# Conformal anomaly and gravitational pair production

M. N. Chernodub  Institut Denis Poisson UMR 7013, Université de Tours, 37200
Tours, France

(August 31, 2024)

###### Abstract

We argue that the rate density of particle pair production $\Gamma$ in
background fields in conformal field theories is determined by the conformal
anomaly and related to anomalous trace of the energy-momentum tensor as
$\Gamma=(\pi/2)\langle T^{\mu}_{\ \mu}\rangle$ if the trace is positive (and
$\Gamma=0$ otherwise). This formula perfectly reproduces (presumably, non-
Hawking) radiation generated by static gravitational fields in the absence of
an event horizon via a new evaporation mechanism suggested recently. Our
relation also correctly describes the one-loop Schwinger pair creation in
massless (scalar and spinor) quantum electrodynamics. It also accurately
points to the Savvidi instability of the gluonic vacuum towards the formation
of the chromomagnetic condensate. Photon and neutrino pair production are also
discussed.

## Introduction.

Signatures of vacuum instability in a strong electric field were first found
in work by Sauter on the Klein paradox Sauter (1931). This effect has been
recognized and developed further by Heisenberg and Euler Heisenberg and Euler
(1936) and later formalized in terms of a pair production process in QED by
Schwinger Schwinger (1951, 1954).

The physical interpretation of this instability, often called the Schwinger
effect, is linked to the quantum vacuum fluctuations in which virtual pairs of
electrons $e^{-}$ and positrons $e^{+}$ are constantly created to be
annihilated shortly later. In a sufficiently strong background electric field,
the created $e^{+}e^{-}$ particles are taken away in opposite directions by
the field. As they are spatially separated, they cannot annihilate and become
real particles. Thus, a sufficiently strong electric field creates matter
($e^{+}e^{-}$ pairs) from the vacuum.

A similar phenomenon exists in gravitational fields near black holes. A black
hole emits the Hawking radiation HAWKING (1974); Hawking (1975), which can be
associated with the particle tunneling process Parikh and Wilczek (2000) in
which one particle from the pair, created in the vicinity of the event
horizon, gets swallowed by the hole while another particle has sufficient
energy escape to infinity. The escaping particles form the outgoing energy
flux, which diminishes the mass of the black hole and, therefore, leads to the
black hole evaporation. Due to a nonlocality of the tunneling process, this
effect operates in an extended vicinity above the black hole event horizon,
thus creating the notion of the quantum atmosphere Giddings (2016) (see also
Eune and Kim (2017)). It was recently suggested that such quantum atmospheres
could possess nontrivial thermodynamic features that can be probed in
condensed matter experiments Bermond  _et al._ (2022).

In 1+1 spacetime dimensions, the Hawking radiation can be related to a
gravitational (Einstein) anomaly which implies a non-conservation of energy-
momentum of a chiral particle in a curved spacetime Robinson and Wilczek
(2005). The anomaly appears due to quantum fluctuations when classical
symmetries are inconsistent with the quantization procedure Bertlmann (2000).
In 1+1 and 3+1 spacetime dimensions, the Hawking effect can also be
interpreted Christensen and Fulling (1977) in terms of conformal (or trace)
anomalies Duff (1994); Capper and Duff (1974a); Capper  _et al._ (1974)111It
is worth mentioning about terminology used in the paper. The terms “scale” and
“conformal” in relation to the symmetries of the system and the quantum
anomalies are often used interchangeably in the literature. Mathematically,
these concepts correspond to different symmetries as the requirement of local
conformal invariance is much stronger than the condition of global scale
invariance. Physically, the distinction between the scale and conformal
concepts is frequently ignored because all physically relevant scale-invariant
field theories in four spacetime dimensions also exhibit conformal invariance
Nakayama (2015). Moreover, since these anomalies are seen as a non-zero trace
of the energy-momentum tensor, they are also called “trace” anomalies..

In addition, one can argue that the particle creation in a static
gravitational field can also produce particles even without the event horizon
Wondrak  _et al._ (2023). In this scenario, the virtual pairs of particles are
separated by local tidal forces and become real particles, similar to what
happens in the Schwinger effect. Some of these real particles will fall to the
gravitating body and will later be recaptured, while other particles will
escape to infinity and create, similarly to the Hawking effect, an outgoing
flux of matter Wondrak  _et al._ (2023).

In our article, we argue that in the off-event-horizon mechanism of Ref.
Wondrak  _et al._ (2023) of particle pair production, the creation rate in the
background gravitational field can be directly related to the conformal
anomaly. We will show that our approach works for the Schwinger pair-
production mechanism in QED and is consistent with the Savvidi vacuum
instability in non-Abelian gauge theories Savvidy (1977).

We set $\hbar=c=1$ everywhere in the article and work in the mostly-plus
metric convention.

## Particle production and effective action.

The rate density of particle production events $dN/dt=\Gamma$ is determined by
the imaginary part, $\Gamma=2\,{\mathrm{Im}}\,{\mathcal{L}}_{\mathrm{eff}}$,
of the Lagrangian ${\mathcal{L}}_{\mathrm{eff}}$ associated with the effective
action Schwinger (1951),

$\displaystyle W=\int d^{4}x\sqrt{-g}{\mathcal{L}}_{\mathrm{eff}}\,,$ (1)

which takes into account quantum corrections. The rate $\Gamma$ has a sense of
non-persistence of vacuum due to pair creation Schwinger (1951, 1954).

In our paper, we argue that in conformal field theories, the pair-creation
rate $\Gamma$ in a background gravitational and gauge (electromagnetic or
gluon) fields can be related to the conformal (trace) anomaly:

$\displaystyle\Gamma=\frac{\pi}{2}{\left\langle T^{\mu}_{\
\mu}\right\rangle}\,,$ (2)

where ${\left\langle T^{\mu}_{\ \mu}\right\rangle}\equiv{\left\langle
T^{\mu}_{\ \mu}\right\rangle}_{\mathrm{an}}$ is the anomalous trace of the
energy-momentum tensor $T^{\mu\nu}$. Relation (2) is quantum because, in
classical conformal theories in an even number of spacetime dimensions, the
trace of the stress-energy tensor vanishes identically, $(T^{\mu}_{\
\mu})_{\mathrm{cl}}\equiv 0$. Quantum fluctuations can violate this identity,
${\left\langle T^{\mu}_{\ \mu}\right\rangle}\neq 0$, hence the term “trace
anomaly” or “conformal anomaly”.

In order to keep Eq. (2) as simple as possible, we used the convention that
this equation has a relation to the pair production if and only if
${\left\langle T^{\mu}_{\ \mu}\right\rangle}\geqslant 0$. Otherwise,
$\Gamma<0$ is equivalent to $\Gamma\equiv 0$ because a negative pair
production rate does not lead to the production of pairs.

The particle production rate of $N$ massless scalar degrees of freedom in the
curved $d=3+1$ dimensional spacetime (described by the metric $g_{\mu\nu}$) in
the presence of the classical electromagnetic field (characterized by the
field strength $F_{\mu\nu}$) has been found in the recent work Wondrak  _et
al._ (2023)222We have slightly re-arranged and combined the original
expressions of Ref. Wondrak  _et al._ (2023) for further convenience.:

$\displaystyle\Gamma_{N\mathrm{sc}}=\frac{N}{32\pi}$
$\displaystyle\biggl{[}\frac{1}{180}\bigl{(}R_{\mu\nu\alpha\beta}R^{\mu\nu\alpha\beta}-R_{\mu\nu}R^{\mu\nu}\bigr{)}$
(3)
$\displaystyle+\frac{1}{2}\left(\frac{1}{6}-\xi\right)^{2}R^{2}-\frac{q^{2}}{12}F_{\mu\nu}F^{\mu\nu}\biggr{]}\,,$

where the curved background is expressed via the Riemann tensor
$R_{\mu\nu\alpha\beta}$, the Ricci tensor $R_{\mu\nu}=R^{\alpha}_{\
\mu\alpha\nu}$, and the scalar curvature $R\equiv R^{\mu}_{\ \mu}$. The
subscript “$N$sc” in Eq. (3) stands for $N$ scalar degrees of freedom (for
example, $N=1$ for a neutral scalar field and $N=2$ for a complex scalar
field).

The quantity $q$ in Eq. (3) is the electric charge of the scalar particle
minimally coupled to electromagnetism. A neutral ($q=0$) scalar field carrying
one degrees of freedom ($N=1$) is described by the following Lagrangian:

$\displaystyle{\mathcal{L}}=-\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{2}\xi
R\phi^{2}-\frac{1}{2}m^{2}\phi^{2}\,,$ (4)

where the parameter $\xi$ controls the local coupling of the Ricci curvature
scalar $R$ to the scalar field. The conformally invariant massless theory
corresponds to $\xi=1/6$. For consistency with previous studies, we also add
to Eq. (4) the mass term, which will be set to zero at the end of our
considerations, $m=0$. The charged (complex) scalar field carrying the
elementary electric charge $q=e$ has $N=2$ degrees of freedom with
corresponding modifications of Eq. (4).

The remarkable feature of the pair-production effect (3) is that it can take
place in static gravitational fields, which immediately suggests that this
effect is a Hawking-type of radiation associated with the presence of an event
horizon HAWKING (1974); Hawking (1975). However, the pair production (3) takes
place even in the absence of an event horizon (that is, not only for a black
hole), thus indicating that this phenomenon is either an addition or a
generalization of Hawking radiation Wondrak  _et al._ (2023).

## Effective action, trace anomaly, and pair production.

It is instructive first to start from the simplest case of the scalar field
for which the trace anomaly has been elaborated in great detail in Ref. Brown
(1977). Our relation (2) between the conformal (trace) anomaly and the off-
event-horizon particle production can be deduced by matching the anomalous
term in the one-loop effective action $W$ represented as an integral over the
proper time $s$ of Ref. Brown (1977) with the representation of the same
action in terms of the spectral parameter $s$ in the heat-kernel approach of
Ref. Wondrak  _et al._ (2023) based on the Barvinsky–Vilkovisky expansion
Barvinsky and Vilkovisky (1990).

The one-loop action functional $W$ is given by a formal divergent expression,
$W=(i/2)\ln\det G^{-1}$, where $G(x,x^{\prime})\equiv{\left\langle
iT\bigl{(}\phi(x)\phi(x^{\prime})\bigr{)}\right\rangle}$ represents the Green
function associated with the quadratic Lagrangian (4):

$\displaystyle\Bigl{(}-\frac{1}{\sqrt{-g}}\partial_{\mu}\sqrt{-g}g^{\mu\nu}\partial_{\nu}+\xi
R$ $\displaystyle\,+m^{2}\Bigr{)}G(x,x^{\prime})$ (5)
$\displaystyle=\delta(x-x^{\prime})/\sqrt{-g}\,.$

The functional $W$ has a close relation to the expectation value of the
energy-momentum tensor, ${\left\langle T^{\mu\nu}\right\rangle}$, in its
response, $W\to W+\delta W$ to the metric variation, $g_{\mu\mu}\to
g_{\mu\nu}+\delta g_{\mu\nu}$ in (even) $D$ spacetime dimensions:

$\displaystyle\delta W=\frac{i}{2}{\mathrm{Tr}}\,G\delta G^{-1}=\int
d^{D}x\,\sqrt{-g}{\left\langle T^{\mu\nu}\right\rangle}\frac{1}{2}\delta
g_{\mu\nu}\,,$ (6)

thus giving access to the evaluation of the trace ${\left\langle T^{\mu}_{\
\mu}\right\rangle}$, allowing us to uncover an eventual conformal (trace)
anomaly.

The variation of the effective action (6) can be expressed in the proper-time
representation of Schwinger and DeWitt Schwinger (1951); DeWitt (1975) (in
notations of Brown (1977)):

$\displaystyle\delta
W=-\frac{i}{2}\delta{\mathrm{Tr}}\,\int_{0}^{\infty}\frac{ids}{is}e^{-isH}\,,$
(7)

via a relativistic Hamiltonian-like operator $H=\Delta+\xi R+m^{2}$, where a
second-order differential operator $\Delta$ represents the kinetic term, the
coupling to the curvature $R$ plays a role of an external potential, and
$m^{2}$ gives the mass term. The correct analytical properties of Eq. (7) and
similar subsequent relations are maintained by an appropriate complex
continuation of the mass term, $m^{2}\to m^{2}(1-i0^{+})$, silently assumed
here.

The effective Lagrangian (1) takes the following form:

$\displaystyle
L_{\mathrm{eff}}=\frac{1}{2}\frac{1}{(4\pi)^{\frac{D}{2}}}\int_{0}^{\infty}\frac{ids}{(is)^{1+\frac{D}{2}}}e^{-im^{2}s}F(x,x;is;D)\,,$
(8)

where $F(x,x^{\prime};is;D)$ is the weight bi-scalar in the proper-time
Green’s function $\langle x,s|x^{\prime},0\rangle=\langle
x|e^{-isH}|x^{\prime}\rangle$ defined in a manner similar to Eq. (8). The
Green’s function satisfies the Schrödinger-like equation:
$-\frac{\partial}{\partial is}\langle x,s|x^{\prime},0\rangle=H\langle
x,s|x^{\prime},0\rangle$, which gives a quantum-mechanical flavor to the whole
proper-time formalism.

We will not dwell on the precise definition of the bi-scalar $F$, which can be
found in detail in Refs. DeWitt (1975); Brown (1977). The key mathematical
point of our arguments is that the function $F$ allows for the power-series
expansion in terms of the proper time $s$ (omitting other arguments):

$\displaystyle F=1+is\,{\mathsf{f}}_{1}+(is)^{2}\,{\mathsf{f}}_{2}+\dots\,,$
(9)

where, in four space-time dimensions, the $O(s^{2})$ term captures the trace
anomaly Brown (1977). On the other hand, the $O(s^{2})$ term in an
identical333Taking into account the signs and $i$-th prefactors arising from
the difference between Minkowski/Euclidean spacetimes employed in Refs. Brown
(1977); Wondrak  _et al._ (2023) one finds that
${\mathsf{f}}_{1}=(\frac{1}{6}-\xi)R$ term in Eq. (A20) of the proper-time
approach of Ref. Brown (1977) coincides precisely with the second, $O(s)$ term
under the integral in Eq. (S.17) of the heat-kernel expansion of Ref. Wondrak
_et al._ (2023). Analogously, ${\mathsf{f}}_{2}$ in Eq. (A24) of Ref. Brown
(1977) coincides precisely with purely gravitational contribution to the
third, $O(s^{2})$ term under the integral in Eq. (S.17) of Ref. Wondrak  _et
al._ (2023). The ${\mathsf{f}}_{2}$ term in series (9) is also reproduced, up
to irrelevant contact term $\Box R$, by the $m=0$ expression in the square
brackets of our Eq. (13) below. Notice that our functions ${\mathsf{f}}_{a}$
in Eq. (9) correspond to $f_{a}$ of Ref. Brown (1977) and not to $f_{a}$ of
Ref. Wondrak  _et al._ (2023). expansion of the same 1-loop effective action
over the proper time $s$ has been shown in Ref. Wondrak  _et al._ (2023) to be
associated with the (off-event-horizon) pair-production rate $\Gamma$. The
mentioned equivalence of the $O(s^{2})$ terms allows us to identify the trace-
anomalous origin of the pair production and eventually leads us to Eq. (2) as
we discuss below.

The renormalized energy-momentum tensor,

$\displaystyle{\left\langle
T^{\mu\nu}\right\rangle}_{\mathrm{ren}}=\frac{1}{4}{\mathcal{A}}_{4}\,g^{\mu\nu}+{\text{non-
anomalous part}},$ (10)

contains the anomalous part given by an ${\mathcal{A}}_{4}$ function and a
non-anomalous part (not shown explicitly). In $D=4$ spacetime dimensions, the
${\mathcal{A}}_{4}$ function in stress-energy tensor (10) is related to the
$O(s^{2})$ prefactor in the power series expansion (9) of the bi-scalar $F$
Brown (1977):

$\displaystyle{\mathcal{A}}_{4}=\frac{1}{2}\frac{1}{(4\pi)^{2}}\left(\frac{\partial}{\partial
is}\right)^{2}\left[e^{-im^{2}s}F(x,x;is,4)\right]{\biggl{|}}_{s=0}.$ (11)

In the massless theory ($m=0$), the last term in Eq. (10) reduces to a
traceless tensor, and the trace of the energy-momentum tensor (10) is fully
determined by the trace (scale) anomaly (11):

$\displaystyle{\left\langle T^{\mu}_{\ \mu}\right\rangle}\equiv
g_{\mu\nu}{\left\langle
T^{\mu\nu}\right\rangle}_{\mathrm{ren}}={\mathcal{A}}_{4},\qquad\
[\mathrm{for}\ m=0]\,,$ (12)

where the short-hand notation ${\left\langle T^{\mu}_{\ \mu}\right\rangle}$ is
used for representational convenience (see also the discussion of Ref. Duff
(1994) on non-commutativity of the regularization operation and the trace
operation).

Finally, combining Eqs. (9), (11), (12) and matching them with the $O(s^{2})$
term in the effective action of Ref. Wondrak  _et al._ (2023) leads us to our
main result (2). In the rest of the paper, we ensure that Eq. (2) is valid for
physical environments where both sides of this equation are known. We also
discuss photon and neutrino pair production.

## A neutral scalar field in curved spacetime.

As the first check of our result (2), we consider a single-component neutral
scalar field of mass $m$ and generic non-conformal coupling $\xi$ to gravity
described by Lagrangian (4). It is well known that quantum fluctuations in
this theory produce the following trace anomaly Deser  _et al._ (1976); Brown
(1977):

$\displaystyle{\left\langle T^{\mu}_{\ \mu}\right\rangle}_{1\mathrm{sc}}$
$\displaystyle=\frac{1}{(4\pi)^{2}}\biggl{[}\frac{1}{180}R_{\mu\nu\alpha\beta}R^{\mu\nu\alpha\beta}-\frac{1}{180}R_{\mu\nu}R^{\mu\nu}$
(13) $\displaystyle+\frac{1}{6}\left(\frac{1}{5}-\xi\right)\Box
R+\frac{1}{2}\left(\frac{1}{6}-\xi\right)^{2}R^{2}+\frac{1}{2}m^{4}\biggr{]}\,.$

We substitute Eq. (13) to our formula (2) and recover exactly the result of
Ref. Wondrak  _et al._ (2023) given in Eq. (3) for the off-horizon pair
creation rate for a single ($N=1$) neutral ($q=0$) massless ($m=0$) scalar
field444These equations correspond to the same physical result since the $\Box
R$ term, present in Eq. (13) and absent in Eq. (3), can be removed by a finite
local counterterm during the renormalization procedure and is, therefore,
physically irrelevant.. The subscript “1sc” in Eq. (13) stresses that this
expression is derived for a one-component scalar field.

## General case.

A quantum field theory of $N_{S}$ scalar degrees of freedom, $N_{F}$ Dirac
fermions (a single Majorana or Weyl fermion contributes half of a Dirac
fermion, $N_{F}=1/2$) and $N_{V}$ species of massless vector fields, the trace
anomaly gets the following form Birrell  _et al._ (1984); Buchbinder and
Shapiro (2021); Duff (1994); Birrell  _et al._ (1984); Buchbinder and Shapiro
(2021):

$\displaystyle{\left\langle T^{\mu}_{\
\mu}\right\rangle}=bC^{2}+b^{\prime}E_{4}+cF_{\mu\nu}F^{\mu\nu}\,,\qquad$ (14)

where

$\displaystyle
C^{2}=R_{\mu\nu\alpha\beta}R^{\mu\nu\alpha\beta}-2R_{\mu\nu}R^{\mu\nu}+\frac{R^{2}}{3},$
(15)

is the Weyl tensor squared and

$\displaystyle
E_{4}=R_{\mu\nu\alpha\beta}R^{\mu\nu\alpha\beta}-4R_{\mu\nu}R^{\mu\nu}+R^{2},$
(16)

is the Euler density (the integrand of the topological, Gauss–Bonnet term) in
$D=4$ dimensions. In Eq. (14), the physically irrelevant $\Box R$ term is
omitted555See also a relevant discussion on $\Box R$ in Ref. Duff (1994). and
the conformal coupling ($\xi=1/6$ for scalars) is assumed. The parameters are
as follows Duff (1994); Capper and Duff (1974a); Capper  _et al._ (1974);
Capper and Duff (1974b); Brown (1977); Christensen and Fulling (1977):

$\displaystyle b$
$\displaystyle=\phantom{-}\frac{1}{120}\frac{1}{(4\pi)^{2}}\left(N_{S}+6N_{F}+12N_{V}\right)\,,$
(17) $\displaystyle b^{\prime}$
$\displaystyle=-\frac{1}{360}\frac{1}{(4\pi)^{2}}\left(N_{S}+11N_{F}+62N_{V}\right)\,.$
(18)

As an immediate check, one finds that in a pure gravitational background
($F_{\mu\nu}=0$), $N_{S}=N$ species of neutral scalar fields (with
$N_{D}=N_{V}=0$), Eq. (14) expectedly reduces to Eq. (13) with a factor $N$
and leads us, via Eq. (2), to the recent result (3).

The last term (14) represents a non-universal (“matter”) part which accounts
for renormalization effects related to the scale dependence of the couplings
of the theory. While Eq. (14) gives the matter term for a vector-field
background, a nontrivial scalar background in an interacting scalar field
theory can also generate a matter-type contribution to the conformal anomaly
which can be found in Ref. Asorey  _et al._ (2023).

For gauge vector fields coupled minimally with matter fields via the electric
coupling $e$, the prefactor

$\displaystyle c=-\frac{\beta(e)}{2e}\,,$ (19)

of the last term in Eq. (14) depends the beta function
$\beta(e)=\mu\,{\mathrm{d}}e/{\mathrm{d}}\mu$ associated with the running of
the coupling $e$. A nonvanishing beta function expresses the fact that
radiative corrections make the electric charge $e=e(\mu)$ dependent on the
renormalization energy scale $\mu$. This effect appears as a result of vacuum
polarization which implies, for example, that the electric charges of a
particle probed at a large distance (by a low-energy photon) does not match
the charge of the same particle at a short distance (probed by a high-energy
photon). Therefore, radiative corrections can break the scale invariance of
the system and naturally contribute to the scale anomaly (14).

Notice that the gravitational part of the anomaly represented by the first two
terms in Eq. (14) is exact in one loop implying that the higher-order
corrections to this expression vanish. This statement is not true for the
third, matter term since radiative corrections exist, generally, in all loops
Shifman (1989).

## Scalar QED in flat spacetime.

A similarity between the gravitational particle production and the Schwinger
pair production in flat spacetime in the background electric field has been
noticed in Ref. Wondrak  _et al._ (2023) on the basis of equation (3). Here we
show that the conformal anomaly plays an essential role in this relation.

Consider a theory of $N_{S}$ species of massless complex scalars coupled to
electromagnetism with the same electric coupling $e$ (a “scalar Quantum
Electrodynamics” or sQED). Since we already established the relation with the
gravitational part of the trace anomaly in this theory, we consider below a
flat spacetime where first two (gravitational) terms in the trace (14) vanish.
However, this theory still possesses the trace anomaly because its one-loop
beta function is non-zero Weisskopf (1936); Dunne (2004),

$\displaystyle\beta_{{\text{sQED}}}^{\mathrm{1loop}}=\frac{N_{S}e^{3}}{48\pi^{2}}\,.$
(20)

Equations (19) and (20) imply that the coefficient in the last term of the
trace anomaly (14) is $c=N_{S}e^{2}/(96\pi^{2})$. Then Eq. (2) gives us
$\Gamma_{\mathrm{sQED}}=-N_{S}e^{2}F_{\mu\nu}F^{\mu\nu}/(192\pi)$, which
exactly coincides with the pair production rate (3) of Ref. Wondrak  _et al._
(2023) if one takes into account that each complex field carries two degrees
of freedom: $N=2N_{S}$.

According to our convention in Eq. (2), there is no particle production for a
negative production rate. Since
$F_{\mu\nu}F^{\mu\nu}=2({\bm{B}}^{2}-{\bm{E}}^{2})$, implies the absence of
particle creation in a pure magnetic field because $\Gamma_{\mathrm{sQED}}<0$.
However, in the electric-field background, one gets the following well-known
result the complex scalar field (reproduced also in Ref. Wondrak  _et al._
(2023) for $N_{S}=1$):

$\displaystyle\Gamma_{\mathrm{sQED}}=N_{S}\frac{e^{2}{\bm{E}}^{2}}{96\pi}\,.$
(21)

This equivalence further supports the validity of Eq. (2) in one loop.

## Spinor QED.

The pair creation rate in the (spinor) QED with a single flavor of massless
Dirac fermion and a single gauge (electromagnetic) field can be derived with
the use of correspondence (2) together with the anomaly relations (14)–(19) by
setting $N_{S}=0$, $N_{F}=1$, and $N_{V}=1$. Taking into account that the one-
loop beta function of the spinor QED is four times bigger (per particle) than
its scalar analogue (22) Dunne (2004):

$\displaystyle\beta_{{\text{QED}}}^{\mathrm{1loop}}=\frac{e^{3}}{12\pi^{2}}\,,$
(22)

one gets the following prediction for the particle production rate:

$\displaystyle\Gamma^{(m=0)}_{\mathrm{QED}}=\frac{1}{11\,520\pi}$
$\displaystyle\biggl{(}-19R_{\mu\nu\alpha\beta}R^{\mu\nu\alpha\beta}+184R_{\mu\nu}R^{\mu\nu}$
(23) $\displaystyle-55R^{2}\biggr{)}-\frac{e^{2}}{48}F_{\mu\nu}F^{\mu\nu}\,,$

Notice that in flat background, the particle production rate in the massless
QED (23) reduces exactly to the well-known QED result Schwinger (1951); Dunne
(2004) in the limit $m\to 0$:

$\displaystyle\Gamma^{(m=0)}_{\mathrm{QED}}=\frac{e^{2}{\bm{E}}^{2}}{24\pi}\qquad\mathrm{[electromagnetic]}\,.$
(24)

To estimate the contribution of the gravitational part to the production rate,
consider now a purely gravitational background given by the static
Schwarzschild spacetime of a body with the mass $M$:

$\displaystyle ds^{2}$
$\displaystyle=-\left(1-\frac{2MG}{r}\right)dt^{2}+\left(1-\frac{2MG}{r}\right)^{-1}dr^{2}$
$\displaystyle+r^{2}(d\theta^{2}+\sin^{2}\theta d\varphi^{2})$ (25)

Given Ricci flatness ($R_{\mu\nu}=0$) of this metric, the gravitational
contribution to the pair-creation rate (23) is provided only by the
Kretschmann scalar:

$\displaystyle
K=R_{\mu\nu\alpha\beta}R^{\mu\nu\alpha\beta}=48\frac{G^{2}M^{2}}{r^{6}}\,.$
(26)

Thus, it appears that the purely gravitational contribution to the pair
production rate (23) is always negative

$\displaystyle\delta\Gamma^{(m=0)}_{\mathrm{QED}}=-\frac{19}{240\pi}\frac{G^{2}M^{2}}{r^{6}}\qquad\mathrm{[gravitational]}\,,$
(27)

implying that our conformal anomaly mechanism alone cannot create particles
outside of the event horizon even in the presence of the strong gravitational
field and even for massless QED. Moreover, our result imply that the ordinary
Schwinger pair production due to background electric field (24) will be
inhibited by the gravitational contribution (27) in curved spacetime.

It is worth here mentioning that the considerations of Ref. Duff (1994) on the
sense of the conformal anomaly in the context of renormalization of quantum
field theories suggest that in conformally-non-invariant theories (for
example, for massive fields), the right-hand-side of Eq. (2) should be
modified: ${\left\langle T^{\mu}_{\ \mu}\right\rangle}\to
g_{\mu\nu}{\left\langle T^{\mu\nu}\right\rangle}_{\mathrm{ren}}-{\left\langle
g_{\mu\nu}T^{\mu\nu}\right\rangle}_{\mathrm{ren}}$. This conjecture implies,
in particular, that the explicitly non-conformal mass term $m^{4}$ will not
enter Eq. (2).

Coming back to the massless case in flat spacetime, the proportionality of the
pair-creation rates for scalar (21) and spinor (23) QED to their beta
functions, Eqs. (20) and (22), respectively, is not surprising given an
intimate relation between the effective Euler-Heisenberg Lagrangian and the
beta function (for an excellent review, see Ref. Dunne (2004)). Since the beta
function also contributes to the trace anomaly, the relation of the trace
anomaly to the pair-creation rate closes the logical triangle, thus
qualitatively supporting Eq. (2) on physical grounds.

Our results (2) suggest that in the flat spacetime, the creation rate of pairs
of massless particles in classical (electromagnetic) background is related to
the respective beta function:

$\displaystyle\Gamma_{\mathrm{flat}}=-\frac{\pi\beta(e)}{4e}F_{\mu\nu}F^{\mu\nu}\,.$
(28)

This result should be valid at least in one-loop order with the already
mentioned reservation that a negative production rate implies no production.

## No photon production.

In realistic QED in weak background electromagnetic fields with the strength
below the Schwinger limit, the four-photon scattering can be neglected Dunne
(2004), so that photon propagation can be described by free Maxwell theory
with simple Lagrangian,
${\mathcal{L}}_{\mathrm{ph}}=-(1/4)F_{\mu\nu}F^{\mu\nu}$. The particle
creation rate corresponds to $N_{F}=0$, $N_{S}=0$ and $N_{V}=1$ and gives us
the following discouraging result $\Gamma_{\mathrm{ph}}=-13G^{2}M^{2}/(120\pi
r^{4})<0$ implying that no photons can created in the gravitational field due
to this conformal anomaly mechanism.

## Neutrino–anti-neutrino pairs.

Similar considerations can also be applied to neutrino–anti-neutrino pair
creation with the appropriate replacement of the spinor degrees of freedom by
the sum of Dirac and Majorana/Weyl neutrino species: $N_{D}\to
N_{\nu}=N_{D}+(1/2)N_{M}$ and taking $N_{V}=N_{S}=0$ in the above expressions.
One gets $\Gamma_{\nu}=7G^{2}M^{2}/(240\pi r^{4})>0$, so that the pairs of
light neutrino can potentially be created by sufficiently strong background
gravitational field. For a Dirac neutrino, the pair production rate
$\Gamma_{\nu}$ is $7/2$ times bigger than the rate of pair production for
scalar particles which is about two times bigger than the one for the Hawking
radiation. The relevant estimations for scalar particles in physically
interesting gravitational fields can be found in Ref. Wondrak  _et al._
(2023).

Finally, one could ask whether these results, derived for massless spinors,
are applicable to particles with mass $m_{\nu}$. For the pure electromagnetic
contribution to the pair creation rate (23), the condition is well known Dunne
(2004): the electric field strength should substantially exceed the critical
Schwinger field, $E\gg E_{c}^{\mathrm{Sch}}=m^{2}_{e}/e\simeq 1.3\times
10^{18}\,\mathrm{V/m}$. Likewise, the same condition can be obtained by
demanding that the gravitational contribution should exceed the
anticipated666Cf. the last term in Eq. (13). $\propto m_{e}^{4}$ term
generated by the explicit breaking of the conformal symmetry. For neutrinos in
the field of a gravitating body with mass $M$, the applicability condition
then reads as $r\ll r_{c}$ with the critical field $r_{c}=\sqrt{GM}/m_{\nu}$
up to a $O(1)$ factor.

## Savvidi magnetic instability in QCD.

In the pure magnetic field, both in scalar QED and spinor QED, the right-hand
of Eq. (2) is a negative quantity and, therefore, no instability associated
with the particle production can occur. Of course, this natural conclusion is
supported by the fact that their beta functions, Eqs. (20) and (22), are
positively defined. But what happens if the beta function is negative?

Consider, for example, Yang-Mills (YM) theory which determines non-
perturbative properties of Quantum Chromodynamics (QCD). The beta function of
$N_{c}$-color YM theory, $\beta_{\mathrm{YM}}(g)=-11N_{c}g^{3}/(48\pi^{2})$,
is a negative function of the strong coupling constant $g$. Adopting Eq. (28)
to non-Abelian fields possessing the field-strengths $F_{\mu\nu}^{a}$, one
gets the following formal perturbative expression for the gluon production
rate:

$\displaystyle\Gamma^{\mathrm{pert}}_{\mathrm{YM}}$
$\displaystyle=\frac{11N_{c}g^{2}}{192\pi}F^{a}_{\mu\nu}F^{a,\mu\nu}$
$\displaystyle\equiv\frac{11N_{c}}{96\pi}\left[(g{\bm{B}}^{a})^{2}-(g{\bm{E}}^{a})^{2}\right]\,,$
(29)

where the sum over the gluon species, $a=1,\dots,N_{c}^{2}-1$, is implicitly
assumed.

Equation (29) represents a formal expression which is not applicable to the
ground state of YM theory because Eq. (29) corresponds to the anomalous
breaking of scale symmetry associated with the perturbative renormalization of
couplings – hence the superscript “pert” in Eq. (29) – while in YM theory, the
conformal symmetry is broken dynamically and non-perturbatively777In QCD, the
magnitude of the dynamical breaking of conformal symmetry, ${\left\langle
T^{\mu}_{\ \mu}\right\rangle}\simeq\Lambda_{\mathrm{QCD}}^{4}$, is determined
by an intrinsic mass scale $\Lambda_{\mathrm{QCD}}$ of the order of a few
hundred MeV.. Despite this fact, Eq. (29) still allows us to make another
interesting relationship with already known effect: the instability of the
perturbative gluonic vacuum. Indeed, since
$\Gamma_{\mathrm{YM}}=11N_{c}(g{\bm{B}}^{a})^{2}/(96\pi)>0$, even the weakest
background gluomagnetic field leads to the creation of gluon pairs and makes
the gluonic vacuum unstable. This observation matches well with the
instability of the perturbative gluon vacuum Nielsen and Olesen (1978) which
drives creation of the magnetic condensate (the Savvidi vacuum Savvidy (1977))
and the formation of the suggested magnetic-spaghetti vacuum state Nielsen and
Olesen (1979); Ambjorn and Olesen (1980) precisely due to the negativeness of
the YM beta function, $\beta_{\mathrm{YM}}(g)<0$ (see also a related critical
discussion in Ref. Yildiz and Cox (1980)).

## Conclusions.

We suggested the simple formula (2) for the off-horizon particle production
rate in curved spacetime proposed in Ref. Wondrak  _et al._ (2023) and argued
that its underlying mechanism is based on the anomalous breaking of the
conformal symmetry. The anomalous particle production can occur in static
gravitational fields and can operate, in particular, above the event horizons
of black holes. These two properties discriminate the anomalous production
from the dynamical Casimir effect in time-dependent backgrounds and the
Hawking mechanism of particle production, which occurs near the black-hole
horizons.

Our formula (2) agrees with known results for the pair production rate in the
gravitational background for a scalar field presented recently in Ref. Wondrak
_et al._ (2023) where the off-horizon pair production has been suggested
first. We also reproduce the known expressions for the Schwinger pair
production in QCD with scalar and spinor particles. Our approach also supports
instability in the perturbative gluonic vacuum in the chromomagnetic field,
thus suggesting the formation of the magnetic condensate in accordance with
widely accepted ideas about the nature of the QCD vacuum.

Our mechanism suggests that the photon pairs cannot be produced in a static
Schwarzschild spacetime. However, our estimations show that a sufficiently
strong gravitational field can create pairs of neutrinos and anti-neutrinos
(as well as other light spinors), thus providing us with another channel for
the evaporation of black holes and other gravitating objects.

## References

  * Sauter (1931) Fritz Sauter, “Uber das Verhalten eines Elektrons im homogenen elektrischen Feld nach der relativistischen Theorie Diracs,” Z. Phys. 69, 742–764 (1931).
  * Heisenberg and Euler (1936) W. Heisenberg and H. Euler, “Consequences of Dirac’s theory of positrons,” Z. Phys. 98, 714–732 (1936), arXiv:physics/0605038 .
  * Schwinger (1951) Julian Schwinger, “On gauge invariance and vacuum polarization,” Physical Review 82, 664–679 (1951).
  * Schwinger (1954) Julian Schwinger, “The Theory of Quantized Fields. 5,” Phys. Rev. 93, 615–628 (1954).
  * HAWKING (1974) S. W. HAWKING, “Black hole explosions?” Nature 248, 30–31 (1974).
  * Hawking (1975) S. W. Hawking, “Particle creation by black holes,” Communications In Mathematical Physics 43, 199–220 (1975).
  * Parikh and Wilczek (2000) Maulik K. Parikh and Frank Wilczek, “Hawking radiation as tunneling,” Physical Review Letters 85, 5042–5045 (2000).
  * Giddings (2016) Steven B. Giddings, “Hawking radiation, the Stefan–Boltzmann law, and unitarization,” Phys. Lett. B 754, 39–42 (2016), arXiv:1511.08221 [hep-th] .
  * Eune and Kim (2017) Myungseok Eune and Wontae Kim, “Proper temperature of the Schwarzschild AdS black hole revisited,” Phys. Lett. B 773, 57–61 (2017), arXiv:1703.00589 [gr-qc] .
  * Bermond  _et al._ (2022) Baptiste Bermond, Maxim Chernodub, Adolfo G. Grushin, and David Carpentier, “Anomalous Luttinger equivalence between temperature and curved spacetime: From black hole’s atmosphere to thermal quenches,” (2022), arXiv:2206.08784 [cond-mat.stat-mech] .
  * Robinson and Wilczek (2005) Sean P. Robinson and Frank Wilczek, “A Relationship between Hawking radiation and gravitational anomalies,” Phys. Rev. Lett. 95, 011303 (2005), arXiv:gr-qc/0502074 .
  * Bertlmann (2000) Reinhold A Bertlmann,  _Anomalies in quantum field theory_, Vol. 91 (Oxford university press, 2000).
  * Christensen and Fulling (1977) S. M. Christensen and S. A. Fulling, “Trace Anomalies and the Hawking Effect,” Phys. Rev. D 15, 2088–2104 (1977).
  * Duff (1994) M. J. Duff, “Twenty years of the Weyl anomaly,” Class. Quant. Grav. 11, 1387–1404 (1994), arXiv:hep-th/9308075 .
  * Capper and Duff (1974a) D. M. Capper and M. J. Duff, “The one loop neutrino contribution to the graviton propagator,” Nucl. Phys. B 82, 147–154 (1974a).
  * Capper  _et al._ (1974) D. M. Capper, M. J. Duff, and L. Halpern, “Photon corrections to the graviton propagator,” Phys. Rev. D 10, 461–467 (1974).
  * Nakayama (2015) Yu Nakayama, “Scale invariance vs conformal invariance,” Phys. Rept. 569, 1–93 (2015), arXiv:1302.0884 [hep-th] .
  * Wondrak  _et al._ (2023) Michael F. Wondrak, Walter D. van Suijlekom, and Heino Falcke, “Gravitational pair production and black hole evaporation,” Physical Review Letters 130 (2023), 10.1103/physrevlett.130.221502.
  * Savvidy (1977) G. K. Savvidy, “Infrared Instability of the Vacuum State of Gauge Theories and Asymptotic Freedom,” Phys. Lett. B 71, 133–134 (1977).
  * Brown (1977) Lowell S. Brown, “Stress-tensor trace anomaly in a gravitational metric: Scalar fields,” Physical Review D 15, 1469–1483 (1977).
  * Barvinsky and Vilkovisky (1990) A. O. Barvinsky and G. A. Vilkovisky, “Covariant perturbation theory. 2: Second order in the curvature. General algorithms,” Nucl. Phys. B 333, 471–511 (1990).
  * DeWitt (1975) Bryce S. DeWitt, “Quantum field theory in curved spacetime,” Physics Reports 19, 295–357 (1975).
  * Deser  _et al._ (1976) S. Deser, M.J. Duff, and C.J. Isham, “Non-local conformal anomalies,” Nuclear Physics B 111, 45–55 (1976).
  * Birrell  _et al._ (1984) Nicholas David Birrell, Nicholas David Birrell, and Paul CW Davies,  _Quantum fields in curved space_ (Cambridge University Press, 1984).
  * Buchbinder and Shapiro (2021) Iosif L Buchbinder and Ilya Shapiro,  _Introduction to quantum field theory with applications to quantum gravity_ (Oxford University Press, USA, 2021).
  * Capper and Duff (1974b) D. M. Capper and M. J. Duff, “Trace anomalies in dimensional regularization,” Nuovo Cim. A 23, 173–183 (1974b).
  * Asorey  _et al._ (2023) Manuel Asorey, Wagno Cesar e Silva, Ilya L. Shapiro, and Públio R. B. do Vale, “Trace anomaly and induced action for a metric-scalar background,” Eur. Phys. J. C 83, 157 (2023), arXiv:2202.00154 [hep-th] .
  * Shifman (1989) Mikhail A. Shifman, “Anomalies and Low-Energy Theorems of Quantum Chromodynamics,” Sov. Phys. Usp. 32, 289–309 (1989).
  * Weisskopf (1936) V. Weisskopf, “The electrodynamics of the vacuum based on the quantum theory of the electron,” Kong. Dan. Vid. Sel. Mat. Fys. Med. 14N6, 1–39 (1936).
  * Dunne (2004) Gerald V. Dunne, “Heisenberg-Euler effective Lagrangians: Basics and extensions,” in  _From fields to strings: Circumnavigating theoretical physics. Ian Kogan memorial collection (3 volume set)_, edited by M. Shifman, A. Vainshtein, and J. Wheater (2004) pp. 445–522, arXiv:hep-th/0406216 .
  * Nielsen and Olesen (1978) N. K. Nielsen and P. Olesen, “An Unstable Yang-Mills Field Mode,” Nucl. Phys. B 144, 376–396 (1978).
  * Nielsen and Olesen (1979) Holger Bech Nielsen and P. Olesen, “A Quantum Liquid Model for the QCD Vacuum: Gauge and Rotational Invariance of Domained and Quantized Homogeneous Color Fields,” Nucl. Phys. B 160, 380–396 (1979).
  * Ambjorn and Olesen (1980) Jan Ambjorn and P. Olesen, “On the Formation of a Random Color Magnetic Quantum Liquid in QCD,” Nucl. Phys. B 170, 60–78 (1980).
  * Yildiz and Cox (1980) Asim Yildiz and Paul H. Cox, “Vacuum behavior in quantum chromodynamics,” Physical Review D 21, 1095–1099 (1980).


# Non-Linear Estimation using the Weighted Average Consensus-Based Unscented
Filtering for Various Vehicles Dynamics towards Autonomous Sensorless Design

Bambang L. Widjiantoro 1, Moh Kamalul Wafi 2*, Katherin Indriawati 3
1,2,3 Department of Engineering Physics, Institut Teknologi Sepuluh Nopember,
Surabaya, Indonesia
Email: 1<EMAIL_ADDRESS>2<EMAIL_ADDRESS>3
<EMAIL_ADDRESS>
*Corresponding Author

###### Abstract

The concerns to autonomous vehicles have been becoming more intriguing in
coping with the more environmentally dynamics non-linear systems under some
constraints and disturbances. These vehicles connect not only to the self-
instruments yet to the neighborhoods components, making the diverse
interconnected communications which should be handled locally to ease the
computation and to fasten the decision. To deal with those interconnected
networks, the distributed estimation to reach the untouched states, pursuing
sensorless design, is approached, initiated by the construction of the
modified pseudo measurement which, due to approximation, led to the weighted
average consensus calculation within unscented filtering along with the
bounded estimation errors. Moreover, the tested vehicles are also associated
to certain robust control scenarios subject to noise and disturbance with some
stability analysis to ensure the usage of the proposed estimation algorithm.
The numerical instances are presented along with the performances of the
control and estimation method. The results affirms the effectiveness of the
method with limited error deviation compared to the other centralized and
distributed filtering. Beyond these, the further research would be the
directed sensorless design and fault-tolerant learning control subject to
faults to negate the failures.

###### Keywords:

autonomous vehicles, estimation method, unscented Kalman filtering, the
weighted average consensus filtering

##  I Introduction

The ideas of autonomous vehicles being operated in certain environment have
been discussed before the 20th century [1], from underwater, land, and air,
however the future challenges with advanced manufacturing are constantly open
to tackle [2] in addition to the current surveys [3]. The breakthrough in
communication technologies along with the more dynamics entities causes this
subject more reality with the guarantees from the stability studies, such as
using neural-network [4] and under some regular switching methods [5].
Moreover, the stability of the discretized autonomous system was also studied
[6] in a connected networked system [7] subject to delays while considering
the stability controls [8, 9]. To give more comprehensive results, the
examined autonomous vehicles in this paper comprise various dynamical systems;
the adaptive cruise control [10, 11], the active suspension system [12, 13],
the electric aircraft [14, 15], and the DC motor drive system [16, 17]. To
deal with the more interconnected systems, the control design is required from
the basic feedback control [18] to intelligent-based control [19, 20] in real-
time [21] with constraints [22].

The stability analysis and control designs, as the discussions in the paper,
then leads to the modest-cost implementation using sensorless design in lieu
of the hardware-sensor from the basic estimation methods. These methods vary
in applications, comprising in standard filtering [23], the adaptive
estimation method [24], and the most current distributed estimation [25],
while this paper focuses on the unscented Kalman filtering (UKF) as the
foundation. This UKF estimation, said to answer the lack of EKF estimation in
terms of the Gaussian Random variable (GRV), has been widely applied in non-
linear systems [26] and the adaptive dynamical systems [27] with considering
the stochastic uncertainties [28]. However, this sensorless design is supposed
to be distributed, therefore the modified UKF estimation, which is the one
applied in this paper, is required to construct as written in [29] with sensor
networks and it would be propagated into the mentioned vehicles. The pseudo
measurement matrix presented in [30] was the basis of this development
mentioning the Markov non-linear system while this was the linearized
approximation [31] and the ultimate consensus-based algorithms in multi-
vehicle implementing cooperative control were studied in [32] and also for the
sake of distributed filtering [33].

From those, this research focuses to study the effectiveness of the proposed
distributed estimation into various vehicles as the basis of sensorless
networks which could be developed into the simpler relaxed computation [34] to
ease the decision. Beyond that, the simplified robust sensorless algorithms in
electric vehicles and drives are our upcoming concerns of research in addition
to fault-tolerant distributed learning control taking into account these
research from [35, 36, 37, 38, 39] and [40, 41, 42, 43, 44, 45, 46, 47].

##  II Mathematical Dynamics

This section focuses on building the dynamics vehicle-related systems being
used the examine the effectiveness of the proposed scenarios. There are five
various vehicle plants, from the car cruise control, the quarter bus-
suspension with disturbance, the longitudinal-pitch of the aeroplane, to the
speed and position of DC motor. Beyond that, the control mechanism along with
the stability discussion under some limited assumptions are written in the
following sections before the algorithms are then proposed, leading to the
sensorless designs.

###  II-A The Car Cruise Control Model

In many recent contemporary vehicles, the development of any advanced self-
acting controls is demanding to guarantee the safety alongside the comfort, no
exception to cruise control. This cruise control is designed to maintain a
stable desired speed regardless arbitrary disturbances, including the
alterations of winds and roads.

Figure 1: Free-body dynamic of the car

Moreover, it is then achieved by comparing the measured and the desired speed,
which regulating the throttle based on the designed control scenario. The
dynamic of the vehicle ($m$) is depicted in Fig.(1) with the force ($u$) being
produced at the road surface, by assuming the perfect control to this force
and neglecting the arbitrary forces acting on producing the force. By contrast
on the mode’s motion, the resistive external forces ($bv$) are implied to be
linearly changed with respect to the velocity ($v$). Eq.(1) is the dynamic of
the Newton’s second law along with the measured system ($y$),

$\displaystyle m\dot{v}+bv=u,\qquad\textrm{and}\qquad y=v$ (1)

and the basic state-space representative constitute as follows,

$\displaystyle\begin{aligned}
\dot{\textbf{x}}=\left[\dot{v}\right]&=\left[\frac{-b}{m}\right]v+\left[\frac{1}{m}\right]u,\\\
y&=1\cdot v\end{aligned}$ (2)

which is from Eq.(2), the transfer function, $\Phi_{n}(s),\forall n=1,\dots$
showing the order of the examined systems, results in Eq.(3),

$\displaystyle\Phi_{1}(s)=\frac{V(s)}{U(s)}=\frac{1}{ms+b}\frac{m}{Ns}$ (3)

###  II-B A Quarter Bus-Suspension Design

The attractive advanced suspension designs are becoming more intriguing, being
linked to autonomous design. This active suspension scenario, with actuator
enabling to produce the control force ($u$) acting on the body motion control,
is derived from a-quarter simplified bus-mode design. The variables are
explained as follows; $M_{1}$ and $M_{2}$ denote the a-quarter body and
suspension mass, the constants of springs ($k_{n}$) and dampers ($b_{n}$),
$\forall n=1,2$, of suspension and wheel in turn along with non-linear
disturbance ($\gamma$). From Newton’s law the equations of motions could be
written as Eq.(6),

Figure 2: A quarter bus suspension

$\displaystyle\begin{gathered}M_{1}\ddot{x}_{1}=-\psi_{b_{1}}-\psi_{k_{1}}+u\\\
M_{2}\ddot{x}_{2}=\psi_{b_{1}}+\psi_{k_{1}}+\psi_{b_{2}}+\psi_{k_{2}}-u\end{gathered}$
(6)

with $\psi_{\bullet}$ defines the associated dynamics of ($\bullet$)-term,
such that

$\displaystyle\begin{aligned}
\psi_{b_{1}}&=b_{1}\left(\dot{x}_{1}-\dot{x}_{2}\right)\quad&\quad\psi_{b_{2}}&=b_{2}\left(\dot{\gamma}-\dot{x}_{2}\right)\\\
\psi_{k_{1}}&=k_{1}(x_{1}-x_{2})\quad&\quad\psi_{k_{2}}&=k_{2}(\gamma-
x_{2})\end{aligned}$

and the Laplacian functions, with zero initial conditions along with the input
($u,\gamma$) and the output ($x_{1}-x_{2}$) are drawn as,

$\displaystyle M_{1}s^{2}X_{1}(s)+\Psi_{b_{1}}(s)+\Psi_{k_{1}}(s)=U(s)$
$\displaystyle
M_{2}s^{2}X_{2}(s)-\Psi_{b_{1}}(s)-\Psi_{k_{1}}(s)-\Psi_{b_{2}}(s)-\Psi_{k_{2}}(s)=-U(s)$

which could be concluded into standard algebraic equation,

$\displaystyle\textbf{F}x=\textbf{g}$ (7)

where the terms of $\textbf{F},x$, and g in Eq.(7) are made of,

F $\displaystyle=\begin{bmatrix}M_{1}s^{2}+b_{1}s+k_{1}&-(b_{1}s+k_{1})\\\
-(b_{1}s+k_{1})&M_{2}s^{2}+(b_{1}+b_{2})s+(k_{1}+k_{2})\end{bmatrix}$
$\displaystyle x$ $\displaystyle=\begin{bmatrix}X_{1}(s)\\\
X_{2}(s)\end{bmatrix},\qquad\textrm{and}\qquad\,\textbf{g}=\begin{bmatrix}U(s)\\\
(b_{2}s+k_{2})\Gamma(s)-U(s)\end{bmatrix}$

and the value of $x$ is obtained from the inverse with slight modification of
matrix ($g$) such that it only appears $U(s)$ and $\Gamma(s)$ as in Eq.(8)
with the detail parameters of $\Delta_{n}$,

$\displaystyle
x=\frac{1}{\det(\textbf{F})}\begin{bmatrix}\Delta_{1}(s)&\Delta_{2}(s)\\\
\Delta_{3}(s)&\Delta_{4}(s)\end{bmatrix}\begin{bmatrix}U(s)\\\
\Gamma(s)\end{bmatrix}$ (8)

where, after being altered, the
$\operatorname{adj}(\textbf{F})\times\textbf{g}$ is then turned into
$\Delta_{n},\forall n=1\to 4$ as described below,

$\displaystyle\Delta_{1}(s)$ $\displaystyle=M_{2}s^{2}+b_{2}s+k_{2}$
$\displaystyle\Delta_{2}(s)$ $\displaystyle=z(s)$ $\displaystyle\Delta_{4}(s)$
$\displaystyle=M_{1}b_{2}s^{3}+M_{1}k_{2}s^{2}+z(s)$
$\displaystyle\Delta_{3}(s)$ $\displaystyle=-M_{1}s^{2}$

with the modification of initial g into the remaining $U(s),\Gamma(s)$

$\displaystyle\begin{bmatrix}U(s)\\\
(b_{2}s+k_{2})\Gamma(s)-U(s)\end{bmatrix}\quad\longrightarrow\quad\begin{bmatrix}U(s)\\\
\Gamma(s)\end{bmatrix}$

where $z(s)$ in $\Delta_{2}(s)$ term is
$b_{1}b_{2}s^{2}+(b_{1}k_{2}+b_{2}k_{1})s+k_{1}k_{2}$. To construct the
transfer functions $\Phi_{2}(s)$, it is required to set the sequence of the
inputs. For $\Phi_{2a}$, the control input ($u$) is taken and the disturbance
($\gamma$) is assumed zero while for $\Phi_{2b}$ is the reciprocal as in
Eq.(9) in turn,

$\displaystyle\begin{aligned} \Phi_{2a}(s)&=\frac{X_{1}(s)-X_{2}(s)}{U(s)}\\\
&=\frac{(M_{1}+M_{2})s^{2}+b_{2}s+k_{2}}{\det(\textbf{F})}&\rightarrow\gamma=0\\\
\Phi_{2b}(s)&=\frac{X_{1}(s)-X_{2}(s)}{\Gamma(s)}\\\
&=\frac{-M_{1}b_{2}s^{3}-M_{1}k_{2}s^{2}}{\det(\textbf{F})}&\rightarrow
u=0\end{aligned}$ (9)

Beyond that, the state-space representative is shown in Eq.(10) with the
respected extended matrices in Eq.(11). Furthermore, the state variables
includes ($x_{1},y_{1}$) and their derivative with $y_{1}=x_{1}-x_{2}$ while
the output $y=y_{1}$, therefore

$\displaystyle\begin{aligned} \dot{\textbf{x}}&=A\textbf{x}+B\textbf{u}\\\
y&=C\textbf{x}+D\textbf{u}\end{aligned}$ (10)

using the following concepts or otherwise the transfer functions modification
as presented,

$\displaystyle\dot{\textbf{x}}$ $\displaystyle=\begin{bmatrix}0&1&0&0\\\
\dfrac{-b_{1}b_{2}}{M_{1}M_{2}}&0&\dfrac{b_{1}}{M_{1}}\left(\dfrac{b_{1}}{M_{1}}+\dfrac{b_{1}}{M_{2}}+\dfrac{b_{2}}{M_{2}}\right)-\dfrac{k_{1}}{M_{1}}&\dfrac{-b_{1}}{M_{1}}\\\
\dfrac{b_{2}}{M_{2}}&0&-\left(\dfrac{b_{1}}{M_{1}}+\dfrac{b_{1}}{M_{2}}+\dfrac{b_{2}}{M_{2}}\right)&1\\\
\dfrac{k_{2}}{M_{2}}&0&-\left(\dfrac{k_{1}}{M_{1}}+\dfrac{k_{1}}{M_{2}}+\dfrac{k_{2}}{M_{2}}\right)&0\end{bmatrix}\textbf{x}+\begin{bmatrix}0&0\\\
\dfrac{1}{M_{1}}&\dfrac{b_{1}b_{2}}{M_{1}M_{2}}\\\\[7.5pt]
0&\dfrac{-b_{2}}{M_{2}}\\\\[7.5pt]
\dfrac{1}{M_{1}}+\dfrac{1}{M_{2}}&\dfrac{-k_{2}}{M_{2}}\end{bmatrix}\textbf{u};\quad
y=\begin{bmatrix}0\\\ 0\\\ 1\\\
0\end{bmatrix}^{\top}\textbf{x}+\begin{bmatrix}0\\\
0\end{bmatrix}^{\top}\textbf{u}$ (11)

$\displaystyle\int_{k}\frac{d^{k}x_{n}}{dt^{k}}\;dt=\int_{k-1}\frac{d^{k-1}x_{n}}{dt^{k-1}}\,dt=x_{n},$
$\displaystyle\frac{1}{M_{n}}\sum_{i=1}^{k}F_{i}=\frac{d^{k}x_{n}}{dt^{k}},\quad\forall
n=1\to 2;k=2$

###  II-C The Aircraft Longitudinal-Pitch Dynamics

The mathematical approaches portraying the aircraft motions with six nonlinear
paired are intricate to deal with yet with appropriate assumptions, the
schemes of decoupling and linearizing into two axes, lateral and longitudinal
perspective, are acceptable. This system focuses on the autonomous aircraft
pitch control

Figure 3: Coordinate dynamics of the aircraft

being administered by the solely longitudinal axis as shown in Fig.(3).
Supposed the steady-cruise occurs in the airplane at certain constants of
speed and altitude, then the force variables of drag, lift, weight and thrust
systematically balance between the two planes, $x$ and $y$. Keep in mind the
more forced assumption of pitch angle alteration, not influencing the
velocity, in arbitrary conditions is also applied to simplify the model. This
leads to the following longitudinal dynamics with the steady-state variables
of attack ($\alpha$) and pitch ($\theta$) angle and pitch rate ($q$) as
written in Eq.(12), Eq.(13), and Eq.(14),

$\displaystyle\dot{\alpha}=\mu\Omega\sigma\left[-\psi_{\alpha_{1}}\alpha+\psi_{\alpha_{2}}q-\psi_{\alpha_{3}}\delta\right]$
(12)

where $\psi_{\alpha_{n}},\forall n=1\to 3$ make of

$\displaystyle\psi_{\alpha_{1}}=\Gamma_{\ell}+\Gamma_{d};\quad\psi_{\alpha_{2}}=\frac{1}{\mu-\Gamma_{\ell}};\quad\psi_{\alpha_{3}}=\Gamma_{w}\sin\gamma+\Gamma_{\ell}$

and the pitch rate ($q$) is written with the following equation of motion

$\displaystyle\dot{q}=\frac{\mu\Omega}{2I_{n}}\left[\psi_{q_{1}}\alpha+\psi_{q_{2}}q+\psi_{q_{3}}\delta\right]\quad\longrightarrow\quad\mu=\frac{\rho\textbf{S}\bar{c}}{4m}$
(13)

where $\psi_{q_{n}},\forall n=1\to 3$ constitute

$\displaystyle\psi_{q_{1}}$
$\displaystyle=\Gamma_{m}-\eta(\Gamma_{\ell}+\Gamma_{d}),$
$\displaystyle\longrightarrow\eta$ $\displaystyle=\mu\sigma\Gamma_{m}$
$\displaystyle\psi_{q_{2}}$
$\displaystyle=\Gamma_{m}+\sigma\Gamma_{m}(1-\mu\Gamma_{\ell}),$
$\displaystyle\longrightarrow\sigma$ $\displaystyle=(1+\mu\Gamma_{\ell})^{-1}$
$\displaystyle\psi_{q_{3}}$ $\displaystyle=\eta\Gamma_{w}\sin\gamma$

The last would be the pitch angle ($\theta$) written as

$\displaystyle\dot{\theta}=\Omega
q\quad\longrightarrow\quad\Omega=\frac{2\textbf{E}_{u}}{\bar{c}}$ (14)

where $\mu,\Omega,\sigma,\eta$ are the constants being then affected by the
following variables;
$\delta,\rho,\textbf{S},\bar{c},m,\textbf{E}_{u},\gamma,I_{n}$ comprise the
deflection angle of elevator, air density, wing area, mean chord length, mass,
speed equilibrium, angle of flight trajectory, and the normalized of moment
inertia respectively. Beyond that the coefficients are also considered as the
coefficient of thrust ($\Gamma_{t}$), drag ($\Gamma_{d}$), lift
($\Gamma_{\ell}$), weight ($\Gamma_{w}$) and pitch moment ($\Gamma_{m}$). To
obtain the dynamics, it is required to form the state-space from the Laplacian
transfer function as in Eq.(15) with the respected $c_{n},\forall n=1\to 7$,

$\displaystyle sA(s)=c_{1}A(s)+c_{2}Q(s)+c_{3}\Delta(s)$ $\displaystyle
sQ(s)=c_{4}A(s)+c_{5}Q(s)+c_{6}\Delta(s)$ (15) $\displaystyle
s\Theta=c_{7}Q(s)$

and after some algebraic formula, it is achieved this function,

$\displaystyle\Phi_{3}(s)=\frac{\Theta(s)}{\Delta(s)}=\frac{1.151s+0.177}{s^{3}+0.739s^{2}+0.921s}$
(16)

with the standard matrices as in Eq.(17) which could be also built from
Eq.(16), therefore

$\displaystyle\begin{bmatrix}\dot{\alpha}\\\\[2.5pt] \dot{q}\\\\[2.5pt]
\dot{\theta}\end{bmatrix}=\begin{bmatrix}c_{1}&c_{2}&0\\\\[2.5pt]
c_{4}&c_{5}&0\\\\[2.5pt]
0&c_{7}&0\end{bmatrix}\begin{bmatrix}\alpha\\\\[2.5pt] q\\\\[2.5pt]
\theta\end{bmatrix}+\begin{bmatrix}c_{3}\\\\[2.5pt] c_{6}\\\\[2.5pt]
0\end{bmatrix}\delta;\qquad y=\theta$ (17)

###  II-D DC Motor Systems - Speed & Position Perspectives

The last dynamical system would be one of the most common actuators being used
in electrical drives, which is focused on the two measured variables of speed
and position. This system illustrates the translational rotating rotor-motion
paired with the wheels as shown in Fig.(4). The rotor plant is supposed to
have the voltage ($V$) input working on the armature of the motor whereas the
outputs capture the two states, the position $\theta$ and the speed
$\dot{\theta}$ of the shaft. Furthermore, the two objects associated to the
input-output are assumed

Figure 4: Free-body rotor design

to be rigid while regarding the resistive-contacting force, the model of the
friction torque linearly parallels to angular velocity. With the steady value
assumption of magnetic field, the torque, having the positive linear
combination with the current and the magnetic field, is then solely
corresponding to the current ($i$) with certain torque parameter
($\kappa_{t}$), such that

$\displaystyle T=\kappa_{t}i$ (18)

where the opposite emf ($e$) is positively affected by the multiplication
between certain electromotive force parameter ($\kappa_{e}$) which equals to
($\kappa_{t}$) and the shaft speed $\dot{\theta}$

$\displaystyle e=\kappa_{e}\dot{\theta}$ (19)

and from Eq.(18) and Eq.(19), the respected models in terms of the force
Newton’s law and the voltage Kirchhoff’s could be written in Eq.(20)

$\displaystyle\begin{aligned} T\ddot{\theta}+b\theta&=\kappa i\\\
L\frac{di}{dt}+Ri&=V-\kappa\dot{\theta}\end{aligned}$ (20)

which are then constructed by the Laplacian functions stated as

$\displaystyle\begin{aligned} s(Js+b)\Theta(s)&=\kappa I(s)\\\
(Ls+R)I(s)&=V(s)-\kappa s\Theta(s)\end{aligned}$ (21)

From Eq.(21), the term of ($s$) in the position $\Theta(s)$ is then mixed as
the output velocity whereas the $I(s)$ becomes the equality variable between
the two equations, making the input voltage $V(s)$ as written in Eq.(22),
therefore

$\displaystyle\Phi_{4}(s)=\frac{\dot{\Theta}(s)}{V(s)}=\frac{\kappa}{(Js+b)(Ls+R)+\kappa^{2}}$
(22)

Either the Laplacian term in Eq.(22) or by constructing from the equation
models could be easily turned into state-space representation with two
measured variables of position and speed as denoted in Eq.(23),

$\displaystyle\begin{aligned} \frac{d}{dt}\begin{bmatrix}\theta\\\\[2.5pt]
\dot{\theta}\\\\[2.5pt] i\end{bmatrix}&=\begin{bmatrix}0&1&0\\\\[2.5pt]
0&\frac{-b}{j}&\frac{\kappa}{j}\\\\[2.5pt]
0&\frac{-\kappa}{L}&\frac{-R}{j}\end{bmatrix}\begin{bmatrix}\theta\\\\[2.5pt]
\theta\\\\[2.5pt] i\end{bmatrix}+\begin{bmatrix}0\\\\[2.5pt] 0\\\\[2.5pt]
\frac{1}{L}\end{bmatrix}V\\\ y&=\begin{bmatrix}1&0&0\\\\[2.5pt]
0&1&0\end{bmatrix}\textbf{x}\longrightarrow\begin{bmatrix}\theta\\\\[2.5pt]
\dot{\theta}\end{bmatrix}\end{aligned}$ (23)

##  III Control Designs

Figure 5: A precompensator state-feedback system

The Control method used is the feedback aided with pre-compensation along with
some disturbance under some desired

condition. Fig.(5) explains the control scenario of the feedback aided
compensation $\bar{N}$ where it requires standard pole placement to obtain
$K$. Regarding the compensation, it has two initial setup of $s_{1}$ as the
length of matrix $A$ and $s_{2}$ defining the ones vector of output in
addition to zeros vector of $s_{1}$, such that,

$\displaystyle s_{1}=\textrm{length}(A)\qquad
s_{2}=[\textrm{zeros}([1,s_{1}]),\;1]$

where the compensation ($\bar{N}$) is constructed using the formulas

$\displaystyle N=\textrm{inv}\left(A,B;C,D\right)\times s_{2}^{\top}$
$\displaystyle\bar{N}=N_{u}+KN_{x}\longrightarrow Nx=N(1:s_{1})\textrm{ and
}N_{u}=N(1+s_{1})$

which could vanish the error. Furthermore, as regards the first plant, the
cruise is designed to be constant in any two different points for 50s. The
first 30s is for 10 m/s as the reference whereas the rest 20s is to maintain 7
m/s with ($-1.5$) pole for finding the gain $K$. The second plant requires the
disturbance $\gamma$ in addition to input signal $u$ and from this, the
control should stabilize the output $y_{1}$ in under 5s. Keep in that while
having two inputs ($u,\gamma$), the state-feedback could only control the
input signal of the first column $B(\textbf{1})$, such that

$\displaystyle\dot{x}=(A-B(\textbf{1})\times K)+B[U,\Gamma]^{\top}$

where the characteristic polynomial of the system is then written as the
$\det[sI-(A-B(\textbf{1})K)]$ instead of the standard $\det[sI-(A-BK)]$ along
with the minimizing compensation $\bar{N}$ algorithm. With respect to the
third plant, the full-rank ($n$) analysis of controllable and observable is
done, ensuring to place the poles around complex $s-$plane. The input $u$
would be the desired pitch angle $\theta_{r}$ and the $K$ times the full-state
measured x however, since this is the higher-order dynamics, the more advanced
technique is applied with the weighted matrices of $R$ and modified $Q$
applying the varied-constant $p$ and matrix $C$. After that, the gain $K$ and
the pre-compensation are ready to be implemented. Finally, the rotor dynamics
have poles in sequence of speed $v$ ($-5\pm i$) and position $\theta$
($-100\pm 100i,-200$) along with the same scenarios of finding the gain $K$
and the minimizing compensation $\bar{N}$. Beyond that, the stability analysis
are discussed further to guarantee the desired outputs along with the proposed
estimation mechanism.

##  IV Stability Analysis

Figure 6: The block diagrams of the tested plants according to the transfer
function $\Phi_{n},\forall n=1\to 4$ as the basis of stability analysis

The design of the control systems $C(s)$ for each plant is shown in Fig.(6)
according to certain transfer function $\Phi_{n},\forall n=1\to 4$ with
$(r,e,d)$ represent the reference, error and disturbance. For PID-typed
control, the signal control would be evaluated through Eq.(24),

$\displaystyle u(t)=K_{p}e(t)+K_{i}\int e(t)dy+K_{d}\frac{de}{dt}$ (24)

while the control transfer functions in Eq.(25) would be adjusted to the plant
$\Phi_{n}$ based on Eq.(3), Eq.(9), Eq.(16), and Eq.(22), especially when
finding the position in $\Phi_{4}(s)$ it is then required to integrate,
therefore

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

Figure 7: The root-locus analysis of: 1) the first dynamics $\Phi_{1}$ with
proportional (a) and lag-controller (b); 2) the second plant $\Phi_{2\alpha}$
(c) with notch filter (d) and its magnifier (e) to the limit along with PID-
control (f); 3) the third system $\Phi_{3}$ with original-induced by
proportional control (g) and the modification of lead-compensation scenario
(h); 4) while the last transfer functions comprise the speed $\Phi_{4}$ with
lag-control (i) and the position $\Phi_{4}$ with proportional (j), the PI
control (k) and the PID-control (l)

$\displaystyle C(s)=\begin{cases}K_{p}E(s),&\rightarrow\textrm{P}\\\\[7.5pt]
(K_{p}+K_{d}s)E(s),&\rightarrow\textrm{PD}\\\\[5.0pt]
\dfrac{K_{P}s+K_{i}}{s}E(s),&\rightarrow\textrm{PI}\\\\[5.0pt]
\dfrac{K_{d}s^{2}+K_{P}s+K_{i}}{s}E(s),&\rightarrow\textrm{PID}\end{cases}$
(25)

Moreover, the damping ratio ($\zeta$) and natural frequency ($\omega_{n}$) for
each plant should be well-defined using Eq.(26), such that

$\displaystyle\omega_{n}\geq\frac{1.8}{t_{r}},\qquad\zeta\geq\sqrt{\dfrac{\ln^{2}(M_{p})}{\pi^{2}+\ln^{2}(M_{p})}}$
(26)

Keep in mind that other than PID combination in Eq.(25), there are lag-control
and lead-compensation being used in analysing the stability. For instance the
lag-control of the first dynamic $\Phi_{1}$ as in Eq.(27),

$\displaystyle
C(s)=\frac{s+z_{0}}{s+p_{0}}\longrightarrow\frac{V(s)}{R(s)}=\frac{s+z_{0}}{ms^{2}+(b+mp_{0})s+bp_{0}}$
(27)

where the closed-loop design constitutes in Eq.(28),

$\displaystyle\frac{V(s)}{U(s)}=\frac{K_{p}(s+z_{0})}{ms^{2}+(b+mp_{0}+K_{p})s+(bp_{0}+K_{p}z_{0})}$
(28)

and the led mechanism could be seen with another additional gain $K_{\ell}$ in
$C(s)$ and putting the value of zero $z_{0}$ less than that of pole $p_{0}$.
While the lag considers the right of $s-$ plane, the counterpart led places on
the left side with results in Fig.(7).

##  V The Modified Unscented Filtering

The common algorithm of Unscented Kalman Filter covers the vital response for
the standard-EKF question in terms of the best-opted Gaussian random variables
(GRV). It gives the broader deterministic sampled-values and this has captured
the posterior mean along with its covariance of the opted GRV when being
matched to the more dynamic non-linear systems,

$\displaystyle\begin{aligned}
\textbf{x}_{k+1}&=f(\textbf{x}_{k},\textbf{u}_{k})+q_{k}\\\
\textbf{y}_{k}&=h(\textbf{x}_{k})+r_{k}\end{aligned}$ (29)

where the matrices of $\textbf{x}_{k},\textbf{u}_{k},f_{k},h_{k},w_{k},v_{k}$
and $\textbf{y}_{k}$ comprise state, control signal, system and measurement
model, process and measurement noise, and measurement vector respectively.
Considering to input a state-variable (x), with the properties of
$\bar{\textbf{x}}$ and $\textbf{P}_{x}$ with $L-$dimension, into the non-
linear model $f(\bullet)$ to gain the measured $\textbf{y}_{k}$, the modified
matrix $\chi_{i}$ with $\forall i=0\to(2L+1)$ sigma values, such that

$\displaystyle\begin{cases}\chi_{0}=\bar{\textbf{x}}&i=0\\\
\chi_{i}=\bar{\textbf{x}}+\left(\sqrt{\left(L+\lambda\right)\textbf{P}_{k-1}}\right)_{i}&i=1,\dots,L\\\
\chi_{i}=\bar{\textbf{x}}-\left(\sqrt{\left(L+\lambda\right)\textbf{P}_{k-1}}\right)_{i}&i=L+1,\dots,2L\end{cases}$
(30)

in which $\lambda=\alpha^{2}(L+\kappa)-L$ explains the weighted value and a
parameter ($\alpha$) as the key factor defines the distribution of the sigma
values $(\chi_{i})$ surrounding the expected value ($\bar{\textbf{x}}$) with
the value in a range of $10^{-4}$ and 1. While ($\kappa$) constitutes the
second parameter, either 0 or $(3-L)$, the third ($\beta$), demonstrating the
previous data of the state, makes of the exact 2 as the optimal. Moreover, the
method of the standard UKF is written and to differ from the preceding
discussion, this applies the $(n)-$th iteration in place of ($k$),

  1. 1.

Initialization setup the inputs $\hat{\textbf{x}}_{0}$ and $\textbf{P}_{0}$

  2. 2.

For $n=1\to\infty$, iterate the algorithms from Eq.($\alpha_{1}$) to
Eq.($\alpha_{11}$). First, calculate the sigma values ($\chi$),

$\displaystyle\chi_{a,n-1}^{(i)}$
$\displaystyle=\begin{bmatrix}\textbf{x}_{a,n-1}^{(i)}&\textbf{x}_{a,n-1}^{(i)}\pm\sqrt{(L+\lambda)\textbf{P}_{a,n-1}}\end{bmatrix}$
($\alpha_{1}$) $\displaystyle\hat{\chi}_{a,n}^{(i)}$
$\displaystyle=\textbf{f}\left(\chi_{a,n-1}^{(i)},\textbf{u}_{n}\right)$
($\alpha_{2}$) $\displaystyle\hat{\textbf{x}}_{a,n}$
$\displaystyle=\sum_{i=0}^{2L}W^{(i)}\hat{\chi}_{a,n}^{(i)}$ ($\alpha_{3}$)
$\displaystyle\hat{\textbf{P}}_{a,n}$
$\displaystyle=\sum_{i=0}^{2L}W^{(i)}\left[\hat{\chi}_{a,n}^{(i)}-\hat{\textbf{x}}_{a,n}\right]\left[\hat{\chi}_{a,n}^{(i)}-\hat{\textbf{x}}_{a,n}\right]^{\top}$
($\alpha_{4}$) $\displaystyle\hat{\bf{y}}_{n}^{(i)}$
$\displaystyle=\textbf{g}\left(\hat{\chi}_{a,n}^{(i)},\textbf{u}_{n}\right)$
($\alpha_{5}$) $\displaystyle\hat{\bf{y}}_{n}$
$\displaystyle=\sum_{i=0}^{2L}W^{(i)}\hat{\bf{y}}_{n}^{(i)}$ ($\alpha_{6}$)
$\displaystyle\hat{\textbf{S}}_{n}$
$\displaystyle=\sum_{i=0}^{2L}W^{(i)}\left[\hat{\bf{y}}_{n}^{(i)}-\hat{\bf{y}}_{n}\right]\left[\hat{\bf{y}}_{n}^{(i)}-\hat{\bf{y}}_{n}\right]^{\top}$
($\alpha_{7}$) $\displaystyle\hat{\textbf{K}}_{n}^{xy}$
$\displaystyle=\sum_{i=0}^{2L}W^{(i)}\left[\hat{\chi}_{n}^{(i)}-\hat{\textbf{x}}_{n}\right]\left[\hat{\bf{y}}_{n}^{(i)}-\hat{\textbf{x}}_{n}\right]^{\top}$
($\alpha_{8}$) $\displaystyle\textbf{W}_{n}$
$\displaystyle=\hat{\textbf{K}}_{n}^{xy}\hat{\textbf{S}}_{n}^{-1}$
($\alpha_{9}$) $\displaystyle\hat{\textbf{x}}_{n}^{+}$
$\displaystyle=\hat{\textbf{x}}_{n}+\textbf{W}_{n}z_{n}\longrightarrow
z_{n}=\textbf{y}_{n}-\hat{\textbf{y}}_{n}^{(i)}$ ($\alpha_{10}$)
$\displaystyle\textbf{P}_{n}$
$\displaystyle=\hat{\textbf{P}}_{n}-\textbf{W}_{n}\hat{\textbf{S}}_{n}\textbf{W}_{n}^{\top}$
($\alpha_{11}$)

with the magnitudes of $W_{i}$ are stated as follows,

Figure 8: The UT propagation of mean and covariance: (left) The actual;
(center) First-linearization of EKF; (right) The UT itself

$\displaystyle\begin{aligned} W_{0}^{(m)}&=\frac{\lambda}{L+\lambda}\\\
W_{0}^{(c)}&=\frac{\lambda}{L+\lambda}+\left(1-\alpha^{2}+\beta\right)\\\
W_{i}^{(m)}&=W_{i}^{(c)}=\frac{1}{2(L+\lambda)},\forall i=1\to
2L\end{aligned}$ (31)

The weighted average comes up inspired by the consensus-based algorithms with
some matrix modification of the pseudo measurement as written in [30].
Nevertheless this was linear approximation of the pseudo matrix [31] and
according to [32] and [33], the consensus was applied by [29] without pseudo
matrix. More specifically, each node ($i$) calculates locally with the
weighted average between the estimated state and error covariance
($\hat{\textbf{x}}_{n}^{i},\textbf{P}_{n}^{i}$) in neighborhood region
$\mathcal{N}_{i}$ with certain magnitude of $\pi^{(1,j)},j\in\mathcal{N}_{i}$.
The coupled $(\hat{\textbf{x}}_{n}^{i},\textbf{P}_{n}^{i})_{i\in\mathcal{N}}$
is then supposed to be weighted average if $\ell\to\infty$ as in Eq.(32),

$\displaystyle\left(\hat{\textbf{x}}_{n}^{i},\textbf{P}_{n}^{i}\right)_{i\in\mathcal{N}}\xrightarrow[]{\textrm{weighted}}\left(\hat{\textbf{x}}_{n}^{\ast},\textbf{P}_{n}^{\ast}\right)=\lim_{\ell\to\infty}\left(\hat{\textbf{x}}_{n,\ell}^{i},\textbf{P}_{n,\ell}^{i}\right)$
(32)

where the coupled-term
$(\hat{\textbf{x}}_{n,\ell}^{i},\textbf{P}_{n,\ell}^{i})_{i\in\mathcal{N}}$
comprises the data provided at ($i$) point at the $\ell-$th cycle, satisfying
Eq.(33)

$\displaystyle\begin{aligned}
\hat{\textbf{x}}_{n,\ell+1}^{i}&=\sum_{j\in\mathcal{N}_{i}}\pi^{(ij)}\hat{\textbf{x}}_{n,\ell}^{j}\\\
\textbf{P}_{n,\ell+1}^{i}&=\sum_{j\in\mathcal{N}_{i}}\pi^{(ij)}\textbf{P}_{n,\ell}^{j}\end{aligned}\rightarrow\pi^{(ij)}\geq
0,\sum_{j\in\mathcal{N}_{i}}\pi^{(ij)}=1$ (33)

and the coupled
$(\hat{\textbf{x}}_{n}^{i},\textbf{P}_{n}^{i})_{i\in\mathcal{N}}$ is then
achieved if the formula $\Pi=\pi^{(1,j)}\mathcal{R}^{n}$ is primitive

$\displaystyle\hat{\textbf{x}}_{n,\ell+1}$ $\displaystyle=\left(\Pi\otimes
I\right)\hat{\textbf{x}}_{n,\ell}$ $\displaystyle=\left(\Pi\otimes
I\right)\dots\left(\Pi\otimes
I\right)\hat{\textbf{x}}_{n,0}=\left(\Pi^{\ell+1}\otimes
I\right)\hat{\textbf{x}}_{n,0}$

and therefore,

$\displaystyle\lim_{\ell\to\infty}\left(\Pi^{\ell+1}\right)=1\textbf{v}^{\top}$

where as $\ell\to\infty$ with the column vector of v,

$\displaystyle\hat{\textbf{x}}_{n,\ell+1}=\left(1\textbf{v}^{\top}\otimes
I\right)\hat{\textbf{x}}_{n,0}$

the estimated state and the error covariance matrix would be,

$\displaystyle\hat{\textbf{x}}_{n,\ell+1}=v_{1}\hat{\textbf{x}}_{n,0}^{1}+v_{1}\hat{\textbf{x}}_{n,0}^{2}+\dots+v_{1}\hat{\textbf{x}}_{n,0}^{k}=\hat{\textbf{x}}_{n}^{\ast}$
$\displaystyle\textbf{P}_{n,\ell+1}=v_{1}\textbf{P}_{n,0}^{1}+v_{1}\textbf{P}_{n,0}^{2}+\dots+v_{1}\textbf{P}_{n,0}^{k}=\textbf{P}_{n}^{\ast}$

Furthermore, the algorithm of the weighted average consensus from
Eq.($\beta_{1}$) o Eq.($\beta_{5}$) is written below as the extension of the
standard UKF,

  1. 1.

For every $i\in\mathcal{N}$, collect the information of
$\hat{\textbf{y}}_{n}^{(i)}$ and find

$\displaystyle\hat{\textbf{x}}_{n}^{i}$
$\displaystyle=\hat{\textbf{x}}_{n}+\textbf{W}_{n}z_{n}\longrightarrow
z_{n}=\textbf{y}_{n}-\hat{\textbf{y}}_{n}^{(i)}$ ($\beta_{1}$)
$\displaystyle\textbf{P}_{n}^{i}$
$\displaystyle=\hat{\textbf{P}}_{n}-\textbf{W}_{n}\hat{\textbf{S}}_{n}\textbf{W}_{n}^{\top}$
($\beta_{2}$)

  2. 2.

Initialize that $\hat{\textbf{x}}_{n}^{i}=\hat{\textbf{x}}_{n,0}^{i}$ and
$\textbf{P}_{n,0}^{i}=\textbf{P}_{n,0}^{i}$

  3. 3.

For the $\ell=0,1,\dots,l$, apply the method of the weighted average
consensus, such that:

    1. (a)

Broadcast the coupled node data $\hat{\textbf{x}}_{n,\ell}^{i}$ and
$\textbf{P}_{n,\ell}^{i}$ to the surrounding neighborhoods
$j\in\mathcal{N}_{i}\setminus(i)$

    2. (b)

Ensure the information of $\hat{\textbf{x}}_{n,\ell}^{j}$ and
$\textbf{P}_{n,\ell}^{j}$ from the whole neighborhoods
$j\in\mathcal{N}_{i}\setminus(i)$

    3. (c)

Collect the coupled data $\hat{\textbf{x}}_{n,\ell}^{j}$ and
$\textbf{P}_{n,\ell}^{j}$ based on

$\displaystyle\left(\hat{\textbf{x}}_{n,\ell+1}^{i},\textbf{P}_{n,\ell+1}^{i}\right)=\sum_{j\in\mathcal{N}_{i}}\pi^{(ij)}\left(\hat{\textbf{x}}_{n,\ell}^{j},\textbf{P}_{n,\ell}^{j}\right)$
($\beta_{3}$)

  4. 4.

Setup the estimated state as,

$\displaystyle\hat{\textbf{x}}_{n}^{i}=\hat{\textbf{x}}_{n+l}^{i}\quad\textrm{and}\quad\textbf{P}_{n}^{i}=\textbf{P}_{n+l}^{i}$
($\beta_{4}$)

  5. 5.

Perform the updated of the prediction error, therefore

$\displaystyle\begin{aligned}
\hat{\textbf{x}}_{a,n}^{+}&=\sum_{i=0}^{2L}W^{(i)}\hat{\chi}_{a,n}^{(i)}\\\
\hat{\textbf{P}}_{a,n}^{+}&=\sum_{i=0}^{2L}W^{(i)}\left[\hat{\chi}_{a,n}^{(i)}-\hat{\textbf{x}}_{a,n}\right]\left[\hat{\chi}_{a,n}^{(i)}-\hat{\textbf{x}}_{a,n}\right]^{\top}\end{aligned}$
($\beta_{5}$)

As for theoretical feasibility of the stochastic boundedness with in-depth
mathematical proof is well-explained in [29]

##  VI Numerical Designs and Findings

Having discussed the various dynamical systems denoted as $P(s)$ along with
some control scenarios $C(s)$ and their stability, the network is portrayed as
in Fig.(9) considering the process $w$ and measurement $v$ noise and the
disturbance $d$. These control scenarios are diverged from the classical
perspective (PID, LQR, etc) to the modern actor-critic reinforcement learning
and other dynamic control to match the condition of the system.

Figure 9: Block diagram of the system

Furthermore, the modified UKF estimation $U(s)$ is then applied to touch the
hidden states of the systems to reach the information leading to sensorless
design. This estimation is then compared to other centralized and distributed
filtering as explained in the followings. The whole dynamics of the tested
systems are constructed according to certain plants $\Phi_{n}(s),\forall
n=1\to 4$ as shown in Fig.(6). As for the first $\Phi_{1}$, the mass $m$ and
the damping parameter $b$ are set to be 1000 kg and 50 N.s/m in turn while the
nominal control $u$ is 500 N with dynamic reference $v_{r}$ in 10 m/s and 7
m/s. The second plant $\Phi_{2}$ comes up with a quarter-body mass $M_{1}$ as
2500 kg and the suspension mass $M_{2}$ as 320 kg whereas the parameters of
the spring with respect to system $k_{1}$ and wheel $k_{2}$ are 80,000 N/m and
500,000 N/m in turn while the damping of the same respected terms, $b_{1}$ and
$b_{2}$, have 350 N.s/m and 15,020 N.s/m respectively. Regarding the third
plant $\Phi_{3}$, according to Eq.(17), the values of $c_{k},\forall k=1\to 7$
constitute $-0.313$, $56.7$, $0.232$, $-0.0139$, $-0.426$, $0.0203$, and
$56.7$ while the fourth $\Phi_{4}$ with speed $\dot{\theta}$ and position
$\theta$ comprises the variables as follows. The moment $J$ is 0.01 kg.m2 with
the friction parameter $b$ of 0.1 N.m.s and the same gain $\kappa$ makes of
0.01 to the system resistance $R$ equals to 1 Ohm and inductance $L$ with 0.5
H. Regarding the parameters of the proposed estimation methods, the time-
sampling ($t_{s}$), covariance matrices of $Q$ and $R$ are

(a) States comparison over certain reference points

(b) Error comparisons of the states

Figure 10: (a) The performance results of speed ($x=v$) using plant $\Phi_{1}$
of the measured ($y$), the proposed estimated ($x_{u}$) and the true ($x_{t}$)
states according to the reference ($r$); (b) while it shows the error
comparisons of the measured and the estimated over the true values

(a) State ($x_{3}$) response due to disturbance

(b) State ($x_{3}$) response due to disturbance

(c) The closed-loop responses of the states

(d) The open-loop response with 0.15 m disturbance

Figure 11: (a), (b) The performance results of ($x_{3}=\varphi$) using plant
$\Phi_{2}$ of the proposed estimated ($x_{u}$) and the true ($x_{t}$) with
some disturbance ($\gamma$); (c) while it highlights the closed-loop response
of the states ($x_{1},x_{2},x_{4},x_{5}$); and (d) the open-loop response if
$\gamma$ is applied to the system

$\displaystyle\begin{array}[]{l|l|l|l}\textrm{Plant}&t_{s}&R&Q\\\\[1.99997pt]
\hline\cr\Phi_{1}=&0.01&0.5&0.1\\\\[1.99997pt]
\Phi_{2}=&0.0005&0.05&10\\\\[1.99997pt]
\Phi_{3}=&0.01&1&pH^{\top}H\\\\[1.99997pt]
\Phi_{4}=&0.01&1&pH^{\top}H\end{array}$

where the measurement noise $v$ of the every system $\Phi_{n}(s)$ is
formulated based on the output matrix $H$ with some unique distribution
$\mathcal{R}_{u}\in(0,1)$ by

$\displaystyle v=\sqrt{R}\times\textrm{randn}[\textrm{size}(H,1),1]$

where (randn) means the normal distribution of GRV while the
$\textrm{size}(\epsilon_{1},\epsilon_{2})$ denotes either the row or column
dimension $\epsilon_{2}\in(1,2)$ in turn of the matrix $\epsilon_{1}$. $p_{0}$
as the arbitrary constant solving the algebraic Riccati equation (ARE) equals
to 50 with eye($\bullet$) and discrete($\bullet$) states the identity matrix
of length ($\bullet$), known as the maximum size or dimension, and the
discretized systems of ($\bullet$), such that it shows

$\displaystyle\textbf{P}_{0}=p_{0}\times\textrm{eye}(F)\longrightarrow
F=\textrm{discrete}(A)$

while the performance of the estimation error is approached with the error as
opposed to the true values, having no noise in the systems,

$\displaystyle e=x_{u}-x_{t}$

Regarding the performance of plant $\Phi_{1}$, the desired references in 50 s
are situated in two different speed values and the control along with the
estimated values could deal with the changes as depicted in Fig.(10(a)) while
the performance errors are written in Fig.(10(b)) showing the zero
convergence. As for the second plant $\Phi_{2}$, the dynamics of the
disturbance-effect systems, the closed-loop and the open-loop are discussed,
saying the capability of handling the external forces in under 5 seconds. The
values $x_{3}\to y_{1}$ between the true and the estimation parallels with
slight difference on the peak and trough as illustrated in Fig.(11(a)) while
the closed-loop in Fig.(11(b)) also performs almost on par with preceding
dynamics. Fig.(11(c)) and Fig.(11(d)) constitute the stabilized and the non-
control performance of the systems. Furthermore, the third dynamics $\Phi_{3}$
describes the control of the pitch angle $\theta$ of an aircraft and it is
built into two divergent reference points, 0.2 and 0.5, within 7 seconds. In
terms of control scenario, the systems could trace the reference while for the
estimated states, the controlled state $x_{3}$ in Fig.(12(b)) and the free-
state $x_{1}$ in Fig.(12(b)) are well-covered by the proposed algorithm as
opposed to the true states.

(a) States comparison of $x_{3}$

(b) States comparison of $x_{1}$

Figure 12: The performance results of pitch angle ($x_{3}=\theta$) using plant
$\Phi_{3}$ of the measured ($y$), the proposed estimated ($x_{u}$) and the
true ($x_{t}$) states according to the reference ($r$) along with the free-
state

(a)

(b)

Figure 13: The performance results of speed ($x_{1}=\dot{\theta}$) in various
dynamics using plant $\Phi_{4}$ of the measured ($y$), the proposed estimated
($x_{u}$) and the true ($x_{t}$) states according to the reference ($r$)

(a)

(b)

(c)

Figure 14: The performance results of speed ($x_{1}=\theta$) in various
dynamics using plant $\Phi_{4}$ of the measured ($y$), the proposed estimated
($x_{u}$) and the true ($x_{t}$) states according to the reference ($r$)

Likewise, the rotor dynamics $\Phi_{4}$ for speed variable $\dot{\theta}$ with
two different reference scenarios in Fig.(13(a)) and Fig.(13(b)) are handled
by the control design and well-estimated by the weighted average consensus
method. Finally, the position-type variable $\theta$ of $\Phi_{4}$ comprises
the same trends with respect to the comparison of the measured, the estimated
and the true states with various dynamics as portrayed in Fig.(14). To
conclude, the control designs from various plants $\Phi_{n},\forall n=1\to 4$
successfully capture the dynamics of the systems while the proposed algorithm
effectively tracks the true values under some disturbances this also leads to
the sensorless design of the future works. The performances are affected
solely on the covariance matrix pairs $(Q,R)$ and the measurements. Beyond
that, the performances of the estimation are also weighed according to the
reviews of distributed estimation [48, 49, 50] along with various applications
in terms of non-linear uncertainties interconnected system [51, 52, 53, 54,
55, 56], and Pareto optimization [57]. The presented results from arbitrary
initial conditions ($x_{0}$), in terms of estimation errors, highlight the
convergence, almost identical as the centralized Kalman with different in the
noise scalability, as compared to distributed estimation studied in [23, 49,
50]. For the more severe faulty unobserved systems, the mechanism to maturely
detect the states is highly required while for large-scale systems, the
information of purely local measurement is also possible as opposed to the
local and its neighborhood.

##  VII Conclusions

The mathematical dynamics of the vehicle systems along with the graphs have
been constructively designed under some disturbance as the object of the
performance results. The control scenarios for the tested plants $\Phi_{n}$
along with some stability analysis have also been discussed comprehensively to
check the observability of the systems. The standard UKF and the proposed of
the weighted average consensus estimation method is written considering some
neighborhoods events as the local collecting information to obtain the more
accurate estimated states. The results of the designs conclude the
effectiveness of the control designs along with the proposed estimation
algorithm to track the hidden states. For further research, the sensorless
design to vehicles dynamics is elaborated leading to the autonomous concepts
along with some fault-tolerant learning control if faults occur to negate the
failured systems.

## Acknowledgment

This research was provided by a funding granted by the Engineering Physics
Department of Institut Teknologi Sepuluh Nopember (ITS), Indonesia with letter
contract number: 1868/PKS/ITS/2022 in May 24, 2022. We thank our colleagues
for the ideas, dedication, and times for the final paper

## References

  * [1] A. Meyrowitz, D. Blidberg, and R. Michelson, “Autonomous vehicles,” Proceedings of the IEEE, vol. 84, no. 8, pp. 1147–1164, 1996.
  * [2] R. Hussain and S. Zeadally, “Autonomous cars: Research results, issues, and future challenges,” IEEE Communications Surveys & Tutorials, vol. 21, no. 2, pp. 1275–1313, 2019.
  * [3] D. Omeiza, H. Webb, M. Jirotka, and L. Kunze, “Explanations in autonomous driving: A survey,” IEEE Transactions on Intelligent Transportation Systems, vol. 23, no. 8, pp. 10142–10162, 2022.
  * [4] H. Yin, P. Seiler, and M. Arcak, “Stability analysis using quadratic constraints for systems with neural network controllers,” IEEE Transactions on Automatic Control, vol. 67, no. 4, pp. 1980–1987, 2022.
  * [5] Y. Wang, N. Roohi, G. E. Dullerud, and M. Viswanathan, “Stability of linear autonomous systems under regular switching sequences,” in 53rd IEEE Conference on Decision and Control, pp. 5445–5450, 2014.
  * [6] I. Al-Darabsah, M. A. Janaideh, and S. A. Campbell, “Stability of connected autonomous vehicle networks with commensurate time delays,” in 2021 American Control Conference (ACC), pp. 3308–3313, 2021.
  * [7] W. M. Haddad and J. Lee, “Finite-time stability of discrete autonomous systems,” in 2020 American Control Conference (ACC), pp. 5188–5193, 2020\.
  * [8] Y. Huang, S. Z. Yong, and Y. Chen, “Stability control of autonomous ground vehicles using control-dependent barrier functions,” IEEE Transactions on Intelligent Vehicles, vol. 6, no. 4, pp. 699–710, 2021.
  * [9] M. K. Wafi, “System identification on the families of auto-regressive with least-square-batch algorithm,” International Journal of Scientific and Research Publications (IJSRP), vol. 11, no. 5, pp. 65–72, 2021.
  * [10] A. Mehra, W.-L. Ma, F. Berg, P. Tabuada, J. W. Grizzle, and A. D. Ames, “Adaptive cruise control: Experimental validation of advanced controllers on scale-model cars,” in 2015 American Control Conference (ACC), pp. 1411–1418, 2015.
  * [11] C. Hu and J. Wang, “Trust-based and individualizable adaptive cruise control using control barrier function approach with prescribed performance,” IEEE Transactions on Intelligent Transportation Systems, vol. 23, no. 7, pp. 6974–6984, 2022.
  * [12] J. Na, Y. Huang, Q. Pei, X. Wu, G. Gao, and G. Li, “Active suspension control of full-car systems without function approximation,” IEEE/ASME Transactions on Mechatronics, vol. 25, no. 2, pp. 779–791, 2020.
  * [13] M. Yu, S. A. Evangelou, and D. Dini, “Position control of parallel active link suspension with backlash,” IEEE Transactions on Industrial Electronics, vol. 67, no. 6, pp. 4741–4751, 2020.
  * [14] G. Buticchi, P. Wheeler, and D. Boroyevich, “The more-electric aircraft and beyond,” Proceedings of the IEEE, pp. 1–15, 2022.
  * [15] P. Wheeler, T. S. Sirimanna, S. Bozhko, and K. S. Haran, “Electric/hybrid-electric aircraft propulsion systems,” Proceedings of the IEEE, vol. 109, no. 6, pp. 1115–1127, 2021.
  * [16] S. Li, X. Liang, and W. Xu, “Modeling dc motor drive systems in power system dynamic studies,” IEEE Transactions on Industry Applications, vol. 51, no. 1, pp. 658–668, 2015.
  * [17] T. Verstraten, R. Furnémont, G. Mathijssen, B. Vanderborght, and D. Lefeber, “Energy consumption of geared dc motors in dynamic applications: Comparing modeling approaches,” IEEE Robotics and Automation Letters, vol. 1, no. 1, pp. 524–530, 2016.
  * [18] B. L. Widjiantoro and M. K. Wafi, “Discrete-time state-feedback controller with canonical form on inverted pendulum (on a chart),” International Journal of Science and Engineering Investigations (IJSEI), vol. 11, no. 120, pp. 16–21, 2022.
  * [19] A. Liniger and J. Lygeros, “Real-time control for autonomous racing based on viability theory,” IEEE Transactions on Control Systems Technology, vol. 27, no. 2, pp. 464–478, 2019.
  * [20] S. Kuutti, R. Bowden, Y. Jin, P. Barber, and S. Fallah, “A survey of deep learning applications to autonomous vehicle control,” IEEE Transactions on Intelligent Transportation Systems, vol. 22, no. 2, pp. 712–733, 2021.
  * [21] Y. V. Pant, H. Abbas, K. Mohta, R. A. Quaye, T. X. Nghiem, J. Devietti, and R. Mangharam, “Anytime computation and control for autonomous systems,” IEEE Transactions on Control Systems Technology, vol. 29, no. 2, pp. 768–779, 2021.
  * [22] M. Liu, K. Chour, S. Rathinam, and S. Darbha, “Lateral control of an autonomous and connected following vehicle with limited preview information,” IEEE Transactions on Intelligent Vehicles, vol. 6, no. 3, pp. 406–418, 2021.
  * [23] M. K. Wafi, “Filtering module on satellite tracking,” AIP Conference Proceedings, vol. 2088, no. 1, p. 020045, 2019.
  * [24] B. L. Widjiantoro, K. Indriawati, and M. K. Wafi, “Adaptive kalman filtering with exact linearization and decoupling control on three-tank process,” International Journal of Mechanical & Mechatronics Engineering, vol. 21, no. 3, pp. 41–48, 2021.
  * [25] M. K. Wafi and B. L. Widjiantoro, “Distributed estimation with decentralized control for quadruple-tank process,” International Journal of Scientific Research in Science and Technology, vol. 9, no. 1, pp. 301–307, 2022\.
  * [26] A. Onat, “A novel and computationally efficient joint unscented kalman filtering scheme for parameter estimation of a class of nonlinear systems,” IEEE Access, vol. 7, pp. 31634–31655, 2019.
  * [27] W. Zhou and J. Hou, “A new adaptive high-order unscented kalman filter for improving the accuracy and robustness of target tracking,” IEEE Access, vol. 7, pp. 118484–118497, 2019.
  * [28] S. Liu, Z. Wang, Y. Chen, and G. Wei, “Protocol-based unscented kalman filtering in the presence of stochastic uncertainties,” IEEE Transactions on Automatic Control, vol. 65, no. 3, pp. 1303–1309, 2020.
  * [29] W. Li, G. Wei, F. Han, and Y. Liu, “Weighted average consensus-based unscented kalman filtering,” IEEE Transactions on Cybernetics, vol. 46, no. 2, pp. 558–567, 2016.
  * [30] W. Li and Y. Jia, “Consensus-based distributed multiple model ukf for jump markov nonlinear systems,” IEEE Transactions on Automatic Control, vol. 57, no. 1, pp. 227–233, 2012.
  * [31] T. Lefebvre, H. Bruyninckx, and J. De Schuller, “Comment on ”a new method for the nonlinear transformation of means and covariances in filters and estimators” [with authors’ reply],” IEEE Transactions on Automatic Control, vol. 47, no. 8, pp. 1406–1409, 2002.
  * [32] W. Ren, R. W. Beard, and E. M. Atkins, “Information consensus in multivehicle cooperative control,” IEEE Control Systems Magazine, vol. 27, no. 2, pp. 71–82, 2007.
  * [33] G. Battistelli, L. Chisci, G. Mugnai, A. Farina, and A. Graziano, “Consensus-based algorithms for distributed filtering,” in 2012 IEEE 51st IEEE Conference on Decision and Control (CDC), pp. 794–799, 2012.
  * [34] J. Kuti, I. J. Rudas, H. Gao, and P. Galambos, “Computationally relaxed unscented kalman filter,” IEEE Transactions on Cybernetics, pp. 1–9, 2022\.
  * [35] M. Alzayed and H. Chaoui, “Efficient simplified current sensorless dynamic direct voltage mtpa of interior pmsm for electric vehicles operation,” IEEE Transactions on Vehicular Technology, pp. 1–10, 2022.
  * [36] S. M. N. Ali, M. J. Hossain, D. Wang, K. Lu, P. O. Rasmussen, V. Sharma, and M. Kashif, “Robust sensorless control against thermally degraded speed performance in an im drive based electric vehicle,” IEEE Transactions on Energy Conversion, vol. 35, no. 2, pp. 896–907, 2020.
  * [37] R. Silva-Ortigoza, E. Hernández-Márquez, A. Roldán-Caballero, S. Tavera-Mosqueda, M. Marciano-Melchor, J. R. García-Sánchez, V. M. Hernández-Guzmán, and G. Silva-Ortigoza, “Sensorless tracking control for a “full-bridge buck inverter–dc motor” system: Passivity and flatness-based design,” IEEE Access, vol. 9, pp. 132191–132204, 2021.
  * [38] D. Xiao, S. Nalakath, S. R. Filho, G. Fang, A. Dong, Y. Sun, J. Wiseman, and A. Emadi, “Universal full-speed sensorless control scheme for interior permanent magnet synchronous motors,” IEEE Transactions on Power Electronics, vol. 36, no. 4, pp. 4723–4737, 2021.
  * [39] R. Yildiz, M. Barut, and E. Zerdali, “A comprehensive comparison of extended and unscented kalman filters for speed-sensorless control applications of induction motors,” IEEE Transactions on Industrial Informatics, vol. 16, no. 10, pp. 6423–6432, 2020.
  * [40] K. Indriawati, A. Jazidie, and T. Agustinah, “Reconfigurable controller based on fuzzy descriptor observer for nonlinear systems with sensor faults,” in Instrumentation and Measurement Systems, vol. 771 of Applied Mechanics and Materials, pp. 59–62, Trans Tech Publications Ltd, 8 2015.
  * [41] W. Wang, Z. Lu, Y. Feng, W. Tian, W. Hua, Z. Wang, and M. Cheng, “Coupled fault-tolerant control of primary permanent-magnet linear motor traction systems for subway applications,” IEEE Transactions on Power Electronics, vol. 36, no. 3, pp. 3408–3421, 2021.
  * [42] M. K. Wafi, “Estimation and fault detection on hydraulic system with adaptive-scaling kalman and consensus filtering,” International Journal of Scientific and Research Publications (IJSRP), vol. 11, no. 5, pp. 49–56, 2021\.
  * [43] Z. Wang, J. Shao, and Z. He, “Fault tolerant sensorless control strategy with multi-states switching method for in-wheel electric vehicle,” IEEE Access, vol. 9, pp. 61150–61158, 2021.
  * [44] M. Ebadpour, N. Amiri, and J. Jatskevich, “Fast fault-tolerant control for improved dynamic performance of hall-sensor-controlled brushless dc motor drives,” IEEE Transactions on Power Electronics, vol. 36, no. 12, pp. 14051–14061, 2021.
  * [45] M. K. Wafi and K. Indriawati, “Fault-tolerant control design in scrubber plant with fault on sensor sensitivity,” The Journal of Scientific and Engineering Research, vol. 9, no. 2, pp. 96–104, 2022.
  * [46] W. He, M. M. Namazi, T. Li, and R. Ortega, “A state observer for sensorless control of power converters with unknown load conductance,” IEEE Transactions on Power Electronics, vol. 37, no. 8, pp. 9187–9199, 2022.
  * [47] M. R. Pinandhito, K. Indriawati, and M. Harly, “Active fault tolerant control design in regenerative anti-lock braking system of electric vehicle with sensor fault,” AIP Conference Proceedings, vol. 2088, no. 1, p. 020024, 2019.
  * [48] C.-Y. Chong, “Forty years of distributed estimation: A review of noteworthy developments,” in 2017 Sensor Data Fusion: Trends, Solutions, Applications (SDF), pp. 1–10, 2017.
  * [49] R. Olfati-Saber, “Distributed kalman filter with embedded consensus filters,” in Proceedings of the 44th IEEE Conference on Decision and Control, pp. 8179–8184, 2005.
  * [50] R. Olfati-Saber, “Distributed kalman filtering for sensor networks,” in 2007 46th IEEE Conference on Decision and Control, pp. 5492–5498, 2007.
  * [51] B. Chen, G. Hu, D. W. Ho, and L. Yu, “Distributed estimation and control for discrete time-varying interconnected systems,” IEEE Transactions on Automatic Control, vol. 67, no. 5, pp. 2192–2207, 2022.
  * [52] S. Knotek, K. Hengster-Movric, and M. Šebek, “Distributed estimation on sensor networks with measurement uncertainties,” IEEE Transactions on Control Systems Technology, vol. 29, no. 5, pp. 1997–2011, 2021.
  * [53] D. Castanon and D. Teneketzis, “Distributed estimation algorithms for nonlinear systems,” IEEE Transactions on Automatic Control, vol. 30, no. 5, pp. 418–425, 1985.
  * [54] C. Freundlich, S. Lee, and M. M. Zavlanos, “Distributed estimation and control for robotic sensor networks,” in 2016 IEEE 55th Conference on Decision and Control (CDC), pp. 3518–3523, 2016.
  * [55] G. Yang, H. Rezaee, and T. Parisini, “Distributed state estimation for a class of jointly observable nonlinear systems,” IFAC-PapersOnLine, vol. 53, no. 2, pp. 5045–5050, 2020.  21st IFAC World Congress.
  * [56] Y. Song, H. Lee, C. Kwon, H.-S. Shin, and H. Oh, “Distributed estimation of stochastic multiagent systems for cooperative control with a virtual network,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, pp. 1–13, 2022.
  * [57] F. Boem, Y. Zhou, C. Fischione, and T. Parisini, “Distributed pareto-optimal state estimation using sensor networks,” Automatica, vol. 93, pp. 211–223, 2018.