# Images of the Ultra-High Energy Cosmic Rays from Point Sources

K. Dolgikh1, A. Korochkin2, G. Rubtsov1, D. Semikoz3, and I. Tkachev1 1
Institute for Nuclear Research of the Russian Academy of Sciences, Moscow,
117312 Russia  2 Université Libre de Bruxelles, CP225 Boulevard du Triomphe,
1050 Brussels, Belgium  3 APC, Université Paris Cité, CNRS/IN2P3, CEA/IRFU,
Observatoire de Paris, 119 75205 Paris, France

###### Abstract

Our latest paper [1] investigates the effects of UHECR propagation in a
turbulent intergalactic magnetic field in the small-angle scattering regime,
specifically focusing on the non-trivial caustic-like pattern that arises with
strong deviation from isotropy. In this paper, we explore the effect of the
observer’s position on the measurement of source flux at a given distance. We
examine three types of source locations, characterized by the density of
cosmic rays from a given source at the observation point, which we call knots,
filaments and voids. We also investigate the energy spectrum in these
different cases and present simulated images of the source as it appears on
the observer’s telescope after propagation in the combination of intergalactic
and Galactic magnetic fields. We show that hot spots in the UHECR data can
arrive due to combined distortions of source images on the intergalactic and
Galactic magnetic fields. Also the fact that flux of most nearby sources is
diluted in the voids affects source population studies.

##  I Introduction

Contrary to photons and neutrinos, Ultra-High Energy Cosmic Rays (UHECR) are
charged particles and they are unavoidably deflected in the Galactic and
intergalactic magnetic fields. Expected large number of UHECR sources in the
Universe together with large deflections of UHECR from source positions
prevents to do astronomy except for the highest energies $E>6\cdot 10^{19}$
eV, where the spectrum has cutoff due to interactions with Cosmic Microwave
Background (CMB). This so called GZK cutoff was predicted by Greizen, Zatsepin
and Kuzmin in 1966 [2, 3] and was observed 40 years later by HiRes experiment
in 2007 [4] and later confirmed with large statistics by modern experiments
Pierre Auger Observatory (Auger) [5] and Telescope Array (TA) [6]. Existence
of GZK cutoff guarantees that UHECR with energies $E>6\cdot 10^{19}$ eV come
from local Universe with significant contribution of sources located in nearby
Large Scale Structures (LSS) with distance to our Milky Way galaxy $R<200$
Mpc. Since LSS at those distances is significantly non-uniform, this gives a
hope to see individual UHECR sources. Astronomy with UHECR at the highest
energies $E>6\cdot 10^{19}$ eV was one of the main motivations for the
construction of Auger and TA experiments.

For cosmic ray protons deflection in the $B_{\mathrm{Gal}}\sim\mu$G Galactic
magnetic field is only few degrees for $E>6\cdot 10^{19}$ eV depending on
direction. Intergalactic magnetic fields order of $B_{\mathrm{IGMF}}\leq n$G
are again not efficient to disturb UHECR proton directions from point sources.
Thus one expect to see point or slightly extended sources of protons $E>6\cdot
10^{19}$ eV in UHECR data. However, even with large exposure Auger and TA have
not found any signature of point sources. This observation is consistent with
statement made by Auger that most of UHECR with $E>6\cdot 10^{19}$ eV are
intermediate and heavy nuclei. In the case of heavy nuclei, the Galactic
magnetic field disturbs and even washes out images of UHECR sources [7, 8].

After years of observation, Auger and TA see several anomalies in the sky that
look like hot spots with a radius of 10-20 degrees. These hot spots can be
images of nearby point sources, disturbed both in Galactic and extragalactic
magnetic fields.

Previous studies have explored the effects of intergalactic magnetic fields on
the trajectories of UHECRs. One such study, [9], investigated the Centaurus A
UHECR excess and its potential connection to the local extragalactic magnetic
field. This study shows that, assuming a proton composition, the strength of
the extragalactic magnetic field should be at least 20 nG. The latter value is
in conflict with existing observational constraints [10, 11, 12]. In present
work we assume a high but realistic magnetic field strength.

Main goal of the present study is to show that the observed 10-20 degree
radius hot spots in UHECR data can be signals from nearby point sources,
disturbed both in Galactic and intergalactic magnetic field. In particular, we
show examples of sources, which can explain TA hot spot. We show that caustic-
like pattern of intergalactic magnetic field amplify small number of sources
in filaments and knots and at the same time reduce contribution of most of
nearby sources.

##  II Simulation

To obtain images and spectra of UHECR sources, we rely on numerical
simulations of UHECR propagation. Our simulation procedure generally consists
of two steps. In the first step, we propagate UHECRs in turbulent
intergalactic magnetic fields, tracing their paths until they reach the edge
of our galaxy. In the second step, we consider the effect of the galactic
magnetic field and construct images at the Earth’s position.

For intergalactic propagation we use 3D Monte Carlo code
CRbeam111https://github.com/okolo/mcray[14] which allows to take into account
UHECR deflections in magnetic fields and interactions with CMB and EBL. The
general setup of the these simulations is the same as in our previous paper
[1]. The main difference is the possibility of emitting particles in the form
of a jet with a given opening angle and axis direction (isotropic source is
thus a special case with opening angle equals $\pi$). In all simulations we
keep flat profile of the jet. Another important modification is that we catch
particles when they hit a small spherical observer which represents our
galaxy. The radius of observer was chosen to be 100, 200 or 1000 kpc, the last
one for void cases. Once the particle hits the observer, we record its energy,
momentum, and position on the sphere. We subsequently use this information to
reconstruct the source image.

CRbeam represents turbulent magnetic field with Kolmogorov’s spectrum as a sum
of plain waves with random directions and phases, following the method of
[15]. The number density of waves is uniform in logarithmic scale by
wavenumber. We fix the number of waves to be 463 with the largest eddy being
100 larger than the smallest. A smaller number of waves may lead to an
unwanted effect when a wave with the longest wavelength dominates UHECR
deflection. On the over hand, bigger number of waves increase calculation
time. In our previous study [1], we carefully verified that existence of the
caustic-like patterns in UHECR spacial distribution does not depend on either
the method of generating the turbulent magnetic field or its spectrum.
Therefore, we believe that our result are stable with respect to the technical
parameters of the magnetic field generators such as number of modes and
maximum and minimum scales of the spectrum. We keep the same random seed for
magnetic field generation in all simulations within the paper and use OpenMP
for multiprocessing acceleration.

The magnetic field strength was set to $B=1$ nG and correlation length
$\lambda_{\mathrm{c}}=1$ Mpc. In most simulations, we propagate protons with
an energy of $E=10$ EeV, but our results remain unchanged if one
proportionally rescales the particle’s charge and energy.

The typical scale of distances we consider here is about tens of Mpc. Given
the fact that the size of the observer is tens of kpc and protons have
nonnegligible deflections in IGMF with $B\propto 1$ nG the problem of aiming
at the observer arises. This is due to both a decrease in the angular size of
the observer with distance and to the fact that statistically more often the
observer finds himself in the region of space where the cosmic ray flux from
point source is even lower than expected from standard inverse-square law [1].
This prompts the need of targeting algorithm.

Optimization of the targeting mechanism was studied earlier, for example in
[16]. The authors used von Mises–Fisher distribution instead of simple cone
jet and solve task for only one distance and energy. We reproduced this
approach and modified it for our case to optimize the performace for our case.

Since we are examining the source images and flux excess/deficit at different
distances or energies, we apply the following procedure for optimization of
calculation time. First, we aim to observer from small distance
$D_{\mathrm{1}}$ where deflection by magnetic field is negligible. After that
we use the directions found on the preceding step at distance $D_{\mathrm{i}}$
for aiming on distance $D_{\mathrm{i+1}}>D_{\mathrm{i}}$
($D_{\mathrm{i+1}}-D_{\mathrm{i}}\sim 0.5-2$ Mpc $\lesssim r_{\mathrm{g}}$ in
our cases where $r_{\mathrm{g}}$ is gyroradius given by
$r_{\mathrm{g}}=\frac{E}{\mid q\mid B}$).

Another simulation parameter – jet angle $\theta$ is chosen as previous
minimal necessary jet angle multiplied by a constant factor. We have
empirically seen that a value of $\sim 3$ for this factor allows to ensure
that the most of the flux have not been missed. We obtain minimal necessary
jet angle from the analysis of particle directions in CRbeam output files. For
the first run we choose $\theta_{1}\sim\frac{\pi}{2}$ or
$\theta_{1}\gtrsim\arctan(\frac{R_{\mathrm{obs}}}{D})$. If simulation has
strong flux deficit, it is restarted with the wide angle of the jet
$\theta\propto\sqrt{D\lambda_{\mathrm{c}}}/E$ (see [1]) to ensure that all
particles which may reach the observed are included into emitted jet.

To estimate the simulation acceleration factor, let’s consider the Fig. 2. The
flux can decrease by more than $\sim$100 times compared to the average at a
distance of $\sim$20 Mpc, at this distance the observer occupies no more than
$10^{-3}$ of the entire manifold of end particles coordinates. So, the
probability of hitting an isotropically emitted particle is less than
$10^{-5}$. Therefore, obtaining sufficient statistics would take a long time.
Using the procedure explained above allows to direct particles into a cone
with an angle of $\theta\sim 0.2$ radians and the geometric probability of
hitting increases by about $2/(1-\cos(\theta))\sim 100$ times. So, the Fig. 4
have been simulated about 100 times faster than if the source was considered
isotropic.

Finally we take into account deflections in galactic magnetic field and obtain
images of the sources as seen by observer at Earth’s position inside the
galaxy. We use galactic magnetic lenses provided by CRPropa3 [17, 18], which
enables efficient utilization of backtracking simulations to consider galactic
deflections in forward simulations of UHECRs. We adopt the JF12 model of
galactic magnetic field with both coherent and turbulent components turned on
[19, 20].

##  III Results

###  III.1 Observed fluxes

Figure 1: Probability of observer to be located in the region with flux higher
or smaller compared to an avarage one as function of the distance to the
source for protons with energy $E=10$ EeV. Figure 2: Example of the relative
flux excess and deficit as function of the distance to the source for protons
with energy $E=10$ EeV. Figure 3: Energy spectrum of the source seen by an
observer located in the knot at the distance $D=20$ Mpc and in the void at
$D=30$ Mpc. Each spectral bin is normalized to the average flux at the same
energy.

In our previous paper [1] we have shown that in the regime of small angle
scattering UHECR from point source form a caustic pattern at distance equal to
tens of coherent length from the source. This pattern consist of regions with
density of UHECRs larger or smaller compared to isotropic averaged. By shape
of those regions on sphere around source we called them knots, filaments and
voids. Typical flux in knots is 10 times larger compared to average, in
filaments at least 2 times larger and in voids at least two times smaller. By
average, we mean the expected flux from the source, assuming that it follows
the inverse-square law.

We studied propagation of UHECR protons with energies $E>10^{19}$ eV in the
turbulent inter-galactic magnetic field with strength $B=1$ nG and coherent
length $\lambda_{\mathrm{c}}=1$ Mpc. For such parameters of magnetic field
non-trivial UHECR pattern exist at all distances in nearby LSS up to 100 Mpc,
from which most of UHECR at highest energies arrive in the Earth detectors.

First, we examine the properties of the observed signal as a function of UHECR
energy and a distance to the source. In Fig. 1 we show probability that at a
given distance from the source UHECR flux differs from an average one in 2 or
10 times. One can see, that most probably the flux from the given source will
be lower compared to the average one. For 10 % of sources it is at least two
times higher while in up to 75% of cases it is 2 times smaller. Only in a few
percent of cases (in knots) flux can be enhanced 10 times and more
$F/F_{\mathrm{av}}>10$. On the other hand in up to 55 % of cases flux can be
reduced in 10 times $F/F_{\mathrm{av}}<10$ in voids. This mean that for most
of sources UHECR flux will stay same or will be reduced, while only for small
fraction of sources it will be significantly increased, giving possibility to
observe ’hot spot’ in the UHECR spectrum.

In Fig. 2 we show example of $E=10$ EeV proton fluxes relative to average one
in the directions to knot, void and filament as function of the distance from
the source. The black solid line marks an average flux $F=F_{\mathrm{av}}$.
One can see that in example of Fig. 2 filament at knot have maximal flux at 12
Mpc from source and disappeared at 25-30 Mpc distance. Moreover, filament
transforms to void between 30 Mpc and 50 Mpc from source.

Finally we constructed the energy spectrum of the source for an observer
located in the knot and in the filament, see Fig. 3. One can see that the
observed flux deviates significantly from the average one at the energies
around 10-30 EeV. These deviations disappear both at low and high energies. At
the highest energies $\sim$ 100 EeV the deflections become too small to
produce strong enough flux variations. On the other hand at low energy end of
the spectrum UHECR propagation becomes diffusive which is also washing out the
filamentary pattern.

###  III.2 Source images

The observer may have a sufficiently large radius compared to the distance to
it, which introduces some distortions in the apparent size of the source when
naively constructing a HealPy222http://healpix.sf.net[22] map from the
momentum of trapped particles. We remove this distortion by taking the
projection of the particle’s momentum on the Oz axis of our basis, which is
parallels to the direction to the source. The second vector of the basis is
chosen manually. We complete the basis by taking the cross product of first
and second vectors.

Figure 4: Source images seen by the observer located at knot and filament
calculated at the edge of the Galaxy and after propagation through the GMF.
Color encodes the number of hits to a given pixel. In both cases the intrinsic
source luminosity is adjusted so that total number of events from the source
observed at Earth matches with the cumulative number of TA events falling
inside the hot spot circle of $25^{\circ}$ (44 events for the 15-year TA SD
data). The blue color corresponds to the background level which is shown as
uniform for clarity.

We have studied two cases of observer location 4: observer in a filament and
observer in knot. We didn’t consider the most likely case of the void (it is
about $\sim 70\%$, see Fig.1) as in this case the flux is extremely small and
hard for detection. Knot case produces a bright smudged spot but this case is
the most unlikely. Thus, if we have many sources of UHECR around us at a
distance of tens Mpc, we see only a few of them. With respect to most sources,
we will be in the voids. Sources at greater distances (more 100 Mpc) give an
isotropic contribution, which decreases with distance and creates a uniform
background.

On top right panel of Fig.4 you can see an image of the source when observer
is located in a filament before propagation through the galactic magnetic
field. in Ihis case the source has enough for detection flux and probability
and an image which is very stretched. The elongation of the image is
perpendicular to the spacial UHECR density filament and disappears when the
energy is doubled. After passing the galactic magnetic field, a smeared spot
could be formed as in bottom right panel Fig.4. The positions of the sources
in the sky were chosen in a way to correspond to TA hotspot direction before
GMF lensing. The final spots after GMF lensing have a size of about 20 degrees
as known hotspots.

We performed analysis for 10-20 EeV protons here. Since UHECR at highest
energies can be dominated by intermediate nuclei according to Auger data,
propagation of protons at those energies would correspond to propagation of
nuclei with Z times larger energies in the same magnetic fields, in particular
UHECR with $E>57$ EeV would correspond to CNO nuclei.

As was shown for example in [23], CNO nuclei of required energies can reach us
from up to 70 Mpc distance.

##  IV Conclusions

In this paper we studied influence of the intergalactic and Galactic magnetic
fields on the images of nearby sources in the UHECR data. In particular we
studied possibility to explain hot spots in UHECR data by contribution of
nearby sources. In our previous paper [1] we have shown that the effects of
UHECR propagation in a turbulent intergalactic magnetic field in the small-
angle scattering regime produce non-trivial caustic-like pattern in the
density of cosmic rays, with knots, filaments and voids in the density profile
at given distance from source. Here we studied dependence of density profile
from distance from source and from energy for fixed properties of magnetic
field with strength 1 nG and coherence length 1 Mpc. We found that for large
range of distances and energies of UHECR caustic-like pattern stays similar in
the sky. This means that for given position of Milky Way galaxy in this
pattern we have get (de-)amplification of original flux of source.

Final source images after propagation in the Galactic magnetic field are shown
in Fig. 4. One can see that both knot and filament structures can provide
final hot spots in the UHECR image, while location in the void significantly
reduce contribution of given source in the observed UHECR flux.

For given class of UHECR sources we expect that most of nearby sources are
located in the voids and we see only few of sources, which are located in the
filaments. With small probability one can see even brighter amplification of
sources flux in the knot.

Thus, in this work we show how hot spots in UHECR data can form. In Fig.4 we
have shown examples of source image in the direction of TA hot spot. We show
that after propagation in Galactic and inter-galactic magnetic fields source
image can look like one observed in UHECR data. We performed analysis for 10
EeV protons, which would corresponds to CNO nuclei for energies of hot sport
in TA and Auger data $E>57$ EeV. Source location at several tens of Mpc is
still for CNO nuclei of those energies.

#### Acknowledgements.

Modeling of propagation of high-energy radiation in the Universe has been
supported by the Russian Science Foundation grant 22-12-00253 (K.D. and G.R.).
The work of K.D. has been supported by the fellowship of Basis Foundation
(grant No. 22-1-5-92-1). The work of A.K. has been supported by the IISN
project No. 4.4501.18. Some of the results in this paper have been derived
using the healpy and HEALPix packages.

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# Robust Event Triggering Control for Lateral Dynamics of Intelligent Vehicles
with Designable Inter-event Times

Xing Chu ID , Zhi Liu ID , Lei Mao, Xin Jin ID , ,
Zhaoxia Peng ID , and Guoguang Wen ID This work is supported by the Youth
Talent Project of Yunnan Province under Grant No.C619300A020, the National
Natural Science Foundation of China under Grant No.62101481, and Key
Laboratory in Software Engineering of Yunnan Province in China (No.2020SE408;
No.2020SE317). (Corresponding author: Guoguang Wen.)Xing Chu, Zhi Liu, and Xin
Jin are with the School of Software, Yunnan University, Kunming 650091, China
(e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>xinxin_jin@163.com)Lei Mao is with
the traffic engineering design and research in Yunnan Communications
Investment & Construction Group Co. Ltd. (e-mail: 1191633510@qq.com)Zhaoxia
Peng is with the School of Transportation Science and Engineering, Beihang
University, Beijing 100191, China (e-mail: pengzhaoxia@buaa.edu.cn).Guoguang
Wen is with the Department of Mathematics, Beijing Jiaotong University,
Beijing 100044, China (e-mail:guoguang.wen@bjtu.edu.cn).

###### Abstract

In this brief, an improved event-triggered update mechanism (ETM) for the
linear quadratic regulator is proposed to solve the lateral motion control
problem of intelligent vehicle under bounded disturbances. Based on a novel
event function using a clock-like variable to determine the triggering time,
we further introduce two new design parameters to improve control performance.
Distinct from existing event-based control mechanisms, the inter-event times
(IETs) derived from the above control framework are designable, meaning that
the proposed ETM can be deployed on practical vehicle more easily and
effectively. In addition, the improved IETs-designable ETM features a global
robust event-separation property that is extremely required for practical
lateral motion control of vehicle subject to diverse disturbances. Theoretical
analysis proves the feasibility and stability of the proposed control strategy
for trajectory tracking under bounded disturbances. Finally, simulation
results verify the theoretical results and show the advantages of the proposed
control strategy.

###### Index Terms:

Intelligent vehicle, lateral motion control, event triggering mechanism,
designable inter-event times, disturbance.

##  I INTRODUCTION

Intelligent vehicle is a promising solution for improving the safety and
efficiency of traffic in modern intelligent transportation systems. In
addition, it makes valuable contributions to reducing air pollution and saving
energy for improving the natural environment and social economy. Thereinto,
lateral motion control, as a crucial problem in intelligent vehicle, still
faces many challenges. This brief focuses on the lateral motion control of
intelligent vehicle.

The core objective of lateral motion control is achieving autonomous driving
along the desired path. This research field has been expanded and innovated
continuously in recent years. The kinematics and dynamics of vehicular lateral
motion control were discussed in [1]. Owing to the importance of lateral
motion control in autonomous driving, diverse control approaches have been
proposed and developed, including PID control [2], $H_{\infty}$ control [3,
4], linear quadratic regulator (LQR) [5], model predictive control (MPC) [6,
7, 8], and sliding-mode control (SMC) [9, 10], and just name a few.

The above review shows rich results of lateral motion control in intelligent
vehicle area. Nevertheless, after these sort of cutting-edge control
algorithms are deployed in the real digital platforms of vehicles, several
practical issues should be further considered. For example, the computing unit
and the bandwidth of controller area network (CAN) bus are shared with other
control loops and extremely limited. Thus, reducing the unnecessary control
computing and data traffic so as to improve the efficiency of resource
utilization and energy consumption, while guaranteeing the original control
performance, is what we desperately need. Under this background, event-based
control mechanism, as shown in [11, 12, 13, 14, 15] and the references
therein, is introduced to the recent varied control researches [16, 17, 18,
19]. Instead of feeding back all the sampling data regardless of the practical
need (for example, stability) as in the current rich results, event-based
control schemes determine the time instant, at which the data of control
system is sampled and sent back to update the control input, based on certain
triggering rules.

Unfortunately, there are still some drawbacks to dealing with real control
problems under traditional event-triggered schemes. For example, choosing a
larger threshold means tolerating a larger state error, which obviously
conserves resources in such a control scheme, but also influences control
performance, such as the accuracy of trajectory tracking. So far, hence,
several researchers have studied adaptive event-based control schemes to solve
this problem. In [8], adaptive event-based model predictive control was
developed for the control of unmanned vehicles. An adaptive-event-trigger-
based fuzzy nonlinear lateral dynamic control algorithm was proposed in [20].
On the other hand, there are still some issues that are less frequently
mentioned, such as the practical constraints of inter-event times (IETs)
imposed on the sensor and communication, which prevent the traditional event-
triggered schemes from being applied flexibly in practical control systems.
More importantly, global robust event-separation property of event-triggered
control system, which determines the feasibility and reliability of the
designed event triggering scheme, cannot be ignored.

A potential solution to cope with the all above-mentioned issues of event-
based control mechanisms was provided in [21]. Specifically, a novel design
and analysis approach of an event function was provided, which guarantees
original control performance through a clock-like triggering signal whose
variable range and evolution rate can be flexibly adjusted to obtain the
desired IETs. Meanwhile, the evet-triggered control studied in [21] holds the
global robust event-separation property. This type of IETs-designable event
triggering mechanism (ETM) is therefore extremely suitable for the deployment
on practical systems, which constructs the main motivation for this work.

In this brief, inspired by the idea from the IET-designable ETM, we aim to
tackle the execution problem of lateral LQR of intelligent vehicle. The
contributions of this brief are the following:

  * •

An event-based execution mechanism for a LQR controller to solve the lateral
motion control problem of intelligent vehicle is designed. Compared with the
time-triggered strategy, this mechanism is proved to reduce unnecessary
communication and control update, so as to reduce energy consumption and
resource occupation, while stabilizing the system.

  * •

By introducing two new design parameters to improve the performance of the
IETs-designable ETM, the IETs can be adjusted more freely than in [21]. Thus,
designability of the IETs is further expanded, which benefits its application
in actual systems.

  * •

Many existing ETMs do not feature a robust event-separation property, so any
tiny disturbance maybe leads to the Zeno-behavior. Considering the bounded
disturbances in the real movement of intelligent vehicle, we verify the
stability and feasibility of the proposed event-based controller with improved
IETs-designable ETM.

##  II PROBLEM FORMULATION

###  II-A VEHICLE DYNAMIC MODEL DESCRIPTION

In the dynamic bicycle model illustrated in Fig. 1, it is assumed that the
vehicle is assumed to be symmetrical, and tire sideslip angles on the same
axle are equal. We neglect the roll and pitch dynamics, and assume the
sideslip $\beta$, yaw $\psi$, and control input steering angle $\delta$ are
small. A linear parameter variation (LPV) model is derived from the linear
tire force model. [1] provides dynamic equations for the bicycle model in
terms of sideslip angle and yaw rate.

$\begin{cases}\begin{array}[]{l}\dot{\beta}=-\frac{\mu\left(C_{f}+C_{r}\right)}{mV_{x}}\beta-\left(1+\frac{\mu\left(l_{f}C_{f}-l_{r}C_{r}\right)}{mV_{x}^{2}}\right)\dot{\psi}+\frac{\mu
C_{f}}{mV_{x}}\delta\vspace{1ex}\\\
\ddot{\psi}=-\frac{\mu\left(l_{f}C_{f}-l_{r}C_{r}\right)}{I_{z}}\beta-\frac{\mu\left(l_{f}^{2}C_{f}+l_{r}^{2}C_{r}\right)}{I_{z}V_{x}}\dot{\psi}+\frac{\mu
l_{f}C_{f}}{I_{z}}\delta\end{array}\end{cases}.$ (1)

The corresponding vehicle parameters and nomenclature are listed in Table I.

Fig. 1: Dynamic bicycle model. TABLE I: PARTIAL PARAMETERS AND NOMENCLATURE Symbol |  Description
---|---
$\beta$ |  Vehicle sideslip angle
$\dot{\psi}$ |  Yaw rate
$\delta$ |  Steering wheel angle
$\mu$ |  Road friction coefficient
$V_{x}$ |  Vehicle longitudinal velocity
$m$ |  Vehicle mass
CG |  Center of gravity of vehicle
$I_{z}$ |  Yaw moment inertia of vehicle
$l_{f}$ |  Front axle-CG distance
$l_{r}$ |  Rear axle-CG distance
$C_{f}$ |  Front tire cornering stiffness
$C_{r}$ |  Rear tire cornering stiffness

###  II-B CONTROL PROBLEM DEFINITION

The objective of lateral control is to minimize a vehicle’s lateral
displacement relative to a specific reference path. The dynamics of a
vehicle’s lateral error displacement $e$ relative to a reference path is
$\ddot{e}=V_{x}(\dot{\beta}+\dot{\psi})-V^{2}_{x}\rho$, where the road
curvature $\rho$ is a constant. And based on (1), system state variables can
be denoted $x=[\beta,\dot{\psi},\dot{e},e]^{\mathrm{T}}$, and the control
input is the steering angle $\delta$. Then, owing to eliminating the lateral
error, for a given road curvature $\rho$ and longitudinal velocity $V_{x}$,
$\dot{e}=e=0$ is the control object. In [1], the desired equilibrium point can
be written as
$x^{\star}=[\beta,\dot{\psi},\dot{e},e]^{\mathrm{T}}=[\beta^{\star},\dot{\psi}^{\star},0,0]^{\mathrm{T}}$,
and the control input $\delta$ at the desired equilibrium point is now
$\beta^{\star}$, where $\beta^{\star}=\left(L_{r}-\dfrac{L_{f}mV_{x}^{2}}{\mu
C_{r}\left(L_{f}+L_{r}\right)}\right)\rho$, $\dot{\psi}^{\star}=V_{x}\rho$,
and
$\delta^{\star}=\left(L_{f}+L_{r}\right)\rho+\dfrac{mV_{x}^{2}\left(L_{r}C_{r}-L_{f}C_{f}\right)}{\mu
C_{f}C_{r}\left(L_{f}+L_{r}\right)}\rho$.

By defining the new error variables $\tilde{\beta}=\beta-\beta^{\star}$,
$\dot{\tilde{\psi}}=\dot{\psi}-\dot{\psi}^{\star}$, and
$\tilde{\delta}=\delta-\delta^{\star}$, the new system state variables become
$\tilde{x}=(\tilde{\beta},\dot{\tilde{\psi}},\dot{e},e)^{\mathrm{T}}$. We
write the error dynamics [22], the origin of which is the equilibrium, as
$\tilde{x}^{\star}=(\tilde{\beta},\dot{\tilde{\psi}},\dot{e},e)^{\mathrm{T}}=(0,0,0,0)^{\mathrm{T}}$
as:

$\dot{\tilde{x}}(t)=A\tilde{x}(t)+B\tilde{\delta}(t),$ (2)

where

$\begin{split}\begin{array}[]{l}A=\left[\begin{array}[]{cccc}-\frac{\mu\left(C_{f}+C_{r}\right)}{mV_{x}}&-1-\frac{\mu\left(l_{f}C_{f}-l_{r}C_{r}\right)}{mV_{x}^{2}}&0&0\vspace{1ex}\\\
-\frac{\mu\left(l_{f}C_{f}-l_{r}C_{r}\right)}{I_{z}}&-\frac{\mu\left(l_{f}^{2}C_{f}+l_{r}^{2}C_{r}\right)}{I_{z}V_{x}}&0&0\vspace{1ex}\\\
-\frac{\mu\left(C_{f}+C_{r}\right)}{m}&-\frac{\mu\left(l_{f}C_{f}-l_{r}C_{r}\right)}{mV_{x}}&0&0\vspace{1ex}\\\
0&0&1&0\end{array}\right],\vspace{1ex}\\\
B=\left[\begin{array}[]{cccc}\frac{\mu C_{f}}{mV_{x}};&\frac{\mu
l_{f}C_{f}}{I_{z}};&\frac{\mu
C_{f}}{m};&0\end{array}\right]^{\mathrm{T}}.\end{array}\end{split}$ (3)

###### Remark 1

The steady-state solution is the desired reference state $x^{\star}$ in this
model. As a result, $\tilde{\delta}$ can ignore the model’s natural transient
dynamics when using this as a reference input. Additional information about
this section can be found in [1, 22, 23].

###  II-C LQR CONTROLLER

To solve the control problem in (2), the LQR controller is designed. The
linear quadratic cost functional index is constructed first:

$J=\int_{t_{0}}^{\infty}\left[\tilde{x}^{\mathrm{T}}(t)Q\tilde{x}(t)+\tilde{\delta}^{\mathrm{T}}(t)R\tilde{\delta}(t)\right]\mathrm{d}t.$
(4)

Then, the control input steering angle $\tilde{\delta}$ is calculated
according to

$\tilde{\delta}(t)=-K\cdot\tilde{x}(t)=-R^{-1}B^{\mathrm{T}}P\cdot\tilde{x},$
(5)

where $P$ is derived by solving the Riccati equation:

$A^{\mathrm{T}}P+PA-{PBR}^{-1}B^{\mathrm{T}}P+Q=0.$ (6)

##  III MAIN RESULTS

###  III-A IET-DESIGNABLE EVENT TRIGGERING MECHANISM

In this subsection, we aim to apply the improved IET-designable ETM to
determine the triggering instant when the feedback control law is updated,
which is a malleable solution to the computational load and hardware
limitations. This mechanism is based on a countdown variable with an upper
bound. Thus, we create an event function $\mathbf{Z}(t)$ whose upper bound
$\mathbf{\bar{Z}}$ can be freely set and apply the following event triggering
rule:

$\displaystyle t_{0}$ $\displaystyle=0,$ (7) $\displaystyle t_{k+1}$
$\displaystyle=inf\left\\{t>t_{k}\mid\mathbf{Z}(t)=0\right\\},$

where $\mathbf{Z}(t)$ is reset as the design parameter $\mathbf{\bar{Z}}>0$ at
every triggering instant. The triggering rule (7) will determine the time
series $\left\\{t_{k}\right\\},k\in{\mathbb{N}^{*}}$. To ensure the
feasibility and stability of the IET-designable ETM, the event function
dynamics is defined as $\dot{\mathbf{Z}}(t)=\omega(\varpi,\varepsilon)$, where
$\varepsilon>0$ is also a design parameter. Detailed descriptions of the event
function are given in the following subsection.

###  III-B STABILITY AND FEASIBILITY ANALYSIS

In the movement process of intelligent vehicle, an inevitable disturbance is
necessary, especially in the case of road curvature $\rho>0$. After adding the
disturbances, the system dynamics (2) become

$\dot{\tilde{x}}(t)=A\tilde{x}(t)+B\tilde{\delta}(t)+G\xi(t),$ (8)

where $A$ and $B$ are shown in (3), and for all initial conditions
$\tilde{x}(0)$ any bounded disturbance $\left|\xi\right|\leq\bar{\xi}$, $G$
denotes a corresponding constant matrix. Moreover, we assume that the
disturbance $\xi(t)$ is convergent to $0$.

To stabilize the system (8), recalling (5), the feedback control law is
$\tilde{\delta}(t)=-K\tilde{x}(t)$, and the closed-loop control system
considering the perturbation $\xi(t)$ can be written as
$\dot{\tilde{x}}(t)=(A-BK)\tilde{x}(t)+G\xi(t)$. If a Lyapunov function
$V=\tilde{x}^{\mathrm{T}}M\tilde{x}$ exists, where $M$ is a symmetric positive
definite matrix,

$(A-BK)^{\mathrm{T}}M+M(A-BK)=-N$ (9)

is satisfied, where $N$ is an arbitrary symmetric positive-definite matrix.
Under the IET-designable ETM (7), the control input becomes

${\tilde{\delta}}(t)=-K\tilde{x}(t_{k}),\,t\in[t_{k},t_{k}+1).$ (10)

Then, define the event triggering error and its derivative as

$\displaystyle\eta(t)$
$\displaystyle=\tilde{x}(t_{k})-\tilde{x}(t),\,t\in\left[t_{k},t_{k+1}\right)$
(11) $\displaystyle\dot{\eta}(t)$ $\displaystyle=-\dot{\tilde{x}}(t).$

We can the express the system (8) as

$\dot{\tilde{x}}(t)=A\tilde{x}(t)-BK\tilde{x}(t)-BK\eta(t)+G\xi(t).$ (12)

###### Theorem 1

For all initial conditions $x(0)$, applying the IET-designable ETM (7) in the
system (12), $\tilde{x}(t)$ asymptotically converges to $\tilde{x}^{\star}$.
Besides, designable IETs with lower-bounded $\tau$ are available:

$\displaystyle\tau$
$\displaystyle=\sqrt{\frac{1}{\sigma\varepsilon}}\left\\{\operatorname{atan}\left[\sqrt{\frac{\sigma}{\varepsilon}}(1+\mathbf{\bar{Z}})\right]-\operatorname{atan}\left[\sqrt{\frac{\sigma}{\varepsilon}}\right]\right\\}>0\vspace{1ex}$
(13) $\displaystyle\sigma$
$\displaystyle=\frac{\theta_{r}^{2}|MBK|^{2}}{\theta_{l}\lambda_{\min}(M)\lambda_{\min}(N)}.$

###### Proof:

We construct the Lyapunov function candidate
$W=\frac{1}{2}\tilde{x}^{\mathrm{T}}M\tilde{x}+\frac{1}{2}\mathbf{Z}\eta^{\mathrm{T}}M\eta$.
Based on (9), (11), and (12), its derivative can be written as

$\displaystyle\dot{W}$ $\displaystyle=$
$\displaystyle\tilde{x}^{\mathrm{T}}M\dot{\tilde{x}}+\frac{1}{2}\omega\eta^{\mathrm{T}}M\eta+\mathbf{Z}\eta^{\mathrm{T}}M\dot{\eta}$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\tilde{x}^{\mathrm{T}}N\tilde{x}-\tilde{x}^{\mathrm{T}}MBK\eta+\tilde{x}MG\xi+\frac{1}{2}\omega\eta^{\mathrm{T}}M\eta$
$\displaystyle-\mathbf{Z}\eta^{\mathrm{T}}M(A-BK)\tilde{x}+\mathbf{Z}\eta^{\mathrm{T}}MBK\eta-\mathbf{Z}\eta^{\mathrm{T}}M\xi.$

By some properties of matrix and norm, we can obtain

$\displaystyle\dot{W}$ $\displaystyle\leq$
$\displaystyle-\frac{1}{2}\lambda_{\min}(N)|\tilde{x}|^{2}+|\tilde{x}||MBK||\eta|+\frac{1}{2}\omega\lambda_{\min}(M)|\eta|^{2}$
$\displaystyle+\mathbf{Z}|\eta||MA||\tilde{x}|+\mathbf{Z}|\eta||MBK||\tilde{x}|+\mathbf{Z}|MBK||\eta|^{2}$
$\displaystyle+\mathbf{Z}|\eta||M||\xi|+|\tilde{x}||MG||\xi|,$

where $\left|\cdot\right|$ denotes the Euclidean norm for vectors and the
induced 2-norm for matrices. We define the variable $\varpi$ as

$\varpi=\frac{\theta_{l}\lambda_{\min}(N)}{\lambda_{\min}(M)}\frac{|\tilde{x}|^{2}}{|\eta|^{2}}-2(1+\mathbf{Z})\frac{\theta_{r}|MBK|}{\lambda_{\min}(M)}\frac{|\tilde{x}|}{|\eta|},$
(14)

where $\theta_{l}>1$ and $\theta_{r}<1$ are two new design parameters.
Therefore, variable $\omega$ further satisfies the inequality $\omega<\varpi$
to guarantee the asymptotic stability of the closed-loop system (12). Using
Young inequality for
$2(1+\mathbf{Z})\frac{\theta_{r}|MBK|}{\lambda_{\min}(M)}\frac{|\tilde{x}|}{|\eta|}$,
we obtain the following inequality:

$-2(1+\mathbf{Z})\frac{\theta_{r}|MBK|}{\lambda_{\min}(M)}\frac{|\tilde{x}|}{|\eta|}\geq-\sigma(1+\mathbf{Z})^{2}-\frac{{\theta_{r}^{2}|MBK|}^{2}}{\sigma{\lambda_{\min}}^{2}(M)},\frac{{|\tilde{x}|}^{2}}{|{\eta|}^{2}},$

which can lead to (14) arriving at

$\varpi\geq-\sigma(1+\mathbf{Z})^{2}+\left(\frac{\theta_{l}\lambda_{\min}(N)}{\lambda_{\min}(M)}-\frac{\theta_{r}^{2}|MBK|^{2}}{\sigma\lambda_{\min}^{2}(M)}\right)\frac{|\tilde{x}|^{2}}{|\eta|^{2}}.$

By choosing
$\sigma=\frac{\theta_{r}^{2}|MBK|^{2}}{\theta_{l}\lambda_{\min}(M)\lambda_{\min}(N)}$,
we have

$\varpi-\varepsilon\geq-\sigma(1+\mathbf{Z})^{2}-\varepsilon,$

where $\varepsilon>0$. Next, we can further design $\omega$ as follows:

$\omega=\left\\{\begin{array}[]{ll}\min(0,\varpi)-\varepsilon,&\eta\neq 0,\\\
-\varepsilon,&\eta=0.\end{array}\right.$ (15)

Analyzing the two instances of value of $\omega$, when $\eta\neq 0$, if
$\varpi<0$, it can always be obtained that
$\omega=\varpi-\varepsilon\geq-\sigma(1+\mathbf{Z})^{2}-\varepsilon$, and if
$\varpi\geq 0$,
$\omega=-\varepsilon\geq-\sigma(1+\mathbf{Z})^{2}-\varepsilon$. In addition,
when $\eta=0$, the inequality also satisfies
$\omega=-\varepsilon\geq-\sigma(1+\mathbf{Z})^{2}-\varepsilon$. Recalling the
dynamics of event function $\dot{\mathbf{Z}}(t)=\omega(\varpi,\varepsilon)$,
we have $\dot{\mathbf{Z}}\geq-\sigma(1+\mathbf{Z})^{2}-\varepsilon$ in all
instances. By defining $\phi\leq\mathbf{Z}$, where $\phi$ is the solution of
$\dot{\phi}\geq-\sigma(1+\phi)^{2}-\varepsilon$ satisfying
$\phi(0)=\mathbf{\bar{Z}}$, the IET is lower-bounded by the time $\tau$ takes
for $\phi$ to evolve from $\mathbf{\bar{Z}}$ to 0. $\hfill\blacksquare$ ∎

Compared with the results in [21], the two new design parameters $\theta_{l}$
and $\theta_{r}$ are introduced. As a result, the evolution rate of the event
function $\mathbf{Z}(t)$ can be further adjusted, meaning that the IETs can be
adjusted more freely.

It is worth noting that the derivation of minimum IET $\tau$ in (13) is
independent of the state $\tilde{x}$ or the event triggering error $\eta$,
thus we can conclude that the IET-designable ETM features the global robust
event-separation property. Meanwhile, solving a strictly positive $\tau$
automatically excludes Zeno behavior.

###### Remark 2

Theorem 1 and its proof give an improved IET-designable ETM for the lateral
dynamics of intelligent vehicle. We can find that a freely designable minimum
IET relies on design parameters $\bar{\mathbf{Z}}$ and $\varepsilon$.
Additionally, a desired minimum IET can be acquired by adjusting the parameter
$\sigma$, which suggests that we can also adjust the matrices $M$ and $N$,
control gain $K$, and parameters $\theta_{l}$ and $\theta_{r}$.

###### Remark 3

If considering that the disturbances $\xi$ is unnecessary, we can obtain the
same results as in Theorem 1. Therefore, we conclude that the derivation
process and the strictly positive minimum IET $\tau$ is guaranteed whether or
not disturbances $\xi(t)$ exists. Furthermore, convergent $\xi(t)$ indicates
convergent $\tilde{x}(t)$ as a result of the exponential stability of the
linear system.

##  IV SIMULATION RESULTS

In simulation, the entire simulation time is $15$ s and the sampling period is
$0.01$ s. We adopt three different strategies to achieve the control object:
an improved IET-designable ETM ($\theta_{l}=8$ and $\theta_{r}=0.1$), IET-
designable ETM in [21] (without $\theta_{l}$ and $\theta_{r}$), and time
triggering (0.01 s). In addition, in an IET-designable ETM,
$\bar{\mathbf{Z}}=1$ and $\varepsilon=1$ are chosen. By [22], some values of
bicycle model parameters are given as follows: $m=1421\mathrm{~{}kg}$,
$\mu=0.6$, $V_{x}=18\mathrm{m}/\mathrm{s}$,
$I_{z}=2570\mathrm{~{}kg}\cdot\mathrm{m}^{2}$, $C_{f}=170,550N/rad$,
$C_{r}=137,844N/rad$, $l_{f}=1.191m$, and $l_{r}=1.513m$. We set the initial
state $\tilde{x}=[0;0;0;0]$ and choose the weight matrix
$Q=\operatorname{diag}\\{30,10,1,1\\}$ and $R=[1000]$ of the LQR controller.
Based on $A$ and $B$ in (3), assuming the matrix $N$ is an identity matrix, we
obtain the matrix $M$ by (9). We next assume the upper-boundedness of the
disturbance $\bar{\xi}=[3\times 10^{-4};1\times 10^{-3};0;0]$ and
$G=\operatorname{diag}\\{1,1,1,1\\}$.

Fig. 2: Sideslip $\beta$, yaw rate $\dot{\psi}$, lateral error displacement
$e$, and control input steering angle $\delta$. Fig. 3: Countdown variable
$\mathbf{Z}(t)$ and IET. Fig. 4: Desired and actual trajectories under
different control schemes.

The simulation results of sideslip $\beta$, yaw rate $\dot{\psi}$, lateral
error $e$, and control input $\delta$ are computed according to the system
state variables $\tilde{x}$ and desired equilibrium point $x^{\star}$, and are
presented in Fig. 2. Fig. 4 displays the actual trajectories of three control
strategies and a reference trajectory. Fig. 3 depicts the countdown variable
$\mathbf{Z}$ and IET at every triggering instant and implies the influence of
the two new parameters. If we adopt the IET-designable ETM in [21], the value
of $\varpi$ will be too large or too small, with the maximum being
approximately $8\times 10^{8}$. Furthermore, this means that we will acquire
the same results for values of $\omega$ and $\mathbf{Z}$ as shown in Fig. 3.
Obviously, this is a heavy computational burden, which contradicts the purpose
of this work.

TABLE II: COMPARISON OF TRIGGERING NUMBERS Strategy | Triggering numbers
---|---
Time triggering | 1500
IET-designable ETM in [21] | 749
IET-designable ETM with $\theta_{l}=8$ and $\theta_{r}=0.1$ | 83

We attempted to change the values of design parameters $\bar{\mathbf{Z}}$ and
$\varepsilon$, and chose different matrices $M$, $N$, $Q$, and $R$, but these
attempts had little effect on the results, implying that the designability of
IET-designable ETM in [21] still has some limitations when applied in actual
systems. To obtain a desirable control performance, we introduced two new
design parameters, $\theta_{l}\geq 1$ and $\theta_{r}\leq 1$. This is a
natural operation that can ensure that $\omega$ further satisfies
$\omega<\varpi$, so that the asymptotic stability of system is guaranteed. We
adopted this improved IET-designable ETM, and the two new design parameters
were chosen as $\theta_{l}=8$ and $\theta_{r}=0.1$. In Fig. 3 and Table II, it
can be easily seen that the triggering numbers decrease significantly after
introducing $\theta_{l}$ and $\theta_{r}$, leading to nearly up to a $94\%$
savings of triggering numbers compared to the time triggering control strategy
and $88\%$ compared to the IET-designable ETM strategy in [21].

It should be noted that when we select different parameters of vehicle models,
some fluctuations in the results of saving triggering numbers occur, but this
approach can save the triggering numbers by more than $60\%$ by adjusting the
values of two new parameters $\theta_{l}$ and $\theta_{r}$, in general.

##  V CONCLUSIONS

This brief focuses on the lateral motion control problem of intelligent
vehicle via a malleable event triggering control mechanism for the LQR
controller. Simulation results verify the performance gain from saving
computational load and communication resources, while achieving the control
objective under bounded disturbance. Furthermore, the two new design
parameters greatly improve the designability of the IET-designable ETM, which
can be developed further to broaden its applicability. In our future work, the
dynamical characteristics of the IETs in the proposed IET-designable ETM will
be further investigated.

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