# Synthesis of Output-Feedback Controllers for Mixed Traffic Systems in
Presence of Disturbances and Uncertainties

Shima Sadat Mousavi†, Somayeh Bahrami‡, and Anastasios Kouvelas† †The authors
are with the Institute for Transport Planning and Systems, ETH Zurich,
Switzerland, {shimaossadat.mousavi<EMAIL_ADDRESS>author is with the Department of Electrical Engineering, Razi University,
Kermanshah, Iran<EMAIL_ADDRESS>
###### Abstract

In this paper, we study mixed traffic systems that move along a single-lane
ring-road or open-road. The traffic flow forms a platoon, which includes a
number of heterogeneous human-driven vehicles (HDVs) together with only one
connected and automated vehicle (CAV) that receives information from several
neighbors. The dynamics of HDVs are assumed to follow the optimal velocity
model (OVM), and the acceleration of the single CAV is directly controlled by
a dynamical output-feedback controller. The ultimate goal of this work is to
present a robust control strategy that can smoothen the traffic flow in the
presence of undesired disturbances (e.g. abrupt deceleration) and parametric
uncertainties. A prerequisite for synthesizing a dynamical output controller
is the stabilizability and detectability of the underlying system.
Accordingly, a theoretical analysis is presented first to prove the
stabilizability and detectability of the mixed traffic flow system. Then, two
$H_{\infty}$ control strategies, with and without considering uncertainties in
the system dynamics, are designed. The efficiency of the two control methods
is subsequently illustrated through numerical simulations, and various
experimental results are presented to demonstrate the effectiveness of the
proposed controller to mitigate disturbance amplification and achieve platoon
stability.

##  I Introduction

In recent decades, thanks to developments in automation, such as the emergence
of automated vehicles or automated infrastructures, a tremendous revolution
has occurred in transportation systems. The transition phase from using only
human-driven vehicles (HDVs) to fully connected and automated vehicles (CAVs)
results in new challenges and creates a strong motivation to study the mixed
traffic systems, that include both HDVs and CAVs (e.g., see [1, 2] and the
references therein). In this direction, new opportunities arise to utilize the
potential abilities of CAVs to control a transportation network, manage
congestion, and promote the efficiency and safety of traffic systems.

More traditional methodologies for controlling traffic flow employ controllers
and actuators at fixed locations, among which variable speed limits (VSLs) and
ramp metering (RMs) can be mentioned [3]. However, the installation of these
actuators is not cost-effective and reduces the flexibility of the control
system. On the other hand, the advent of CAVs as mobile actuators–so-called
Lagrangian actuators–paves the way for applying traffic flow control in a more
effective and flexible manner. For instance, if all the involved vehicles are
CAVs, efficient control strategies, such as adaptive cruise control (ACC) and
cooperative adaptive cruise control (CACC), can be employed and lead to a
desirable system performance [4, 5, 6, 7]. Nevertheless, in a mixed traffic
system, where the penetration rate of CAVs is less than one, new challenges
are introduced that require further theoretical and experimental analyses.

In this direction, we consider a mixed traffic system, including one CAV and
numerous HDVs, for the two common scenarios of a ring-road and an open-road
and reveal how the single CAV is capable of controlling the entire network. In
the following, we first present a review of some relevant works in the
literature, and subsequently, we discuss the main contributions of this work.

###  I-A Literature Review

There are a few experimental studies that verify the emergence of stop-and-go
waves in traffic flow systems. For instance, in the study in [8], the outcome
of a practical experiment on a single-lane ring-road demonstrated that a
platoon consisting only of HDVs has the potential to initiate stop-and-go
waves. These waves that travel upstream along the road make a uniform flow
unstable and create a so-called _phantom traffic jam_. This phenomenon of
instability has been studied in the literature from macroscopic [9], cellular
automaton [10], and microscopic [11] point of view. The emerging nonlinear
waves can be amplified by some effects such as stochastic behaviour of human
drivers, lane changing, road characteristics, and ramps, to name a few. In
[12], a field experiment was conducted to show that utilizing a single CAV in
a platoon on a circular roadway can dissipate the undesired waves. Moreover,
in [13], through some theoretical analysis, the capability of a single CAV to
control the traffic flow on a ring-road was investigated.

In fact, since the platoon is connected, and neighboring vehicles can
interact, a sparse number of CAVs–that act as mobile actuators–can influence
the whole network and stabilize the traffic system. The notions of string and
ring stability have been employed here for the stability analysis of
interconnected vehicles on a string and on a ring roadway, respectively [14,
15, 16]. In the same direction, and for mixed traffic systems, the string
stability of a mixed platoon of infinite length has been analyzed in [17].
Furthermore, in [18], a linear stability condition has been stated in terms of
the penetration rate and spatial distribution of CAVs.

In order to dissipate stop-and-go waves in a mixed traffic system, an
appropriate control strategy can be provided that is applied to the CAVs as
the controllers. To establish a theoretical analysis for these systems, we
should first derive a mathematical model that represents the dynamical
behaviour of HDVs. In this direction, there are different car-following
models, among which the optimal-velocity-follow-the-leader (OV-FTL) model [11]
and the intelligent driver model (IDM) [19] can be indicated. These models are
nonlinear in principle, but most of studies in the field of mixed traffic
systems utilize a linearized version of the nonlinear dynamics around the
equilibrium flow (see e.g., [20, 21, 16, 22, 23]).

A fundamental network property that should be checked before designing a
controller is its controllability or stabilizability [24, 25, 26]. If a linear
system is controllable, then a control signal can be designed to steer the
system from any initial state to any final state within finite time. A weaker
condition that is necessary for the existence of a controller is the
stabilizability of a system. A linear system is stabilizable if with a
suitable choice of control signals, all the states remain bounded or converge
to constant values [27]. Recently, some works in the literature have focused
on providing a rigorous controllability and stabilizability analysis for ring-
road mixed traffic systems with one single CAV.

In [13, 22], it is assumed that all HDVs in the platoon are homogeneous, which
is a strong assumption for practical scenarios. In fact, the problem where
heterogeneity of HDVs is considered is closer to reality, but theoretically
more challenging. In [28, 23], a controllability analysis for mixed traffic
systems, including one CAV and a number of heterogeneous HDVs, is provided. It
is stated that under a restrictive condition on parameters of the dynamic
model, all nonzero eigenvalues of the system are controllable; while there
exists only one uncontrollable eigenvalue at origin. More recently, in [29],
by considering the similar condition on the parameters of the system, the
controllability of a so-called “$1+n$” mixed platoon, forming a string at a
signalized intersection, is provided. Moreover, in [30], by defining a new
notion of leading cruise control, the controllability of a platoon along a
string is analyzed. None of these works provide the controllability analysis
of a mixed platoon in the most general case and without assuming any
constraint on the system parameters.

There are numerous works in the literature, proposing various control
strategies to stabilize a mixed platoon on an open-road or a ring-road [31,
32, 13, 33, 12, 34, 35] . However, in most of these works, the communication
topology of the network which represents the capability of the CAV to receive
information from its neighboring vehicles has been neglected [31, 36, 20, 37,
35], and it has been assumed that it can be connected to any vehicle in the
platoon. More recently, in [28], the issue of limited communication has been
considered, and a _structured_ optimal control has been proposed, which is in
general computationally intractable [38] and results in a sub-optimal
solution. In addition, the HDVs usually do not follow deterministic dynamic
models, and there exists uncertainties in the model parameters of HDVs, that
can affect the efficiency of the control strategies. Accordingly, it is needed
to provide robust control methods that dissipate the perturbations of the
traffic flow in the presence of system uncertainties (see e.g., [39, 40, 41]).

###  I-B Contributions

In this paper, we consider a mixed traffic system with one CAV and numerous
heterogeneous HDVs. The mixed platoon is studided in two cases of a ring-road
and an open-road. The traffic system is modeled at a microscopic scale, and
the dynamics of HDVs are represented by the optimal velocity model.
Furthermore, the acceleration of the CAV is directly controlled. For this
system, we first establish a stabilizability analysis for the two cases of a
ring-road and an open-road. Moreover, since the goal is to synthesize an
output-feedback controller, the detectability analysis of the system is also
necessary, which is presented subsequently. As a real-world scenario, we also
consider the limited communication capability of the CAV in this work. In
order to deal with the topological communication constraints, we propose an
output-feedback controller that employs the information of a sparse set of
vehicles in the control signal design. Our proposed method offers a robust
$H_{\infty}$ controller, that not only increase the efficiency of the CAV, but
it also dampens the disturbances appearing as nonlinear waves and improves the
performance in the behavior of the entire traffic network.

In summary, the main contributions of this work are listed as follows:

  * •

We analyze the controllability and observability of a mixed traffic system
with one CAV and numerous heterogeneous HDVs for both common scenarios of a
ring-road and an open-road in the most general case. In fact, unlike the
existing works in the literature, such as [23], that investigates the
stabilizability under restrictive parameter constraints, we prove that, for
any value of system parameters, the mixed platoon is stabilizable. This
analysis verifies the ability of the single CAV to make the states of all HDVs
converge to desired values. Further, since we aim to synthesize a dynamic
output controller that can utilize the states of only a subset of HDVs, we
prove in this work that the mixed traffic system is also detectable.

  * •

In order to dampen the undesired perturbations occurring in a mixed traffic
flow where there is no uncertainties in the system parameters, we propose a
solution based on synthesizing an $H_{\infty}$ output dynamic feedback
controller. This controller also tackles the issue of the CAV’s communication
constraints. To the best of our knowledge, this is the first time in the
literature that an output dynamic controller is utilized for the control of a
mixed platoon. Unlike some existing control strategies, e.g., [7, 42], that
aim to increase the local efficiency around the CAVs, our method offers a
controller that improves the performance in the behavior of the entire traffic
network. More importantly, as we consider a higher degree of freedom in
designing the control strategy, our proposed control method leads to a global
optimal solution, while the structural control method in [23] results is a
sub-optimal one.

  * •

As the next and the more practical scenario, we consider the model mismatch
and parametric uncertainties in the dynamics of HDVs and provide a robust
control strategy that can smoothen the traffic flow in the presence of
disturbances and uncertainties.

###  I-C Outline

The rest of this paper is organized as follow. In Section II, the model of a
mixed traffic system is presented, and the main problems of this work are
formulated. Section III discusses the stabilizability and detectability of a
mixed platoon with a single CAV for both ring-road and open-road setups. In
section IV, we propose an output dynamic controller for a mixed traffic flow
that has no uncertainty in the system parameters. Section V establishes a
control strategy that smoothen the traffic flow in the presence of
uncertainties. In Section VI, numerical validations of the results as well as
a numerical comparison with some of existing works in the literature are
provided. Finally, Section VII concludes the paper.

##  II Preliminaries

In this section, we first present a dynamic model for a mixed traffic system
on a single-lane ring-road, and then we formulate the problem.

We denote the set of real and complex numbers by $\mathbb{R}$ and
$\mathbb{C}$, respectively. For $a\in\mathbb{C}$, $\mathrm{Re}(a)$ represents
its real part. We denote the transpose of matrix $M$ by $M^{T}$. Also,
$\mathrm{det}(M)$ represents its determinant. For a vector space
$\mathcal{V}$, $\mathrm{dim}(\mathcal{V})$ indicates its dimension. The
identity matrix is denoted by $I$, and its $j$-th column is designated by
$e_{j}$. Also, $0_{n\times m}$ represents an $n\times m$ zero matrix. We also
show by $0$ a zero matrix of an appropriate dimension. For
$\alpha_{1},\ldots,\alpha_{n}\in\mathbb{R}$,
$A=\mathrm{diag}(\alpha_{1},\ldots,\alpha_{n})$ is an $n\times n$ diagonal
matrix whose diagonal elements are $\alpha_{1},\ldots,\alpha_{n}$. For a
matrix $M\in\mathbb{R}^{n\times n}$, where $M=M^{T}$, $M\succ 0$ (resp.,
$M\succeq 0$) implies that $M$ is a positive definite (resp., positive semi-
definite) matrix. $||.||$ denotes the Euclidean norm of vectors and the
induced norm of matrices. Also, $||.||_{\mathcal{F}}$ denotes Frobenius norm
of matrices.

###### Fact 1

[43] For any matrix $A\in\mathbb{R}^{n\times n}$, we have
$||A||<||A||_{\mathcal{F}}$, where
$||A||_{\mathcal{F}}=(\sum_{i=1}^{n}\sum_{j=1}^{n}a_{ij}^{2})^{\frac{1}{2}}$.

###### Fact 2

[43] For any matrix $A\in\mathbb{R}^{n\times n}$, $||A||\leq 1$ if and only if
$A^{T}A\preceq I$.

###  II-A Modeling a Mixed Traffic System

We study a mixed traffic system that is a network of $n$ vehicles, including
one CAV and $n-1$ HDVs. We consider two cases of a single-lane ring-road and a
single-lane open-road with length $D$. In Fig. 1(a), a schematic diagram of
this network is illustrated, where the red car denotes the CAV, indexed with
$1$, and all others are HDVs. The position and velocity of vehicle $i$ are
denoted by $p_{i}$ and $v_{i}$, respectively. We define as
$s_{i}=p_{i-1}-p_{i}$ the back-to-back distance of the $i$-th vehicle from the
$i-1$-th vehicle.

There are different models in the literature to represent the car-following
dynamics of human-driven vehicles (see e.g., [19, 21, 11]). For instance, the
optimal velocity model (OVM) [11] can be described as:

$\displaystyle\dot{s}_{i}(t)$ $\displaystyle=v_{i-1}(t)-v_{i}(t)$ (1)
$\displaystyle\dot{v}_{i}(t)$
$\displaystyle=H_{i}(v_{i}(t),s_{i}(t),\dot{s}_{i}(t)),$

where $H_{i}(\cdot)$ is the acceleration of vehicle $i$, that is a nonlinear
function of its velocity $v_{i}$, the spacing $s_{i}$, and the relative
velocity $\dot{s}_{i}$. Note that unlike most of the works in literature
(e.g., [13, 31, 2]), in this paper, HDVs are assumed to be heterogeneous, and
thus, the dynamics of each vehicle $i$ is described by a distinct nonlinear
function $H_{i}(\cdot)$. One can see that at the equilibrium point of dynamics
(1), all vehicles have the same velocity $v^{*}$. Moreover, since we have
$\dot{v}^{*}=0$, the spacing $s^{*}_{i}$ is computed by
$0=H_{i}(v^{*},s^{*}_{i},0)$.

Figure 1: (a)–(b) Schematic of a mixed traffic system on a ring-road and the
corresponding graph; (c)–(d) Schematic of a mixed traffic system on an open-
road and the corresponding graph.

Now, let us define the state error
$x_{i}^{T}=\begin{bmatrix}\bar{s}_{i}&\bar{v}_{i}\end{bmatrix}=\begin{bmatrix}s_{i}-s^{*}_{i}&v_{i}-v^{*}\end{bmatrix}$.
Then, by linearization of (1) around the equilibrium point
$\begin{bmatrix}s^{*}_{i}&v^{*}\end{bmatrix}^{T}$, for $i=2,\ldots,n$, one can
derive a linear time-invariant (LTI) model for the $i$-th HDV as:

$\begin{aligned} \dot{\bar{s}}_{i}(t)&=\bar{v}_{i-1}(t)-\bar{v}_{i}(t)\\\
\dot{\bar{v}}_{i}(t)&=\beta_{i1}\bar{s}_{i}(t)-\beta_{i2}\bar{v}_{i}(t)+\beta_{i3}\bar{v}_{i-1}\end{aligned},$

where

$\beta_{i1}=\dfrac{\partial{H_{i}}}{\partial{s_{i}}},\ \
\beta_{i2}=\dfrac{\partial{H_{i}}}{\partial{\dot{s}_{i}}}-\dfrac{\partial{H_{i}}}{\partial{v_{i}}},\
\ \beta_{i3}=\dfrac{\partial{H_{i}}}{\partial{\dot{s}_{i}}},$

computed at the equilibrium point. Due to some physical constraints imposed by
the behavior of HDVs in practice [13], one should consider

$\beta_{i1}>0,\>\>\>\beta_{i2}>0,\>\>\>\beta_{i3}>0.$ (2)

Now, for the dynamics of the single CAV, we consider two cases of a ring-road
and an open-road: 1) Corresponding to the case of a ring-road, the dynamics of
the single CAV whose acceleration can be directly controlled are given by:

$\begin{aligned} \dot{\bar{s}}_{1}(t)&=\bar{v}_{n}(t)-\bar{v}_{1}(t)\\\
\dot{\bar{v}}_{1}(t)&=u(t)\end{aligned},$ (3)

where $u(t)\in\mathbb{R}$ is the control signal. 2) In the case of an open
road, we have:

$\begin{aligned} \dot{\bar{s}}_{1}(t)&=u_{1}(t)-\bar{v}_{1}(t)\\\
\dot{\bar{v}}_{1}(t)&=u_{2}(t)\end{aligned},$ (4)

where $u_{1}(t),u_{2}(t)\in\mathbb{R}$ are two external inputs, and
$u(t)=\begin{bmatrix}u_{1}&u_{2}\end{bmatrix}^{T}\in\mathbb{R}^{2}$. In fact,
in the case of an open-road, both acceleration and velocity of the CAV are
directed by external inputs.

Now, by defining the aggregated vector of states of all vehicles as
$x=\begin{bmatrix}x_{1}^{T}&x_{2}^{T}&\ldots&x_{n}^{T}\end{bmatrix}^{T}\in\mathbb{R}^{2n}$,
one can derive the following LTI dynamics for the overall system:

$\dot{x}(t)=Ax(t)+Bu(t),$ (5)
$A=\begin{bmatrix}J_{1}&0&\ldots&\ldots&0&J_{2}\\\
A_{21}&A_{22}&0&\ldots&\ldots&0\\\ 0&A_{31}&A_{32}&0&\ldots&0\\\
\vdots&\ddots&\ddots&\ddots&\ddots&\vdots\\\
0&\ldots&\ldots&0&A_{n1}&A_{n2}\end{bmatrix},\>\>\>B=\begin{bmatrix}B_{1}\\\
B_{2}\\\ B_{2}\\\ \vdots\\\ B_{2}\end{bmatrix},$ (6)

with $J_{1}=\begin{bmatrix}0&-1\\\ 0&0\end{bmatrix}$, and for $i=2,\ldots,n$,
we have:

$A_{i1}=\begin{bmatrix}0&1\\\
0&\beta_{i3}\end{bmatrix},\>\>\>A_{i2}=\begin{bmatrix}0&-1\\\
\beta_{i1}&-\beta_{i2}\end{bmatrix}.$ (7)

Now, for the case of a ring-road, we have $B\in\mathbb{R}^{n\times 1}$ and
define:

$J_{2}=\begin{bmatrix}0&1\\\ 0&0\end{bmatrix},\>\>\>B_{1}=\begin{bmatrix}0\\\
1\end{bmatrix},\>\>\>B_{2}=\begin{bmatrix}0\\\ 0\end{bmatrix}.$ (8)

Moreover, for an open-road, we have $B\in\mathbb{R}^{n\times 2}$, and $J_{2}$,
$B_{1}$, and $B_{2}$ are defined as:

$J_{2}=\begin{bmatrix}0&0\\\
0&0\end{bmatrix},\>\>\>B_{1}=\begin{bmatrix}1&0\\\
0&1\end{bmatrix},\>\>\>B_{2}=\begin{bmatrix}0&0\\\ 0&0\end{bmatrix}.$ (9)

Finally, we note that the CAV can receive the state information associated
with only a number of HDVs, due to its communications constraints. For
example, in Fig. 1, the green links show HDVs whose information is available
to the CAV. In this direction, an output vector $y(t)\in\mathbb{R}^{2m}$ is
defined, that includes the states of the CAV together with the states of all
HDVs whose information is accessible by the CAV. For $k=1,\ldots,m$, let
$j_{k}$ be the index of the vehicle whose state information, i.e.
$\bar{s}_{j_{k}}$ and $\bar{v}_{j_{k}}$, can be directly measured and
observed. Consequently, we have:

$y(t)=Cx(t),$ (10)

where
$C^{T}=\begin{bmatrix}e_{(2*j_{1}-1)}&e_{2*j_{1}}&\ldots&e_{(2*j_{m}-1)}&e_{2*j_{m}}\end{bmatrix}$.
Note that we can have $j_{1}=1$, since the CAV has access to its own
information.

In this paper, we design a dynamical output-feedback controller that has the
following dynamics:

$\begin{aligned} \dot{x}_{k}(t)&=A_{k}x_{k}(t)+B_{k}y(t)\\\
u(t)&=C_{k}x_{k}(t)\end{aligned},$ (11)

where $A_{k}\in\mathbb{R}^{2n\times 2n}$, $B_{k}\in\mathbb{R}^{2n\times 2m}$
and $C_{k}\in\mathbb{R}^{1\times 2m}$, and $x_{k}\in\mathbb{R}^{2n}$ is state
of the controller.

Problem 1: In order to ensure the existence of an output-feedback controller,
based on the separation principle [27], the first goal of this paper is to
prove the stabilizability of pair $(A,B)$ and detectability of pair $(A,C)$ in
(5) and (10).

Pair $(A,B)$ is stabilizable if the uncontrollable modes are all stable.
Similarly, pair $(A,C)$ is detectable if the unobservable modes are all
stable. We can use the Popov-Belevitch-Hautus (PBH) test for checking the
controllability and observability of a specific eigenvalue.

######  Property 1 ([44])

An eigenvalue $\lambda$ of $A$ for a pair $(A,B)$ (resp., $(A,C)$) is
controllable (resp., observable) if and only if for all nonzero $\rho$ for
which $\rho^{T}A=\lambda\rho^{T}$ (resp., $A\rho=\lambda\rho$), $\rho^{T}B\neq
0$ (resp., $C\rho\neq 0$).

Problem 2: The next problem of this work is dedicated to computing matrices
$A_{k}$, $B_{k}$, $C_{k}$, and $D_{k}$ in (11), for the mixed traffic system
described in (5) and (10), such that the undesired perturbations in the
traffic flow are dissipated. We first solve this problem without considering
any uncertainty in the parameters of heterogeneous vehicles. Subsequently, we
assume that there might be uncertainties in the dynamical model of HDVs, and
we compute a robust output-feedback controller in the presence of disturbance
and parametric uncertainty.

Accordingly, in the next section, we analyze the stabilizability and
detectability of the mixed traffic system (5) and (10), and subsequently, we
design a dynamic controller.

##  III Stabilizability and Detectability

In this section, the stabilizability and detectability of dynamical system (5)
and (10) are discussed.

###  III-A Stabilizability Analysis

In order to prove the stabilizability of the mixed traffic system, we first
consider a ring-road and prove that there is only one uncontrollable
eigenvalue at the origin.

###### Property 2

In the case of a ring-road, the pair $(A,B)$ in (5), where $B$ is defined in
(8), has only one uncontrollable eigenvalue at the origin.

_Proof:_ Let
$\rho=\begin{bmatrix}\rho_{1}^{T}&\ldots&\rho_{n}^{T}\end{bmatrix}\in\mathbb{R}^{2n}$,
where $\rho_{i}^{T}=\begin{bmatrix}\rho_{i1}&\rho_{i2}\end{bmatrix}^{T}$.
Considering the expression of $A$ in (6), equation $\rho^{T}A=0$ leads to

$\begin{aligned}
&\rho^{T}_{i}A_{i2}+\rho^{T}_{i+1}A_{(i+1)1}=0,\>\>\>i=2,\ldots,n-1\\\
&\rho_{1}^{T}J_{1}+\rho_{2}^{T}A_{21}=0\\\
&\rho_{1}^{T}J_{2}+\rho_{n}^{T}A_{n2}=0\end{aligned}.$ (12)

Now, from (12), one can derive

$\displaystyle\rho_{i2}=0,\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>i=2,\ldots,n$
$\displaystyle\rho_{i1}=\rho_{(i+1)1},\>\>\>\>\>i=1,\ldots,n-1.$

Thus, a left eigenvector $\rho$ of $A$ associated with $\lambda=0$ can be
written as $\rho=[\alpha,\beta,\alpha,0,\ldots,\alpha,0]^{T}$. Now, assume
that $\lambda=0$ is an uncontrollable eigenvalue. Thus, from Proposition 1,
one can see that $\rho^{T}B=0$, which essentially means $\rho_{12}=0$. Now,
define $V_{u}=\\{\rho\in\mathbb{R}^{2n}:\rho^{T}A=0,\rho^{T}B=0\\}$. Then, one
can see that
$V_{u}=\\{\rho\in\mathbb{R}^{2n}:\rho=[\alpha,0,\alpha,0,\ldots,\alpha,0]^{T},\mathrm{for\>\>some}\>\;\alpha\neq
0\\}$. Thus, $\mathrm{dim}(V_{u})=1$, which implies that there is only one
uncontrollable eigenvalue of $A$ at the origin. $\blacksquare$

Now, consider an open-road. Next, we show that in this case, there is no
uncontrollable eigenvalue at the origin.

###### Property 3

In the case of an open-road, the pair $(A,B)$ in (5), where $B$ is defined in
(9), has no uncontrollable eigenvalue at the origin.

_Proof:_ In this case, from (12), we have:

$\displaystyle\rho_{i2}=0,\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>\>i=2,\ldots,n$
$\displaystyle\rho_{i1}=\rho_{(i+1)1},\>\>\>\>\>i=1,\ldots,n-1$
$\displaystyle\rho_{n1}=0.$

Therefore, a nonzero left eigenvector $\rho$ of $A$ associated with
$\lambda=0$ has a form as
$\rho=\begin{bmatrix}0&\beta&0&\ldots&0\end{bmatrix}^{T}$. Assume that we have
an uncontrollable eigenvalue at the origin. Thus, from Proposition 1,
$\rho^{T}B=0$ leads to $\rho_{11}=\rho_{12}=0$, implying that $\rho=0$, that
is a contradiction. $\blacksquare$

Next, it suffices to show that all unstable eigenvalues of $A$ in (5), if any,
are controllable.

###### Lemma 1

Let $\lambda\in\mathbb{C}$ with $\mathrm{Re}(\lambda)>0$, and for
$i=2,\ldots,n$, let $A_{i2}$ be defined as (7). Then, the matrix $(\lambda
I-A_{i2})$ is nonsingular.

_Proof:_ For $i=2,\ldots,n$, define
$\mathcal{S}_{i}(\lambda)=\mathrm{det}(\lambda
I-A_{i2})=\lambda^{2}+\beta_{i2}\lambda+\beta_{i1}$. Based on this equation,
the sum of eigenvalues of $A_{i2}$ equals $-\beta_{i2}$, and their product is
$\beta_{i1}$. If the roots of $\mathcal{S}_{i}(\lambda)$ are real, then, since
$\beta_{i1},\beta_{i2}>0$ from (2), both the roots are negative. Moreover, if
the roots are complex and are written as $a+ib$ and $a-ib$, then since
$2a=-\beta_{i2}$, we have $a<0$. Therefore, because we have
$\mathrm{Re}(\lambda)>0$, one can conclude that $\mathcal{S}_{i}(\lambda)\neq
0$, implying that for $i=2,\ldots,n$, $(\lambda I-A_{i2})$ is nonsingular.
$\blacksquare$

###### Theorem 1

For both cases of a ring-road and an open-road, the pair $(A,B)$ associated
with the mixed traffic system described in (5) is stabilizable.

_Proof:_ Let $\lambda$ be an eigenvalue of $A$ with $\mathrm{Re}(\lambda)>0$,
and let also
$\rho=\begin{bmatrix}\rho_{1}^{T}&\ldots&\rho_{n}^{T}\end{bmatrix}^{T}\in\mathbb{R}^{2n}$
be its nonzero left eigenvector, where
$\rho_{i}^{T}=\begin{bmatrix}\rho_{i1}&\rho_{i2}\end{bmatrix}$. Define
$A_{11}=J_{2}$ and $A_{12}=J_{1}$. Then, for $i=1,\ldots,n$, equation
$\rho^{T}A=\lambda\rho^{T}$ implies that

$\rho_{i}^{T}(\lambda I-A_{i2})=\rho_{i+1}^{T}A_{(i+1)1}.$ (13)

Now, one can see that $\mathrm{det}(\lambda I-A_{12})=\lambda^{2}$, which is
nonzero, since $\lambda\neq 0$. Moreover, since $\mathrm{Re}(\lambda)>0$,
Lemma 1 implies that $(\lambda I-A_{i2})$ is nonsingular. Accordingly, for
$i=1,\ldots,n$, one can rewrite (13) as

$\rho_{i}^{T}=\rho_{i+1}^{T}A_{(i+1)1}(\lambda I-A_{i2})^{-1}.$ (14)

Let $L_{i}=(\lambda I-A_{i2})^{-1}$. Now, by recursively employing equation
(14) for $i=1,\ldots,n$, we can derive

$\displaystyle\rho_{1}^{T}$ $\displaystyle=\rho_{2}^{T}A_{21}L_{1}$ (15)
$\displaystyle=\rho_{3}^{T}A_{31}L_{2}A_{21}L_{1}$ $\displaystyle=\ldots$
$\displaystyle=\rho^{T}_{n}A_{n1}L_{n-1}A_{(n-1)1}\ldots L_{2}A_{21}L_{1}$
$\displaystyle=\rho^{T}_{1}A_{11}(L_{n}A_{n1})\ldots(L_{2}A_{21})L_{1}.$

In the case an open-road, since $A_{11}=0$, one can conclude from (15) that
$\rho_{1}=0$. Now, we want to prove that $\rho=0$ holds for a ring-road as
well. Note that we have

$L_{i}A_{i1}=\dfrac{1}{s^{i}_{1}}\begin{bmatrix}0&s^{i}_{2}\\\
0&s^{i}_{3}\end{bmatrix},\>\>\>\>i=2,\ldots,n,$ (16)

where $s^{i}_{1}=\lambda^{2}+\beta_{i2}\lambda+\beta_{i1}$,
$s^{i}_{2}=\lambda+\beta_{i2}-\beta_{i3}$, and
$s^{i}_{3}=\beta_{i3}\lambda+\beta_{i1}$. As shown before, $s_{1}^{i}\neq 0$.
Now, we substitute (16) into (15), and for the case of a ring-road, we obtain

$\begin{bmatrix}\rho_{11}&\rho_{12}\end{bmatrix}=\dfrac{\Pi_{i=2}^{n}s^{i}_{3}}{\lambda\Pi_{i=2}^{n}s^{i}_{1}}\begin{bmatrix}\rho_{11}&\rho_{12}\end{bmatrix}\begin{bmatrix}0&1\\\
0&0\end{bmatrix}.$ (17)

Equation (17) leads to $\rho_{11}=0$ and
$\rho_{12}=\frac{\Pi_{i=2}^{n}s^{i}_{3}}{\lambda\Pi_{i=2}s^{i}_{1}}\rho_{11}=0$.
Thus, $\rho_{1}=0$. Now, for both cases of a ring-road and an open-road, by
recursively applying (14) for $i=n,n-1,\ldots,2$, one can conclude that
$\rho_{i}=0$. Therefore, we have $\rho=0$ that contradicts the assumption.
Thus, $A$ has no eigenvalue which lies on the right half-plane. In addition,
from Proposition 2, for the case of a ring-road, there is only one
uncontrollable eigenvalue at origin. Furthermore, Proposition 3 implies that
in the case of an open-road, there is no uncontrollable eigenvalue at origin.
Hence, in both cases, the system is stabilizable. $\blacksquare$

###### Remark 1

In [23], it has been stated that the mixed traffic system described in (5) is
stabilizable if for all $i,k\in\\{1,2,\ldots,n\\}$, we have
$\beta_{k1}^{2}-\beta_{i2}\beta_{k1}\beta_{k3}+\beta_{i1}\beta_{k3}^{2}\neq
0$. Moreover, through a different approach, it has been shown that there is
only one uncontrollable eigenvalue at origin. However, in Theorem 1, without
assuming any restrictive constraint, we have demonstrated the stabilizability
of a mixed traffic system with heterogeneous HDVs for both cases of a ring-
road and an open-road and proved that the system is stabilizable even if for
some $i,k\in\\{1,2,\ldots,n\\}$, we have
$\beta_{k1}^{2}-\beta_{i2}\beta_{k1}\beta_{k3}+\beta_{i1}\beta_{k3}^{2}=0$.

###  III-B Detectability Analysis

Here, the detectability of the mixed traffic system is studied, and we show
that by observing only the states of the single CAV, the detectability of the
whole system can be ensured.

###### Property 4

The zero eigenvalue of the mixed traffic system described in (5) and (10), in
both cases of a ring-road and an open-road, is observable even if the CAV has
access to only its own states.

_Proof:_ Let $C=\begin{bmatrix}e_{1}&e_{2}\end{bmatrix}$, and assume that
$\lambda=0$ is not observable. Then, from Proposition 1, $A$ has a nonzero
right eigenvector $\rho$, where we have both $A\rho=0$ and $C\rho=0$. Let
$\rho=\begin{bmatrix}\rho_{1}^{T}&\ldots&\rho_{n}^{T}\end{bmatrix}^{T}\in\mathbb{R}^{2n}$,
where $\rho_{i}^{T}=\begin{bmatrix}\rho_{i1}&\rho_{i2}\end{bmatrix}$. From
equation $A\rho=0$, one can write:

$\begin{aligned} &\rho_{i2}-\rho_{(i+1)2}=0,&i=2,\ldots,n\\\
&\beta_{i3}\rho_{(i-1)2}+\beta_{i1}\rho_{i1}-\beta_{i2}\rho_{i2}=0,&i=2,\ldots,n\end{aligned}.$
(18)

Moreover, from $C\rho=0$, one can conclude that $\rho_{11}=\rho_{12}=0$. Thus,
(18) leads to $\rho=0$ in both cases of a ring-road and an open-road, which is
a contradiction. Therefore, the zero eigenvalue is observable. $\blacksquare$

###### Theorem 2

In the both cases of a ring-road and an open-road, the mixed traffic system
described in (5) and (10) is detectable even if only the states of the single
CAV are directly observed.

_Proof:_ Let $C=\begin{bmatrix}e_{1}&e_{2}\end{bmatrix}$. Assume that the
system is not detectable. Then, $A$ has an eigenvalue $\lambda$ on the right
half-plane with a nonzero right eigenvector $\rho$ such that $C\rho=0$. Denote
$\rho=\begin{bmatrix}\rho_{1}^{T}&\ldots&\rho_{n}^{T}\end{bmatrix}^{T}\in\mathbb{R}^{2n}$,
where $\rho_{i}^{T}=\begin{bmatrix}\rho_{i1}&\rho_{i2}\end{bmatrix}$. Then, we
should have $\rho_{11}=\rho_{12}=0$. Define $A_{11}=J_{2}$ and $A_{12}=J_{1}$.
Now, from equation $A\rho=\lambda\rho$, for $i=1,\ldots,n$, one can write:

$(\lambda I-A_{i2})\rho_{i}=A_{i1}\rho_{i-1}.$

Since $\mathrm{Re}(\lambda)>0$, from Lemma 1, one can see that, for
$i=2,\ldots,n$, $(\lambda I-A_{i2})$ is invertible. Moreover,
$\mathrm{det}(\lambda I-A_{12})=\lambda^{2}\neq 0$. Let $L_{i}=(\lambda
I-A_{i2})^{-1}$. Then, for $i=1,\ldots,n$, we can write:

$\rho_{i}=L_{i}A_{i1}\rho_{i-1}.$ (19)

Now, since $\rho_{1}=0$, recursively using equation (19) for $i=2,\ldots,n$
leads to $\rho=0$, which contradicts the assumption. In addition, based on
Proposition 4, the zero eigenvalue is observable. Thus, in summary, one can
conclude that the system is detectable. $\blacksquare$

##  IV Controller Synthesis: Without Uncertainties

In this section, we aim to design a dynamic output-feedback controller for the
mixed traffic system (5) to dissipate undesired perturbations. In the first
step, we assume there is no uncertainty in the model of the traffic system and
design an output-feedback controller.

###  IV-A Disturbances and Performance Outputs

In fact, the perturbations may appear due to lane changes or merges or the
stochastic behavior of HDVs in ring-roads with no bottlenecks [8, 23].

Perturbations are modeled as disturbance signals added to the acceleration of
each vehicle. Thus, by defining
$d=\begin{bmatrix}d_{1}(t)&\ldots&d_{n}(t)\end{bmatrix}^{T}\in\mathbb{R}^{n}$
as the disturbance vector and the matrix

$B_{d}=\begin{bmatrix}b_{d}&0&\ldots&0\\\ 0&b_{d}&\ldots&\vdots\\\
\vdots&\ddots&\ddots&0\\\ 0&\ldots&0&b_{d}\end{bmatrix}\in\mathbb{R}^{2n\times
n},$

with $b_{d}=[0,1]^{T}$, the dynamics of the mixed traffic system in the
presence of disturbances can be written as

$\begin{aligned} \dot{x}(t)&=Ax(t)+Bu(t)+B_{d}d(t)\\\
z(t)&=C_{z}x(t)+D_{z}u(t)\\\ y(t)&=Cx(t)\end{aligned},$ (20)

where for the case of a ring-road, one can define the controlled output
(performance output) $z(t)\in\mathbb{R}^{2n+1}$ as

$z(t)=\begin{bmatrix}\gamma_{s}\bar{s}_{1}(t)&\gamma_{v}\bar{v}_{1}&\ldots&\gamma_{s}\bar{s}_{n}(t)&\gamma_{v}\bar{v}_{n}&\gamma_{u}u\end{bmatrix}^{T}.$
(21)

The parameters $\gamma_{s},\gamma_{v},\gamma_{u}>0$ denote the penalties
associated with the spacing error, the velocity deviation, and the control
energy, respectively. Thus,
$C_{z}=\begin{bmatrix}\mathcal{T}^{\frac{1}{2}}&0_{2n\times
1}\end{bmatrix}^{T}$ includes the weights of the states, with
$\mathcal{T}^{\frac{1}{2}}=\mathrm{diag}(\gamma_{s},\gamma_{v},\ldots,\gamma_{s},\gamma_{v})$,
and $D_{z}=\begin{bmatrix}0_{1\times
2n}&\mathcal{Q}^{\frac{1}{2}}\end{bmatrix}^{T}$ is the weight of the control
input, with $\mathcal{Q}^{\frac{1}{2}}=\gamma_{u}$. In the case of an open-
road, we define
$z(t)=\begin{bmatrix}\gamma_{s}\bar{s}_{1}(t)&\gamma_{v}\bar{v}_{1}&\ldots&\gamma_{s}\bar{s}_{n}(t)&\gamma_{v}\bar{v}_{n}&\gamma_{u_{1}}u_{1}&\gamma_{u_{2}}u_{2}\end{bmatrix}^{T}\in\mathbb{R}^{2n+2}$.
In this case, one has
$C_{z}=\begin{bmatrix}\mathcal{T}^{\frac{1}{2}}&0_{2n\times
2}\end{bmatrix}^{T}$ and $D_{z}=\begin{bmatrix}0_{2\times
2n}&\mathcal{Q}^{\frac{1}{2}}\end{bmatrix}^{T}$, with
$\mathcal{Q}^{\frac{1}{2}}=\mathrm{diag}(\gamma_{u_{1}},\gamma_{u_{2}})$.

###  IV-B Output-feedback controller

Consider an output-feedback controller with the dynamics described in (11). We
aim to design a dynamic controller that stabilizes the system (20) and
minimizes the influence of the disturbance $d$ on the performance output $z$.

By applying the controller (11) to (20), the closed-loop system is expressed
as

$\begin{aligned} \dot{\bar{x}}(t)&=\bar{A}\bar{x}(t)+\bar{B}d(t)\\\
z(t)&=\bar{C}\bar{x}(t)\end{aligned},$ (22)

where $\bar{x}=\begin{bmatrix}{x}(t)\\\ {x}_{k}(t)\end{bmatrix}$,
$\bar{A}=\begin{bmatrix}A&BC_{k}\\\ B_{k}C&A_{k}\end{bmatrix}$,
$\bar{B}=\begin{bmatrix}B_{d}\\\ 0\end{bmatrix}$ and
$\bar{C}=\begin{bmatrix}C_{z}&D_{z}C_{k}\end{bmatrix}$.

###  IV-C $H_{\infty}$ Control Problem

Let $T_{zd}$ be the closed-loop transfer function from the disturbance $d$ to
the performance output $z$.

_Problem:_ Find the matrices $A_{k}$, $B_{k}$, and $C_{k}$ for the controller
(11) such that the closed-loop system (22) satisfies the inequality

$||T_{zd}||_{\infty}=\max_{d(t)\neq
0}\frac{||z(t)||_{2}}{||d(t)||_{2}}<\gamma,$ (23)

and $\gamma>0$ is minimized.

Note that $||T_{zd}||_{\infty}$ denotes the $H_{\infty}$ norm of $T_{zd}$,
which measures the largest input-output gain for energy or power input signals
[45].

_Problem solution:_ Based on the bounded real lemma (BRL) [46],
$||T_{zd}||_{\infty}$ is smaller than $\gamma$ if and only if there exists a
positive definite matrix $P\succ 0$, and matrices $A_{k}$, $B_{k}$, and
$C_{k}$ satisfying

$\begin{bmatrix}\bar{A}^{T}P+P\bar{A}&P\bar{B}&\bar{C}^{T}\\\
\bar{B}^{T}P&-\gamma^{2}I&0\\\ \bar{C}&0&-I\end{bmatrix}\prec 0.$ (24)

Now, note that the inequality (24) is not a linear matrix inequality (LMI)
with respect to the variables $P$, $A_{k}$, $B_{k}$, and $C_{k}$, because it
is not linear with respect to these variables. In order to extract an LMI for
computing the controller parameters $A_{k}$, $B_{k}$, and $C_{k}$, we apply a
method that has been presented in [45] in details, and we provide a summarized
description of this method in the following.

Let us partition the matrices $P$ and $P^{-1}$ as the following form:

$P=\begin{bmatrix}Y&N\\\
N^{T}&*\end{bmatrix},\;\;\;P^{-1}=\begin{bmatrix}X&M\\\ M^{T}&*\end{bmatrix},$

where $X,Y\in\mathbb{R}^{2n\times 2n}$. Note that since $P\succ 0$, we have
$X,Y\succ 0$. From the equation $PP^{-1}=I$, one can deduce that
$NM^{T}+YX=I$. Further, one can find that $P=\Lambda_{2}\Lambda_{1}^{-1}$,
where

$\Lambda_{1}=\begin{bmatrix}X&I\\\
M^{T}&0\end{bmatrix},\>\>\>\Lambda_{2}=\begin{bmatrix}I&Y\\\
0&N^{T}\end{bmatrix}.$

Therefore, we have

$\Lambda_{1}^{T}P\Lambda_{1}=\Lambda_{1}^{T}\Lambda_{2}=\begin{bmatrix}X&I\\\
I&Y\end{bmatrix}\succ 0.$

Substituting $P=\Lambda_{2}\Lambda_{1}^{-1}$ in (24), we obtain

$\Omega_{1}=\begin{bmatrix}\bar{A}^{T}\Lambda_{2}\Lambda_{1}^{-1}+\Lambda_{1}^{-T}\Lambda_{2}^{T}\bar{A}&\Lambda_{1}^{-T}\Lambda_{2}^{T}\bar{B}&\bar{C}^{T}\\\
\bar{B}^{T}\Lambda_{2}\Lambda_{1}^{-1}&-\gamma^{2}I&0\\\
\bar{C}&0&-I\end{bmatrix}\prec 0.$ (25)

Now, let us define a matrix $\Omega_{2}$ by the following congruent
transformation of $\Omega_{1}$

$\displaystyle\Omega_{2}$ $\displaystyle=\begin{bmatrix}\Lambda_{1}^{T}&0&0\\\
0&I&0\\\ 0&0&I\end{bmatrix}\Omega_{1}\begin{bmatrix}\Lambda_{1}&0&0\\\
0&I&0\\\ 0&0&I\end{bmatrix}$ (26)
$\displaystyle=\begin{bmatrix}\Lambda_{1}^{T}\bar{A}^{T}\Lambda_{2}+\Lambda_{2}^{T}\bar{A}\Lambda_{1}&\Lambda_{2}^{T}\bar{B}&\Lambda_{1}^{T}\bar{C}^{T}\\\
\bar{B}^{T}\Lambda_{2}&-\gamma^{2}I&0\\\
\bar{C}\Lambda_{1}&0&-I\end{bmatrix}\prec 0.$

Now, substituting the values of $\Lambda_{1}$, $\Lambda_{2}$, $\bar{A}$,
$\bar{B}$ and $\bar{C}$, we obtain the following matrix inequality:

$\displaystyle\Omega_{2}$
$\displaystyle=\begin{bmatrix}\Omega_{11}&\Omega_{12}&\Omega_{13}&\Omega_{14}\\\
\Omega_{21}&\Omega_{22}&\Omega_{23}&\Omega_{24}\\\
\Omega_{31}&\Omega_{32}&\Omega_{33}&\Omega_{34}\\\
\Omega_{41}&\Omega_{42}&\Omega_{43}&\Omega_{44}\end{bmatrix}\prec 0,$ (27)

where

$\displaystyle\Omega_{11}=AX+XA^{T}+BC_{k}M^{T}+MC_{k}^{T}B^{T}$
$\displaystyle\Omega_{12}=\Omega^{T}_{21}=MA_{k}^{T}N^{T}+XC^{T}B_{k}^{T}N^{T}+MC_{k}^{T}B^{T}Y$
$\displaystyle\quad\quad\quad\quad\>\>\>+XA^{T}Y+A$
$\displaystyle\Omega_{22}=A^{T}Y+YA+NB_{k}C+C^{T}B_{k}^{T}N^{T}$
$\displaystyle\Omega_{13}=\Omega^{T}_{31}=B_{d},\;\Omega_{14}=\Omega^{T}_{41}=XC_{z}^{T}+MC_{k}^{T}D^{T}_{z},$
$\displaystyle\Omega_{23}=\Omega^{T}_{32}=YB_{d},\;\Omega_{24}=\Omega^{T}_{42}=C_{z}^{T},\;\Omega_{33}=-\gamma^{2}I,$
$\displaystyle\Omega_{34}=\Omega_{43}=0,\;\Omega_{44}=-I.$

Finally, by defining a new set of variables as

$\begin{aligned} &\hat{A}=NA_{k}M^{T}+NB_{k}CX+YBC_{k}M^{T}+YAX\\\
&\hat{B}=NB_{k}\\\ &\hat{C}=C_{k}M^{T}\\\ &\eta=\gamma^{2}\end{aligned},$ (28)

we obtain the LMIs described in (29) with respect to variables
$\eta,X,Y,\hat{A},\hat{B},\hat{C}$.

$\displaystyle\min_{\eta,X,Y,\hat{A},\hat{B},\hat{C}}\eta$ (29) subject to:
$\displaystyle\begin{bmatrix}X&I\\\ I&Y\end{bmatrix}\succ 0$
$\displaystyle\begin{bmatrix}AX+XA^{T}+B\hat{C}+\hat{C}^{T}B^{T}&\hat{A}^{T}+A&B_{d}&XC_{z}^{T}+\hat{C}^{T}D_{z}^{T}\\\
*&A^{T}Y+YA+\hat{B}C+C^{T}\hat{B}^{T}&YB_{d}&C_{z}^{T}\\\ *&*&-\eta I&0\\\
*&*&*&-I\end{bmatrix}\prec 0$

Therefore, if the optimization problem (29) is feasible, then we can find
$\eta$, $X$, $Y$, $\hat{A}$, $\hat{B}$, and $\hat{C}$ and solve the matrix
equation $NM^{T}=I-YX$ for the non-singular matrices $M$ and $N$. Moreover,
matrices $A_{k}$, $B_{k}$, and $C_{k}$ in the state-space realization of the
output-feedback controller can be derived based on (28) as follows:

$\displaystyle A_{k}=N^{-1}(\hat{A}-NB_{k}CX-YBC_{k}M^{T}-YAX)M^{-T},$ (30)
$\displaystyle B_{k}=N^{-1}\hat{B},$ $\displaystyle C_{k}=\hat{C}M^{-T}.$

$\blacksquare$

##  V Controller Synthesis: With Uncertainties

In this section, we consider parametric uncertainties in the dynamic model of
the HDVs. In fact, we assume that $\beta_{i1}$, $\beta_{i2}$, and
$\beta_{i3}$, appearing in (7), are unknown parameters for the mixed traffic
system. Then, we synthesize an output-feedback controller with dynamics (11)
that stabilizes the entire closed-loop system in the presences of disturbances
and parametric uncertainties.

After the linearization of the overall dynamics of the system, the uncertainty
is assumed to appear in the system matrix $A$ in (6) as

$A=A_{\mathcal{N}}+\Delta{A}.$ (31)

The matrix $A_{\mathcal{N}}$ is the mean-valued matrix that is constant and
known. One can calculate $A_{\mathcal{N}}$ as
$A_{\mathcal{N}}=\begin{bmatrix}\frac{a_{ij,\min}+a_{ij,\max}}{2}\end{bmatrix}$,
where $a_{ij,\min}$ (resp., $a_{ij,\max}$) is the minimum (resp., maximum)
value that an entry of $A$ in its $i$th row and $j$th column can have. On the
other hand, $\Delta{A}$ represents parametric uncertainties, that is assumed
to be structurally bounded. One can write $\Delta A$ as

$\Delta{A}=LFR,$

where the matrices $L\in\mathbb{R}^{2n\times 2n}$ and
$R\in\mathbb{R}^{2n\times 2n}$ are known constant matrices. Moreover,
$F\in\mathbb{R}^{2n\times 2n}$ is an unknown matrix, which satisfies the
following condition:

$F^{T}F\preceq I.$ (32)

###### Remark 2

(Computing $L$ and $R$) We can choose $L$ and $R$ as $L=\varrho I$ and $R=\rho
I$. Therefore, $\Delta A=LFR=\rho\varrho F$, and based on Fact 1, we get
$||F||=(\rho\varrho)^{-1}||\Delta A||\leq(\rho\varrho)^{-1}||\Delta
A||_{\mathcal{F}}$. On the other hand, $\Delta
A=A-A_{\mathcal{N}}=\begin{bmatrix}\tilde{a}_{ij}\end{bmatrix}$, where
$|\tilde{a}_{ij}|\leq\frac{1}{2}(a_{ij,\max}-a_{ij,\min})$. Therefore,
$||F||\leq(2\rho\varrho)^{-1}(\sum_{i=1}^{2n}\sum_{j=1}^{2n}(a_{ij,\max}-a_{ij,\min})^{2})^{\frac{1}{2}}$.
Now, if one sets
$\rho\varrho=\frac{1}{2}(\sum_{i=1}^{2n}\sum_{j=1}^{2n}(a_{ij,\max}-a_{ij,\min})^{2})^{\frac{1}{2}}$,
then $||F||\leq 1$, and the condition (32) is satisfied.

In the following, we show how we can ensure the existence of an output-
feedback controller that attenuates the effect of the disturbance on the
performance output, while stabilizing the closed-loop system.

###### Theorem 3

Consider a system with dynamics (20), which has uncertainties that are
modelled as (31). Then, there exists an output-feedback controller that
stabilizes the closed-loop system (22) and minimizes $\gamma$ in the
inequality (23) if the optimization problem (33) feasible.

$\displaystyle\min_{\eta,\epsilon_{1},\epsilon_{2},\epsilon_{3},X,Y,\hat{A},\hat{B},\hat{C}}\eta$
(33) subject to: $\displaystyle\begin{bmatrix}X&I\\\ I&Y\end{bmatrix}\succ 0$
$\displaystyle\begin{bmatrix}\Gamma_{11}&\Gamma_{12}\\\
*&\Gamma_{22}\end{bmatrix}\prec 0$ $\displaystyle\Gamma_{11}=$
$\displaystyle\begin{bmatrix}A_{\mathcal{N}}X+XA_{\mathcal{N}}^{T}+B\hat{C}+\hat{C}^{T}B^{T}+L(\epsilon_{1}+\epsilon_{2})L^{T}&\hat{A}^{T}+A_{\mathcal{N}}&B_{d}&XC_{z}^{T}+\hat{C}^{T}D_{z}^{T}\\\
*&A_{\mathcal{N}}^{T}Y+YA_{\mathcal{N}}+\hat{B}C+C^{T}\hat{B}^{T}+\epsilon_{3}R^{T}R&YB_{d}&C_{z}^{T}\\\
*&*&-\eta I&0\\\ *&*&*&-I\end{bmatrix}$ $\displaystyle\Gamma_{12}=$
$\displaystyle\begin{bmatrix}XR^{T}&0&0&YL&0\\\ 0&R^{T}&YL&0&XR^{T}\\\
0&0&0&0&0\\\ 0&0&0&0&0\end{bmatrix}$ $\displaystyle\Gamma_{22}=$
$\displaystyle-\mathrm{diag}(\epsilon_{1}I,\epsilon_{2}I,\epsilon_{3}I,I,I)$

_Proof of Theorem 3:_ See Appendix.

###### Remark 3

By solving the optimization problem (33), $\eta$,
$\epsilon_{1},\epsilon_{2},\epsilon_{3}$, $\hat{A}$, $\hat{B}$, $\hat{C}$,
$X$, and $Y$ can be computed. We note that $\eta=\gamma^{2}$, where $\gamma$
is an upper bound of the $||T_{zd}||_{\infty}$. In addition, we can solve the
matrix equation $NM^{T}=I-YX$ for the non-singular matrices $M$ and $N$. Then,
the matrices $A_{k}$, $B_{k}$, and $C_{k}$ in the state-space realization of
the output-feedback controller can be obtained as

$\displaystyle A_{k}=N^{-1}(\hat{A}-NB_{k}CX-
YBC_{k}M^{T}-YA_{\mathcal{N}}X)M^{-T}$ $\displaystyle B_{k}=N^{-1}\hat{B}$
$\displaystyle C_{k}=\hat{C}M^{-T}.$

##  VI Simulation Results

In this section, the efficiency of the proposed control strategies are
validated through numerical simulations. The minimization problems (29) and
(33) are solved in YALMIP interface for MATLAB with Sedumi solver. For a
better comparison of the results, we simulate an experimental setup that is
similar to the ones considered in [28, 23], where a ring-road with a
circumference $D=400\>\mathrm{m}$ and $20$ vehicles has been studied (since
the results for the case of an open-road is analogous to the case of a ring-
road, we illustrate here the simulation results only for a ring-road). We
assume that the 1st vehicle can be a CAV that has access to the state
information of the five HDVs ahead and the five HDVs behind.

###  VI-A Simulation setup

Through using an optimal velocity model (OVM), any nonlinear function
$H_{i}(\cdot)$ in (1), $i=2,\ldots,20$, that describes the acceleration
function of the $i$th HDV is written as
$H_{i}(\cdot)=\alpha_{i}(\mathcal{V}_{i}(s_{i}(t))-v_{i}(t))+\theta_{i}\dot{s}_{i}(t)$,
where $\alpha_{i}$ and $\theta_{i}$ are sensitivity coefficent in an OVM.
Moreover, $\mathcal{V}_{i}(s_{i})$ is the desired speed of HDV $i$ that is a
function of spacing $s_{i}$. Due to heterogeneity of HDVs, we set
$\alpha_{i}=0.6+U[-0.1,0.1]$ and $\theta_{i}=0.9+U[-0.1,0.1]$, where $U[a,b]$
represents a uniform distribution function that take values from the interval
$[a,b]$. Moreover, we define $\mathcal{V}_{i}(s_{i})$ as a piecewise function

$\mathcal{V}_{i}(s_{i})=\begin{cases}0,&s_{i}\leq s_{i,st},\\\
h_{i,v}(s_{i}),&s_{i,st}<s_{i}<s_{i,go},\\\ v_{i,\max},&s_{i}\geq
s_{i,go},\end{cases}$

where we set $s_{i,st}=5$, $s_{i,go}=35+U[-5,5]$, $v_{i,\max}=30$, and
$h_{i,v}(s_{i})$ is a nonlinear function chosen as [31]

$h_{i,v}(s_{i})=\dfrac{v_{i,\max}}{2}(1-\cos(\pi\dfrac{s_{i}-s_{i,st}}{s_{i,go}-s_{i,st}})).$

In order to ensure the safety and prevent collisions, every vehicle is also
assumed to be equipped with an automatic braking system, described as

$\dot{v}(t)=a_{\min}\;\;\;\;\;\;\text{if}\;\;\;\;\;\;\dfrac{v_{i}^{2}(t)-v^{2}_{i-1}(t)}{2s_{i}(t)}\geq|a_{\min}|,$

where $a_{\min}=-5\>\mathrm{m}/\mathrm{s}^{2}$. Moreover, we set the maximum
acceleration of any vehicle as $a_{\max}=2\>\mathrm{m}/\mathrm{s}^{2}$.

###  VI-B Stabilizability Verification

As the first scenario, assume that all vehicles are randomly distributed along
the ring-road and start their movement with initial velocity $v_{i}(0)$ from
the distribution $15+U[-4,4]\>\mathrm{m}/\mathrm{s}$. Notice that we consider
no uncertainties in the dynamic model of the system in this case. First,
assume that all vehicles of the mixed traffic system are HDVs. In Fig. 2(a),
it can be seen that multiple perturbations occur in this system, which are
amplified over time and generate an unstable nonlinear wave moving upstream
the traffic flow.

Next, assume that the 1st vehicle is a CAV that is controlled by an output-
feedback controller described in Section IV-C. First, we set the equilibrrium
velocity as $v^{*}=15\>\mathrm{m}/\mathrm{s}$. Then, the equilibrium spacing
of vehicle $i$ is obtained by solving the equation $0=H_{i}(15,s_{i}^{*},0)$.
Note that in a circular path, the sum of spacing of all vehicles should equal
to the circumference $D$. In other words, we should have
$\sum_{i=1}^{2}s_{i}(t)=\sum s_{i}^{*}=400$. Now, one can see in Fig. 2(b)
that the perturbations can be attenuated within a short time. As the next
experiments, we change the equilibrium velocity to $16\>\mathrm{m}/\mathrm{s}$
and $14\>\mathrm{m}/\mathrm{s}$, respectively. Notice that in this case, the
parameters of the linearized system (5) that are computed around the
equilibrium point will be changed. In these two cases, one can observe in
Figs. 2(c,d) that the single CAV is still capable of stabilizing traffic flow
and steering it towards the new equilibrium points, which verifies the
stabilizability of the mixed traffic system with a single CAV.

Figure 2: (a) Velocity profile of vehicles when they are all HDVs; (b)–(d)
Velocity profile of vehicles when the 1st vehicle is a CAV under the proposed
control in Section IV-C, and $v^{*}$ is 15, 16, and 14 for (b), (c), and (d),
respectively.

###  VI-C Robustness Against Disturbances

In this part, we aim to illustrate the performance of the output-feedback
controller, proposed in Section IV-C, to dampen the undesired disturbances.

We note that the parameters $\gamma_{s}$, $\gamma_{v}$, and $\gamma_{u}$ in
the performance output $z(t)$ in (21), should be selected such that the
optimization problem (29) is feasible. Moreover, we should prevent rapid
oscillations in the output of the system. If we choose smaller values for the
parameters $\gamma_{s}$ and $\gamma_{v}$, the system oscillations increase,
while the system response converges more slowly that is not desirable. As a
result, there is a trade-off between the convergence rate of the system
response and its quality in terms of the amplitude of the oscillations.
Accordingly, adjusting the values of the parameters $\gamma_{s}$ and
$\gamma_{v}$ is an important part of the controller design. In order to have
an appropriate output behaviour of the traffic flow system, we choose
$\gamma_{s}=0.03$, $\gamma_{v}=0.15$, and $\gamma_{u}=1$.

As the next experiment, we assume that, at $t=20\>s$, the 7th vehicle is
decelerated at $-3\>\mathrm{m}/\mathrm{s}^{2}$ for $3\>s$ (this perturbation
can be due to road bottlenecks). In Figs. 3 (a,b), the trajectory and the
velocity profile of all vehicles when there is no CAV in the system are
illustrated. One can see in these figures that the perturbation does not
vanish, and a nonlinear wave appears that propagates against the traffic flow.
On the other hand, when one CAV is added to the traffic system and is
controlled by the proposed strategy in Section IV-C, one can see in Figs. 4
(a,b) that the stop-and-go-wave can be quickly dissipated, and the traffic
flow is stabilized to the equilibrium point.

Figure 3: (a) Trajectory of all vehicles (the trajectory of the perturbed
vehicle and vehicle no. 20 are, respectively, shown by black and blue lines)
when there is no CAV in the platoon (a disturbance is added to the
acceleration of vehicle 7 at $t=20\>s$); (b) Velocity profile of vehicles.
Figure 4: (a) Trajectory of all vehicles (the trajectory of the perturbed
vehicle and vehicle no. 20 are, respectively, shown by black and blue lines)
when there is one CAV under the proposed control is Section IV-C (a
disturbance is added to the acceleration of vehicle 7 at $t=20\>s$); (b)
Velocity profile of vehicles.

###  VI-D Comparison with Existing Results

In this part, we compare the efficiency of the output-feedback controller
proposed in Section IV-C with some of the existing control strategies in the
literature. In particular, we compare our result with an optimal control
strategy proposed in [23] and the two heuristic control methods presented in
[12], that is, Follower-Stopper and PI with Saturation.

At any stage of the new experiment, we assume that one of the HDVs is
decelerated at $-3\>\mathrm{m}/\mathrm{s}^{2}$ for $3\>s$. Thus, the
disturbance signal is a pulse with a time duration of $3\>s$ and an amplitude
of $-3\>\mathrm{m}/\mathrm{s}^{2}$. The initial velocity of all vehicles is
also $15\>\mathrm{m}/\mathrm{s}$.

In Fig. 5 (a), after applying different control strategies, we illustrate the
maximum absolute value of the spacing error during the overall process with
respect to the index of a single perturbed HDV. Then, one can observe that
with our proposed controller, the absolute value of the spacing error is much
less than the two heuristic methods [12] and the optimal control strategy in
[23]. In fact, with our controller, the spacing remains close to the
equilibrium spacing over the whole process, and its fluctuations around the
desired spacing are considerably smaller compared to the other strategies.

An another performance measure, in Fig.5 (b), associated with any control
strategy, a linear quadratic cost that is defined as the energy of the
performance output $z(t)$ in (21) with respect to the different position of
the perturbation is illustrated. As is evident in this figure, thanks to
considering a system-level performance, our control methodology not only leads
to a much better efficiency compared to the heuristic strategies in [12], but
it has also a substantially smaller cost than the optimal controller proposed
in [23]. In fact, in [23], a sub-optimal solution for synthesizing a static
controller has been developed, while in this work, a global optimal dynamic
controller with more degrees of freedom is designed.

Figure 5: Comparison of the results for four different control strategies; (a)
$\max_{t}|s_{1}(t)-s_{1}^{*}|$ w.r.t. the index of the perturbed HDV; (b)
Energy of the performance output, i.e.,
$||z(t)||_{2}=\int_{t=0}^{\infty}x^{T}(t)\mathcal{T}x(t)+u^{T}(t)\mathcal{Q}u(t)$,
w.r.t. the index of the perturbed HDV.

###  VI-E Robustness against Disturbances and Uncertainties

As the last experiment, we assume that some parameters appearing in the
dynamic model of HDVs are uncertain. However, in our case study, the nominal
values of the parameters are the same for all HDVs. In fact, instead of
considering heterogeneous HDVs, we assume that there are a number of vehicles
with homogeneous nominal dynamic models that may also include uncertainties.
In order to dampen the perturbations of the mixed traffic system in the
presence of parametric uncertainties, we design an output-feedback controller,
using the procedure proposed in Section V.

In this experiment, we assume that the parameters $\alpha_{i}$, $\theta_{i}$,
and $s_{i,go}$ (see Section VI-A) are unknown. The uncertain parameters are
distributed around the nominal values $\alpha_{\mathcal{N}}=0.6$,
$\theta_{\mathcal{N}}=0.9$, and $s_{\mathcal{N},go}=35$. Then, one can define
$\alpha_{i}=\alpha_{\mathcal{N}}+\Delta\alpha$,
$\theta_{i}=\theta_{\mathcal{N}}+\Delta\theta$,
$s_{i,go}=s_{\mathcal{N},go}+\Delta s_{go}$, where $-0.1\leq\Delta\alpha\leq
0.1$, $-0.1\leq\Delta\theta\leq 0.1$, and $-5\leq\Delta s_{go}\leq 5$.

We first assume in this experiment that the initial velocity of the vehicles
is randomly chosen from the distribution $15+U[-4,4]\>\mathrm{m}/\mathrm{s}$.
We compute the controller parameters associated with three different
equilibrium velocities, that is, $14$, $15$, and $16\>\mathrm{m}/\mathrm{s}$.
As observed in Figs. 6 (a–c), in these three cases, by using the control
strategy proposed in Section V, the perturbations occurring in the traffic
flow system are dampened, and the velocities of all vehicles converge to the
equilibrium points within a short time.

Next, we assume that, at $t=20\>s$, the vehicle no. 7 brakes at
$-3\>\mathrm{m}/\mathrm{s}^{2}$ for $3\>s$, and the single CAV is under the
proposed control in Section V. In Fig. 7 (a,b), the velocity profile of all
vehicles from a 3-dimensional (3D) and 2-dimensional (2D) perspective are
illustrated. One can see the with the proposed output-feedback controller, the
perturbation is attenuated very quickly in this traffic system. Moreover, as
observed in Fig. 8, the spacing of HDVs and the CAV from the preceding
vehicles converges to the equilibrium value in a few seconds.

Figure 6:  (a)–(c) Velocity profile of vehicles when there is one CAV under
the proposed control in Section V in the presence of disturbances and the
parametric uncertainties, and $v^{*}$ is 14, 15, and 16
$\mathrm{m}/\mathrm{s}$ for (a), (b), and (c), respectively. Figure 7:
(a)–(b) Velocity profile of all vehicles over time from 3D and 2D perspectives
in the presence of disturbances and parametric uncertainties when the single
CAV is under the proposed control in Section V (the blue (resp., black) line
represents the spacing of the CAV (resp., the perturbed vehicle), i.e.
$s_{1}(t)$ (resp., $s_{7}(t)$), and the grey lines denote the spacing of other
HDVs).  Figure 8:  (a)–(b) Spacing profile of all vehicles over time from 3D
and 2D perspective in the presence of disturbances and parametric
uncertainties when the single CAV is under the proposed control in Section V
(the blue (resp., black) line represents the spacing of the CAV (resp., the
perturbed vehicle), i.e. $s_{1}(t)$ (resp., $s_{7}(t)$), and the grey lines
denote the spacing of other HDVs).

##  VII Conclusion

In this work, the stabilizability and detectability of a mixed traffic system
along a ring-road and an open-road has been studied and analyzed. It has been
shown that in both cases, the system under investigation is stabilizable when
there is one single CAV in the platoon. Moreover, we have demonstrated that
the system is detectable even if the state information of only one vehicle is
directly measured. Furthermore, by considering limited communication ability
of the CAV to receive state information from its neighboring vehicles, an
$H_{\infty}$ dynamic output-feedback controller has been designed, that can
provably smoothen the traffic flow even in the presences of abrupt and large
disturbances (e.g. sharp deceleration downstream). We have also presented an
$H_{\infty}$ control method in the presence of parametric uncertainties in the
dynamics of HDVs. The effectiveness of the proposed methods to achieve traffic
flow stability has been verified and demonstrated through numerical simulation
experiments. Finally, the numerical results are compared to another approach
from the literature. Future work is going to deal with the generalization of
these results to more complex traffic patterns.

We first present two results that will be used in the proof of Theorem 3 and
then discuss the proof.

###### Lemma 2

[47] (Young inequality) Given matrices $D,F$, and $S$ of appropriate
dimensions, the inequality $DFS+(DFS)^{T}\preceq\epsilon^{-1}DD^{T}+\epsilon
S^{T}S$ holds for some scalar $\epsilon>0$ if we have $F^{T}F\preceq I$.

###### Lemma 3

[48] (Schur complement) Consider the matrices $W_{1}$, $W_{2}$, and $W_{3}$
with the appropriate dimensions, where $W_{1}=W_{1}^{T}\succ 0$. Then, the
matrix inequality $W_{1}+W_{3}^{T}W_{2}^{-1}W_{3}\prec 0$ is equivalent to

$\begin{bmatrix}W_{1}&W_{3}^{T}\\\ W_{3}&-W_{2}\end{bmatrix}\prec 0.$

_Proof of Theorem 3:_ As shown in section (IV-C), based on the bounded real
lemma (BRL), if the matrix inequality (27) is satisfied, then
$||T_{zd}||_{\infty}<\gamma$. Incorporating $A$ from (31) to (27), we get

$\Omega_{4}=\Omega_{3}+\Delta\Omega\prec 0,$

where

$\begin{aligned}
\Omega_{3}=\begin{bmatrix}\Omega_{11}&\Omega_{12}&\Omega_{13}&\Omega_{14}\\\
\Omega_{21}&\Omega_{22}&\Omega_{23}&\Omega_{24}\\\
\Omega_{31}&\Omega_{32}&\Omega_{33}&\Omega_{34}\\\
\Omega_{41}&\Omega_{42}&\Omega_{43}&\Omega_{44}\end{bmatrix}\end{aligned},$

with

$\displaystyle\Omega_{11}=A_{\mathcal{N}}X+XA_{\mathcal{N}}^{T}+BC_{k}M^{T}+MC_{k}^{T}B^{T},$
$\displaystyle\Omega_{12}=\Omega^{T}_{21}=MA_{k}^{T}N^{T}+XC^{T}B_{k}^{T}N^{T}+MC_{k}^{T}B^{T}Y,$
$\displaystyle\quad\quad\quad\quad+XA_{\mathcal{N}}^{T}Y+A_{\mathcal{N}}$
$\displaystyle\Omega_{22}=A_{\mathcal{N}}^{T}Y+YA_{\mathcal{N}}+NB_{k}C+C^{T}B_{k}^{T}N^{T},$
$\displaystyle\Omega_{13}=\Omega^{T}_{31}=B_{d},\;\;\Omega_{14}=\Omega^{T}_{41}=XC_{z}^{T}+MC_{k}^{T}D^{T}_{z},$
$\displaystyle\Omega_{23}=\Omega^{T}_{32}=YB_{d},\;\;\Omega_{24}=\Omega^{T}_{42}=C_{z}^{T},\;\;\Omega_{33}=-\gamma^{2}I,$
$\displaystyle\Omega_{34}=\Omega_{43}=0,\;\;\Omega_{44}=-I,$ and
$\displaystyle\Delta\Omega=\begin{bmatrix}\Delta AX+X\Delta A^{T}&\Delta
A+X\Delta A^{T}Y&0&0\\\ \Delta A^{T}+Y\Delta AX&\Delta A^{T}Y+Y\Delta A&0&0\\\
0&0&0&0\\\ 0&0&0&0\end{bmatrix}.$

By substituting $\Delta A=LFR$, and applying condition (32) and Lemma 2,
$\Omega_{4}$ can be bounded as

$\Omega_{4}\preceq\Omega_{3}+\Sigma_{i=1}^{4}\Upsilon_{i}\prec 0,$

where

$\displaystyle\Upsilon_{1}=\begin{bmatrix}L\\\ 0\\\ 0\\\
0\end{bmatrix}\epsilon_{1}\begin{bmatrix}L^{T}&0&0&0\end{bmatrix}+\begin{bmatrix}XR^{T}\\\
0\\\ 0\\\
0\end{bmatrix}\epsilon_{1}^{-1}\begin{bmatrix}RX&0&0&0\end{bmatrix},$
$\displaystyle\Upsilon_{2}=\begin{bmatrix}L\\\ 0\\\ 0\\\
0\end{bmatrix}\epsilon_{2}\begin{bmatrix}L^{T}&0&0&0\end{bmatrix}+\begin{bmatrix}0\\\
R^{T}\\\ 0\\\
0\end{bmatrix}\epsilon_{2}^{-1}\begin{bmatrix}0&R&0&0\end{bmatrix},$
$\displaystyle\Upsilon_{3}=\begin{bmatrix}XR^{T}\\\ 0\\\ 0\\\
0\end{bmatrix}\delta\begin{bmatrix}RX&0&0&0\end{bmatrix}+\begin{bmatrix}0\\\
YL\\\ 0\\\ 0\end{bmatrix}\delta^{-1}\begin{bmatrix}0&L^{T}Y&0&0\end{bmatrix},$
$\displaystyle\Upsilon_{4}=\begin{bmatrix}0\\\ R^{T}\\\ 0\\\
0\end{bmatrix}\epsilon_{3}\begin{bmatrix}0&R&0&0\end{bmatrix}+\begin{bmatrix}0\\\
YL\\\ 0\\\
0\end{bmatrix}\epsilon_{3}^{-1}\begin{bmatrix}0&L^{T}Y&0&0\end{bmatrix}.$

Now, by defining
$\hat{A}=NA_{k}M^{T}+NB_{k}CX+YBC_{k}M^{T}+YA_{\mathcal{N}}X$,
$\hat{B}=NB_{k}$, $\hat{C}=C_{k}M^{T}$, $\eta=\gamma^{2}$ and selecting
$\delta=1$, we obtain

$\Omega_{5}+\Gamma_{12}\Gamma^{-1}_{22}\Gamma^{T}_{12}\prec 0$

where

$\displaystyle\Omega_{5}=\begin{bmatrix}\Omega_{11}&\Omega_{12}&\Omega_{13}&\Omega_{14}\\\
\Omega_{21}&\Omega_{22}&\Omega_{23}&\Omega_{24}\\\
\Omega_{31}&\Omega_{32}&\Omega_{33}&\Omega_{34}\\\
\Omega_{41}&\Omega_{42}&\Omega_{43}&\Omega_{44}\end{bmatrix},$

with

$\displaystyle\Omega_{11}=A_{\mathcal{N}}X+XA_{\mathcal{N}}^{T}+B\hat{C}+\hat{C}^{T}B^{T}+(\epsilon_{1}+\epsilon_{2})LL^{T},$
$\displaystyle\Omega_{12}=\Omega^{T}_{21}=\hat{A}^{T}+A_{\mathcal{N}}$
$\displaystyle\Omega_{22}=A_{\mathcal{N}}^{T}Y+YA_{\mathcal{N}}+\hat{B}C+C^{T}\hat{B}^{T}+\epsilon_{3}R^{T}R,$
$\displaystyle\Omega_{13}=\Omega^{T}_{31}=B_{d},\;\;\Omega_{14}=\Omega^{T}_{41}=XC_{z}^{T}+\hat{C}^{T}D^{T}_{z},$
$\displaystyle\Omega_{23}=\Omega^{T}_{32}=YB_{d},\;\;\Omega_{24}=\Omega^{T}_{42}=C_{z}^{T},\;\;\Omega_{33}=-\eta
I,$ $\displaystyle\Omega_{34}=\Omega_{43}=0,\;\;\Omega_{44}=-I.$

Moreover, we have

$\displaystyle\Gamma_{12}=$ $\displaystyle\begin{bmatrix}XR^{T}&0&0&YL&0\\\
0&R^{T}&YL&0&XR^{T}\\\ 0&0&0&0&0\\\ 0&0&0&0&0\end{bmatrix},$
$\displaystyle\Gamma_{22}=$
$\displaystyle-\mathrm{diag}(\epsilon_{1}I,\epsilon_{2}I,\epsilon_{3}I,I,I).$

Finally, by applying Lemma 3, the LMIs in (33) are obtained.

## Acknowledgement

This work was supported in part by the Swiss National Science Foundation
(SNSF) under the project RECCE, “Real-time traffic estimation and control in a
connected environment”, contract No. 200021-188622.

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