follower is here considered to correctly measure the distance and speed of its leading vehicles, using for example radar and LIDAR. The leader acceleration is communicated only to its follower. To improve control performance, macroscopic information is supposed to be obtained and communicated either from the road infrastructure (V2I) or from the whole platoon (V2V). Both technologies have strengths and weaknesses. For example, V2V technology requires the macroscopic information to be propagated through the vehicles, and possibly estimated in a distributed manner by each one. On the other hand, V2I technology may provide a more reliable information at the cost of allocating several sensors along the way and computing the quantities in a centralized manner, implying a high computation request to the central computer. A thorough analysis of pros and cons of the two communication typologies is out of the scope of the present paper. 

We target a platoon composed by autonomous vehicles implementing CACC, but the framework we propose is suitable for including autonomous vehicles implementing simple ACC or even human-driven vehicles as part of the platoon. 

The framework we propose is based on sharing macroscopic quantities along the platoon. 



The use of those quantities aims at increasing the ability of each car-following situation to counteract the disturbances by providing an anticipatory behaviour capable to absorb traffic jam. The idea of using macroscopic quantities, mainly the density, for microscopic traffic control has already been introduced in the literature, resulting in a mesoscopic modeling. In <cit.>, <cit.> and in the references therein, the focus is on simulation aspects and real data analysis. Several works are now focusing on a mesoscopic modeling for traffic control purposes (see <cit.>, <cit.> <cit.>, <cit.>, <cit.>).




The controller we present in this paper considers  macroscopic information and ensures Asymptotic String Stability. 

The adopted nonlinear spacing policy relies on the family of nonlinear spacing strategies introduced in <cit.> and <cit.>. 
Similarly to <cit.>, the result is obtained through an inductive method exploiting Input-to-State Stability (ISS). The main difference is that 

ISS is ensured with respect to the leader-follower situation and the ahead vehicles of each predecessor.  


Simulations show the improvements on the whole traffic throughput producing an anticipatory behaviour and oscillations reduction, and providing a better transient harmonization while maintaining String Stability properties.





The paper is organized as follows. Section <ref> introduces the considered framework, while Section 
<ref> the needed control tools. Control laws are derived and stability analysis is performed in Section <ref>. Simulations are carried out in Section  <ref>. Some concluding remarks are outlined in Section <ref>.





Notation -  ℝ^+ is the set of non-negative real numbers. For a vector x∈ℝ^n, |x|=√(x^T x) is its Euclidean norm. The ℒ_∞ signal norm is defined as |x(·)|_∞^[t_0,t] = sup_t_0≤τ≤ t|x(τ)|. 
We refer to <cit.> for the definition of Lyapunov functions, and functions 𝒦, 𝒦_∞ and  𝒦ℒ.





§ MODELING AND STRING STABILITY DEFINITIONS





 §.§ Platoon modeling

We consider a cluster of N+1 vehicles, N∈ℕ, proceeding in the same direction on a single lane road, as in Fig.<ref>. We make the following 


    All the vehicles are equal, with the same length L∈ℝ^+ and have the common goal of maintaininig a strictly positive distance among them, while keeping the same speed.


We denote with i = 0 the first vehicle of the platoon and with ℐ_N = {1,2,...,N} the set of follower vehicles. The set including all the vehicles is ℐ_N^0 = ℐ_N ∪{0}.



Similarly to <cit.>, each vehicle i∈ℐ_N^0 is assumed to satisfy the following longitudinal dynamics:


    [              ṗ_i = f_p(ξ_i); ξ̇_i = f_ξ(ξ_i)+g_ξ(ξ_i)u_i ]


where p_i∈ℝ^+ is the position of vehicle i, v_i=ṗ_i (0< v_i≤ v_max,  v_max∈ℝ^+) is its velocity and the acceleration u_i is the vehicle control input. Variable ξ_i∈ℝ^n-1 represents the remaining dynamics of the vehicle, such as actuators dynamics.  



According to <cit.> and <cit.>, the dynamics in (<ref>) can be simplified:

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    ṗ_i = v_i,   v̇_i = u_i



 where ξ_i=v_i, the functions f_p(ξ_i)=v_i, f_ξ(ξ_i)=0 and g_ξ(ξ_i)=1; u_i is the acceleration of vehicle i (|u_i|≤ a_max, a_max∈ℝ^+). The introduced double integrator model is widely used in literature for string stability analysis purposes. Moreover, field experiments adopting control laws developed with respect to this model have shown satisfactory behaviors <cit.>. 
To provide a  global description of the platoon, we adopt the leader-follower model that describes the inter-vehicular interaction  (see <cit.>, <cit.>). We define the state of each vehicle i∈ℐ_N^0 as

    x_i = [ p_i   v_i ]^T

and the state of each car-following situation among the leading vehicle i-1 and the following one i as


    χ_i = x_i-x_i-1 
        = [
            [ Δ p_i; Δ v_i ]] =  [
            [ p_i-p_i-1; v_i-v_i-1 ]].


Positions, speed and acceleration of each leading vehicle are supposed to be perfectly known, either measured or communicated to the following one. Consequently, the obtained dynamical model is:


    χ̇_i = [
        [  Δṗ_i; Δv̇_i ]] =  [
        [      Δv_i; u_i-u_i-1 ]],    i∈ℐ_N,


or, shortly,


    χ̇_i  = f(χ_i,u_i,u_i-1),    i∈ℐ_N.








To derive the dynamics of the first vehicle i=0 of the platoon, in the same form of (<ref>), a non-autonomous non-communicating  virtual leader i=-1 is considered to precede the set of vehicles, with dynamical model


    ẋ_-1 = [
        [  ṗ_-1; v̇_-1 ]] = [
        [ v_-1; u_-1 ]].


Then, the car-following dynamics with respect to vehicle i=0 can be described by:


    χ̇_0 = [
        [  Δṗ_0; Δv̇_0 ]] =  [
        [     Δv_0; u_0-u_-1 ]].


It follows that the dynamics in (<ref>) is valid ∀  i∈ℐ_N^0. Since we consider i=-1 to represent a virtual vehicle, p_-1(t) = ∫_0^t v_-1(τ)dτ,  t≥ 0, is a dummy state. Moreover, we consider Δ p_0(t) = -Δp,  ∀ t≥ 0, where Δp > 0 is the constant desired inter-vehicular distance. Since p_i<p_i-1, the desired distance has to be negative.

In accordance to <cit.>, <cit.>, <cit.>, we consider the  widely accepted hypothesis of constant speed for the virtual leader i=-1, that precedes the entire cluster. Then, we have


    p_-1(t) = v· t,  v_-1(t) = v,  u_-1(t) = 0,  ∀ t≥ 0


where v>0 is a constant speed. 









For vehicular platoons, the constant speed assumption defines the equilibrium point for all the vehicles in the cluster. Consequently, when all i∈ℐ_N^0 have equal speed and are at the same desired distance Δp, the equilibrium point for the i-th system of dynamics (<ref>) is


    χ_e,i = χ = [ -Δp   0  ]^T.



Let χ be the lumped state of the entire vehicle platoon:


    χ=[χ_0^T  χ_1^T  ...  χ_N^T]^T.


Then, for u_-1=0 it follows that 


    χ_e = [ χ^T  χ^T  ...  χ^T]^T.





 §.§ String Stability definitions



String Stability is a concept originally developed for the analysis of line graph interconnected systems <cit.>. Since vehicular platoons fall into this category of system, this notion is strictly linked with their analysis. From a general point of view, String Stability relies on the concept that disturbances acting on an agent of the cluster must no amplify backward in the string [ref!]. In the case of vehicular platoon, disturbances can be due to reference speed variation, external inputs acting on each vehicle, wrong modelization, etc. In this framework, String Stability is strictly linked to the safety issue; in fact, String Instability lead to unpredictable oscillations of the vehicles, that could cause collisions or platoon disruption. Moreover, the String Stability property is also an important feature for the energy saving issue, because a smoother velocity profile leads to reduced fuel consumption.

Recently, in <cit.> is investigated the possibility to extend the notion of String Stability to the general class of networks of interconnected systems. In fact, being a platoon of vehicles an interconnected cascaded system, it is a special case in which the head agent (namely the first vehicle) has no incoming links. Moreover, the head vehicle is the one who set the equilibrium point of the entire cluster. Then, the authors introduce the concept of scalable String Stability to study the performance of a network, with general topology, with respect to the propagation of the perturbations. The main property of interest is the possibility to guarantee the String Stability independently of the network size.

In this work we exploit the information related to the traffic flow, so called macroscopic, to guarantee String Stability of a vehicular platoons. We show that we are able to build an interconnected system with a more complete information path. 



    

    


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Let the model in (<ref>) describe a platoon, and its equilibrium be (<ref>). The control input u_i is generated by the following dynamic controller


    ρ̇_i = ω_i( ρ_i , χ ) 
    
            u_i = h_i( ρ_i , χ , χ_e,i , u_i-1 )


where ρ_i ∈ℝ^r, r ≥ 1, is the state vector with dimension r of the dynamic controller;  ω_i : ℝ^r ×ℝ^2N→ℝ^r is the vector field describing the evolution of the controller state; h_i : ℝ^r ×ℝ^2N×ℝ^2 ×ℝ→ℝ is the output function of the system that corresponds to the control input u_i. The inputs of the dynamical system (<ref>) are χ,  χ_e,i,  u_i-1. The resulting closed loop system is denoted in the sequel by P_cl, where the state vector of the i-th vehicle and the corresponding equilibrium point are


    χ_i = [  χ_i^T   ρ_i^T   ]^T,   χ_e,i = [  χ^T    0_r^T   ]^T,   ∀  i ∈ℐ_N^0,


with 0_r∈ℝ^r a null column vector. 

We now recall the notions of String Stability and Asymptotic String Stability from <cit.> and <cit.>.

(String Stability)
    The equilibrium χ_e,i,   i∈ℐ_N^0, of P_cl is said to be String Stable if, for any ϵ>0, there exists δ>0 such that, for all N∈ℕ,
    
    
    max_i∈ℐ_N^0|χ_i(0)-χ_e,i|<δ⇒max_i∈ℐ_N^0|χ_i(t)-χ_e,i|<ϵ,  ∀ t≥ 0.

    


(Asymptotic String Stability)
    The equilibrium χ_e,i=χ,  i∈ℐ_N^0, of P_cl is said to be Asymptotically String Stable if it is String Stable and, for all N∈ℕ,
    
    
    lim_t→∞|χ_i(t)-χ_e,i|=0,   ∀ i∈ℐ_N^0.

    






Let the model in (<ref>) describe a platoon, and its equilibrium be (<ref>). The input u_i is generated by the following dynamic controller


    ϑ̇_i = ω_i( ϑ_i , χ , χ_e,i ) 
    
            u_i = h_i( ϑ_i , χ , χ_e,i , u_i-1 )


taking as inputs χ,  χ_e,i,  u_i-1.
The resulting closed loop system is denoted in the sequel by P_cl. We recall the notions of String Stability and Asymptotic String Stability according to <cit.> and <cit.>.


(String Stability)
    The equilibrium χ_e,i=χ,  i∈ℐ_N^0, of P_cl is said to be String Stable if, for any ϵ>0, there exists δ>0 such that, for all N∈ℕ,
    
    
    max_i∈ℐ_N^0|χ_i(0)-χ|<δ⇒max_i∈ℐ_N^0|χ_i(t)-χ|<ϵ,  ∀ t≥ 0.

    


(Asymptotic String Stability)
    The equilibrium χ_e,i=χ,  i∈ℐ_N^0, of P_cl is said to be Asymptotically String Stable if it is String Stable and, for all N∈ℕ,
    
    
    lim_t→∞|χ_i(t)-χ|=0,   ∀ i∈ℐ_N^0.

    



































§ CONTROL TOOLS

The goal of this paper is to design a controller (<ref>) that adopts mesoscopic quantities and ensures asymptotic string stability of P_cl. To this purpose, a proper spacing policy  and a function describing macroscopic information are introduced. 



 §.§ Spacing policy








  §.§.§ Constant spacing policy
 
A constant spacing strategy (see <cit.>) consists in tracking a constant desired inter-vehicular distance:


    Δ p^r = -Δp


where Δp>0 is a constant value. Usually, the main drawback of such policy is that it does not guarantee String Stability if the control input of vehicle i exploits only measurements with respect to its preceding vehicle i-1 (see <cit.>). To stability purposes, additional information about other preceding vehicles must be shared. For example, the authors in <cit.> develop a control law based on both the information of vehicle i-1 and vehicle i=0. The same methodology is used to develop the controller in <cit.>, where a time-delayed spacing policy is proposed  to follow a constant inter-vehicular distance. As also discussed in <cit.>, assuming that the first vehicle of the platoon is allowed to send information to all the other vehicles is not practical. 




Several spacing policies have been introduced in the literature (see <cit.>, <cit.>). 







We adopt a variable time spacing policy, which consists in tracking a variable inter-vehicular desired distance and allows for string stability and a low inter-vehicular spacing at steady-state (see <cit.> and <cit.>).



We define a mesoscopic time varying trajectory for the distance policy Δ p^r_i of the i-th vehicle with respect to its leader i-1:


    Δ p_i^r(t) = -Δp - ρ^M_i(t),  t≥ 0


where Δp>0 is the desired constant inter-vehicular distance and ρ^M_i(t) is a function describing macroscopic information. Our goal is to show that, by using the macroscopic information, transient harmonization when traffic conditions vary is obtained while maintaining the platoon equilibrium in (<ref>) in steady-state.



    

















 §.§ Macroscopic information

Here we define proper macroscopic functions taking into account microscopic distance and speed variance, similarly to <cit.>. 




Given the generic vehicle i∈ℐ_N, let μ_Δ p,i and σ^2_Δ p,i be the inter-vehicular distance mean and variance computed from vehicle 0 to vehicle i, respectively:


    μ_Δ p,i = 1/i+1∑_j = 0^iΔ p_j ,   σ_Δ p,i^2 = 1/i+1∑_j = 0^i(Δ p_j-μ_Δ p_i)^2.


Let μ_Δ v,i and σ^2_Δ v,i be the velocity tracking error mean and variance computed from vehicle 0 to vehicle i, respectively:


    μ_Δ v,i = 1/i+1∑_j = 0^iΔ v_j,   σ_Δ v,i^2 = 1/i+1∑_j = 0^i(Δ v_j-μ_Δ v_i)^2 .


Let ψ_Δ p^i : ℝ×ℝ^+ →ℝ be the distance macroscopic function

and ψ_Δ v^i : ℝ×ℝ^+ →ℝ the speed tracking error macroscopic function, defined as






    ψ_Δ p^i = γ_Δ p sign(Δp+μ_Δ p,i)√(σ^2_Δ p,i),



    ψ_Δ v^i = γ_Δ v sign(μ_Δ v,i)√(σ^2_Δ v,i),


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where γ_Δ p,γ_Δ v > 0 are constant parameters, μ_Δ p,i, μ_Δ v,i, σ^2_Δ p,i and σ^2_Δ v,i are defined in (<ref>) and (<ref>), and


    sign(y) = 
        
            1,        y > 0 
    
            0,     y = 0 
    
            -1,     y < 0




 

Functions (<ref>) and (<ref>) connect the macroscopic density function with the variance of the microscopic distance and speed difference. These functions catch the distance of the system with respect to its equilibrium.
Instead of considering the whole set of leader-follower situations, they allow for a complexity reduction of the considered interconnected system without reducing the level of available information.

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We embed the macroscopic information given by (<ref>) and (<ref>) in the macroscopic function denoted by 


    ρ_i = [ ρ_1,i  ρ_2,i ]^T, 


the evolution of which is given by the controller dynamics (<ref>), where we choose a state dimension r = 2 and an asymptotically stable dynamics:





    ρ̇_1,i = -λ_1 ρ_1,i + ρ_2,i
    ρ̇_2,i = -λ_2 ρ_2,i + a ψ_Δ p^i-1 + b ψ_Δ v^i-1
    ρ_1,i(0) = ρ_2,i(0) = 0










where a,b≥ 0 are chosen parameters, and λ_1,λ_2 > 0.

The superscript i-1 in ψ^i-1_Δ p and ψ^i-1_Δ v means that we consider the macroscopic information calculated up to the preceding vehicle. Since no macroscopic information is available to vehicle 0 we define ψ^-1_Δ p = ψ^-1_Δ v = 0. Different macroscopic functions can be proposed, as in <cit.>. The macroscopic function ρ_i^M in (<ref>) is defined as a linear combination of the components of ρ_i.  

Its role is to incorporate the whole macroscopic information of the platoon avoiding complexity calculation explosion by the control law due to the state explosion. 





If the functions (<ref>) and (<ref>) were to be obtained through V2V communications, a demanding exchange of information could be necessary. However, those functions can be easily calculated by the road infrastructure, and then communicated to the vehicles via V2I communications. Consequently, the function ρ_i in (<ref>) can be obtained without V2V communication.  












































§ MESOSCOPIC CONTROL LAW


In this section, the control law adopting mesoscopic quantities for a single car-following situation is introduced. Then, String Stability and Asymptotic String Stability as in Definitions <ref> and <ref> are ensured when the control laws are implemented for each leader-follower situation along the platoon. 
The control law implements the variable spacing policy in (<ref>) while considering the function ρ_i in (<ref>). Each vehicle is modeled according to dynamics (<ref>) and each car-following situation according to χ_i in (<ref>) and (<ref>). 

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    To  analyze  the  String  Stability  of  the  closed  loop system, we consider the extended state (<ref>) that includes the dynamics in (<ref>).
    
    
    
    
    
    
    
    
    
    
    
    Defining g_cl,0(χ̂_-1) = 0, we can describe the closed loop dynamics of (<ref>) as 
    
    
    χ̇̂̇_0    = f_cl(χ̂_0),  i=0, 
    
            χ̇̂̇_i    = f_cl(χ̂_i) + g_cl,i(χ̂_i-1,χ̂_i-2,...,χ̂_0),  ∀  i∈ℐ_N.
  
    
    where f_cl:ℝ^4 →ℝ^4 is the vector field describing the evolution dynamics of each isolated subsystem, and g_cl,i:ℝ^4×···×ℝ^4_i  times→ℝ^4 is the interconnection term.


We assume that the virtual leader i=-1 has a constant speed v_-1 = v>0,  u_-1 = 0. 










The assumption of Δ p_0(t) = -Δp,  ∀ t≥ 0 is considered. 








 §.§ Control strategy for constant spacing policy

We introduce here a control law for implementing the constant spacing policy in (<ref>). The controller associated to the i-th vehicle, ∀ i∈ℐ_N^0, is given by 



    u_i = u_i-1 + Δv̇_i^r - K_Δ v(Δ v_i - Δ v_i^r) - (Δ p_i + Δp)+ρ_i-1


with given constant gains K_Δ p,K_Δ v>0, equal for each i∈ℐ_N^0,


    Δ v_i^r    = -K_Δ p(Δ p_i + Δp), 
    Δv̇_i^r    = -K_Δ pΔ v_i.


and ρ_i defined in (<ref>) and (<ref>).


We remark that the controller in (<ref>) considers both microscopic and macroscopic information, resulting to describe a mesoscopic framework.






Considering the extended state in (<ref>), we introduce the corresponding equilibrium point as:


    χ̂_e,i = [ 
            [ -Δp              0               0 ]]^T,  ∀ i∈ℐ_N^0.


The closed loop dynamics with respect to (<ref>) and (<ref>) results to be, respectively:


    χ̇̂̇_0 = [  
            [  Δṗ_0; Δv̇_0; ρ̇_-1 ]] = [  
            [               Δ v_0; -(K_Δ p+K_Δ v)Δ v_0;                   0 ]]



    χ̇̂̇_i     = [  
            [   Δṗ_i;  Δv̇_i; ρ̇_i-1 ]] 
    
            = [  
            [                                                   Δ v_i; Δv̇_i^r - K_Δ v(Δ v_i - Δ v_i^r) - (Δ p_i + Δp) + ρ_i-1;                       -λρ_i-1 + aψ_Δ p^i-1 + bψ_Δ v^i-1 ]]









 


The equations (<ref>) and (<ref>) explicit the formalism in (<ref>) and (<ref>), where



    g_cl,i(χ_i-1,χ_i-2,...,χ_0) = [ 
            [                       0;                       0; aψ_Δ p^i-1 + bψ_Δ v^i-1 ]].





Given χ_i = χ̂_i-χ̂_e,i, the following result is proven:


    Consider the closed loop system in   (<ref>) and (<ref>). Then, there exist functions β of class 𝒦ℒ and γ of class 𝒦_∞ such that
    
    
    |χ_i(t)| ≤β( |χ_i(t_0)| , t-t_0 ) + γ( max_j=0,...,i-1 |χ_j(·)|_∞^[t_0,t])

    
    ∀ t≥ t_0 , and γ(s) = γs, s≥ 0, γ∈ℝ^+ constant.



    Let us consider the candidate Lyapunov function (see <cit.>) W_i = W(χ_i) for the i-th dynamical system χ_i, for i∈ℐ_N^0:
    
    
    W(χ_i)    = 1/2(Δ p_i + Δp)^2 + 1/2(Δ v_i + K_Δ p(Δ p_i + Δp))^2 
    
                  +1/2ρ^2_i-1.

    
    It satisfies 
    
    
    α|χ_i|^2 ≤ W(χ_i) ≤α|χ_i|^2,

    
    where
    
    
    α = 1/2min{ K^2_Δ p , 1 },   α  = 1/2(1 + K^2_Δ p).

    
    More details on inequalities in (<ref>) are provided in Appendix.
    
    Then, the time derivative of W_i in  (<ref>) is:
    
    
    Ẇ_i     =  -K_Δ p( Δ p_i + Δp )^2 - K_Δ v( Δ v_i 
    
                  + K_Δ p( Δ p_i+Δp ) )^2 -λρ_i-1^2 
    
                  + ρ_i-1( Δ v_i+K_Δ p(Δ p_i+Δp) 
    
                   + aψ_Δ p^i-1 + bψ_Δ v^i-1 ) 
    
               ≤ -α |χ_i|^2 + |ρ_i-1|( k|χ_i| + a|ψ_Δ p^i-1| + b|ψ_Δ v^i-1| )

    
    where 
    
    
    α   = min{ K_Δ v , K_Δ p+K_Δ v K_Δ p^2 , λ}, 
    
            
            
            k   = √(3)max( K_Δ p , 1 ).

    
    Calculations are provided in Appendix.

    Let l∈{1,...,m} and y_l∈ℝ,  -∞ < y_l < ∞ ∀ l. Then, the variance with respect to the set of values y_l satisfies the property 
    
    
    σ_y^2 ≤1/4(max_l y_l - min_l y_l)^2.

    
    From  (<ref>) it follows that
    
    
    a|ψ^i_Δ p| + b|ψ^i_Δ v| ≤ (aγ_Δ p+bγ_Δ v)max_j=0,...,i|χ_j|.

    
    Let us define d = aγ_Δ p+bγ_Δ v > 0, α = α-k and Υ∈(0,1). Then
    
    
    Ẇ_i    ≤ -α|χ_i|^2 + d|χ_i|·max_j=0,...,i-1|χ_j| 
    
               ≤ -(1-Υ)α|χ_i|^2,   ∀  |χ_i|≥d/αΥ·max_j=0,...,i-1|χ_j|.

    
    With α > k, the condition in (<ref>) satisfies the ISS requirements. Consequently,  the inequality in (<ref>) is verified according to <cit.>. Moreover, 
    
    
    γ(s) = γs  ∀  s ≥ 0,    γ = √(α/α)d/αΥ > 0.

    



    The closed loop system described by (<ref>) and (<ref>) satisfying the condition in Lemma <ref> with γ∈(0,1), is String Stable.



    The proof is based on the forward recursive application of the ISS property in Lemma (<ref>)  through an inductive method.
    For i=0:
    
    
    |χ_0(t)|    ≤β(|χ_0(0)|,t),  ∀ t ≥ 0.

    
    For i=1:
    
    
    |χ_1(t)| ≤β(|χ_1(0)|,t)+γ|χ_0(·)|^[0,t]_∞,  ∀ t ≥ 0,

    
    where |χ_0(·)|^[0,t]_∞≤β(|χ_0(0)|,0). Defining |χ_M(0)| = max{ |χ_0(0)| , |χ_1(0)| }, then for both i=0 and i=1:
    
    
    |χ_0(t)|    ≤β(|χ_M(0)|,0),  ∀ t ≥ 0, 
    
            |χ_1(t)|    ≤β(|χ_M(0)|,0)(1+γ),  ∀ t ≥ 0.

    
    For i=2:
    
    
    |χ_2(t)|    ≤β(|χ_2(0)|,t) 
    
                  +γmax_j=0,1|χ_j(·)|^[0,t]_∞,  ∀ t ≥ 0.

    
    Defining |χ_M'(0)| = max_j=0,1,2{ |χ_j(0)| }, since γ > 0, then
    
    
    |χ_0(t)| ≤β(|χ_M'(0)|,0)(1+γ),  ∀ t ≥ 0, 
    
            |χ_1(t)| ≤β(|χ_M'(0)|,0)(1+γ),  ∀ t ≥ 0 ,

    
    and
    
    
    |χ_2(t)| ≤β(|χ_M'(0)|,0)(1+γ+γ^2),  ∀ t ≥ 0.

    
    By recursively applying these steps, and since γ∈(0,1) for hypothesis, for each i∈ℐ_N^0 we state:
    
    
    |χ_i(t)|    ≤β( max_j=0,...,i|χ_j(0)|,0 )∑_j=0^i γ^j 
    
               ≤β( max_j=0,...,i|χ_j(0)|,0 )∑_j=0^∞γ^j 
    
               ≤1/1-γβ( max_j=0,...,i|χ_j(0)|,0 ),  ∀ t ≥ 0.

    
    Then
    
    
    max_i∈ℐ_N^0|χ_i(t)| ≤1/1-γβ( max_i∈ℐ_N^0|χ_i(0)|,0 ),  ∀ t ≥ 0.

    
    Let us define ω(s) = β(s,0),  s ≥ 0. By definition of 𝒦ℒ functions, ω is 𝒦_∞ and hence invertible. Since (<ref>) holds for any t ≥ 0, then 
    
    
    δ = ω^-1((1-γ)ϵ),   ∀ ϵ≥ 0.

    
    The value of δ in (<ref>) does not depend on the system dimension. From (<ref>), (<ref>) and (<ref>), String Stability is ensured according to Definition <ref>.


Similarly to <cit.>, the result is obtained through an inductive method exploiting ISS. However, the main difference is that ISS is ensured among each leader-follower situation in <cit.>, while in our approach ISS is insured with respect to the leader-follower situation and the ahead vehicles of each leader, as shown by (<ref>). According to (<ref>), the proposed solution does
not depend on the number of vehicles along the platoon. 
We focus now on the possibility to ensure Asymptotic String Stability. 


    The closed loop system described by (<ref>) and (<ref>)  satisfying the String Stability condition in Theorem <ref> is Asymptotically String Stable.



    The proof is based on a composition of Lyapunov functions (see <cit.>). Let us consider the function W_i in (<ref>) that is associated with the i-th dynamical system, for i∈ℐ_N^0. It satisfies the inequalities in (<ref>) and (<ref>).
    black
    Given its time derivative in (<ref>), 
        by exploiting norm inequalities it follows that
        
        
    a|ψ_Δ p^i-1|+b|ψ_Δ v^i-1| ≤∑_j=0^i-1k_j|χ_j|.

        
        See calculations in Appendix for more details.
        Then
        
        
    Ẇ_i ≤ -α |χ_i|^2 + |χ_i| ∑_j=0^i k_j|χ_j| ,

        
        where 
        
        
    k_i   = k > 0, 
    
                k_j   = √(3/i)max{aγ_Δ p,bγ_Δ v} > 0 ,  j<i.

        
    
    
    
    Let χ̂ and χ̂_e be the extended lumped state of the platoon and the extended equilibrium point, respectively, that are defined similarly to (<ref>) and (<ref>). Let us consider the state χ = χ̂-χ̂_e and the parameters d_i>0. We define the composite function W_c(χ) as:
    
    
    W_c(χ) = ∑_i=0^N d_i W(χ_i) .

    
    It clearly verifies
    
    
    α_c|χ|^2 ≤ W_c(χ) ≤α_c|χ|^2

    
    where
    
    
    α_c = min_i∈ℐ_N^0{d_i}α,   α_c=max_i∈ℐ_N^0{d_i}α.

    
    The time derivative of W_c in (<ref>) satisfies the inequality
    
    
    Ẇ_c(χ) ≤∑_i=0^N d_i [ -α |χ_i|^2 + ∑_j=0^ik_j |χ_j||χ_i| ].

    
    black
        Let us introduce the operator  ϕ:ℝ^2N+1→ℝ^N+1, defined as
        
        
    ϕ(χ) = [ |χ_0|,|χ_1|,...,|χ_N| ]^T.

        
    
    
    
    
    
    
    
    Then, equation (<ref>) can be rewritten as
    
    
    Ẇ_c(χ) ≤ -1/2ϕ(χ)^T(DS+S^TD)ϕ(χ)

    
    where
    
    
    D =  diag(d_0,d_1,...,d_N)

    
    and S is an N× N matrix whose elements are
    
    
    s_ij = 
            α - k_i      i = j 
    
                -k_j      i < j 
    
                0       i > j

    
    For α = α-k_i > 0, each leading principal minor of S is positive and hence it is an M-matrix. By <cit.> there exists a matrix D such that DS+S^TD is positive definite. Consequently, Ẇ_c in (<ref>) is negative definite. 

    
    black
        It follows that W_c in (<ref>) is a Lyapunov function for the overall platoon system described by (<ref>) and (<ref>). Furthermore, there exists a 𝒦ℒ function β_c : ℝ^+×ℝ^+ →ℝ^+ such that
        
        
    |χ(t)| ≤β_c(|χ(0)|,t),   ∀ t ≥ 0.

         
        Condition (<ref>) ensures asymptotic stability. Consequently:
        
        
    lim_t→∞|χ_i(t)| = 0,   ∀ i∈ℐ_N^0.

        
        
        The platoon system is proved to be String Stable by Theorem <ref>. Consequently, for each i∈ℐ_N^0 the state evolution |χ_i| is constrained by a bound that is independent from the system dimension. Furthermore, from (<ref>) Asymptotic String Stability is ensured according to Definition <ref>.
        
        
        
        
        
        
        
        
    
    


Then, Asymptotic String Stability according to Definition <ref> is ensured by Theorem <ref> and Corollary <ref>.



















 §.§ Control strategy for variable spacing policy


We present a control law obtained by backstepping, see <cit.>), for implementing the variable spacing policy in (<ref>) with ρ^M_i = ρ_1,i. The controller associated to the i-th vehicle, ∀ i∈ℐ_N^0, is given by: 











    u_i    = u_i-1 - (Δ p_i + Δp + ρ_1,i) - λ_1(λ_1ρ_1,i - ρ_2,i) 
    
              + λ_2ρ_2,i - K_Δ p(Δ v_i - λ_1ρ_1,i +ρ_2,i) 
    
              - aψ^i-1_Δ p - bψ^i-1_Δ v - K_Δ v(Δ v_i - λ_1ρ_1,i + ρ_2,i
    
              +K_Δ p(Δ p_i + Δp + ρ_1,i)) 
    
           = u_i-1 - (Δ p_i - Δ p_i^r) - K_Δ v(Δ v_i - Δ v_i^r) 
    
              + (K_Δ p - λ_1)(λ_1ρ_1,i - ρ_2,i) +λ_2ρ_2,i
    
              -K_Δ pΔ v_i - aψ^i-1_Δ p - bψ^i-1_Δ v,


with


    Δ v_i^r = λ_1ρ_1,i-ρ_2,i-K_Δ p(Δ p_i-Δ p_i^r)


and given constant gains K_Δ p,K_Δ v>0 equal for each i∈ℐ_N^0, Δ p_i^r = -Δp-ρ_1,i, 










and ρ_1,i,ρ_2,i as defined in (<ref>). 


Note that the controller in (<ref>) considers both microscopic and macroscopic information, leading to a mesoscopic framework. 


To analyze the String Stability of the closed loop system, we consider the extended leader-follower state vector χ̂_i and its corresponding equilibrium point χ_e,i in (<ref>). 













The closed loop dynamics for each i∈ℐ_N^0 results to be:






















    χ̇̂̇_i =  [  
            [   Δṗ_i;  Δv̇_i; ρ̇_1,i; ρ̇_2,i ]] = [  
        [                               Δ v_i;                                 (*);                   -λ_1ρ_1,i + ρ_2,i; -λ_2ρ_2,i + aψ_Δ p^i-1 + bψ_Δ v^i-1 ]]


with 


    (*)    = - (Δ p_i - Δ p_i^r) - K_Δ v(Δ v_i - Δ v_i^r) 
    
              + (K_Δ p - λ_1)(λ_1ρ_1,i - ρ_2,i) +λ_2ρ_2,i-1
    
              -K_Δ pΔ v_i - aψ^i-1_Δ p - bψ^i-1_Δ v.


We remark that ψ^-1_Δ p = ψ^-1_Δ v = 0. Then, we can rewrite the system in 
(<ref>) as (<ref>) and (<ref>), 

where g_cl,i(χ̂_i-1,...,χ̂_0) is 


    g_cl,i(χ̂_i-1,χ̂_i-2,...,χ̂_0) = [ 
            [                          0; -(aψ_Δ p^i-1 + bψ_Δ v^i-1);                          0;    aψ_Δ p^i-1 + bψ_Δ v^i-1 ]].





 §.§ String Stability analysis

 




Set χ_i = χ̂_i - χ̂_e,i. Then, the following result holds:


    Consider the closed loop system described by 
    (<ref>). Then, there exist functions β of class 𝒦ℒ and γ of class 𝒦_∞ such that, if K_Δ p, K_Δ v, λ_1, λ_2 > 0
    
    
    
    
    
    then,
    
    
    |χ_i(t)| ≤β( |χ_i(0)| , t ) + γ( max_j=0,...,i-1 |χ_j(·)|_∞^[0,t])

    
    ∀ t≥ 0, and γ(s) = γs, s≥ 0, γ∈ℝ^+. Moreover, there exist 
    
     a and b in (<ref>) such that γ∈ (0,1). 



    See Appendix <ref> in <cit.>.




    
    Let us consider the candidate Lyapunov function (see <cit.>) W_i = W(χ_i) for the i-th dynamical system χ_i, for i∈ℐ_N^0:
        
        
    W(χ_i)    = 1/2(Δ p_i - Δ p_i^r)^2 + 1/2(Δ v_i - Δ v_i^r)^2 
    
                      + 1/2ρ_1,i-1^2 + 1/2ρ_2,i-1^2.

        
        
        
        
        
        
        
        
        
        
        It satisfies 
        
        
    α|χ_i|^2 ≤ W(χ_i) ≤α|χ_i|^2,

        
        where
        
        
    α = 1/2,   α  = 1/2max{ 1+K_Δ p^2 , 2+(λ_1-K_Δ p)^2 },
 (see Appendix <ref> in <cit.> for more details). 
        Then, the time derivative of W_i in (<ref>) verifies:
        
        
    Ẇ_i    = -K_Δ p(Δ p_i - Δ p_i^r)^2 -K_Δ v(Δ v_i - Δ v_i^r)^2  
    
                      -λ_1ρ^2_1,i-1 - λ_2ρ^2_2,i-1 + ρ_1,i-1ρ_2,i-1
    
                       + ρ_2,i-1(aψ^i-1_Δ p + bψ^i-1_Δ v) 
    
                   ≤ -α|χ_i|^2 + |χ_i|(a|ψ^i-1_Δ p| + b|ψ^i-1_Δ v|)

        
        where 
        
        
    α   = min{ c_1 , c_2 , c_3 , c_4 },

        
        c_1 = K_Δ p(1+K_Δ pK_Δ v),  c_2 = K_Δ v,  c_3 = K_Δ p+λ_1+K_Δ v(λ_1-K_Δ p)^2,  c_4 = λ_2+K_Δ v. See Appendix <ref> in <cit.> for more details on the calculations in (<ref>). 
        
       
        Let l∈{1,...,m} and y_l∈ℝ. Then, the variance with respect to the set of values y_l satisfies the property 
        
        
    σ_y^2 ≤1/4(max_l y_l - min_l y_l)^2.

        
        From  (<ref>) it follows that
        
        
    a|ψ^i_Δ p| + b|ψ^i_Δ v| ≤ (aγ_Δ p+bγ_Δ v)max_j=0,...,i|χ_j|.

        
        See Appendix <ref> in <cit.> for more details on the calculations in (<ref>). Define 
        
        
    d = aγ_Δ p+bγ_Δ v > 0,    Υ∈(0,1).

        
        Then
        
        
        
        
    Ẇ_i    ≤ -α|χ_i|^2 + d|χ_i| max_j=0,...,i-1|χ_j| + Υα|χ̃_i|^2 - Υα|χ̃_i|^2 
    
                   ≤ -(1-Υ)α|χ_i|^2,   ∀ |χ_i| ≥d/αΥmax_j=0,...,i-1|χ_j|.

        
        Since α > 0, the inequality in (<ref>) satisfies the Input-to-State Stability (ISS) condition (see <cit.>). According to <cit.>, the inequality in (<ref>) is verified. Moreover, 
        
        
    γ(s) = γs   ∀ s ≥ 0,  γ = √(α/α)d/αΥ > 0.

        
        Since the parameters a,b ≥ 0 in the dynamics of ρ_i in (<ref>) can be arbitrarily selected, the constant d defined in (<ref>) can be chosen such that γ in (<ref>) belongs to (0,1).
        
        

            □
        

        
        
        
    
    





On the basis of Lemma <ref>, Asymptotic String Stability of the platoon can be obtained by an appropriate choice of the parameters in (<ref>), 

as shown in the following:


    
    
    The closed loop system described by 
    (<ref>) where the parameters K_Δ p, K_Δ v, λ_1, λ_2 > 0 and parameters a,b are such that γ̃∈ (0,1), is Asymptotically String Stable.



    See Appendix <ref> in <cit.>.




    
     The first part of the proof is based on the forward recursive application of the ISS property in Lemma (<ref>) through an inductive method.
        
        For i=0:
        
        
    |χ_0(t)|    ≤β(|χ_0(0)|,t),  ∀ t ≥ 0.

        
        For i=1:
        
        
    |χ_1(t)| ≤β(|χ_1(0)|,t)+γ|χ_0(·)|^[0,t]_∞,  ∀ t ≥ 0,

        
        where |χ_0(·)|^[0,t]_∞≤β(|χ_0(0)|,0). Defining |χ_M(0)| = max{ |χ_0(0)| , |χ_1(0)| }, then for both i=0 and i=1:
        
        
    |χ_0(t)|    ≤β(|χ_M(0)|,0),  ∀ t ≥ 0, 
    
                |χ_1(t)|    ≤β(|χ_M(0)|,0)(1+γ),  ∀ t  ≥ 0.

        
        For i=2:
        
        
    |χ_2(t)|    ≤β(|χ_2(0)|,t) 
    
                      +γmax_j=0,1|χ_j(·)|^[0,t]_∞,  ∀ t ≥ 0.

        
        Defining |χ_M'(0)| = max_j=0,1,2{ |χ_j(0)| }, since γ > 0, then
        
        
    |χ_0(t)| ≤β(|χ_M'(0)|,0)(1+γ),  ∀ t ≥ 0, 
    
                |χ_1(t)| ≤β(|χ_M'(0)|,0)(1+γ),  ∀ t ≥ 0 ,

        
        and
        
        
    |χ_2(t)| ≤β(|χ_M'(0)|,0)(1+γ+γ^2),  ∀ t ≥ 0.

        
        By recursively applying these steps, and since γ∈(0,1) for hypothesis, for each i∈ℐ_N^0 we state:
        
        
    |χ_i(t)|    ≤β( max_j=0,...,i|χ_j(0)|,0 )∑_j=0^i γ^j 
    
                   ≤β( max_j=0,...,i|χ_j(0)|,0 )∑_j=0^∞γ^j 
    
                   ≤1/1-γβ( max_j=0,...,i|χ_j(0)|,0 ),  ∀ t ≥ 0.

        
        Then
        
        
    max_i∈ℐ_N^0|χ_i(t)| ≤1/1-γβ( max_i∈ℐ_N^0|χ_i(0)|,0 ),  ∀ t ≥ 0.

        
        Define ω(s) = β(s,0),  s ≥ 0. By definition of 𝒦ℒ functions, ω is 𝒦_∞ and hence invertible. Since (<ref>) holds for any t ≥ 0, then 
        
        
    δ = ω^-1((1-γ)ϵ),   ∀ ϵ≥ 0.

        
        The value of δ in (<ref>) does not depend on the system dimension. From (<ref>), (<ref>) and (<ref>), String Stability is ensured according to Definition <ref>.
    
    
    
    We focus now on the possibility to ensure Asymptotic String Stability.
    
    
    
    
    
    
        This second part of the proof is based on a composition of Lyapunov functions (see <cit.>). We consider the function W_i associated with the i-th dynamical system, for i∈ℐ_N^0, that is described in (<ref>) and  satisfies the condition in (<ref>).
        
       Let us consider the time derivative of W_i in (<ref>).
        black
            By exploiting norm inequalities, it follows 
            
            
    a|ψ_Δ p^i-1|+b|ψ_Δ v^i-1| ≤∑_j=0^i-1k_j|χ_j|.

            
            See Appendix <ref> in <cit.> for more details on the calculations in (<ref>). Then
            
            
    Ẇ_i ≤ -α |χ_i|^2 + |χ_i| ∑_j=0^i-1k_j|χ_j| ,

            
            where 
            
            
    k_j   = √(3/i)max{aγ_Δ p,bγ_Δ v} > 0,  j<i.

            
        
        
        
        Let χ̂ and χ̂_e be the extended lumped state of the platoon and the extended equilibrium point respectively, defined in a similar way as (<ref>) and (<ref>). Let us consider χ = χ̂-χ̂_e and the parameters d_i>0 to define a composite function W_c(χ):
        
        
    W_c(χ) = ∑_i=0^N d_i W(χ_i) .

        
        It clearly verifies
        
        
    α_c|χ|^2 ≤ W_c(χ) ≤α_c|χ|^2

        
        where
        
        
    α_c = min_i∈ℐ_N^0{d_i}α,   α_c=max_i∈ℐ_N^0{d_i}α.

        The time derivative of W_c in (<ref>) satisfies the inequality
        
        
    Ẇ_c(χ) ≤∑_i=0^N d_i [ -α |χ_i|^2 + ∑_j=0^i-1k_j |χ_j||χ_i| ].

        
        black
            Let us introduce the operator  ϕ:ℝ^2N+1→ℝ^N+1, defined as
            
            
    ϕ(χ) = [ |χ_0|,|χ_1|,...,|χ_N| ]^T.

            
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        
        Then, equation (<ref>) can be rewritten as
        
        
    Ẇ_c(χ) ≤ -1/2ϕ(χ)^T(DS+S^TD)ϕ(χ)

        
        where
        
        
    D =  diag(d_0,d_1,...,d_N)

        
        and S is an N× N matrix whose elements are
        
        
    s_ij = 
                α      i = j 
    
                    -k_j      i < j 
    
                    0       i > j

        
        For α > 0, each leading principal minor of S is positive and hence it is an M-matrix. By <cit.> there exists a matrix D such that DS+S^TD is positive definite. Consequently, Ẇ_c in (<ref>) is negative definite. 
        
        black
            It follows that W_c in (<ref>) is a Lyapunov function for the overall platoon system described by (<ref>) and (<ref>). Therefor, there exists a 𝒦ℒ function β_c : ℝ^+×ℝ^+ →ℝ^+ such that
            
            
    |χ(t)| ≤β_c(|χ(0)|,t),   ∀ t ≥ 0.

            
            Condition in  (<ref>) ensures the asymptotic stability:
            
            
    lim_t→∞|χ_i(t)| = 0,   ∀ i∈ℐ_N^0.

            
            The platoon system is proved to be String Stable by  (<ref>) and (<ref>). Consequently, for each i∈ℐ_N^0 the state evolution |χ_i| is constrained by a bound that is independent from the system dimension. Furthermore, from (<ref>) Asymptotic String Stability is ensured according to Definition <ref>.
            
            
            
            
            
            
            
            
            

                □
            

            
        
    









§ SIMULATIONS


The introduced control strategy is simulated in Matlab&Simulink. Based on the modeling in (<ref>), we consider a platoon of N+1=11 vehicles. 
The initial conditions for each vehicle are randomly generated in a neighborhood of the equilibrium point. It results μ_Δ p≠-Δp,μ_Δ v≠ 0 σ^2_Δ p,σ^2_Δ v≠ 0. The reference distance is Δp = 10m and the initial desired speed of the leading vehicle is v = 14m/s. Vehicle speed is  0 < v_i≤ 36 [m/s] and the acceleration is bounded by -4≤ u_i≤ 4 [m/s^2]. The control parameters are introduced in Table <ref>, with a resulting γ=0.5. To better stress the advantages of the proposed controller, we analyze the behavior of the system when a disturbance acts on the acceleration of vehicle i=0, and it is not communicated to vehicle i=1. 








The simulation time is 1 minute. We split it into three phases:

























    
  * From t=t_0=0s to t=t_1=10s: the vehicles start with initial conditions that are different from the desired speed and the desired distance. No disturbance is acting on the leader vehicle, and its desired speed is the initial one, i.e. v = 14m/s.
    
    
  * From t=t_1=10s to t=t_2=30s: a disturbance acts on the acceleration of the first vehicle i=0. At t_1 a positive pulse of amplitude 4m/s^2 and length 5s is considered, while a similar pulse with negative amplitude is considered at t=15s. The control input of i=0 being saturated, i=0 succeeds to properly counteract to it but it is not able to operate the needed corrective action to return to the desired speed. Since the disturbance is an external input, it is not communicated to the follower and can propagate along the platoon.
    
    
  * From t=t_2=30s to t=t_3=60s: the leader tracks a variable speed reference. From t=30s to t=45s the desired speed is v=30m/s, while from t=45s to t=60s it is v=20m/s. 
    











 §.§ Control strategy for constant spacing policy


Figures <ref>, <ref> and <ref> show, respectively, the inter-vehicular distance, speed and acceleration profiles for each vehicle the platoon when the control input (<ref>) is implemented. In this first phase, the vehicles are shown to converge to the desired speed and the desired distance in short time. 

In the second phase, the controller of i=1 does not know the correct value of u_0. Also, the macroscopic variable is not available to it: for these reasons, it does not succeed to perfectly track the desired distance neither in case of positive disturbance between t=10s to  t=15s nor in case of negative one between t=15s to  t=20s (see Figure <ref>). However, it converges to the same speed of i=0 after a small transient of three seconds in both cases (see Figure <ref>). Finally, at t=20s the disturbance is not active anymore and the leader can restore its desired speed. Also, i=1 receives correct information about its leader acceleration and is able to return to the ideal distance. The dynamical evolution of the remaining vehicles in the platoon during the generated transients after t=10s, t=15s and t=20s catch the contribution of the macroscopic information. To this purpose, let us consider the speed dynamics of the last vehicle in Figure <ref>. It is possible to remark an anticipatory behaviour due to the macroscopic information resulting in a higher speed between t=10s and t=11s with respect to the leading vehicles. In this increasing speed phase, the vehicles along the platoon scale to intensify their accelerations and speeds. Moreover, the same anticipatory behaviour is shown when the leading vehicles are converging to the same speed. Here, the decreasing of the vehicles' speed along the platoon scales with respect to their position along the platoon, resulting more stressed in the last vehicles (see between t=12s and t=13s). The same anticipatory behaviour is shown in Figure <ref> with respect to the accelerations of the leading vehicles. The same applies for transients taking place after t=15s and t=20s, which are generated by the fast reaction of the leading vehicle to the disturbance. 
 
In the third phase, since there is no unknown perturbation acting on the platoon, the vehicles succeed to track the variable speed profile and to remain at the desired distance. no oscillations are shown by the proposed control law, even if the desired speed profile has high steps.
 











   
    
   
    
   














 

 







Figures <ref>, <ref> and <ref> show, respectively, the inter-vehicular distance, speed and acceleration profiles for each vehicle of the platoon when the control input in (<ref>) is implemented. In the first phase, the vehicles are shown to quickly converge to the desired speed and the desired distance. 



In the second phase, the controller of i=1 does not know the correct acceleration value of vehicle i=0. 
Also, the macroscopic variable is not available to it: for these reasons, it does not succeed to perfectly track the desired distance neither in case of positive disturbance between t=10s to  t=15s nor in case of negative one between t=15s to  t=20s (see Figure <ref>). However, it converges to the same speed of i=0 after a small transient of three seconds in both cases (see Figure <ref>). Finally, at t=20s the disturbance is not active anymore and the leader can restore its desired speed. Also, i=1 receives correct information about its leader acceleration and is able to return to the ideal distance. The dynamical evolution of the remaining vehicles in the platoon during the generated transients after t=10s, t=15s and t=20s catches the contribution of the macroscopic information. To this purpose, let us consider the speed dynamics of the last vehicle in Figure <ref>. It is possible to remark an anticipatory behaviour due to the macroscopic information resulting in a higher speed between t=10s and t=12s with respect to the leading vehicles
. Then,  the decreasing of vehicles' speed along the platoon scales with respect to their position, resulting more stressed in the last vehicles (see between t=12s and t=14s). The same anticipatory behaviour is shown in Figure <ref> with respect to the accelerations of the leading vehicles. The acceleration profiles better show how the vehicles along the platoon scale to intensify their accelerations and speeds, both for increasing and decreasing speed phases. An anticipatory behaviour is shown, both when the leading vehicles are accelerating and converging to the same speed. 
The same applies for transients taking place after t=15s and t=20s, which are generated by the fast reaction of the leading vehicle to the disturbance.  
An anticipatory harmonizing acceleration for each vehicle scales along the platoon with respect to their knowledge of the macroscopic quantities, as shown in Figure <ref> and <ref>. 

In the third phase, since there is no unknown perturbation acting on the platoon, the vehicles succeed to track the variable speed profile and to remain at the desired distance. No oscillations are shown by the proposed control law, even if the desired speed profile has high steps.

















    














The proposed control laws in (<ref>) exploit the information resulting from the macroscopic variable and safely control a platoon of vehicles. The control inputs provide transient harmonization on the whole traffic throughput while ensuring Asymptotic String Stability properties. The dynamical evolution results in a reduction of the oscillations propagation along the platoon, both in nominal case and in the presence of an active external disturbance. The utilization of the macroscopic information results to be a powerful tool. 








§ CONCLUSIONS

This paper introduces macroscopic variables for ensuring the String Stability of a platoon of CACC autonomous vehicles. As the variance of microscopic quantities is related to the macroscopic density, the proposed stability analysis opens to the possibility of properly controlling a platoon by propagating only macroscopic density information. 

A control law based on information obtained by V2V communication has been proposed. The improvements resulting from taking into account macroscopic information are shown by simulation results. The proposed mesoscopic control law produces an anticipatory behaviour, which provides a better transient harmonization. Future work will focus on 

extending the proposed framework in a mixed traffic situation with non-communicating vehicles. Also, due to the satisfactory results,  the next objective is to extend our approach to more complex models including non-idealities such as actuation and communication delays.



















ieeetr
  






§ APPENDICES









 §.§ Proof of Lemma <ref>


Let us consider the candidate Lyapunov function (see <cit.>) W_i = W(χ_i) for the i-th dynamical system χ_i, for i∈ℐ_N^0:


    W(χ_i)    = 1/2(Δ p_i - Δ p_i^r)^2 + 1/2(Δ v_i - Δ v_i^r)^2 
    
              + 1/2ρ_1,i^2 + 1/2ρ_2,i^2.











It can be proven that (<ref>) is bounded by two quadratic functions as follows


    α|χ_i|^2 ≤ W(χ_i) ≤α|χ_i|^2.


To derive the constants α > 0 and α > 0 we replace the expressions of Δ p_i^r defined in (<ref>) and of Δ v_i^r in (<ref>) in the definition of W_i in (<ref>). Then, 


    W(χ_i) = 1/2(Δ p_i + Δp + ρ_1,i)^2 
    
            + 1/2(Δ v_i - λ_1ρ_1,i + ρ_2,i + K_Δ p(Δ p_i + Δp + ρ_1,i))^2 
    
            + 1/2ρ^2_1,i + 1/2ρ^2_2,i
    
            = 1/2χ_i^T [ 
        [       1+K^2_Δ p             p_1             p_2             p_1;               0               1             p_3               2;               0               0 2+(λ_1-K_Δ p)^2             p_3;               0               0               0               2 ]]_P_Wχ_i


where


    p_1 = 2K_Δ p,  p_2 = 2(1+K^2_Δ p-λ_1 K_Δ p),
        
        
       p_3 = 2(K_Δ p-λ_1).


By defining λ_min(P),λ_max(P) respectively the minimum and maximum eigenvalues of the generic matrix P, then,


    1/2λ_min(P_W)|χ_i|^2 ≤ W(χ_i) ≤1/2λ_max(P_W)|χ_i|^2.


Since K_Δ p,λ_1,λ_2 > 0, then 


    λ_min(P_W) = 1,  λ_max(P_W) = max{ 1+K^2_Δ p , 2+(λ_1-K_Δ p)^2 }.


From (<ref>) and (<ref>) it follows


    α = 1/2,   α  = 1/2max{ 1+K_Δ p^2 , 2+(λ_1-K_Δ p)^2 }.
 



The time derivative of W_i in (<ref>) verifies:


    Ẇ_i    = -K_Δ p(Δ p_i - Δ p_i^r)^2 -K_Δ v(Δ v_i - Δ v_i^r)^2  
    
              -λ_1ρ^2_1,i - λ_2ρ^2_2,i + ρ_1,iρ_2,i
    
               + ρ_2,i(aψ^i-1_Δ p + bψ^i-1_Δ v) 
    
            = - χ_i^T [ 
        [         q_1 2K_Δ pK_Δ v         q_2 2K_Δ pK_Δ v;           0       K_Δ v         q_3      2K_Δ v;           0           0         q_4         q_5;           0           0           0   λ_2+K_Δ v ]]_Q_Wχ_i 
    
            + |χ_i|(a|ψ^i-1_Δ p| + b|ψ^i-1_Δ v|) 
    
           ≤ -α|χ_i|^2 + |χ_i|(a|ψ^i-1_Δ p| + b|ψ^i-1_Δ v|)


where 


    α   = λ_min(Q_W) = min{ q_1 , K_Δ v , q_4 , λ_2+K_Δ v},


with


    q_1 = K_Δ p(1+K_Δ pK_Δ v), 
    
            q_2 = 2K_Δ p(1+K_Δ v(K_Δ p-λ_1)), 
    
            q_3 = 2K_Δ v(K_Δ p-λ_1), 
    
            q_4 = K_Δ p+λ_1+K_Δ v(λ_1-K_Δ p)^2, 
    
            q_5 = 1-2K_Δ v(K_Δ p-λ_1).





We proceed to prove some inequalities with respect to functions ψ_Δ p^i and ψ^i_Δ v by exploiting the variance property given below. Let l∈{1,...,m} and y_l∈ℝ. Then, the variance with respect to the set of values y_l satisfies the property 


    σ_y^2 ≤1/4(max_l y_l - min_l y_l)^2.


Since we consider the dynamics in (<ref>) and (<ref>) with respect to χ_i = χ̂_i-χ̂_e,i, let us define Δp_i = Δ p_i -  Δp, then


    |ψ^i_Δ p|    ≤γ_Δ p√(σ^2_Δ p)
    
           ≤1/2γ_Δ p|max_j=0,...,iΔp_j - min_j=0,...,iΔp_j| 
    
           ≤γ_Δ pmax_j=0,...,i|Δp_j| 
    
        
        
           ≤γ_Δ pmax_j=0,...,i|χ_j|


where we have exploited the relationship


    |max_j Δp_j|    ≤max_j |Δp_j| 
    
        |min_j Δp_j|    ≤max_j |Δp_j|


By applying the same methodology, is proven the inequality


    |ψ^i_Δ v| ≤γ_Δ vmax_j=0,...,i|χ_j|


Then,


    a|ψ^i_Δ p| + b|ψ^i_Δ v| ≤ (aγ_Δ p+bγ_Δ v)max_j=0,...,i|χ_j|.




Define 


    d = aγ_Δ p+bγ_Δ v > 0,    Υ∈(0,1),


then for the time derivative of W_i in (<ref>) the following holds




    Ẇ_i    ≤ -α|χ_i|^2 + d|χ_i| max_j=0,...,i-1|χ_j| + Υα|χ̃_i|^2 - Υα|χ̃_i|^2 
    
           ≤ -(1-Υ)α|χ_i|^2,   ∀ |χ_i| ≥d/αΥmax_j=0,...,i-1|χ_j|.


Since α > 0, the inequality in (<ref>) satisfies the Input-to-State Stability (ISS) condition (see <cit.>). According to <cit.>, the inequality in (<ref>) is verified. Moreover, 


    γ(s) = γs   ∀ s ≥ 0,  γ = √(α/α)d/αΥ > 0.


Since the parameters a,b ≥ 0 in the dynamics of ρ_i in (<ref>) can be arbitrarily selected, the constant d defined in (<ref>) can be chosen such that γ in (<ref>) belongs to (0,1).



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 §.§ Proof of Theorem <ref>


The first part of the proof is based on the forward recursive application of the ISS property in Lemma <ref> through an inductive method.

For i=0:


    |χ_0(t)|    ≤β(|χ_0(0)|,t),  ∀ t ≥ 0.


For i=1:


    |χ_1(t)| ≤β(|χ_1(0)|,t)+γ|χ_0(·)|^[0,t]_∞,  ∀ t ≥ 0,


where |χ_0(·)|^[0,t]_∞≤β(|χ_0(0)|,0). Defining |χ_M(0)| = max{ |χ_0(0)| , |χ_1(0)| }, then for both i=0 and i=1:


    |χ_0(t)|    ≤β(|χ_M(0)|,0),  ∀ t ≥ 0, 
    
        |χ_1(t)|    ≤β(|χ_M(0)|,0)(1+γ),  ∀ t  ≥ 0.


For i=2:


    |χ_2(t)|    ≤β(|χ_2(0)|,t) 
    
              +γmax_j=0,1|χ_j(·)|^[0,t]_∞,  ∀ t ≥ 0.


Defining |χ_M'(0)| = max_j=0,1,2{ |χ_j(0)| }, since γ > 0, then


    |χ_0(t)| ≤β(|χ_M'(0)|,0)(1+γ),  ∀ t ≥ 0, 
    
        |χ_1(t)| ≤β(|χ_M'(0)|,0)(1+γ),  ∀ t ≥ 0 ,


and


    |χ_2(t)| ≤β(|χ_M'(0)|,0)(1+γ+γ^2),  ∀ t ≥ 0.


By recursively applying these steps, and since γ∈(0,1) for hypothesis, for each i∈ℐ_N^0 we state:


    |χ_i(t)|    ≤β( max_j=0,...,i|χ_j(0)|,0 )∑_j=0^i γ^j 
    
           ≤β( max_j=0,...,i|χ_j(0)|,0 )∑_j=0^∞γ^j 
    
           ≤1/1-γβ( max_j=0,...,i|χ_j(0)|,0 ),  ∀ t ≥ 0.


Then


    max_i∈ℐ_N^0|χ_i(t)| ≤1/1-γβ( max_i∈ℐ_N^0|χ_i(0)|,0 ),  ∀ t ≥ 0.


Define ω(s) = β(s,0),  s ≥ 0. By definition of 𝒦ℒ functions, ω is 𝒦_∞ and hence invertible. Since (<ref>) holds for any t ≥ 0, then 


    δ = ω^-1((1-γ)ϵ),   ∀ ϵ≥ 0.


The value of δ in (<ref>) does not depend on the system dimension. From (<ref>), (<ref>) and (<ref>), String Stability is ensured according to Definition <ref>.



We focus now on the possibility to ensure Asymptotic String Stability.






This second part of the proof is based on a composition of Lyapunov functions (see <cit.>). We consider the function W_i associated with the i-th dynamical system, for i∈ℐ_N^0, that is described in (<ref>) and  satisfies the condition in (<ref>).

Let us consider the time derivative of W_i in (<ref>).

black
    Since we consider the dynamics in (<ref>) and (<ref>) with respect to χ_i = χ̂_i-χ_e,i, we define Δp_i = Δ p_i -  Δp, then for the macroscopic functions ψ^i_Δ p and ψ^i_Δ v, the following inequalities are proved:
    
    
    |ψ^i_Δ p|    ≤γ_Δ p√(σ^2_Δ p,i)
    
               = γ_Δ p( 1/i+1∑_j=0^iΔp_j^2 - 1/(i+1)^2( ∑_j=0^iΔp_j )^2  )^1/2
    
               ≤γ_Δ p1/√(i+1)( ∑_j=0^iΔp_j^2 )^1/2
    
               ≤γ_Δ p1/√(i+1)∑_j=0^i|Δp_j|

    
    where we have used the inequality |x|_2 ≤ |x|_1. In the same way we can prove that 
    
    
    |ψ^i_Δ v| ≤γ_Δ v1/√(i+1)∑_j=0^i |Δ v_j|

    
    Then,
    
    
    a|ψ^i_Δ p|    + b|ψ^i_Δ v| ≤1/√(i+1)( aγ_Δ p∑_j=0^i|Δp_j| + bγ_Δ v∑_j=0^i|Δ v_j| ) 
    
               = 1/√(i+1)∑_j=0^i | [
            [ aγ_Δ p      0      0      0;      0 bγ_Δ v      0      0;      0      0      0      0;      0      0      0      0 ]] 
            
            
            
            
            
            
            
            χ_j
            |_1 
    
               ≤2/√(i+1)max{aγ_Δ p,bγ_Δ v}∑_j=0^i |χ_j|

    
    
    
    
    
    
    
    
    
    
    



Let χ̂ and χ̂_e be the extended lumped state of the platoon and the extended equilibrium point respectively, defined in a similar way as (<ref>) and (<ref>). Let us consider χ = χ̂-χ̂_e and the parameters d_i>0 to define a composite function W_c(χ):


    W_c(χ) = ∑_i=0^N d_i W(χ_i) .


It clearly verifies


    α_c|χ|^2 ≤ W_c(χ) ≤α_c|χ|^2


where


    α_c = min_i∈ℐ_N^0{d_i}α,   α_c=max_i∈ℐ_N^0{d_i}α.

The time derivative of W_c in (<ref>) satisfies the inequality


    Ẇ_c(χ) ≤∑_i=0^N d_i [ -α |χ_i|^2 + ∑_j=0^i-1k_i |χ_j||χ_i| ].


where 


    k_0   = 0 , 
    
        k_i   = 2/√(i)max{aγ_Δ p,bγ_Δ v} > 0,  i ≥ 1.


black
    Let us introduce the operator  ϕ:ℝ^2N+1→ℝ^N+1, defined as
    
    
    ϕ(χ) = [ |χ_0|,|χ_1|,...,|χ_N| ]^T.

    















Then, equation (<ref>) can be rewritten as


    Ẇ_c(χ) ≤ -1/2ϕ(χ)^T(DS+S^TD)ϕ(χ)


where


    D =  diag(d_0,d_1,...,d_N)


and S is an N× N matrix whose elements are


    s_ij = 
        α      i = j 
    
            -k_i      i < j 
    
            0       i > j


For α > 0, each leading principal minor of S is positive and hence it is an M-matrix. By <cit.> there exists a matrix D such that DS+S^TD is positive definite. Consequently, Ẇ_c in (<ref>) is negative definite. 

black
    It follows that W_c in (<ref>) is a Lyapunov function for the overall platoon system described by 
    (<ref>). Therefor, there exists a 𝒦ℒ function β_c : ℝ^+×ℝ^+ →ℝ^+ such that
    
    
    |χ(t)| ≤β_c(|χ(0)|,t),   ∀ t ≥ 0.

    
    Condition in  (<ref>) ensures the asymptotic stability:
    
    
    lim_t→∞|χ_i(t)| = 0,   ∀ i∈ℐ_N^0.

    
    The platoon system is proved to be String Stable by  (<ref>) and (<ref>). Consequently, for each i∈ℐ_N^0 the state evolution |χ_i| is constrained by a bound that is independent from the system dimension. Furthermore, from (<ref>) Asymptotic String Stability is ensured according to Definition <ref>.
    
    
    
    
    
    
    
    
    
    
    
    






























































