# Mean square stability conditions for platoons with lossy inter-vehicle
communication channels111This work was supported by the Chilean National
Agency for Research and Development (ANID) through the scholarship program
“Doctorado Nacional/2020-21202404” and Fondecyt Iniciación Project 11221365.
The material in this paper was not presented at any conference. Corresponding
author F.J. Vargas.

Marco A. Gordon<EMAIL_ADDRESS>Francisco J. Vargas
<EMAIL_ADDRESS>Andrés A. Peters<EMAIL_ADDRESS>Electronic
Engineering Department, Universidad Técnica Federico Santa María, 2390123,
Valparaíso, Chile  Faculty of Engineering and Sciences, Universidad Adolfo
Ibáñez, Santiago 7941169, Chile

###### Abstract

This paper studies the mean-square stability of heterogeneous LTI vehicular
platoons with inter-vehicle communication channels affected by random data
loss. We consider a discrete-time platoon system with predecessor following
topology and constant time-headway spacing policy. Lossy channels are modeled
by Bernoulli processes allowed to be correlated in space. We make use of a
class of compensation strategies to reduce the effect of data loss. Necessary
and sufficient conditions are derived to guarantee the convergence of the mean
and variance of the tracking errors, which depend not only on the controller
design but also on the compensation strategy and the probabilities of
successful transmission. Through numerical simulations, we illustrate the
theoretical results, describing different platoon behaviors. We also provide
insights on the mean-square stability as a necessary condition for string
stability in this stochastic setting.

###### keywords:

Mean square stability; vehicle platoons; lossy channels.

##  1 Introduction

Vehicular platooning, where a set of vehicles travel at a cruise velocity,
while maintaining a desired inter-vehicle distance, is a research topic with
potential for achieving increased safety and better performance in roads [34,
10, 30]. A simple case considers that each vehicle follows its immediate
predecessor. An independent leader, usually following an arbitrary trajectory
of constant speed, is located at the front forming a string. The leading agent
then determines the global trajectory and speed for the collection of
vehicles. When the platoon is assumed to be in a networked environment, where
agents exchange information through wireless channels, the study is extended
to the field of Cooperative Adaptive Cruise Control (CACC) [17, 38]. CACC may
be able to better enhance the safety and efficiency of road networks [41, 33].
Platoons possess several characteristics. For instance, the dynamical model of
the vehicles can be assumed to be linear or non-linear, or defined in
continuous or discrete-time [16, 39, 14]. If all the vehicles have the same
dynamic model, the platoon is called homogeneous [23, 3, 16]; otherwise, it is
called heterogeneous [17]. The desired inter-vehicle distance can be constant,
or variable. The employment of a constant time-headway spacing policy
increases the inter-vehicle spacing as the speeds of the vehicles increase
[42, 24, 25, 22]. The information flow topology of a platoon specifies how the
information is exchanged from vehicle to vehicle. A commonly used topology is
predecessor following (PF), where the communication is unidirectional and
occurs only between two consecutive vehicles [10]. Most of the literature
considers that the vehicle-to-vehicle communication channels are not affected
by random issues. However, in reality, wireless channels are subject to
problems such as random delays, noise, random data loss, etc. In such cases,
the platoon becomes stochastic due to the nature of the communication
channels. In this paper, we focus on platooning with lossy inter-vehicle
communication channels.

The performance of a platoon is analyzed through the tracking error that
measures both platoon tasks, position tracking, and maintaining the inter-
vehicle distance under constant speeds in steady state. The platoon tracking
errors then must converge in time as a primary objective. However, due to the
concatenation of vehicles, it is also important to ensure that disturbances do
not amplify along the string. This property is known as string stability and
guarantees the convergence and scalability of the platoon [29, 20, 13]. For
stochastic platoon setups, the string stability analysis is currently
incipient [4], and presents a lack for a consistent formal definition of
stochastic string stability. However, some works present an approach based on
the statistics of the signal of interest (see for instance [37, 25, 1, 35, 8,
7]). It is natural then, as an initial step, to focus on studying the
convergence properties of the platoon statistics. In particular, the analysis
of the convergence in time can be done through the notion of mean square
stability (MSS) [9].

MSS has been widely studied in the field of Networked Control Systems (NCS)
for several types of communication constraints, such as channels subject to
signal to noise ratio constraints, random delays, multiplicative noise and
data dropouts, among others [36, 6, 27, 15, 19]. In multi-agent systems (MAS)
applications, MSS has also been studied, mostly associated with the notion of
mean-square consensus [5, 40, 43]. In the field of vehicular platooning, MSS
literature is less extensive. In [25], platoons with stochastic delays induced
by data dropouts are studied. Sufficient conditions for stability are obtained
by analyzing the mean and covariance dynamics. The authors of [32] obtain
sufficient conditions for mean-square consensus based on LMIs with a topology
more complex than PF but considering some specific type of homogeneous
platoons. Platoons with communication affected by additive noise are
considered in [7], where necessary and sufficient conditions were obtained for
mean square convergence. In [42], topologies where the information of multiple
predecessors are received by a follower are considered, and sufficient
conditions for MSS are provided for heterogeneous time-independent random
packet loss. Platoons with communication delays and packet dropouts are
considered in [3], where sufficient conditions for mean square consensus
stability are derived. In [1], the authors propose a controller designed to
guarantee the convergence of the expected trajectory and minimize its variance
in a platoon with lossy channels.

In a platooning setting, when dealing with random packet loss in a
communication channel, the local controller in the closed-loop of each vehicle
should not wait for the lost data to make a decision. In practice, it is
required to use a policy or a data-loss compensation strategy to deal with the
missing information [11]. Common strategies in the field of NCS consider
replacing the missing information, for instance, with an estimate [15], or
with previous available data [1] (to-hold strategy), or to set the signal to
zero (to-zero strategy) [26]. In the context of platooning with lossy
channels, most of the available works use one specific compensation strategy.
For instance, [32, 39, 42, 25] use the last available measurement when the
inter-vehicle communication fails. In [35] and [3], a to-zero type protocol
was applied to the controller input and the plant input respectively. In [8],
a numerical study was performed with several types of data-loss compensation
strategies adapted to a platooning setup, and it was observed that the
convergence of the tracking error statistics depends on the chosen strategy,
although these results lack an analytical explanation. In the present paper,
we analyze the MSS of heterogeneous platoons of vehicles modeled as linear
discrete-time systems with PF topology and a constant time-headway spacing
policy. Inter-vehicle communication is considered to be affected by lossy
channels, which are modeled as spatially correlated Bernoulli processes. The
main contributions of this work are as follows:

  * 1.

We obtain necessary and sufficient conditions for a platoon to be mean square
stable, providing a characterization of the mean and variance of the tracking
errors in the platoon under analysis, and studying their convergence
properties. Such conditions depend not only on the controller design and time-
headway policy, but also on the channels arrival rates and on the compensation
strategy. More precise conditions are derived for the case with independent
lossy channels.

  * 2.

Our analysis is valid for vehicles having different dynamical models,
controllers, time headway values, and compensations strategies. Additionally,
the inter-vehicle communication channels are also heterogeneous, and can be
spatially correlated. The case of independent channels for both heterogeneous
and homogeneous platoons are studied as a special cases.

  * 3.

We consider not only one, but a wide class of compensation strategies to deal
with missing data. We provide explicit conditions on the transfer function
resulting from including such strategies, that ensure the platoon is MSS
compatible. Our results also show that the chosen data-loss strategy plays a
crucial role for MSS. This provides an analytical explanation to the
performance of the strategies analyzed in [8].

  * 4.

We present a set of numerical examples to illustrate our results, support our
findings and enhance the related discussion. Here we also discuss about the
implication of MSS on future research on stochastic string stability.

The paper has the following outline: Section 2 presents the platoon setup
under analysis. In Section 3 we propose a general state-space representation
of the platoon. Section 4 presents necessary and sufficient conditions for
MSS. In Section 5 we show numerical simulations and discuss about the
implication of our results on string stabilization problems. Finally,
conclusions and future work are discussed in Section 6.

###  1.1 Notation and preliminaries

Let $M\in\mathbb{R}^{m\times m}$ be a real square matrix with eigenvalues
$\lambda_{1},...,\lambda_{m}$. We use
$\rho(M)\triangleq\textrm{max}\\{|\lambda_{1}|,...,|\lambda_{m}|\\}$ to denote
the spectral radius of $M$. We will refer to $M$ as stable if $\rho(M)<1$.
Given three matrices $A\in\mathbb{R}^{p\times q}$, $B\in\mathbb{R}^{q\times
r}$ and $C\in\mathbb{R}^{r\times s}$, the following property holds [2]

$vec(ABC)=(C^{\top}\otimes A)\ vec(B),$ (1)

where $vec$ denotes the vectorization of a matrix and $\otimes$ represents the
Kronecker product. Let $Y\in\mathbb{R}^{m\times n}$ and
$y\in\mathbb{R}^{mn\times 1}$ such that $vec(Y)=y$, the inverse of the $vec$
operator is such that $vec^{-1}(vec(Y))=Y$. Let $z\in\mathbb{R}^{l\times 1}$
be a column vector of length $l$, $M\in\mathbb{R}^{l\times l},\
N\in\mathbb{R}^{l\times l}$ and $D_{z}\in\mathbb{R}^{l\times l}$ a diagonal
matrix $D_{z}=diag(z)$, then

$\displaystyle D_{z}M{D_{z}}^{\top}$ $\displaystyle=(zz^{\top})\odot M,$ (2)
$\displaystyle vec(M\odot N)$ $\displaystyle=diag(vec(M))vec(N)$ (3)

hold [2], where $\odot$ denotes the Hadamard product. Let $x(k)$ and $y(k)$ be
random variables of a stochastic process at the discrete time
$k\in\mathbb{Z}$. The mean $\mu_{x}(k)$, covariance $P_{x}(k)$, and cross
covariance $P_{xy}(k)$ are defined as
$\mu_{x}(k)\triangleq\mathcal{E}\left\\{x(k)\right\\},\
P_{x}(k)\triangleq\mathcal{E}\left\\{\bar{x}(k)\bar{x}(k)^{\top}\right\\},\
P_{xy}(k)\triangleq\mathcal{E}\left\\{\bar{x}(k)\bar{y}(k)^{\top}\right\\}$,
where $\mathcal{E}\left\\{\cdot\right\\}$ denotes the expectation operator and
$\bar{x}(k)\triangleq x(k)-\mathcal{E}\left\\{x(k)\right\\}.$

##  2 Platooning Setup

This section presents the characteristics of the platooning setup under
consideration for the analysis.

###  2.1 Platoon with ideal communication

We consider an interconnection of autonomous vehicles that consists of a
leading vehicle and $N\in\mathbb{N}$ followers, all modeled as discrete-time
linear time-invariant (LTI) systems. Each vehicle is identified with labels
$i=0,\ldots,N$ that refer to their place within the string; $i=0$ represents
the leader. The collection of vehicles is heterogeneous, having a plant model
$G_{i}(z)$ and a local controller $K_{i}(z)$. The platoon is then controlled
in a decentralized fashion, that is, each vehicle has access to limited
information, and the control of each agent is designed locally. The platoon is
assumed to have wireless communication capabilities in each vehicle and to
move in a straight line with a predecessor-following topology. This
configuration means that the vehicle-to-vehicle (V2V) communication is
unidirectional and occurs only between an agent and its nearest follower. We
consider that through the wireless channel, each vehicle has access to the
position of its predecessor.

We use $y_{i}(k)$ to denote the position of the $i$-th agent at the discrete
time instant $k$. For simplicity in the exposition, vehicles will be
considered to have zero length [18], and thus $y_{i}(k)$ represents the
position of a point in the real line. This assumption has no impact on the
controller design nor on the subsequent stability analysis [23, 41]. The
inter-vehicle distance $\ell_{i}(k)$ between the $i$-th agent and its
predecessor is given by $\ell_{i}(k)=y_{i-1}(k)-y_{i}(k)$. An illustrative
scheme of the platoon is given in Fig. 1.

Figure 1: Platoon formation of three consecutive vehicles

The performance signal is the tracking error, denoted as $\zeta_{i}$. It is
defined as the difference between the measured inter-vehicle distance
$\ell_{i}$ and the desired separation $r_{i}$

$\zeta_{i}(k)=\ell_{i}(k)-r_{i}(k).$ (4)

In platooning designing it is common to consider configurations that achieve
string stability, which is an important property for this type of application
[38, 10]. A platoon with PF topology can be compatible with string stability
by implementing a 2-dof architecture [17]. This requirement is commonly
achieved by modifying the desired inter-vehicle distance. For instance, the
constant time-headway spacing policy increases the separation between two
consecutive vehicles according to the speed at which the $i$-th vehicle
travels [23, 31, 42, 21]. With this in mind, the desired inter-vehicle
distance is set as

$r_{i}(k)=\epsilon_{i}+h_{i}(y_{i}(k)-y_{i}(k-1))$ (5)

where $\epsilon_{i}$ is the safety distance and $h_{i}>0$ is a value known as
the time-headway constant that weighs the measured discrete-time
representation of the speed, i.e. $\nu_{i}(k)=y_{i}(k)-y_{i}(k-1)$. Notice
that $\nu_{i}(0)$ is the initial speed of the $i$-th vehicle. Starting from
rest we have that $y_{i}(-1)=y_{i}(0)$. Without loss of generality, the safety
distance is set to zero for simplicity [1, 17]. Consequently, the tracking
error is calculated as

$\zeta_{i}(k)=y_{i-1}(k)-w_{i}(k)$ (6)

where $w_{i}(k)=(h_{i}+1)y_{i}(k)-h_{i}\ y_{i}(k-1)$. This error measures the
performance of both platoon tasks, namely, maintaining the separation
distance, and tracking the leader position. The control loop of one vehicle is
shown in Fig. 2 where the vehicle dynamic of the $i$-th agent is given by
$G_{i}(z)$. The transfer function $H_{i}(z)$ incorporates the time-headway
constant and it is defined as

$H_{i}(z)=(1+h_{i})-h_{i}\ {z}^{-1}.$ (7)

Note that the output of $H_{i}$ corresponds to the feedback signal $w_{i}(k)$
introduced in (6). We use $u_{i}(k)$ to denote the output of the local
controller $K_{i}$. In Fig. 2, $e_{i}(k)$ represents the local control error,
which is equal to the tracking error $\zeta_{i}(k)$ if the communication is
perfect. In the next subsection, the distinction between these two types of
errors becomes relevant. For the given two degree-of-freedom architecture, the
complementary sensitivity function $T_{i}(z)$ is

$\displaystyle T_{i}(z)=\frac{G_{i}(z)K_{i}(z)}{1+G_{i}(z)H_{i}(z)K_{i}(z)}.$
(8)

Note that we are not interested in designing the controllers $K_{i}$, nor the
time-headway $h_{i}$. Instead, we assume that they are properly designed in an
ideal communication scenario and focus on analyzing the effect that lossy
channels may have on the platoon.

###### Assumption 1

For $i=\\{0,1,\ldots,N\\}$ we assume that:

  1. 1.

$T_{i}(z)$ is stable and strictly proper.

  2. 2.

$T_{i}(z)$ has no unstable cancellations in the product
$G_{i}(z)H_{i}(z)K_{i}(z)$.

  3. 3.

The position of the leader $y_{0}(k)$ converges to a ramp signal.

  4. 4.

The product $K_{i}(z)G_{i}(z)$ must have at least double integral action (two
poles at $z=1$).

The first two are standard for the design of a closed loop system $T_{i}(z)$.
The third one is made to consider a platoon moving at a cruise velocity in
steady-state. This requirement does not remove the existence of disturbance
changes in the speed (acceleration and breaking). The last assumption provides
zero steady-state tracking error with a ramp reference and can be easily
guaranteed through the controller design.

Figure 2: Closed loop of the $i$-th vehicle

###  2.2 Platooning with data-loss

The transfer function between the output of the $(i-1)$-th agent and the
output of its immediate follower ($i$-th agent), corresponds to $T_{i}(z)$ in
the case of ideal inter-vehicle communication channels. However, when random
data loss is considered, the signals involved may be stochastic, and the model
of the communication channels affects the interconnected system [28]. The use
of policies or compensation strategies to deal with data-loss becomes
unavoidable, and may modify the aforementioned transfer function [39]. The
trajectories of the vehicles are not expected to be smooth in this scenario
due to the effect of the random losses. A particular realization may exhibit
erratic behavior and underperform in a practical sense. However, the effect of
the unreliable channels may be mitigated by designing the dynamical features
of the mean and variance of the tracking error.

Figure 3: Scheme of the platoon interconnected through lossy channels

Fig. 3 presents a general scheme of a platoon with lossy inter-vehicle
communication channels where $f_{i}(\cdot)$ denotes the resulting feedback
loop function after including a data-loss compensation scheme. The effect of
the adopted strategies on the definition of $f_{i}(\cdot)$ is discussed in the
next subsection.

Figure 4: Block diagram of a vehicle with (a) measurement compensation
strategy, (b) error compensation strategy, (c) mixed strategies (error and
control signal based).

Lossy channels are modeled as erasure channels defined by a Bernoulli
stochastic process $\theta_{i}\in\\{1,0\\}$ that describes whether the
transmitted data is received or lost. When $\theta_{i}(k)=1$, the data is
received successfully. Conversely, when $\theta_{i}(k)=0$ the data is
considered lost (not available). The channel output is denoted as
$\tilde{y}_{i-1}$ with

$\tilde{y}_{i-1}(k)=\theta_{i}(k)y_{i-1}(k).$ (9)

###### Assumption 2

We assume that each individual lossy channel $\theta_{i}$, with
$i=1,2,\dots,N$ is an independent and identically distributed processes
(i.i.d) with successful communication probability $p_{i}$. Thus, the mean
$\mu_{\theta_{i}}$, and variance, $P_{\theta_{i}}$, of each channel are
constant, and given by

$\displaystyle\mu_{\theta_{i}}=p_{i},\qquad\qquad
P_{\theta_{i}}=p_{i}(1-p_{i}).$ (10)

Additionally, we assume that $\theta_{i}(k)$ could be correlated with
$\theta_{j}(k)$, for $i\neq j$. Thus, the covariance $P_{\Theta}$ of the
vector $\Theta(k):=\left[\theta_{1}(k),\cdots,\theta_{N}(k)\right]^{\top}$
could be non diagonal.

Assumption 2 implies that the channels are uncorrelated in time but possibly
correlated in space. Thus, there could be a statistical link between two pairs
of channels given by $\theta_{i}$ and $\theta_{j}$. The case where channels
are spatially independent is given as a special case in Section 4. Clearly,
the value of $p_{i}$ depends on several factors, but strongly on the
transmitting and receiving devices of the $(i-1)$-th and $i$-th vehicles
respectively.

The local error $e_{i}$ should not be confused with the tracking error
$\zeta_{i}$. Indeed, only for ideal communication, both errors are equivalent.
Given that $e_{i}$ utilizes the channel output signal (9), it does not measure
the true inter-vehicle distance error. Thus, it is suitable to use $\zeta_{i}$
(see (6)) as the performance signal in this setting, since it is calculated
using the true position of the vehicles.

####  2.2.1 Strategies to deal with data-loss

A compensation strategy refers to the policy or action that is taken by the
receiver when the information transmitted by the predecessor does not arrive.
Commonly used strategies either hold the previous value of a signal or replace
it with zero when dropouts occur [11]. Besides, more than one strategy can be
used simultaneously and be applied to the measured position, the local error,
or the control signal [8]. The use of an appropriate compensation strategy may
have a positive impact on the MSS and also on the string stability of the
platoon [8]. Said strategies modify the closed-loop structure and the
resulting state-space representation of the interconnected system. To
illustrate this, Fig. 4 presents three types of control strategies applied to
different signals of the control loop. In Fig. 4$(a)$ a measurement
compensation strategy is included after the channel output to estimate the
position. This strategy is described as

$\hat{y}_{i-1}(k)=\tilde{y}_{i-1}(k)-\eta_{i}(k)\theta_{i}(k)+\eta_{i}(k).$
(11)

where $\eta_{i}(k)=2\hat{y}_{i-1}(k-1)-\hat{y}_{i-1}(k-2)$ is the estimated
distance traveled in a lapse of time between $k-1$ and $k$. An error
compensation strategy like the one in Fig. 4$(b)$ sets the error to zero when
the position is lost. This type of strategy is described as

$\hat{e}_{i}(k)=e_{i}(k)\theta_{i}(k)$ (12)

Fig. 4$(c)$ shows a combination of strategies. The error is replaced with its
previous value [35], and the control signal is replaced with its previous
estimated value in events of data loss. These strategies are described as

$\displaystyle\hat{e}_{i}(k)$
$\displaystyle=\left[e_{i}(k)-\hat{e}_{i}(k-1)\right]\theta_{i}(k)+\hat{e}_{i}(k-1),$
(13) $\displaystyle\hat{u}_{i}(k)$
$\displaystyle=\left[u_{i}(k)-u_{i}(k-1)\right]\theta_{i}(k)+u_{i}(k-1)$ (14)

For all compensation strategies, their initial conditions must be specified
with a given policy, especially those depending on past values. For instance,
we can set $\hat{e}_{i}(-1)=0$ and $\hat{u}_{i}(-1)=0$ for the strategy in
Fig. 4$(c)$. The above strategies are particular examples, but our analysis is
not restricted to them. The structure of the strategies directly affect the
resulting functions $f_{i}(\cdot)$ that relate $\tilde{y}_{i-1}(k)$ with
$y_{i}(k)$. This motivates the use of a general representation of the closed-
loop system.

###### Assumption 3

We consider strategies that can be written as linear feedback loops subject to
multiplicative Bernoulli channels, such as those in Fig. 4.

####  2.2.2 Problem Statement

In the present paper, we study the time-convergence of the platoon. We first
consider that each closed loop $T_{i}$ is properly designed and internally
stabilizes the platoon assuming ideal communication. Then, we assume lossy
communication in each inter-vehicular channel and analyze the stability of the
platoon. The stability analysis of linear systems with random data loss can be
performed through the property of mean square stability which guarantees the
convergence of the mean and variance of the signal of interest. The signal of
interest in our case is the tracking error vector defined as
$\bm{\zeta}(k)=\begin{bmatrix}\zeta_{1}(k)&\ldots&\zeta_{N}(k)\end{bmatrix}^{\top}.$

###### Definition 1

The platoon under analysis is mean square stable if and only if the mean and
covariance matrix of the tracking error vector $\bm{\zeta}(k)$ are such that

$\lim_{k\to\infty}\mu_{\bm{\zeta}}(k)=\mu_{\bm{\zeta}}\quad\text{and}\quad\lim_{k\to\infty}P_{\bm{\zeta}}(k)=P_{\bm{\zeta}},$
(15)

for some finite $\mu_{\bm{\zeta}}$ and $P_{\bm{\zeta}}.$

Our interest is to determine necessary and sufficient conditions for a platoon
to be mean square stable.

Figure 5: Alternative platoon representation

##  3 State-space representation

Given Assumption 3, the platoon can be represented as depicted in Fig. 5 where
the lossy channels $\theta_{i}$ are separated from the remaining linear system
$F_{i}$, which is defined by $G_{i}$, $K_{i}$, $H_{i}$ and the applied data-
loss compensation mechanism. Hence, a platoon with PF topology can be viewed
as a cascade interconnection of systems, although there could be a statistical
link among these systems beyond the cascade structure. It is convenient to use
this representation as it is independent of the internal structure of $F_{i}$
and provides flexibility for the inclusion, for example, of several data
compensation strategies (see for instance [8]).

In this representation each $F_{i}$, for $i\geq 1$, is assumed to have the
following minimal state-space description

$\displaystyle x_{i}(k+1)$ $\displaystyle=A_{i}\
x_{i}(k)+B_{i}\tilde{v}_{i}(k),\quad x_{i}(0)={x_{i}}_{0}$ (16a)
$\displaystyle y_{i}(k)$ $\displaystyle=C_{y_{i}}\ x_{i}(k)$ (16b)
$\displaystyle\zeta_{i}(k)$ $\displaystyle=C_{\zeta_{i}}\
x_{i}(k)+D_{\zeta_{i}}\ y_{i-1}(k)$ (16c) $\displaystyle v_{i}(k)$
$\displaystyle=C_{v_{i}}\ x_{i}(k)+D_{v_{i}}\ y_{i-1}(k)$ (16d)

where $x_{i}(k)\in\mathbb{R}^{n_{x}}$ is the system state,
$v_{i}(k)\in\mathbb{R}^{n_{v}}$ and $\tilde{v}_{i}(k)\in\mathbb{R}^{n_{v}}$
are the input and the output of the $i$-th Bernoulli link, and $A_{i}$,
$B_{i}$, $C_{y_{i}}$, $C_{\zeta_{i}}$, $C_{v_{i}}$, $D_{\zeta_{i}}$,
$D_{v_{i}}$ are real matrices of appropriate dimensions. For some particular
compensation strategies, $\zeta_{i}(k)$ could be equal to $v_{i}(k)$, but in
general these are different signals. Indeed, $\zeta_{i}(k)$ is a scalar
signal, while $v_{i}(k)$ could be not, e.g., when using strategy in Fig.
4$(c)$, where $v_{i}(k)=[v_{i_{a}}(k)\,\,v_{i_{b}}(k)]^{\top}$.

###### Remark 1

Naturally, the behavior of the systems $F_{i}$ depends on the arrival rate
$p_{i}$, however it can also depend on other channels statistics. For
instance, we can define $\tau_{i}=(p_{i}+p_{i-1}+\cdots p_{i-\ell})/\ell$ and
use it as a measure of the communication quality of $\ell$ vehicles ahead to
define, with a given criteria, the time headway $h_{i}$, the controller gain,
a parameter of the compensation strategy, etc.

The concatenation of the systems $F_{i}$ can also be represented using a
generalized feedback loop with a zero-mean multiplicative MIMO channel. Since
the channels are considered temporally independent, we first define the matrix
$\Theta_{d}$ and its mean $\Upsilon$ as follows:

$\displaystyle\Theta_{d}(k)$
$\displaystyle=diag(\Theta(k))=diag(\theta_{1}(k),\cdots,\theta_{N}(k))$ (17)
$\displaystyle\Upsilon$ $\displaystyle=diag(p_{1},\dots,p_{N}).$ (18)

We also define the zero-mean multiplicative channel matrix
$\bar{\Theta}_{d}(k)=\Theta_{d}(k)-\Upsilon$. This allows us to write the
platoon dynamics using the representation of Fig. 6, where $\mathbf{M}$
represents a linear time invariant system with state vector denoted by
$\mathbf{x}$ and where $\mathbf{v}$ and $\breve{\mathbf{v}}$ represent the
channel input and output respectively. The state and the channel input vectors
are formed by the concatenation of the corresponding terms of each vehicle,
that is,

$\displaystyle\mathbf{x}(k)$
$\displaystyle=\begin{bmatrix}x_{1}(k)&\ldots&x_{N}(k)\end{bmatrix}^{\top},$
$\displaystyle\mathbf{v}(k)$
$\displaystyle=\begin{bmatrix}v_{1}(k)&\ldots&v_{N}(k)\end{bmatrix}^{\top},\;\,\breve{\mathbf{v}}=\bar{\Theta}_{d}(k)\mathbf{v}.$

Figure 6: General feedback closed loop of the platoon system.

###### Lemma 1

The closed loop dynamics of the system $\mathbf{M}(z)$ can be written as

$\displaystyle\mathbf{x}(k+1)$ $\displaystyle=\mathbf{A}\
\mathbf{x}(k)+\mathbf{B}\ y_{0}(k)+\mathcal{B}\breve{\mathbf{v}}(k)$ (19a)
$\displaystyle\bm{\zeta}(k)$ $\displaystyle=\mathbf{C}_{\bm{\zeta}}\
\mathbf{x}(k)+\mathbf{D}_{\bm{\zeta}}\ y_{0}(k)$ (19b)
$\displaystyle\mathbf{v}(k)$ $\displaystyle=\mathbf{C_{v}}\
\mathbf{x}(k)+\mathbf{D_{v}}y_{0}(k)$ (19c)

with the following state-space matrices:

$\displaystyle\mathbf{A}$ $\displaystyle=\begin{bmatrix}\alpha_{1}&\ &\ &\ \\\
\gamma_{2}&\alpha_{2}&\ &\ \\\ \ &\ddots&\ddots&\ \\\ \ &\
&\gamma_{N}&\alpha_{N}\\\
\end{bmatrix},\;\mathbf{B}=\begin{bmatrix}p_{1}B_{1}D_{v_{1}}\\\ 0\\\ \vdots\
\\\ 0\\\ \end{bmatrix},$ $\displaystyle\mathcal{B}$
$\displaystyle=diag(B_{1},\cdots B_{N}),$
$\displaystyle\mathbf{C}_{\bm{\zeta}}$
$\displaystyle=\begin{bmatrix}C_{\zeta_{1}}&\ &\ &\ \\\
D_{\zeta_{2}}C_{y_{1}}&C_{\zeta_{2}}&\ &\ \\\ \ &\ddots&\ddots&\ \\\ \ &\
&D_{\zeta_{N}}C_{y_{N-1}}&C_{\zeta_{N}}\\\ \end{bmatrix},\
\mathbf{D}_{\bm{\zeta}}=\begin{bmatrix}D_{\zeta_{1}}\\\ 0\\\ \vdots\ \\\ 0\\\
\end{bmatrix},$ $\displaystyle\mathbf{C_{v}}$
$\displaystyle=\begin{bmatrix}C_{v_{1}}&\ &\ &\ \\\
D_{v_{2}}C_{y_{1}}&C_{v_{2}}&\ &\ \\\ \ &\ddots&\ddots&\ \\\ \ &\
&D_{v_{N}}C_{y_{N-1}}&C_{v_{N}}\\\ \end{bmatrix},\
\mathbf{D_{v}}=\begin{bmatrix}D_{v_{1}}\\\ 0\\\ \vdots\ \\\ 0\\\
\end{bmatrix}$

where $\alpha_{i}=A_{i}+p_{i}B_{i}C_{v_{i}}$ and
$\gamma_{i}=p_{i}B_{i}D_{v_{i}}C_{y_{i-1}}$.

###### Proof 1

See appendix A.1.

Given this representation, we can partition $\mathbf{M}$ as follows

$\displaystyle\begin{bmatrix}\bm{\zeta}\\\
\mathbf{v}\end{bmatrix}=\underbrace{\begin{bmatrix}\mathbf{M}_{11}&&\mathbf{M}_{12}\\\
\mathbf{M}_{21}&&\mathbf{M}_{22}\end{bmatrix}}_{\mathbf{M}}\begin{bmatrix}y_{0}\\\
\breve{\mathbf{v}}\end{bmatrix},$ (20)

and write the transfer function from $y_{0}$ to $\bm{\zeta}$ and $\mathbf{v}$
as

$\displaystyle\mathbf{M}_{11}(z)$
$\displaystyle=\mathbf{C}_{\bm{\zeta}}(z\mathbf{I}-\mathbf{A})^{-1}\mathbf{B}+\mathbf{D}_{\bm{\zeta}}$
$\displaystyle\mathbf{M}_{21}(z)$
$\displaystyle=\mathbf{C}_{\mathbf{v}}(z\mathbf{I}-\mathbf{A})^{-1}\mathbf{B}+\mathbf{D}_{\mathbf{v}}.$

###### Remark 2

As long as the platoon has an equivalent representation like in Fig. 6 with a
linear and time invariant $\mathbf{M}$, this general feedback loop scheme
allows flexibility to analyze different platoon setups. For instance,
$\mathbf{M}$ can describe homogeneous and heterogeneous platoons. This
representation allows vehicles with different dynamic models, controllers,
channel successful transmission probabilities, or compensation strategies.
Note also that, the current mathematical description may be suitable for
describing interconnections of systems with local controllers, that may not
necessarily represent vehicular systems. In that regard, the following results
might have some use in multi-agent applications different from platooning.

##  4 MSS conditions

In order to derive our main results, we first present in Lemma 2 below the
first and second order statistics of the signals of interest.

###### Lemma 2

Consider a platoon described by the representation given in Lemma 1. The mean
of the state, tracking error, and channel input, respectively, satisfy

$\displaystyle\mu_{\mathbf{x}}(k+1)$ $\displaystyle=\mathbf{A}\
\mu_{\mathbf{x}}(k)+\mathbf{B}\ y_{0}(k)$ (21a)
$\displaystyle\mu_{\bm{\zeta}}(k)$ $\displaystyle=\mathbf{C}_{\bm{\zeta}}\
\mu_{\mathbf{x}}(k)+\mathbf{D}_{\bm{\zeta}}\ y_{0}(k)$ (21b)
$\displaystyle\mu_{\mathbf{v}}(k)$ $\displaystyle=\mathbf{C_{v}}\
\mu_{\mathbf{x}}(k)+\mathbf{D_{v}}\ y_{0}(k),$ (21c)

and the corresponding covariance matrices satisfy

$\displaystyle P_{\mathbf{x}}(k+1)$ $\displaystyle=\mathbf{A}\
P_{\mathbf{x}}(k)\ \mathbf{A}^{\top}+\mathcal{B}\left[P_{\Theta}\odot
P_{\mathbf{v}}(k)\right]{\mathcal{B}}^{\top}$
$\displaystyle+\mathcal{B}\left[P_{\Theta}\odot(\mu_{\mathbf{v}}(k)\mu_{\mathbf{v}}(k)^{\top})\right]{\mathcal{B}}^{\top}$
(22a) $\displaystyle P_{\bm{\zeta}}(k)$
$\displaystyle=\mathbf{C}_{\bm{\zeta}}\ P_{\mathbf{x}}(k)\
\mathbf{C}_{\bm{\zeta}}^{\top}$ (22b) $\displaystyle P_{\mathbf{v}}(k)$
$\displaystyle=\mathbf{C_{v}}\ P_{\mathbf{x}}(k)\ {\mathbf{C_{v}}}^{\top}.$
(22c)

###### Proof 2

See appendix A.2.

The following theorem is the main result of this paper and presents necessary
and sufficient conditions for MSS of platoons subject to random data-loss as
the one described above.

###### Theorem 4

Consider the platoon described in Section 2, with the state-space
representation given in Section 3. Then,

  1. 1.

the mean $\mu_{\bm{\zeta}}(k)$ converges if and only if

$\mathbf{M}_{11}(1)=0\;\text{ and }\;\rho(\mathbf{A})<1,$ (23)

  2. 2.

the variance $P_{\bm{\zeta}}(k)$ converges if and only if $\rho(\mathbf{A})<1$
and

$\mathbf{M}_{21}(1)=0\;\text{ and
}\;\rho((\mathbf{A}\otimes\mathbf{A})+\Delta)<1,$ (24)

where $\Delta=(\mathcal{B}\otimes\mathcal{B})\ diag(vec(P_{\Theta}))\
(\mathbf{C_{v}}\otimes\mathbf{C_{v}})$.

Consequently, the platoon is mean square stable if and only if (23) and (24)
are met.

###### Proof 3

We define an alternative state
$\mathbf{x}^{*}(k)=\mathbf{x}(k)-\mathbf{x}(k-1)$. From (21a) we have that

$\mu_{\mathbf{x}^{*}}(k+1)=\mathbf{A}\ \mu_{\mathbf{x}^{*}}(k)+\mathbf{B}\
m_{0}(k),$ (25)

where $m_{0}(k)=y_{0}(k)-y_{0}(k-1)$. Given Assumption 1, the position input
$y_{0}(k)$ converges to a ramp signal. Then, $m_{0}(k)$ converges to a
constant value which is the cruise velocity (slope of the position at instant
$k$). From (21b) and (21c) we have

$\displaystyle\mu_{\bm{\zeta}}(k)$
$\displaystyle=\mu_{\bm{\zeta}}(k-1)+\mathbf{C}_{\bm{\zeta}}\
\mu_{\mathbf{x}^{*}}(k)+\mathbf{D}_{\bm{\zeta}}\ m_{0}(k)$ (26)
$\displaystyle\mu_{\mathbf{v}}(k)$
$\displaystyle=\mu_{\mathbf{v}}(k-1)+\mathbf{C}_{\mathbf{v}}\
\mu_{\mathbf{x}^{*}}(k)+\mathbf{D}_{\mathbf{v}}\ m_{0}(k).$ (27)

On the other hand, replacing (22c) in (22), and using the linearity property
of the Hadamard product it follows that

$\displaystyle P_{\mathbf{x}}(k+1)$
$\displaystyle=\mathbf{A}P_{\mathbf{x}}(k)\mathbf{A}^{\top}+\mathcal{B}\left[P_{\Theta}\odot(\mathbf{C_{v}}P_{\mathbf{x}}(k)\
{\mathbf{C_{v}}}^{\top})\right]{\mathcal{B}}^{\top}$
$\displaystyle\quad+\mathcal{B}\left[P_{\Theta}\odot\left(\mu_{\mathbf{v}}(k)\
\mu_{\mathbf{v}}(k)^{\top}\right)\right]{\mathcal{B}}^{\top}.$ (28)

Applying the $vec$ operator and property (1) in (3) yields

$\displaystyle\mathbf{X}(k+1)$
$\displaystyle=(\mathbf{A}\otimes\mathbf{A})\mathbf{X}(k)+(\mathcal{B}\otimes\mathcal{B})Z(k)+S(k),$

where $Z(k)=vec(P_{\Theta}\odot(\mathbf{C_{v}}\ P_{\mathbf{x}}(k)\
{\mathbf{C_{v}}}^{\top}))$,
$S(k)=vec\left[\mathcal{B}(P_{\Theta}\odot(\mu_{\mathbf{v}}(k)\mu_{\mathbf{v}}(k)^{\top}))\mathcal{B}^{\top}\right]$
and $\mathbf{X}(k)=vec(P_{\mathbf{x}}(k))$. Using property (3), we write

$\displaystyle Z(k)$ $\displaystyle=vec(P_{\Theta})\odot vec(\mathbf{C_{v}}\
P_{\mathbf{x}}(k){\mathbf{C_{v}}}^{\top})$
$\displaystyle=diag(vec(P_{\Theta}))(\mathbf{C_{v}}\otimes\mathbf{C_{v}})\mathbf{X}(k).$

Let $\Delta=(\mathcal{B}\otimes\mathcal{B})diag(vec(P_{\Theta}))\
(\mathbf{C_{v}}\otimes\mathbf{C_{v}})$. Then,

$\displaystyle\mathbf{X}(k+1)$
$\displaystyle=\left[(\mathbf{A}\otimes\mathbf{A})+\Delta\right]\mathbf{X}(k)+S(k).$
(29)

Necessity: Consider the platoon MSS. Thus, we conclude from (26) that there
exists $\mu_{\bm{\zeta}}=\lim_{k\rightarrow\infty}\mu_{\bm{\zeta}}(k)$
satisfying $\mu_{\bm{\zeta}}=\mu_{\bm{\zeta}}+\mathbf{C}_{\bm{\zeta}}\
\mu_{\mathbf{x}^{*}}+\mathbf{D}_{\bm{\zeta}}\ m_{0}$, where
$\mu_{\mathbf{x}^{*}}$ and $m_{0}$ are the stationary values of
$\mu_{\mathbf{x}^{*}}(k)$ and $m_{0}(k)$, respectively. This implies that
$\mathbf{C}_{\bm{\zeta}}\ \mu_{\mathbf{x}^{*}}+\mathbf{D}_{\bm{\zeta}}\
m_{0}=0$. From (25) we have $\mu_{\mathbf{x}^{*}}=\mathbf{A}\
\mu_{\mathbf{x}^{*}}+\mathbf{B}\ m_{0}=(I-\mathbf{A})^{-1}\ \mathbf{B}\
m_{0}$, and thus $\mathbf{C}_{\bm{\zeta}}\
\mu_{\mathbf{x}^{*}}+\mathbf{D}_{\bm{\zeta}}\
m_{0}=\left(\mathbf{C}_{\bm{\zeta}}(I-\mathbf{A})^{-1}\
\mathbf{B}+\mathbf{D}_{\bm{\zeta}}\right)m_{0}=\mathbf{M}_{11}(1)m_{0}$.
Hence, MSS implies $\mathbf{M}_{11}(1)=0$. Also, it is clear that
$\mu_{\mathbf{x}^{*}}=(I-\mathbf{A})^{-1}\ \mathbf{B}\
m_{0}=\sum_{k=1}^{\infty}\mathbf{A}^{k}\mathbf{B}\ m_{0}$, which implies
$\rho(\mathbf{A})<1$, otherwise the summation will not converge. This shows
that (23) is necessary for the mean convergence.

MSS assumption also implies that there exists
$P_{\bm{\zeta}}=\lim_{k\rightarrow\infty}P_{\bm{\zeta}}(k)$, which requires
$P_{\mathbf{x}}(k)$ to converge to a constant matrix $P_{\mathbf{x}}$
satisfying

$\displaystyle P_{\mathbf{x}}$
$\displaystyle=\mathbf{A}P_{\mathbf{x}}\mathbf{A}^{\top}+\mathcal{B}\left[P_{\Theta}\odot(\mathbf{C_{v}}P_{\mathbf{x}}\
{\mathbf{C_{v}}}^{\top})\right]{\mathcal{B}}^{\top}$
$\displaystyle\quad+\mathcal{B}\left[P_{\Theta}\odot\left(\mu_{\mathbf{v}}\
\mu_{\mathbf{v}}^{\top}\right)\right]{\mathcal{B}}^{\top},$ (30)

where $\mu_{\mathbf{v}}$ is the stationary value of $\mu_{\mathbf{v}}(k)$.
From (27), and mimicking the analysis above for $\mu_{\bm{\zeta}}$, it is easy
to conclude that MSS implies $\mathbf{M}_{21}(1)=0$.

Since $P_{\mathbf{x}}(k)\rightarrow P_{\mathbf{x}}$, then the vector
$\mathbf{X}(k)$ converges too. It is clear from (29) that the stationary value
$\mathbf{X}$ is given by
$\mathbf{X}=(I-(\mathbf{A}\otimes\mathbf{A})-\Delta)^{-1}\
S=\sum_{k=1}^{\infty}((\mathbf{A}\otimes\mathbf{A})+\Delta)^{k}S,$ where
$S=vec\left[\mathcal{B}(P_{\Theta}\odot(\mu_{\mathbf{v}}\mu_{\mathbf{v}}^{\top}))\mathcal{B}^{\top}\right]$.
We then conclude that if the platoon is MSS, then the spectral radius of
$(\mathbf{A}\otimes\mathbf{A})+\Delta$ is less than one. This proves (24) is
necessary for the covariance convergence and thus for MSS.

Sufficiency: Assume conditions in (23) and (24) are met. From (25) we have

$\mu_{\mathbf{x}^{*}}(k)=\mathbf{A}^{k}\
\mu_{\mathbf{x}^{*}}(0)+\sum_{i=1}^{k}\mathbf{A}^{i-1}\mathbf{B}\ m_{0}(k-i).$
(31)

Since $\rho(\mathbf{A})<1$, then $\lim_{k\rightarrow\infty}\mathbf{A}^{k}=0$.
It is known that $m_{0}=\lim_{k\rightarrow\infty}m_{0}(k)$, thus the summation
in (31) converges and, hence, $\mu_{\mathbf{x}^{*}}(k)$ converges to a
constant value $\mu_{\mathbf{x}^{*}}$. Such value can be written as
$\mu_{\mathbf{x}^{*}}=(I-\mathbf{A})^{-1}\ \mathbf{B}\ m_{0}$. Thus, from (26)
we have that
$\lim_{k\rightarrow\infty}\left(\mu_{\bm{\zeta}}(k)-\mu_{\bm{\zeta}}(k-1)\right)=\mathbf{M}_{11}(1)m_{0}$.
Since $\mathbf{M}_{11}(1)=0$, we conclude that $\mu_{\bm{\zeta}}(k)$ converges
to a constant value $\mu_{\bm{\zeta}}$, proving that (23) is sufficient for
the mean convergence. Similarly, given that $\mathbf{M}_{21}(1)=0$, it is easy
to see from (27) that $\mu_{\mathbf{v}}(k)$ also converges. The latter implies
that $S(k)$ converges to $S$. Finally, from (29) we have

$\displaystyle\mathbf{X}(k)=$
$\displaystyle\left[(\mathbf{A}\otimes\mathbf{A})+\Delta\right]^{k}\mathbf{X}(0)$
$\displaystyle+\sum_{i=1}^{k}\left[(\mathbf{A}\otimes\mathbf{A})+\Delta\right]^{i-1}S(k-i).$
(32)

Since $\rho((\mathbf{A}\otimes\mathbf{A})+\Delta)<1$, and $S(k)$ converges, we
conclude that $\mathbf{X}(k)$ converges and so does $P_{\mathbf{x}}(k)$, which
proves that both (23) and (24) are sufficient for MSS.

Theorem 4 presents necessary and sufficient conditions for the platoons of
interest to be stable in the mean square sense. Such conditions depend not
only on the plant, controller dynamics, and time-headway, but also on the
applied compensation strategy, the probabilities of successful transmission,
and the correlation between channels. It is expected for the quality of the
channels to have a relevant impact on the MSS of the platoon. Section 5
provides a simulation-based analysis about the influence that the
probabilities $p_{i}$ may have on MSS for a two-follower platoon.

The conditions for $\mathbf{M}_{11}(z)$ and $\mathbf{M}_{21}(z)$ in Theorem 4
are equivalent to requiring such systems to have at least one zero at $z=1$.
$\mathbf{M}_{11}(z)$ and $\mathbf{M}_{21}(z)$ strongly depend on the chosen
strategy to deal with data loss. It is important to note that, although $T(z)$
is required to have two zeros at $z=1$ to be able to track ramp signals, such
zeros could not be part of $\mathbf{M}_{11}(z)$ and $\mathbf{M}_{21}(z)$,
since the adopted protocol to deal with data loss could cancel one or both
zeros. This reveals that the strategy to deal with data loss must be carefully
chosen. The MSS convergence of the error could be to values different from
zero for the mean and the variance. In the following corollary, we specify
conditions to ensure that the mean and variance of the error converge to zero.

###### Corollary 5

Consider a platoon satisfying the conditions in Theorem 4. Then,

  1. 1.

if $\mathbf{M}_{11}(z)$ has two or more zeros at $z=1$, $\mu_{\bm{\zeta}}(k)$
converges to zero. Instead, if $\mathbf{M}_{11}(z)$ has only one zero at
$z=1$, then $\mu_{\bm{\zeta}}(k)$ converges to a non-zero value given by

$\mu_{\bm{\zeta}}=-m_{0}{\mathbf{C}_{\bm{\zeta}}}(I-\mathbf{A})^{-2}\mathbf{B},$
(33)

  2. 2.

if $\mathbf{M}_{21}(z)$ has two or more zeros at $z=1$, then
$P_{\bm{\zeta}}(k)$ converges to zero. Instead, if $\mathbf{M}_{21}(z)$ has
only one zero at $z=1$ then $P_{\bm{\zeta}}(k)$ converges to a non-zero value
given by

$\displaystyle P_{\bm{\zeta}}=$
$\displaystyle\mathbf{C}_{\bm{\zeta}}\,vec^{-1}\left[\left(I-(\mathbf{A}\otimes\mathbf{A})-\Delta\right)^{-1}S\right]\mathbf{C}_{\bm{\zeta}}^{\top},$
(34)

where
$S=vec\left[\mathcal{B}(P_{\Theta}\odot(\mu_{\mathbf{v}}\mu_{\mathbf{v}}^{\top}))\mathcal{B}^{\top}\right]$,
with $\mu_{\bf{v}}=-m_{0}{\mathbf{C_{v}}}(I-\mathbf{A})^{-2}\mathbf{B}$.

###### Proof 4

For the first statement, we recall that $\mathbf{M}_{11}(z)$ is the transfer
function from $y_{0}$ to $\mu_{\bm{\zeta}}$. Since the input $y_{0}$ converges
to a ramp signal with slope $m_{0}$, we can use the final value theorem to
conclude that, if $\mathbf{M}_{11}(z)$ has two or more zeros at $z=1$, then

$\mu_{\bm{\zeta}}=\lim_{z\rightarrow
1}(z-1)\mathbf{M}_{11}(z)\frac{m_{0}z}{(z-1)^{2}}=0.$

On the other hand, if $\mathbf{M}_{11}(z)$ has only one zero at $z=1$, then

$\mu_{\bm{\zeta}}=\lim_{z\rightarrow
1}(z-1)\mathbf{M}_{11}(z)\frac{m_{0}z}{(z-1)^{2}}=m_{0}\left.\frac{\mathbf{M}_{11}(z)}{z-1}\right|_{z=1}.$

We can write

$\displaystyle\frac{\mathbf{M}_{11}(z)}{z-1}=$
$\displaystyle{\mathbf{D}_{\bm{\zeta}}}z^{-1}+({\mathbf{D}_{\bm{\zeta}}}+{\mathbf{C}_{\bm{\zeta}}}\mathbf{B})z^{-2}$
$\displaystyle+$
$\displaystyle({\mathbf{D}_{\bm{\zeta}}}+{\mathbf{C}_{\bm{\zeta}}}\mathbf{B}+{\mathbf{C}_{\bm{\zeta}}}\mathbf{A}\mathbf{B})z^{-3}$
$\displaystyle+$
$\displaystyle({\mathbf{D}_{\bm{\zeta}}}+{\mathbf{C}_{\bm{\zeta}}}\mathbf{B}+{\mathbf{C}_{\bm{\zeta}}}\mathbf{A}\mathbf{B}+{\mathbf{C}_{\bm{\zeta}}}\mathbf{A}^{2}\mathbf{B})z^{-4}+\cdots$

Evaluating at $z=1$, and reordering terms we can write

$\displaystyle\left.\frac{\mathbf{M}_{11}(z)}{z-1}\right|_{z=1}=$
$\displaystyle\lim_{k\rightarrow\infty}\left[k({\mathbf{D}_{\bm{\zeta}}}+\sum_{i=1}^{k-1}{\mathbf{C}_{\bm{\zeta}}}\mathbf{A}^{i-1}\mathbf{B})-\sum_{i=1}^{k-1}i\;{\mathbf{C}_{\bm{\zeta}}}\mathbf{A}^{i-1}\mathbf{B}\right].$

Note that
${\mathbf{D}_{\bm{\zeta}}}+\sum_{i=1}^{k-1}{\mathbf{C}_{\bm{\zeta}}}\mathbf{A}^{i-1}\mathbf{B}$
converges exponentially fast to
${\mathbf{C}_{\bm{\zeta}}}(I-\mathbf{A})^{-1}\mathbf{B}+{\mathbf{D}_{\bm{\zeta}}}=0$
when $k\rightarrow\infty$. Hence, we have that

$\mu_{\bm{\zeta}}=-{\mathbf{C}_{\bm{\zeta}}}\sum_{i=1}^{\infty}i\;\mathbf{A}^{i-1}\mathbf{B}m_{0}=-m_{0}{\mathbf{C}_{\bm{\zeta}}}(I-\mathbf{A})^{-2}\mathbf{B}.$
(35)

For the second statement, we recall that $\mathbf{M}_{21}(z)$ is the transfer
function from $y_{0}$ to $\mu_{\mathbf{v}}$. Thus, an analogous analysis to
the one above allows to conclude that, if $\mathbf{M}_{21}(z)$ has two or more
zeros at $z=1$, then $\mu_{\mathbf{v}}=0$, and if $\mathbf{M}_{21}(z)$ has
only one zero at $z=1$, then
$\mu_{\mathbf{v}}=-m_{0}{\mathbf{C_{v}}}(I-\mathbf{A})^{-2}\mathbf{B}.$ Also,
since the platoon is MSS, $S(k)$ in (29) converges to
$S=vec\left[\mathcal{B}(P_{\Theta}\odot(\mu_{\mathbf{v}}\mu_{\mathbf{v}}^{\top}))\mathcal{B}^{\top}\right],$
which is zero when $\mu_{\mathbf{v}}=0$ and thus, the recursion in (29)
converges to zero since $\rho((\mathbf{A}\otimes\mathbf{A})+\Delta)<1$ holds.
Given that $\mathbf{X}=vec\\{P_{\mathbf{X}}\\}$, and
$P_{\bm{\zeta}}=\mathbf{C}_{\bm{\zeta}}P_{\mathbf{X}}\mathbf{C}_{\bm{\zeta}}^{\top}$
we conclude that $P_{\bm{\zeta}}=0$ in that case. On the other hand, if
$\mu_{\mathbf{v}}\neq 0$, then from (29) we have that $\mathbf{X}(k)$
converges to $\mathbf{X}$, given by
$\mathbf{X}=\left[I-(\mathbf{A}\otimes\mathbf{A})-\Delta\right]^{-1}S$, with
$S$ as above. From (22), and given that $\mathbf{X}=vec\\{P_{\mathbf{X}}\\}$,
we obtain (34).

Corollary 5 states conditions for which the statistics of the error converge
or not to zero. We also explicitly provide such stationary values for both,
mean and covariance matrix. It is clear then that the chosen strategy to deal
with data loss plays an important role since it could affect the MSS and the
convergence properties of the platoon statistics. Naturally, it is preferable
to choose a strategy that ensures zero stationary statistics for the error.
This was observed via simulation results in [8]. Corollary 5 provides an
analytical explanation for such phenomena and the specific limiting values.

###  4.1 Mutually independent channels

In this section, we consider the case where the communication channels are
assumed to be independent and derive simplified conditions for MSS. Hence, we
consider the following assumption:

###### Assumption 6

We assume that the random processes $\theta_{i}$, for $i=1,2,\dots N$, are
mutually independent. Thus, the covariance matrix of $\Theta(k)$ is given by
$P_{\Theta}=diag(p_{1}(1-p_{1}),\dots,p_{N}(1-p_{N})).$

####  4.1.1 Heterogeneous platoon analysis

To present our result we define $\alpha_{i}=A_{i}+p_{i}B_{i}C_{v_{i}}$, and
the transfer functions
$M_{a_{i}}(z)=C_{\zeta_{i}}(zI-\alpha_{i})^{-1}B_{i}D_{v_{i}}p_{i}+{D_{\zeta_{i}}}$
and
$M_{b_{i}}(z)=C_{v_{i}}(zI-\alpha_{i})^{-1}B_{i}D_{v_{i}}p_{i}+{D_{v_{i}}}$.

###### Theorem 7

Consider the platoon described in Section 2, with the representation given in
Section 3 and consider Assumption 6. Then

  1. 1.

$\mu_{\bm{\zeta}}(k)$ converges if and only if, $\forall
i\in\\{1\dots,N\\}\;$,

$M_{a_{i}}(1)=0,\;\text{ and }\;\max_{i}\rho(\alpha_{i})<1,$ (36)

  2. 2.

$P_{\bm{\zeta}}(k)$ converges if and only if $\max_{i}\rho(\alpha_{i})<1$ and,
$\forall i\in\\{1\dots,N\\}\;$,

$M_{b_{i}}(1)=0,\;\text{ and
}\;\max_{i}\rho(\alpha_{i}\otimes\alpha_{i}+\delta_{i})<1,$ (37)

where $\delta_{i}=p_{i}(1-p_{i})(B_{i}\otimes B_{i})(C_{v_{i}}\otimes
C_{v_{i}})$.

Consequently, the platoon is mean square stable if and only if (36) and (37)
are met.

###### Proof 5

See appendix A.3

####  4.1.2 Homogeneous platoon analysis

Here we consider a homogeneous case, that is, a platoon where every agent has
the same dynamics. In a lossy communication setup, homogeneity is not only
considered in the dynamics of the vehicles but also in the communication
channels’ statistics. Hence, here we consider the following assumption:

###### Assumption 8

The system described in (19a), (19b) and (19c) holds with $A_{i}=A$,
$B_{i}=B$, $C_{\zeta_{i}}=C_{\zeta}$, $C_{y_{i}}=C_{y}$, $C_{v_{i}}=C_{v}$,
$D_{v_{i}}=D_{v}$ and $D_{\zeta_{i}}=D_{\zeta}$, that is, every vehicle is
described by the same constant matrices ($F_{i}=F$). Additionally, every
inter-vehicle communication channel has the same probability of successful
transmission, that is, we set $p_{i}=p$, where $0<p\leq 1$.

For the homogeneous setup, we define $\alpha=A+pBC_{v}$ and the transfer
functions $M_{a}(z)=C_{\zeta}(zI-\alpha)^{-1}BD_{v}p+{D_{\zeta}}$ and
$M_{b}(z)=C_{v}(zI-\alpha)^{-1}BD_{v}p+{D_{v}}$.

###### Corollary 9

Consider the platoon described in Section 2, with the representation given in
Section 3, and considering Assumption 6 and 8. Then

  1. 1.

$\mu_{\bm{\zeta}}(k)$ converges if and only if

$M_{a}(1)=0\;\text{ and }\;\rho(\alpha)<1,$ (38)

  2. 2.

$P_{\bm{\zeta}}(k)$ converges if and only if $\rho(\alpha)<1$ and

$M_{b}(1)=0\;\text{ and }\;\rho(\alpha\otimes\alpha+\delta)<1,$ (39)

where $\delta=p(1-p)(B\otimes B)(C_{v}\otimes C_{v})$.

Consequently, the platoon is mean square stable if and only if (38) and (39)
are met.

###### Proof 6

Straightforward from Theorem 7.

Conditions in Theorem 7 and Corollary 9 are similar in form to those in
Theorem 4, but applied to simpler matrices. The results in Corollary 5 are
also applicable for the setup with independent channels. It is not difficult
to see that the transfer functions required to have two zeros at $z=1$ in
order to ensure convergence to zero for the statistics of the tracking error
are $M_{a_{i}}(z)$ and $M_{b_{i}}(z)$ for the heterogeneous case and
$M_{a}(z)$ and $M_{b}(z)$ for the homogeneous one.

##  5 Numerical Examples

In this section, we present numerical tests of the MSS conditions derived in
Section 4. Through simulations, different platoon behaviors are presented and
discussed. Also, we briefly discuss the relationship between MSS and string
stability. Here we use admissible controllers that stabilize each closed-loop
system (with perfect communication). However, in view of the posterior string
stability discussion, we also require the platoon with ideal communication to
be string stable in the deterministic case, which requires that
$||T_{i}(z)||_{\infty}\leq 1$ (see e.g. [12]).

The platoon behaviors are presented in figures that use a color code. The
first follower is marked in dark blue, while the last follower is marked in
dark red. The vehicles in between follow the pattern shown in the color bar at
the top of each figure. Each color (from left to right) is mapped with the
corresponding increasing index of the vehicles along the string.

---
Figure 7: Behavior of a platoon composed of: leader (black), vehicle $V_{A}$
(red), and vehicle $V_{B}$ (blue).

###  5.1 MSS analysis with two followers

Influence of the vehicles location: Here we consider a heterogeneous platoon
with two followers (labeled as $V_{A}$ and $V_{B}$) and a leading vehicle
(labeled as $V_{L}$). Each vehicle has the following plant and controller

$\displaystyle G_{V_{A}}(z)$ $\displaystyle=\frac{1}{z-1},\quad
K_{V_{A}}(z)=\frac{z}{(z-1)(z-0.7)}$ $\displaystyle G_{V_{B}}(z)$
$\displaystyle=\frac{1.2}{z-1},\quad
K_{V_{B}}(z)=\frac{1.33z}{(z-1)(z-0.88)}.$

For simplicity, we assume that the arrival rates only depend on the
transmitting devices and thus we set $p_{1}=0.87$. In both vehicles, the
strategy used is as in Fig. 4.c) and the time headway is set to $h=3.8$. The
lead vehicle starts from rest and then it applies an acceleration of 2
[$m/s^{2}$] till it reaches a constant speed of 20 [$m/s$]. Finally, from
instant $k=50$ the leader brakes to a stop with a deceleration of $-1.33$
[$m/s^{2}$]. In Fig. 7, vehicle $V_{A}$ is in position $i=1$ and vehicle
$V_{B}$ is in position $i=2$. With this setup, the mean and variance of the
tracking error converge and the means of the position, speed and acceleration
exhibit a good performance. On the other hand, if the followers switch
positions, the variance of the tracking error does not converge in the mean
square sense. This behavior is shown in Fig. 8. The MSS of the platoon is
affected by the location of the vehicles in the string, this is because the
inter-vehicle communication between two vehicles depends on the transmitting
and receiving devices of such vehicles. Changing the vehicles’ location
changes, in general, the communication features.

---
Figure 8: Behavior of a platoon composed of: leader (black), vehicle $V_{B}$
(blue), and vehicle $V_{A}$ (red).

Influence of the arrival rates: We consider a platoon with the strategy in
Fig. 4.b) and time headway $h=3.8$. Here we analyze the effect on MSS of the
probabilities $p_{i}$ for two cases. The first one considers spatially
independent channels and systems $F_{i}$ not depending on other channel
statistics. The plant and controller are those for Vehicle A above. The second
case considers spatially correlated channels and systems $F_{i}$ depending on
other channel statistics. For this case we set
$G_{1}(z)=G_{2}(z)=G_{V_{A}}(z)$, $K_{1}(z)=K_{V_{A}}(z)/p_{1}$ and
$K_{2}(z)=2K_{V_{A}}(z)/(p_{1}+p_{2})$. In the first column of Fig. 9 (case 1)
we present the results for the independent channels case. Two surfaces reveal
the influence between the arrival rates $p_{1}$ and $p_{2}$ on the spectral
ratio conditions for MSS for the mean and the variance. The spectral radius is
larger for low values of $p_{1}$ and $p_{2}$, and decreases as long as the
probabilities approach to 1, which is expected. In the third graphic, a region
in the plane $p_{1}$ vs $p_{2}$ where MSS is achieved is presented. It can be
seen that there is no interplay between $p_{1}$ and $p_{2}$ in this case. We
can also notice that there are minimum values for $p_{1}$ and $p_{2}$ below
which MSS is not possible. The second case results are given in the second
column of Fig. 9. The resulting surfaces reveal an important interplay between
$p_{1}$ and $p_{2}$. Again, higher probabilities reduce the spectral radius
value. The MSS region now clearly exhibits an interaction between channels
statistics to achieve MSS.

Figure 9: MSS surface for: case 1 (left column) - independent case, case 2
(right column) - dependent case.

###  5.2 MSS analysis of homogeneous platoons

Consider a homogeneous platoon of $N=10$ followers. We set the time-headway
$h=4$, $G(z)=(z-1)^{-1}$ and

$K(z)=\frac{0.27z(z+0.88)}{(z-1)(z-0.79)(z-0.8)}.$

The compensation strategy implemented is the one in Fig. 4$(c)$. In Fig. 10
the leader reference is a ramp (vehicle moving with constant speed of
$35\left[m/s\right]$). With this setup, for every $0<p\leq 1$, conditions
$M_{a}(1)=0$ and $M_{b}(1)=0$ are achieved (see Corollary 9). For instance,
with $p=0.9$ we have

$\displaystyle{M}_{a}(z)$
$\displaystyle=\frac{(z-1)^{2}(z+0.79)(z-0.1)}{(z+0.39)(z-0.85)(z^{2}-0.84z+0.56)}$
$\displaystyle{M}_{b}(z)$
$\displaystyle=\begin{bmatrix}\frac{(z-1)^{3}(z+0.79)}{(z+0.39)(z-0.86)(z^{2}-0.84z+0.56)}\\\
\frac{0.24z(z-1)^{2}(z-0.88)}{(z+0.39)(z-0.8)(z-0.86)(z^{2}-0.84z+0.56)}\end{bmatrix}.$

It can be observed that both ${M}_{a}(z)$ and ${M}_{b}(z)$ have at least two
zeros at $z=1$.

Figure 10: MSS analysis of homogeneous platoons. Plots in the $i$-th row refer
to the results for case $i$.

By varying the arrival rate we have different results. In Fig. 10 we show
three different cases. When the probability of success is set to $p=0.9$, the
mean and variance of the tracking error converge uniformly as time approaches
infinity. The spectral radius conditions are met with the values
$\rho(\alpha)=0.8586$ and $\rho(\alpha\otimes\alpha+\delta)=0.8417$. Also, the
system achieves zero error in steady-state, which is also verified through
Corollary 5. When the probability of success is diminished to $p=0.8$, the
platoon becomes mean square unstable. In this example $\rho(\alpha)=0.8597$
and thus, the convergence of the mean is ensured, but the variance does not
converge since $\rho(\alpha\otimes\alpha+\delta)=1.0106\nless 1$. Thus, the
instability is due to the variance behavior. Since the platoon is homogeneous,
the unstable behavior of the variance can be observed from the first follower.
For the third case, neither the mean nor the variance converge. This behavior
is forced by choosing a poor channel with a probability of success $p=0.47$.
The spectral radius conditions are greater than one ($\rho(\alpha)=1.0046$,
$\rho(\alpha\otimes\alpha+\delta)=1.2948$).

###  5.3 MSS analysis of a heterogeneous platoon

Consider a heterogeneous platoon of $N=10$ followers with independent
channels, where each vehicle has its own dynamic and controller, but share the
same structure. The model of the vehicles is characterized by
$G_{i}(z)=g_{i}/(z-1)$, where $g_{i}$ is the gain of the plant. The
controllers structure is as follow

$K_{i}(z)=\frac{(k_{i}/(1+h))z\ (z+c_{i})}{(z-1)(z+\lambda_{i})(z-(h/(1+h)))}$
(40)

where $k_{i}$, $c_{i}$ and $\lambda_{i}$ are the gain, zero and pole of the
controller, respectively. The leader reference is a ramp (moving with constant
speed of $35\left[m/s\right]$). All agents use the compensation strategy in
Fig. 4$(c)$, time-headway with $h=4$, and the channels have the same arrival
rate $p$. In Fig. 11 we present three different cases of convergence for
different values of $p$. For $p=0.9$, the statistics of the tracking error
converge to zero in the mean square sense. Here we have
$\max_{i}\rho(\alpha_{i})=0.8517$ and
$\max_{i}\rho(\alpha_{i}\otimes\alpha_{i}+\delta_{i})=0.8474$, respectively.
In contrast with the homogeneous platoon, the statistics of each follower have
different transient and peak responses. By decreasing the probability of
success to $p=0.82$, the vehicle in the position $i=8$ becomes mean square
unstable. The instability occurs only in the variance of the tracking error.
The conditions obtained from the mean square analysis are
$\max_{i}\rho(\alpha_{i})=0.8531$ and
$\max_{i}\rho(\alpha_{i}\otimes\alpha_{i}+\delta_{i})=1.0109$. Notice that the
non-convergence of the $8-th$ vehicle forces the non-convergence of the
upcoming vehicles. Hence, the whole platoon becomes mean square unstable. For
$p=0.55$, both conditions $\max_{i}\rho(\alpha_{i})=1.0065$ and
$\max_{i}\rho(\alpha_{i}\otimes\alpha_{i}+\delta_{i})=1.3003$ exceed the value
of one. Therefore, the mean and variance of the tracking error dramatically
increase in amplitude.

A platoon with non-constant time headway or whose communication channels have
different probabilities of success is also a heterogeneous platoon. The
expected behavior is similar to the one presented above.

Figure 11: MSS analysis of heterogeneous platoons. Plots in the $i$-th row
refer to the results for case $i$.

###  5.4 Analysis of strategies for data-loss compensation

Now we will illustrate the fact that even if the ideal communication platoon
is designed to reach zero steady-state error according to Assumption 1, when
data-loss occurs, the influence of the chosen strategy could change the
convergence properties. We consider the controller and the plant of the
homogeneous case in Section 5.2.

Figure 12: Example of compensation strategies with non-zero steady-state
statistics.

Example of a non-convergent strategy: Consider a homogeneous platoon with a
data-loss compensation strategy in which the missing data is replaced by zero
(see first row of Fig. 12). In Fig. 13 we show the simulation results for
$N=10$ vehicles, $p=0.98$ and $h=4$. The transfer function ${M}_{a}(z)$
obtained is

${M}_{a}(z)=\frac{(z+0.78)(z^{2}-1.99z+0.988)}{(z-0.853)(z^{2}-0.356z+0.446)}$

and $M_{b}(z)=1$. Even if the channel is almost ideal, and the spectral radius
conditions are compatible with MSS, the system cannot track the reference
because of the missing zeros at $z=1$, as predicted by Corollary 5, and MSS is
not achieved.

Example of a non-zero convergent strategy: Consider a platoon with a to-hold
compensation strategy in the measurement signal as depicted in the second row
of Fig. 12. In Fig. 13 we show the numerical results of a homogeneous platoon
with $p=0.95$ and $h=4$. The conditions obtained are $\rho(\alpha)=0.8535$ and
$\rho(\alpha\otimes\alpha+\delta)=0.7284$. The transfer functions obtained are

${M}_{a}(z)=\frac{(z-1)(z-0.995)(z+0.71)(z+0.027)}{(z-0.05)(z-0.853)(z^{2}-0.356z+0.446)}$

and ${M}_{b}(z)=(z-1)(z-0.05)^{-1}$. As seen, the spectral radius conditions
are satisfied, however, $M_{a}$ and $M_{b}$ have only one zero at $z=1$.
Consequently, the statistics converge to a value different from zero.

Example of a zero convergent strategy: Any of the compensation strategies
employed in section 5.1, 5.2 and 5.3 are zero convergent.

Figure 13: First row: behavior of a platoon with non-convergent strategies.
Second row: behavior of a platoon with non-zero convergent strategies.

###  5.5 String Stability discussion

String stability is a property that guarantees that disturbances are not
amplified along the string of vehicles as they propagate. String stability is
a stronger property when compared with internal stability, since, in addition
to the convergence in time, it demands the signals of interest to remain
uniformly bounded when the number of agents increases. Although in a
deterministic setting string stability has been extensively studied, results
are scarce for stochastic scenarios [40, 11, 4, 9]. Nevertheless, some works
suggest that a platoon with stochastic signals involved is string stable if
the mean and the variance of the error exhibit a string stable behavior [37,
3, 42]. For instance, the first row in Fig. 10 where a homogeneous platoon is
considered, shows a string stable behavior for both the mean and the variance.
Conversely, if we use the parameters $N=15$, $p=0.95$ and $h=3$ we have a
string unstable behavior, as shown in Fig. 14. Note that both the mean and
variance in Fig. 14 are mean square stable and the statistics converge to
zero, however, the string unstable behavior can be appreciated in the peak
magnitude of the statistics. As more vehicles are added to the string, said
amplitudes reach higher peak values. Of course, this behavior is undesirable
in platooning.

Figure 14: String unstable behavior

Some interesting results about platooning with random data loss can be found
in [1], where the authors first impose a string stability behavior for the
mean and then they minimize the corresponding variance. However, this two-step
approach should be applied judiciously because there are cases in which
depending on the platooning setup, the mean behaves as string stable, but the
variance is not even mean square stable, such as in the second row of Fig. 10.
We believe that, for a proper approach to study string stabilization in an
stochastic setting, both the mean and variance of the tracking error must be
required to behave in string stable fashion. Hence, the necessary and
sufficient conditions for MSS derived in this work would be necessary
conditions for string stabilization of platooning over lossy channels.

##  6 Conclusions and future work

In this article, we studied the mean square stabilization of platoons of
vehicles with lossy V2V communication channels. When the dropout occurs, a
compensation strategy was used to replace the missing data or modify an
internal signal of the closed-loop. We derived necessary and sufficient
conditions for MSS considering heterogeneous platoons and spatially correlated
communication channels, guaranteeing the convergence of the statistics of the
tracking errors. We found that the channels’ qualities and the employed
compensation strategy play a crucial role for MSS. Additionally, we derived a
condition that guarantees that said convergence reaches zero in steady-state.
We also present simplified conditions for the case when the channels are
independent. Through simulation examples, we tested the performance of
homogeneous and heterogeneous platoons with different probabilities of success
and compensation strategies. We also provide a brief discussion about the
importance of considering MSS as a basic requirement in order to study string
stabilization over lossy channels.

As part of our future work, we will consider extending the MSS analysis to
platoons with different communication topologies (e.g. leader-predecessor
following) and control schemes (e.g. speed control). Using the MSS conditions
to properly design controllers and compensation strategies is also part of the
future work. Obtaining conditions for string stabilization over lossy channels
is also in the scope of our future work.

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##  Appendix A Appendix

###  A.1 Proof of Lemma 1

By replacing the input position of each agent (16b) in (16c) and (16d), it is
not difficult to show that the open loop of the concatenated system can be
described by the following state-space representation

$\begin{bmatrix}\mathbf{x}(k+1)\\\ \bm{\zeta}(k)\\\
\mathbf{v}(k)\end{bmatrix}=\begin{bmatrix}\mathcal{A}&&\bf{0}&&\mathcal{B}\\\
\mathbf{C}_{\bm{\zeta}}&&\mathbf{D}_{\bm{\zeta}}&&\bf{0}\\\
\mathbf{C_{v}}&&\mathbf{D_{v}}&&\bf{0}\\\ \end{bmatrix}\
\begin{bmatrix}\mathbf{x}(k)\\\ y_{0}(k)\\\
\tilde{\mathbf{v}}(k)\end{bmatrix},$ (41)

where $\mathcal{A}=diag(A_{1},\ldots,A_{N})$ and
$\tilde{\mathbf{v}}(k)=\Theta_{d}(k)\mathbf{v}(k).$ Notice that some matrices
of the state-space representation are matrices of zeros due to transfer
functions being strictly proper. The cascade structure of the platoon is
reflected in the state-space matrices, which are block diagonal
($\mathcal{A}$, $\mathcal{B}$) or lower block bidiagonal
($\mathbf{C}_{\bm{\zeta}}$, $\mathbf{C_{v}}$) matrices. Due to only the first
follower having access to the leader position, matrices
$\mathbf{D}_{\bm{\zeta}}$ and $\mathbf{D_{v}}$ have only one element different
from zero. Then, considering that
$\bar{\Theta}_{d}(k)=\Theta_{d}(k)-\Upsilon$,
$\breve{\mathbf{v}}(k)=\bar{\Theta}_{d}(k)\mathbf{v}(k)$, and defining
$\mathbf{A}=\mathcal{A}+\mathcal{B}\Upsilon\mathbf{C_{v}}$,
$\mathbf{B}=\mathcal{B}\Upsilon\mathbf{D_{v}}$; the lemma is proven
straightforwardly from (41).

###  A.2 Proof of Lemma 2

The three means in (21) are obtained directly by applying the expectation
operator to (19) and considering the fact that the input signal $y_{0}(k)$ is
deterministic. The variance of the state is obtained as follows:

$\displaystyle P_{\mathbf{x}}(k+1)$
$\displaystyle=\mathcal{E}\big{\\{}\left[\mathbf{A}\bar{\mathbf{x}}(k)+\mathcal{B}\bar{\Theta}_{d}(k)\mathbf{v}(k)\right]\left[\mathbf{A}\bar{\mathbf{x}}(k)+\mathcal{B}\bar{\Theta}_{d}(k)\mathbf{v}(k)\right]^{\top}\big{\\}}$
$\displaystyle=\mathbf{A}\
\mathcal{E}\left\\{\bar{\mathbf{x}}(k)\bar{\mathbf{x}}(k)^{\top}\right\\}\mathbf{A}^{\top}+\mathcal{B}\
\mathcal{E}\left\\{\bar{\Theta}_{d}(k)\mathbf{v}(k)\bar{\mathbf{x}}(k)^{\top}\right\\}\mathbf{A}^{\top}$
$\displaystyle\quad+\mathbf{A}\
\mathcal{E}\left\\{\bar{\mathbf{x}}(k)\mathbf{v}(k)^{\top}\bar{\Theta}_{d}(k)^{\top}\right\\}{\mathcal{B}}^{\top}$
$\displaystyle\quad+\mathcal{B}\
\mathcal{E}\left\\{\bar{\Theta}_{d}(k)\mathbf{v}(k)\mathbf{v}(k)^{\top}\bar{\Theta}_{d}(k)^{\top}\right\\}{\mathcal{B}}^{\top}.$

Given the assumption of independence (in time) of the channels in $\Theta(k)$,
we see that

$\displaystyle\mathcal{E}\left\\{\bar{\Theta}_{d}(k)\mathbf{v}(k)\bar{\mathbf{x}}(k)^{\top}\right\\}=\mathcal{E}\left\\{\bar{\Theta}_{d}(k)\right\\}\mathcal{E}\left\\{\mathbf{v}(k)\bar{\mathbf{x}}(k)^{\top}\right\\}=0,$

since $\mathcal{E}\left\\{\bar{\Theta}_{d}(k)\right\\}=0$. Likewise,
$\mathcal{E}\left\\{\bar{\mathbf{x}}(k)\mathbf{v}(k)^{\top}\bar{\Theta}_{d}(k)^{\top}\right\\}=0$.
Using the property in (2), we have

$\displaystyle\mathcal{E}\left\\{\bar{\Theta}_{d}(k)\mathbf{v}(k)\mathbf{v}(k)^{\top}\bar{\Theta}_{d}(k)^{\top}\right\\}=P_{\Theta}\odot\mathcal{E}\left\\{\mathbf{v}(k)\mathbf{v}(k)^{\top}\right\\}.$

Since $y_{0}(k)$ is deterministic, the covariance matrix of the tracking error
and the channel input can be written as

$\displaystyle P_{\bm{\zeta}}(k)$
$\displaystyle=\mathcal{E}\left\\{\bar{\bm{\zeta}}(k)\
\bar{\bm{\zeta}}(k)^{\top}\right\\}=\mathbf{C}_{\bm{\zeta}}\
\mathcal{E}\left\\{\bar{\mathbf{x}}(k)\bar{\mathbf{x}}(k)^{\top}\right\\}{\mathbf{C}_{\bm{\zeta}}}^{\top}$
$\displaystyle P_{\mathbf{v}}(k)$
$\displaystyle=\mathcal{E}\left\\{\bar{\mathbf{v}}(k)\bar{\mathbf{v}}(k)^{\top}\right\\}=\mathbf{C_{v}}\
\mathcal{E}\left\\{\bar{\mathbf{x}}(k)\bar{\mathbf{x}}(k)^{T}\right\\}{\mathbf{C_{v}}}^{\top}.$

Which completes the proof

###  A.3 Proof of Theorem 7

In this proof, we slightly modify the notation to explicitly include the
number of vehicles of the platoon. This will allow us to write the platoon
dynamics recursively in terms of smaller platoons to use mathematical
induction and prove our claim. Hence, given the representation in Lemma 1 we
write

$\displaystyle\mathbf{A}\rightarrow\mathbf{A}_{N}$
$\displaystyle=\left[\begin{array}[]{c|c}\mathbf{A}_{N-1}&\bm{0}\\\
\hline\cr\Psi_{N}&\alpha_{N}\end{array}\right]$
$\displaystyle\mathbf{B}\rightarrow\mathbf{B}_{N}$
$\displaystyle=\begin{bmatrix}\beta_{1}^{\top}&\bm{0}&\cdots&\bm{0}\end{bmatrix}^{\top}=\left[\begin{array}[]{c|c}\mathbf{B}_{N-1}&\bm{0}\end{array}\right]^{\top}.$

where $\beta_{i}=p_{i}B_{i}D_{v_{i}}$ and $\Psi_{N}=[\bm{0}\;\;\gamma_{N}]$
for $N\geq 3$, and $\Psi_{\bf{2}}=\gamma_{2}$. We also define, for $N\geq 3$,

$\displaystyle\varPhi_{\bm{\zeta}_{N}}$
$\displaystyle=\begin{bmatrix}\bm{0}&&D_{\zeta_{N}}C_{y_{N-1}}\end{bmatrix},\textrm{
with }\varPhi_{\bm{\zeta}_{2}}=D_{\zeta_{2}}C_{y_{1}},$
$\displaystyle\varPhi_{\mathbf{v}_{N}}$
$\displaystyle=\begin{bmatrix}\bm{0}&&D_{v_{N}}C_{y_{N-1}}\end{bmatrix},\textrm{
with }\varPhi_{\mathbf{v}_{2}}=D_{v_{2}}C_{y_{1}},$

where $\bf{0}$ is a zero matrix of appropriate dimensions. We also explicitly
include the length of a platoon in the concatenated state, the error and
channel input vectors, and write $\mathbf{x_{N}}(k)$, $\bm{\zeta_{N}}(k)$ and
$\mathbf{v_{N}}(k)$ respectively. The terms $x_{N}(k)$ $\zeta_{N}(k)$ and
$v_{N}(k)$ refer to the state, the error and the channel input of the $N$-th
vehicle. With this notation, we can write a representation of a platoon with
$N$ vehicles in terms of the set of $N-1$ predecessor vehicles, allowing us to
analyze the effects of adding a new vehicle to the string. Indeed, it is
straightforward to see that

${\mu_{\mathbf{x_{N}}}}(k)=\left[\begin{array}[]{c | c}{\mu_{\mathbf{x_{N-1}}}}(k)^{\top}&\;{\mu_{x_{N}}}(k)^{\top}\end{array}\right]^{\top},$

and similarly for the error and channel input means. Thus, from (25), the mean
of the alternative state
$\mathbf{x_{N}}^{*}(k)=\mathbf{x_{N}}(k)-\mathbf{x_{N}}(k-1)$ satisfies
${\mu_{\mathbf{x_{N}^{*}}}}(k+1)=\mathbf{A}_{\bf{N}}{\mu_{\mathbf{x_{N}^{*}}}}(k)+\mathbf{B}_{\bf{N}}m_{0}(k).$
For the first follower we can write
${\mu_{\mathbf{x_{1}^{*}}}}(k+1)=\alpha_{1}{\mu_{\mathbf{x_{1}^{*}}}}(k)+\beta_{1}m_{0}(k)$
which converges if and only if the condition $\rho(\alpha_{1})<1$ holds.
Furthermore, we have that (see proof of Theorem 4)

$\displaystyle\mu_{\bm{\zeta}_{1}}$
$\displaystyle=\mu_{\bm{\zeta}_{1}}+\left(C_{\zeta_{1}}(I-\alpha_{1})^{-1}B_{1}D_{v_{1}}p_{1}+{D_{\zeta_{1}}}\right)m_{0}.$
$\displaystyle=\mu_{\bm{\zeta}_{1}}+M_{a_{1}}(1)m_{0}.$ (42)

Clearly, the equality in (42) requires $M_{a_{1}}(1)=0$ to be valid. For $N=2$
we can write

${\mu_{x_{2}^{*}}}(k+1)=\alpha_{2}{\mu_{x_{2}^{*}}}(k)+\Psi_{2}{\mu_{\mathbf{x_{1}}}}^{*}(k).$
(43)

Consequently, ${\mu_{x_{2}^{*}}}(k)$ converges when $\rho(\alpha_{2})<1$ and
$\rho(\alpha_{1})<1$. Specifically, ${\mu_{x_{2}^{*}}}(k)$ converges to
${\mu_{x_{2}^{*}}}=(I-\alpha_{2})^{-1}\Psi_{2}{\mu_{\mathbf{x_{1}^{*}}}}$.
From (26), the corresponding follower error can be written as
$\mu_{{\zeta}_{2}}(k)=\mu_{{\zeta}_{2}}(k-1)+C_{\zeta_{2}}\mu_{x_{2}^{*}}(k)+\varPhi_{\bm{\zeta}_{2}}\mu_{\mathbf{x}_{1}^{*}}(k).$
In the limit
$\mu_{{\zeta}_{2}}:=\lim_{k\rightarrow\infty}\mu_{{\zeta}_{2}}(k)$ satisfies

$\displaystyle\mu_{{\zeta}_{2}}$
$\displaystyle=\mu_{{\zeta}_{2}}+C_{\zeta_{2}}(I-\alpha_{2})^{-1}\Psi_{\bf{2}}{\mu_{\mathbf{x_{1}^{*}}}}+\varPhi_{\bm{\zeta}_{2}}\mu_{\mathbf{x}_{1}^{*}}$
$\displaystyle=\mu_{{\zeta}_{2}}+M_{a_{2}}(1)C_{y_{1}}\mu_{{x}_{1}^{*}}.$

Since
$M_{a_{2}}(1)=C_{\zeta_{2}}(I-\alpha_{2})^{-1}B_{2}D_{v_{2}}p_{2}+{D_{\zeta_{2}}}=0$,
the convergence is ensured. For an arbitrary $N$ we have

${\mu_{x_{N}^{*}}}(k+1)=\alpha_{N}{\mu_{x_{N}^{*}}}(k)+\Psi_{\bf{N}}{\mu_{\mathbf{x^{*}_{N-1}}}}(k).$
(44)

which additionally requires $\rho(\alpha_{N})<1$ to converge, yielding
$\mu_{{\zeta}_{N}}=\mu_{{\zeta}_{N}}+C_{\zeta}\mu_{x^{*}_{N}}+\varPhi_{\bm{\zeta}_{N}}\mu_{\mathbf{x}_{N-1}^{*}}=\mu_{{\zeta}_{N}}+M_{a_{N}}(1)C_{y_{N-1}}\mu_{{x}_{N-1}^{*}}$,
which requires $M_{a_{N}}(1)=0$ to ensure the existence of the stationary
value $\mu_{{\zeta}_{N}}$. Clearly $\mu_{{\zeta}_{N}}$ converges if and only
if $\max_{i}\rho(\alpha_{i})<1$ and $M_{a_{i}}(1)=0$ for every $i=1,2,\dots
N$, which proof the first statement in Theorem 7.

It is easy to prove, mutatis mutandis, that the convergence of
$\mu_{\mathbf{v}_{N}}$ is guaranteed if and only if
$\max_{i}\rho(\alpha_{i})<1$ and $M_{b_{i}}(1)=0$ for every $i=1,2,\dots N$.

On the other hand, the covariance matrix of the state can be written as

$P_{\bf{x_{N}}}(k)=\left[\begin{array}[]{c|c}P_{\bf{x_{N-1}}}(k)&P_{{\bf{x}}_{N-1}{x}_{N}}(k)\\\
\hline\cr P_{{x}_{N}{\bf{x}}_{N-1}}(k)&P_{x_{N}}(k)\end{array}\right],$

Thus, the convergence of the error covariance matrix
$P_{\bf{\zeta}_{N}}(k)=\mathbf{C}_{\bm{\zeta}_{N}}P_{\bf{x_{N}}}(k)\mathbf{C}_{\bm{\zeta}_{N}}^{\top},$
requires the convergence of $P_{\bf{x_{N-1}}}(k)$, $P_{x_{N}}(k)$ and
$P_{{\bf{x}}_{N-1}{x}_{N}}(k)=P_{{x}_{N}{\bf{x}}_{N-1}}^{\top}(k)$. To ease
the notation we define

$\displaystyle\Omega_{N}(k)=$ $\displaystyle
B_{N}\left[P_{\theta_{N}}\odot(\mu_{v_{N}}(k)\
{\mu_{v_{N}}(k)}^{\top})\right]B_{N}^{\top}$ $\displaystyle R_{N}(k)=$
$\displaystyle\Psi_{\bf{N}}P_{\mathbf{x_{N-1}}}(k){\Psi_{\bf{N}}}^{\top}$
$\displaystyle+B_{N}\left[P_{\theta_{N}}\odot(\varPhi_{\mathbf{v}_{N}}P_{\mathbf{x_{N-1}}}(k)\
{\varPhi_{\mathbf{v}_{N}}}^{\top})\right]{B_{N}}^{\top}$
$\displaystyle\Lambda_{N}(k)=$
$\displaystyle\alpha_{N}P_{x_{N}{\bf{x_{N-1}}}}(k){\Psi_{\bf{N}}}^{\top}+\Psi_{\bf{N}}P_{{\bf{x_{N-1}}}x_{N}}(k)\alpha_{N}^{\top}$
$\displaystyle+B_{N}\left[P_{\theta_{N}}\odot(C_{v_{N}}P_{x_{N}{\bf{x_{N-1}}}}(k)\
{\varPhi_{\mathbf{v}_{N}}}^{\top})\right]{B_{N}}^{\top}$
$\displaystyle+B_{N}\left[P_{\theta_{N}}\odot(\varPhi_{\mathbf{v}_{N}}P_{{\bf{x_{N-1}}}x_{N}}(k)\
{C_{v_{N}}}^{\top})\right]{B_{N}}^{\top}.$

It must be noted that $\Omega_{N}(k)$, $R_{N}(k)$ and $\Lambda_{N}(k)$
converge if and only if $\mu_{v_{N}}(k)$, $P_{\mathbf{x_{N-1}}}(k)$ and
$P_{x_{N},{\bf{x_{N-1}}}}(k)$ converge, respectively.

Given the heterogeneous setup with mutually independent channels, we notice
that, for $N=1$

$\displaystyle P_{x_{1}}(k+1)=$ $\displaystyle\;\alpha_{1}\ P_{x_{1}}(k)\
\alpha_{1}^{\top}+\Omega_{1}(k)$ (45)
$\displaystyle+B_{1}\left[P_{\theta_{1}}\odot({C_{v_{1}}}\
P_{x_{1}}(k){{C_{v_{1}}}}^{\top})\right]B_{1}^{\top}.$

Using a similar reasoning as in the proof of Theorem 4 for the variance
convergence, we notice that, given $\rho(\alpha_{1})<1$ and $M_{b_{1}}(1)=0$,
$\Omega_{1}(k)$ converges and thus the matrix $P_{x_{1}}(k)$ also converges if
$\rho((\alpha_{1}\otimes\alpha_{1})+\delta_{1})<1$, where $\delta_{1}$ is as
in Theorem 7.

For $N=2$, we have to analyze $P_{x_{1}}(k)$, $P_{x_{2}}(k)$ and the cross
term $P_{x_{1}x_{2}}(k)$. Since $P_{\Theta_{N}}$ is diagonal, then the cross
covariance can be written as

$\displaystyle P_{x_{1}x_{2}}(k+1)$
$\displaystyle=\alpha_{1}P_{x_{1}}(k)\gamma_{2}^{\top}+\alpha_{1}P_{x_{1}x_{2}}(k)\alpha_{2}^{\top},$
(46)

where it is possible to see that $P_{x_{1}x_{2}}(k)$ converges only if
$\rho(\alpha_{2})<1$. Note that $P_{x_{1}x_{2}}(k)$ convergence requires
$P_{x_{1}}(k)$ to also converge, which is guaranteed with
$\rho(\alpha_{1})<1$, $\rho((\alpha_{1}\otimes\alpha_{1})+\delta_{1})<1$ and
$M_{b_{1}}(1)=0$. For $P_{x_{2}}$, the diagonal structure of $P_{\Theta_{N}}$
allow us to write the variance of the state for the second follower as

$\displaystyle P_{x_{2}}(k+1)=$
$\displaystyle\Omega_{2}(k)+R_{2}(k)+\Lambda_{2}(k)+\alpha_{2}P_{x_{2}}(k)\alpha_{2}^{\top}$
$\displaystyle+B_{2}\left[P_{\theta_{2}}\odot({C_{v_{2}}}\
P_{x_{2}}(k){{C_{v_{2}}}}^{\top})\right]B_{2}^{\top}.$

We can see that the convergence of $P_{x_{2}}(k)$ requires that
$\Omega_{2}(k)$,$R_{2}(k)$ and $\Lambda_{2}(k)$ converge. $\Omega_{2}(k)$
converge if and only if $\rho(\alpha_{1})<1$, $\rho(\alpha_{2})<1$,
$M_{b_{1}}(1)=0$ and $M_{b_{2}}(1)=0$. $R_{2}(k)$ converge if and only if
$P_{x_{1}}$ does, which is analyzed above. $\Lambda_{2}(k)$ converge if and
only if $P_{x_{1}x_{2}}(k)$ converges, which is also analyzed above. Thus, it
is not difficult to see that, for $P_{x_{2}}(k)$ to converge, the additionally
condition $\rho((\alpha_{2}\otimes\alpha_{2})+\delta_{2})<1$ must be included.

For any $N>1$ we have that the cross covariance between the last vehicle and
the rest of the platoon is given as

$\displaystyle P_{{\bf{x_{N-1}}}x_{N}}(k+1)$
$\displaystyle=\mathbf{A}_{\bf{N-1}}P_{\bf{x_{N-1}}}(k)\Psi_{\bf{N}}^{\top}$
$\displaystyle\quad\quad+\mathbf{A}_{\bf{N-1}}P_{{\bf{x_{N-1}}}x_{N}}(k)\alpha_{N}^{\top}.$
(47)

Assuming that $P_{\bf{x_{N-1}}}(k)$ converges, then we see that we require
$\rho(\mathbf{A}_{\bf{N-1}}\otimes\alpha_{N})<1$ for
$P_{{\bf{x_{N-1}}}x_{N}}(k)$ to converge. Given the structure of
$\mathbf{A}_{\bf{N-1}}$ we notice that
$\rho(\mathbf{A}_{\bf{N-1}}\otimes\alpha_{N})<1\Leftrightarrow\max(\rho(\mathbf{A}_{\bf{N-1}}),\rho(\alpha_{N}))<1$.
This is implies that $\rho(\alpha_{N})<1$. The covariance matrix of the $N$-th
vehicle can then be written as

$\displaystyle P_{x_{N}}(k+1)$
$\displaystyle=\Omega_{N}(k)+R_{N}(k)+\Lambda_{N}(k)+\alpha_{N}P_{x_{N}}(k)\alpha_{N}^{\top}$
$\displaystyle+B_{N}$
$\displaystyle\left[P_{\theta_{N}}\odot(C_{v_{N}}P_{x_{N}}(k)\
C_{v_{N}}^{\top})\right]B_{N}^{\top}$ (48)

$\Lambda_{N}(k)$ converge if $P_{{\bf{x_{N-1}}}x_{N}}(k)$ does, which requires
$\rho(\alpha_{N})<1$. If additionally we require $M_{b_{N}}(1)=0$, then we
ensure $\Omega_{N}(k)$ to converge. $R_{N}(k)$ converge given that
$P_{\bf{x_{N-1}}}(k)$ does. Thus, it is not difficult to see from (48) that
additionally requiring that $\rho(\alpha_{N}\otimes\alpha_{N}+\delta_{N})<1$
ensures the convergence of $P_{x_{N}}(k)$.

Hence, applying these individual conditions recursively yields the second
statement of Theorem 7.