# Consensus Formation Tracking for Multiple AUV Systems Using Distributed
Bioinspired
Sliding Mode Control

Tao Yan, , Zhe Xu, , Simon X. Yang This work was supported by the Natural
Sciences and Engineering Research Council (NSERC) of Canada.The authors are
with Advanced Robotics and Intelligent Systems (ARIS) Laboratory, School of
Engineering, University of Guelph, Guelph, ON. N1G2W1, Canada (e-mail: tyan03;
zxu02; syang@uoguelph.ca).Corresponding author: Simon X. Yang

###### Abstract

Consensus formation tracking of multiple autonomous underwater vehicles (AUVs)
subject to nonlinear and uncertain dynamics is a challenging problem in
robotics. To tackle this challenge, a distributed bioinspired sliding mode
controller is proposed in this paper. First, the conventional sliding mode
controller (SMC) is presented, and the consensus problem is addressed on the
basis of graph theory. Next, to tackle the high frequency chattering issue in
SMC scheme and meanwhile improve the robustness to the noises, a bioinspired
approach is introduced, in which a neural dynamic model is employed to replace
the nonlinear sign or saturation function in the synthesis of conventional
sliding mode controllers. Furthermore, the input-to-state stability of the
resulting closed-loop system is proved in the presence of bounded lumped
disturbance by the Lyapunov stability theory. Finally, simulation experiments
are conducted to demonstrate the effectiveness of the proposed distributed
formation control protocol.

###### Index Terms:

Bioinspired approach, consensus formation tracking, distributed sliding mode
control, graph theory, multiple AUVs.

##  I Introduction

Autonomous underwater vehicles (AUVs) are a class of marine mechatronic
systems capable of operating underwater automatically for a long period of
time. This type of machine enables human beings to have the abilities to
explore and exploit the marine resources. A great number of applications both
of commercial and military purposes are surged by employing such vehicles,
including oceanographic seabed mapping, gas and oil exploration, payload
delivery, etc. Thus, developing high performance AUV systems has been a
pressing topic in both control and ocean engineering communities [1, 2, 3, 4,
5, 6].

In recent decades, the number of AUVs is becoming larger in order to adapt to
more complex marine missions, such as large-scale mine sweeping, undersea
sampling and mapping, oil and gas exploration, etc. In addition to that,
multiple autonomous vehicle systems are proven to be able to collect more
robust data and exhibit more fault-tolerant capabilities than the single
vehicles [7]. Therefore, the coordination problem among multiple AUVs has
attracted much attention. Particularly, the formation control as the
fundamental problem in multiple vehicle coordination has been widely studied
[8]. However, due to the nonlinear and uncertain nature of AUV’s
characteristics as well as the complexity of underwater situations, steering a
fleet of unmanned marine vehicles to keep a desired formation configuration
precisely may be in front of many challenges [9, 10].

A large amount of work has been done to develop an effective formation control
protocol for a team of AUVs. The coordination strategies can typically be
classified as behavior-based approach, virtual structure, leader-follower
structure, and artificial potential approaches. In behavioral approaches [8,
11], decentralized control implementations are used to design local
controllers for each individual, where the control law is of the weighted form
to meet various control objectives, such as formation keeping, goal seeking,
path following, etc. Because of the separative feature of such approaches, it
provides a more flexible and natural way to synthesize controllers to satisfy
the collective behaviors. Nonetheless, the main drawback relies on the fact
that quite often it is hard to formulate the problem and provide the
theoretical guarantees. In virtual structure [12, 13], the desired formation
shape is viewed as a single rigid body, and the reference points are derived
for individuals according to the overall formation, by which the coordination
problem could be cast into a series of trajectory tracking problems for each
vehicle. Although this facilitates the design of formation controllers, the
cooperative performance is greatly sacrificed as a whole. To improve
cooperation among vehicles, the leader-follower structure is proposed, where
one or several vehicle is selected as the leader which is able to access the
desired formation and reference trajectory, and the rest of the vehicles are
treated as the followers with the sole objective to maintain the desired
posture (position and orientation) to the leader [14, 15]. Due to its
simplicity as well as the scalability to a larger group, such a structure has
been extensively employed in multi-robot system’s formation control. Another
class of widely used approach in multiple vehicle cooperation is referred to
as the artificial potential field, first developed to address the problems of
goal seeking as well as collision avoiding, and then adapted to the formation
control by designing proper potential functions for individuals which may
produce the corresponding potential fields enforcing the vehicles to maintain
a relative pose with their neighbors [16, 17]. Nevertheless, the biggest issue
of such approach is that it has chance to enter into the local minimum, which
may render the entire system malfunction.

In addition to the coordination strategies, there also are many of efforts
being put to improve the performance of motion control of AUVs in view of
strong nonlinear characteristics as well as complicated marine environments.
The robust optimal formation control for a fleet of marine vehicles subject to
communication delays was addressed based on the leader-follower structure
[18]. Adaptive robust formation control of AUVs was studied to handle the
time-varying environmental disturbances [19]. An adaptive sliding mode
formation control scheme was proposed based on the communication consensus to
allow for the variable added mass as well as the communication constraints
[20]. To tackle the chattering in sliding mode control, a novel reaching law
was designed, in which an adaptive gain was incorporated based on the
variation of the sliding surface [21]. Furthermore, taking into account the
unavailable velocity measure as well as reduction of the conservativeness, a
equivalent output injection adaptive sliding mode observer based terminal
sliding mode control approach was proposed for trajectory tracking of
underwater vehicles [22]. Formation control for underactuated AUVs was
addressed based on the leader-follower backstepping scheme; moreover, both
parametric uncertainty and unknown disturbance arising in dynamic model were
handled [23]. To further improve the formation accuracy, neural adaptive
formation control for multi-underactuated marine vehicles was studied via
employing a sliding mode controller with neural networks [24]. By mean of the
graph theory, a distributed consensus formation control protocol was proposed
[25]. Researchers developed a formation control policy based on a self-
organizing map neural network to obtain the coordinated behavior [26].
Nevertheless, the foregoing methods either addressed the 2-dimensional
simplified scenarios, or employed the kinematic or reduced dynamic model for
formation control synthesis. Considering higher dimensional and more
complicated dynamics, the simulation study on formation control for multiple
marine vehicles with full dynamics was addressed, and the gravity uncertainty
was also taken into account [7]. To alleviate the effects caused by data
transmission delays and packet dropouts and reduce the mutual information
exchange among the vehicles, line-of-sight measurement-based leader-follower
formation control was studied, and the backstepping controller was then
developed using Lyapunov’s method [27]. The receding horizontal formation
tracking control constrained with underactuated dynamics was investigated, and
the corresponding receding horizon controller was presented [28]. Formation
control for multiple underactuated underwater vehicles was addressed based on
the small gain method which allows to design decentralized controllers with
stability guarantees [29]. In order to achieve finite-time convergence and
meanwile adapt to the changeable dynamic parameters, a robust adaptive fast
terminal sliding mode formation controller was proposed for multi-
underactuated AUV systems [30].

Almost all of aforementioned research utilized the leader-following
coordinating structure to design the formation controller for each vehicle. As
we mentioned earlier, this type of approach was highly dependent on the
performance of the leading vehicle and was not able to obtain a good
coordination among the neighboring vehicles. As such, in order to further
improve the coordination performance, it is quite necessary to incorporate the
information of neighboring vehicles when synthesizing the control law.
Different from the existing work, this paper is concerned with consensus
formation tracking control in 3-dimensional space, where the vehicles’
formation is controlled not only relying on the information from leader, but
also their neighbors, and meanwhile it is also assumed that each vehicle may
suffer from the unknown disturbances either because of hydrodynamic-related
phenomena or time-varying marine environments. On the other hand, it is well
known that the sliding mode control (SMC) owing to its robustness and
simplicity has been attracted much attention in the area of robotics and
networked systems, as shown in above literature. The readers are recommended
to refer to the survey paper [31] for more detailed review on potential
applications of SMC schemes in networked systems. As a result, the goal of
this paper is to develop a distributed sliding mode control protocol for
multi-AUV systems to achieve a good formation tracking performance, and
particularly taking into consideration that the conventional SMC scheme is
always suffering from the chattering issue as well as the vulnerability to the
noises, a novel bioinspired handling method is employed. The main
contributions of this paper are listed as follows:

  1. 1.

Distributed formation tracking control of multi-AUVs subject to the
hydrodynamic parameter uncertainties and unknown exogenous disturbances in
3-dimensional space is addressed, and particularly full dynamics of the AUVs
is adopted.

  2. 2.

Consensus problem is accounted for in the design of distributed formation
controller to improve the coordination performance of the overall formation
system.

  3. 3.

A novel distributed bioinspired sliding mode control protocol is developed to
achieve the considered control objectives, where the control law is realized
in a fully distributed manner. Meanwhile, the chattering phenomena and non-
smoothness arising in the conventional SMC scheme are completely eliminated,
rendering a more practical control activity, and the robustness to the noises
is also enhanced.

  4. 4.

Rigorous stability analysis is provided based on the Lyapunov theory to
theoretically guarantee the anticipated performance of proposed distributed
formation control protocol.

The rest of the paper is arranged as follows. The problem formulation and
preliminary are presented in the next section. In Section III, bioinspired
sliding mode controller is derived, and the stability analysis of resulting
system is discussed. The simulation results are shown in Section IV. Section V
draws the conclusion.

##  II PRELIMINARY AND PROBLEM FORMULATION

In this section, the mathematical model of AUVs in three-dimensional space is
presented, and the objective of formation control for multi-AUV system is
described. Moreover, the preliminary knowledge on graph theory is also
presented.

###  II-A Preliminary on graph theory

For the considered group of underwater vehicles, it is assumed that each
vehicle can be viewed as a node, and the communication topology associated
with the networked system among multiple AUVs can then be represented by a
weighted directed graph $G=\\{{V,E,A}\\}$. As for a simple time-invariant
graph $G$, it is composed of the vertex set
$V=\\{{\nu_{1},\nu_{2},\ldots,\nu_{N}}\\}$ , the edge set $E\subseteq V\times
V$, and the weighted adjacency matrix
$A=\left[{{a_{ij}}}\right]\in\mathbb{R}^{\text{N}\times\text{N}}$. The element
$\nu_{i}$ in vertex set $V$ denotes vehicle $i$, and the index $i$ belongs to
a finite index set $\Gamma=\left\\{{1,\ldots,N}\right\\}$. If there exists the
information exchange between AUV $i$ and AUV $j$, then, say, there is a edge
between AUVs $i$ and $j$, i.e., $\left({{\nu_{i}},{\nu_{j}}}\right)\in E$, and
${a_{ij}}={a_{ji}}>0$. Particularly, call vehicle $j$ a neighbor of vehicle
$i$, and the set of neighbors is represented by
${N_{i}}=\left\\{{j|{\left({{\nu_{i}},{\nu_{j}}}\right)\in E}}\right\\}$.
Otherwise, there is no edge among them, and $a_{ij}=a_{ji}=0$. Moreover, we
define $a_{ii}=0$ for all $i\in\Gamma$, and the out-degree $d_{i}=\sum_{j\in
N_{i}}{a_{ij}}$ associated with the node $i$. Afterwards, the degree matrix
together with the Laplacian matrix of the graph $G$ can then be defined as
$D=\text{diag}\left\\{{d_{1},\ldots,d_{N}}\right\\}\in\mathbb{R}^{\text{N}\times\text{N}}$
and $L=D-A$, respectively. A path in graph is a sequence that is consisted of
a set of successive adjacent nodes, starting from node $i$ and ending at node
$j$. If any two nodes in a graph $G$ have at least one path, then, say, graph
$G$ is connected.

In order to make the group of AUVs move along with the desired path as a
whole, a reference trajectory must be defined ahead of time. The availability
to the information of reference trajectory for $i$-th AUV is indicated by a
parameter $b_{i}$; that is, if AUV $i$ is able to access this information,then
$b_{i}>0$; $b_{i}=0$, otherwise. Let
$B=\text{diag}\left\\{{b_{1},\ldots,b_{N}}\right\\}$.

######  Assumption 1.

For the considered multi-AUV formation control network, graph $G$ is
connected, and moreover there is at least one AUV able to receive the
information of reference trajectory, i.e., the elements of $B$ are not all
equal to zero.

######  Lemma 1.

if Assumption 1 holds, then matrix $L+B$ is positive definite.

Figure 1: Schematic diagram of $i$-th AUV .

###  II-B Problem formulation

The formation control of $N$ numbers of autonomous underwater vehicles is
investigated in this article. As presented in the work of Yang et al. [32],
the dynamics of $i$-th AUV $(i\in\Gamma)$ can be modeled as

$\begin{split}&\dot{\eta}_{i}=J_{i}(\eta_{2,i})v_{i},\\\
&M_{i}\dot{v}_{i}+C_{i}(v_{i})v_{i}+D_{i}(v_{i})v_{i}=\tau_{i}+d_{i},\end{split}$
(1)

where $\eta_{i}=\left[\eta_{1,i}\eta_{2,i}\right]^{\rm
T}\in\mathbb{R}^{6\times 1}$, $\eta_{1,i}=\left[x_{i},y_{i},z_{i}\right]^{\rm
T}\in\mathbb{R}^{3\times 1}$,
$\eta_{2,i}=\left[\phi_{i},\theta_{i},\psi_{i}\right]^{\rm
T}\in\mathbb{R}^{3\times 1}$ denote the position and orientation of $i$-th
AUV, respectively, which are expressed in the Earth-fixed frame $\hat{E}^{\rm
I}=\left\\{{\hat{e}_{o}^{\rm I},\hat{e}_{x}^{\rm I},\hat{e}_{y}^{\rm
I},\hat{e}_{z}^{\rm I}}\right\\}$, and
$v_{i}=\left[v_{1,i},v_{2,i}\right]^{\rm T}\in\mathbb{R}^{6\times 1}$,
$v_{1,i}=\left[v_{x,i},v_{y,i},v_{z,i}\right]^{\rm T}\in\mathbb{R}^{3\times
1}$, $v_{2,i}=\left[\omega_{x,i},\omega_{y,i},\omega_{z,i}\right]^{\rm
T}\in\mathbb{R}^{3\times 1}$ are the $i$-th AUV’s translational and rotational
velocities, respectively, which are described in vehicle’s body-fixed frame
$\hat{E}^{\rm B}=\left\\{{\hat{e}_{o,i}^{\rm B},\hat{e}_{x,i}^{\rm
B},\hat{e}_{y,i}^{\rm B},\hat{e}_{z,i}^{\rm B}}\right\\}$. The schematic of
$i$-th AUV is illustrated in Fig. 1. $J_{i}(\eta_{2,i})\in\mathbb{R}^{6\times
6}$ denotes the coordinate-transform Jacobian matrix whose entries are time-
varying. $M_{i}\in\mathbb{R}^{6\times 6}$ is the inertia matrix;
$C_{i}(v_{i})\in\mathbb{R}^{6\times 6}$ is the Coriolis and centripetal
matrix, and $D_{i}(v_{i})\in\mathbb{R}^{6\times 6}$ is the hydrodynamic
damping matrix. $\tau_{i}\in\mathbb{R}^{6\times 1}$ denotes the generalized
control input vector of $i$-th AUV, and $d_{i}\in\mathbb{R}^{6\times 1}$ is
the lumped disturbance, describing both the hydrodynamic parameter
uncertainties as well as the maritime disturbances induced by the wind, waves
and ocean currents. Jacobian matrix is defined as
$J_{i}(\eta_{2,i})=\text{diag}\left\\{{J_{1,i}(\eta_{2,i}),J_{2,i}(\eta_{2,i})}\right\\}$,
and it satisfies

$J_{1,i}(\eta_{2,i})=\left[\begin{array}[]{cc}\cos{\theta_{i}}\cos{\psi_{i}}&\sin{\phi_{i}}\sin{\theta_{i}}\cos{\psi_{i}}-\cos{\phi_{i}}\sin{\psi_{i}}\\\
\cos{\theta_{i}}\sin{\psi_{i}}&\cos{\phi_{i}}\cos{\psi_{i}}+\sin{\phi_{i}}\sin{\theta_{i}}\sin{\psi_{i}}\\\
-\sin{\theta_{i}}&\sin{\phi_{i}}\cos{\theta_{i}}\end{array}\right.$
$\left.\begin{array}[]{c}\sin{\phi_{i}}\sin{\psi_{i}}+\cos{\phi_{i}}\sin{\theta_{i}}\cos{\psi_{i}}\\\
\cos{\phi_{i}}\sin{\theta_{i}}\sin{\psi_{i}}+\cos{\phi_{i}}\sin{\psi_{i}}\\\
\cos{\phi_{i}}\cos{\theta_{i}}\end{array}\right],$
$J_{2,i}(\eta_{2,i})=\begin{bmatrix}1&\sin{\phi_{i}}\tan{\theta_{i}}&\cos{\phi_{i}}\tan{\theta_{i}}\\\
0&\cos{\phi_{i}}&-\sin{\phi_{i}}\\\
0&\sin{\phi_{i}}/\cos{\theta_{i}}&\cos{\phi_{i}}/\cos{\theta_{i}}\end{bmatrix}.$

In order to circumvent the singularity of the Jacobian matrix, it is required
to guarantee $\left|{\theta_{i}}\right|\neq\pi/2$. The vessels’ inertia matrix
$M_{i}$ with added mass and the hydrodynamic damping matrix $D_{i}$ are both
diagonal and satisfy

$\displaystyle M_{i}$
$\displaystyle=\text{diag}\left\\{{m_{i}-\beta_{\dot{v}x,i},m_{i}-\beta_{\dot{v}y,i},m_{i}-\beta_{\dot{v}z,i},I_{x,i},I_{y,i},I_{z,i}}\right\\},$
$\displaystyle D_{i}$
$\displaystyle=-\text{diag}\left\\{{\beta_{vx,i},\beta_{vy,i},\beta_{vz,i},\beta_{\omega
x,i},\beta_{\omega y,i},\beta_{\omega z,i}}\right\\},$

where $m_{i}$ and $I_{x,i}$, $I_{y,i}$, $I_{z,i}$ are the mass and rotational
inertia of the $i$-th AUV, respectively; $\beta_{\dot{v}x,i}$,
$\beta_{\dot{v}y,i}$, $\beta_{\dot{v}z,i}$, $\beta_{vx,i}$, $\beta_{vy,i}$,
$\beta_{vz,i}$, $\beta_{\omega x,i}$, $\beta_{\omega y,i}$ and $\beta_{\omega
z,i}$ are the hydrodynamic parameters. The matrix $C_{i}(v_{i})$ is made up of
the Coriolis and centripetal matrix $C_{R,i}(v_{i})$ as well as the
hydrodynamic extra term $C_{A,i}(v_{i})$, defined as

$\displaystyle C_{i}(v_{i})$ $\displaystyle=C_{R,i}(v_{i})+C_{A,i}(v_{i}),$
$\displaystyle=\begin{bmatrix}\mathbf{0}_{3\times 3}&C_{R1,i}(v_{i})\\\
C_{R1,i}(v_{i})&C_{R2,i}(v_{i})\end{bmatrix}+\begin{bmatrix}\mathbf{0}_{3\times
3}&C_{A1,i}(v_{i})\\\ C_{A1,i}(v_{i})&C_{A2,i}(v_{i})\end{bmatrix},$

where

$\displaystyle C_{R1,i}(v_{i})$
$\displaystyle=\begin{bmatrix}0&m_{i}v_{z,i}&-m_{i}v_{y,i}\\\
-m_{i}v_{z,i}&0&m_{i}v_{x,i}\\\ m_{i}v_{y,i}&-m_{i}v_{x,i}&0\end{bmatrix},$
$\displaystyle C_{R2,i}(v_{i})$
$\displaystyle=\begin{bmatrix}0&I_{z,i}\omega_{z.i}&-I_{y,i}\omega{y,i}\\\
-I_{z,i}\omega_{z.i}&0&I_{x,i}\omega{x,i}\\\
I_{y,i}\omega{y,i}&-I_{x,i}\omega{x,i}&0\end{bmatrix},$ $\displaystyle
C_{A1,i}(v_{i})$
$\displaystyle=\begin{bmatrix}0&\beta_{\dot{v}z,i}v_{z,i}&-\beta_{\dot{v}y,i}v_{y,i}\\\
-\beta_{\dot{v}z,i}v_{z,i}&0&\beta_{\dot{v}x,i}v_{x,i}\\\
\beta_{\dot{v}y,i}v_{y,i}&-\beta_{\dot{v}x,i}v_{x,i}&0\end{bmatrix},$
$\displaystyle C_{A2,i}(v_{i})$
$\displaystyle=\begin{bmatrix}0&-\beta_{\dot{\omega}z,i}\omega_{z,i}&\beta_{\dot{\omega}y,i}v_{y,i}\\\
\beta_{\dot{\omega}z,i}\omega_{z,i}&0&-\beta_{\dot{\omega}x,i}\omega_{x,i}\\\
-\beta_{\dot{\omega}y,i}\omega_{y,i}&\beta_{\dot{\omega}x,i}\omega_{x,i}&0\end{bmatrix},$

and ${\beta_{\dot{\omega}x,i}}$, ${\beta_{\dot{\omega}y,i}}$,
${\beta_{\dot{\omega}z,i}}$ are the hydrodynamic parameters.

######  Remark 1.

It is worthwhile noting that the dynamics of AUV is a typical nonlinear system
as described in (1), having 6 degree of freedom (DOF), i.e., 3 translational
DOF and 3 rotational DOF, and subject to the external disturbances as well as
numerous uncertain hydrodynamic related parameters.

In formation control problem, a desired formation pattern of group of AUVs is
formed uniquely through the specific relative postures, i.e., positions and
orientations, between the vehicles $i$ and $j$, $(i$, $j\in\Gamma)$; let the
desired postures
$\Delta_{ij}=\left[\delta_{x,ij},\delta_{y,ij},\delta_{z,ij},\delta_{\phi,ij},\delta_{\theta,ij},\delta_{\psi,ij}\right]^{\rm
T}\in\mathbb{R}^{6\times 1}$. In most practical applications, the orientation
of each AUV is required to be aligned; that is, the relative attitudes
$\left[\delta_{\phi,ij},\delta_{\theta,ij},\delta_{\psi,ij}\right]^{\rm T}$ of
AUVs are set to $\mathbf{0}_{3\times 1}$. In addition to the formation
maintenance, a team of vehicles are also employed to track a desired
trajectory in many maneuver missions. Let
$\eta_{1}^{d}=\left[x^{d},y^{d},z^{d}\right]^{\rm T}\in\mathbb{R}^{3\times 1}$
be the prescribed differentiable trajectory of the formation center, which can
be viewed as a virtual leader in the group. Based on the defined formation
pattern, i.e., $\Delta_{ij}$, each vehicle’s desired trajectory
$\eta_{1,i}^{d}$ associated with the virtual leader can then be derived
accordingly. $\eta_{2}^{d}=\left[\phi^{d},\theta^{d},\psi^{d}\right]^{\rm
T}\in\mathbb{R}^{3\times 1}$ denotes the desired differentiable attitude of
the virtual leader, and combining position and orientation, let
$\eta^{d}=\left[\eta_{1}^{d},\eta_{2}^{d}\right]^{\rm T}$ be the desired
posture of virtual leader. The objective of this paper is aimed to design a
distributed formation control law for each individual in the group with full
nonlinear dynamics and affected by the disturbances which are described in (1)
so that the coordinated motion among AUVs can be achieved; in particular, the
following control requirements are satisfied: 1) The AUVs can maintain a
desired formation shape. 2) The group of AUVs can follow up a predefined
trajectory as a whole.

##  III FORMATION CONTROL PROTOCOL DESIGN

In this section, systematic design procedures for achieving solution for
consensus formation control are presented. First, the distributed sliding
model controller is designed. Next, to eliminate the impacts of high-frequency
oscillation that is caused by the employment of sign function, a bioinspired
neurodynamics is introduced to avoid the use of nonlinear operation in control
design. The main advantage of proposed controller is that it can provide
smoother control inputs, which is much more significant for the actuators to
generate realistic forces or moments. At the same time, a good performance in
terms of disturbance rejection is still preserved. Finally, the input-to-state
stability of overall closed-loop system is proved.

###  III-A Sliding mode control design

The consensus cooperative tracking errors for $i$-th AUV $(i\in\Gamma)$ are
defined as

$\begin{split}e_{1,i}&=\sum\limits_{j\in
N_{i}}{a_{ij}(\eta_{i}-\eta_{j}-\Delta_{ij})+b_{i}(\eta_{i}-\eta_{i}^{d})},\\\
\dot{e}_{1,i}&=\sum\limits_{j\in
N_{i}}{a_{ij}(\dot{\eta}_{i}-\dot{\eta}_{j})+b_{i}(\dot{\eta}_{i}-\dot{\eta}_{i}^{d})}.\end{split}$
(2)

Here, the non-negative constant $a_{ij}$ indicates the information exchange
between $i$-th vehicle and its neighbors $j\in N_{i}$, and another non-
negative constant $b_{i}$ denotes whether or not $i$-th vehicle is able to
access the information of the desired reference trajectory. $\Delta_{ij}$
denotes the relative posture (position and orientation) between $i$-th AUV and
its neighbors, which determines the desired formation pattern needed to
maintain. Note that since the whole group is intended to track the common
reference trajectory, the desired velocities and accelerations of AUVs are the
same; that is, $\dot{\eta}_{i}^{d}=\dot{\eta}^{d}$, and
$\ddot{\eta}_{i}^{d}=\ddot{\eta}^{d}$, $(i\in\Gamma)$, where $\dot{\eta}^{d}$
and $\ddot{\eta}^{d}$ represent the desired velocities and accelerations of
the whole group, respectively.

Taking the time derivative of $\dot{e}_{1,i}$, the following error dynamics
for $i$-th AUV can be obtained

${\ddot{e}_{1,i}}=\sum\limits_{j\in{N_{i}}}{{a_{ij}}\left({{{\ddot{\eta}}_{i}}-{{\ddot{\eta}}_{j}}}\right)+{b_{i}}\left({{{\ddot{\eta}}_{i}}-\ddot{\eta}^{d}}\right)},$
(3)

where $\ddot{\eta}_{i}$ and $\ddot{\eta}_{j}$ represent the dynamics of $i$-th
AUV and its neighbors $j\in{N_{i}}$, respectively. According to the $i$ AUV’s
kinematic and dynamic model as described in (1), the expression of
$\ddot{\eta}_{i}$ can be derived as follows

${\ddot{\eta}_{i}}={A_{i}}\left({{v_{i}},{\eta_{i}}}\right){v_{i}}+{J_{i}}\left({{\eta_{i}}}\right){B_{i}}{\tau_{i}}+{J_{i}}\left({{\eta_{i}}}\right){B_{i}}{d_{i}},$
(4)

where

${A_{i}}\left({{v_{i}},{\eta_{i}}}\right)={\dot{J}_{i}}\left({{\eta_{i}}}\right)-{J_{i}}\left({{\eta_{i}}}\right){B_{i}}{C_{i}}\left({{v_{i}}}\right)-{J_{i}}\left({{\eta_{i}}}\right){B_{i}}{D_{i}}\left({{v_{i}}}\right),$

and ${B_{i}}={M_{i}}^{-1}$.

######  Assumption 2.

The disturbance term ${J_{i}}\left({{\eta_{i}}}\right){B_{i}}{d_{i}}$ can be
bounded as

$\left\|{{J_{i}}\left({{\eta_{i}}}\right){B_{i}}{d_{i}}}\right\|\leq{\tilde{\lambda}_{i}},\quad
i\in\Gamma,$ (5)

where $\tilde{\lambda}_{i}$ is a known positive constant. This parameter
typically depends on the marine situations.

Letting

$\begin{split}{{\bar{e}}_{1}}&={\left[{{e_{1,1}},{e_{1,2}},\ldots,{e_{1,N}}}\right]^{\rm
T}},\\\
{{\bar{e}}_{2}}&={\left[{{{\dot{e}}_{1,1}},{{\dot{e}}_{1,2}},\ldots,{{\dot{e}}_{1,N}}}\right]^{\rm
T}},\end{split}$ (6)

and

$\ddot{\eta}={\left[{{{\ddot{\eta}}_{1}},{{\ddot{\eta}}_{2}},\ldots,{{\ddot{\eta}}_{N}}}\right]^{\rm
T}},$ (7)

the global error dynamics of overall system can be written as

$\begin{split}{{\dot{\bar{e}}}_{1}}&={{\bar{e}}_{2}},\\\
{{\dot{\bar{e}}}_{2}}&=\left({L+B}\right)\left({\ddot{\eta}-{{\mathbf{1}}_{N}}{{\ddot{\eta}}^{d}}}\right),\end{split}$
(8)

where matrices $L$ and $B$ are defined in II-A, which describes the
communication topology of entire system. ${{\mathbf{1}}_{N}}$ represents a
$N$-dimensional vector whose entries are all equal to $1$.

The consensus formation control objective, now, is cast into designing a
distributed control law so that the global error dynamics (8) can be
stabilized at the origin. Next, sliding mode control design is employed and
can be divided into the following two steps.

Step1: Define the sliding mode surface

Let

$s={\left[{{s_{1}},{s_{2}},\ldots{s_{N}}}\right]^{\rm T}},$ (9)

and the sliding mode variable $s$ is defined as

$s={k_{1}}\left({L+B}\right){\bar{e}_{1}}+{\bar{e}_{2}},$ (10)

where $k_{1}$ is a positive constant needed to be designed. As we can see, so
long as the sliding mode variable $s$ is able to reach to $\mathbf{0}$ in
finite time, i.e., $s\left(t\right)\equiv 0$, $t\geq T$, say, system reached
the sliding mode surface, the behavior of $\bar{e}_{1}$ then will be solely
governed by the sliding mode surface
$s=\dot{\bar{e}}_{1}+{k_{1}}\left({L+B}\right){\bar{e}_{1}}=0$ in all future
time, which leads to $\bar{e}\left(t\right)\to 0$, as $t\to\infty$. In other
words, the system’s behaviour will be firmly restricted into the surface and
sliding on it until $\bar{e}\left(t\right)\to 0$. Most importantly, as we
observed there is no disturbance terms enforced on the sliding mode surface
while the original system dynamics may be subject to all kinds of
disturbances, and thus robustness properties can be achieved once system
enters into the sliding mode surface. At next step, our goal is to design
sliding mode control law to render system reach to the sliding mode surface
and stay on it in all finite time.

Step2: Design sliding mode control law

In this step, it is shown that the sliding mode surface can be reached in
finite time, and the system will stay at the surface for all future time.

Choose the following Lyapunov function candidate

${V_{s}}=\frac{1}{2}{s^{\rm T}}{\left({L+B}\right)^{-1}}s,$ (11)

whose time derivative is calculated as

${\dot{V}_{s}}={s^{\rm T}}{\left({L+B}\right)^{-1}}\dot{s}.$ (12)

Based on the definition of sliding mode variable $s$, as shown in (10),
$\dot{s}$ can be obtained

$\dot{s}={k_{1}}(L+B){\bar{e}_{2}}+{\dot{{\bar{e}}}_{2}}.$ (13)

Due to the global error dynamics (8), yield

$\dot{s}={k_{1}}(L+B){\bar{e}_{2}}+\left({L+B}\right)\left({\ddot{\eta}-{{\mathbf{1}}_{N}}{{\ddot{\eta}}^{d}}}\right).$
(14)

By substituting (14) into (12), we have

${\dot{V}_{s}}={s^{\rm
T}}{\left({L+B}\right)^{-1}}\left[{{k_{1}}\left({L+B}\right){{\bar{e}}_{2}}+\left({L+B}\right)\left({\ddot{\eta}-{{\mathbf{1}}_{N}}{{\ddot{\eta}}^{d}}}\right)}\right].$
(15)

Let

$\displaystyle\bar{A}\left({v,\eta}\right)=\text{diag}\left\\{{{A_{1}}\left({{v_{1}},{\eta_{1}}}\right),\ldots,{A_{1}}\left({{v_{N}},{\eta_{N}}}\right)}\right\\},$
$\displaystyle\bar{B}=\text{diag}\left\\{{{B_{1}},\ldots,{B_{N}}}\right\\},$
$\displaystyle\bar{J}\left(\eta\right)=\text{diag}\left\\{{{J_{1}}\left({{\eta_{1}}}\right),\ldots,{J_{N}}\left({{\eta_{N}}}\right)}\right\\},$
$\displaystyle\tau={\left[{{\tau_{1}},{\tau_{2}},\ldots{\tau_{N}}}\right]^{\rm
T}},$

and propose the following distributed control law

$\tau={\left[{\bar{J}\left(\eta\right)\bar{B}}\right]^{-1}}\left[{-\bar{A}\left({v,\eta}\right)v+\tau^{\prime}}\right],$
(16)

where $\tau^{\prime}$ is an auxiliary control variable and is designed as

$\tau^{\prime}=-{k_{1}}{\bar{e}_{2}}+{{\bf{1}}_{N}}{\ddot{\eta}^{d}}-{\beta_{0}}\text{sign}\left(s\right),$
(17)

where $\beta_{0}$ is a positive designable constant whose value is dependent
on the upper bound of disturbances and is needed to satisfy with
${\beta_{0}}\geq{\sup_{i\in\Gamma}}\left\\{{{{\tilde{\lambda}}_{i}}}\right\\}+\varepsilon$,
where $\varepsilon$ is an arbitrary positive constant.

Combing with AUV’s dynamics (4) as well as (7) and plugging (16) into the (15)
result in

$\dot{V}_{s}={s^{\rm
T}}\left[{{k_{1}}{{\bar{e}}_{2}}-{{\bf{1}}_{N}}{{\ddot{\eta}}^{d}}+\tau^{\prime}+\bar{J}\left(\eta\right)\bar{B}d}\right],$
(18)

where $d={\left[{{d_{1}},\ldots{d_{N}}}\right]^{\rm T}}$. Substituting the
auxiliary control variable (17) into (18), we may end up with

$\begin{split}{{\dot{V}}_{s}}&={s^{\rm
T}}\left[{-{\beta_{0}}sign\left(s\right)+\bar{J}\left(\eta\right)\bar{B}d}\right]\\\
&\leq-\sum\limits_{i=1}^{N}{({\beta_{0}}-\left\|{{J_{i}}\left({{\eta_{i}}}\right){B_{i}}{d_{i}}}\right\|)\left|{{s_{i}}}\right|}\\\
&\leq-\sum\limits_{i=1}^{N}\varepsilon\left|{{s_{i}}}\right|\leq-\sqrt{\frac{2}{{{\lambda_{\max}}{{\left({L+B}\right)}^{-1}}}}}{V^{1/2}}.\end{split}$
(19)

By taking integral on both sides, we can conclude

$\sqrt{{V_{s}}\left(t\right)}\leq-\sqrt{\frac{1}{{2{\lambda_{\max}}{{\left({L+B}\right)}^{-1}}}}}t+{V_{s}}\left(0\right).$
(20)

Above inequality shows that under proposed sliding mode control law (16) and
(17) the sliding variable $s$ will reach to zero in finite time and stay there
for all future time, i.e., $s\equiv 0$, $t\geq T$, where

$T=\sqrt{2{\lambda_{\max}}{{\left({L+B}\right)}^{-1}}}{V_{s}}\left(0\right).$
(21)

Once system is sliding on the surface $s\equiv 0$, the overall behavior of the
system is then determined by the linear differential equation

${\dot{\bar{e}}}_{1}+{k_{1}}\left({L+B}\right){\bar{e}_{1}}=0.$

Moreover, as the Lemma 1 shows $L+B$ is a positive definite matrix, it follows
that ${\bar{e}_{1}}\to 0$ as $t\to\infty$. When ${\bar{e}_{1}}\to 0$,
${\dot{\bar{e}}}_{1}$ will also tend to $0$; then, according to (8),
${\bar{e}_{2}}\to 0$ as well. That is, the error dynamics (8) is
asymptotically stable and its converging rate is dependent on the $k_{1}$.

###  III-B Bioinspired control design

Last section we presented the design process of sliding mode control for
consensus formation control of multiple AUV systems, and the asymptotical
stability is achieved even in the presence of unknown bounded disturbances,
which actually relies on the fact that the nonlinear sign function is employed
in SMC control law. Theoretically, this type of control is able to handle any
matched bounded disturbances enforced on the system. However, the resulting
high frequency non-smooth control activities can hardly be realized in reality
due to the physical limitations on actuators. Thus, the theoretical robust
performance cannot always be obtained in applications; furthermore, the
stability of overall system might be affected due to the excitation of high-
frequency unmodeled dynamics. As a result, a neural dynamics-based control
approach is proposed and studied to address this critical issue encountered in
conventional sliding mode control.

In this section a neurodynamic model, termed shunting model, is employed to
take place of the classic nonlinear sign function from which the chattering
originates in conventional SMC scheme so that the use of discontinuous control
is avoided and chattering phenomenon is completely vanishing. In fact, we will
show that due to the integration with the shunting model many favorable
properties can be obtained in the resulting formation system.

Shunting model is one of the most popular bioinspired dynamic models that was
first proposed in the work of [33], which was built to try to characterize the
adaptive behavior of membrane to the ever changing environment. Due to its
desirable characteristics, in the past few decades it has been incorporated in
various robotic systems to develop efficient algorithms for real-time path
planning as well as feedback control [34, 35, 36, 37, 38]. The shunting
equation can be described as follows

${\dot{x}_{i}}=-{a_{i}}{x_{i}}+\left({{b_{i}}-{x_{i}}}\right)s_{i}^{+}-\left({{d_{i}}+{x_{i}}}\right)s_{i}^{-},$
(22)

where $x_{i}$ represents the neural activity of $i$-th neuron, associated with
some nonnegative constant parameters $a_{i}$, $b_{i}$ and $d_{i}$ indicating
the rate of response, the maximum and minimum values of the neuron activities,
respectively. Signals $s_{i}^{+}$ and $s_{i}^{-}$ represent the effects of
environmental changes, i.e., the corresponding excitatory and inhibitory
inputs to the neuron, respectively.

######  Remark 2.

It is interesting to note that the presented shunting equation describes a
dynamic characteristic of a neuron, and thus a more consistent behavior of the
neural activities can be obtained, while the environmental changes may exhibit
a highly evident discontinuity or non-smoothness. In particular, the transient
stage can be adjustable by the parameter $a_{i}$, and furthermore, due to the
nonlinearity nature of the shunting model larger environmental changes may
result in a faster transient process. Besides of that, the outputs of the
shunting model, i.e., the neural activities, are able to be bounded upper by
the parameter $b_{i}$ and lower by the parameter $d_{i}$. It will be shown, in
what follows, that such properties of the shunting model can be applied to
significantly improve the performance of the conventional SMC scheme.

In order to apply the shutting model to formation control design, let

$\begin{split}s_{i}^{+}&=\begin{cases}s_{i},&\text{if}\ s_{i}\geq{0},\\\
{0},&{\text{otherwise}}.\end{cases}\\\
s_{i}^{-}&=\begin{cases}-s_{i},&\text{if}\ s_{i}<{0},\\\
{0},&{\text{otherwise}}.\end{cases}\end{split}$ (23)

Define ${s^{+}}={\left[{s_{1}^{+},\ldots s_{N}^{+}}\right]^{\rm T}}$ and
${x_{s}}={\left[{{x_{1}},\ldots{x_{N}}}\right]^{\rm T}}$, and the shunting
model-based SMC control law can be proposed as follows

$\begin{split}&\tau={\left[{\bar{J}\left(\eta\right)\bar{B}}\right]^{-1}}\left[{-\bar{A}\left({v,\eta}\right)v+\tau^{\prime}}\right],\\\
&\tau^{\prime}=-{k_{1}}{{\bar{e}}_{2}}+{{\bf{1}}_{N}}{{\ddot{\eta}}^{d}}-{\beta_{0}}{x_{s}},\\\
&{{\dot{x}}_{s}}=-{A_{s}}{x_{s}}+\left({{B_{s}}-{x_{s}}}\right){s^{+}}-\left({{D_{s}}+{x_{s}}}\right){s^{-}},\\\
&s={k_{1}}\left({L+B}\right){{\bar{e}}_{1}}+{{\bar{e}}_{2}}.\end{split}$ (24)

Here, ${A_{s}}=\text{diag}\left\\{{{a_{1}},\ldots,{a_{N}}}\right\\}$,
${B_{s}}=\text{diag}\left\\{{{b_{1}},\ldots,{b_{N}}}\right\\}$ and
${D_{s}}=\text{diag}\left\\{{{d_{1}},\ldots,{d_{N}}}\right\\}$.

######  Remark 3.

It is observed that the output of the shunting model, i.e., $x_{s}$, is used
in the auxiliary control law $\tau^{\prime}$, instead of the original
switching function as shown in (17). As a result, due to the benefits of the
dynamic characteristic of the shunting equation, as mentioned previously in
Remark 2, a more consistent and smoother control activities can be rendered,
and thus the chattering issue is avoided completely. Additionally, the output
of the shunting model is also allowed to be bounded. Such observations
demonstrate the significant improvements for the conventional SMC approach for
real-world applications, that is, the proposed bioinspired SMC scheme can lead
to a chattering-free behavior and yield more practical control actions for
actual actuators to implement.

######  Remark 4.

It is worth noting that while the robust asymptotical stability of the SMC
scheme along the sliding mode surface is no longer held, owing to the
employment of shunting model, the resulting control strategy can still allow
to achieve a good robustness against the bounded disturbances as well as the
noises, which can be seen clearly in the next section on stability analysis of
the proposed formation system.

###  III-C Stability analysis

To analyze the stability of overall AUV formation system, plugging the
proposed distributed controller (24) into the global error dynamics (8)
together with (4), (6) and (7), the following closed-loop system then can be
obtained

$\displaystyle{\dot{\bar{e}}}_{1}=-{k_{1}}\left({L+B}\right){\bar{e}_{1}}+s,$
(25)
$\displaystyle\dot{s}=-\left({L+B}\right){\beta_{0}}{x_{s}}+\left({L+B}\right)\bar{J}\left(\eta\right)\bar{B}d,$
(26)
$\displaystyle{\dot{x}_{s}}=-{A_{s}}{x_{s}}+\left({{B_{s}}-{x_{s}}}\right){s^{+}}-\left({{D_{s}}+{x_{s}}}\right){s^{-}}.$
(27)

It can be found that the resulting closed-loop system is a coupled perturbed
system with state vectors
$\left({{{\bar{e}}_{1}},\dot{s},{{\dot{x}}_{s}}}\right)$ as well as
disturbances $d$. To be more specific, subsystem (25) and (26) are cascaded
while subsystem (26) and (27) are of interconnected architecture. Hence, based
on this feature the proof may proceed with two steps. That is, first step
shows subsystem (25) is input-to-state stable, regarding $s$ as the input;
second, the interconnected subsystems (26) and (27) are also input-to-state
stable with the impacts of disturbances. Finally, on the basis of such two
results one may conclude that the overall system (25)-(27) is input-to-state
stable.

Step1: Stability of subsystem (25)

Due to the characteristic of linearity, the solution of subsystem (25) can be
readily written as

${\bar{e}_{1}}\left(t\right)={e^{-{k_{1}}\left({L+B}\right)t}}{\bar{e}_{1}}\left(0\right)+\int_{0}^{t}{{e^{-{k_{1}}\left({L+B}\right)\left({t-\tau}\right)}}sd\tau}.$
(28)

It follows that

$\begin{split}\left\|{{{\bar{e}}_{1}}\left(t\right)}\right\|&\leq{e^{-\tilde{k}t}}\left\|{{{\bar{e}}_{1}}\left(0\right)}\right\|+\int_{0}^{t}{{e^{-{k_{1}}\left({L+B}\right)\left({t-\tau}\right)}}\left\|{s\left(\tau\right)}\right\|d\tau}\\\
&\leq{e^{-\tilde{k}t}}\left\|{{{\bar{e}}_{1}}\left(0\right)}\right\|+\frac{1}{{\tilde{k}}}\mathop{\sup}\limits_{0\leq\tau\leq
t}\left\|{s\left(\tau\right)}\right\|,\end{split}$ (29)

where $\tilde{k}={\lambda_{\min}}\left({{k_{1}}\left({L+B}\right)}\right)$ .
It can be concluded from (29) the ultimate bound of
$\left\|{{{\bar{e}}_{1}}\left(t\right)}\right\|$ is proportional to the bound
of $\left\|{s\left(t\right)}\right\|$, from which it is shown that the
subsystem (25) is input-to-state stable; moreover, the larger $k_{1}$ is , the
less ultimate bound will be.

Step2: Stability of interconnected subsystems (26) and (27)

Because of the employment of piecewise function $s^{+}$ and $s^{-}$ in
subsystem (27), the analysis shall proceed with two cases. First, when $s\geq
0$, the subsystem (27) can be rewritten as

${\dot{x}_{s}}=-\left({{A_{s}}+S}\right){x_{s}}+{B_{s}}s,$ (30)

where $S=\text{diag}\left\\{s_{1},s_{2},\ldots,s_{N}\right\\}$.

Let $\tilde{x}=\left[{s,x_{s}}\right]^{\rm T}$, and choose the following
Lyapunov function candidate

${V_{1}}=\frac{1}{2}\tilde{x}^{\rm T}Q\tilde{x},$ (31)

where

$Q=\begin{bmatrix}\frac{2}{\beta_{0}}{\left({L+B}\right)^{-1}}&{-\left({A_{s}+S}\right)}^{-1}\\\
{-\left({A_{s}+S}\right)}^{-1}&{B_{s}^{-1}}\end{bmatrix},$ (32)

and the following condition is assumed to be satisfied

$\frac{2}{{{\beta_{0}}}}{\left({L+B}\right)^{-1}}B_{s}^{-1}-{\left({{A_{s}}^{-1}}\right)^{2}}\succ
0,$ (33)

which guarantees $V_{1}$ to be positive definite. Then, taking the time
derivative of $V_{1}$ along the subsystems (26) and (30) together with (32)
yields

$\begin{split}{{\dot{V}}_{1}}&=-{s^{\rm T}}{\left({{A_{s}}+S}\right)^{-1}}s\\\
&\quad-x_{s}^{\rm
T}\left[{B_{s}^{-1}\left({{A_{s}}+S}\right)-{\beta_{0}}{{\left({{A_{s}}+S}\right)}^{-1}}\left({L+B}\right)}\right]{x_{s}}\\\
&\quad+\frac{2}{{{\beta_{0}}}}{s^{\rm
T}}\bar{J}\left(\eta\right)\bar{B}d-x_{s}^{\rm
T}{\left({{A_{s}}+S}\right)^{-1}}\left({L+B}\right)\bar{J}\left(\eta\right)\bar{B}d\\\
&\leq-{s^{\rm T}}{\left({{A_{s}}+S}\right)^{-1}}s\\\ &\quad-x_{s}^{\rm
T}\left[{B_{s}^{-1}{A_{s}}-{\beta_{0}}{A_{s}}^{-1}\left({L+B}\right)}\right]{x_{s}}+\frac{2}{{{\beta_{0}}}}{s^{\rm
T}}\bar{J}\left(\eta\right)\bar{B}d\\\ &\quad-x_{s}^{\rm
T}{\left({{A_{s}}+S}\right)^{-1}}\left({L+B}\right)\bar{J}\left(\eta\right)\bar{B}d.\end{split}$
(34)

Since $A_{s}$ and $S$ are both positive definite diagonal matrices, the first
term in (34) is already negative definite with respect to $s$. As for the
second term, it is assumed that the following matrix inequality holds

$B_{s}^{-1}{A_{s}}-{\beta_{0}}{A_{s}}^{-1}\left({L+B}\right)\succ 0.$ (35)

Define $P=B_{s}^{-1}{A_{s}}-{\beta_{0}}{A_{s}}^{-1}\left({L+B}\right)$. The
rest of the terms indicating the effects from disturbances can be addressed
using the bounded approach. Applying the bounded disturbance condition (5),
the derivative of $V_{1}$ can be rewritten as

$\begin{split}{{\dot{V}}_{1}}&\leq-{\zeta_{1}}{\left\|s\right\|^{2}}-{\zeta_{2}}{\left\|{{x_{s}}}\right\|^{2}}+\frac{{2}}{{{\beta_{0}}}}{\left\|\tilde{\lambda}\right\|}\left\|s\right\|+{\zeta_{3}}{\left\|\tilde{\lambda}\right\|}\left\|{{x_{s}}}\right\|\\\
&=-\left({1-\theta}\right){\zeta_{1}}{\left\|s\right\|^{2}}-\left({1-\vartheta}\right){\zeta_{2}}{\left\|{{x_{s}}}\right\|^{2}}\\\
&\quad-\theta{\zeta_{1}}{\left\|s\right\|^{2}}-\vartheta{\zeta_{2}}{\left\|{{x_{s}}}\right\|^{2}}+\frac{{2}}{{{\beta_{0}}}}{\left\|\tilde{\lambda}\right\|}\left\|s\right\|+{\zeta_{3}}{\left\|\tilde{\lambda}\right\|}\left\|{{x_{s}}}\right\|\\\
&\leq-\left({1-\theta}\right){\zeta_{1}}{\left\|s\right\|^{2}}-\left({1-\vartheta}\right){\zeta_{2}}{\left\|{{x_{s}}}\right\|^{2}},\\\
&\qquad\left(\forall\left\|s\right\|\geq\frac{{2}}{{\theta{\beta_{0}}{\zeta_{1}}}}{\left\|\tilde{\lambda}\right\|},\forall\left\|{{x_{s}}}\right\|\geq\frac{{{\zeta_{3}}}}{{\vartheta{\zeta_{2}}}}{\left\|\tilde{\lambda}\right\|}\right)\end{split}$
(36)

where
$\tilde{\lambda}={\left[{{{\tilde{\lambda}}_{1}},\ldots,{{\tilde{\lambda}}_{N}}}\right]^{\rm
T}}$,
${\zeta_{1}}={\lambda_{\min}}\left({{{\left({{A_{s}}+S}\right)}^{-1}}}\right)$,
${\zeta_{2}}={\lambda_{\min}}\left({P}\right)$ and
${\zeta_{3}}={\lambda_{\max}}\left({{{\left({{A_{s}}+S}\right)}^{-1}}\left({L+B}\right)}\right)$.
$\theta$ and $\vartheta$ are two numbers in the interval $\left({0,1}\right)$.

Equation (36) shows that ${\dot{V}_{1}}<0$, when
$\left\|s\right\|\geq\frac{{2\tilde{\lambda}}}{{\theta{\beta_{0}}{\zeta_{1}}}}$
and
$\left\|{{x_{s}}}\right\|\geq\frac{{{\zeta_{3}}\tilde{\lambda}}}{{\vartheta{\zeta_{2}}}}$,
from which it can be concluded that the interconnected subsystems (26) and
(30) are input-to-state stable.

Likewise, when $s<0$, the subsystem (27) can be reduced to

${\dot{x}_{s}}=-\left({{A_{s}}-S}\right){x_{s}}+{D_{s}}s.$ (37)

The analysis procedure is very similar to the previous case. First, choose
Lyapunov function candidate as

${V_{1}}=\frac{1}{2}\tilde{x}^{\rm T}Q^{\prime}\tilde{x},$ (38)

where

$Q^{\prime}=\begin{bmatrix}\frac{2}{\beta_{0}}{\left({L+B}\right)^{-1}}&{-\left({A_{s}-S}\right)}^{-1}\\\
{-\left({A_{s}-S}\right)}^{-1}&{D_{s}^{-1}}\end{bmatrix},$ (39)

and assume the following condition is satisfied

$\frac{2}{{{\beta_{0}}}}{\left({L+B}\right)^{-1}}D_{s}^{-1}-{\left({{A_{s}}^{-1}}\right)^{2}}\succ
0,$ (40)

which renders $V_{2}$ to be positive definite. Then, the derivative of $V_{2}$
along the subsystems (26) and (37) can be obtained

$\begin{split}{{\dot{V}}_{2}}&\leq-{s^{\rm
T}}{\left({{A_{s}}-S}\right)^{-1}}s\\\ &\quad-x_{s}^{\rm
T}\left[{D_{s}^{-1}{A_{s}}-{\beta_{0}}{A_{s}}^{-1}\left({L+B}\right)}\right]{x_{s}}\\\
&\quad+\frac{2}{{{\beta_{0}}}}{s^{\rm T}}\bar{J}\left(\eta\right)\bar{B}d\\\
&\quad-x_{s}^{\rm
T}{\left({{A_{s}}-S}\right)^{-1}}\left({L+B}\right)\bar{J}\left(\eta\right)\bar{B}d.\end{split}$
(41)

The term $A_{s}-S$ is already positive definite due to the fact that $s<0$,
and we also need the following matrix inequality to be satisfied

$D_{s}^{-1}{A_{s}}-{\beta_{0}}{A_{s}}^{-1}\left({L+B}\right)\succ 0.$ (42)

Define
$P^{\prime}=D_{s}^{-1}{A_{s}}-{\beta_{0}}{A_{s}}^{-1}\left({L+B}\right)$.
Similarly, as above ${\dot{V}_{2}}$ can be rewritten as

$\begin{split}{\dot{V}_{2}}\leq-\left({1-\theta}\right){\zeta^{\prime}_{1}}{\left\|s\right\|^{2}}-\left({1-\vartheta}\right){\zeta^{\prime}_{2}}{\left\|{{x_{s}}}\right\|^{2}},\\\
\left(\forall\left\|s\right\|\geq\frac{{2\tilde{\lambda}}}{{\theta{\beta_{0}}{{\zeta^{\prime}}_{1}}}},\forall\left\|{{x_{s}}}\right\|\geq\frac{{{{\zeta^{\prime}}_{3}}\tilde{\lambda}}}{{\vartheta{{\zeta^{\prime}}_{2}}}}\right)\end{split}$
(43)

where
${\zeta^{\prime}_{1}}={\lambda_{\min}}\left({{{\left({{A_{s}}-S}\right)}^{-1}}}\right)$,
${\zeta^{\prime}_{2}}={\lambda_{\min}}\left({P^{\prime}}\right)$ and
${\zeta^{\prime}_{3}}={\lambda_{\max}}\left({{{\left({{A_{s}}-S}\right)}^{-1}}\left({L+B}\right)}\right)$.
$\theta$ and $\vartheta$ are two numbers in the interval $\left({0,1}\right)$.

It follows from (43) that ${\dot{V}_{2}}<0$, when
$\left\|s\right\|\geq\frac{{2\tilde{\lambda}}}{{\theta{\beta_{0}}{{\zeta^{\prime}}_{1}}}}$
and
$\left\|{{x_{s}}}\right\|\geq\frac{{{{\zeta^{\prime}}_{3}}\tilde{\lambda}}}{{\vartheta{{\zeta^{\prime}}_{2}}}}$,
which shows that the interconnected subsystems (26) and (37) are input-to-
state stable.

Since the subsystem (25) and interconnected subsystems (26) and (27) are all
input-to-state stable, so long as the controller parameters are properly
chosen such that the conditions (33), (35), (40) as well as (42) are
satisfied, it is readily shown that the overall closed-loop system (25)-(27)
is input-to-state stable. This completes the proof.

######  Remark 5.

It should be noted that due to the introducing of shunting equations that
effectively handles the SMC’s chattering issue and makes control activities
more consistent and practical, strictly speaking, the two-phase characteristic
of SMC scheme is no longer existing, i.e., reaching phase and sliding mode
phase, but the resultant system is of a desirable cascaded interconnection
with modular architecture consisted of several subsystems, which actually
results from the design procedure of sliding mode control. Such a modularity
property is very helpful and beneficial for the control system analysis and
parameter tuning in practice.

######  Remark 6.

It is observed from equations (25)–(27) that the order of the system is
extended under the proposed shunting model based SMC scheme, from which we may
say that our methodology can be regarded as a variant of the higher order
sliding mode control (HOSMC) techniques. Generally speaking, in conventional
HOSMC scheme an integral standard form is typically used to extend the
relative degree of the system with the aims to smoothen the control inputs.
Nonetheless, such an approach cannot totally render a chattering-free
behavior, as the switching control law is still applied to satisfy the
reaching condition. Furthermore, in order to synthesize such a controller, the
assumption of bounded time derivative of disturbances is required, thus
leading to a more conservative design. In a stark contrast, bioinspired HOSMC
scheme completely gets rid of the chattering problem and doesn’t rely on such
an assumption. Most importantly, due to the filtering properties of the
shunting model, a more robust behavior can be obtained with respect to the
bounded disturbances as well as the noises without using the high-gain
control.

##  IV SIMULATION RESULTS

To validate the effectiveness of proposed formation control protocol,
simulation experiments are conducted, in which four underwater vehicles
steered by their own onboard controllers are employed to track a prescribed
trajectory in 3-dimensional space, and meanwhile, a quadrilateral formation
pattern is required to be maintained for the whole group of AUVs. Moreover,
the comparison between conventional sliding model control and proposed
bioinspired control is made to show the practical improvements in terms of the
smoothness of control activities, chattering suppression and disturbance
rejections.

Figure 2: The communication topology graph for 4 AUVs formation control.

Each vehicle’s dynamics used in simulations is described by (1), and the
corresponding system and hydrodynamic parameters are selected as follows with
international units: $m_{i}=20$, $I_{x,i}=20$, $I_{y,i}=30$, $I_{z,i}=35$,
$\beta_{vx,i}=-8$, $\beta_{vy,i}=-10$, $\beta_{vz,i}=-9$,
$\beta_{\dot{v}x,i}=-7$, $\beta_{\dot{v}y,i}=-8$, $\beta_{\dot{v}z,i}=-6$,
$\beta_{\omega x,i}=-0.2$, $\beta_{\omega y,i}=-0.25$, $\beta_{\omega
z,i}=-0.15$, $\beta_{\dot{\omega}x,i}=-20$, $\beta_{\dot{\omega}y,i}=-30$,
$\beta_{\dot{\omega}z,i}=-35$,
$\left({i\in\left\\{{1,2,3,4}\right\\}}\right)$. The information exchange
among the vehicles is illustrated in Fig. 2, and the weights of the undirected
edges are given as ${a_{12}}={a_{21}}={a_{23}}={a_{23}}={a_{34}}={a_{43}}=1$,
and it is assumed that only the AUV $1$ can assess the information of desired
trajectory, i.e., ${b_{1}}=1$. The desired trajectory is defined as
$\eta_{1}^{d}\left(t\right)={\left[{30-30{e^{-t}},5t,2t}\right]^{\rm T}}$, and
each AUV’s posture in group is required to align with
$\eta_{2}^{d}\left(t\right)={\left[{0,0,0}\right]^{\rm T}}$. The relative
distances between AUVs that are designed to form the desired quadrilateral
formation pattern are given by ${\delta_{12}}={\left[{0,10,0}\right]^{\rm
T}}$, ${\delta_{21}}={\left[{0,-10,0}\right]^{\rm T}}$,
${\delta_{23}}={\left[{-10,0,0}\right]^{\rm T}}$,
${\delta_{32}}={\left[{10,0,0}\right]^{\rm T}}$,
${\delta_{34}}={\left[{0,-10,0}\right]^{\rm T}}$ and
${\delta_{43}}={\left[{0,10,0}\right]^{\rm T}}$. The initial positions of four
vehicles are set as follows: ${\eta_{1}}={\left[{3,3,3,0.3,0,0.2}\right]^{\rm
T}}$, ${\eta_{2}}={\left[{2.5,3.5,3,0.2,0,0.25}\right]^{\rm T}}$,
${\eta_{3}}={\left[{2,3,3,0.3,0,0.2}\right]^{\rm T}}$ and
${\eta_{4}}={\left[{3,3,2,0.3,0,0.2}\right]^{\rm T}}$; the initial
translational and rotational velocities of AUVs are all set to
${v_{i}}={\mathbf{0}_{6\times
1}},\left({i\in\left\\{{1,2,3,4}\right\\}}\right)$. The periodic external
disturbances to simulate the impacts of wind, waves as well as ocean currents
are given by
$d_{i}=\left[{{2.5\sin\left(t\right),2.5\cos\left(t\right),2.5\sin\left(t\right)}}\right.$,
$\left.{{0.5\sin\left(t\right),0.5\cos\left(t\right),0.5\sin\left(t\right)}}\right]$,
$\left({i\in\left\\{{1,2,3,4}\right\\}}\right)$. The parameters of sliding
model controller are chosen as
${k_{1}}=\text{diag}\left\\{{10,10,10,10}\right\\}$,
${\beta_{0}}=\text{diag}\left\\{{100,100,100,100}\right\\}$, and the
bioinspired controller’s parameters are selected as
${A_{s}}=\text{diag}\left\\{{100,100,100,100}\right\\}$,
${B_{s}}=\text{diag}\left\\{{30,30,30,30}\right\\}$,
${D_{s}}=\text{diag}\left\\{{30,30,30,30}\right\\}$,
${k_{1}}=\text{diag}\left\\{{10,10,10,10}\right\\}$ and
${\beta_{0}}=\text{diag}\left\\{{10,10,10,10}\right\\}$. It is worth noting
that considering the practical saturation issue that quite often occurs in
real-world actuators, the amplitudes of controller’s outputs are artificially
confined into the interval $\left[{-300,300}\right]$. Moreover, because of the
high-frequency switching nature of the conventional sliding mode control that
makes the control signals oscillate extremely fast, to moderate this
phenomenon saturation functions are used in simulation to take place of the
sign functions that were employed in foregoing derivation of SMC controller.

Figure 3: The actual tracking trajectories and formation shape of four AUVs
under bioinspired control approach. Figure 4: The consensus position error of
AUVs under bioinspired control approach. Figure 5: The consensus attitude
error of AUVs under bioinspired control approach. Figure 6: Translational
control inputs of AUVs under bioinspired control approach. Figure 7:
Rotational control inputs of AUVs under bioinspired control approach.

The formation performance using the proposed distributed bioinspired control
protocol is shown in Fig.s 3-7. Fig. 3 shows that the control objectives are
perfectly achieved, i.e., under the proposed controller overall multi-AUV
system can form the desired quadrilateral formation pattern, and at same time
the prescribed trajectory is followed by the group of AUVs even in the
presence of external disturbances. It can also be seen from Fig.s 4 and 5 that
the corresponding consensus tracking errors of all AUVs in each degree of
freedom are convergent uniformly asymptotically to a very small region of the
origin, which also indicates that the resulting system has good robustness
property. In addition to that, it can be observed that the consensus tracking
errors of each vehicle almost tend to zero simultaneously, suggesting that a
good coordination performance is also achieved. Fig.s 6 and 7 show the control
signals generated by the bioinspired controller, from which one can find that
the chattering issue is totally disappeared in proposed control scheme due to
employment of shunting model, and the control actions, thus, become much
smoother, which is far more critical for the practical applications. It should
be noted that because of the transient response of the shunting model some
control saturations can be occurred at the beginning of time, but few moments
later the control behaviors soon get into normal.

The next case as the comparison shows the formation performance using the
conventional sliding mode control protocol. It can be seen from Fig.s 8-10
that the SMC approach can also achieve the prescribed control objectives.
However, the corresponding control signals as shown in Fig.s 11 and 12 exhibit
apparent high-frequency oscillations and the non-smoothness, even though the
saturation function is employed in place of the classic sign function which
undoubtedly will further exacerbate this phenomenon. As an immediate
consequence, behavior of sliding mode controller is almost unable to be
realized by the actual actuators, since any physical device has its own
response frequency and bandwidth. Besides, it may also excite the system’s
high frequency unmodeled dynamics. Therefore, the actual formation performance
by applying such control law may not be as good as the simulation results;
instead, it may even render the closed-loop system unstable. In an obvious
contrast, it can be seen from Fig.s 6 and 7 that the proposed bioinspired SMC
scheme completely overcomes these drawbacks; that is, there is no high-
frequency switching behavior arising in the control process, and the control
signals are much smoother than the conventional SMC approach.

Figure 8: The actual tracking trajectories and formation shape of four AUVs
under sliding model control. Figure 9: The consensus position error of AUVs
under sliding model control. Figure 10: The consensus attitude error of AUVs
under sliding model control. Figure 11: Translational control inputs of AUVs
under sliding model control. Figure 12: Rotational control inputs of AUVs
under sliding model control.

##  V CONCLUSION

The formation tracking control of multiple AUV systems in 3-dimensional space
subject to the hydrodynamic parameter uncertainties and unknown environmental
disturbances is addressed in this paper, where the overall system is not only
required to maintain a specified formation pattern, but is aimed to follow a
desired trajectory as a whole. In order to obtain a good coordination
performance in shape forming, consensus problem is also considered between
neighbor vehicles. To this end, a novel distributed bioinspired sliding mode
control protocol is proposed based on the graph theory. In particular, due to
the favorable dynamic characteristic of the shunting model, the chattering
phenomena are completely avoided in the present SMC scheme, the amplitude of
the control is able to be bounded, and the generated control actions are more
consistent and smoother compared to the conventional one. Most importantly,
owing to the filtering nature of the shunting equation, more robust behavior
can be obtained with respect to the bounded lumped disturbance as well as the
noises without needing the large control efforts. Moreover, the stability
analysis of the resulting closed-loop system is proved using the Lyapunov
direct method. Finally, several simulation experiments are carried out to
demonstrate the efficiency and effectiveness of proposed distributed formation
control protocol. In future work, more practical constraints on multiple AUV
formation systems need to be further taken into account and addressed, such as
the impacts of the communication imperfections (e.g., time delays, package
loss, and even misleading data) as well as the nonholonomic kinematics.

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|  Tao Yan (S’22) received the B.Sc. degree from the North China Institute of
Aerospace Engineering, Langfang, China, in 2016; the M.Sc. degree from the
Zhejiang University of Technology, Hangzhou, China, in 2020. He is currently
pursuing his Ph.D. degree at the University of Guelph, ON, Canada. His
research interests include the intelligent control, distributed control and
optimization, and networked underwater vessel systems.
---|---
|  Zhe Xu (S’20) received the B.Eng. degree in mechanical engineering and the
M.A.Sc. degree in engineering systems and computing from the University of
Guelph, in 2018 and 2019, respectively, where he is currently pursuing the
Ph.D. degree in engineering systems and computing with the School of
Engineering, under the supervision of Prof. S. X. Yang. His research interests
include tracking control, estimation theory, robotics, and intelligent
systems.
---|---
|  Simon X. Yang (S’97–M’99–SM’08) received the B.Sc. degree in engineering
physics from Beijing University, Beijing, China, in 1987, the first of two
M.Sc. degrees in biophysics from the Chinese Academy of Sciences, Beijing, in
1990, the second M.Sc. degree in electrical engineering from the University of
Houston, Houston, TX, in 1996, and the Ph.D. degree in electrical and computer
engineering from the University of Alberta, Edmonton, AB, Canada, in 1999. Dr.
Yang is currently a Professor and the Head of the Advanced Robotics and
Intelligent Systems Laboratory at the University of Guelph, Guelph, ON,
Canada. His research interests include robotics, intelligent systems, sensors
and multi-sensor fusion, wireless sensor networks, control systems, machine
learning, fuzzy systems, and computational neuroscience. Prof. Yang he has
been very active in professional activities. He serves as the Editor-in- Chief
of International Journal of Robotics and Automation, and an Associate Editor
of IEEE Transactions on Cybernetics, IEEE Transactions of Artificial
Intelligence, and several other journals. He has involved in the organization
of many international conferences.
---|---


# Redefining entanglement entropy - can it solve the information paradox?

Madhur Mehta<EMAIL_ADDRESS>Department of Physics, The Ohio State
University.

###### Abstract

Qubit Models which aim to resolve the information paradox must not redefine
entanglement entropy as a redefinition does not address the correct issue of
the information paradox. The purpose of this letter is to understand why any
redefinition of entanglement entropy is not relevant to the paradox. Also,
conditions required for a bit model to resolve the information paradox are
listed, motivating the community to look for such conditions as a consistency
check in their qubit models.

##  I Introduction

Recently, models which establish the rising and then falling of the Page curve
are being considered as solutions to the Information Paradox, and many efforts
have been made to show how can the Page curve of a semiclassical black hole be
modified to decrease. One such effort is the idea of the _Qubit_ Island
Entropy Neuenfeld (2021) which provides a bit model to describe the Islands
approach for a black hole to bring down its Page curve. The model redefines
the Entanglement Entropy for the Hawking radiation which substantially differs
from the von Neumann entropy of radiation at later times of evaporation. In
this letter, I argue why such a resolution fails to address Hawking’s paradox
while showing how it describes Hawking’s original calculation. Models which
redefine entanglement entropy to resolve the information paradox are not
relevant to the paradox. In this letter, I show how any redefinition of
entanglement entropy could lead to a rising and then falling Page curve, and
thus does not solve the paradox.
We begin in section 2, where I present the Hawking’s paradox by contrasting
the two Page curves produced by a piece of coal and a semiclassical black
hole. In section 3, I attempt to understand the Qubit Island Entropy Model in
detail and how it differs from the original Hawking’s computation. In section
4, I argue how such models do not answer Hawking’s paradox. At last, I discuss
what conditions must be fulfilled by a correct _qubit_ model to do so and end
with acknowledgements.

##  II The two Page curves and the Qubit Model

###  II.1 The paradox

Hawking’s black hole information paradox Hawking (1976) can be cast into the
language of Page curvesPage (1993a). The paradox is the apparent conflict
between the Page curve of an ordinary quantum system, say a piece of coal, and
the Page curve of a black hole. In this section, I discuss the Page curve
produced by the burning of a piece of coal and compare it to the Page curve
produced by Hawking’s computation of a semiclassical black hole.

####  II.1.1 Burning Coal

Let us start with a piece of coal(_N_) which has _n_ quantum bits or _qubits_.
Let each _qubit_ have 2 states, so the total states in the piece of coal
becomes $2^{n}$.

Figure 1: Burning of a piece of coal (system $N$). The set of radiated qubits
is labelled as the subsystem $A$ and the rest of the coal is $A^{c}$

We define the qubits emitted out during the burning of the coal, as the
subsystem $A$ and we wish to plot the entanglement entropy versus time as the
qubits in this subsystem are emitted.

The von Neumann entropy of the two subsystems $A$ and $A^{c}$ was calculated
in Page (1993b),

$S_{A}=\log{\mathcal{N}}+O(1/2^{m})$ (II.1)

where $\mathcal{N}=\mathrm{min}(N_{A},N_{A^{c}})$ and
$N_{A}=2^{m},N_{A^{c}}=2^{n-m}$ are the number of states in $A$ and $A^{c}$
respectively. The plot of $S_{A}$ with the _qubits_ $m$ gives the Page curve.
See Figure 2(a)

$\displaystyle\begin{split}S_{A}\quad=\qquad m\qquad\qquad m\leq n/2\\\
n-m\qquad\qquad m\geq n/2\end{split}$ (II.2)

Figure 2: (a)Page curve of a piece of coal which falls down following
equation(II.1).This is how any quantum system should behave. (b) Page curve of
a semiclassical black hole which keeps on rising and does not behave correctly
as a quantum system.

####  II.1.2 Hawking’s Black Hole

While the Page curve of any quantum system rises and then falls, the Page
curve of a semi-classical black hole differs significantly from this and it is
this that leads to Hawking’s paradox.
The semiclassical black hole differs from a piece of coal in that it has a
horizon where particle pairs are created. The emitted radiation, called
Hawking radiation, is entangled with the black hole and as the black hole
evaporates, more and more qubits are radiated. The entanglement entropy is a
direct measure of the number of entangled qubits, as the black hole
evaporates, the entanglement entropy continues to increase. Hence, the Page
curve of the semiclassical black hole never falls back to zero(Figure 2(b)).

As opposed to a burning coal, where the radiated qubits carry the information
of the internal qubits away at later times, the interior of a black hole is
inaccessible and no information can be extracted; therefore, the Page curve
never drops. This is one of the forms of the _information paradox_.

###  II.2 The Qubit Island Entropy Model(QIEM)

A resolution of the information paradox in the _qubit_ language means a model
which brings the Page curve of the semiclassical black hole back to zero as
expected from a conventional quantum system. Recent work on _Islands_ has
tried to show this using replica wormholes Penington  _et al._ (2020). The bit
model in Ref Neuenfeld (2021) is an attempt to look at Hawking’s results and
understand the recent Island proposals Almheiri  _et al._ (2021) which
reproduce a rising and then falling Page curve using _qubits_ and hence a
potential resolution to the information paradox.
In this section, we closely follow RefNeuenfeld (2021) and rederive Hawking’s
result using the black hole radiation (name it $N$) along with its interior
$I$ and understand the QIEM.
The model and its prescription, as we understand, are as follows:

  1. 1.

Consider the black hole radiation $N$ with $n$ qubits and along with this
another system $I$ with $k$ qubits ($k>n$) as in the Figure 3;

  2. 2.

Mix the systems $N$ and $I$ using some random unitary matrix($U$) and get a
system $N\cup I$;

  3. 3.

Now, we calculate the entanglement entropy, once using the usual von Neumann
method and, once with the QIEM prescription, and compare the two results.

####  II.2.1 Paradox in the qubit language

In Ref Neuenfeld (2021), Hawking’s paradox is described in the following
sense. When the two systems are scrambled, we get a system $N\cup I$ with
$n+k$ bits. When one looks at a subsystem $A$, it is entangled with its
complement $\bar{A}$ which consists of bits from _N_ and bits from _I_ , and
the plot of von Neumann entropy $S^{vN}_{A}(\tilde{\rho})$ gives the Page
curve.

$\displaystyle\begin{split}S^{vN}_{A}(\tilde{\rho})\quad=\qquad m\qquad\qquad
m\leq(n+k)/2\\\ n+k-m\qquad m\geq(n+k)/2\end{split}$ (II.3)

but since $n<k\rightarrow n<(n+k)/2$ and we see that

$S^{vN}_{A}(\tilde{\rho})\quad=\qquad m\qquad\qquad m<n$ (II.4)

and the Page curve keeps rising for the first $n$ bits just like the Hawking’s
paradox as in Figure 2(b). Hence, the addition of the interior resembles
Hawking’s result.

####  II.2.2 Resolution in the qubit language

Once, Hawking’s result has been cast into the _qubit_ language, the QIEM
computes the quantity which reproduces the falling Page curve:

$\tilde{S}_{island}(\tilde{\rho})=\mathop{min}_{B\subset I}{S^{vN}_{A\cup
B}(\tilde{\rho})}$ (II.5)

here $B$ represents an arbitrary set of bits in $I$.

To summarize:
The entanglement entropy of the radiation is redefined using this
prescription:

  1. 1.

Choose a system _I_ with more bits than the radiation system _N_ ;

  2. 2.

Define a small variable part of _I_ as _B_ ;

  3. 3.

Find the von Neumann entropy of the system $A\cup B$ instead of _A_ ;

  4. 4.

Then find the QIEM entropy by choosing that _B_ which minimizes the above von
Neumann entropy

This gives us

$\displaystyle\begin{split}\tilde{S}_{island}(\tilde{\rho})\quad=\qquad
m\qquad\qquad m<n/2\\\ n-m\qquad n/2\leq m\leq n\end{split}$ (II.6)

This prescription is understood as a resolution as its prediction resembles
the Page curve of a piece of coal.

Figure 3: Subsets A of N and B of I are shown. The _Qubit_ Island Entropy
attains minimum when B is empty($\phi$) for $m\leq n/2$ and when B is all of I
for $n/2\leq m\leq n$ where $m$ is the number of bits in A.

##  III How the model fails to address the right question

The above _qubit_ islands entropy model describes a falling Page curve which
is very interesting, but a closer look at it reveals an important aspect of
the information paradox.

####  III.0.1 Observable Replacement - the problem

The QIEM aims to understand how semi-classical gravity emerges, provided there
is a true quantum gravitational picture of a black hole. Reproducing the
correct Page curve behavior, rising and returning back to zero is understood
as the emergence of semi-classical gravity which does not seem correct.
Although the QIEM reproduces the falling Page curve for a black hole, it does
not resolve the information paradox.

The QIEM prescription does not reproduce a falling Page curve of the von
Neumann entropy, rather, it redefines the entanglement entropy as the _Qubit_
Island Entropy which produces a Page curve with the desired behavior. The
information paradox asks why the von Neumann entropy of a black hole differs
significantly from that of a conventional quantum system. Hence, a change to
the definition of the entropy for a black hole does not resolve this paradox.

Let us say, we have two systems $A$(which is the coal in our case) and
$B$(which is the black hole) and we wish to measure quantity $S_{1}$(entropy
for our case) for $A$ and $B$. Suppose the graphs for $S_{1}$ for $A$ and
$S_{1}$ for $B$ differ and we wish to resolve this difference. The resolution
of such a difference cannot be done by redefining $S_{1}$ with $S_{2}$ for
system $B$.
If otherwise, one could just define another type of entropy which would
produce a new curve different from both a quantum system and a semiclassical
system and call it new Physics for that.
So far we have understood that a redefinition of entanglement entropy can
resemble a resolution to the information paradox. And now we would see how
such a redefinition is not unique and hence cannot be a resolution to the
information paradox.

####  III.0.2 Different definitions of Entanglement Entropy

Consider again the model as defined in Section II.2 with two systems namely
the radiated qubits _N_ and the interior qubits _I_. Let us now look at what
another redefinition of the entanglement entropy looks like:

Define:

$\displaystyle\begin{split}S_{new}(\tilde{\rho})=\mathop{min}_{B\subset
I}\frac{({S^{vN}_{A\cup B}(\tilde{\rho})})^{2}}{S^{vN}_{A}}\end{split}$
(III.1)

This definition when applied to find the entropy gives us:

$\displaystyle\begin{split}S_{new}(\tilde{\rho})\quad\quad=\qquad
m\qquad\qquad m<n/2\\\ \qquad\quad\frac{(n-m)^{2}}{m}\qquad n/2\leq m\leq
n,\end{split}$ (III.2)

which gives a rising and falling Page curve but is significantly different
from the _Qubit_ Island entropy at later times as seen in Figure 4.

Figure 4: Two kind of Page curves constructed by two different definitions of
entanglement entropy. Both the definitions produce a falling Page curve so
choosing the correct prescription becomes an issue. A redefinition of
entanglement entropy, thus, cannot address the information paradox.

Many such Page curves can be constructed by redefining the entanglement
entropy of the radiation and black hole and the above definition is one such
example of that. One may equally argue that $S_{new}$ is the correct
entanglement entropy as the Page curve rises at earlier times and falls at
later times. A redefinition of entanglement entropy, thus, cannot address the
paradox. One must follow the von Neumann entropy of a quantum system with no
change in the definition of entropy while finding a resolution to the paradox
through _qubit_ models.

##  IV Further Discussion

Hawking’s black hole information paradox is the apparent loss of unitary
evolution during the evaporation of a black hole. Resolutions to protect
unitarity can be created using bit models but they must target the right
issues of the paradox.
Recent _Islands_ resolutions which aim to tackle the paradox either lead to
non-localities or violate the small corrections theorem as observed in Ref Guo
_et al._ (2021); Mathur (2009). The issue of the Page curve of a black hole is
the inconsistency between the von Neumann entropy of a burning piece of coal
and the von Neumann entropy of a black hole. If this is to be addressed using
the _qubit_ language, the model must be consistent across both systems (coal
and black hole). This leads us to enumerate some requirements which are
necessary for a correct _qubit_ model which wishes to resolve the information
paradox.

#### Requirements for a correct _Qubit_ Model to resolve the information
paradox

Any correct model of resolution for the black hole information paradox should
contain the following :

  1. 1.

The _qubit_ model which contains a black hole must have unitary evolution
inside the interior of the black hole.

  2. 2.

If the _qubit_ model, which consists of a horizon, aims to resolve the
paradox, it must show how the effective pair creation at the horizon takes
place without violating the effective small corrections theorem Guo  _et al._
(2021).

  3. 3.

The _qubit_ model must lead to “normal”, everyday Physics (classical gravity
and thermodynamics) with negligible quantum gravity effects at far away
distances from the black hole.

  4. 4.

The _qubit_ model must generate a Page curve of von Neumann entropy which
rises at early times and then falls down to zero after evaporation.

  5. 5.

The entanglement entropy should be consistent across all systems and it should
not be defined differently for different systems.

The black hole information paradox is a simple yet mysterious problem.
Understanding this paradox and working towards a resolution enables its
learner to work in a breeding ground of ideas that connect many different
branches of Physics together. The aim of this letter is to motivate the
community to understand the correct form of the paradox and create models that
tackle the relevant issue.

###### Acknowledgements.

I wish to thank Prof. Samir D. Mathur for his valuable remarks and motivation.
I wish to extend my special thanks to Dominik Neuenfeld and Bin Guo for
various discussions and comments. My family which includes Lalit Mehta, Ranu
Mehta, Tanvi Mehta and Manvi Mehta was a constant source of motivation. My
friends Abhijeet Mishra, Ayan Gupta, Brandon Manley and Naeem Bharmal also
helped me to work with dedication and gave important remarks. Support from
Kevin Ingles was crucial in making this letter much more comprehensible.
Discussions with Marcel Hughes, Sami Rawash, Davide Bufalini, David Turton and
Bidisha Chakrabarty were also helpful.

## References

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Shanti…