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\DOI{10.5802/crphys.184}
\datereceived{2023-05-26}
\daterevised{2024-01-17}
\dateaccepted{2024-03-19}
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\dateposted{2024-05-02}
\begin{document}

\begin{noXML}

%\makeatletter
%\def\TITREspecial{\relax}
%\def\cdr@specialtitle@english{Energy in the heart of EM waves: modelling, measurements and management}
%\def\cdr@specialtitle@french{L'\'energie au c{\oe}ur des ondes \'electromagn\'etiques : mod\'elisation, mesures et gestion}
%\makeatother

\CDRsetmeta{articletype}{review}

\title{Matching of an observed event and its virtual model in relation
to smart theories, coupled models and supervision of complex
procedures---A review{\vspace*{-1pt}}}

\alttitle{Appariement d'un \'{e}v\`{e}nement observ\'{e} et de son
mod\`{e}le virtuel en relation aux th\'{e}ories intelligentes, aux
mod\`{e}les coupl\'{e}s et \`{a} la supervision de proc\'{e}dures
complexes --- Bilan{\vspace*{-1pt}}}

\author{\firstname{Adel} \lastname{Razek}\CDRorcid{0000-0003-0038-1934}}
\address{Group of Electrical Engineering -- Paris (GeePs), CNRS,
University of Paris-Saclay and Sorbonne University, F91190 Gif sur
Yvette, France}
\email{adel.razek@centralesupelec.fr}

\begin{abstract} 
This contribution aims to illustrate the nature of the
observation--modeling (or real--virtual) link, the importance of the
exact model (or coupled model) in the matching involved in this link
and the use of this link in the supervision of complex procedures. This
involves offline and real-time matching practices. The offline case is
mainly about the management and ruling of elegant theories and
computational tools mimicking physical paradigms. Real-time pairing
notably concerns natural phenomena, autonomous automated systems and
complex procedures. The paper assesses, analyzes and discusses the
different elements mentioned. This is aided by a literature review.
{\vspace*{-1pt}}
\end{abstract} 

\begin{altabstract} 
Cette contribution vise \`{a} illustrer la nature du lien 
observation--mod\'{e}lisation (ou r\'{e}el--virtuel), l'importance du
mod\`{e}le exact (ou mod\`{e}le coupl\'{e}) dans l'appariement
impliqu\'{e} dans ce lien et l'utilisation de ce lien dans la
supervision de proc\'{e}dures complexes. Cela implique des pratiques de
mise en correspondance hors ligne et en temps r\'{e}el. Le cas hors
ligne concerne principalement la gestion de th\'{e}ories
\'{e}l\'{e}gantes et d'outils informatiques imitant des paradigmes
physiques. L'appariement en temps r\'{e}el concerne notamment les
ph\'{e}nom\`{e}nes naturels, les syst\`{e}mes automatis\'{e}s autonomes
et les proc\'{e}dures complexes. Le document \'{e}value, analyse et
discute les diff\'{e}rents \'{e}l\'{e}ments mentionn\'{e}s. Cela est
assist\'{e} par une revue de la litt\'{e}rature. 
{\vspace*{-2pt}}
\end{altabstract} 

\keywords{\kwd{Matching}\kwd{Coupled models}\kwd{Electromagnetic
systems}\kwd{Complex producers}\kwd{Supervision}}

\altkeywords{\kwd{Appariement}\kwd{Mod\`{e}les
coupl\'{e}s}\kwd{Syst\`{e}mes
\'{e}lectromagn\'{e}tiques}\kwd{Producteurs
complexes}\kwd{Supervision}}

\def\thanksname{Note}
\thanks{This article follows the URSI-France workshop held on 21 and 22
March 2023 at Paris-Saclay.}

\maketitle

\end{noXML}

\section{Introduction}\label{sec1}

Cognitive inference or virtual modeling can account for the observation
of an object, phenomenon or procedure. Pairing or mirroring an
observable and its virtual image has been, and still is done in many
natural and man-made situations. Humankind, other creatures and natural
elements often exercise the practice of observation, experience or
sensory manipulation. At the same time, from this practice, they will
eventually use deductive (or mimesis) skills to manage their evolution,
self-protection, comfort, and survival. The activity of deduction
associated with observation is one of the first natural duties born in
the world. Deduction, prediction, or reasoning (modeling) associated
with observation may be encountered in inherent natural events or
manufactured procedures. Such a couple often works according to a
process of pairing or imitation. For example, in nature, based on
observation, cases of mimetic simulation (imitation strategy) are very
frequent allowing camouflage~\cite{1}. This permits creatures to blend
into their surroundings. This could involve simple matching or dynamic
(adaptive) matching. 

Both cases of link observation--modeling involving offline and online
matching can be used in distinct matching categories. The offline one
can be practiced in managing universal elegant theories involving their
validation, explanation and unification. The online matching procedures
of the link observation--modeling are practiced in various natural
processes and artificial modern applications related to the supervision
of automated and complex systems. In these applications, we need to
reduce the involved uncertainties to achieve an optimized supervision.
Such reduction is mostly needed in the virtual side of the link. Thus,
we need accurate realistic system models, which can be obtained by
reintegrating neglected items committed in idealizing for elegance of
theories. Such coupled models permit optimized matching of the link
observation--modeling. We see that the matching in the link is closely
associated to elegant theories and coupled models, respectively for
offline theories managing and online systems supervision.

Indeed, the foundation of basic research is built on elegant and
consistent theories, which is essential for science. Let us illustrate
the notion of elegance in the theories that belong to fundamental
science. When a theory or model clearly and directly describes a
phenomenon, it is said to be elegant. Additionally, an
easy-to-understand enterprise can capture a lot of information and
answer many questions. Therefore, the definition of elegance as
simplicity plus greater capacity seems fair. Note that this last
statement is valid only when the theory is applied in its strict scope.
One of the most famous elegant unified theories is Maxwell's set of
equations~\cite{2}. The case of Maxwell's equations illustrated the
interest of the concept of elegance. However, later in this article in
the application to electromagnetic systems, we will see that in real
systems, Maxwell's equations could not always be applied immediately
and such elegance could conflict with real applications. In such cases,
one must make a volte-face from elegance to reality by reconsidering
the corresponding committed approximations. We are therefore inclined
to modify the model based on the theory of the main field, by combining
the secondary fields in a modified coupled model~\cite{3}. Such a
modified model resulting from ``retrograde postulations'' paradoxically
seems to represent the real context. Note that coupled models belong to
applied science.

Many recent innovative technological processes use the concept of
matching physical (observable) operations with their (virtual) mirror
models. Matching depth is closely related to the fidelity of the
virtual model to the real physical object. Such consistency implies the
nature and ability of the model to take into account the variation of
the physical element due to its operational and environmental
conditions~\cite{4}. Therefore, a complete model taking into account
all the phenomena governing these conditions becomes necessary and the
model uncertainty involved in such a circumstance will be of knowledge
type. Currently, in very promising fields, where the large number of
creations and the growing importance of digital components in automated
{\unskip\break}assemblies offer an opportunity to reach higher levels of
production~\cite{5}.  The practice of digital technologies allows the
virtual projection of products and processes~\cite{6}. The combination
of physical and virtual elements can be achieved through the concept of
matching physical operations with their mirror models---digital twin
(DT). DT is gradually being studied as a means of improving the
functioning of physical units by taking advantage of the computational
practices made possible by those of virtual pairing. Bidirectional
links feed data from the physical element to its virtual image, and
process it from the latter to the physical element~\cite{7}. This
matching sequence (pairing) is a kind of mirroring of real and virtual
elements. The virtual one allows various specific tasks of simulation,
test, optimization\,{\ldots}~\cite{8}. Since Michael Grieves introduced
the concept of digital twins in 2002, which has quickly taken hold in
various fields; the number of publications on its applications has
increased significantly.

This contribution aims to illustrate the nature of the
observation--modeling link and its relation with elegant theories and
coupled models. First, we analyze the role of this link in the managing
of elegant universal theories. Then we discuss the relation of these
smart theories and coupled models. At the last part of the paper, we
illustrate the importance of coupled models in the real-time matching
involved in this link and its use in the supervision of complex
procedures.

\section{Characteristics of the link observation--modeling}
\label{sec2}

This section aims to examine how the two elements of observation and
theory each support and mutually form a duo. Thus, we examine how they
are complementary and evaluate their actions in the management of
universal theories involving validation, explanation and unifying
capacities. Finally, we discuss advanced computational tools mimicking
the physical paradigms ruled by the duo. All the observation--theory duo
activities discussed in this section fall under offline matching
practices.

\subsection{Managing of smart theories}\label{ssec21}
\subsubsection{Observation and theory complementarity}
\label{sssec211}

Observation or theoretical modeling can be self-ruling in areas of
investigation that are consistently seen as standards. However, in
widespread cases, we use the two items in a complementary way.
Therefore, yet in a domain that customarily necessitates observation,
it is generally not autonomous and it requires modeling for further
investigation. Structural research in social sciences is a typical
example in this category; see e.g.~\cite{9}, in addition, in a field
currently requiring theoretical modeling, it is not regularly either
autonomous and it requires to be validated by observation, simply to be
reliable~\cite{10}, as we will see in the next lines.

\subsubsection{Validating or invalidating a theory by observation}
\label{sssec212}

In general, a theory is only thought to be established after it has
been verified by observation. Furthermore, such a theory stays true
until inconsistency with another observation. 

\paragraph{Validation of the theory of superposition states in
quantum mechanics}

Considering the case of the ``theory of superposition states'' in
quantum mechanics proposed by Schr\"{o}dinger in 1926~\cite{11}, (Nobel
1933). In this theory, the wave function provides the probability of
locating a particle at a specific position. Wineland's ion
traps~\cite{12} and the cavity quantum electrodynamics of
Haroche~\cite{13} validated this theory a little before 2000 (Nobel
2012: for revolutionary experimental methods, which make it possible to
measure and manipulate individual quantum systems). It was only after
such validation that this theory was established until a possible
future invalidation.

\paragraph{Partial invalidation of the treatise of JC Maxwell by
the Hall effect}

Concerning the ``Hall Effect'' proposed by Hall in 1879 that resulting
from an experiment; it concerns the relation between the force and the
current in a conductor. It invalidates part of the ``treatise on
electricity and magnetism'' proposed by Maxwell in 1873~\cite{14}. Hall
revealed and experimentally confirmed in his thesis work, the effect of
force on current (distribution) in a conductor immersed in a magnetic
field~\cite{15}. Maxwell thought there was no such effect.

\subsubsection{Observation confirmed and explained later by theory}
\label{sssec213}

One can meet the situation of first reaching a finding from experiments
and then establishing the theory explaining and confirming such
discovery. Generally, we come across such a situation in a ``serendipity
condition'': we find something while looking for another. A typical
illustration is the revealing of the superconductivity phenomenon by
Kamerlingh Onnes (1853--1926), (Nobel 1913: for his investigations on
the properties of matter at low temperatures which led, inter alia, to
the production of liquid helium)~\cite{16}. In this context, he was
studying the problems connecting to the effects of low temperatures on
electronics. He could not imagine the phenomenon he observed. All the
theories confirming and explaining the superconductivity phenomenon
followed his discovery.

\subsubsection{Generalizing and amalgamating observations by a theory}
\label{sssec214}

Several characteristics can distinguish intelligence of theories such
as enhancement, generalization, and fusion. An example of such
intelligence can be seen in Maxwell's equations, which are an
illustration of the highest elegant composite theories. These equations
originated by James Clerk Maxwell (1831--1879) incorporate an
association of three laws that are obtained experimentally, discovered
by three of his predecessors. They are Carl Friedrich Gauss
(1777--1855), Andr\'{e}-Marie Amp\`{e}re (1775--1836) and Michael Faraday
(1791--1867). The unification of Maxwell's equations was possible only
because Maxwell remarked how to progress from the three experimental
laws, introducing into one equation a missing link, the announced
displacement current, the occurrence of which guarantees the
consistency of the integrated organization~\cite{2,14}.

\subsection{Innovative computing tools imitating physical paradigms}
\label{ssec22}

Neuromorphic and quantum computing technologies are two constructed
tools based on imitations of physical systems. These two modeling tools
originate straight from two paradigms belong to neurosciences and
quantum physics.

\subsubsection{Neuromorphic computing}\label{sssec221}

The brain is an exceptionally intricate organization that performs
tasks much quicker than the swiftest digital computers. Neuromorphic
computing uses inspired models of the brain built on biologically
replicated or artificial neural networks. Neuromorphic computers can
perform complex calculations quicker, with greater power efficiency and
lesser size than traditional architectures. They have the capacity to
expand trained real-time learning algorithms to work online like real
brains. This showed potential due to the similarities of biological and
artificial neural networks (BNN and ANN)~\cite{17}. The rising request
of deep learning and neural networks has stimulated a sprint to advance
artificial intelligence (AI) hardware devoted to neural network
calculations~\cite{18}. These tools are broadly operated in
optimization, diagnostics, images, machine learning, AI, etc.

\subsubsection{Quantum computing}\label{sssec222}

The notion of states in quantum mechanics is the base of ``quantum
computers'', a term created by Richard Feynman~\cite{19}. A typical
computer uses transistors to process information in sequences of zeros
and ones (binary mode). A quantum computer uses qubits according to the
rules of quantum mechanics connecting to particle states. For a qubit,
a particle can be in several states simultaneously, as well, a
different phenomenon affects particle states called entanglement. This
means that when two qubits in a superposition meet; signifying the
state of one depends on the state of the other. Due to these phenomena,
a quantum computer can achieve 0, 1, or both states at the same time
for a qubit or a qubit entanglement. Thus, an n-qubit quantum computer
can work instantaneously on the 2n possibilities; however, a standard
computer with n bits can only operate on one of these 2n possibilities
at a time. Therefore, the former gives us more processing power.
Scientists agree that quantum computers are theoretically exponentially
faster and much smarter at cracking codes that are apparently
unfeasible for classical technology~\cite{20,21}.

\section{Idealized smart theories and coupled models}
\label{sec3}

This section aims to analyze and discuss the characteristics of elegant
theories and realistic coupled models. Often, the notion of elegance
belongs to the philosophy of science. On the other hand, in the present
article we specify that the use of the term of elegance of theories
concern fundamental sciences whereas that of coupled models concerns
applied sciences. Let us clarify the notion of elegance in the theories
of fundamental sciences. When a theory describes a phenomenon in a
clear and direct way, it is said to be elegant. Additionally, an
easy-to-understand construct can provide a large amount of information
and answer many questions. Therefore, the definition of elegance as
simplicity and greater capacity seems right. Note that this last
statement is only valid when the theory is employed within its strict
scope of{\unskip\break} application.

\subsection{Smart theories and postulations}\label{ssec31}

Let us consider a real physical problem which could be represented by
the field $A$, which is the union of the functions B, C, D\,{\ldots}\ 
which depend on the variables $x$, $y$, $z$\,{\ldots}. Each of these functions
relates to a different domain of science. On the other hand, often a
domain is more concerned by the problem studied than the others are,
let us call it the main domain and represent it by the function B in
(\ref{eq1}). If we allow that the main domain B can represent the real
problem, Equation~(\ref{eq2}) will give this approximation $A_{1}$. Moreover,
founding coherent and elegant theories usually requires postulations
that compress and idealize the real context resulting in $A_{2}$ given
by~(\ref{eq3}).
{\begin{eqnarray}
A: && \mathrm{B(x,y)} \cup \mathrm{C} (\mathrm{y}) \cup \mathrm{D}
(\mathrm{z}) \ldots \label{eq1} \Seqnsplit
A_{1}: && \mathrm{B(x, y)} \label{eq2} \Seqnsplit
A_{2}: && \mathrm{B(x)} \label{eq3}
\end{eqnarray}}\unskip
Note that the validation of this elegant theory given by $A_{2}$, which
allows its foundation, must also be done under these postulation
conditions.

Therefore, when we model a real problem using the main domain idealized
theory, the result would often be erroneous. This is due to committing
two approximations. The first is relative to overlooking the other
domains influences (replacing $A$ by $A_{1}$) and the second is due to
the use of idealizing postulations (replacing $A_{1}$ by $A_{2}$). The
more these two approximations are unfounded vis-\`{a}-vis the real
setting, the obtained results will be far from the reality. In such a
case, in order to adjust this situation, we have to track a reverse
procedure that to re-integer in the model, via coupling, all the
ignored aspects subsequent to the used approximations. Concerning the
reduction from expressions (\ref{eq1})--(\ref{eq3}), one can study a
given problem from different aspects corresponding to different
reductions involving different approximations. This depends, for a
multi-domain problem, on the investigated domain. For example, we will
consider a problem involving thermal and biological domains. When
studying thermal performance, one may tend to introduce biological
approximations for reduction and reciprocally.

\subsection{Revised coupled models and solution strategy}
\label{ssec32}

The reverse procedure mentioned in the last section will go through a
kind of revised model comprising the main theory associated with the
other theories involved and reintegrating into the model all the
characteristics ignored in the idealizing action. In general, coupled
problem schemes involve the mathematical solution of equations
governing different natural or artificial phenomena belonging to
distinct branches of the theoretical sphere. The nature of the
behaviors of these phenomena and their interdependence as well as the
proximity of their temporal evolution (time constants) are directly
linked to the approach of solving the corresponding governing
equations. Each of these behaviors can be linear or non-linear and have
a low or high time evolution. Moreover, these behaviors can be
independent or interdependent, which may or may not be linear. At one
extreme, we have the case of independent linear behaviors with very
distant time constants. In this case, we can solve the governing
equations individually. At the other extreme, we have the case of
nonlinear and, nonlinearly interdependent, behaviors with very close
time constants. In this situation, we need a strongly coupled
simultaneous solution of equations. Between these two extremes, the
equations can be solved in consecutive progression mode by iteration
according to the severity and the degree of complexity of the{\unskip\break}
behaviors.

Moreover, in general, the nature of the source, the behavior of the
matter and the geometry concerned in the real problems are more complex
than, those envisaged in a smart theory. Therefore, the spatial and
temporal behaviors of the different variables in the corresponding
equations are also more complicated compared to elegant theories. Such
a complex system of equations does not allow analytical solutions. In
order to apply the theories correctly, it is often necessary to
consider a discretized form in space and time of the equations. In this
case, the theories will operate locally in finite discrete domains for
which the global assembly solution will operate in the discretized time
domain. Spatial local non-linearity is considered by iterative
procedures.

\subsection{Case of electromagnetic and energy conversion systems}
\label{ssec33}

For a better understanding of the problem addressed in the last
section, we will consider an application in the field of
electromagnetic systems (EMS) including energy conversion drives. These
are present in many societal applications such as mobility, health,
security, communication, etc. In these systems, the intelligent
management, conversion and supervision of energy involve the use of an
accurate realistic representation of the arrangement concerned. A
revised realistic coupled model achieves this goal through its use in
system design, optimization, and control. The main field in such a case
is electromagnetic (EM), which is governed by Maxwell's equations.
However, EMS generally behave in four territories: electrical,
magnetic, mechanical and{\unskip\break} thermal.

\subsubsection{Maxwell equations}\label{sssec331}

This system of equations can be formulated mathematically in different
forms depending on the problem under consideration. One of the most
common is the basic full-wave electromagnetic formulation given by:
{\begin{eqnarray}
&& \bnabla \times \mathbf{H} = \mathbf{J}
\label{eq4} \Seqnsplit
&& \mathbf{J} = \sigma \mathbf{E} + \mathrm{j} \omega \mathbf{D} + \mathbf{J}\mathbf{e}
\label{eq5} \Seqnsplit
&& \mathbf{E} = {-} \bnabla \mathrm{V} - \mathrm{j} \omega \mathbf{A}
\label{eq6} \Seqnsplit
&& \mathbf{B} = \bnabla \times \mathbf{A}
\label{eq7}
\end{eqnarray}}\unskip
where $\mathbf{H}$ and $\mathbf{E}$ are the magnetic and electric
fields, $\mathbf{B}$ and $\mathbf{D}$ are the magnetic and electric
inductions, $\mathbf{A}$ and V are the magnetic vector and electric
scalar potentials. $\mathbf{J}$ and $\mathbf{J}\mathbf{e}$ are the
total and source current densities, $\sigma$ is the electric
conductivity and $\omega$ is the frequency pulsation. The symbol
$\bnabla$ is a vector of partial derivative operators, and its three
possible implications are gradient (product with a scalar field),
divergence and curl (dot and cross products respectively with a vector
field). The magnetic and electric behavior laws respectively between
$\mathbf{B}$/$\mathbf{H}$ and $\mathbf{D}$/$\mathbf{E}$ are
characterized respectively by the permeability $\mu$ and the
permittivity $\varepsilon$.

The solution of Equations (\ref{eq4})--(\ref{eq7}) permits to
determine in a system the concerns of electromagnetic fields for a
frequency pulsation accounting for the magnetic materials behaviors
through the permeability, for eddy currents in electric conductors
through the electric conductivity and for behavior of dielectrics
through the permittivity. Often, EMS involve other fields than EM. In
some cases, the influence of these other fields could be negligible and
it will be then possible to solve the problem correctly with only the
Maxwell's equations. In general, to model an EMS we need to account for
other fields in addition to EM field through the coupling of the
corresponding governing equations. As mentioned before, EMS behave
under four phenomena: electrical, magnetic, mechanical, and thermal.
The first three have small and relatively near-time constants while the
thermal phenomenon has a relatively higher time constant. The different
mixtures of these phenomena can be classified into causal (system
behavior), integrated (electrical and magnetic) and intrinsic material
(functional). The last mainly concerns intelligent materials such as
magnetostrictive, electrostrictive, shape-memory, thermoelectric\,{\ldots}.

\subsubsection{Coupling and solution of equations in EMS}
\label{sssec332}

The solution of the equations of the events involved must take into
account different specifications. The nature of the behavior of the
system concerned, involves analyses either in the frequency domain or
in the time domain. The fact that EMS often have complex geometries and
involve materials with nonlinear laws of behavior implies going through
a local distribution of variables such as, fields, potentials\,{\ldots}.
For this purpose, we use 2D or 3D discretized geometric cells, with
conditions defined on the boundaries of the discretized domains, see
e.g.~\cite{22,23}. The above-mentioned categories of couplings are
detailed as follows.

\paragraph{Integrated coupling}

Generally, in EMS the current is delivered by a voltage source through
an external electric circuit. The general relation between the voltage
$v$ and the current $i$ in the circuit is given by:
{\begin{equation}\label{eq8}
v = 1/C\cdot \int i \,\mathrm{d}t + r i + L\cdot \,\mathrm{d}i/\mathrm{d}t + 
\mathrm{d}\Psi/\mathrm{d}t + \entmath{1D15}{\text{\Taurus}}
\end{equation}}\unskip
In this expression $r$ is the total resistance of the circuit, $L$ a
linear inductance, $C$ a capacitance, \entmathtext{1D15}{\Taurus} a
non-linear voltage drop (e.g., a diode) in the electrical circuit and
${\Psi}$ the implied flux linkage.

This circuit equation should be solved coupled with the EM equations.
Therefore, the equations to solve are (\ref{eq4})--(\ref{eq8}).  This
coupling between the EM domain and the external electric circuit is
particular regarding other couplings with other domains than EM because
it represents a ``correction'' inside the EM domain. We call it
integrated coupling. Generally, the coupling of EM domain with the
external electric domain needs simultaneous strong solution of the
equations due to non-linearity of behaviors and closeness of the
magnetic and electric time constants, see e.g.~\cite{24}.

\paragraph{Causal couplings}

This class of couplings is related to system behavior. Typical
situations in this category are the EMS that governed by EM domain,
where the operation, the source or the outcome is directly related to
another domain. Most EMS related to energy conversion stand in this
category; for instance, the mechanical source or outcome respectively
in electric generators, e.g.~\cite{25} or motors e.g.~\cite{26}. These
cases may involve behavioral alterations in the EM and other areas
modified by each other. This happens when the behaviors are
interdependent. The solution of the equations for a given EMS would be
separate, iterative or strongly coupled, as mentioned before, depending
on the severity of the behaviors.

\medskip
\noindent \textit{EM and mechanical coupled problem.}
We consider the case of an EMS where beside EM the mechanical domain is
involved in forms of displacement or deformation. Let us consider the
example of the typical electromagnet given in Figure~\ref{fig1}, which
is a characteristic EMS involving electro--magneto--mechanical aspects
permitting to illustrate the consideration of these different 
domains~\cite{27}. It consists of a stationary part constituted of
non-conducting magnetic material ($\mu$) and a mobile armature made of
conducting magnetic material ($\mu$, $\sigma$). A coil fed by a voltage
source excites the stationary part. The mobile armature is connected to
a spring, a damper and an external force. The equations governing such
a system are:
{\begin{eqnarray}
m\cdot \,\mathrm{d}^{2}X/\mathrm{d}t^{2} + c\cdot \,\mathrm{d}X/\mathrm{d}t + k
X &=& F_{\mathrm{mag}} + F_{\mathrm{ext}} \label{eq9} \Seqnsplit
\mathrm{d}\Psi/\mathrm{d}t + r I &=& U \label{eq10}
\end{eqnarray}}\unskip
In these equations, $U$ is the source voltage and $I$ the current in
the exciting coil. $X$ is the displacement, $F_{\mathrm{mag}}$ and
$F_{\mathrm{ext}}$ are the magnetic and external forces, $m$, $c$ and
$k$ are respectively the mass of the moving object, the damping
coefficient and the stiffness of the spring. It may be noted that
(\ref{eq10}) is a particular case of (\ref{eq8}).

\begin{figure}
\includegraphics{fig01}
\caption{\label{fig1}Schematic of an electromagnet involving
electro--magneto--mechanical aspects~\cite{3}.}
\end{figure}

We consider for example in the system in Figure~\ref{fig1} a step
source voltage in the exciting coil. The unknown variables are the
current $I$ across the coil and the displacement $X$ of the mobile
armature. The magnetic linkage flux $\Psi$ and the magnetic force
$F_{\mathrm{mag}}$ generally could be nonlinear function of the
magnetic saturation and the mechanical motion. To solve the problem we
have to consider Equations (\ref{eq4})--(\ref{eq7}) with the
mechanical and circuit equations (\ref{eq9})--(\ref{eq10}). Generally,
the coupling of EM domain with the mechanical domain needs simultaneous
strong solution of the equations due to non-linearity of behaviors and
closeness of the time constants. 

\medskip
\noindent\textit{EM and thermal coupled problem.}
We consider the case of EMS where in addition to EM the thermal domain
is present in the form of heating production~\cite{28} or resulting
undesirable heating~\cite{29}. Heat production by means of EMS can be
magnetic induction heating by eddy currents in conducing metals owning
high conductivity or electric induction microwave heating in dielectric
materials possessing high permittivity~\cite{30}. The coupling of EM and
thermal domains involves phenomena with very different time constants.
Moreover, the problem may include non-linear behaviors and/or variables
that are interdependent. Here we need a weak separately iterative
coupling.

\paragraph{Material intrinsic couplings}

This class of couplings is relative to functional nature regarding
material intrinsic interactions. These concern mainly smart materials
that each linking two phenomena: magnetostrictive (magnetic--mechanic),
electrostrictive (electric--mechanic), shape-memory
(thermic--mechanic), and thermoelectric (thermic--electric).

The couplings in these cases relate to two groups. The first reflects
linear behavior (electrostrictive) and/or very different time constants
(shape-memory, thermoelectric). In this case, we can practice separate
solutions or coupled iterative solutions for respectively independent
or interdependent behavior~\cite{31,32}. The second concerns non-linear
behavior and/or close time constants (magnetostrictive). In function of
the complexity of the nonlinear relationships, we use strong coupling
or multiscale methodologies~\cite{33,34}.

\subsubsection{Supervised energy conversion systems}
\label{sssec333}

Energy conversion drives are frequently used in a wide range of
applications ranging from small household appliances of a few watts to
heavy industrial needs in megawatts, including mobility, medical,
robotics applications\,{\ldots}. These drives are supervised in several
ways depending on the nature of the application in terms of required
accuracy and required response time, ranging from slow to
instantaneous; see e.g.~\cite{35,36,37}. In any case, we need the most
accurate model of the drive involved in the control, which allows
efficient and robust supervision. These energy conversion devices can
be involved in simple automated systems or in complex supervised
adaptive and dynamic procedures. This topic will be discussed in the
next section.

\section{Online matching of the observation--modeling pair}
\label{sec4}

In Section~\ref{sec2}, we surveyed the virtues of offline
observation--modeling pairing. This duo is actively involved in many
natural and artificial processes operating in online (real-time)
pairing mode. This concerns both simple automated systems and complex
procedures. 

\subsection{Automated procedures}\label{ssec41}

Automated systems are used in various fields related to energy,
industrial manufacturing, mobility, health\,{\ldots}. In various
automated procedures, sensors are commonly used to determine specific
operating variables and system parameters. However, in some situations,
estimation can be used for variables or parameters that are difficult
to measure. Accurate parameter estimation plays a crucial role in the
operation of automated systems. The implementation of an estimation
algorithm on an embedded controller platform requires the
simplification of the mathematical model of the system. That is why we
often have to do this estimation offline to get reasonable accuracy.
For this, one can use Computer Aided Design (CAD) tools based on
complete models representing the systems in their environments (see
Section~\ref{ssec33}). In such a case, the matching of the estimated
parameters with the actual parameters would be successful. However, the
problem is that pairing cannot be instantaneous with the system
running. Various studies have proposed a compromise between the
precision of the estimation and the speed of the matching by
implementing, more sophisticated algorithms, on specialized platforms
of embedded controllers~\cite{35,36,37}. For this, in automated
systems, different types of observers, state filters and controllers
are offered as estimators. The robustness of the controller is
supported by the use of adaptive methods. Large-capacity
microcontrollers can improve controller board design and software
required for estimation, which iteratively targets the match
simultaneously. 

\subsection{Observation--modeling pairing in complex procedures}
\label{ssec42}

Real-time pairings in complex processes are present in different
natural circumstances practiced or involved in functions. In addition,
online matching of complex procedures is used in many innovative
applications.

\subsubsection{Modeling matching observation in natural processes}
\label{sssec421}

As mentioned in Section~\ref{sec1}, creatures often engage in the
practice of sensory observation and simultaneously use deductive skills
to manage their natural lives. Also that the activities of deduction
and prediction associated with observation are one of the first natural
duties born in the world. In this section, we will discuss and analyze
two natural processes, the dynamic adaptive camouflage in ecology and
the Bayesian Brain theory in neuroscience.

\paragraph{Dynamic camouflage}

In nature, based on observation, cases of mimetic simulation (imitation
strategy) are very frequent allowing camouflage~\cite{1}. This permits
creatures to blend into their surroundings. It may be a predation
strategy or an anti-predation adaptation. It relates to camouflage and
imitation that may involve visual, olfactory or auditory cover-up
through sensory systems. There are two main categories of camouflage. A
form of camouflage consists in the selection of a support, of the
environment on which to ``land'' or/and ``disappear''. The second form
of camouflage is that of dynamic metamorphosis. The first corresponds
to choose a matched environment in a single step, and the second
corresponds to a self-adapting (transfiguration) dynamic matching.
Thus, we have an offline matching resulting from a single observational
imitation in the first case and an online dynamic adapting matching in
the second. There is a significant literature regarding the
multiplicity, processes, roles, and evolution of camouflage, which is
measured by the sensory systems of predators targeted by camouflage;
see e.g.~\cite{38,39}. This depends on the ability of predators to
identify the impacts of predation-enforced selection, where changes in
environmental characteristics can be quantified. The victim needs, even
if it is complex, to identify changes in the visual systems, cognition
and behavior of predators. Just as any victim often uses multiple forms
of encryption; it is likely that their predators have multiple ways to
defeat them, in response to multiple types of prey. Indeed, the mimetic
victim individual adopts the appearance and colors of its environment
and remains motionless so as not to be detected by its predators. In
addition to color, some organisms are also able to take on the shape of
their viewpoint object. Many insects can thus take on the appearance of
branches or leaves. This defensive imitation gives the individual
protection against predation. There are also cases of offensive
imitation, which allows the mimetic individual to chase his victim
without being noticed. The junction between observation capacities and
mimetic capacities is practiced in a successive way, which allows the
improvement of these capacities.

\paragraph{Bayesian brain theory}

The Bayesian theory of the brain in neuroscience is widely recognized
when it comes to brain function. This theory briefly indicates that
after a cerebral sensory observation (vision, smell, hearing, etc.),
the predictive model of the brain generates, from the learned data,
cerebral perspectives of the observed phenomenon or object. Note that
in this case, the predictive model is managed by a sophisticated
supercomputer (Human brain: 10\tsup{11} neurons each linked to
10\tsup{4} others). Bayesian brain theory explains the cognitive
abilities of the brain to work under circumstances of uncertainty to
reach the optimum advocated by Bayesian methodologies~\cite{40}. It is
assumed that neural structure retains inner probabilistic patterns
revised by sensory information via neural processing~\cite{41}.
Bayesian inference works at the level of cortical macrocircuits, which
are structured according to a hierarchy that mirrors the observable
object scenes around us. The brain encodes a model of these objects and
makes predictions about their sensory input: predictive coding. The
corresponding areas of brain activity will be near the top hierarchy.
The links from the upper zones to the lower zones then convert a model
describing the scenes. The lowest-level predictions are compared to the
sensory inputs and the prediction error is distributed up the
hierarchy. This happens simultaneously at all hierarchical levels.
Predictions are sent and prediction errors are returned in a dynamic
process. The prediction error indicates that the actual model did not
fully account for the input. The next level readjustment can increase
the accuracy and reduce the prediction error~\cite{42,43}. It is clear
that the observation--prediction duo works in a real-time two-way
matching process. 

\subsubsection{Matching twins in complex procedures}
\label{sssec422}

In this section, matched twins in complex procedures will be examined,
which helps to expose the concept of digital twin. In
Section~\ref{ssec41}, we examined the role of the matching of
estimated and actual parameters in automated procedures. This
illustrated the need to improve the matching of virtual models to their
real procedures. We have seen that the nature of a real system and the
uncertainty of the emulation process often makes it difficult to build
a realistic virtual system and that we need a compromise between
estimation accuracy and speed adaptation in automated systems. These
two remarks are related to the improvement of the matching of virtual
models to their real procedures. Such an action depends on the
qualities of the virtual model and its interaction with the real
procedure. The quality of the virtual model is associated with its
ability to account for the environmental phenomena involved in the
actual procedure. The characteristic of the ``real--virtual'' link is
connected to detection, processing and control capabilities. The weight
of the matching improvement becomes particularly crucial in compound
procedures where the complexity concerns the various incorporated
components accounting for the physical phenomena involved (the notion
of complexity will be discussed in the next paragraph). To handle such
complex procedures, one can practice the Internet of Things (IoT) which
intensely deliberates in the physical domain via direct real-time data
collection, or Computer Aided Design (CAD), which focuses exclusively
on digital territory. However, it is essential to temper and control
the irregular and unnecessary behaviors that occur in these complex
procedures. Achieving such a goal requires a matched observation--model
twin practiced in the relevant procedure~\cite{44}. A consistent
representation of such a matched twin is shown in Figure~\ref{fig2}.
Such a twin differs from both IoT and CAD by focusing on both the
physical and digital spheres. This twin requires the practice of
different skills mainly involving detection (observation side),
calculation (model side) and the information and control link (between
observation side and model side). Detection on the observation side
concerns the various recognitions of the sensors. Model-side
computation could involve simulation, optimization, design, diagnosis,
prediction, and testing. These operations can use learned collected
data in addition to sensor data. The link between the observation side
and the model side is bidirectional. The observation part provides
sensor measurements in processed form to the model part while the
latter sends process and control information to the observation part.

\begin{figure}
\includegraphics{fig02}
\caption{\label{fig2}Schematics of a real-time bidirectional matched
observation--model twin in a complex procedure.}
\end{figure}

\subsubsection{Complex systems}\label{sssec423}

Generally, in the so-called complex procedure, the complexity concerns
the components and the physical phenomena involved. Complexity can be
defined in terms of interactions~\cite{45}. These can be classified
into three forms: simple, complicated and complex interactions. The
former simply behaves in a direct or linear manner, complicated
interactions are linear and loosely coupled while complex interactions
with tightly coupled links would be characteristic of a complex system
or procedure. Such a classification is reminiscent of the one mentioned
previously in Section~\ref{ssec33}, relating to the coupling of different
phenomena.

Coupling in a complex system involves its various components. This
could represent an oversized model and we can use model reduction
techniques, see e.g.~\cite{46}, while preserving accuracy depending on
the application concerned (modeling, design, optimization or online
supervision).  

\subsubsection{Digital twin concept}\label{sssec424}

The twin described in Section~\ref{sssec422} (Figure~\ref{fig2})
corresponds to the Digital Twin DT. Grieves~\cite{44} first
introduced this concept in 2002. It is distinguished by a beneficial
two-way communication between the digital and physical spheres. The
three components of a DT are a paired physical observable, a real-time
replicated digital element, and their sensory, processing, control, and
pairing links. The physical element dynamically adjusts its behavior in
real time according to the recommendations made by the digital element.
While the digital item correctly reproduces the real state of the
territory of the physical product. Thus, DT offers an intelligent
alliance of the physical and digital domains. Thus, in DT technology,
physical observation and virtual modeling are interconnected in a
reciprocal exchange in real time. The observed element corrects the
virtual error and the virtual element corrects the observed sensory
data. This iterative process leads to a more objective and intelligent
association. The DT concept is mainly used for fault diagnosis,
predictive maintenance, performance analysis and product
design~\cite{47}. This concerns various fields and innovative
industrial devices such as energy and utilities, aerospace and defense,
automotive transport, machinery manufacturing, healthcare and consumer
goods.

Note that similar uses of the concept of DT existed~\cite{48} before
its introduction in 2002 by Grieves~\cite{44}. As early as 1993, in
``Mirror Worlds'', David Gelernter evoked a similar concept, the
possibilities of software models, which represent a portion of
reality~\cite{49}. However, even before that, NASA used complex
simulations to monitor spacecraft safety~\cite{50}; then came the
unexpected explosion of the oxygen tank of the Apollo 13 mission in
1970~\cite{51}. Following this accident, the mission modified several
high-fidelity simulators to adapt them to the real conditions of the
damaged spacecraft and used them to land safely~\cite{52}. This was
probably one of the first real applications of a DT. This involved
several basic features of a DT, although it was not a familiar concept
in 1970.

\subsubsection{Examples of applications of DT}\label{sssec425}

Given the huge number of publications on DT, and to illustrate the
range of applications, from manufacturing to smart cities, we will
provide several examples from different areas of this work. One of DT's
most widely used businesses is industrial manufacturing and product
design. For example, the pairing of physical and virtual products can
be used for the iterative redesign of an existing product or for the
creation of a new product. Such DT-based product design can guide
manufacturers to support the product design process, see
e.g.~\cite{53,54}. Additionally, the integration of manufacturing data
and sensory data in the development of DT virtual products that can
enhance cyber-physical manufacturing capabilities can be
valuable~\cite{55}. Another activity of DT concerns predictive
maintenance, which is used in many fields. In the context of industrial
procedures, predictive maintenance has become an important concern; the
main objective is to optimize the maintenance schedule by predicting
system and process failures. Such an approach will result in a
reduction in unplanned system downtime and severe outages. In addition,
the advantages are the minimization of costs and the reduction of the
substitution of fundamental elements of the system, see
e.g.~\cite{56,57,58,59}. Additionally, we can mention healthcare services
using DT technology as an exciting and encouraging approach that can
promote progress efforts in medical innovations and improve clinical
and societal health outcomes~\cite{60,61}. In addition, DT's security
business as Cyber DT designed for cybersecurity protection~\cite{62}
and security of DT-based industrial automation and control
systems~\cite{63}. Also in control, DT technology is used for
applications in control centers of electrical systems and in
mechatronic systems~\cite{64}. Another activity concerns the
application of DT technology in EV smart electric vehicles. This
concerns various aspects such as autonomous navigation control, driver
assistance systems, vehicle health monitoring, battery management
systems, electronics and electric drive systems~\cite{65,66}. In
addition to the mentioned examples of using DT, we can mention some
innovative applications. The application of DT in the livestock sector
to improve large-scale precision farming practices, machinery and
equipment use, and the health and well-being of a wide variety of
animals~\cite{48}. Moreover, the application of DT in smart cities to
ensure smart aspects in real estate, transportation, construction,
health system, building, home, transportation and parking~\cite{67}.

\section{Conclusions}\label{sec5}

The analysis, discussion and evaluation carried out in this
contribution have brought to light the following points. Offline
matching in the observation--theory link has proven effective in
managing and governing elegant theories. We can synthesize the
characters of this duo as follows, a mathematical theory simply needs
observation to be credible and observation needs a theory to be
universal allowing further research. Real-time pairing in the
observation--modeling link governs natural phenomena and requires
comprehensive models in the supervision of automated and complex
physical procedures to behave in the most advantageous manner. The DT
concept has shown a wide range of innovative applications with
promising capabilities in various modern everyday uses.

\section*{Declaration of interests}

The authors do not work for, advise, own shares in, or receive funds
from any organization that could benefit from this article, and have
declared no affiliations other than their research organizations.

\section*{Funding}

This research received no external funding.

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