# On the Constrained CAV Platoon Control Problem

MirSaleh Bahavarnia†, Junyi Ji†¶ Ahmad F. Taha†, and Daniel B. Work†¶ †The
authors are with the Department of Civil and Environmental Engineering,
Vanderbilt University, 2201 West End Avenue, Nashville, TN 37235, USA.
¶Institute for Software Integrated Systems. Emails:
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>work is supported by
the National Science Foundation, United States under grants ECCS 2151571, CMMI
2152450 and 2152928, and CNS 2135579.

###### Abstract

The main objective of the connected and automated vehicle (CAV) platoon
control problem is to regulate CAVs’ position while ensuring stability and
accounting for vehicle dynamics. Although this problem has been studied in the
literature, existing research has some limitations. This paper presents two
new theoretical results that address these limitations: (i) the synthesis of
unrealistic high-gain control parameters due to the lack of a systematic way
to incorporate the lower and upper bounds on the control parameters, and (ii)
the performance sensitivity to the communication delay due to inaccurate
Taylor series approximation. To be more precise, taking advantage of the well-
known Padé approximation, this paper proposes a constrained CAV platoon
controller synthesis that (i) systematically incorporates the lower and upper
bounds on the control parameters, and (ii) significantly improves the
performance sensitivity to the communication delay. The effectiveness of the
presented results is verified through conducting extensive numerical
simulations. The proposed controller effectively attenuates the stop-and-go
disturbance—a single cycle of deceleration followed by
acceleration—amplification throughout the mixed platoon (consisting of CAVs
and human-driven vehicles). Modern transportation systems will benefit from
the proposed CAV controls in terms of effective disturbance attenuation as it
will potentially reduce collisions.

###### Index Terms:

Car-following models, connected and automated vehicles, $\mathcal{H}_{\infty}$
control, local stability, string stability, time-delay systems.

##  I Introduction and Paper Contributions

Traffic oscillations, also known as stop-and-go disturbances, in congested
traffic have attained an increasing amount of attention in recent years (see
[1] and the relevant references therein). The term stop-and-go disturbance
refers to a single cycle of deceleration followed by acceleration. Various
approaches have been developed to mostly control human-driven vehicles. One
promising approach is via variable speed limit (VSL) control [2, 3]. Another
one utilizes the emerging connected and automated vehicle (CAV) technology
which facilitates the effective control of CAVs leading to significant
enhancements in traffic flow capacity and stability [4, 5, 1, 6].
Specifically, the main objective of the CAV platoon control problem is to
regulate CAVs’ position while ensuring stability and accounting for vehicle
longitudinal dynamics [1].

CAV control strategies can mainly be classified into two controller types: (i)
model predictive control (MPC) [7, 8, 9, 10, 5], and (ii) linear controller
[11, 12, 13, 1, 6]. Tab. I reflects the pros and cons corresponding to such
controller types in terms of their ability to incorporate two crucial
features: (i) local stability and string stability (two main stability
constraints to be thoroughly detailed in Section II-A), and (ii) explicit
constraints (e.g., lower and upper bounds on the control parameters). Inspired
by Tab. I, in the current study, we choose linear controller as the controller
type and propose a systematic procedure to effectively incorporate both (i)
local stability and string stability (as stability constraints) and (ii) box
constraints (i.e., the lower and upper bounds) on the control parameters (as a
main subclass of explicit constraints).

TABLE I: The general schematic structure of feature incorporation for two main controller types in the CAV control strategies: (i) MPC, and (ii) Linear Controller. Feature Controller Type | MPC | Linear Controller
---|---|---
Local Stability and String Stability | ✗ | ✓
Explicit Constraints | ✓ | ✗

In CAV-centered studies, vehicle platooning [14] emerges as a pivotal
application with significant potential in both the immediate and distant
future. This technology, crucial for enhancing road safety and efficiency,
heavily relies on the efficacy of vehicle-to-everything (V$2$X) communication
systems. Research, including findings from [15], underscores that both
dedicated short-range communications and cellular vehicle-to-everything
technologies can competently support safety applications necessitating end-to-
end latency around $100$ milliseconds, provided that vehicle density remains
within manageable limits. However, a critical challenge arises as traffic
density escalates, leading to a marked increase in V$2$X communication delay
due to the communication channel congestion, as noted in several studies [16,
17, 18]. This surge in latency, particularly in congested scenarios, is a
matter of concern.

As highlighted in [18], effective vehicle platooning must accommodate maximum
latencies ranging from $10$ to $500$ milliseconds. This necessitates the
development of advanced control systems capable of adapting to these varying
delay conditions, thereby ensuring the reliability and safety of vehicle
platooning in diverse traffic environments. This research aims to address the
escalating delays in high-density traffic scenarios, emphasizing the need for
advanced controller synthesis that can effectively manage these challenges in
vehicle platooning applications.

Paper Objectives and Contributions. The main objective of the CAV platoon
control problem is to regulate CAVs’ position while ensuring stability and
accounting for vehicle longitudinal dynamics [1]. Although such a problem has
been studied thoroughly in the literature, the existing research work [1] has
some limitations. Two main limitations include (i) the synthesis of
unrealistic high-gain control parameters due to the lack of a systematic way
to incorporate the lower and upper bounds on the control parameters, and (ii)
the performance sensitivity to the communication delay due to inaccurate
Taylor series approximation. The paper’s contributions can be summarized as
follows:

  * •

To effectively address such limitations, taking advantage of the well-known
Padé approximation, this paper proposes a constrained CAV platoon controller
synthesis that (i) systematically incorporates the lower and upper bounds on
the control parameters, and (ii) significantly improves the performance
sensitivity to the communication delay.

  * •

Given box constraints on the control parameters, we first parameterize the
locally stabilizing feedback and feedforward gains. Deriving a necessary
condition to ensure the string stability criterion additionally, we then
obtain more representative parameterized locally stabilizing feedback and
feedforward gains. Utilizing the Padé approximation, we more accurately ensure
the string stability criterion in the presence of communication delay compared
to the widely utilized Taylor series approximation. Furthermore, in the case
of a sufficiently small communication delay, the Padé approximation attains a
better near-optimality and in the case of a large communication delay, it
successfully obtains a near-optimal CAV platoon controller synthesis while the
widely utilized Taylor series approximation counterpart becomes unusable.

  * •

Considering the mixed vehicular platoon scenario and minimizing the
$\mathcal{H}_{\infty}$ norm of the Padé approximated transfer function over an
interval defined by predominant acceleration frequency boundaries of human-
driven vehicles, we solve for a near-optimal constrained CAV platoon
controller synthesis via nonlinear optimization tools. The effectiveness of
the presented results is verified through conducting extensive numerical
simulations. The proposed CAV platoon controller synthesis can effectively
attenuate the stop-and-go disturbance amplification throughout the mixed
vehicular platoon. Modern transportation systems will potentially benefit from
the proposed CAV control strategy in terms of effective stop-and-go
disturbance attenuation as it will potentially reduce collisions.

The remainder of the paper is structured as follows: Section II formally
presents the preliminaries and states the problem to be solved. Section III
contains the main results followed by numerical simulations illustrated by
Section IV. Finally, Section V ends the paper with a few concluding remarks.

Notation. The uppercase and lowercase letters denote the matrices and vectors,
respectively. We represent the supremum and maximum by $\sup$ and $\max$,
respectively. The set of $n$-dimensional real-valued vectors is symbolized by
$\mathbb{R}^{n}$. For a vector $v\in\mathbb{R}^{n}$, we denote its
$\ell_{\infty}$ norm (i.e., $\max_{i}|v_{i}|$) by $\|v\|_{\ell_{\infty}}$. We
represent the vector of all ones with $\mathbf{1}$. To symbolize the imaginary
unit, we use $j=\sqrt{-1}$. For a square matrix $M$, we denote the set of its
eigenvalues with $\lambda(M)$ and show its spectral abscissa (i.e., the
maximum real part of its eigenvalues) by $\mathbf{sa}(M)$. A square matrix $M$
is said to be Hurwitz if $\mathbf{sa}(M)<0$ holds. The time derivative of a
signal $\chi(t)$ is represented by $\dot{\chi}(t)$. For a function
$f(\omega)$, the first derivative and the second derivative with respect to
$\omega$ are respectively denoted by $\frac{df(\omega)}{d\omega}$ and
$\frac{d^{2}f(\omega)}{d\omega^{2}}$. Also, for brevity, we alternatively use
$f^{\prime}(\omega)$ and $f^{\prime\prime}(\omega)$ to refer to the first and
second derivatives, respectively. To represent the limit of a function
$f(\omega)$ as $\omega$ tends to $a$, we utilize the notation $\lim_{\omega\to
a}f(\omega)$.

##  II Preliminaries and Problem Statement

This section is divided into two main parts: (a) preliminaries in Section
II-A, and (b) problem statement in Section II-B. Preliminaries consist of (i)
the state-space representation, (ii) the local stability, and (iii) the string
stability of the CAV system.

Built upon the preliminaries, we next state a control problem to propose a
constrained CAV platoon controller synthesis subject to the lower and upper
bounds on the control parameters to prevent the synthesis of unrealistic high-
gain control parameters. Furthermore, we will observe that communication delay
can effectively be handled by utilizing a more accurate approximation for the
delay-dependent (exponential) term.

According to the travel direction, we consider a vehicular platoon consisting
of $N_{\mathrm{vp}}$ vehicles with vehicle $1$ and vehicle $N_{\mathrm{vp}}$
as the most preceding (the first) and the most following (the last) vehicles,
respectively.

###  II-A Preliminaries

####  II-A1 State-space representation

Let us consider the state-space representation of the CAV system $i$ as
follows [19, 20, 1]:

$\displaystyle\dot{\chi}_{i}(t)$
$\displaystyle=A_{i}{\chi}_{i}(t)+B_{i}u_{i}(t)+Da_{i-1}(t),$ (1)

with

$\displaystyle A_{i}$ $\displaystyle=\begin{bmatrix}0&1&-\tau^{\ast}_{i}\\\
0&0&-1\\\ 0&0&-\frac{1}{T_{i}}\end{bmatrix},B_{i}=\begin{bmatrix}0\\\ 0\\\
\frac{K_{i}}{T_{i}}\end{bmatrix},D=\begin{bmatrix}0\\\ 1\\\ 0\end{bmatrix},$

where ${\chi}_{i}(t)=\begin{bmatrix}\sigma_{i}(t)&\Delta
v_{i}(t)&a_{i}(t)\end{bmatrix}^{\top}$, $\sigma_{i}(t)$, $\Delta
v_{i}(t):=v_{i-1}(t)-v_{i}(t)$, $a_{i}(t)$, and $u_{i}(t)$ denote the state
vector, the deviation from equilibrium spacing, the speed difference with the
preceding vehicle (i.e., vehicle $i-1$), the realized acceleration, and the
control input, respectively. Moreover, $\tau^{\ast}_{i}$, $T_{i}$, and $K_{i}$
represent the predefined constant time gap, time-lag for vehicle $i$ to
realize the acceleration, and ratio of demanded acceleration that can be
realized, respectively. We can find an equilibrium state $\chi_{i,e}(t)$ by
setting $\chi_{i}(t)=\chi_{i,e}(t)=\begin{bmatrix}0&0&0\end{bmatrix}^{\top}$.
Also, note that $a_{i-1}(t)$ is treated as an external disturbance for CAV
system $i$ because it is not controllable by CAV system $i$.

As highlighted in [1], to bypass multiple delay accumulation, we utilize the
following standard decentralized linear control strategy:

$\displaystyle u_{i}(t)$
$\displaystyle=\begin{bmatrix}k_{si}&k_{vi}&k_{ai}\end{bmatrix}{\chi}_{i}(t)+k_{fi}a_{i-1}(t-\theta),$
(2)

where $k_{si}$, $k_{vi}$, and $k_{ai}$ respectively represent the feedback
gains associated with the derivation from equilibrium spacing $\sigma_{i}$,
the speed difference $\Delta v_{i}$, and the acceleration $a_{i}$. Moreover,
$k_{fi}$ and $\theta$ denote the feedforward gain and V$2$V/V$2$I
communication delay, respectively. For the sake of brevity, we alternatively
utilize $k=\begin{bmatrix}k_{1}&k_{2}&k_{3}&k_{4}\end{bmatrix}$ to denote
$k=\begin{bmatrix}k_{si}&k_{vi}&k_{ai}&k_{fi}\end{bmatrix}$. Similarly, we
simply denote $A_{i}$, $B_{i}$, $K_{i}$, $T_{i}$, and $\tau_{i}^{\ast}$ by
$A$, $B$, $K$, $T$, and $\tau$, respectively.

The main objective of the linear control strategy (2) is to regulate CAVs’
position while ensuring stability and accounting for vehicle longitudinal
dynamics [1]. The linear control strategy (2) has widely been utilized in the
literature [11, 13, 1]. As highlighted in Tab. I, one important advantage of
the linear control strategy (2) over the MPC control strategy is that it
enables us with stability guarantees [1]. Furthermore, later on, we show that
by taking advantage of the parameterization of stability regions of the CAV
system in (1) governed by the control strategy in (2) subject to imposed box
constraints on the control parameters, we effectively embed the constraints
and minimize the $\mathcal{H}_{\infty}$-based objective function similar to
the MPC control strategy.

####  II-A2 Local stability

Local stability (also known as internal stability) means that a disturbance
(deviation from an equilibrium point) can locally be resolved by a system. In
the context of a CAV system, it means that deviation from an equilibrium
spacing, speed difference, and acceleration can locally be resolved in a
vehicle [1].

######  Definition 1 (Local stability [21, 1]).

A CAV platoon following a linear control strategy (e.g., the control strategy
in (2)) is called locally stable with respect to an equilibrium point
$\chi_{e}$ if and only if $\mathbf{sa}(A+Bk)<0$ holds ($A+Bk$ is Hurwitz).

As Proposition $1$ in [1] states, the CAV system in (1) governed by the
control strategy in (2) is locally stable if the following inequalities are
satisfied:

$\displaystyle k_{1}>0,$ (3a) $\displaystyle\tau k_{1}+k_{2}>0,$ (3b)
$\displaystyle\frac{1}{K}-k_{3}>0,$ (3c)
$\displaystyle\bigg{(}\frac{1}{K}-k_{3}\bigg{)}(\tau
k_{1}+k_{2})-\frac{T}{K}k_{1}>0.$ (3d)

It is noteworthy that inequalities (3) are derived by applying the Hurwitz
criterion [22] to the cubic polynomial
$\det(sI-(A+Bk))=\frac{T}{K}s^{3}+(\frac{1}{K}-k_{3})s^{2}+(\tau
k_{1}+k_{2})s+k_{1}$. Applying the Hurwitz criterion [22], ensures that all
the elements of $\lambda(A+Bk)$ lie on the LHS of the imaginary axis, i.e.,
$\mathbf{sa}(A+Bk)<0$ holds ($A+Bk$ is Hurwitz).

####  II-A3 String stability

Strict string stability (also known as $\mathcal{H}_{2}$ norm string
stability) means that the magnitude of a disturbance is not amplified for each
leader-follower pair through a vehicular string [1].

######  Definition 2 (Strict string stability [12, 1]).

A CAV vehicular string is strict string stable if and only if

$\displaystyle\frac{\|a_{i}(s)\|_{2}}{\|a_{i-1}(s)\|_{2}}\leq 1,~{}\forall
i\in\mathcal{I}_{\mathrm{CAV}},$

hold where $\mathcal{I}_{\mathrm{CAV}}$ denotes the set of CAVs and
$\|a_{i}(s)\|_{2}$ represents the $\mathcal{H}_{2}$ norm of $a_{i}(s)$ which
is defined as
$\|a_{i}(s)\|_{2}:=\sqrt{\int_{0}^{\infty}|a_{i}(j\omega)|^{2}d\omega}$.
($\|a_{i-1}(s)\|_{2}$ follows the similar notation/definition.)

As expressed by [1], applying the following Cauchy inequality [12]:

$\displaystyle\frac{\|a_{i}(s)\|_{2}}{\|a_{i-1}(s)\|_{2}}\leq\bigg{\|}\frac{a_{i}(s)}{a_{i-1}(s)}\bigg{\|}_{\infty},$

a sufficient condition to guarantee the string stability can be derived as

$\displaystyle\|F_{i}(s)\|_{\infty}$
$\displaystyle:=\underset{\omega>0}{\sup}~{}|F_{i}(j\omega)|\leq 1,$ (4)

where $F_{i}(s)=\frac{a_{i}(s)}{a_{i-1}(s)}$ and $\|F_{i}(s)\|_{\infty}$
denote the transfer function capturing disturbance propagation through the
vehicular string in the frequency domain and its $\mathcal{H}_{\infty}$ norm
(i.e., the upper bound of the disturbance propagation ratio through the
vehicular string), respectively. As a reminder, $a_{i-1}(t)$ (equivalently,
$a_{i-1}(s)$ in frequency domain) is treated as an external disturbance which
facilitates the $\mathcal{H}_{\infty}$ norm applications in this context.

By taking the Laplace transform of the closed-loop system consisting of (1)
and (2), one can obtain the following expression for $F_{i}(s)$:

$\displaystyle F_{i}(s)=\frac{K(k_{4}s^{2}e^{-\theta
s}+k_{2}s+k_{1})}{c_{3}s^{3}+c_{2}s^{2}+c_{1}s+c_{0}},$ (5) $\displaystyle
c_{3}=T,~{}c_{2}=-Kk_{3}+1,~{}c_{1}=K(\tau k_{1}+k_{2}),~{}c_{0}=Kk_{1}.$

Also, it can be verified that $|F_{i}(j\omega)|$ can be computed as

$\displaystyle|F_{i}(j\omega)|=\sqrt{\frac{N(\omega)}{D(\omega)}},$ (6)
$\displaystyle
N(\omega)=n_{4}\omega^{4}+(n_{2}+g_{\theta}(\omega))\omega^{2}+n_{0},$
$\displaystyle
D(\omega)=d_{6}\omega^{6}+d_{4}\omega^{4}+d_{2}\omega^{2}+d_{0},$
$\displaystyle
n_{4}=K^{2}k_{4}^{2},~{}n_{2}=K^{2}k_{2}^{2},~{}n_{0}=K^{2}k_{1}^{2},$
$\displaystyle
g_{\theta}(\omega)=2K^{2}k_{4}(-k_{1}\cos(\theta\omega)+k_{2}\omega\sin(\theta\omega)),$
$\displaystyle d_{6}=T^{2},~{}d_{4}=-2KT(\tau k_{1}+k_{2})+(Kk_{3}-1)^{2},$
$\displaystyle d_{2}=K^{2}(\tau
k_{1}+k_{2})^{2}+2Kk_{1}(Kk_{3}-1),~{}d_{0}=K^{2}k_{1}^{2}.$

It is noteworthy that the exponential term associated with the communication
delay $\theta$ in $F_{i}(s)$ in (5), i.e., $e^{-\theta s}$, adds more
complexity to the $\mathcal{H}_{\infty}$ controller synthesis. In other words,
we need to utilize approximation techniques to efficiently compute the
approximate value of $\|F_{i}(s)\|_{\infty}$. To that end, there exist various
approximations. For instance, the authors in [1], built upon a sufficient
condition (i.e., utilizing the Taylor series approximations of
$\cos(\theta\omega)\approx 1-\frac{\theta^{2}\omega^{2}}{2}$ and
$\sin(\theta\omega)\approx\theta\omega-\frac{\theta^{3}\omega^{3}}{6}$ when
$\theta$ is sufficiently small) and imposing the sufficient condition (4),
derive a set of inequalities presented by Proposition $2$ therein as

$\displaystyle~{}T^{2}+\frac{K^{2}k_{4}k_{2}\theta^{3}}{3}\geq 0,$ (7a)
$\displaystyle-2KT(\tau k_{1}+k_{2})+(Kk_{3}-1)^{2}$
$\displaystyle-K^{2}k_{4}(k_{1}\theta^{2}+2k_{2}\theta+k_{4})\geq 0,$ (7b)
$\displaystyle~{}2Kk_{1}\bigg{(}K\bigg{(}k_{4}+k_{3}+\tau
k_{2}+\frac{\tau^{2}k_{1}}{2}\bigg{)}-1\bigg{)}\geq 0,$ (7c)

to guarantee the string stability of the CAV system in (1) governed by the
control strategy in (2) when $\theta$ is sufficiently small. The proof of
Proposition $2$ in [1] relies on a strong condition. In other words, to impose
the non-negativeness of the corresponding quartic polynomial, namely,
$p\omega^{4}+q\omega^{2}+r\geq 0$ for which $p$, $q$, and $r$ respectively
denote the expressions on the LHS of (7a)–(7c), they require all coefficients
$p$, $q$, and $r$ to be non-negative as expressed by inequalities (7a)–(7c).
However, the feasibility set associated with the string stability can be
expanded by additionally including the following case:

$\displaystyle p\geq 0,~{}q<0,~{}r\geq 0,~{}q^{2}-4pr\leq 0,$ (8)

which is missing in [1]. Such an inclusion helps us to search a broader space
and potentially find a better near-optimal solution which was not achievable
in [1] previously.

###  II-B Problem statement

It is noteworthy that the ultimate goal is to apply the CAV platoon controller
synthesis results to the mixed vehicular platoon. Due to the string unstable
behavior of human-driven vehicles, we have no direct control over them.
However, applying the (fully) CAV platoon controller synthesis results to the
mixed vehicular platoon can effectively attenuate the stop-and-go disturbance
amplification throughout the mixed vehicular platoon. In the case of the mixed
vehicular platoon, the frequency of human-driven vehicle acceleration
throughout traffic oscillations is bounded and typically shows a predominant
range [23, 1]. Then, for the sake of practicality, we focus on minimizing
$\|F_{i}(s)\|_{\infty}$ over an interval defined by predominant acceleration
frequency boundaries of human-driven vehicles (see [1] for more details).

Given the transfer function $F_{i}(s)=\frac{a_{i}(s)}{a_{i-1}(s)}$ and the
predominant acceleration frequency boundaries of human-driven vehicles,
namely, $\omega_{2}>\omega_{1}>0$, let us define its $\mathcal{H}_{\infty}$
norm over $\omega\in[\omega_{1},\omega_{2}]$, namely,
$\mathcal{H}_{\infty}^{[\omega_{1},\omega_{2}]}$, as [1]

$\displaystyle\|F_{i}(s)\|_{\infty}^{[\omega_{1},\omega_{2}]}$
$\displaystyle:=\underset{\omega\in[\omega_{1},\omega_{2}]}{\sup}~{}|F_{i}(j\omega)|.$
(9)

Observe that based on definitions (4) and (9), the following inequalities:

$\displaystyle\|F_{i}(s)\|_{\infty}^{[\omega_{1},\omega_{2}]}\leq\|F_{i}(s)\|_{\infty}\leq
1,$

are satisfied if (4) holds.

Defining the $\mathcal{H}_{\infty}$ norm over the predominant acceleration
frequency boundaries of human-driven vehicles, we are now ready to state the
main problem to be tackled in this work.

######  Problem 1.

Given the CAV system in (1) governed by the control strategy in (2), the
predominant acceleration frequency boundaries of human-driven vehicles
$\omega_{2}>\omega_{1}>0$, and the following box constraints (i.e., the lower
and upper bounds) on the feedback and feedforward gains
$\begin{bmatrix}k_{1}&k_{2}&k_{3}&k_{4}\end{bmatrix}$:

$\displaystyle k_{1}^{l}\leq k_{1}\leq k_{1}^{u},$ (10a) $\displaystyle
k_{2}^{l}\leq k_{2}\leq k_{2}^{u},$ (10b) $\displaystyle k_{3}^{l}\leq
k_{3}\leq k_{3}^{u},$ (10c) $\displaystyle k_{4}^{l}\leq k_{4}\leq k_{4}^{u},$
(10d)

synthesize an $\mathcal{H}_{\infty}^{[\omega_{1},\omega_{2}]}$ optimal control
strategy with an optimal $\mathcal{H}_{\infty}^{[\omega_{1},\omega_{2}]}$ norm
of $\gamma$ ($\gamma\leq 1$).

Solving Problem 1 enables us to effectively attenuate the stop-and-go
disturbance amplification throughout the mixed vehicular platoon over the
predominant acceleration frequency boundaries of human-driven vehicles.
Furthermore, such a controller synthesis (i) facilitates the effective stop-
and-go disturbance attenuation for the scenario with a large communication
delay, and (ii) systematically incorporates the lower and upper bounds arising
from the physics of the problem. It is noteworthy that these objectives are
not addressable via the controller synthesis proposed by [1].

##  III Main Results

To effectively investigate Problem 1, the current section is divided into
three main parts: (a) parameterized locally stabilizing feedback and
feedforward gains subject to box constraints, (b) string stability via the
Padé approximation, and (c) a near-optimal solution to Problem 1. By
parameterizing locally stabilizing feedback and feedforward gains subject to
box constraints, first, we systematically incorporate the local stability and
the box constraints. Second, we additionally ensure the string stability via
the Padé approximation. Finally, taking advantage of the previous two stages,
we efficiently solve for a near-optimal solution to Problem 1.
To enhance the readability/tractability, we move all the proofs to Appendices
A–C.

###  III-A Parameterized locally stabilizing gains

In this section, we elaborate on how to effectively guarantee local stability
subject to the satisfaction of the box constraints via handy
parameterizations.

The set of feedback gains $\begin{bmatrix}k_{1}&k_{2}&k_{3}\end{bmatrix}$
satisfying the local stability (3) can be parameterized via the parameters
$\begin{bmatrix}x&y&z\end{bmatrix}$ as

$\displaystyle k_{1}(x)$ $\displaystyle=x,$ (11a) $\displaystyle k_{2}(x,y)$
$\displaystyle=-\tau x+y,$ (11b) $\displaystyle k_{3}(x,y,z)$
$\displaystyle=\frac{-Tx+y}{Ky}-z,$ (11c)

where $x$, $y$, and $z$ are all positive parameters.
Imposing box constraints (10a)–(10c) to the parameterization (11), we obtain
the following parameterization of the parameters $x$, $y$, and $z$ in (11):

######  Proposition 1.

The parameters $\begin{bmatrix}x&y&z\end{bmatrix}$ in (11) satisfying box
constraints (10a)–(10c) can be parameterized via the parameters
$\begin{bmatrix}\psi_{1}&\psi_{2}&\psi_{3}\end{bmatrix}$ as

$\displaystyle x=\texttt{x}(\psi_{1})=(1-\psi_{1})x^{l}+\psi_{1}x^{u},$ (12a)
$\displaystyle
y=\texttt{y}(\psi_{1},\psi_{2})=(1-\psi_{2})y^{l}_{\psi_{1}}+\psi_{2}y^{u}_{\psi_{1}},$
(12b) $\displaystyle
z=\texttt{z}(\psi_{1},\psi_{2},\psi_{3})=(1-\psi_{3})z^{l}_{\psi_{1},\psi_{2}}+\psi_{3}z^{u}_{\psi_{1},\psi_{2}},$
(12c) with $\displaystyle x^{l}=\max\\{\epsilon,k_{1}^{l}\\},$ (12d)
$\displaystyle x^{u}=k_{1}^{u},$ (12e) $\displaystyle
y^{l}_{\psi_{1}}=\max\bigg{\\{}\epsilon,\tau\texttt{x}(\psi_{1})+k_{2}^{l},\frac{T\texttt{x}(\psi_{1})}{-Kk_{3}^{l}+1-K\epsilon}\bigg{\\}},$
(12f) $\displaystyle y^{u}_{\psi_{1}}=\tau\texttt{x}(\psi_{1})+k_{2}^{u},$
(12g) $\displaystyle
z^{l}_{\psi_{1},\psi_{2}}=\max\bigg{\\{}\epsilon,\frac{-T\texttt{x}(\psi_{1})+\texttt{y}(\psi_{1},\psi_{2})}{K\texttt{y}(\psi_{1},\psi_{2})}-k_{3}^{u}\bigg{\\}},$
(12h) $\displaystyle
z^{u}_{\psi_{1},\psi_{2}}=\frac{-T\texttt{x}(\psi_{1})+\texttt{y}(\psi_{1},\psi_{2})}{K\texttt{y}(\psi_{1},\psi_{2})}-k_{3}^{l},$
(12i)

where $\psi_{i}\in[0,1]$ holds for all $i\in\\{1,2,3\\}$ and $\epsilon>0$ is
an infinitesimal value.

For the proof, the reader is referred to Appendix A.

Similar to Proposition 1, the feedforward gain $k_{4}$ satisfying box
constraint (10d) can simply be parameterized via the parameter $\psi_{4}$ as

$\displaystyle k_{4}$ $\displaystyle=(1-\psi_{4})k_{4}^{l}+\psi_{4}k_{4}^{u},$
(13)

where $\psi_{4}\in[0,1]$ holds. It is noteworthy that in parameterizations
(12) and (13), the form of $\psi_{i}$’s can be chosen via any arbitrary
sigmoid function, e.g., the logistic function

$\displaystyle\psi(\beta)$ $\displaystyle=\frac{1}{1+e^{-\zeta\beta}}\,,$

where $\zeta>0$ denotes the logistic growth rate. Merging (11)–(13) along with
sigmoid functions, we obtain the following parameterization of the locally
stabilizing feedback and feedforward gains
$\begin{bmatrix}k_{1}&k_{2}&k_{3}&k_{4}\end{bmatrix}$ subject to box
constraints (10):

######  Corollary 1.

The locally stabilizing feedback and feedforward gains
$\begin{bmatrix}k_{1}&k_{2}&k_{3}&k_{4}\end{bmatrix}$ subject to box
constraints (10) can be parameterized via the parameters
$\begin{bmatrix}\kappa_{1}&\kappa_{2}&\kappa_{3}&\kappa_{4}\end{bmatrix}$ as

$\displaystyle k_{1}(\kappa_{1})=\texttt{x}(\psi(\kappa_{1})),$ (14a)
$\displaystyle
k_{2}(\kappa_{1},\kappa_{2})=-\tau\texttt{x}(\psi(\kappa_{1}))+\texttt{y}(\psi(\kappa_{1}),\psi(\kappa_{2})),$
(14b) $\displaystyle
k_{3}(\kappa_{1},\kappa_{2},\kappa_{3})=\frac{-T\texttt{x}(\psi(\kappa_{1}))+\texttt{y}(\psi(\kappa_{1}),\psi(\kappa_{2}))}{K\texttt{y}(\psi(\kappa_{1}),\psi(\kappa_{2}))}$
$\displaystyle-\texttt{z}(\psi(\kappa_{1}),\psi(\kappa_{2}),\psi(\kappa_{3})),$
(14c) $\displaystyle
k_{4}(\kappa_{4})=(1-\psi(\kappa_{4}))k_{4}^{l}+\psi(\kappa_{4})k_{4}^{u},$
(14d)

where $\kappa_{i}\in\mathbb{R}$ holds for all $i\in\\{1,2,3,4\\}$ and
$\texttt{x}()$, $\texttt{y}()$, and $\texttt{z}()$ represent the same
functions expressed in (12).

######  Lemma 1.

If the sufficient condition on the string stability (4) holds for the transfer
function $F_{i}(s)$ in (5), then inequality (7c) holds for the control
parameters $\begin{bmatrix}k_{1}&k_{2}&k_{3}&k_{4}\end{bmatrix}$.

The proof is in Appendix B.

It can easily be verified that utilizing the necessary condition (associated
with the string stability) stated by Lemma 1, i.e., inequality (7c), and
noting that $K>0$ and $k_{1}>0$ hold, the feedforward gain $k_{4}$ can be
parameterized via the parameters $\begin{bmatrix}x&y&z&w\end{bmatrix}$ as

$\displaystyle k_{4}(x,y,z,w)$ $\displaystyle=\frac{\tau^{2}x}{2}-\tau
y+\frac{Tx}{Ky}+z+w,$ (15)

where $x$, $y$, and $z$ are as expressed in (11) and $w$ is a non-negative
parameter. Imposing box constraints (10) to the parameterizations (11) and
(15), we obtain the following parameterization of the parameters $x$, $y$,
$z$, and $w$ in (11) and (15):

######  Proposition 2.

The parameters $\begin{bmatrix}x&y&z&w\end{bmatrix}$ in (11) and (15)
satisfying box constraints (10) can be parameterized via the parameters
$\begin{bmatrix}\psi_{1}&\psi_{2}&\psi_{3}&\psi_{4}\end{bmatrix}$ as

$\displaystyle~{}x=\texttt{x}(\psi_{1})=(1-\psi_{1})x^{l}+\psi_{1}x^{u},$
(16a)
$\displaystyle~{}y=\texttt{y}(\psi_{1},\psi_{2})=(1-\psi_{2})y^{l}_{\psi_{1}}+\psi_{2}y^{u}_{\psi_{1}},$
(16b)
$\displaystyle~{}z=\texttt{z}(\psi_{1},\psi_{2},\psi_{3})=(1-\psi_{3})z^{l}_{\psi_{1},\psi_{2}}+\psi_{3}z^{u}_{\psi_{1},\psi_{2}},$
(16c) $\displaystyle~{}w=\texttt{w}(\psi_{1},\psi_{2},\psi_{3},\psi_{4})$
$\displaystyle=(1-\psi_{4})w_{\psi_{1},\psi_{2},\psi_{3}}^{l}+\psi_{4}w_{\psi_{1},\psi_{2},\psi_{3}}^{u},$
(16d) with $\displaystyle~{}x^{l}=\max\\{\epsilon,k_{1}^{l}\\},$ (16e)
$\displaystyle~{}x^{u}=k_{1}^{u},$ (16f)
$\displaystyle~{}y^{l}_{\psi_{1}}=\max\bigg{\\{}\epsilon,\tau\texttt{x}(\psi_{1})+k_{2}^{l},\frac{T\texttt{x}(\psi_{1})}{-Kk_{3}^{l}+1-K\epsilon},$
$\displaystyle\frac{\xi(\texttt{x}(\psi_{1}))+\sqrt{\xi(\texttt{x}(\psi_{1}))^{2}+\frac{4T\tau\texttt{x}(\psi_{1})}{K}}}{2\tau}\bigg{\\}},$
(16g) $\displaystyle~{}y^{u}_{\psi_{1}}=\tau\texttt{x}(\psi_{1})+k_{2}^{u},$
(16h)
$\displaystyle~{}z^{l}_{\psi_{1},\psi_{2}}=\max\bigg{\\{}\epsilon,\frac{-T\texttt{x}(\psi_{1})+\texttt{y}(\psi_{1},\psi_{2})}{K\texttt{y}(\psi_{1},\psi_{2})}-k_{3}^{u}\bigg{\\}},$
(16i)
$\displaystyle~{}z^{u}_{\psi_{1},\psi_{2}}=\min\bigg{\\{}\frac{-T\texttt{x}(\psi_{1})+\texttt{y}(\psi_{1},\psi_{2})}{K\texttt{y}(\psi_{1},\psi_{2})}-k_{3}^{l},$
$\displaystyle\frac{-\tau^{2}\texttt{x}(\psi_{1})}{2}+\tau\texttt{y}(\psi_{1},\psi_{2})+\frac{-T\texttt{x}(\psi_{1})}{K\texttt{y}(\psi_{1},\psi_{2})}+k_{4}^{u}\bigg{\\}},$
(16j)
$\displaystyle~{}w_{\psi_{1},\psi_{2},\psi_{3}}^{l}=\max\bigg{\\{}0,\frac{-\tau^{2}\texttt{x}(\psi_{1})}{2}+\tau\texttt{y}(\psi_{1},\psi_{2})$
$\displaystyle+\frac{-T\texttt{x}(\psi_{1})}{K\texttt{y}(\psi_{1},\psi_{2})}-\texttt{z}(\psi_{1},\psi_{2},\psi_{3})+k_{4}^{l}\bigg{\\}},$
(16k)
$\displaystyle~{}w_{\psi_{1},\psi_{2},\psi_{3}}^{u}=\frac{-\tau^{2}\texttt{x}(\psi_{1})}{2}+\tau\texttt{y}(\psi_{1},\psi_{2})$
$\displaystyle+\frac{-T\texttt{x}(\psi_{1})}{K\texttt{y}(\psi_{1},\psi_{2})}-\texttt{z}(\psi_{1},\psi_{2},\psi_{3})+k_{4}^{u},$
(16l)

where $\psi_{i}\in[0,1]$ holds for all $i\in\\{1,2,3,4\\}$ and
$\xi(\texttt{x}(\psi_{1}))=\frac{\tau^{2}\texttt{x}(\psi_{1})}{2}+\epsilon-
k_{4}^{u}$.

See Appendix C for the proof.

Merging (11), (15), and (16) along with sigmoid functions, we obtain the
following parameterization of the locally stabilizing feedback and feedforward
gains $\begin{bmatrix}k_{1}&k_{2}&k_{3}&k_{4}\end{bmatrix}$ subject to box
constraints (10), given in the following corrollary.

######  Corollary 2.

The locally stabilizing feedback and feedforward gains
$\begin{bmatrix}k_{1}&k_{2}&k_{3}&k_{4}\end{bmatrix}$ subject to box
constraints (10) can be parameterized via the parameters
$\begin{bmatrix}\kappa_{1}&\kappa_{2}&\kappa_{3}&\kappa_{4}\end{bmatrix}$ as

$\displaystyle k_{1}(\kappa_{1})=\texttt{x}(\psi(\kappa_{1})),$ (17a)
$\displaystyle
k_{2}(\kappa_{1},\kappa_{2})=-\tau\texttt{x}(\psi(\kappa_{1}))+\texttt{y}(\psi(\kappa_{1}),\psi(\kappa_{2})),$
(17b) $\displaystyle
k_{3}(\kappa_{1},\kappa_{2},\kappa_{3})=\frac{-T\texttt{x}(\psi(\kappa_{1}))+\texttt{y}(\psi(\kappa_{1}),\psi(\kappa_{2}))}{K\texttt{y}(\psi(\kappa_{1}),\psi(\kappa_{2}))}$
$\displaystyle-\texttt{z}(\psi(\kappa_{1}),\psi(\kappa_{2}),\psi(\kappa_{3})),$
(17c) $\displaystyle
k_{4}(\kappa_{1},\kappa_{2},\kappa_{3},\kappa_{4})=\frac{\tau^{2}\texttt{x}(\psi(\kappa_{1}))}{2}-\tau\texttt{y}(\psi(\kappa_{1}),\psi(\kappa_{2}))$
$\displaystyle+\frac{T\texttt{x}(\psi(\kappa_{1}))}{K\texttt{y}(\psi(\kappa_{1}),\psi(\kappa_{2}))}+\texttt{z}(\psi(\kappa_{1}),\psi(\kappa_{2}),\psi(\kappa_{3}))$
$\displaystyle+\texttt{w}(\psi(\kappa_{1}),\psi(\kappa_{2}),\psi(\kappa_{3}),\psi(\kappa_{4})),$
(17d)

where $\kappa_{i}\in\mathbb{R}$ holds for all $i\in\\{1,2,3,4\\}$ and
$\texttt{x}()$, $\texttt{y}()$, $\texttt{z}()$, and $\texttt{w}()$ represent
the same functions expressed in (16).

######  Corollary 3.

Given the locally stabilizing feedback and feedforward gains
$\begin{bmatrix}k_{1}&k_{2}&k_{3}&k_{4}\end{bmatrix}$ subject to box
constraints (10) and utilizing the parameterization (17), the corresponding
parameters
$\begin{bmatrix}\kappa_{1}&\kappa_{2}&\kappa_{3}&\kappa_{4}\end{bmatrix}$ can
be extracted as

$\displaystyle\kappa_{1}(k_{1})=\frac{1}{\zeta}\ln\bigg{(}\frac{\phi_{1}(k_{1})-x^{l}}{x^{u}-\phi_{1}(k_{1})}\bigg{)},$
(18a)
$\displaystyle\kappa_{2}(k_{1},k_{2},\kappa_{1})=\frac{1}{\zeta}\ln\bigg{(}\frac{\phi_{2}(k_{1},k_{2})-y^{l}_{\psi(\kappa_{1})}}{y^{u}_{\psi(\kappa_{1})}-\phi_{2}(k_{1},k_{2})}\bigg{)},$
(18b) $\displaystyle\kappa_{3}(k_{1},k_{2},k_{3},\kappa_{1},\kappa_{2})=$
$\displaystyle\frac{1}{\zeta}\ln\bigg{(}\frac{\phi_{3}(k_{1},k_{2},k_{3})-z^{l}_{\psi(\kappa_{1}),\psi(\kappa_{2})}}{z^{u}_{\psi(\kappa_{1}),\psi(\kappa_{2})}-\phi_{3}(k_{1},k_{2},k_{3})}\bigg{)},$
(18c)
$\displaystyle\kappa_{4}(k_{1},k_{2},k_{3},k_{4},\kappa_{1},\kappa_{2},\kappa_{3})=$
$\displaystyle\frac{1}{\zeta}\ln\bigg{(}\frac{\phi_{4}(k_{1},k_{2},k_{3},k_{4})-w^{l}_{\psi(\kappa_{1}),\psi(\kappa_{2}),\psi(\kappa_{3})}}{w^{u}_{\psi(\kappa_{1}),\psi(\kappa_{2}),\psi(\kappa_{3})}-\phi_{4}(k_{1},k_{2},k_{3},k_{4})}\bigg{)},$
(18d) with $\displaystyle\phi_{1}(k_{1})=k_{1},$ (18e)
$\displaystyle\phi_{2}(k_{1},k_{2})=\tau k_{1}+k_{2},$ (18f)
$\displaystyle\phi_{3}(k_{1},k_{2},k_{3})=\frac{-Tk_{1}}{K(\tau
k_{1}+k_{2})}+\frac{1}{K}-k_{3},$ (18g)
$\displaystyle\phi_{4}(k_{1},k_{2},k_{3},k_{4})=\frac{\tau^{2}}{2}k_{1}+\tau
k_{2}+k_{3}+k_{4}-\frac{1}{K},$ (18h)

where

$\displaystyle
y^{l}_{\psi(\kappa_{1})},~{}y^{u}_{\psi(\kappa_{1})},~{}z^{l}_{\psi(\kappa_{1}),\psi(\kappa_{2})},~{}z^{u}_{\psi(\kappa_{1}),\psi(\kappa_{2})},$
$\displaystyle
w^{l}_{\psi(\kappa_{1}),\psi(\kappa_{2}),\psi(\kappa_{3})},~{}w^{u}_{\psi(\kappa_{1}),\psi(\kappa_{2}),\psi(\kappa_{3})},$

are evaluated by replacing $\psi_{i}$ (in (16)) with $\psi(\kappa_{i})$ for
all $i\in\\{1,2,3\\}$.

As a summary, (i) Corollary 2 will be utilized as a cornerstone to
parameterize the locally stabilizing feedback and feedforward gains
$\begin{bmatrix}k_{1}&k_{2}&k_{3}&k_{4}\end{bmatrix}$ via the parameters
$\begin{bmatrix}\kappa_{1}&\kappa_{2}&\kappa_{3}&\kappa_{4}\end{bmatrix}$, and
(ii) Corollary 3 will facilitate the extraction of the parameters
$\begin{bmatrix}\kappa_{1}&\kappa_{2}&\kappa_{3}&\kappa_{4}\end{bmatrix}$
given the locally stabilizing feedback and feedforward gains
$\begin{bmatrix}k_{1}&k_{2}&k_{3}&k_{4}\end{bmatrix}$. The former is a useful
feature in the case of searching for the near-optimal solution to Problem 1
while the latter is a key tool to the case of opting (extracting) an initial
feasible point for the main optimization problem associated with Problem 1.

###  III-B String stability via the Padé approximation

In this section, we shed light on incorporating the string stability into the
locally stabilizing box-constrained gains parameterized in the previous
section.

As mentioned earlier, one needs to overcome the complexity of the
communication delay in the $\mathcal{H}_{\infty}$ controller synthesis. In
that regard, unlike the Taylor series approximation utilized by [1], we employ
the Padé approximation [24] approach to approximate the transfer function
$F_{i}(s)$ in (5), namely, $\hat{F}_{i}(s)$. To that end, we replace the
exponential term associated with the communication delay $\theta$ in
$F_{i}(s)$ in (5), i.e., $e^{-\theta s}$ with its Padé approximation which
poses the following rational form:

$\displaystyle e^{-\theta s}$
$\displaystyle\approx\frac{P^{Pad\acute{e}}(s)}{Q^{Pad\acute{e}}(s)},$ (19)

where $P^{Pad\acute{e}}(s)$ and $Q^{Pad\acute{e}}(s)$ denote the $N$-th order
($N$: Padé approximation order) numerator and denominator polynomials,
respectively. Inspired by the sufficient condition (4), to more accurately
ensure the string stability of the CAV system in (1) governed by the control
strategy in (2) in the presence of communication delay, we utilize the
following condition:

$\displaystyle\|\hat{F}_{i}(s)\|_{\infty}\leq 1.$ (20)

Notice that in our case, deriving a set of inequalities similar to the
inequalities (7) proposed by Proposition $2$ in [1] is impossible as we do not
limit ourselves to the case that $\theta$ is sufficiently small in the current
paper. Specifically, the authors in [1] take advantage of the specific
properties of the corresponding quartic polynomial
$p\omega^{4}+q\omega^{2}+r\geq 0$ resulting from the aforementioned
simplifying assumption regarding the $\theta$ that is no longer applicable to
our case.

Centering around (20), we incorporate the string stability into the locally
stabilizing box-constrained gains by utilizing the Padé approximation (19).
Also, later on, we empirically observe that the Padé approximation provides a
more accurate approximation than the Taylor series approximation, leading to a
potentially better near-optimal solution. We emphasize that although there
exists an analytical explicit formula for $F_{i}(j\omega)$ in (4) and (9)
(according to (6)), utilizing the approximate forms of (4) and (9), i.e.,
(19), is inevitable as (4) and (9) cannot be utilized directly due to the
computational complexity of computing $\|F_{i}(s)\|_{\infty}$ and
$\|F_{i}(s)\|_{\infty}^{[\omega_{1},\omega_{2}]}$, respectively. Then, we
utilize the Padé approximation-based counterparts
$\|\hat{F}_{i}(s)\|_{\infty}$ and
$\|\hat{F}_{i}(s)\|_{\infty}^{[\omega_{1},\omega_{2}]}$ in the next section to
effectively solve Problem 1 for a near-optimal solution.

###  III-C A near-optimal solution to Problem 1

Here, we propose a two-stage procedure to solve Problem 1 for a near-optimal
solution. Such two stages are as follows:

  1. 1.

Finding an initial stabilizing feasible point satisfying box constraints (10)
and both local stability (3) and string stability (4).

  2. 2.

Solving Problem 1 for a near-optimal solution starting from the obtained
initial stabilizing feasible point in the first stage.

Substituting the parameterized locally stabilizing feedback and feedforward
gains (17) provided by Corollary 2 into $F_{i}(s)$ in (5), we obtain the
parameterized $F(s;\kappa)$ and denote its Padé approximation by
$\hat{F}(s;\kappa)$. Then, defining the optimization variable $\kappa$ as
$\kappa:=\begin{bmatrix}\kappa_{1}&\kappa_{2}&\kappa_{3}&\kappa_{4}\end{bmatrix}^{\top}$,
we consider the following optimization problem:

$\displaystyle\underset{\kappa\in\mathbb{R}^{4}}{\mathrm{Minimize}}~{}$
$\displaystyle\|\hat{F}(s;\kappa)\|_{\infty}^{[\omega_{1},\omega_{2}]},$ (21a)
$\displaystyle\mathrm{subject~{}to:}~{}$
$\displaystyle\|\hat{F}(s;\kappa)\|_{\infty}\leq 1,$ (21b)

to solve Problem 1 for a near-optimal solution (Remember that
$\|\hat{F}(s;\kappa)\|_{\infty}^{[\omega_{1},\omega_{2}]}$ denotes the
$\mathcal{H}_{\infty}^{[\omega_{1},\omega_{2}]}$ norm of $\hat{F}(s;\kappa)$).
The aforementioned two-stage procedure takes the following form:

  1. 1.

We first search for an initial stabilizing feasible point $\kappa^{0}$ that
satisfies constraint (21b).

  2. 2.

We then solve optimization problem (21) for a near-optimal solution
$\kappa^{\ast}$ starting from the obtained initial stabilizing feasible point
in the first stage, i.e., $\kappa^{0}$. Finally, one can compute $k^{\ast}$ by
substituting $\kappa^{\ast}$ into the parameterization (17) provided by
Corollary 2 as $k^{\ast}=k(\kappa^{\ast})$.

In the sequel, we delve into each stage thoroughly.

####  III-C1 First stage

Notably, one can take advantage of (18) provided by Corollary 3 to extract
$\kappa^{0}$ from $k^{0}$. Such a fact motivates us to: first, alternatively
search for an initial stabilizing feasible point $k^{0}$ and second, extract
the corresponding initial stabilizing feasible point $\kappa^{0}$ from
$k^{0}$. To that end, first, we consider the following parameterization:

$\displaystyle k_{1}(\mu_{1})$
$\displaystyle=(1-\rho(\mu_{1}))\max\\{\epsilon,k_{1}^{l}\\}+\rho(\mu_{1})k_{1}^{u},$
(22a) $\displaystyle k_{2}(\mu_{2})$
$\displaystyle=(1-\rho(\mu_{2}))k_{2}^{l}+\rho(\mu_{2})k_{2}^{u},$ (22b)
$\displaystyle k_{3}(\mu_{3})$
$\displaystyle=(1-\rho(\mu_{3}))k_{3}^{l}+\rho(\mu_{3})k_{3}^{u},$ (22c)
$\displaystyle k_{4}(\mu_{4})$
$\displaystyle=(1-\rho(\mu_{4}))k_{4}^{l}+\rho(\mu_{4})k_{4}^{u},$ (22d)

with

$\displaystyle\rho(\beta)$ $\displaystyle=\frac{1}{1+\nu\beta^{2}},$ (23)

where $\nu>0$ can arbitrarily be chosen and second, defining
$\mu:=\begin{bmatrix}\mu_{1}&\mu_{2}&\mu_{3}&\mu_{4}\end{bmatrix}^{\top}$ and
substituting the parameterized feedback and feedforward gains (22) into
$F_{i}(s)$ in (5), we obtain the parameterized $G(s;\mu)$ and denote its Padé
approximation by $\hat{G}(s;\mu)$.

Then, to search for an initial stabilizing feasible point $\mu^{0}$ that
satisfies $\|\hat{G}(s;\mu)\|_{\infty}\leq 1$ (an equivalent constraint to
constraint (21b)), we solve the following standard $\mathcal{H}_{\infty}$
optimization problem:

$\displaystyle\underset{\mu\in\mathbb{R}^{4}}{\mathrm{Minimize}}~{}$
$\displaystyle\|\hat{G}(s;\mu)\|_{\infty},$ (24)

for $\mu^{0}$, utilizing a well-developed standard $\mathcal{H}_{\infty}$
problem solver built upon [25], i.e., hinfstruct solver proposed by [26, 27].
Afterwards, according to the parameterization (22), we get $k^{0}=k(\mu^{0})$
and plug $k^{0}$ to (18) provided by Corollary 3 to extract $\kappa^{0}$.

######  Remark 1.

It is noteworthy that due to the fundamental limitations of the implementation
of hinfstruct, one cannot utilize the most representative sophisticated
parameterization (17) provided by Corollary 2 at this stage, to solve (24) for
$\mu^{0}$. That is why we alternatively employ the simple parameterization
(22) along with (23) to that end.

####  III-C2 Second stage

Motivated by (20), let us define the following function:

$\displaystyle h(\kappa)$
$\displaystyle:=\left\\{\begin{array}[]{ll}\|\hat{F}(s;\kappa)\|_{\infty}^{[\omega_{1},\omega_{2}]}&\|\hat{F}(s;\kappa)\|_{\infty}\leq
1\\\ \alpha&\|\hat{F}(s;\kappa)\|_{\infty}>1\\\ \end{array},\right.$ (27)

where $\alpha>1$ can arbitrarily be chosen. Now, we can alternatively solve
the following unconstrained optimization problem:

$\displaystyle\underset{\kappa\in\mathbb{R}^{4}}{\mathrm{Minimize}}~{}$
$\displaystyle h(\kappa),$ (28)

to find a solution $\kappa^{\ast}$ to optimization problem (21). Finally, one
can compute $k^{\ast}$ by substituting $\kappa^{\ast}$ into the
parameterization (17) provided by Corollary 2 as $k^{\ast}=k(\kappa^{\ast})$.
Regarding the initialization, we utilize $\kappa^{0}$ obtained from the first
stage. Furthermore, due to the non-convex and non-smooth nature of the
function $h(\tau)$ in (27), we need to utilize non-convex and non-smooth
optimization tools to solve the unconstrained optimization problem (28). For
instance, we can employ fminsearch solver developed based on Nelder–Mead
simplex method [28] in that regard. It is noteworthy that given a non-convex
non-smooth optimization problem, finding a globally optimal solution is
generally a challenging task. As a result, we utilize the term near-optimal
instead of the term optimal in the current paper.

Procedure Procedure 1 summarizes the two-stage $\mathcal{H}_{\infty}$
controller synthesis procedure.

2 Input:
$\tau$,$T$,$K$,$\theta$,$\omega_{1}$,$\omega_{2}$,$\\{k_{i}^{l}\\}_{i=1}^{4}$,$\\{k_{i}^{u}\\}_{i=1}^{4}$,$\alpha$,$\zeta$,$\nu$,$N$

3 First stage:

4 Construct $G(s;\mu)$ via the parameterization (22).

5 Get $\hat{G}(s;\mu)$ via the Padé approximation of $G(s;\mu)$.

6 Solve (24) for $\mu^{0}$ via hinfstruct solver.

7 Compute $k^{0}=k(\mu^{0})$ via the parameterization (22).

8 Extract $\kappa^{0}$ from $k^{0}$ via (18).

9 Second stage:

10 Construct $F(s;\kappa)$ via the parameterization (17).

11 Get $\hat{F}(s;\kappa)$ via the Padé approximation of $F(s;\kappa)$.

12 Initialize (28) with $\kappa^{0}$ obtained from the first stage.

13 Solve (28) for $\kappa^{\ast}$ via fminsearch solver.

14 Compute $k^{\ast}=k(\kappa^{\ast})$ via the parameterization (17).

Output: $k^{\ast}$.

Procedure 1 Two-stage $\mathcal{H}_{\infty}$ Controller Synthesis

######  Remark 2.

Observe that for the parameterized $F(s;\kappa)$, on the one hand, according
to the supremum-based definition of the $\mathcal{H}_{\infty}$ norm expressed
in (4), we have $\|F(s;\kappa)\|_{\infty}\geq 1$ based on $\lim_{\omega\to
0^{+}}F(j\omega;\kappa)=1$, $\lim_{\omega\to
0^{+}}F^{\prime}(j\omega;\kappa)=0$, and $\lim_{\omega\to
0^{+}}F^{\prime\prime}(j\omega;\kappa)\leq 0$ (See Appendix B for more
details). On the other hand, according to (4), we impose
$\|F(s;\kappa)\|_{\infty}\leq 1$ to obtain the parameterized locally
stabilizing feedback and feedforward gains that guarantee the string
stability. Thus, we conclude that for such feedback and feedforward gains
$\|F(s;\kappa)\|_{\infty}=1$ holds. Particularly, for $\kappa^{0}$ and
$\kappa^{\ast}$, the equalities $\|F(s;\kappa)\|_{\infty}=1$ and
$\|F(s;\kappa)\|_{\infty}=1$ are satisfied, respectively.

######  Remark 3.

The first stage presented in Procedure Procedure 1 can be devised differently.
In other words, various alternatives exist to choose an appropriate
$\kappa^{0}$. As an effective alternative, one can think of sampling the
$4$-dimensional search space $\mathbb{R}^{4}$ equipped with a pre-assumed
distribution (e.g., uniform distribution). Denoting the set of samples
$\tilde{\kappa}$ by $\mathcal{S}$, for any sample
$\tilde{\kappa}\in\mathcal{S}$, we can check that if
$\|F(s;\tilde{\kappa})\|_{\infty}\leq 1$ holds or not. Collecting all those
samples satisfying $\|F(s;\tilde{\kappa})\|_{\infty}\leq 1$ and sorting the
corresponding $\mathcal{H}_{\infty}^{[\omega_{1},\omega_{2}]}$ norm values,
i.e.,
$h(\tilde{\kappa})=\|\hat{F}(s;\tilde{\kappa})\|_{\infty}^{[\omega_{1},\omega_{2}]}$,
in an ascending order, we can set $\kappa^{0}$ as
$\kappa^{0}=\operatorname*{arg\
min}_{\tilde{\kappa}\in\mathcal{S}}h(\tilde{\kappa})$. Notice that at the
expense of a denser sampling (higher computational time), one expects to
obtain a more representative choice of such $\kappa^{0}$ leading to a better
near-optimal solution $k^{\ast}$. However, the first stage presented in
Procedure Procedure 1 significantly saves computational time while mostly
generating a high-quality choice of $\kappa^{0}$ leading to an acceptable
near-optimal solution $k^{\ast}$.

##  IV Numerical Simulations

Throughout this section, we assess the effectiveness of the theoretical
results presented in the paper. Depending on how small the communication delay
is, the section is divided into two main parts: (a) Case $1$: a sufficiently
small communication delay $\theta$ and (b) Case $2$: a large communication
delay $\theta$.

In Case $1$, since the communication delay $\theta$ is sufficiently small, we
can set the unconstrained $\mathcal{H}_{\infty}$ controller synthesis proposed
in [1], namely, $k^{\mathrm{unc}}$, as a benchmark for comparative analysis
purposes. In that regard, to have a fair comparison, we impose the box
constraints on $k^{\ast}$, based on $\|k^{\mathrm{unc}}\|_{\ell_{\infty}}$. In
other words, we consider a hypercube that encompasses $k^{\mathrm{unc}}$. In
Case $2$, since $\theta$ is not sufficiently small anymore, the Taylor series
approximation-based method proposed by [1] becomes unusable.

We conduct all the numerical experiments in MATLAB R$2022$b on a MacBook Pro
with a $3.1$ GHz Intel Core i$5$ and memory $8$ GB $2133$ MHz. To create the
continuous-time transfer function models, we employ tf. We also utilize
getPeakGain (developed built upon [29]) to evaluate the
$\mathcal{H}_{\infty}^{[\omega_{1},\omega_{2}]}$/$\mathcal{H}_{\infty}$ norm
values. Regarding the Padé approximation, we utilize pade equipped with $N$
which denotes the Padé approximation order. To utilize hinfstruct, we take
advantage of realp, creating real-valued tunable parameters $\mu_{1}$,
$\mu_{2}$, $\mu_{3}$, and $\mu_{4}$. Furthermore, we initialize those real-
valued parameters via rand.

TABLE II: Value setting for the experimental $\mathcal{H}_{\infty}$ controller synthesis [1]. Time quantities are all in seconds. Parameter | Value
---|---
$\tau$ | $1$
$T$ | $0.45$
$K$ | $1$
$\theta$ | $0.1$
$\omega_{2}$ | $2.5$

###  IV-A Case 1: a sufficiently small communication delay

In this section, considering the case of sufficiently small $\theta$, we
conduct a comparative analysis between the unconstrained and constrained
$\mathcal{H}_{\infty}$ syntheses $k^{\mathrm{unc}}$ [1] and $k^{\ast}$.

Similar to [1], we consider the value setting in Tab. II for the experimental
$\mathcal{H}_{\infty}$ controller synthesis. Time quantities are all in
seconds. Setting $\omega_{1}=0.5$,
$k^{l}=-\|k^{\mathrm{unc}}\|_{\ell_{\infty}}\begin{bmatrix}0&1&1&1\end{bmatrix}^{\top}$,
$k^{u}=\|k^{\mathrm{unc}}\|_{\ell_{\infty}}\mathbf{1}$, $\alpha=1.05$,
$\zeta=5$, $\nu=5$, $N=5$ and running Procedure Procedure 1, we obtain the
two-stage $\mathcal{H}_{\infty}$ controller synthesis illustrated by Tab. III
for which the $\mathcal{H}_{\infty}^{[\omega_{1},\omega_{2}]}$ norm values
corresponding to the unconstrained and constrained $\mathcal{H}_{\infty}$
syntheses $k^{\ast}$ and $k^{\mathrm{unc}}$ [1] are
$\|F(s)\|_{\infty}^{[\omega_{1},\omega_{2}]}=0.6758$ and
$\|F(s)\|_{\infty}^{[\omega_{1},\omega_{2}]}=0.8667$, respectively.
Accordingly, Fig. 1 visualizes the $\omega$-dependent $|F(j\omega)|$ values
corresponding to the unconstrained and constrained $\mathcal{H}_{\infty}$
syntheses $k^{\mathrm{unc}}$ and $k^{\ast}$ with $\theta=0.1$ and
$\omega_{1}=0.5$ for $\omega\in[\omega_{1},\omega_{2}]$ on the left and
$\omega\in[0.01,5.01]$ on the right. According to Tab. III,
$\lambda(A+Bk^{\ast})=\\{-8.0215,-0.4595-0.3606j,-0.4595+0.3606j\\}$, and Fig.
1, we observe that box constraints (10) and both local stability (3) and
string stability (4) are satisfied for $k^{\ast}$. Also, Fig. 2 depicts the
relative error percentages
$100\times\frac{|F(j\omega)|-|\hat{F}(j\omega)|}{|F(j\omega)|}$ (%) associated
with the Taylor series approximation-based method and the Padé approximation-
based method with $\theta=0.1$ and $\omega_{1}=0.5$ for
$\omega\in[\omega_{1},\omega_{2}]$ on the left and $\omega\in[0.01,5.01]$ on
the right. As Fig. 2, the Padé approximation-based method attains a more
accurate approximation than the Taylor series approximation-based counterpart.

Figure 1: The $\omega$-dependent $|F(j\omega)|$ values corresponding to the
unconstrained and constrained $\mathcal{H}_{\infty}$ syntheses
$k^{\mathrm{unc}}$ and $k^{\ast}$ with $\theta=0.1$ and $\omega_{1}=0.5$ for
$\omega\in[\omega_{1},\omega_{2}]$ on the left and $\omega\in[0.01,5.01]$ on
the right.

Figure 2: The relative error percentages $100\times\frac{|F(j\omega)|-|\hat{F}(j\omega)|}{|F(j\omega)|}$ (%) associated with the Taylor series approximation-based method and the Padé approximation-based method with $\theta=0.1$ and $\omega_{1}=0.5$ for $\omega\in[\omega_{1},\omega_{2}]$ on the left and $\omega\in[0.01,5.01]$ on the right. TABLE III: The two-stage $\mathcal{H}_{\infty}$ controller synthesis for the value setting in Tab. II, $\omega_{1}=0.5$, $k^{l}=-\|k^{\mathrm{unc}}\|_{\ell_{\infty}}\begin{bmatrix}0&1&1&1\end{bmatrix}^{\top}$, $k^{u}=\|k^{\mathrm{unc}}\|_{\ell_{\infty}}\mathbf{1}$, $\alpha=1.05$ , $\zeta=5$, $\nu=5$, $N=5$. $k^{\mathrm{unc}}$: the unconstrained $\mathcal{H}_{\infty}$ controller synthesis [1]. Vector | Value
---|---
$k^{\ast}$ | $\begin{bmatrix}0.4212&0.4775&-1.0078&1.3197\end{bmatrix}^{\top}$
$\kappa^{\ast}$ | $\begin{bmatrix}-0.1516&-0.0237&1.7065&-0.7647\end{bmatrix}^{\top}$
$\kappa^{0}$ | $\begin{bmatrix}0.0918&-0.0378&-0.2983&-0.1611\end{bmatrix}^{\top}$
$k^{0}$ | $\begin{bmatrix}0.8089&0.3191&0.3611&0.3492\end{bmatrix}^{\top}$
$\mu^{0}$ | $\begin{bmatrix}0.3555&0.3495&0.3377&0.3411\end{bmatrix}^{\top}$
$k^{\mathrm{unc}}$ | $\begin{bmatrix}0.9200&1.3200&-0.9200&0.7200\end{bmatrix}^{\top}$
TABLE IV: The $\omega_{1}$-dependency of the $\mathcal{H}_{\infty}^{[\omega_{1},\omega_{2}]}$ norm values corresponding to the unconstrained and constrained $\mathcal{H}_{\infty}$ syntheses $k^{\mathrm{unc}}$ [1] and $k^{\ast}$ for predominant acceleration frequency boundary of human-driven vehicles $\omega_{1}\in\\{0.1,0.3,0.5,0.7\\}$. $\omega_{1}$ | $k$ | $\|F(s)\|_{\infty}^{[\omega_{1},\omega_{2}]}$
---|---|---
0.1 | $k^{\mathrm{unc}}$ | $0.9739$
0.1 | $k^{\ast}$ | $0.9628$
0.3 | $k^{\mathrm{unc}}$ | $0.9001$
0.3 | $k^{\ast}$ | $0.8207$
0.5 | $k^{\mathrm{unc}}$ | $0.8667$
0.5 | $k^{\ast}$ | $0.6758$
0.7 | $k^{\mathrm{unc}}$ | $0.7304$
0.7 | $k^{\ast}$ | $0.5669$

Fig. 3 visualizes the $3$D search space (fixing $\kappa_{3}=\kappa_{3}^{\ast}$
and $\kappa_{4}=\kappa_{4}^{\ast}$) corresponding to the constrained
$\mathcal{H}_{\infty}$ controller synthesis $k^{\ast}$ with $\theta=0.1$ and
$\omega_{1}=0.5$.

Figure 3: The $3$D search space (fixing $\kappa_{3}=\kappa_{3}^{\ast}$ and
$\kappa_{4}=\kappa_{4}^{\ast}$) corresponding to the constrained
$\mathcal{H}_{\infty}$ controller synthesis $k^{\ast}$ with $\theta=0.1$ and
$\omega_{1}=0.5$.

For predominant acceleration frequency boundary of human-driven vehicles
$\omega_{1}\in\\{0.1,0.3,0.5,0.7\\}$, Tab. IV reflects the
$\omega_{1}$-dependency of the
$\mathcal{H}_{\infty}^{[\omega_{1},\omega_{2}]}$ norm values corresponding to
the unconstrained and constrained $\mathcal{H}_{\infty}$ syntheses
$k^{\mathrm{unc}}$ and $k^{\ast}$. According to Tab. IV, the Padé
approximation-based box-constrained solution outperforms the Taylor series
approximation-based unconstrained solution [1] in terms of the
$\mathcal{H}_{\infty}^{[\omega_{1},\omega_{2}]}$ norm. As another observation
that holds for both unconstrained and constrained $\mathcal{H}_{\infty}$
syntheses $k^{\mathrm{unc}}$ and $k^{\ast}$, the smaller predominant
acceleration frequency boundary of human-driven vehicles $\omega_{1}$, the
larger $\mathcal{H}_{\infty}^{[\omega_{1},\omega_{2}]}$ norm. Such a
monotonous trend is aligned with the fact that decreasing the value of
predominant acceleration frequency boundary of human-driven vehicles
$\omega_{1}$ in (9) leads to a larger interval $[\omega_{1},\omega_{2}]$,
i.e., an expanded searching space. Note that as predominant acceleration
frequency boundary of human-driven vehicles $\omega_{1}$ tends to $0$, the
$\mathcal{H}_{\infty}^{[\omega_{1},\omega_{2}]}$ norm value converges to $1$
(due to $\lim_{\omega\to 0^{+}}|F(j\omega)|=1$, and based on definitions (4)
and (9)), i.e., the same value for the $\mathcal{H}_{\infty}$ norm value.

###  IV-B Case 2: a large communication delay

In this section, we corroborate the usefulness of the two-stage
$\mathcal{H}_{\infty}$ controller synthesis procedure presented by Procedure
Procedure 1 in the case of dealing with large $\theta$. Repeating the
experiments for $\theta=1.5$, we obtain the two-stage $\mathcal{H}_{\infty}$
controller synthesis illustrated by Tab. V for which the
$\mathcal{H}_{\infty}^{[\omega_{1},\omega_{2}]}$ norm value corresponding to
$k^{\ast}$ is $\|F(s)\|_{\infty}^{[\omega_{1},\omega_{2}]}=0.8669$.
Accordingly, Fig. 4 visualizes the $\omega$-dependent $|F(j\omega)|$ values
corresponding to the constrained $\mathcal{H}_{\infty}$ controller synthesis
$k^{\ast}$ for $\omega\in[\omega_{1},\omega_{2}]$ on the left and
$\omega\in[0.01,5.01]$ on the right. According to Tab. V,
$\lambda(A+Bk^{\ast})=\\{-3.2621,-0.2376-0.3644j,-0.2376+0.3644j\\}$, and Fig.
4, we observe that box constraints (10) and both local stability (3) and
string stability (4) are satisfied for $k^{\ast}$.

TABLE V: The two-stage $\mathcal{H}_{\infty}$ controller synthesis for the value setting in Tab. II except for $\theta$ with $\theta=1.5$, $\omega_{1}=0.5$, $k^{l}=-2\begin{bmatrix}0&1&1&1\end{bmatrix}^{\top}$, $k^{u}=2\mathbf{1}$, $\alpha=1.05$, $\zeta=5$, $\nu=5$, $N=5$. Vector | Value
---|---
$k^{\ast}$ | $\begin{bmatrix}1.9696&1.9953&-0.2273&0.0234\end{bmatrix}^{\top}$
$\kappa^{\ast}$ | $\begin{bmatrix}0.8341&1.3187&-0.1138&-0.0214\end{bmatrix}^{\top}$
$\kappa^{0}$ | $\begin{bmatrix}-0.2638&0.5087&0.1669&-0.3410\end{bmatrix}^{\top}$
$k^{0}$ | $\begin{bmatrix}0.4219&1.8308&-1.1174&0.3717\end{bmatrix}^{\top}$
$\mu^{0}$ | $\begin{bmatrix}-0.8649&-0.0940&0.8405&0.3706\end{bmatrix}^{\top}$

Figure 4: The $\omega$-dependent $|F(j\omega)|$ values corresponding to the
constrained $\mathcal{H}_{\infty}$ controller synthesis $k^{\ast}$ for
$\omega\in[\omega_{1},\omega_{2}]$ on the left and $\omega\in[0.01,5.01]$ on
the right.

##  V Concluding Remarks

To effectively address the limitations arising from the unconstrained CAV
platoon controller synthesis in the literature, this work proposes a (near-
optimal) constrained CAV platoon controller synthesis subject to the lower and
upper bounds on the control parameters. To that end, given box constraints on
the control parameters, parameterizing the set of feedback and feedforward
gains to ensure the local stability of the CAV platoon and deriving a
necessary condition to ensure the string stability criterion additionally, we
obtain more representative parameterized locally stabilizing feedback and
feedforward gains. Then, we employ the Padé approximation to more accurately
ensure the string stability criterion in the presence of communication delay.

We empirically observe that such an approximation outperforms the Taylor
series approximation in the literature. Due to the string unstable behavior of
human-driven vehicles, we have no direct control over them. However, applying
the (fully) CAV platoon controller synthesis results to the mixed vehicular
platoon can effectively attenuate the stop-and-go disturbance amplification
throughout the mixed vehicular platoon. Considering the mixed vehicular
platoon scenario and minimizing the $\mathcal{H}_{\infty}$ norm of the Padé
approximated transfer function over an interval defined by predominant
acceleration frequency boundaries of human-driven vehicles, we solve for a
near-optimal constrained CAV platoon controller synthesis via non-convex and
non-smooth optimization tools. Conducting extensive numerical experiments
corroborates that (i) in the case of a sufficiently small communication delay,
the Padé approximation-based method outperforms the Taylor series
approximation in terms of the $\mathcal{H}_{\infty}$ norm over an interval
defined by predominant acceleration frequency boundaries of human-driven
vehicles and (ii) in the case of a large communication delay, the Padé
approximation-based method successfully proposes an $\mathcal{H}_{\infty}$
controller synthesis while the Taylor series approximation-based method in the
literature becomes unusable.

Modern transportation systems will potentially benefit from the proposed CAV
control strategy in terms of effective stop-and-go disturbance attenuation as
it will potentially reduce collisions. Furthermore, taking advantage of the
sensory data, data-driven CAV platoon controllers can be synthesized to
enhance the disturbance attenuation performance. Specifically, in the case of
inaccurate models, such data-driven counterparts will show their potential
superiority over model-based synthesis in terms of disturbance attenuation
performance.

Limitations: Although the near-optimal performance of the proposed controller
synthesis is satisfactory, its near-optimality level can be sensitive to the
first stage (stable initialization) of the proposed two-stage
$\mathcal{H}_{\infty}$ controller synthesis procedure. It is noteworthy that
enhancing the quality of the non-convex and non-smooth optimization tools also
affects the quality of the proposed controller synthesis in terms of the near-
optimality level. As another limitation, in the current study, we have assumed
that the vehicle dynamics and the communication delay are deterministic and
time-invariant, which is unrealistic. Such a limitation necessitates the
consideration of the uncertain and time-varying nature of those elements [30,
1] to develop a robust and adaptive CAV platoon controller synthesis.
Elaboration on such a generalized synthesis is left as a pertinent future
direction.

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##  Appendix A Proof of Proposition 1

Imposing box constraints (10a)–(10c) to the parameterization (11), we obtain

$\displaystyle k_{1}^{l}\leq x\leq k_{1}^{u},$ (29a) $\displaystyle
k_{2}^{l}\leq-\tau x+y\leq k_{2}^{u},$ (29b) $\displaystyle
k_{3}^{l}\leq\frac{-Tx+y}{Ky}-z\leq k_{3}^{u}.$ (29c)

Since $0<x$, $0<y$, and $0<z$ hold, we can equivalently consider them as
$\epsilon\leq x$, $\epsilon\leq y$, and $\epsilon\leq z$, respectively, for an
infinitesimal $\epsilon>0$. Then, (29) along with $\epsilon\leq x$,
$\epsilon\leq y$, $\epsilon\leq z$, and $K>0$ implies that

$\displaystyle\epsilon$ $\displaystyle\leq x,$ (30a) $\displaystyle k_{1}^{l}$
$\displaystyle\leq x,$ (30b) $\displaystyle x$ $\displaystyle\leq k_{1}^{u},$
(30c) $\displaystyle\epsilon$ $\displaystyle\leq y,$ (30d) $\displaystyle\tau
x+k_{2}^{l}$ $\displaystyle\leq y,$ (30e)
$\displaystyle\frac{Tx}{-Kk_{3}^{l}+1-K\epsilon}$ $\displaystyle\leq y,$ (30f)
$\displaystyle y$ $\displaystyle\leq\tau x+k_{2}^{u},$ (30g)
$\displaystyle\epsilon$ $\displaystyle\leq z,$ (30h)
$\displaystyle\frac{-Tx+y}{Ky}-k_{3}^{u}$ $\displaystyle\leq z,$ (30i)
$\displaystyle z$ $\displaystyle\leq\frac{-Tx+y}{Ky}-k_{3}^{l},$ (30j)

hold. Notice that (30f) is obtained from the combination of (30h) and (30j),
and then simplifying the resulting inequality
$\epsilon\leq\frac{-Tx+y}{Ky}-k_{3}^{l}$ via $K>0$. Thus, utilizing (30), the
$x$, $y$, and $z$ in (11) satisfying box constraints (10a)–(10c) can be
parameterized as (12) and the proof is complete.

##  Appendix B Proof of Lemma 1

Utilizing the derivative rules in classic Calculus, it can be verified that

$\displaystyle\frac{d|F_{i}(j\omega)|}{d\omega}=\frac{D(\omega)N^{\prime}(\omega)-D^{\prime}(\omega)N(\omega)}{2D(\omega)\sqrt{D(\omega)N(\omega)}},$

holds. Since

$\displaystyle\lim_{\omega\to 0^{+}}N(\omega)=n_{0},~{}\lim_{\omega\to
0^{+}}D(\omega)=d_{0},$ $\displaystyle\lim_{\omega\to
0^{+}}N^{\prime}(\omega)=0,~{}\lim_{\omega\to 0^{+}}D^{\prime}(\omega)=0,$

are all satisfied, we obtain

$\displaystyle\lim_{\omega\to 0^{+}}\frac{d|F_{i}(j\omega)|}{d\omega}$
$\displaystyle=0.$ (31)

It can similarly be verified that (drop $\omega$ for the brevity)

$\displaystyle\frac{d^{2}|F_{i}(j\omega)|}{d\omega^{2}}=$
$\displaystyle\frac{D^{2}(2NN^{\prime\prime}-(N^{\prime})^{2})+N^{2}(3(D^{\prime})^{2}-2DD^{\prime\prime})-2DND^{\prime}N^{\prime}}{4D^{2}N\sqrt{DN}},$

holds. Since

$\displaystyle\lim_{\omega\to 0^{+}}N(\omega)=n_{0},~{}\lim_{\omega\to
0^{+}}D(\omega)=d_{0},$ $\displaystyle\lim_{\omega\to
0^{+}}N^{\prime}(\omega)=0,~{}\lim_{\omega\to 0^{+}}D^{\prime}(\omega)=0,$
$\displaystyle\lim_{\omega\to
0^{+}}N^{\prime\prime}(\omega)=2(n_{2}+\lim_{\omega\to
0^{+}}g_{\theta}(\omega)),~{}\lim_{\omega\to
0^{+}}D^{\prime\prime}(\omega)=2d_{2},$ $\displaystyle\lim_{\omega\to
0^{+}}g_{\theta}(\omega)=-2K^{2}k_{4}k_{1},~{}n_{0}=d_{0}=K^{2}k_{1}^{2},$

are all satisfied, we obtain

$\displaystyle\lim_{\omega\to 0^{+}}\frac{d^{2}|F_{i}(j\omega)|}{d\omega^{2}}$
$\displaystyle=\frac{n_{2}-n_{0}-2K^{2}k_{4}k_{1}}{n_{0}}=\frac{-\eta}{K^{2}k_{1}^{2}},$
(32)

where $\eta$ denotes the LHS of (7c). According to (31) and (32), if (4)
(i.e., a sufficient condition to guarantee the string stability) holds for
$F_{i}(s)$ in (5), then $\lim_{\omega\to
0^{+}}\frac{d^{2}|F_{i}(j\omega)|}{d\omega^{2}}=\frac{-\eta}{K^{2}k_{1}^{2}}\leq
0$ is satisfied (proof by contradiction incorporating the second-derivative
test [31]) and consequently, $\eta\geq 0$ holds. Thus, inequality (7c) holds
for the control parameters
$\begin{bmatrix}k_{1}&k_{2}&k_{3}&k_{4}\end{bmatrix}$ and the proof is
complete.

##  Appendix C Proof of Proposition 2

Imposing box constraints (10) to the parameterizations (11) and (15), we
obtain

$\displaystyle k_{1}^{l}\leq x\leq k_{1}^{u},$ (33a) $\displaystyle
k_{2}^{l}\leq-\tau x+y\leq k_{2}^{u},$ (33b) $\displaystyle
k_{3}^{l}\leq\frac{-Tx+y}{Ky}-z\leq k_{3}^{u},$ (33c) $\displaystyle
k_{4}^{l}\leq\frac{\tau^{2}x}{2}-\tau y+\frac{Tx}{Ky}+z+w\leq k_{4}^{u}.$
(33d)

Since $0<x$, $0<y$, and $0<z$ hold, we can equivalently consider them as
$\epsilon\leq x$, $\epsilon\leq y$, and $\epsilon\leq z$, respectively, for an
infinitesimal $\epsilon>0$. Then, (33) along with $\epsilon\leq x$,
$\epsilon\leq y$, $\epsilon\leq z$, $0\leq w$, and $K>0$ implies that

$\displaystyle\epsilon\leq x,$ (34a) $\displaystyle k_{1}^{l}\leq x,$ (34b)
$\displaystyle x\leq k_{1}^{u},$ (34c) $\displaystyle\epsilon\leq y,$ (34d)
$\displaystyle\tau x+k_{2}^{l}\leq y,$ (34e)
$\displaystyle\frac{Tx}{-Kk_{3}^{l}+1-K\epsilon}\leq y,$ (34f)
$\displaystyle\frac{\xi(x)+\sqrt{\xi(x)^{2}+\frac{4T\tau x}{K}}}{2\tau}\leq
y,$ (34g) $\displaystyle y\leq\tau x+k_{2}^{u},$ (34h)
$\displaystyle\epsilon\leq z,$ (34i)
$\displaystyle\frac{-Tx+y}{Ky}-k_{3}^{u}\leq z,$ (34j) $\displaystyle
z\leq\frac{-Tx+y}{Ky}-k_{3}^{l},$ (34k) $\displaystyle
z\leq\frac{-\tau^{2}x}{2}+\tau y+\frac{-Tx}{Ky}+k_{4}^{u},$ (34l)
$\displaystyle 0\leq w,$ (34m) $\displaystyle\frac{-\tau^{2}x}{2}+\tau
y+\frac{-Tx}{Ky}-z+k_{4}^{l}\leq w,$ (34n) $\displaystyle
w\leq\frac{-\tau^{2}x}{2}+\tau y+\frac{-Tx}{Ky}-z+k_{4}^{u},$ (34o)

hold. Notice that (34f) is obtained from the combination of (34i) and (34k).
Similarly, (34l) is obtained from the combination of (34m) and (34o).
Furthermore, (34g) is obtained from the combination of (34i) and (34l), and
then equivalently imposing the non-negativity of the quadratic polynomial
$\mathcal{P}(y)=\tau y^{2}-\xi(x)y-\frac{Tx}{K}$ (as $0<y$ holds) with
$\xi(x)=\frac{\tau^{2}x}{2}+\epsilon-k_{4}^{u}$. Notice that for such a
quadratic polynomial, $\Delta=\xi(x)^{2}+\frac{4T\tau x}{K}>0$ holds, and
since $\big{(}-\frac{Tx}{K}\big{)}/\tau=-\frac{Tx}{K\tau}<0$ is satisfied, the
polynomial has a positive real root (i.e., the larger root)
$0<r_{+}=\frac{\xi(x)+\sqrt{\xi(x)^{2}+\frac{4T\tau x}{K}}}{2\tau}$ and a
negative real root (i.e., the smaller root)
$r_{-}=\frac{\xi(x)-\sqrt{\xi(x)^{2}+\frac{4T\tau x}{K}}}{2\tau}<0$. For
$y\in\mathbb{R}$, we know that $0\leq\mathcal{P}(y)$ holds if and only if
$y\leq r_{-}$ or $r_{+}\leq y$ holds. However, $y\leq r_{-}$ is not feasible
as $0<y$ and $r_{-}<0$ hold. Thus, $0\leq\mathcal{P}(y)$ holds if and only if
$r_{+}\leq y$, i.e., (34g) holds. Thus, utilizing (34), the $x$, $y$, $z$, and
$w$ in (11) and (15) satisfying box constraints (10) can be parameterized as
(16) and the proof is complete.