#  Effect of density control in partially observable asymmetric-exit
evacuation under guidance: Strategic suggestion under time delay

Fengqiang Gao  Yuyue Yan<EMAIL_ADDRESS>Zhihao Chen  Linxiao Zheng
Huan Ren  School of Aerospace Engineering, Xiamen University, Fujian 361102,
China  School of Information Science Technology, Xiamen University Tan Kah Kee
College, Fujian 363105, China  Department of Systems and Control Engineering,
Tokyo Institute of Technology, Tokyo 152-8552, Japan  College of Information
Science and Engineering, Huaqiao University, Fujian 362021, China

###### Abstract

To enhance the evacuation efficiency in partially observable asymmetric-exit
evacuation under guidance, a general framework of the dynamic guiding
assistant system is presented to investigate the effect of density control. In
this framework, several evacuation assistants are established to observe the
partial information of pedestrians’ location and adjust the guiding signals of
the dynamic guiding assistant systems. A simple on-off-based density control
algorithm is proposed for the evacuation assistants according to the delayed
data of the observed information (i.e., pedestrian densities in the observed
regions near the corresponding exits). This paper provides strategic
suggestions on how to set the observed region and the target density by
involving a force-driven cellular automaton model. It is observed that the
proposed density control algorithm can control (positively affect) the global
distribution of the pedestrians’ locations and suppress arching phenomena in
the evacuation process even using the observed partial information under time
delays. By imposing a moderate target density, the dynamic guiding assistant
system also suppresses the triggers of collisions around the exits and avoids
inefficiently separating the pedestrians. To enhance evacuation efficiency, we
reveal an interesting fact without loss of generality that we only need to
observe the pedestrians’ location from a small region near the exit instead of
a large region when the time delay of the observed information is slight
enough. Our numerical findings are expected to provide new insights into
designing computer-aided guiding strategies in real evacuations.

###### keywords:

Pedestrian flow; guided crowd; computer-aided technology; evacuation
efficiency; evacuation simulation.

##  1 Introduction

In the last decades, the efficiency estimation problem has been a key aspect
of evaluating safety performance for pedestrian evacuations. During the
occurred emergency events (such as fire, earthquake, and terrorist attacks),
since the pedestrians in the evacuation space are usually myopic and many
irrational behaviors may inherently and negatively affect the pedestrians’
decision making (which may eventually lead to an efficiency loss causing
massive casualties) in the evacuation process [1, 2, 3, 4], some evacuation
strategies are often beforehand designed to enhance the evacuation efficiency.
In the literature, those evacuation strategies may include the choices on room
layout [5], number of exits [6], location of obstacles [7], to name but a few.
To effectively evaluate evacuation efficiency, establishing mathematical
models to describe the pedestrians’ decision-making behavior and the resulting
phenomena in the evacuation process should be of prime importance [8, 9, 10,
11, 12, 13, 14, 15, 16, 17, 18, 19]. For example, a cellular automaton (CA)
model describing the change of cooperative behavior in heterogeneous
pedestrians was investigated in [20] to examine how the pedestrians’
dependency relationship influences evacuation efficiency. A floor field model
investigating agitated behaviors with elastic characteristics was presented in
[21], whereas a fine discrete field CA model integrating anisotropy,
heterogeneity, and time-dependent characteristics, was discussed in [22]. A
multi-grid CA model capturing the turning behaviors was proposed in [23].
Furthermore, an extended floor-field CA model describing the group behaviors
in crowd evacuation was found in [24], whereas a modified social force model
and a force-driven CA model were proposed in [25] and [26] to reflect the
evacuation process under guidance.

With the development of modern technology, especially in the aspect of the
imaging process, communication networks, and IoT technologies, some evacuation
assistants (or equipment) can be adopted in the (computer-aided) evacuation
process to provide accurate route or exit information to the evacuees and
hence help (guide) them to escape as efficiently as possible. In the related
works with guiding behaviors [27, 28, 29, 30, 31, 32, 33, 34], Yang et al.
[27] are the first researchers to claim the necessity of using guides (dynamic
leaders) in the evacuation process, whereas the effects of leadership in
single-exit rooms were investigated in [28] with limited visible range. The
authors in [28] further stated that guiding behavior may yield a negative
effect when too many pedestrians are neighbor to the leader. Considering the
situation where the dynamic leader attracts the crowds and moves together with
them towards one of the exits, the optimal number and positions for those
leaders were derived in [34] and [35]. Even though the literature [28] and
[34] have claimed that the density of pedestrians plays an important role in
the evacuation process, to our best knowledge, our previous study in [26] is
the first work indicating the possibility and goodness of controlling the
pedestrians’ density. It was found that for a symmetric-exit evacuation space,
unbalanced pedestrian distributions may be yielded because of the mutual
attractions among the pedestrians, but those unbalanced distributions can be
suppressed by imposing some control mechanisms based on the data of pedestrian
density from an observed region. Given the constant observed region, by
adjusting the guiding signals according to the pedestrian densities around the
exits, [26] showed preliminary evidence that imposing control mechanisms may
be able to pursue a more balanced pedestrian distribution and an efficient
evacuation process. However, the effects of the size of the observed regions
are not yet revealed. Furthermore, it is essential to note that the results in
[26] did not consider the influence of data time delay in the control
mechanisms, which is unnatural for a realistic computer-aided evacuation
process.

Contributions: Different from [26] and the other existing results, the main
contribution of this paper is focused on finding the optimal size of the
observed regions under different data time delays for a dynamic guiding
assistant system in partially observable asymmetric-exit evacuation. To this
end, inspired by control theory, we propose a general framework of dynamic
guiding assistant systems with a density control algorithm by assuming that
the evacuation assistants can observe only a few parts of the evacuation
spaces. We consider a simple on-off-based density control algorithm for
evacuation assistants with delayed data of the observed information, i.e.,
pedestrian densities in the regions near the corresponding exits. By involving
a force-driven cellular automaton model, we discuss strategic suggestions on
setting the observed region in this paper by presenting the numerical
simulation results under homogeneous and heterogeneous pedestrians with
respect to the size of the visual field. Our numerical findings contribute to
providing some new insights on designing computer-aided (control-based)
guiding strategies in actual evacuations.

This paper is organized as follows: A framework of dynamic guiding assistant
system with density control algorithm is proposed in Section missing 2. In
Section missing 3, we first introduce the necessity of using dynamic guiding
assistance in the asymmetric-exit evacuation process and then investigate the
optimal size of the observed regions for various scenarios with a slight data
delay and a large data delay. Finally, the conclusion is given in Section
missing 4.

_Notation:_ We use $\mathbb{R}_{+}$ for the set of positive real numbers,
$\mathbb{N}_{+}$ for the positive integers, and $\emptyset$ for the empty set.

##  2 System Description and Problem Formulation

###  2.1 System Description

Consider an evacuation system $\Pi$ consisting of several pedestrians in a
multi-exits evacuation space. Each pedestrian dynamically changes its location
over time to escape from the evacuation space through one of the
$m\in\mathbb{Z}_{+}$ number of exits according to its personal information,
e.g., visible exit, communication with the other evacuees. We denote the set
of exits by $\mathcal{N}_{\rm ex}\triangleq\\{1,2,\ldots,m\\}$. As myopic
decision-makers with limited information, the pedestrians may move in the same
direction even when congestions exist around the target exit. In such a case,
the exits may not be used efficiently. To help pedestrians better efficiently
escape from space, in this paper, we consider a guiding assistant system for
the aforementioned evacuation system with $\varepsilon\in\mathbb{N}_{+}$
evacuation assistants (EAs), where the set of EAs is given by
$\mathcal{N}_{\rm EA}\triangleq\\{1,2,\ldots,\varepsilon\\}$. In this system,
each evacuation assistant $i\in\mathcal{N}_{\rm EA}$ is assigned to a
corresponding exit $j\in\mathcal{N}_{\rm ex}$, and sends a _guiding signal_ to
the pedestrians to attract them towards its own assigned exit. We assume that
the guiding signal is dynamically adjustable by the evacuation assistant
according to the real-time situation of the evacuation process. Henceforth, we
denote the guiding assistant system by $\mathcal{G}$.

Figure 1: Structure of the guiding assistant-based evacuation system. Each
evacuation assistant observes the partial information of agents’ location and
adjusts the guiding signal according to the observed information. Involving
the density controller, the classic (static) guiding assistant system becomes
a dynamic guiding assistant system where the guiding signals of $\mathcal{G}$
are understood as the inputs (feedback) to the evacuation system $\Pi$. With a
well-designed control algorithm (e.g., density control law), the states
(location) of the pedestrians and the evacuation efficiency may be controlled
(positively affected). The detailed control algorithm and pedestrian
evacuation model considered in this paper are given in Sections 2.2 and 2.3
below.

According to the control theory, it is essential to notice that the guiding
signal of the guiding assistant system $\mathcal{G}$ can be understood as the
_input_ to the evacuation system so that the states (location) of the
pedestrians are controlled (affected) by the EAs. This fact is characterized
independently of the underlying mathematical models that describe the
decision-making process of pedestrians under guidance (guiding signals).
However, as a common issue in control systems, the states (location) of the
pedestrians may not be fully observed. Inspired by the classic control theory,
we note that an unobservable system may still be controllable under some well-
designed control algorithms [36, 37]. Consequently, in this paper, we assume
that EAs may observe partial information on the locations of the pedestrians
(e.g., the pedestrian density $\rho_{i}$ near exit $i\in\mathcal{N}_{\rm ex}$)
using some computer-aided technologies. For example, those EAs can observe the
regions near exits and adjust the intensity of guiding signals according to
the observed information. A fundamental question is how to set the size of the
observed regions and control the guiding signals to enhance the evacuation
efficiency as much as possible. To address this problem, we introduce a force-
driven CA model under density control algorithm in the next section to
implement the structure of the guiding assistant-based evacuation system
illustrated in Fig. missing 1.

Figure 2: Preferred moving directions for a given social force $F_{n}$. In
(a), the sequence of preferred moving directions is given by upper right,
right, upward and bottom right. In (b), as the cells in the upper right and
bottom right are occupied, the sequence of preferred moving directions is
given by right and upward, i.e., $P_{n}^{3}=P_{n}^{4}=\emptyset$. Supposing
that the pedestrians are only allowed to compete for 2 rounds in terms of
preferred moving directions (i.e., $k=2$), it is understood that the
pedestrian $n$ in (a) (resp., in (b)) first competes for the upper right
direction (resp., the right direction). If he loses in the competition, then
the next competition target is the right direction (resp., upward direction)
in (a) (resp., in (b)).

###  2.2 Brief Introduction of Force-driven CA Model Under Density Control

In this section, we characterize the force-driven CA model to describe the
guided evacuation process with the two-dimensional Moore cells, where each
cell may be occupied by only one of the pedestrians with eight neighbor cells
as the possible moving directions for the next time instant (see Fig. missing
2(a)). Specifically, to be consistent with the literature in guided evacuation
[27, 28, 29, 34, 35], we consider a square evacuation space and divide the
space to $N\times N$ number of cells. Letting $\mathbb{RM}(t)$ denote the set
of the pedestrians remaining in the evacuation system $\Pi$ at time instant
$t=0,1,2,\ldots$, we suppose that the remaining pedestrians simultaneously
update their locations according to a social force-based evolution rule given
in the following sections.

####  2.2.1 Force Definitions

Before we present the detailed expression of the evolution rule, we define the
social force consisting of a guiding force from an evacuation assistant, the
mutual forces among visible pedestrians, and the attractive forces to visible
exits for each pedestrian $n\in\mathbb{RM}(t)$, i.e.,

$\displaystyle{F_{n}={\color[rgb]{0,0,1}w_{1}F_{\rm
guide}^{n}+w_{2}\\!\sum\nolimits_{m\in\mathbb{VP}_{n}}{F_{\rm
mutual}^{n,m}}+w_{3}\\!\sum\nolimits_{i\in\mathbb{VE}_{n}}F_{\rm
exit}^{n,i},}}$ (1)

where $w_{1}$, $w_{2}$ and $w_{3}\in\mathbb{R}_{+}$ denote the positive
weighting factors depending on the evacuation scenario, and $\mathbb{VP}_{n}$
(resp., $\mathbb{VE}_{n}$) represents the set of visible pedestrians (resp.,
visible exits) for the pedestrian $n\in\mathbb{RM}(t)$.

Note that in Eq. (missing) 1, $F_{\rm guide}^{n}$ represents the guiding force
pointing to the exit whose EA’s guiding signal possesses a maximal _signal to
interference ratio_ , i.e.,

$F_{\rm guide}^{n}=u_{I_{n}}D\vec{r}_{n,I_{n}},\quad
I_{n}\triangleq\arg\max_{i\in\mathcal{N}_{\rm
EA}}\left(\frac{u_{i}}{1+\sum_{j\in\mathcal{N}_{\rm
EA}\setminus\\{i\\}}(r_{n,i}^{2}/r_{n,j}^{2})}\right),$ (2)

where $u_{i}\in[0,1]\subset\mathbb{R}$ refers to the intensity coefficient
controlled by the evacuation assistant $i\in\mathcal{N}_{\rm EA}$,
$D\in\mathbb{R}_{+}$ refers to the maximal intensity of the guiding signal,
$r_{n,i}$ refers to the distance (steps) to the exit $i$, and
$\vec{r}_{n,I_{n}}$ refers to the unit vector pointing from the pedestrian $n$
to the exit $I_{n}$ (see Ref. [26]); $F_{\rm mutual}^{n,m}$ represents a
mutual force from the visible pedestrian $m$ [38, 26], i.e.,

$\displaystyle F_{\rm
mutual}^{n,m}=\begin{cases}{\color[rgb]{0,0,1}{\eta_{1}}\vec{r}_{m,n}},&\mbox{if
$r_{n,m}=1$}\\\ {\eta_{2}}/{r_{n,m}^{2}}\vec{r}_{n,m},&\mbox{if $1<r_{n,m}\leq
w$}\end{cases},\quad m\in\mathbb{VP}_{n},$ (3)

where $\eta_{1}\in\mathbb{R}_{+}$ refers to the repulsion coefficient,
$\eta_{2}\in\mathbb{R}_{+}$ refers to the attraction coefficient, $w$ refers
to the length of the visual field, ${r}_{n,m}$ refers to the distance (steps)
between the pedestrians $m$ and $n$, and $\vec{r}_{n,m}$ refers to the unit
vector pointing to the pedestrian $m$; $F_{\rm exit}^{n,i}$ represents an
attractive force to a visible exit [38, 26], i.e.,

$F_{\rm exit}^{n,i}=E\vec{r}_{n,i},\quad
i\in\mathbb{VE}_{n}\triangleq\\{i\in\mathcal{N}_{\rm ex}:r_{n,i}\leq w\\},$
(4)

where $E\in\mathbb{R}_{+}$ refers to the constant intensity, $r_{n,i}$ and
$\vec{r}_{n,i}$ respectively refer to the distance and the unit vector to the
exit $i\in\mathcal{N}_{\rm ex}$.

######  Remark 1.

_ It is necessary to notice that the repulsions between the pedestrian and
obstacle may need to be further added in the expression of the social force
(1) when the obstacle appears in the visual field. The expression of
repulsions between the pedestrian and obstacle can be found in [38]. To be
consistent with the literature in guided evacuation (see [27, 28, 29, 34,
35]), we do not consider an evacuation space with any obstacle and hence those
forces are excluded in this paper. On the other hand, when the heterogeneity
on quality of the pedestrians [38] is considered, the expressions of the
repulsions and attractions should be changed to $F_{\rm
mutual}^{n,m}=\begin{cases}{\eta_{1}}q_{m}q_{n}\vec{r}_{m,n},&\mbox{if
$r_{n,m}=1$}\\\ {\eta_{2}}q_{m}q_{n}/{r_{n,m}^{2}}\vec{r}_{n,m},&\mbox{if
$1<r_{n,m}\leq w$}\end{cases}$, where $q_{m}$ is the quality of the pedestrian
$m\in\mathbb{VP}_{n}$ and $q_{n}$ is the quality of the pedestrian $n$._

After defining the notion of social forces, the preferences of moving
direction candidates are consequently derived by sorting the magnitude of the
decomposed forces of _F_ n at _x_ , _y_ , _s_ , and _f_ -axis (see Fig.
missing 2). Henceforth, we denote the sequence of ordered preference target
cells by $P_{n}=[P_{n}^{1},P_{n}^{2},P_{n}^{3},P_{n}^{4}]$, where $P_{n}^{i}$
refers to the $i$-th preferred target.

####  2.2.2 Social Force-Based Evolution Rule

Given an initial distribution of the pedestrians $\mathbb{RM}(0)$ and the
intensity coefficients $u_{i}(0)$, $i\in\mathcal{N}_{\rm EA}$ of the guiding
signals from the guiding assistant system $\mathcal{G}$, the social force-
based evolution rule of the force-driven CA model is summarized as follows:

  1. (i)

Calculate the social force according to Eq. (missing) 1 and determine the
sequence of ordered preferred target cells
$P_{n}=[P_{n}^{1},P_{n}^{2},P_{n}^{3},P_{n}^{4}]$ for all the remaining
pedestrians $n\in\mathbb{RM}(t)$. If some of the desired targets have already
been occupied, then some elements of $P_{n}$ are understood as empty. For
example, the cells in the upper right and bottom right directions in Fig.
missing 2(b) are not considered as the preferred target cells and hence
$P_{n}^{3}=P_{n}^{4}=\emptyset$ because those cells have already been
occupied.

  2. (ii)

Collisions and target competition. We suppose that each pedestrian
$n\in\mathbb{RM}(t)$ possesses at most $k$ chances to compete for the target
cells with the other pedestrians according to its own sequence $P_{n}$. If a
cell is targeted as the preferred moving direction by several pedestrians in
one of the competition rounds, it turns out that _collisions_ may happen [39]
so that none of the conflicting pedestrians wins in the competition. Here, we
suppose that the probability that _collision_ happens is $\varphi\in[0,1)$,
whereas the probability that one of the conflicting pedestrians wins (and
hence the target cell is assigned to him) is $1-\varphi\in(0,1]$. In the case
where a collision does not happen, the winner is _randomly_ selected. If the
pedestrian $n$ loses in one of the competitions, then he continues to compete
for the next un-assigned preferred target cell until all the $k$ chances are
used. If the pedestrians lose in all the $k$ rounds of competitions, they do
not change their locations next time instant 111According to our observation
and personal experience in the real world, when two pedestrians are targeting
the same location, one is likely to change his direction when the collision is
about to happen. This change of direction may happen almost at the same time
when another pedestrian successfully comes to the targeted location.
Therefore, in the rest of the paper, we suppose that each pedestrian has one
chance to alter his target, i.e., $k=2$..

  3. (iii)

Update the locations for all the pedestrians $n\in\mathbb{RM}(t)$. Since the
pedestrians who reached one of the exits are understood as the ones who have
successfully escaped from the evacuation system $\Pi$, release the escaped
pedestrians from the simulation data set, let $t=t+1$, and update the set
$\mathbb{RM}(t)$.

  4. (iv)

Calculate and output the pedestrian density $\rho_{i}(t)$ observed at the exit
$i\in\mathcal{N}_{\rm ex}$.

  5. (v)

Require the new information of $u_{i}(t)$, $i\in\mathcal{N}_{\rm EA}$, and go
to (i) to continue until all of the pedestrians left the evacuation system
$\Pi$ (i.e., $\mathbb{RM}=\emptyset$).

######  Remark 2.

_It is important to note that the proposed evacuation model requires the
inputs (i.e., the intensity coefficient $u_{i}$, $i\in\mathcal{N}_{\rm EA}$)
from the guiding assistant system $\mathcal{G}$ and offers the real-time data
(partial information) to the density controller for redesigning the inputs.
Consequently, when a density controller is established, the guiding signals
are understood as the _feedback_ to the evacuation system for enhancing the
efficiency of the evacuation process (see Fig. missing 1)._

###  2.3 Density Control Algorithm

In this section, we consider an on-off-based density control algorithm as the
feedback control law for the guiding assistant system to adjust the guiding
signals and hence affect the evacuation efficiency. Specifically, we suppose
that all of the EAs only observe the values of pedestrian density
$\rho_{j}(t)$ around their own assigned exit $j\in\mathcal{N}_{\rm ex}$ based
on some computer-aided technologies (e.g., image processing) and use the
observed information to update $u_{i}$, $i\in\mathcal{N}_{\rm EA}$ for the
guiding signals. The density control algorithm considered in this paper is
given by

$\displaystyle
u_{i}(t)=\left\\{\begin{array}[]{c}1,\quad\rho_{j}(t-\zeta)\leq\rho^{\rm
aim}_{i},\\\ 0,\quad\rho_{j}(t-\zeta)>\rho^{\rm
aim}_{i},\end{array}\right.\quad i\in\mathcal{N}_{\rm EA},\quad
t=1,2,3,\ldots,\vspace{-4pt}$ (7)

where $\zeta$ denotes the constant time step delay of the data (observed
information), $j$ refers to the exit index assigned to the evacuation
assistant $i$, and $\rho^{\rm aim}_{i}$ refers to the target pedestrian
density set by the evacuation assistant $i$. In this case, the evacuation
assistant $i$ turns on (resp., turns off) the guiding signal when the current
density around its exit is less (resp., larger) than the target density 222If
$\rho^{\rm aim}_{i}=1$ holds for some $i\in\mathcal{N}_{\rm EA}$, then it is
understood that those EAs never stop sending the guiding signals for
attracting the pedestrian towards their own assigned exits. Alternatively, if
$\rho^{\rm aim}_{i}=0$ holds for some $i\in\mathcal{N}_{\rm EA}$, then it is
understood that those EAs only begin to send the guiding signals when all the
pedestrians around their assigned exits have already escaped from the
evacuation system $\Pi$.. It is important to note that when $\rho^{\rm
aim}_{i}=1$, $i\in\mathcal{N}_{\rm EA}$, the guiding signal is always opened
and hence the guidance from the guiding assistant system $\mathcal{G}$ is
understood as the classical (_static_) guiding assistance. Alternatively, when
$\rho^{\rm aim}_{i}\not=1$, $i\in\mathcal{N}_{\rm EA}$, it is understood that
the classical (_static_) guidance from the guiding assistant system
$\mathcal{G}$ changes to _dynamic_ guidance.

######  Remark 3.

_ Note that a continuous-function control can be considered as the density
control algorithm to enhance the evacuation efficiency if the guiding signals
$u_{i}$, $i\in\mathcal{N}_{\rm EA}$ are quantifiable. However, establishing
quantifiable guiding signals relies on the structure (equipment) of the
guidance system. For example, a PI control algorithm can be used in the
evacuation process when the strength of the guiding signals is shown in the
IoT devices to the pedestrians. Since the opening and closing of the guiding
effect do not rely on the structure of the guidance system, we use the on-off-
based density control in this paper._

_Problem_ : Consider the guiding assistant system $\mathcal{G}$ with the
density control algorithm shown in Eq. (missing) 7. The main question
investigated in this paper is how to set the observed regions and the target
density in the guiding assistant system to enhance the evacuation efficiency
as much as possible. Since the data collection (e.g., imaging processing) and
data transmission take time, the time step delay $\zeta$ may not be very
small, and this parameter may affect the evacuation efficiency under the
guiding assistant system $\mathcal{G}$. In the following sections, we first
consider the case where the time step delay $\zeta$ is very small, and then we
discuss the situation when considering a larger time step delay $\zeta$.

We evaluate the evacuation efficiency by using the nation of _travel time_
$T_{\rm end}$, which is defined as the time instant when all the pedestrians
successfully escaped from the evacuation system, i.e., $\mathbb{RM}(t)=0$ at
$t=T_{\rm end}$. The dynamical guiding assistant system (or, the density
control algorithm) is said to be effective if the travel time is shorter than
the one under static guidance (i.e., $\rho^{\rm aim}_{i}=1$,
$i\in\mathcal{N}_{\rm EA}$).

##  3 Simulation and Analysis

Limited by the difficulty of obtaining real data under the COVID pandemic, the
validity of the force-driven CA model is verified by the context of the
fundamental results in [38, 40, 26]. The main objective of this paper is to
obtain the optimal control strategies (e.g., target density, observed region)
for the guiding assistant system $\mathcal{G}$ through analyzing simulation
results. In this section, we present and analyze our simulation results for an
evacuation system $\Pi$ with $m=4$ exits and $\varepsilon=4$ evacuation
assistants where every evacuation assistant is located at the exit and is
observing the pedestrian density near its exit to turn on or shut down the
guiding signal according to Eq. (missing) 7. Specifically, letting $N=23$, we
divide the indoor evacuation space to $23\times 23$ cells so that the
evacuation system $\Pi$ may contain at most 529 pedestrians. We set
$w_{1}=w_{2}=w_{3}=1$ in Eq. (missing) 1, $D=30$ in Eq. (missing) 2, $E=60$ in
Eq. (missing) 4, $\eta_{1}=10$, and $\eta_{2}=20$ in Eq. (missing) 3. We
consider the evacuation space with asymmetric exits layout as shown in Fig.
missing 3.

Figure 3: Evacuation space with asymmetric exits layout with $N=23$ and
$\mathcal{N}_{\rm ex}=\mathcal{N}_{\rm EA}=\\{1,2,3,4\\}$, where the four
colored regions around exits represent the observed region for calculating
pedestrians’ density with the radius being $d$. Two pedestrians possessing the
field of view of $w=3$ and $w=4$ are invisible to each other. Since none of
the two pedestrians appear in the observed regions, their information
(location) is unknown to the evacuation assistants.

######  Remark 4.

_ In general, the effect of exit distribution (e.g., the number of exits and
asymmetric degree of exits) may influence the evacuation efficiency (see the
results in [41]). For example, different from the main contribution of this
paper, the effect of density control with a symmetric exit distribution is
revealed in [26] without considering data time delay. In terms of the
asymmetric exit distribution, it turns out that the tendency of simulation
results shown in the following sections still can be similarly found when we
alter the distributions of those exits. Therefore, the evacuation space shown
in Fig. missing 3 is considered without loss of generality in the rest of this
paper. _

For simplicity, we let $\rho^{\rm aim}_{i}=\rho_{\rm aim}$ for all
$i\in\mathcal{N}_{\rm ex}$ and define the observed region as a rectangular
area expanded from the exit $i\in\mathcal{N}_{\rm ex}$ with
$d\in\mathbb{N}_{+}$ steps (see the typical example of the four observed
regions with $d=3$ in Fig. missing 3, where each of the EAs is able to observe
at most 21 pedestrians near its exits). Recalling the fact that the time step
delay $\zeta$ of the observed data depends on the data collection structure,
we first assume that the time step delay $\zeta$ is very small in Section
missing 3.1 and then relax this assumption in Section missing 3.2 to discuss
how the result changes when we have a more significant time step delay
$\zeta$. It is important to note that as one of the pedestrians’ inherent
properties, the size of the visual field may be diverse from each other and
hence the pedestrians existing in the evacuation system $\Pi$ may be
homogeneous or heterogeneous. As the simulation result might be strongly
connected to the mentioned inherent property (i.e., size of the visual field),
to derive a strategic conclusion on how to set the target density and observed
region without loss of generality, we may consider the evacuation processes
with heterogeneous pedestrians in the following sections.

###  3.1 Small Time Step Delay in Density Control Algorithm

####  3.1.1 Necessity of Using Dynamic Guiding Assistances

In the beginning, to indicate the necessity of using dynamic guiding
assistance in the asymmetric-exits evacuation process, we use the initial
distribution shown in Fig. missing 4(a) in the simulation where 317
pedestrians are distributed randomly in the evacuation space. The field of
view is given by $w=3$ for all the pedestrians so that each one of them is
able to observe at most 48 pedestrians around its location (see the grey
circle in Fig. missing 3 representing a pedestrian with $w=3$). Letting the
collision probability be $\varphi=40\%$, the distributions of the pedestrians
under _static_ guiding assistance (i.e., $\rho_{\rm aim}=1.0$) are shown in
Fig. missing 4. It can be seen from the figure that since the EAs never stop
attracting the pedestrians, whatever how large the density is in its observed
region, the pedestrians are suddenly separated in the horizontal direction,
and hence a blank zone appears in the middle of the evacuation space from the
time instant $t=20$. The blank zone expands along with time and the
utilization of the three exits in the left-hand side of the evacuation space
is almost vacant in the later stage of the evacuation, i.e., the remaining
pedestrians _only_ locate around exit 4 after time instant $t=60$. As a
result, it is observed that a lot of evacuation efficiency is wasted in the
evacuation process under the static guiding assistance.

Figure 4: Distributions of homogeneous pedestrians under _static_ guiding
assistance. Then guiding signal is always opened in the evacuation process
since $\rho_{\rm aim}=1$. Pedestrians immediately huddle near the exits and
never change the moving direction to the reverse direction. A large group of
pedestrians huddle around one of the exits (exit 4) even though the other 3
exits are not utilized at all, which makes the evacuation process run in an
inefficient way.  Figure 5: Distributions of homogeneous pedestrians under
_dynamic_ guiding assistance with $\rho_{\rm aim}=0.3$ and $d=3$, $\zeta=1$.
The 4 colored regions represent the regions at which the evacuation assistants
observe the crowd density. The evacuation assistant turns on or shuts down the
guiding signal according to the real-time data of the observed crowd density.
Pedestrians do not huddle near the exits but stay in the evacuation space’s
central area until the pedestrians in the observed regions are all evacuated.
The 4 exits are almost utilized efficiently even at the time instant $t=100$.
Compared to the simulation results under static guidance in Fig. missing 4,
this example indicates the significance of using _dynamic_ guiding assistance
in the asymmetric-exits evacuation process.

Next, we show a typical example of pedestrian distributions, which reveals the
effectiveness and significance of using _dynamic_ guiding assistance.
Specifically, in the simulation, we let $\rho_{\rm aim}=0.3$, $\zeta=1$, $d=3$
(see the colored areas representing the observed regions in Fig. missing 3).
The distributions of the pedestrians are illustrated in Fig. missing 5.
Compared to the case with _static_ guiding assistance, the amounts of the
remaining pedestrians at the time instants $t=20$, $40$, $60$, $80$, and $100$
under _dynamic_ guiding assistance in Fig. missing 5 are _certainly_ less than
the ones in Fig. missing 4. It can be easily found in Fig. missing 5 that the
pedestrian densities in the four observed regions under control are much less
than the ones in Fig. missing 4 so that the possibility of collisions and
congestions happening around the exits is suppressed, which make the
evacuation run more efficiently. Another direct and interesting observation
found in the comparison is that the pedestrians are no longer separated into
two crowds in the horizontal direction but stay in the central region under
_dynamic_ guiding assistance (see Fig. missing 5(b)–(e)). Those simulation
results capture the fact that the dynamic guiding assistant system with only
partially observable data may still positively control (affect) the location
of all pedestrians in the multi-exit evacuation process by involving the
density control algorithm.

Figure 6: Travel time versus the target density $\rho_{\rm aim}$ in density
control when $\zeta=1$ with different sizes of the observed regions. To avoid
misinterpretation caused by the stochastic process existing in the evacuation
model, all the data of average travel times are generated from more than 30
simulations. The initial distribution is shown on the left-hand side of the
figure. The fitting results are generated by the “Smoothing Spline” function
of the Curve Fitting box in Matlab with the goodness of fit (R-square) being
larger than 0.995. The optimal size of the observed region is found as 2. That
is to say, the observed region is suggested to be small for the case where the
time step delay $\zeta$ is small. The suggested target density for the dynamic
evacuation assistant system ranges from 0.3 to 0.65.  Figure 7: Distributions
of homogeneous pedestrians under _dynamic_ guiding assistance with different
target densities at the time instant $t=40$ when $\zeta=1$, $d=6$. It is
observed that the pedestrians are slowly attracted towards the exits when the
target density is small, and an arching phenomenon occurs when the target
density is large. The pedestrians are obviously separated in the horizontal
direction for $\rho_{\rm aim}\geq 0.7$, which should be avoided as discussed
in Section missing 3.1.1. The number of pedestrians trapped in the arching
phenomenon increases when the target density increases.

####  3.1.2 Influence of Target Density and Size of Observed Regions

In this section, we investigate the influence of the target density of the
dynamic guiding assistant system on the travel time of the evacuation process
and hence derive the optimal size of the observed regions. Letting $\rho_{\rm
aim}\in[0.1,1]$, the response curves of travel time of the evacuation process
on target density are shown in Fig. missing 6 with several different sizes of
the observed regions where the collision probability is still set to
$\varphi=40\%$. Note that the presented curves are yielded from the average
value of the data from more than 30 times simulations and show similar
tendencies with respect to $\rho_{\rm aim}$. On the one hand, it can be seen
from the figure that compared to the case with static guidance (i.e., when
target density equals 1.0), the evacuation efficiency is significantly
improved when the target density $\rho_{\rm aim}$ is moderate in the density
control algorithm Eq. (missing) 7. Without loss of generality, letting the
size of observed regions be $d=3$, Fig. missing 7 shows the pedestrian
distributions at the time instant $t=40$ when different target densities are
used in the simulations. This figure provides evidence to support that the
target density for the dynamic evacuation assistant system should be a
moderate value. In particular, it is observed from Fig. missing 7 that
pedestrians are slowly attracted towards the exits when the target density is
small, and an arching phenomenon occurs when the target density is
large333Arching phenomenon refers to the scene in which a large group of
pedestrians huddle around the exit in the evacuation process. This phenomenon
leads to serious conflicts and congestions among the pedestrians, and
eventually reduce the traffic efficiency of the exits. During designing the
dynamic guiding assistant system, the arching phenomenon should be suppressed
as much as possible to avoid conflicts and congestions.. On the other hand, it
can be further seen from Fig. missing 6 that except for the case with $d=2$,
the efficiency improvement made by the density control algorithm drops off
when the size of the observed regions increases. The optimal size $d$ for the
observed regions and the optimal target density $\rho_{\rm aim}$ are found as
$d=2$ and $\rho_{\rm aim}\in[0.3,0.7]$, respectively. Note that this
observation reveals an interesting fact that to enhance the evacuation
efficiency, we only need to observe the pedestrians’ location from a small
region near the exit instead of a large region when the time step delay in the
density control algorithm (7) is very small.

Figure 8:  Amount of collisions (that trigged around the exits) versus target
density with different sizes of the observation region when $\zeta=1$. The
collision amount is reduced when we narrow the observation region for the
dynamic guiding assistant system. Decreasing the target density of the on-off-
based density control can significantly reduce the collisions around the
exits.  Figure 9:  Proportion of use of each exit versus target density with
different sizes of the observation region when $\zeta=1$. The use of exits is
more balanced by narrowing the observation region for the dynamic guiding
assistant system. The use of exits is extremely imbalanced under static
guidance.

To further understand the intrinsic reason why the target density and the size
of the observed region matter in the evacuation process, we illustrate the
response curves of collision amounts (that trigged around the exits) on the
target density $\rho_{\rm aim}$ in Fig. missing 8. It can be observed from the
figure that the collision amount is reduced when we narrow the observation
region for the dynamic guiding assistant system. Moreover, it can be seen that
decreasing the target density of the on-off-based density control (7) can
significantly reduce the collisions around the exits in the evacuation
process. On the other hand, the proportion of use of each exit versus target
density with different sizes of the observation region is demonstrated in Fig.
missing 9. It can be seen from this figure that the use of exits is extremely
imbalanced under static guidance and the density control can significantly
balance the use of the four asymmetric exits, especially when the target
density is set to $\rho_{\rm aim}\in(0.3,0.7)$. Also, it can be observed that
the use of exits is more balanced by narrowing the observation region for the
dynamic guiding assistant system.

Now, it is important to note that the above observations of the optimal size
$d$ are without loss of generality when we consider different initial
distributions or sizes of the visual field. For example, considering the
situation where the pedestrians in the initial time instant may be mixed by
some taller pedestrians possessing a larger field of view, we present the
simulation results with heterogeneous pedestrians holding the visual field
with $w=3$ and $w=4$ under different mixing ratios in Fig. missing 10. Even
though the tendencies of the curves are slightly different from Fig. missing
6, we note that the size $d=2$ remains the optimal size for the observed
regions in all of the simulations under different mixing ratios. Moreover,
when $d\geq 3$, it is still observed that the efficiency improvement made by
the density control algorithm drops off when the size of the observed regions
increases for each of the sub-figures. Even though the mixing ratios change
the tendency of the response curves, the travel time under the target density
$\rho_{\rm aim}\in(0.5,0.6)$ with $d=2$ is always the minimum value. As a
result, we conclude that no matter how much the mixing ratio is in the
simulation, the observed region is suggested to be small for the case where
the time step delay $\zeta$ is small.

Figure 10: Response of the target density on travel time under density control
with mixed or pure pedestrians possessing different or the same view sizes
when $\zeta=1$. (a) 20 percent of the pedestrians possess a bigger visual
field in the simulation. (b) 50 percent of pedestrians possess a bigger visual
field in the simulation. (c) 80 percent of pedestrians possess a bigger visual
field in the simulation. (d) All pedestrians possess a bigger visual field in
the simulation. The initial distributions set in the simulations are shown the
left-hand side of the sub-figures where the blue circles denote the
pedestrians with a larger visual field (i.e., $w=4$ instead of $w=3$). All of
the fitting results are generated by the “Smoothing Spline” function of the
Curve Fitting box in Matlab with the goodness of fit (R-square) being larger
than 0.995. In all of the cases, the optimal size of the observed region is
found as 2. That is to say, the observed region is suggested to set as a small
one around the exit for the case with small time step delay $\zeta$.

###  3.2 Influence of Data Time Delay

Figure 11: Travel time versus the target density $\rho_{\rm aim}$ in density
control with different sizes of the observed regions and different time delay
$\zeta$. The initial distribution considered in the simulations is the same
one that we used in Fig. missing 6. The travel time is reduced by slightly
increasing the delay. However, when the time delay $\zeta$ is too large, the
density control algorithm may no longer enhance the evacuation efficiency.

Recalling the fact that the time step delay $\zeta$ of the density control
algorithm is a given positive parameter depending on the data collection
structure of the guiding assistant system, it is important to discuss how the
optimal size changes when the given time step delay $\zeta$ changes. The
travel time versus the target density $\rho_{\rm aim}$ in density control with
different sizes of the observed region and time delays is shown in Fig.
missing 11, where we used Fig. missing 6 as the initial distribution in all
the simulations. It can be seen from Fig. missing 11 that the travel time can
be reduced by slightly increasing the delay. When the given time step delay
$\zeta$ is larger, setting up a larger region to observe the pedestrians’
location is better. Those observations are natural in the sense of physical
meaning, since the data with a large delay are not able to represent the
characteristics of the pedestrians’ current situation for a small observed
region, but for a large observed region, those data may be able to represent
the partial characteristics of the pedestrians’ real situation because of the
intrinsic inertia. However, when the time delay $\zeta$ is too large, the
density control algorithm may no longer obviously enhance the evacuation
efficiency (see Fig. missing 11). As a result, the time step delay allowed in
our proposed density control algorithm is bounded. The above numerical
findings give important insights on designing computer-aided (control-based)
guiding strategies in real evacuations.

##  4 Conclusion

To enhance the evacuation efficiency in partially observable asymmetric-exit
evacuation under guidance, a general framework of the dynamic guiding
assistant system was proposed to investigate the effect of density control. In
the characterized system, multiple evacuation assistants are established to
observe the partial information of pedestrians’ location and adjust the
guiding signals of the dynamic guiding assistant systems according to the
observed information (i.e., pedestrian densities in the observed regions near
the corresponding exits). Specifically, a simple on-off-based density control
algorithm associated with a target density is considered for evacuation
assistants based on the delayed data of (observed) pedestrian densities to
meet the physical challenges on data collection, transmission, and
implementation which often exist in the realistic computer-aided evacuation
process. By involving a force-driven CA model, we presented the simulation
results in accordance with data delays to eventually give a strategic
suggestion on how to set the observed region and the target density.

According to our numerical simulations, we first revealed the necessity of
using dynamic guiding assistance in the asymmetric-exit evacuation process. It
was found that the proposed density control algorithm can control (positively
affect) the global distribution of the pedestrians’ locations and suppress
arching phenomena in the evacuation process even using partially observed
information under time delays. To derive the optimal size of the observed
regions, we investigated the influence of the target density of the dynamic
guiding assistant system on the travel time of the evacuation process. It was
found that the dynamic guiding assistant system with only partially observable
data can suppress the triggers of collisions around the exits and avoid
inefficiently separating the pedestrians in the evacuation process. After
showing various simulation results, we revealed an interesting fact without
loss of generality that to enhance the evacuation efficiency, we only need to
observe the pedestrians’ location from a small region near the exit instead of
a large region when the time step delay in the density control algorithm is
very small. Furthermore, we found that the time step delay allowed in our
proposed density control algorithm is bounded since it may be impossible to
significantly enhance the evacuation efficiency by using the density control
algorithm when the time step delay is too large. For both the small delay case
and large delay case, we suggested the target density of the density control
algorithm to be a moderate value. Our numerical results are expected to
provide insights into designing the computer-aided guiding strategies in real
evacuations.

However, due to the difficulty of inviting human volunteers under the COVID
pandemic, conducting an actual experiment is expected in future. Moreover,
there may be many different heterogeneities in the evacuation process, such as
the psychological considerations (risk-averse/risk-seeking), quality of the
pedestrians, will on sharing visual field, to name but a few. The analysis on
those factors may be promised necessary future research directions. As a
preliminary study connecting the control theory and pedestrian evacuation
theory, we only used a simple and intuitive control algorithm for the
evacuation assistant system. However, there may exist possible future research
directions when we apply some novel control algorithms to the evacuation
process, such as model predictive control [42], system identification [43],
Q-learning for optimal control [44], noise elimination, etc.

## CRediT Authorship Contribution Statement

Fengqiang Gao: Investigation; formal analysis; writing- original draft
preparation, reviewing (equal). Zhihao Chen: Software; validation; reviewing
(equal). Yuyue Yan: Conceptualization; methodology; investigation; writing-
review $\&$ editing (lead). Linxiao Zheng: Visualization; reviewing (equal).
Huan Ren: Group management; reviewing (equal).

## Acknowledgments

This work was supported jointly by Program for Young Excellent Talents in
University of Fujian Province (201847) and China Scholarship Council
(201908050058). We thank Xie Chen for participating in a discussion in the
early state of this work.

## References

  * [1] Q. Meng, M. Zhou, J. Liu, H. Dong, Pedestrian evacuation with herding behavior in the view-limited condition, IEEE Trans. Comput. Soc. Syst. 6 (3) (2019) 567–575.  doi:https://doi.org/10.1109/TCSS.2019.2915772.
  * [2] Y. Cheng, X. Zheng, Emergence of cooperation during an emergency evacuation, Appl. Math. Comput. 320 (2018) 485–494.  doi:https://doi.org/10.1016/j.amc.2017.10.011.
  * [3] M. Haghani, E. Cristiani, N. W. Bode, M. Boltes, A. Corbetta, Panic, irrationality, and herding: three ambiguous terms in crowd dynamics research, J. Adv. Transp. 2019.  doi:https://doi.org/10.1155/2019/9267643.
  * [4] J. Guan, K. Wang, Towards pedestrian room evacuation with a spatial game, Appl. Math. Comput. 347 (2019) 492–501.  doi:https://doi.org/10.1016/j.amc.2018.11.003.
  * [5] W. Liao, X. Zheng, L. Cheng, Y. Zhao, Y. Cheng, Y. Wang, Layout effects of multi-exit ticket-inspectors on pedestrian evacuation, Saf. Sci. 70 (2014) 1–8.  doi:https://doi.org/10.1016/j.ssci.2014.04.015.
  * [6] T. Liu, X. Yang, Q. Wang, M. Zhou, S. Xia, A fuzzy-theory-based cellular automata model for pedestrian evacuation from a multiple-exit room, IEEE Access 8 (2020) 106334–106345.  doi:https://doi.org/10.1109/ACCESS.2020.3000606.
  * [7] A. Varas, M. Cornejo, D. Mainemer, B. Toledo, J. Rogan, V. Munoz, J. Valdivia, Cellular automaton model for evacuation process with obstacles, Physica A 382 (2) (2007) 631–642.  doi:https://doi.org/10.1016/j.physa.2007.04.006.
  * [8] G. G. Løvås, Modeling and simulation of pedestrian traffic flow, Transp. Res. Part B 28 (6) (1994) 429–443.  doi:https://doi.org/10.1016/0191-2615(94)90013-2.
  * [9] D. Helbing, I. Farkas, T. Vicsek, Simulating dynamical features of escape panic, Nature 407 (6803) (2000) 487–490.  doi:https://doi.org/10.1038/35035023.
  * [10] F. Müller, O. Wohak, A. Schadschneider, Study of influence of groups on evacuation dynamics using a cellular automaton model, Transp. Res. Rec. 2 (2014) 168–176.  doi:https://doi.org/10.1016/j.trpro.2014.09.022.
  * [11] D. Yanagisawa, K. Nishinari, Mean-field theory for pedestrian outflow through an exit, Phys. Rev. E 76 (2007) 061117.  doi:https://doi.org/10.1103/PhysRevE.76.061117.
  * [12] K. Huang, X. Zheng, Y. Cheng, Y. Yang, Behavior-based cellular automaton model for pedestrian dynamics, Appl. Math. Comput. 292 (2017) 417–424.  doi:https://doi.org/10.1016/j.amc.2016.07.002.
  * [13] R. Alizadeh, A dynamic cellular automaton model for evacuation process with obstacles, Saf. Sci. 49 (2) (2011) 315–323.  doi:https://doi.org/10.1016/j.ssci.2010.09.006.
  * [14] Z. Fu, X. Zhou, Y. Chen, J. Gong, F. Peng, Z. Yan, T. Zhang, L. Yang, The influence of random slowdown process and lock-step effect on the fundamental diagram of the nonlinear pedestrian dynamics: An estimating-correction cellular automaton, Commun. Nonlinear. Sci. Numer. Simul. 20 (3) (2015) 832–845.  doi:https://doi.org/10.1016/j.cnsns.2014.06.034.
  * [15] Z. Shahhoseini, M. Sarvi, Pedestrian crowd flows in shared spaces: investigating the impact of geometry based on micro and macro scale measures, Transp. Res. Part B 122 (2019) 57–87.  doi:https://doi.org/10.1016/j.trb.2019.01.019.
  * [16] X. Yang, X. Yang, F. Pan, Y. Kang, J. Zhang, The effect of passenger attributes on alighting and boarding efficiency based on social force model, Physica A 565 (2021) 125566.  doi:https://doi.org/10.1016/j.physa.2020.125566.
  * [17] M. Chraibi, A. Seyfried, A. Schadschneider, Generalized centrifugal-force model for pedestrian dynamics, Phys. Rev. E 82 (4) (2010) 046111.  doi:https://doi.org/10.1103/PhysRevE.82.046111.
  * [18] Y. Hao, Z. Bin-Ya, S. Chun-Fu, X. Yan, Exit selection strategy in pedestrian evacuation simulation with multi-exits, Chin. Phys. B 23 (5) (2014) 050512.  doi:https://doi.org/10.1088/1674-1056/23/5/050512.
  * [19] P. Hrabak, M. Bukacek, Influence of agents heterogeneity in cellular model of evacuation, J. Comput. Sci. 21 (2017) 486 – 493.  doi:https://doi.org/10.1016/j.jocs.2016.08.002.
  * [20] K. Huang, X. Zheng, A weighted evolving network model for pedestrian evacuation, Appl. Math. Comput. 298 (2017) 57–64.  doi:https://doi.org/10.1016/j.amc.2016.10.040.
  * [21] D.-F. Xie, Z.-Y. Gao, X.-M. Zhao, D. Z.-W. Wang, Agitated behavior and elastic characteristics of pedestrians in an alternative floor field model for pedestrian dynamics, Physica A 391 (7) (2012) 2390–2400.  doi:https://doi.org/10.1016/j.physa.2011.11.053.
  * [22] Z. Fu, Q. Jia, J. Chen, J. Ma, K. Han, L. Luo, A fine discrete field cellular automaton for pedestrian dynamics integrating pedestrian heterogeneity, anisotropy, and time-dependent characteristics, Transp. Res. Part C 91 (2018) 37–61.  doi:https://doi.org/10.1016/j.trc.2018.03.022.
  * [23] D. Miyagawa, G. Ichinose, Cellular automaton model with turning behavior in crowd evacuation, Physica A: Statistical Mechanics and its Applications (2020) 124376doi:https://doi.org/10.1016/j.physa.2020.124376.
  * [24] L. Lu, C.-Y. Chan, J. Wang, W. Wang, A study of pedestrian group behaviors in crowd evacuation based on an extended floor field cellular automaton model, Transp. Res. Part C 81 (2017) 317–329.  doi:https://doi.org/10.1016/j.trc.2016.08.018.
  * [25] X. Yang, H. Dong, Q. Wang, Y. Chen, X. Hu, Guided crowd dynamics via modified social force model, Physica A 411 (2014) 63–73.  doi:https://doi.org/10.1016/j.physa.2014.05.068.
  * [26] H. Ren, Y. Yan, F. Gao, Variable guiding strategies in multi-exits evacuation: Pursuing balanced pedestrian densities, Applied Mathematics and Computation 397 (2021) 125965.  doi:https://doi.org/10.1016/j.amc.2021.125965.
  * [27] X. Yang, H. Dong, X. Yao, X. Sun, Q. Wang, M. Zhou, Necessity of guides in pedestrian emergency evacuation, Physica A 442 (2016) 397–408.  doi:https://doi.org/10.1016/j.physa.2015.08.020.
  * [28] Y. Ma, R. K. K. Yuen, E. W. M. Lee, Effective leadership for crowd evacuation, Physica A 450 (2016) 333–341.  doi:https://doi.org/10.1016/j.physa.2015.12.103.
  * [29] Y. Ma, E. W. M. Lee, M. Shi, Dual effects of guide-based guidance on pedestrian evacuation, Mod. Phys. Lett. 381 (22) (2017) 1837–1844.  doi:https://doi.org/10.1016/j.physleta.2017.03.050.
  * [30] M. Zhou, H. Dong, P. A. Ioannou, Y. Zhao, F.-Y. Wang, Guided crowd evacuation: approaches and challenges, IEEE/CAA Journal of Automatica Sinica 6 (5) (2019) 1081–1094.  doi:https://doi.org/10.1109/JAS.2019.1911672.
  * [31] J. C. Chu, A. Y. Chen, Y.-F. Lin, Variable guidance for pedestrian evacuation considering congestion, hazard, and compliance behavior, Transp. Res. Part C 85 (2017) 664–683.  doi:https://doi.org/10.1016/j.trc.2017.10.009.
  * [32] M. Zhou, H. Dong, Y. Zhao, P. A. Ioannou, F.-Y. Wang, Optimization of crowd evacuation with leaders in urban rail transit stations, IEEE Trans. Intell. Transp. Syst. 20 (12) (2019) 4476–4487.  doi:10.1109/TITS.2018.2886415.
  * [33] S. Long, D. Zhang, S. Li, S. Yang, B. Zhang, Simulation-based model of emergency evacuation guidance in the metro stations of China, IEEE Access 8 (2020) 62670–62688.  doi:https://doi.org/10.1109/ACCESS.2020.2983441.
  * [34] X. Yang, X. Yang, Q. Wang, Y. Kang, F. Pan, Guide optimization in pedestrian emergency evacuation, Appl. Math. Comput. 365 (2020) 124711.  doi:https://doi.org/10.1016/j.amc.2019.124711.
  * [35] X. Yang, X. Yang, Q. Wang, Pedestrian evacuation under guides in a multiple-exit room via the fuzzy logic method, Commun. Nonlinear. Sci. Numer. Simul. 83 (2020) 105138.  doi:https://doi.org/10.1016/j.cnsns.2019.105138.
  * [36] T. Liu, Z.-P. Jiang, Event-based control of nonlinear systems with partial state and output feedback, Automatica 53 (2015) 10 – 22.  doi:https://doi.org/10.1016/j.automatica.2014.12.027.
  * [37] L. Wang, C. P. Chen, H. Li, Event-triggered adaptive control of saturated nonlinear systems with time-varying partial state constraints, IEEE Trans. Cybern. 50 (4) (2018) 1485–1497.  doi:https://doi.org/10.1109/TCYB.2018.2865499.
  * [38] C. K. Chen, J. Li, D. Zhang, Study on evacuation behaviors at a T-shaped intersection by a force-driving cellular automata model, Physica A 391 (7) (2012) 2408–2420.  doi:https://doi.org/10.1016/j.physa.2011.12.001.
  * [39] J. Tanimoto, A. Hagishima, Y. Tanaka, Study of bottleneck effect at an emergency evacuation exit using cellular automata model, mean field approximation analysis, and game theory, Physica A 389 (24) (2010) 5611–5618.  doi:https://doi.org/10.1016/j.physa.2010.08.032.
  * [40] F. Gao, Y. Yan, C. Xu, L. Lin, H. Ren, J. Pei, A new cellular automata model for pedestrian evacuation considering guiding effects, Journal of Transportation Systems Engineering and Information Technology 16 (6) (2016) 60–66.  doi:http://manu01.magtech.com.cn/Jwk_jtys/CN/Y2016/V16/I6/60.
  * [41] H. Yue, H. Guan, C. Shao, X. Zhang, Simulation of pedestrian evacuation with asymmetrical exits layout, Physica A 390 (2) (2011) 198–207.  doi:https://doi.org/10.1016/j.physa.2010.10.003.
  * [42] J. Köhler, R. Soloperto, M. A. Muller, F. Allgower, A computationally efficient robust model predictive control framework for uncertain nonlinear systems, IEEE Trans. Automat. Contr. 66 (2) (2020) 794–801.  doi:https://doi.org/10.1109/TAC.2020.2982585.
  * [43] A. Mauroy, J. Goncalves, Koopman-based lifting techniques for nonlinear systems identification, IEEE Trans. Automat. Contr. 65 (6) (2019) 2550–2565.  doi:https://doi.org/10.1109/TAC.2019.2941433.
  * [44] D. Lee, J. Hu, Primal-dual q-learning framework for LQR design, IEEE Trans. Automat. Contr. 64 (9) (2018) 3756–3763.  doi:https://doi.org/10.1109/TAC.2018.2884649.