# Estimation of electron temperature in heated metallic nanoparticle

N.I. Grigorchuk

(Received May 12, 2017, in final form June 19, 2017)

###### Abstract

A method is proposed for determining the temperature of hot electrons in a
metallic nanoparticle embedded in a dielectric matrix under ultrashort laser
pulses irradiation. The amplitude and power of the longitudinal spherical
acoustic oscillations as functions of density and elastic properties of the
medium, the laser pulse duration, the electron temperature, radii of
particles, and the electron-phonon coupling constant are obtained. The
efficiency of the electron energy transfer from heated noble nanoparticles to
a surrounding environment is estimated for different electron temperatures.

Key words: electron temperature, metal nanoparticle, ultrashort laser pulses

PACS: 78.67.Bf, 68.49.Jk, 73.23.-b, 52.25.Os, 36.40.Vz

###### Abstract

Çàïðîïîíîâàíî ìåòîä âèçíàåííÿ òåìïåðàòóðè ãàðÿèõ åëåêòðîíâ â ìåòàëåâé
íàíîàñòèíö, ùî çíàõîäèòüñÿ â ñåðåäîâèù ïä äþ óëüòðàêîðîòêèõ ëàçåðíèõ ìïóëüñâ.
Îäåðæàí àìïëòóäà ïîòóæíñòü ïîçäîâæíèõ ñôåðèíèõ àêóñòèíèõ êîëèâàíü ÿê ôóíêöÿ
ãóñòèíè ïðóæíèõ âëàñòèâîñòåé ñåðåäîâèùà, òðèâàëîñò ëàçåðíîãî ìïóëüñó, ðàäóñó
àñòèíêè, åëåêòðîí-ôîíîííî¿ êîíñòàíòè çâ’ÿçêó òà åëåêòðîííî¿ òåìïåðàòóðè.
Çðîáëåíî îöíêó åôåêòèâíîñò ïåðåäà åíåðã¿ åëåêòðîíâ âä ïäãðòèõ áëàãîðîäíèõ
íàíîàñòèíîê â îòîóþå ¿õ ñåðåäîâèùå çà ðçíèõ òåìïåðàòóð åëåêòðîíâ.

Ключов слова: åëåêòðîííà òåìïåðàòóðà, ìåòàëåâà íàíîàñòèíêà, óëüòðàêîðîòê
ëàçåðí ìïóëüñè

##  1 Introduction

Metallic nanoparticles (MNs) are mainly studied due to their unique optical
properties [1] and extensive practical applications [2, 3, 4, 5]. Over the
last decade, the acoustic oscillations of MNs excited by ultrashort laser
pulses have been under intense study [1, 2, 3, 4, 5, 7, 8, 9, 6]. This is due
to the availability of important data relating to the elastic properties of
these particles and their mechanical coupling to the surrounding medium, which
forms the foundation for the design of elasticity sensors in the nanometer
range [10]. The generation of sound waves by MNs was originally observed
experimentally for noble nanoparticles [2, 3, 4].

When MNs are excited by ultrashort pulses, the energy pulse is initially
transferred to the gas of free electrons which collide with one another as
well as with the lattice vibrations, and redistribute this energy (being
thermalized) over a short time (of the order of tens of picoseconds in a 100 Å
particle [3]). The electron gas in the particle is immediately heated. Due to
a low heat capacity of the electron gas (compared to the lattice), there
originates a short (but strong) electron pressure burst [11]. It is equivalent
to a short mechanical impact upon the particle surface over an infinitely
small time interval and can generate spherical sound waves in the medium
surrounding the particle [12, 13].

In this article, we proposed a method for determining the temperature of hot
electron gas in a metallic nanoparticle embedded in a dielectric matrix under
ultrashort laser pulses irradiation. To accomplish this, the amplitude and
power of the longitudinal sound wave generated by the excess pressure of an
electron gas inside the MN driven by ultrashort laser pulses are calculated.
So far, this problem has been little studied theoretically.

##  2 Initial principles and model

Let an ultrashort laser pulse of duration $\tau_{0}$ be incident on a
spherical MN embedded in an infinite, elastically isotropic, dielectric
medium. Energy transfer from the laser to the lattice of a nanoparticle
depends much on the relationship between the particle sizes, the electron mean
free path in the particle, and the Debye length
$l_{\text{D}}=\piup\upsilon_{\text{F}}/\omega_{\text{D}}$, which plays an
important role in the energy exchange between the hot electrons and the
lattice [14]. Its values for several metals are listed below in table 2. We
shall consider here the particles with radii $R<l_{\text{D}}/2$.

The additional pressure that develops during heating of the electron gas by
the laser pulse can produce oscillations of the nanoparticle surface, which in
turn, generate longitudinal acoustic waves in the elastically isotropic space
around the particle.

The equation for the propagation of longitudinal acoustic oscillations in this
space is given by [15]

$\nabla^{2}{\bf
u}_{\text{L}}(t)-\frac{1}{s^{2}_{\text{L}}}\frac{\partial^{2}}{\partial
t^{2}}{\bf u}_{\text{L}}(t)=0,$ (2.1)

where $s_{\text{L}}$ is the longitudinal sound speed in the medium. Elastic
displacement waves ${\bf u}_{\text{L}}$ are accompanied by bulk compressions
and expansions of the surrounding medium.

As boundary conditions for equation (2.1), we take a condition that all forces
applied to the particle surface ($r=R$) are equal to zero. This equality can
be presented as [15]

$\left.\left[\sigma_{rr}-\frac{2{\cal{E}}_{\text{s}}}{R^{2}}u_{r}(t)\right]\right|_{r=R}=-\delta
p(t).$ (2.2)

Here, $u_{r}=|{\bf u}_{r}|$ is the radial component of the displacement, $R$
is the particle radius, ${\cal{E}}_{\text{s}}$ is the surface energy density,
$\delta p(t)$ is the time dependent additional pressure of the hot electrons,
and $\sigma_{rr}$ is the stress tensor, whose components are expressed through
the moduli $K$, $\mu$ of the uniform compression and rigidity, which one can
find elsewhere (e.g., in [15]). For spherical acoustic oscillations, one can
assume that $u_{\text{L}}(t)=u_{r}(t)$.

The pressure of the electron gas on the surface of the nanoparticle can be
written as the sum

$p(t)=p_{0}+\delta p(t).$ (2.3)

Here, the first term represents the pressure of a degenerate Fermi gas at
$T_{\text{e}}=0$ K, $p_{0}=\frac{2}{5}n\mu_{0}$, where $n=N/V$ is the electron
density, $V$ is the particle volume, and $\mu_{0}$ is the limiting value of
the chemical potential at $T_{\text{e}}=0$ K
($\mu_{0}=\varepsilon_{\text{F}}$, where $\varepsilon_{\text{F}}$ is the Fermi
energy). The second term, $\delta p(t)$, represents the additional time
dependent gas pressure owing to the electron mobility at temperatures
$T_{\text{e}}>0$. It can be written as [16]

$\delta p(t)=\alpha
T^{2}_{\text{e}}(t),\qquad\alpha=\frac{\piup^{2}}{6}n\frac{k^{2}_{\text{B}}}{\mu_{0}}\,.$
(2.4)

##  3 Amplitude of the sound wave

To solve equation (2.1) with the boundary condition (2.2), we use the
potential method [15] ${\bf u}({\bf r},t)\equiv\boldsymbol{\nabla}\varphi({\bf
r},t)$. Then, representing of function $\varphi({\bf r},t)$ in the form of
spherical waves expanding from particle center $\varphi({\bf
r},t)=\phi\big{(}t-(r-R)/s_{\text{L}}\big{)}/r$, where $\phi$ is an arbitrary,
twice differentiable function and using the boundary condition (2.2) at the
nanoparticle surface, we found for $\phi$ an inhomogeneous differential
equation

$\frac{\partial^{2}}{\partial t^{2}}\phi(t)+2\gamma\frac{\partial}{\partial
t}\phi(t)+\omega^{2}_{0}\,\phi(t)=-\delta p(t)\frac{R}{\rho}\,,$ (3.1)

in which $\rho$ is the mass density of the medium surrounding the
nanoparticle, and

$\gamma=\frac{1}{s_{\text{L}}R}\left(2s^{2}_{\text{T}}+\frac{{\cal{E}}_{\text{s}}}{\rho
R}\right),\qquad\omega^{2}_{0}=\frac{2}{R^{2}}\left(2s^{2}_{\text{T}}+\frac{{\cal{E}}_{\text{s}}}{\rho
R}\right).$ (3.2)

Here, $s_{\text{T}}$ is a transverse sound speed [15]. The parameters
$\omega_{0}$ and $\gamma$ are the oscillator eigenfrequency in the absence of
friction and the damping decrement, respectively. Note that when a solid
medium surrounds the nanoparticle, we must set ${\cal{E}}_{\text{s}}=0$ in
expressions (3.2). However, if the nanoparticle is embedded in a liquid
medium, then only the second terms in expressions (3.2) must be retained. The
quantity $1/\gamma$ specifies the time over which the oscillations are
completely damped. It depends on the nanoparticle radius [17].

Equation (3.1) formally describes the motion of the oscillator under the
action of an external driving force of the form $\delta p(t)R/\rho$. We will
consider that there are no oscillator eigenmodes without an external force,
and that they are generated only by an external force. If the driving force
acting on the oscillator, for example, is $\delta$-shaped

$\delta p(t)=p_{\text{im}}\delta(t-\tau_{0}),$ (3.3)

then the solution of equation (3.1) with the initial conditions $\phi=0$ and
$\phi^{\prime}=-Rp_{\text{im}}/\rho$ for $t=\tau_{0}$ is given by

$\phi(t)=-\mathrm{e}^{-\gamma(t-\tau_{0})}\sin\left[\omega(t-\tau_{0})\right]Rp_{\text{im}}/{\rho\omega},$
(3.4)

where $\omega=\sqrt{\omega^{2}_{0}-\gamma^{2}_{0}}$, provided that
$\omega_{0}>\gamma$.

When the time dependence of the function $T_{\text{e}}(t)$ is known, the value
of $p_{\text{im}}$ can be determined explicitly. Comparing expressions (2.4)
and (3.3), we find

$\alpha\int_{0}^{\infty}T^{2}_{\text{e}}(t-\tau_{0})\,\mathrm{d}t=p_{\text{im}}\,.$
(3.5)

The variation of the electron temperature with time can be modelled or
determined analytically.

Let us suppose that initially (at $t=\tau_{0}$) the oscillator is not
displaced, i.e., is at rest, and obeys the initial conditions
$\phi(\tau_{0})=0$ and $\phi^{\prime}(\tau_{0})=0$. Then, the solution of
equation (3.1) for an arbitrary form of the driving force can be presented in
an integral form [18]

$\phi(t)=-\frac{R\mathrm{e}^{-\gamma t}}{\rho\omega}\int_{\tau_{0}}^{t}\delta
p(\tau)\,\mathrm{e}^{\gamma\tau}\sin\left[\omega(t-\tau)\right]\mathrm{d}\tau,$
(3.6)

with the conditions that $\delta p(\tau)\neq 0$ and $t>0$.

Equation (3.6) easily answers the question of how the oscillator behaves after
an excess pressure $\delta p(\tau)$ has acted on it over a short time interval
$(\tau_{0},\tau_{1})$. The upper limit in equation (3.6) for this case,
evidently, will be the quantity $\tau_{1}$. Then, using the mean-value theorem
in equation (3.6), we obtain

$\phi(t)=-RI\mathrm{e}^{-\gamma t}\mathrm{e}^{\gamma\xi\tau_{1}}\sin(\omega
t)/{\rho\omega},$ (3.7)

where $\tau_{0}/\tau_{1}<\xi<1$, while $\int_{\tau_{0}}^{\tau_{1}}\delta
p(\tau)\mathrm{d}\tau\equiv I$ presents the impulse of the excess pressure of
the electron gas. If $\omega_{0}\tau_{1}\xi\ll 2\piup$, then the explicit form
of $\delta p(\tau)$ is not essential; an important point is only the value of
$I$. When $\omega_{0}\gg\gamma$, the maximum displacement $u_{\text{max}}$
caused by the excess pressure “impact”, with using potential method and
equation (3.7), will be

$u_{\text{max}}=I/{\rho\omega_{0}R}.$ (3.8)

Taking into account here equation (3.2) for the case ${\cal{E}}_{\text{s}}=0$,
we obtain

$u_{\text{max},\,{\cal{E}}_{\text{s}}=0}=I/{2\sqrt{\mu\rho}},$ (3.9)

i.e., the magnitude of $u_{\text{max},\,{\cal{E}}_{\text{s}}=0}$ is
independent of the particle radius (and is more independent since the rigidity
modulus and mass density of the nanoparticle are smaller). On the other hand,
if the ${\cal{E}}_{\text{s}}>0$, then a dependence on the particle radius
shows up, and the magnitude of $u_{\text{max}}$ is decreased as $R$ is
reduced. The maximum displacement over the time $\tau_{1}-\tau_{0}$ with
account for the finite damping, has the form:

$u_{\text{max},\,{\cal{E}}_{\text{s}}=0}=\frac{I(1-R\gamma/s_{\text{L}})}{2\sqrt{\mu\rho}\sqrt{1-(s_{\text{T}}/s_{\text{L}})^{2}}}\mathrm{e}^{-\gamma(\tau_{1}-\tau_{0})}.$
(3.10)

Therefore, it is possible to determine the pressure impulse $I$ by measuring
the maximum displacement of the physical object.

##  4  Power radiated by nanoparticle surface

A certain part of the accumulated by MN’s energy is carried away from the
particle into the surrounding matrix by sound waves. The instantaneous elastic
energy flux from the surface of the sphere (for the case of
${\cal{E}}_{\text{s}}=0$) can be calculated using a formula [19]

${\cal W}(t)=S\,\delta p(t)\,\upsilon_{r}(R,t).$ (4.1)

Here, $S$ is the area of the particle surface and $\upsilon_{r}$ is the radial
or vibrational velocity.

In terms of time, conventionally, two stages in this process can be
identified: one is an impetuous rise in $\delta p$ during an ultrashort time
spell ($t\sim\tau_{0}$) and another one is a slower decrease (under
$t>\tau_{0}$) owing to a transfer of the electron energy to the surrounding
matrix. The dependence of the $\delta p(t)$ within the second time spell, can
be well described by expression (2.4).

In order to take an explicit account of the behavior of the pressure in the
initial time spell, the function $\delta p(t)$ can be presented as the product
of two functions,

$\delta p(t)\rightarrow\theta(t-\tau_{0})\cdot\alpha T_{\text{e}}^{2}(t),$
(4.2)

where we use equation (2.4), while $\theta(t-\tau_{0})$ is the Heaviside unit-
step function that specifies the pressure behavior when
$t\rightarrow\tau_{0}$. Let us also suppose that the electron temperature
varies with time in accordance with the rule

$T_{\text{e}}(t)=T_{\text{e}}(\tau_{0})\,\mathrm{e}^{-\beta(t-\tau_{0})},\qquad\beta=\frac{g_{R}}{C_{\text{e}}}\,,\qquad
C_{\text{e}}=3\alpha T_{\text{e}}\,,$ (4.3)

where $C_{\text{e}}\equiv C_{\text{e}}\big{(}T_{\text{e}}(\tau_{0})\big{)}$ is
the heat capacity of the electron gas, which depends on the electron
temperature taken at the instant of time $t=\tau_{0}$. The electron-phonon
coupling constant in equation (4.3) can be computed from the formula

$g_{R}=\frac{27}{16}\frac{n}{m}\frac{\alpha}{k_{\text{B}}}\frac{1}{\rho_{p}R}\left(\frac{\piup\hbar}{a}\right)^{3}\left(\frac{U_{1}}{A}\right)^{2}.$
(4.4)

Here, $a$ is the lattice constant for the MN, $\rho_{p}$ denotes the mass
density of the MN, $U_{1}$ is the energy required to detatch the first
electron from the neutral unexcited atom, and $A$ refers to the electron work
function of the metal. Formula (4.4) follows from equation (115) of the work
[20] using the equation (2.4) and the following relationships

$\omega_{\text{D}}\simeq\sqrt{\frac{\sigma}{\rho_{p}}\left(\frac{\piup}{a}\right)^{3}},\qquad\upsilon_{\text{F}}=\frac{\hbar}{m}\big{(}3n\piup^{2}\big{)}^{1/3},$
(4.5)

where $\omega_{\text{D}}$ is the Debye frequency and $\sigma$ is the surface
energy density.

Taking into account equations (4.2) and (4.3), after integrating with respect
to $\tau$ within limits $0<\tau<t$ and some transformations, we finally obtain

$\displaystyle{\cal W}(t)$ $\displaystyle=\frac{4\piup R^{2}}{\rho
s_{\text{L}}}\frac{\delta p(t)\,\delta
p_{m}(t)}{\omega^{2}_{0}+4\beta(\beta-\gamma)}\bigg{(}-2\beta\left(\frac{s_{\text{L}}}{R}-2\beta\right)+\mathrm{e}^{(2\beta-\gamma)t_{0}}\bigg{\\{}\left[\omega^{2}_{0}+2\beta\left(\frac{s_{\text{L}}}{R}-2\gamma\right)\right]\cos\left(\omega
t_{0}\right)$
$\displaystyle\quad+\left[(\omega^{2}_{0}-2\beta\gamma)\left(\frac{s_{\text{L}}}{R}-\gamma\right)-2\beta\omega^{2}\right]\frac{\sin\left(\omega
t_{0}\right)}{\omega}\bigg{\\}}\bigg{)},$ (4.6)

where $t_{0}=t-\tau_{0}$. Equation (4) gives the energy per unit time carried
out by spherical sound waves from the nanoparticle into the surrounding
medium.

The analysis show that the power of the sonic signal ${\cal W}(t)$ is of the
form of damped oscillations. The number of oscillations depends much on the
density of the matrix material. The amplitude of the sound power is increased
with an increase in the radius of MN, and it is fully damped over longer
times. For different metals (but with the same radius) the amplitude of ${\cal
W}(t)$ will be greater for MNs with a higher coupling constant $g_{R}$.

The maximum power of the acoustic signal is reached at the time instant
$t_{0}=0$. From equation (4), it is equal to

${\cal W}_{\text{max}}\big{|}_{t_{0}=0}=4\piup R^{2}[\delta
p(\tau_{0})]^{2}/\rho s_{\text{L}}.$ (4.7)

The temperature decrease of the electron gas is proportional to the product
$\beta T_{\text{e}}$, which characterizes the cooling rate of the electron
gas. Since the fall of the electron temperature occurs much more slowly than
the electron’s heating, we can assume that the influence of this mechanism on
the sound power is negligible in the first approximation. Formally, this
allows us to direct $\beta\rightarrow 0$ in equation (4). In this case, the
sound signal will correspond to exponentially damped oscillations.

The time dependence of the excess pressure in the case of $\beta\rightarrow
0$, can also be estimated using the equation

$\delta p(t)\big{|}_{\beta\rightarrow
0}\approx-2g_{R}\,t\,T_{\text{e}}(\tau_{0})/3,$ (4.8)

which follows from the energy balance equation for the electrons. The maximum
power of the signal [equation (4.7)], with the account for approximation
(4.8), is

${\cal W}_{\text{max}}\big{|}_{t=\tau_{0},\,\beta\rightarrow
0}=\frac{16}{9}\frac{\piup}{\rho
s_{\text{L}}}[g_{R}\,\tau_{0}\,R\,T_{\text{e}}(\tau_{0})]^{2},$ (4.9)

where the dependence on the laser pulse duration appears explicitly.

##  5 Total energy of the oscillations of the metal nanoparticle

We now determine the total energy transferred from the pulsating spherical
surface to the surrounding matrix. We will consider the case $\beta\neq 0$,
when the power ${\cal W}(t)$ is given by equation (4). Integrating equation
(4) with respect to time over the interval $0<t<\infty$, in view of
expressions (2.4) and (3.2) at ${\cal{E}}_{\text{s}}=0$, we obtain

$E=\int_{\tau_{0}}^{\infty}{\cal W}(t)\,\mathrm{d}t=\frac{2\piup R}{\rho
s_{\text{L}}}\frac{s_{\text{L}}+2\beta
R}{\omega^{2}_{0}+4\beta(\beta+\gamma)}\alpha^{2}T^{4}_{\text{e}}.$ (5.1)

Equation (5.1) shows that the total energy is proportional to the fourth power
of the electron temperature. In the case $\beta=0$, it is easy to reveal that
this energy depends on the volume of the particle and is reciprocal to the
rigidity modulus of the medium.

On the other hand, the work necessary to shift the spherical surface at the
maximal distance $u_{\text{max}}$ [defined by equation (3.10)], is

$E=8\piup Ru_{\text{max}}^{2}\alpha T^{2}_{\text{e}}.$ (5.2)

Comparing it with equation (5.1), one finds

$T_{\text{e}}=2u_{\text{max}}\sqrt{\frac{\rho
s_{\text{L}}}{\alpha}\frac{\omega^{2}_{0}+4\beta(\beta+\gamma)}{s_{\text{L}}+2\beta
R}}\,.$ (5.3)

Another way to determine $T_{\text{e}}$ is as follows. The total energy
obtained by the electron gas in the MN is

$E_{\text{tot}}=3V\alpha T^{2}_{\text{e}}/2.$ (5.4)

The efficiency of the energy transfer $\eta$ from the pulsating spherical MN
to the sound oscillations can be estimated taking the ratio of equations (5.1)
and (5.4). It has the simplest form in the case $\beta\rightarrow 0$

$\eta(T)\equiv E|_{\beta\rightarrow 0}/E_{\text{tot}}=\alpha
T^{2}_{\text{e}}/4\mu.$ (5.5)

The temperature dependence of $\eta(T)$ for noble nanoparticles embedded in a
plexiglass matrix is plotted in figure 1. The calculations were done using
equations (5.1) and (5.4), with account for the characteristics both of the
matrix and the MNs given in tables 1 and 2. The efficiency increases as the
square of the temperature. Actually, it increases if the rigidity modulus of
the medium in which the nanoparticle is embedded gets smaller. So, knowing
from experiment the efficiency $\eta$, it is easy to estimate the electron
temperature in MN. The highest possible $\eta_{T}$ for noble metal
nanoparticles with $R=100$ Å embedded in a plexiglass matrix are presented in
table 2 at $T_{\text{e}}=10^{4}$.

Figure 1: (Color online) Dependence of the sound signal power efficiency on electron temperature for Cu nanoparticle (solid curve), Ag (dashed curve), and Au (dotted curve), embedded in plexiglass. Table 1: Constants for the plexiglass [21]. Medium | $K$ | $\mu$ | $\rho$ | $s_{\text{L}}$ | $s_{\text{T}}$
---|---|---|---|---|---
(dyn/cm2) | (dyn/cm2) | (g/cm${}^{3})$ | (cm/s) | (cm/s)
Plexiglass | 5.83$\cdot 10^{10}$ | 1.48$\cdot 10^{10}$ | 1.18 | 2.57 $\cdot 10^{5}$ | 1.12$\cdot 10^{5}$
Table 2: Physical parameters of the noble MNs. Metal |  $l_{\text{D}}$ (Å) |  $\alpha$ $\left(\frac{\text{erg}}{\text{cm}^{3}~{}\text{K}^{2}}\right)$ |  $n$ (cm-3) [22] |  $\eta_{10^{4}}$ (%)
---|---|---|---|---
Cu | $1197$ | $235.5$ | $8.45\cdot 10^{22}$ | $39.5$
Ag | $1552$ | $208.35$ | $5.85\cdot 10^{22}$ | $35.15$
Au | $1968$ | $208.9$ | $5.90\cdot 10^{22}$ | $35.3$

##  6 Conclusions

The method has been proposed for the estimation of hot electron temperature in
metallic nanoparticles embedded in a dielectric matrix under irradiation of
ultrashort laser pulses. Analytic expressions have been derived for the
amplitude and power of longitudinal spherical sound oscillations as function
of the density and elastic properties of the medium, the laser pulse duration,
electron temperature, nanoparticle radius, and electron-phonon coupling
constants.

The maximal displacement of any oscillator in the MN’s material is due to the
“impact” of the excess pressure of the electron gas. The magnitude of the
sound signal power at the moment of the laser pulse ending can be used to
estimate the maximal electron temperature in the MN. The latter is determined
by the cooling rate of the electron gas and the rate of its pressure change.

The efficiency with which the energy of the hot electrons is carried away by
sound oscillations has been examined for the noble metals. It has been shown
that the sound energy transfer efficiency is considerably higher in the medium
with smaller rigidity moduli.

## Acknowledgements

Author is indebted to DFFD of Ukraine for single financial support of this
work.

## References

  * [1] Bohren C.F., Huffman D.R., Absorption and Scattering of Light by Small Particles, Wiley & Sons, Weinheim, 2004.
  * [2] Crut A., Maioli P., Del Fatti N., Vallée F., Chem. Soc. Rev., 2014, 43, 3921, doi:10.1039/C3CS60367A.
  * [3] Hodak J.H., Martini I., Hartland G.V., J. Chem. Phys., 1998, 108, 9210, doi:10.1063/1.476374.
  * [4] Thibodeaux C.A., Kulkarni V., Chang W.-S., Neumann O., Cao Y., Brinson B., Ayala-Orozco C., Chen C.-W., Morosan E., Link S., Nordlander P., Halas N.J., J. Phys. Chem. B, 2014, 118, 14056, doi:10.1021/jp504467j.
  * [5] Alabastri A., Tuccio S., Giugni A., Toma A., Liberale C., Das G., De Angelis F., di Fabrizio E., Zaccaria R.P., Materials, 2013, 6, 4879, doi:10.3390/ma6114879.
  * [6] Jiang L., Tsai H.-L., Int. J. Heat Mass Transfer, 2007, 50, 3461, doi:10.1016/j.ijheatmasstransfer.2007.01.049.
  * [7] Karpov V.G., Nardone M., Grigorchuk N.I., Phys. Rev. B, 2012, 86, 075463, doi:10.1103/PhysRevB.86.075463.
  * [8] Grigorchuk N.I., Tomchuk P.M., Phys. Rev. B, 2009, 80, 155456, doi:10.1103/PhysRevB.80.155456.
  * [9] Grigorchuk N.I., Condens. Matter Phys., 2013, 16, 33706, doi:10.5488/CMP.16.33706 .
  * [10] Dissanayake D.W. (Ed.), Acoustic Waves, Sciyo, Rijeka, 2000.
  * [11] Perner M., Gresillon S., M$\ddot{\rm a}$rz J., von Plessen G., Feldmann J., Porstendorfer J., Berg K.-J., Berg G., Phys. Rev. Lett., 2000, 85, 792, doi:10.1103/PhysRevLett.85.792.
  * [12] Voisin C., Christofilos D., Loukakos P.A., Del Fatti N., Vallée F., Lermé J., Gaudry M., Cottancin E., Pellarin M., Broyer M., Phys. Rev. B, 2004, 69, 195416, doi:10.1103/PhysRevB.69.195416.
  * [13] Plech A., Kotaidis V., Lorenc M., Wulff M., Chem. Phys. Lett., 2005, 401, 565, doi:10.1016/j.cplett.2004.11.072.
  * [14] Hartland G.V., Annu. Rev. Phys. Chem., 2006, 57, 403, doi:10.1146/annurev.physchem.57.032905.104533.
  * [15] Landau L.D., Lifshitz E.M., Theory of Elasticity, Pergamon Press, New York, 1986.
  * [16] Koroljuk S., Mel’nychuk S., Val’ O., Fundamentals of Statistical Physics and Thermodynamics, Knygy-XXI, Chernivtsi, 2004, (in Ukrainian).
  * [17] Halté V., Bigot J.-Y., Palpant B., Broyer M., Prével B., Pérez A., Appl. Phys. Lett., 1999, 75, 3799,
doi:10.1063/1.125460.

  * [18] Kamke E., Differentialgleichungen: Lösungsmetoden und Lösungen, Teubner, Stuttgart, 1983.
  * [19] Fedorov F.I., Theory of Elastic Waves in Crystals, Plenum Press, New York, 1968.
  * [20] Fedorovich R.D., Naumovets A.G., Tomchuk P.M., Phys. Rep., 2000, 328, 73,
doi:10.1016/S0370-1573(99)00094-0.

  * [21] Lide D.R. (Ed.), CRC Handbook of Chemistry and Physics, 92th Ed., CRC Press, Boca Raton, 2011.
  * [22] Kittel Ch., Introduction to Solid State Physics, Wiley & Sons, New York, 2005.

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# Distributed Interval Observers for Bounded-Error LTI Systems

Mohammad Khajenejad∗, Scott Brown∗, and Sonia Martínez
∗Equal contribution. M. Khajenejad, S. Brown and S. Martínez are with the
Mechanical and Aerospace Engineering Department of the Jacobs School of
Engineering, University of California, San Diego, La Jolla, San Diego, CA,
USA. (e-mail: {mkhajenejad, sab007, soniamd}@ucsd.edu.)This work is partially
funded by AFOSR and ONR grants FA9550-19-1-0235 and N00014-19-1-2471.

###### Abstract

This paper proposes a novel distributed interval observer design for linear
time-invariant (LTI) discrete-time systems subject to bounded disturbances. In
the  proposed observer algorithm, each agent in a networked group exchanges
locally-computed framers or interval-valued state estimates with neighbors,
and coordinates its update via an intersection operation. We show that the
proposed framers are guaranteed to bound the true state trajectory of the
system by construction, i.e., without imposing any additional assumptions or
constraints. Moreover, we provide necessary and sufficient conditions for the
collective stability of the distributed observer, i.e., to guarantee the
uniform boundedness of the observer error sequence. In particular, we show
that such conditions can be tractably satisfied through a constructive and
distributed approach.  Moreover, we provide an algorithm to verify some
structural conditions for a given system, which guarantee the existence of the
proposed observer. Finally, simulation results demonstrate the effectiveness
of our proposed method compared to an existing distributed observer in the
literature.

##  I Introduction

Many large scale cyber-physical systems, such as electric power grids [1],
intelligent transportation systems [2], and industrial infrastructures [3],
are equipped with sensor networks, providing in situ and diverse measurements
to monitor them. This makes possible the construction of system state
estimates, which are essential to guarantee the safe and effective operation
of these critical applications. Motivated by this, an intense research
activity on the analysis and design of distributed estimation algorithms has
ensued. In this way, each sensor, equipped with local communication and
processing capabilities, interacts with neighboring nodes to compute joint
estimates cooperatively.

A way to obtain such estimates is to use a centralized observer, by which a
super node collects all measurements from the nodes and fuses them in an
optimal way. The ubiquitous Kalman filter [4] and related approaches have been
used extensively for this purpose. However, these algorithms do not scale well
as the size of the network increases and are vulnerable to single-point
failures. This spawned research on the design of distributed estimation
filters (for systems subject to known stochastic disturbances) for sensor
networks communicating only locally over a possibly time-varying network [5].
While these methods are more scalable and robust to communication failures
than their centralized counterparts, they generally have comparatively worse
estimation error. An important class of algorithms that aim to approach the
estimation performance of their Kalman filter counterparts, are Kalman-
consensus filters, which combine a local Kalman-like update with average
consensus to align agents’ estimates [6, 7]. When stochastic characterization
of disturbances is not available, however, other techniques that leverage
alternative information should be considered.

In case the disturbances are known to be bounded, interval observers are a
popular method for obtaining robust, guaranteed estimates of the state, due to
their simplicity and computational efficiency [8, 9, 10]. Hence, various
approaches to design centralized interval observers for various classes of
dynamical systems have been proposed [11, 12, 13, 14, 15, 16, 17]. The main
idea in most of the aforementioned designs is to synthesize appropriate
centralized observer gains to obtain a robustly stable and positive observer
error system for all realizations of the existing uncertainties[11, 14]. This
strategy, which usually boils down to solving centralized semi-definite
programs (SDP) subject to large numbers of constraints, leads to theoretical
and computational difficulties, and thus infeasible solutions, especially for
large-scale systems [15, 16, 17]. In addition to computational issues, the
communication complexity of the centralized approach does not scale well as
the size of the network increases. A recent study [18] proposes a distributed
interval observer for block-diagonalizable linear time-invariant (LTI)
systems, which requires a certain structure on the dynamics and the output of
the system. Another work [19] designs an observer for LTI systems under
denial-of-service attacks. In addition, [20] proposes an internally positive
representation (IPR)-based robust distributed interval observer for
continuous-time LTI systems. However, the proposed design relies on similarity
transformations and the satisfaction of certain algebraic constraints, which
could lead to moderately-performing results. Furthermore, all of the
aforementioned works use average consensus to share estimates throughout the
network, which limits the effectiveness of the proposed methods with respect
to time of convergence and estimation quality.

Contributions. To overcome the aforementioned drawbacks, this work contributes
to bridging the gap between interval observer design approaches and
distributed estimation algorithms in the presence of distribution-free
uncertainties. We introduce a novel method for synthesizing scalable
distributed interval observers for discrete-time LTI systems subject to
bounded additive disturbances. We provide necessary and sufficient conditions
for the stability of our proposed observer. Our observer is correct by
construction, and we leverage this correctness to intersect interval estimates
between neighboring nodes, ensuring that the best possible (agent) estimate is
adopted by consensus in a finite number of iterations. Furthermore, we
introduce the intuitive notion of “collective positive detectability over
neighborhoods” (CPDN) which,  is sufficient to tractably compute gains that
satisfy the aforementioned stability requirement in a distributed manner. This
approach involves the solution to local and feasible linear programs (LP),
which is potentially less conservative and computationally more efficient than
SDP-based approaches.  Finally, we provide an algorithm to verify if CPDN
holds for a given system.

##  II Preliminaries

This section introduces basic notation, preliminary concepts and graph theory
notions used throughout the paper.

Notation. Let n, n×p, ${\mathbb{N}}$, $\mathbb{Z}_{\geq 0}$, and
$\mathbb{R}_{\geq 0}$ denote the $n$-dimensional Euclidean space, the sets of
$n$ by $p$ matrices, natural numbers, nonnegative integers, and nonnegative
real numbers, respectively. For $M\in^{n\times p}$, let $M_{i}$ and $M_{ij}$
denote the $i^{\textup{th}}$ row of $M$, and the $(i,j)^{\textup{th}}$ entry
of $M$, respectively. Furthermore, for $M\in^{n\times p}$, we define
$M^{+}\in^{n\times p}$, such that $M^{+}_{ij}\triangleq\max\\{M_{ij},0\\}$,
$M^{-}\triangleq M^{+}-M$, and $|M|\triangleq M^{+}+M^{-}$. In addition,
$M\succ 0$ (or, $M\succeq 0$, resp.) denote that $M$ is positive definite
(semi-definite, resp.), and $\rho(M)$ is used to denote the spectral radius of
$M$. All the inequalities $\leq,\geq$ are considered element-wise. As usual,
$\mathbf{e}_{i}$ denotes the $i^{\textup{th}}$ vector of the standard basis of
n. Finally, for $A^{1},\dots,A^{N}\in^{n\times n}$,
$\operatorname{diag}(A^{1},\dots,A^{N})\in^{nN\times nN}$ denotes the block-
diagonal matrix with block-diagonal elements being
$A^{i},i\in\\{1,\dots,N\\}$.

Graph-theoretic Notions. Next, we recall some definitions from Graph Theory. A
directed graph (digraph) $\mathcal{G}$ is a set of nodes $\mathcal{V}$ and a
set of directed edges $\mathcal{E}\subseteq\mathcal{V}\times\mathcal{V}$. The
set of neighbors of node $i$, denoted $\mathcal{N}_{i}$, is the set of all
nodes $j$ for which there is an edge $(i,j)\in\mathcal{E}$. We will assume
that $i\in\mathcal{N}_{i}$. A path from node $i$ to node $j$ is a sequence of
nodes starting with $i$ and ending with $j$, such that any two consecutive
nodes are joined by a directed edge. The $d$-hop neighbors of agent $i$,
denoted $\mathcal{N}_{i}^{d}$, is the set of nodes connected to $i$ by a path
of length no more than $d$. The diameter of a graph is the largest distance
between any two nodes, i.e.
$\operatorname{diam}\mathcal{G}\triangleq\max_{i,j}d(i,j)$, where $d(i,j)$
denotes the length of the shortest path between $i$ and $j$.

Multi-dimensional Intervals. Finally, we introduce some definitions and
results regarding multi-dimensional intervals. A (multi-dimensional) interval
$\mathcal{I}\triangleq[\underline{s},\overline{s}]\subset^{n}$ is the set of
all vectors $x\in^{n}$ that satisfy $\underline{s}\leq x\leq\overline{s}$.

######  Proposition 1.

[17, Lemma 1] Let $A\in^{p\times n}$ and $\underline{x}\leq
x\leq\overline{x}\in^{n}$. Then, $A^{+}\underline{x}-A^{-}\overline{x}\leq
Ax\leq A^{+}\overline{x}-A^{-}\underline{x}$. As a corollary, if $A$ is non-
negative, $A\underline{x}\leq Ax\leq A\overline{x}$.

##  III Problem Formulation

System Assumptions. Consider a multi-agent system  (MAS) consisting of
$\mathcal{V}\triangleq\\{1,\dots,N\\}$ agents, which interact over a time-
invariant communication graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$. The
agents are able to obtain distributed measurements of a target as described by
the following LTI dynamics:

$\displaystyle\begin{array}[]{ll}\mathcal{P}:\begin{cases}x_{k+1}=Ax_{k}+Bw_{k},&\\\
y^{i}_{k}=C^{i}x_{k}+D^{i}v^{i}_{k},&i\in\mathcal{V},\quad
k\in\mathbb{Z}_{\geq 0},\end{cases}\end{array}$ (2)

where $x_{k}\in^{n}$ is the continuous state of the target system and
$w_{k}\in\mathcal{I}_{w}\triangleq[\underline{w},\overline{w}]\subset^{n_{w}}$
is bounded process disturbance. Furthermore, at time step $k$, every agent
$i\in\mathcal{V}$ takes a measurement $y^{i}_{k}\in^{m_{i}}$, known only to
itself, which is perturbed by
$v^{i}_{k}\in\mathcal{I}_{v}^{i}\triangleq[\underline{v}^{i},\overline{v}^{i}]\subset^{n_{v}^{i}}$,
a bounded sensor (measurement) noise signal. Finally, $A\in^{n\times
n},B\in^{n\times n_{w}},C^{i}\in^{m_{i}\times n}$ and $D^{i}\in^{m_{i}\times
n^{i}_{v}}$ are system matrices known to all agents. The  MAS’s goal is to
estimate the trajectories of (2) in a distributed manner, when they are
initialized in an interval
$\mathcal{I}_{x}\triangleq[\underline{x}_{0},\overline{x}_{0}]\subset^{n}$,
with $\underline{x}_{0},\overline{x}_{0}$ known to all agents. Next, we define
the notions of framers, correctness, and stability, used throughout the paper.

######  Definition 1 (Framers).

For an agent $i\in\mathcal{V}$, the sequences
$\\{\overline{x}^{i}_{k}\\}_{k\geq 0}$ and $\\{\underline{x}^{i}_{k}\\}_{k\geq
0}\subseteq^{n}$ are called upper and lower individual framers for (the state
of) $\mathcal{P}$ if $\underline{x}^{i}_{k}\leq
x_{k}\leq\overline{x}^{i}_{k},\quad$ for all $k\geq 0$.  Moreover, we define
the individual lower and upper framer errors  as follows:

$\displaystyle\underline{e}^{i}_{k}\triangleq
x_{k}-\underline{x}^{i}_{k},\quad\overline{e}^{i}_{k}\triangleq\overline{x}^{i}_{k}-x_{k},\quad\forall
k\geq 0.$ (3)

Given an MAS with target system $\mathcal{P}$ and communication graph
$\mathcal{G}$, a distributed interval framer is a distributed algorithm over
$\mathcal{G}$ that allows each agent $i$ to cooperatively compute upper and
lower individual framers for $\mathcal{P}$. Finally,

$\displaystyle
e_{k}\triangleq\begin{bmatrix}(\underline{e}^{1}_{k})^{\top}&(\overline{e}^{1}_{k})^{\top}&\cdots&(\underline{e}^{N}_{k})^{\top}&(\overline{e}^{N}_{k})^{\top}\end{bmatrix}^{\top}\in^{2Nn}.$
(4)

is called the collective framer error, which is the vector of all individual
lower and upper framer errors.

######  Definition 2 (Distributed Interval Observer).

A _distributed interval framer_ is input-to-state (ISS) stable if the
collective framer error is bounded as follows:

$\displaystyle\|e_{k}\|\leq\beta(\|e_{0}\|,k)+{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\gamma}\big{(}\max_{0\leq
l\leq k}|\Delta_{l}|\big{)},\quad\forall k\in\mathbb{Z}_{\geq 0},$

where $\Delta_{l}\triangleq[w_{l}^{\top}\
(v^{1}_{l})^{\top}\cdots(v^{N}_{l})^{\top}]^{\top}\in^{n_{w}+Nn_{v}}$, and
$\beta$ and $\gamma$ are functions of classes $\mathcal{KL}$ and
$\mathcal{K}_{\infty}$, respectively. An ISS distributed interval framer is a
distributed interval observer.

The observer design problem can be stated as follows:

######  Problem 1.

Given a multi-agent system and the LTI system in (2), design a distributed
interval observer for $\mathcal{P}$.

##  IV Proposed Distributed Interval Observer

In this section, we describe our novel distributed interval observer design, a
necessary and sufficient condition for stability of the proposed observer, and
an LP-based distributed procedure for computing stabilizing observer gains.

###  IV-A Distributed Observer and its Correctness

To address Problem 1, we propose a two-step distributed interval framer (cf.
Definition 1) for $\mathcal{P}$. The $\operatorname{DIO}$ Algorithm 1 provides
a pseudocode description of our observer, the details of which are further
explained in this section  as follows.

i) Propagation and Measurement Update: At every $k+1\in\mathbb{Z}_{\geq 0}$,
given $\underline{x}^{i}_{k}$, $\overline{x}^{i}_{k}$, $y^{i}_{k}$, and
$y^{i}_{k+1}$, each agent $i\in\mathcal{V}$ performs a state propagation and a
local measurement update step using observer gains
$L^{i},\Gamma^{i}\in^{n\times m_{i}}$ which will be designed to satisfy
desired observer properties:

$\displaystyle\begin{split}\underline{x}^{i,0}_{k+1}&=\tilde{A}^{i+}\underline{x}_{k}^{i}-\tilde{A}^{i-}\overline{x}_{k}^{i}+(T^{i}B)^{+}\underline{w}-(T^{i}B)^{-}\overline{w}+L^{i}y^{i}_{k}+\Gamma^{i}y^{i}_{k+1}\\\
&+((L^{i}D^{i})^{+}+(\Gamma^{i}D^{i})^{+})\overline{v}-((L^{i}D^{i})^{-}+(\Gamma^{i}D^{i})^{-})\underline{v},\\\
\overline{x}^{i,0}_{k+1}&=\tilde{A}^{i+}\overline{x}_{k}^{i}-\tilde{A}^{i-}\underline{x}_{k}^{i}+(T^{i}B)^{+}\overline{w}-(T^{i}B)^{-}\underline{w}+L^{i}y^{i}_{k}+\Gamma^{i}y^{i}_{k+1}\\\
&+((L^{i}D^{i})^{+}+(\Gamma^{i}D^{i})^{+})\underline{v}-((L^{i}D^{i})^{-}+(\Gamma^{i}D^{i})^{-})\overline{v},\end{split}$
(5)

where $T^{i}\triangleq I_{n}-\Gamma^{i}C^{i}$ and $\tilde{A}^{i}\triangleq
T^{i}A-L^{i}C^{i}$.  Further, $\underline{e}^{i,0}_{k}\triangleq
x_{k}-\underline{x}^{i,0}_{k}$ and
$\overline{e}^{i,0}_{k}\triangleq\overline{x}^{i,0}_{k}-x_{k}$ are the
corresponding errors, and $e^{0}_{k}$ is the vector of all agents’ errors, as
in (4).

ii) Network Update: After the measurement update, each agent $i$ iteratively
shares its interval estimate with its neighbors in the network,  and updates
it by taking the tightest interval from all neighbors via intersection:

$\displaystyle\begin{split}\underline{x}^{i,t}_{k}=\max_{j\in\mathcal{N}_{i}}\underline{x}^{j,t-1}_{k},\quad\underline{x}^{i}_{k}=\underline{x}^{i,d}_{k},\\\
\overline{x}^{i,t}_{k}=\min_{j\in\mathcal{N}_{i}}\overline{x}^{j,t-1}_{k},\quad\overline{x}^{i}_{k}=\overline{x}^{i,d}_{k},\end{split}$
(6)

$\forall t\in\\{1,\dots,d\\}$, where $d\in{\mathbb{N}}$ is the number of
network-update iterations. Note that in case $d>1$, this iterative procedure
computes the intersection of intervals with the $d$-hop neighbors of each
agent.  Consequently, each agent $i$ obtains the  following information:

$\displaystyle\underline{x}^{i}_{k}=\max_{j\in\mathcal{N}_{i}^{d}}\underline{x}^{j,0}_{k}\quad\text{and}\quad\overline{x}^{i}_{k}=\min_{j\in\mathcal{N}_{i}^{d}}\overline{x}^{j,0}_{k}.$

We will use this fact as a compact representation of the network update (6).
Next, we show that the proposed $\operatorname{DIO}$ algorithm constructs a
distributed interval framer in the sense of Definition 1 for the plant
$\mathcal{P}$.

Algorithm 1 Distributed Interval Observer ($\operatorname{DIO}$) at node $i$.

1:$\underline{x}^{i}_{0}$, $\overline{x}^{i}_{0}$, $d$; Output:
$\\{\underline{x}^{i}_{k}\\}_{k\geq 0}$, $\\{\overline{x}^{i}_{k}\\}_{k\geq
0}$;

2:Compute $L^{i}$ $\Gamma^{i}$, and $T^{i}$ using (15);

3:loop

4: Compute $\underline{x}^{i,0}_{k+1}$ and $\overline{x}^{i,0}_{k+1}$ using
(5);

5: for $t=1$ to $d$ do

6: Send $\underline{x}^{i,t-1}_{k+1}$ and $\overline{x}^{i,t-1}_{k+1}$ to
$j\in\mathcal{N}_{i}$;

7:
$\displaystyle\underline{x}^{i,t}_{k+1}\leftarrow\,\max_{j\in\mathcal{N}_{i}}\underline{x}^{j,t-1}_{k+1};\,\,\overline{x}^{i,t}_{k+1}\leftarrow\,\min_{j\in\mathcal{N}_{i}}\overline{x}^{j,t-1}_{k+1};$

8: end for

9: $\underline{x}^{i}_{k+1}\leftarrow\,\underline{x}^{i,d}_{k+1}$;
$\overline{x}^{i}_{k+1}\leftarrow\,\overline{x}^{i,d}_{k+1}$; $k\leftarrow
k+1$;

10:end loop

11:return $\\{\underline{x}^{i}_{k}\\}_{k\geq 0}$,
$\\{\overline{x}^{i}_{k}\\}_{k\geq 0}$

######  Lemma 1 (Distributed Framer Construction).

The $\operatorname{DIO}$ algorithm is a distributed interval framer for (2).

###### Proof.

From (2) and the fact that $T^{i}=I_{n}-\Gamma^{i}C^{i}$, we have

$\displaystyle
x_{k+1}=(\Gamma^{i}C^{i}+T^{i})x_{k+1}=T^{i}(Ax_{k}+Bw_{k})+\Gamma^{i}C^{i}x_{k+1}.$
(7)

Plugging $C^{i}x_{k+1}=y^{i}_{k+1}-D^{i}v^{i}_{k+1}$ into, as well as adding
the _zero term_ $L^{i}(y^{i}_{k}-C^{i}x_{k}-D^{i}v^{i}_{k})$ to the right hand
side of (7) results in

$\displaystyle
x_{k+1}=\tilde{A}^{i}x_{k}+T^{i}Bw_{k}+\Gamma^{i}(y^{i}_{k+1}-D^{i}v^{i}_{k+1})+L^{i}(y^{i}_{k}-D^{i}v^{i}_{k}).$
(8)

Applying Proposition 1 to all the uncertain terms in the right hand side of
(8) shows that for each $i\in\mathcal{V}$,

$\displaystyle\underline{x}^{i}_{k}\leq
x_{k}\leq\overline{x}^{i}_{k}\implies\underline{x}^{i,0}_{k+1}\leq
x_{k+1}\leq\overline{x}^{i,0}_{k+1},$

where $\underline{x}^{i,0}_{k+1},\overline{x}^{i,0}_{k+1}$ are given in (5).
This means that individual framers/interval estimates are correct. When the
framer condition is satisfied for all nodes, the intersection of all the
individual estimates of neighboring nodes (cf. (6)) also results in correct
interval framers, i.e.

$\displaystyle\underline{x}^{i,0}_{k}\leq
x_{k}\leq\overline{x}^{i,0}_{k},\;\forall
i\in\mathcal{V}\implies\underline{x}^{i}_{k}\leq
x_{k}\leq\overline{x}^{i}_{k},\;\forall i\in\mathcal{V}.$

Since the initial interval is known to all $i$, by induction (5)-(6)
constructs a correct distributed interval framer for (2). ∎

###  IV-B Collective Input-to-State Stability

In this subsection, we investigate conditions on the observer gains $L^{i}$,
$T^{i}$, and $\Gamma^{i}$, as well as the communication graph $\mathcal{G}$,
that lead to an ISS distributed observer (cf. Definition 2), which
equivalently results in a uniformly bounded observer error sequence
$\\{e_{k}\\}_{k\geq 0}$ (given in (3)–(4)), in the presence of bounded noise.
For ease of exposition, in what follows, we ignore noise terms and focus on
asymptotic stability of the noiseless error dynamics, which we will show
implies collective input-to-state stability.

Switched System Perspective. We begin by stating a preliminary result that
expresses the observer error dynamics in the form of a specific switched
system.

######  Lemma 2.

The collective error $e_{k}$ has dynamics

$\displaystyle e_{k+1}=H_{k}\hat{A}e_{k},$ (9)

where $\hat{A}\triangleq\operatorname{diag}(\hat{A}^{1},\dots,\hat{A}^{N})$,
$\hat{A}^{i}\triangleq\begin{bmatrix}(\tilde{A}^{i})^{+}&(\tilde{A}^{i})^{-}\\\
(\tilde{A}^{i})^{-}&(\tilde{A}^{i})^{+}\end{bmatrix}$.
$H_{k}\in\\{0,1\\}^{2Nn\times 2Nn}$ is a binary matrix which selects a single
minimizer or maximizer of the framers, i.e.,

$\displaystyle\begin{split}(H_{k})_{\underline{\operatorname{id}}(i,s),\underline{\operatorname{id}}(j^{*},s)}=1&\Leftrightarrow
j^{*}=\min(\operatorname*{argmax}_{j\in\mathcal{N}_{i}^{d}}(\underline{x}^{j,0}_{k})_{s}),\\\
(H_{k})_{\overline{\operatorname{id}}(i,s),\overline{\operatorname{id}}(j^{*},s)}=1&\Leftrightarrow
j^{*}=\min(\operatorname*{argmin}_{j\in\mathcal{N}_{i}^{d}}(\overline{x}^{j,0}_{k})_{s}),\\\
\end{split}$ (10)

for $s\in\\{1,\dots,n\\}$ and $i\in\mathcal{V}$, where
$\underline{\operatorname{id}}(i,s)=2n(i-1)+s-1$ and
$\overline{\operatorname{id}}(i,s)=2n(i-1)+s+n-1$ encode the indices
associated with the upper and lower framers of state $s$ at node $i$.
Furthermore, $H_{k}\hat{A}\in\mathcal{F}\subseteq^{2Nn\times 2Nn}$, where

$\displaystyle\mathcal{F}$
$\displaystyle\triangleq\Big{\\{}\begin{bmatrix}\underline{a}_{1,1}^{\top}&\overline{a}_{1,1}^{\top}&\underline{a}_{1,2}^{\top}&\overline{a}_{1,2}^{\top}&\cdots&\underline{a}_{N,n}^{\top}&\overline{a}_{N,n}^{\top}\end{bmatrix}^{\top}$
$\displaystyle\quad\quad:\,\underline{a}_{i,s}\in\underline{\mathcal{F}}^{i}_{s},\overline{a}_{i,s}\in\overline{\mathcal{F}}^{i}_{s},\
s\in\\{1,\dots,n\\},\ i\in\mathcal{V}\Big{\\}},$
$\displaystyle\underline{\mathcal{F}}^{i}_{s}$
$\displaystyle\triangleq\left\\{\mathbf{e}_{j}^{\top}\otimes[(\tilde{A}^{j})^{+}\
(\tilde{A}^{j})^{-}]_{s}\in^{1\times 2Nn}\ :\
{j\in\mathcal{N}_{i}^{d}}\right\\},$
$\displaystyle\overline{\mathcal{F}}^{i}_{s}$
$\displaystyle\triangleq\left\\{\mathbf{e}_{j}^{\top}\otimes[(\tilde{A}^{j})^{-}\
(\tilde{A}^{j})^{+}]_{s}\in^{1\times 2Nn}\ :\
{j\in\mathcal{N}_{i}^{d}}\right\\}.$

###### Proof.

Notice that $\tilde{A}^{i}=\tilde{A}^{i+}-\tilde{A}^{i-}$ in (8). Rewriting in
terms of the error by adding (or subtracting) (8) to (5) and (6), then setting
the noise to zero, we obtain
${\color[rgb]{1,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,1}\pgfsys@color@cmyk@stroke{0}{1}{0}{0}\pgfsys@color@cmyk@fill{0}{1}{0}{0}e^{0}_{k+1}}=\hat{A}e_{k}$
and $e_{k}=H_{k}e^{0}_{k}$. Combining these yields (9). ∎

Recall that the switching in (9) depends on the state according to (10) and
always creates the smallest possible error. In order to take the advantage of
this property note that the set $\mathcal{F}$ has a specific structure known
as _independent row uncertainty_ , formally defined below.

######  Definition 3 (Independent Row Uncertainty [21]).

A set  of matrices $\mathcal{M}\subset^{n\times n}$ has independent row
uncertainties if

$\displaystyle\mathcal{M}=\left\\{\begin{bmatrix}a_{1}^{\top}&\cdots&a_{n}^{\top}\end{bmatrix}^{\top}\
:\ a_{i}\in\mathcal{M}_{i},\ i\in\\{1,\dots,n\\}\right\\},$

where all sets $\mathcal{M}_{i}\subset^{1\times n}$ are compact.

Next, we restate the following lemma on the spectral properties of the sets
with independent row uncertainties, that will be used later in our stability
analysis of system (9).

######  Lemma 3.

[21, Lemma 2] Suppose $\mathcal{M}\subset^{n\times n}$ has independent row
uncertainties.  Then, there exists $M_{*}\in\mathcal{M}$ such that
$\rho(M_{*})=\min_{M\in\mathcal{M}}\rho(M)=\lim\limits_{k\to\infty}\big{(}\min\limits_{M_{i}\in\mathcal{M},\\\
i\in\\{1,\dots,k\\}}\|M_{1}\cdots M_{k}\|^{\frac{1}{k}}\big{)}.$ The latter is
known as the _lower spectral radius_ of  the set of matrices $\mathcal{M}$.

We can now state our first main stability result.

######  Theorem 1 (Necessary and Sufficient Conditions for Stability).

The error system (9) is globally exponentially stable  if and only if there
exists  $H_{*}\in\mathbb{R}^{2nN\times 2nN}$ which can be constructed
according to (10) for some values of $\underline{x}$ and $\overline{x}$, such
that the matrix $H_{*}\hat{A}$ is Schur stable. Consequently, the
$\operatorname{DIO}$ algorithm is ISS  if and only if such an $H_{*}$ exists.

###### Proof.

We first prove sufficiency and then necessity. Assume $H_{*}\hat{A}$ is Schur
stable. Consider the comparison system
$\tilde{e}_{k+1}=H_{*}\hat{A}\tilde{e}_{k}$ with initial condition
$\tilde{e}_{0}=e_{0}$. By the construction of $H_{k}$ in (10), which implies
$H_{*}\hat{A}e_{k}\geq H_{k}\hat{A}e_{k}$, $\tilde{e}_{k}\geq e_{k}\geq 0$ for
all $k\geq 0$ by induction. Therefore, by comparison, (9) is globally
exponentially stable. To prove necessity, note that (9) is asymptotically
stable only if the lower spectral radius of $\mathcal{F}$ is less than 1. By
Lemma 3, this implies existence of a stable $F_{*}=H_{*}\hat{A}$. Finally,
having studied stability of the noise-free system, we now study the ISS
property of the noisy system:

$\displaystyle e_{k+1}=H_{k}\hat{A}e_{k}+H_{k}(W_{k}+V_{k}),\ \text{ where}$
(11)

$W_{k}\triangleq[(\Lambda^{1}_{k})^{\top}\dots(\Lambda^{N}_{k})]^{\top},\Lambda_{k}^{i}\triangleq\begin{bmatrix}(T^{i}B)^{+}\underline{s}_{k}+(T^{i}B)^{-}\overline{s}_{k}\\\
(T^{i}B)^{-}\underline{s}_{k}+(T^{i}B)^{+}\overline{s}_{k}\end{bmatrix},$ with
$\underline{s}_{k}\triangleq w_{k}-\underline{w}$,
$\overline{s}_{k}\triangleq\overline{w}-w_{k}$ and $V_{k}$ is defined
similarly to $W_{k}$, with $(L^{i}D^{i})^{*}+(\Gamma^{i}D^{i})^{*}$ replacing
$(T^{i}B)^{*}$, for $*\in\\{+,-\\}$ and $\underline{v}^{i}$,
$\overline{v}^{i}$, and $v^{i}_{k}$ replacing $\underline{w}$, $\overline{w}$,
and $w_{k}$ respectively. As before, we can use the comparison system

$\displaystyle\tilde{e}_{k+1}=H_{*}\hat{A}\tilde{e}_{k}+H_{*}(W_{k}+V_{k}),\quad\tilde{e}_{0}=e_{0}$
(12)

It is well known that stable LTI systems are ISS [22]. Again, (10) guarantees
$\tilde{e}_{k}\geq e_{k}\geq 0\ \forall k\geq 0$ by induction, regardless of
the values of $W_{k}$ and $V_{k}$. By this comparison, the ISS property of the
system (12) implies that (11) is ISS. ∎

Theorem 1 is only an _existence_ result. It does not provide a method for
constructing $H_{*}$, which could be a difficult combinitorial problem.
Therefore, in the next section we provide  a tractable approach that allows
for the computation of stabilizing gains and the corresponding $H_{*}$ in
Theorem 1.

###  IV-C Distributed Stabilizing of the Error Dynamics

In this subsection, we show that the ISS property formalized in Theorem 1 can
be tractably verified in a constructive and distributed manner. The approach
is motivated by the representation (9), in which $H_{k}$ exchanges rows of
$\hat{A}$ to achieve the best possible estimate. The main idea is that each
agent, depending on its observation of the system, contributes to stabilizing
the state trajectory in _some_ , not necessarily all, dimensions. To guarantee
stability using our design approach, we will use the following assumption,
which characterizes the interplay between the agents’ local observations and
their communication over the network.

######  Assumption 1 (Collective Positive Detectability over Neighborhoods
(CPDN)).

There is a $d^{*}\in{\mathbb{N}}$ such that for every state
$s\in\\{1,\dots,n\\}$ and every agent $i\in\mathcal{V}$, there is an agent
$\ell(i,s)\in\mathcal{N}_{i}^{d^{*}}$ such that there exist gains
$T^{\ell(i,s)}$, $L^{\ell(i,s)}$, and $\Gamma^{\ell(i,s)}$ satisfying
$\|(T^{\ell(i,s)}A-L^{\ell(i,s)}C^{\ell(i,s)})_{s}\|_{1}<1$.

Assumption 1 captures a broad range of conditions on the system and graph
structure that can result in a stable observer.  It is  also different from
the well-known notion of “collective observability” [20], which is not
sufficient here due to the non-negativity of the error dynamics in the
interval observer design. Furthermore,  the conditions in Assumption 1, can be
verified using a distributed procedure described in Algorithm 2,  summarized
here. First, each node $i\in\mathcal{V}$ independently solves the  following
linear program:

$\displaystyle\begin{array}[]{rl}&\min\limits_{\\{E^{i},L^{i},T^{i},\Gamma^{i}\\}}\quad\sum_{j=1}^{n}\sum_{t=1}^{n}E^{i}_{jt}\\\
&\mathrm{s.t.}\ -E^{i}\leq T^{i}A-L^{i}C^{i}\leq
E^{i},T^{i}=I_{n}-\Gamma^{i}C^{i}.\end{array}$ (15)

Then, nodes exchange their $\tilde{A}^{i}$ matrices with increasingly larger
neighborhoods until stabilizing agents are found for every state.  The
following lemma formalizes this result.

Algorithm 2 $\operatorname{DIO}$ initialization at node $i$.

1:$A$, $C^{i}$, $\mathcal{N}_{i}$; Output: $L^{i}$, $T^{i}$, $\Gamma^{i}$,
$d^{*}$

2:Compute $L^{i,*}$ $\Gamma^{i,*}$, and $T^{i,*}$ by solving (15);

3:$\mathcal{Q}_{i}\leftarrow\\{T^{i,*}A-L^{i,*}C^{i,*}\\}$; $d^{*}\leftarrow
1$;

4:while $d^{*}\leq\operatorname{diam}\mathcal{G}$ do

5: if $\forall s\in\\{1,\dots,n\\}$, $\exists P\in\mathcal{Q}_{i}$ s.t.
$\|(P)_{s}\|_{1}<1$ then

6: break

7: end if

8: Send $\mathcal{Q}_{i}$ to $j\in\mathcal{N}_{i}$ and recieve
$\mathcal{Q}_{j}$ from $j\in\mathcal{N}_{i}$;

9: $\mathcal{Q}_{i}\leftarrow\bigcup_{j\in\mathcal{N}_{i}}Q_{j}$;
$d^{*}\leftarrow d^{*}+1$;

10:end while

11:for $t=1$ to $\operatorname{diam}\mathcal{G}$ do
$d_{i}^{*}\leftarrow\max_{j\in\mathcal{N}_{i}}d_{j}^{*}$ end for

12:return $L^{i,*}$, $T^{i,*}$, $\Gamma^{i,*}$, $d_{i}^{*}$

######  Lemma 4.

Assumption 1 holds if and only if Algorithm 2 returns
$d^{*}\leq\operatorname{diam}\mathcal{G}$.

###### Proof.

Assume Assumption 1 does not hold for some agent $i$. Then the condition in
line 4 of Algorithm 2 will never be met, resulting in
$d_{i}^{*}=\operatorname{diam}{\mathcal{G}}+1$. After the max consensus on
line 10, all agents will return $d^{*}=\operatorname{diam}\mathcal{G}+1$. On
the other hand, if Assumption 1 holds, the condition in line 4 will be met
after less than $\operatorname{diam}{\mathcal{G}}$ iterations for every node.
∎

Finally, we show that the solutions to the LP in (15) are the corresponding
stabilizing observer gains.

######  Theorem 2 (Distributed Interval Observer Design).

Suppose Assumption 1 holds. Then the $\operatorname{DIO}$ algorithm is ISS
with $d=d^{*}$ and the corresponding observer gains ${L}^{*,i}$, ${T}^{*,i}$,
and ${\Gamma}^{*,i}$ that are solutions to (15).

###### Proof.

We will construct $H_{*}$, which by Theorem 1 is sufficient for the ISS
property to hold. For each node $i\in\mathcal{V}$ and state
$s\in\\{1,\dots,n\\}$, using $\ell(i,s)$ from Assumption 1,

$\displaystyle(H_{*})_{\overline{\operatorname{id}}(i,s),\overline{\operatorname{id}}(\ell(i,s),s)}=1,\
\
(H_{*})_{\underline{\operatorname{id}}(i,s),\underline{\operatorname{id}}(\ell(i,s),s)}=1,$

and all other entries are zero. Since $\ell(i,s)\in\mathcal{N}_{i}^{d^{*}}$,
then $H_{*}$ could be constructed according to (10) for some $\underline{x}$
and $\overline{x}$. With $H_{*}$ defined as such, rows
$\underline{\operatorname{id}}(i,s)$ and $\overline{\operatorname{id}}(i,s)$
of $H_{*}\hat{A}$ are equal to rows
$\underline{\operatorname{id}}(\ell(i,s),s)$ and
$\overline{\operatorname{id}}(\ell(i,s),s)$ of $\hat{A}$, respectively (cf.
Lemma 2). From the definition of $\hat{A}^{i}$, it is clear that
$\|(\tilde{A}^{i})_{s}\|_{1}$ = $\|(\hat{A}^{i})_{s}\|_{1}$ =
$\|(\hat{A}^{i})_{s+n}\|_{1}$. Note that the gains $T^{i}$ and $L^{i}$ are
computed by (15), which independently minimize the $1$-norm of each row of
$\tilde{A}^{i}$, since the $s^{\textup{th}}$ rows of $T^{i}$ and $L^{i}$ only
affect the $s^{\textup{th}}$ row of  $\tilde{A}^{i}\triangleq
T^{i}A-L^{i}C^{i}$. Moreover, Assumption 1 guarantees
$\|(\tilde{A}^{\ell(i,s)})_{s}\|_{1}<1$ for each $s$. All of this implies
$\|(H_{*}\hat{A})_{\underline{\operatorname{id}}(i,s)}\|_{1}=\|(H_{*}\hat{A})_{\overline{\operatorname{id}}(i,s)}\|_{1}<1$.
Since this holds for every row of the matrix $H_{*}\hat{A}$, then
$\rho(H_{*}\hat{A})\leq\|H_{*}\hat{A}\|_{\infty}\triangleq\max\limits_{1\leq
i\leq 2nN}\sum_{s=1}^{2nN}|(H_{*}\hat{A})_{ij}|<1$. ∎

Even if Assumption 1 doesn’t hold, or if the algorithm is run with $d<d^{*}$,
it is possible that solving (15) will result in stabilizing gains which can be
verified by Theorem 1.

##  V Numerical Examples

This section demonstrates the effectiveness of our distributed interval
observer applied to two LTI target systems.
Example 1: Consider a system from [18] in the form of (2):

$\displaystyle\begin{array}[]{c}N=3,\\\ n=4,\\\
B=I_{4},\end{array}A=\begin{bmatrix}1&0&1&0\\\ 0&1&0&1\\\ 0&0&1&0\\\
0&0&0&1\end{bmatrix},\begin{array}[]{c}C^{1}=\begin{bmatrix}1&0&0&0\end{bmatrix},\\\\[3.0pt]
C^{2}=\begin{bmatrix}0&1&0&0\end{bmatrix},\\\\[3.0pt]
C^{3}=\begin{bmatrix}1&1&0&0\end{bmatrix}.\end{array}$ (22)

The network is fully connected, so $d=d^{*}=1$. The initial state $x_{0}$ and
the process noise $w_{k}$ are bounded by
$\underline{x}_{0}=\begin{bmatrix}-20&-15&-0.5&0\end{bmatrix}^{\top},\overline{x}_{0}=\begin{bmatrix}10&25&2&3\end{bmatrix}^{\top},\underline{w}=\begin{bmatrix}-0.1&-0.1&-1&-1\end{bmatrix}^{\top}$
and $\overline{w}=\begin{bmatrix}0.1&0.1&1&1\end{bmatrix}^{\top}$, while
$v^{i}_{k}$ is bounded by $\underline{v}^{i}=-i$, $\overline{v}^{i}=i$, for
$i=1,2,3$. To show the advantage of including a network update in the observer
design, we also implemented individual local observers for each agent $i$,
i.e., using only (5) and local output $y^{i}_{k}$. The gains of the local
observers are the same as in the distributed observer. Using CVX [23, 24] to
solve the design LPs in (15), we obtained gains for each node:

$\displaystyle
L^{1}=\begin{bmatrix}0&0&-1&0\end{bmatrix}^{\top},L^{2}=\begin{bmatrix}0&0&0&-1\end{bmatrix}^{\top},L^{3}=-\begin{bmatrix}0&0&.5&.5\end{bmatrix}^{\top},$
$\displaystyle\Gamma^{1}=\begin{bmatrix}1&0&1&0\end{bmatrix}^{\top},\Gamma^{2}=\begin{bmatrix}0&1&0&1\end{bmatrix}^{\top},\Gamma^{3}=\begin{bmatrix}.5&.5&.5&.5\end{bmatrix}^{\top},$
$\displaystyle T^{1}=\begin{bmatrix}0&0&0&0\\\ 0&1&0&0\\\ -1&0&1&0\\\
0&0&0&1\end{bmatrix},T^{2}=\begin{bmatrix}1&0&0&0\\\ 0&0&0&0\\\ 0&0&1&0\\\
0&-1&0&1\end{bmatrix},T^{3}=\begin{bmatrix}.5&-.5&0&0\\\ -.5&.5&0&0\\\
-.5&-.5&1&0\\\ -.5&-.5&0&1\\\ \end{bmatrix}.$

Figure 1 shows the resulting framer trajectories and interval widths for each
observer. All observers maintain a _correct_ interval estimate, but some
framers of the individual observers are clearly unstable. On the other hand,
the distributed observer is able to maintain a tight interval around the true
state trajectory, despite the unstable individual observers.
Example 2: Consider the following multi-agent system (a discretized version of
the example from [20], with $\Delta_{t}=0.01$) in the form of (2) with $N=6$,
$n=12$, $A=I_{6}\otimes\begin{bmatrix}1&0.01\\\ 0&1\end{bmatrix}$,
$B=I_{6}\otimes\begin{bmatrix}0.01&0.0001\\\ 0&.01\end{bmatrix}$, and
$C^{i}=\mathbf{e}_{i}\otimes\begin{bmatrix}1&0\end{bmatrix}$. The
communication graph $\mathcal{G}$ is a directed ring with nodes
$\mathcal{V}=\\{1,\dots,6\\}$ and edges
$\mathcal{E}=\\{(i,(i+1\bmod{6}))\,:\,i\in\mathcal{V}\\}$. Moreover, the
initial state is
$x_{0}=\mathbf{1}_{6}\otimes\begin{bmatrix}0.7032&0.0457\end{bmatrix}$ and the
process noise is given by
$w_{k}=\mathbf{1}_{4}\otimes\begin{bmatrix}\frac{1}{2}\sin(0.01k)&0&\frac{1}{5}\cos(0.01k)\end{bmatrix}^{\top}$,
$\overline{w}=\mathbf{1}_{4}\otimes\begin{bmatrix}\frac{1}{2}&0&\frac{1}{5}\end{bmatrix}^{\top}$,
$\underline{w}=\mathbf{1}_{4}\otimes\begin{bmatrix}-\frac{1}{2}&0&-\frac{1}{5}\end{bmatrix}^{\top}$.
The measurement noise is given by
$v^{1}_{k}=v^{4}_{k}=\frac{1}{5}\sin^{2}(0.01k)$, $v^{2}_{k}=v^{5}_{k}=0$, and
$v^{3}_{k}=v^{6}_{k}=\frac{1}{3}\cos(0.02k)$, with bounds
$\overline{v}^{1}=\overline{v}^{4}=\frac{1}{5}$,
$\overline{v}^{2}=\overline{v}^{5}=0$,
$\overline{v}^{3}=\overline{v}^{6}=\frac{1}{3}$,
$\underline{v}^{1}=\underline{v}^{2}=\underline{v}^{4}=\underline{v}^{5}=0$,
and $\underline{v}^{3}=\underline{v}^{6}=-\frac{1}{3}$. The
$\operatorname{DIO}$ gains for each node $i\in\mathcal{V}$ are
$L^{i}=\mathbf{e}_{i}\otimes[0\
-100]^{\top},\quad\Gamma^{i}=\mathbf{e}_{i}\otimes[1\
100]^{\top},T^{i}=I_{12}+\mathbf{e}_{i}\mathbf{e}_{i}^{\top}\otimes\begin{bmatrix}-1&0\\\
-100&0\end{bmatrix}$. Assumption 1 is satisfied with $d^{*}=5$, but $d=1$
still results in a stable observer. Figure 2 shows the state trajectories and
interval widths from the DIO algorithm  with $d=1$. For comparison, we also
plot the results of the continuous time distributed observer proposed in the
work  in [20]. As can be seen, our observer is able to maintain tighter
intervals around the true states, which can be due to the advantage of using
the equivalent system representation (8), solving LPs instead of SDPs and/or
leveraging a more general observer structure.

##  VI Conclusion and Future Work

In this paper, a novel distributed interval observer synthesis was introduced
for linear time-invariant discrete-time systems that are subject to bounded
noise. At each time step, agents first individually compute upper and lower
framers for the system state, based on their own observation of the system.
Then, they communicate to their neighbors, exchanging and updating their
estimates iteratively by selecting the tightest intervals among their agents’
estimates. The true state trajectory of the system was guaranteed to be
bounded by the updated framers. Furthermore, necessary and sufficient
conditions assuring collective stability were provided. Such conditions were
shown to be collectively achievable by contribution of all agents through
solving tractable linear programs in a distributed manner.  In addition, we
provided an algorithm to verify if such sufficient conditions hold for a given
system. Finally, the advantage of the proposed distributed design compared to
its centralized version and also versus a benchmark observer in the literature
was demonstrated through simulation examples. In future work, extension to
nonlinear settings with time-varying networks, and the effect of adversarial
agents and communication and measurement failures will be considered.

Figure 1: States $x_{1}$ through $x_{3}$ (black), along with the upper (red)
and lower (blue) framers (left) and interval widths
$\delta_{i}\triangleq\overline{x}_{i}-\underline{x}_{i}$ (right) from the
observers, for system (22). Superscripts  $1$–$3$ denote framers from
individual agents  $1$–$3$, respectively, while the green marker (left)
denotes the resulting estimate by DIO. Figure 2: State framers and interval
widths from all agents for Example 2, from $\operatorname{DIO}$ (red) and the
observer from [20] (blue).

## References

  * [1] J. Zhao, A. Gomez-Exposito, M. Netto, L. Mili, A. Abur, V. Terzija, I. Kamwa, B. Pal, A.K. Singh, J. Qi, Z. Huang, and A.P. Meliopoulos.  Power system dynamic state estimation: Motivations, definitions, methodologies and future work.  IEEE Transactions on Power Systems, 34:3188–3198, 07 2019.
  * [2] Y. Sun and H. Song.  Secure and trustworthy transportation cyber-physical systems.  Springer, 2017.
  * [3] B. Cheng, J. Zhang, G.P. Hancke, S. Karnouskos, and A.W. Colombo.  Industrial cyber-physical systems: Realizing cloud-based big data infrastructures.  IEEE Industrial Electronics Magazine, 12(1):25–35, 2018.
  * [4] R.E. Kalman.  A new approach to linear filtering and prediction problems.  Transactions of the ASME–Journal of Basic Engineering, 82(Series D):35–45, 1960.
  * [5] F.F.C. Rego, A. M. Pascoal, A. P. Aguiar, and C. N. Jones.  Distributed state estimation for discrete-time linear time invariant systems: A survey.  Annual Reviews in Control, 48:36–56, 2019.
  * [6] R. Olfati-Saber.  Kalman-consensus filter: Optimality, stability, and performance.  In Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference, pages 7036–7042, 2009.
  * [7] A. T. Kamal, J. A. Farrell, and A. K. Roy-Chowdhury.  Information weighted consensus filters and their application in distributed camera networks.  IEEE Transactions on Automatic Control, 58(12):3112–3125, 2013\.
  * [8] Y. Wang, D.M. Bevly, and R. Rajamani.  Interval observer design for LPV systems with parametric uncertainty.  Automatica, 60:79–85, 2015.
  * [9] F. Mazenc, T-N. Dinh, and S-I. Niculescu.  Robust interval observers and stabilization design for discrete-time systems with input and output.  Automatica, 49(11):3490–3497, 2013.
  * [10] M. Khajenejad, F. Shoaib, and S.Z. Yong.  Interval observer synthesis for locally Lipschitz nonlinear dynamical systems via mixed-monotone decompositions.  In American control Conference (ACC), accepted, https://arxiv.org/pdf/2202.11689.pdf, 2022.
  * [11] A.M. Tahir and B. Açıkmeşe.  Synthesis of interval observers for bounded Jacobian nonlinear discrete-time systems.  IEEE Control Systems Letters, 2021.
  * [12] M. Khajenejad, Z. Jin, and S.Z. Yong.  Interval observers for simultaneous state and model estimation of partially known nonlinear systems.  In American Control Conference (ACC), pages 2848–2854, 2021.
  * [13] M. Khajenejad and S.Z. Yong.  Simultaneous input and state interval observers for nonlinear systems with full-rank direct feedthrough.  In IEEE Conference on Decision and Control, pages 5443–5448, 2020\.
  * [14] F. Mazenc and O. Bernard.  Interval observers for linear time-invariant systems with disturbances.  Automatica, 47(1):140–147, 2011.
  * [15] E. Chambon, L. Burlion, and P. Apkarian.  Overview of linear time-invariant interval observer design: towards a non-smooth optimisation-based approach.  IET Control Theory & Applications, 10(11):1258–1268, 2016.
  * [16] F. Cacace, A. Germani, and C. Manes.  A new approach to design interval observers for linear systems.  IEEE Transactions on Automatic Control, 60(6):1665–1670, 2014.
  * [17] D. Efimov, T. Raïssi, S. Chebotarev, and A. Zolghadri.  Interval state observer for nonlinear time varying systems.  Automatica, 49(1):200–205, 2013.
  * [18] D. Li, J. Chang, W. Chen, and T. Raissi.  IPR-based distributed interval observers design for uncertain LTI systems.  ISA Transactions, 121, 03 2021.
  * [19] X. Li, G. Wei, and D. Ding.  Distributed resilient interval estimation for sensor networks under aperiodic denial-of-service attacks and adaptive event-triggered protocols.  Applied Mathematics and Computation, 409:126371, 2021.
  * [20] X. Wang, H. Su, F. Zhang, and G. Chen.  A robust distributed interval observer for LTI systems.  IEEE Transactions on Automatic Control, 2022.
  * [21] V. D. Blondel and Y. Nesterov.  Polynomial-time computation of the joint spectral radius for some sets of nonnegative matrices.  SIAM Journal on Matrix Analysis and Applications, 31(3):865–876, 2010.
  * [22] E.D. Sontag.  Input to state stability: Basic concepts and results.  In Nonlinear and optimal control theory, pages 163–220. Springer, 2008.
  * [23] CVX Research, Inc.  CVX: Matlab software for disciplined convex programming, version 2.0.  http://cvxr.com/cvx, August 2012.
  * [24] M. Grant and S. Boyd.  Graph implementations for nonsmooth convex programs.  In V. Blondel, S. Boyd, and H. Kimura, editors, Recent Advances in Learning and Control, Lecture Notes in Control and Information Sciences, pages 95–110. Springer-Verlag Limited, 2008.


$\psi(T(\sigma(x)))\subset B(\varepsilon)$, this follows from our choice of
$\varepsilon$. See also the end of the proof of Theorem B in §4.4.

It remains to show that
$\psi(\Theta_{{\mathfrak{m}}}(a))\sim\psi(\Theta_{{\mathfrak{m}}^{\prime}}(a))$
for all Markov triples ${\mathfrak{m}},{\mathfrak{m}}^{\prime}$, meaning that
these provide an answer to Question 1.5. Indeed, recall from Remark 3.9 that
$T_{{\mathfrak{m}}}\sim T_{(1,1,1)}$, which implies, by lifting the
corresponding Lagrangian isotopy to $\mathbb{C}^{3}$, that
$\Theta_{{\mathfrak{m}}}(a)\sim
T\left(\frac{a}{3},\frac{a}{3},\frac{a}{3}\right)\subset\mathbb{C}^{3}$ for
every Markov triple ${\mathfrak{m}}$. Since this Lagrangian isotopy is
supported in $H^{-1}(a)$, it can be carried out in the image of $\psi$ and the
claim follows. $\Box$

###  4.7. Proof of Theorem D

In the same vein as above, we first construct exotic tori in $\mathbb{C}^{n}$
and then prove that they have the locality property with respect to the
displacement energy germ. Let $n>3$. For a Markov triple ${\mathfrak{m}}$ and
$a^{\prime},a_{4},\ldots,a_{n}>0$, let,

$\Theta_{{\mathfrak{m}}}^{n}=\Theta_{{\mathfrak{m}}}^{n}(a,a_{4},\ldots,a_{n})=\Theta_{{\mathfrak{m}}}(a)\times
T(a_{4},\ldots,a_{n})\subset\mathbb{C}^{n}.$ (100)

######  Lemma 4.11.

For $\frac{a}{3}\leqslant\min\\{a_{4},\ldots,a_{n}\\}$, the displacement
energy germ of $\Theta_{{\mathfrak{m}}}^{n}$ is given by

${\mathcal{E}}_{\Theta_{{\mathfrak{m}}}^{n}}(B)\sim\min_{i\in I}\left\\{d_{\rm
IA}(\overline{b},\partial\Delta_{{\mathfrak{m}}}),\frac{a}{3}+b_{i}\right\\},\quad
I=\left\\{i\in\\{4,\ldots,n\\}\,\left|\,a_{i}=\frac{a}{3}\right.\right\\},$
(101)

where
$B=(b_{1},\ldots,b_{n})=(\overline{b},b_{4},\ldots,b_{n})\in\mathbb{R}^{n}$.

As in the proof of Proposition 1.6, see in particular (85), this can be
written as

${\mathcal{E}}_{\Theta_{{\mathfrak{m}}}^{n}}(B)\sim\frac{a}{3}+\min_{i\in
I}\\{\langle B,U\rangle,\langle B,V\rangle,\langle B,W\rangle,\langle
B,e_{i}\rangle\\},$ (102)

where
$U=(\overline{u},0),V=(\overline{v},0),W=(\overline{w},0)\in\mathbb{Z}^{n}$,
where $\overline{u},\overline{v},\overline{w}\in\mathbb{Z}^{3}$ are defined as
in the proof surrounding (85). Again, we find $D_{\rm IA}(U,V)=\alpha,D_{\rm
IA}(U,V)=\beta,D_{\rm IA}(U,V)=\gamma$, meaning that
$\Theta_{{\mathfrak{m}}}^{n}(a,a_{4},\ldots,a_{n})\cong\Theta_{{\mathfrak{m}}^{\prime}}^{n}(a^{\prime},a_{4}^{\prime},\ldots,a_{n}^{\prime})$
implies ${\mathfrak{m}}={\mathfrak{m}}^{\prime}$ whenever
$\frac{a}{3}\leqslant\min\\{a_{4},\ldots,a_{n}\\}$ and
$\frac{a^{\prime}}{3}\leqslant\min\\{a_{4}^{\prime},\ldots,a_{n}^{\prime}\\}$.
In other words, we find infinitely many non-equivalent tori in
$\mathbb{C}^{n}$. Recall from §4.6 that $\Theta_{{\mathfrak{m}}}(a)\sim
T(\frac{a}{3},\frac{a}{3},\frac{a}{3})$, which implies that

$\Theta_{{\mathfrak{m}}}^{n}(a,a_{4},\ldots,a_{n})\sim
T\left(\frac{a}{3},\frac{a}{3},\frac{a}{3},a_{4},\ldots,a_{n}\right),$ (103)

meaning the $\Theta_{{\mathfrak{m}}}^{n}$ provide infinitely many examples
answering question 1.5. In case $a_{4}=\ldots=a_{n}=\frac{a}{3}$, the torus
$\Theta_{{\mathfrak{m}}}^{n}$ is monotone.

######  Remark 4.12.

Note that (100) defines a $(n-2)$-parametric family of exotic tori. This
family lies at the intersection of multiple $(n-1)$-parametric families of
exotic tori. Indeed, recall that the monotone Vianna torus
$T_{{\mathfrak{m}}}$ lies a the intersection of one-parameter families of
exotic tori, see Example 3.7 and [47, Section 7], which lift to two-parameter
families of exotic tori in $\mathbb{C}^{3}$, see Remark 1.7. Taking products
with product tori, one obtains $(n-1)$-parameter families of exotic tori in
$\mathbb{C}^{n}$.

Proof of Lemma 4.11.  Let $v_{\Theta_{{\mathfrak{m}}}(a)}$ be the versal
deformation of $\Theta_{{\mathfrak{m}}}(a)$ used in the proof of Lemma 4.9 in
§4.5, where we can assume, up to a change of basis, that
$\psi=\operatorname{id}$. By taking the product with the natural versal
deformation of product tori from Example 2.15, we obtain the following product
versal deformation of
$\Theta_{{\mathfrak{m}}}^{n}=\Theta^{n}_{{\mathfrak{m}}}(a,a_{4},\ldots,a_{n})$,

$v_{\Theta^{n}_{{\mathfrak{m}}}}(B)=v_{\Theta_{{\mathfrak{m}}}(a)}(\overline{b})\times
T(a_{4}+b_{4},\ldots,a_{n}+b_{n}),\quad B=(\overline{b},b_{4},\ldots,b_{n}).$
(104)

From (96), we deduce, on an open dense subset,

$v_{\Theta^{n}_{{\mathfrak{m}}}}(B)\cong
T\left(\frac{a+b_{3}}{3}+\tau_{1}(b),\frac{a+b_{3}}{3}+\tau_{2}(b),\frac{a+b_{3}}{3}-\tau_{1}(b)-\tau_{2}(b),a_{4}+b_{4},\ldots,a_{n}+b_{n}\right),$
(105)

where $\tau$ is the piecewise integral affine transformation coming from
mutations (see §§3.2-3.3) mapping $\Delta_{{\mathfrak{m}}}$ to the toric
moment polytope $\Delta_{(1,1,1)}$. As in the proof of Lemma 4.9, we obtain

$e(v_{\Theta^{n}_{{\mathfrak{m}}}}(B))\sim\min\left\\{d_{\rm
IA}(\overline{b},\partial\Delta_{{\mathfrak{m}}}),a_{4}+b_{4},\ldots,a_{n}+b_{n}\right\\}.$
(106)

Taking the germ in the case $\frac{a}{3}\leqslant\min\\{a_{4},\ldots,a_{n}\\}$
proves the claim. $\Box$

######  Remark 4.13.

The condition $\frac{a}{3}\leqslant\min\\{a_{4},\ldots,a_{n}\\}$ is necessary
for our methods to work, since otherwise, the $d_{\rm
IA}(\cdot,\partial\Delta_{{\mathfrak{m}}})$ does not appear in the germ of
(106). We do not know whether it is a necessary condition for the tori to be
inequivalent.

Proof of Theorem D.  Let $(X,\omega)$ be a geometrically bounded symplectic
$2n$-manifold and $U\subset X$ and open subset. Furthermore, we fix a
symplectic ball embedding $\psi\colon B(R)\hookrightarrow U$. We choose
$\Theta_{{\mathfrak{m}}}^{n}(a,a_{4},\ldots,a_{n})$ such that the product tori
appearing in its versal deformation have property (CS) with respect to $\psi$.
This allows us to apply Lemma 1.15 and deduce that
$\psi(\Theta_{{\mathfrak{m}}}^{n})$ has the locality property. Let
$a,a_{4},\ldots,a_{n}>0$ such that
$\frac{a}{3}\leqslant\min\\{a_{4},\ldots,a_{n}\\}$ and

$\frac{4a}{3}+a_{4}+\ldots+a_{n}<R,\quad\frac{a}{3}<\lambda_{S}(X,\psi).$
(107)

The versal deformation (104) induces the versal deformation
$v_{\psi(\Theta_{{\mathfrak{m}}}^{n})}=\psi\circ
v_{\Theta_{{\mathfrak{m}}}^{n}}$. By (107), we can choose the domain $V$ of
this versal deformation small enough such that

$\displaystyle\frac{4a}{3}+a_{4}+\ldots$
$\displaystyle+a_{n}+b_{3}+\ldots+b_{n}$ $\displaystyle+\min_{i\in
I}\left\\{\frac{b_{3}}{3}+\tau_{1}(b),\frac{b_{3}}{3}+\tau_{2}(b),\frac{b_{3}}{3}-\tau_{1}(b)-\tau_{2}(b),b_{i}\right\\}<R,$
(108) $\displaystyle\frac{a}{3}$ $\displaystyle+\min_{i\in
I}\left\\{\frac{b_{3}}{3}+\tau_{1}(b),\frac{b_{3}}{3}+\tau_{2}(b),\frac{b_{3}}{3}-\tau_{1}(b)-\tau_{2}(b),b_{i}\right\\}<\lambda_{S}(X,\psi),$
(109)

for all $B=(b_{1},\ldots,b_{n})=(b,b_{3},\ldots,b_{n})\in V$. Since the
support of the Hamiltonian isotopy (105) can be chosen to lie in an
arbitrarily small neighbourhood of $S^{5}(a+b_{3})\times
T(a_{4}+b_{4},\ldots,a_{n}+b_{n})$, the condition (108) implies that
$v_{\psi(\Theta_{{\mathfrak{m}}}^{n})}(B)\cong\psi(T(y))$, with support in
$\psi(B(R))$, where $y$ is given by (105). Furthermore (108) and (109) imply
that the versal deformations of $\Theta_{{\mathfrak{m}}}^{n}$ have property
(CS). This means that we can apply Lemma 1.15 to conclude that
$\psi(\Theta_{{\mathfrak{m}}}^{n})$ has the locality property. Together with
(103), this allows us to deduce that if
${\mathfrak{m}}\neq{\mathfrak{m}}^{\prime}$, then

$\psi(\Theta_{{\mathfrak{m}}}^{n}(a,a_{4},\ldots,a_{n}))\ncong\psi(\Theta_{{\mathfrak{m}}^{\prime}}^{n}(a,a_{4},\ldots,a_{n})),\quad\psi(\Theta_{{\mathfrak{m}}}^{n}(a,a_{4},\ldots,a_{n}))\sim\psi(\Theta_{{\mathfrak{m}}^{\prime}}^{n}(a,a_{4},\ldots,a_{n})),$
(110)

which proves Theorem D. $\Box$

##  5\. Questions and outlook

###  5.1. Questions

The following question came up in discussions with L. Polterovich.

######  Question 5.1.

Are exotic tori always unstable?

As mentioned in §1.6, we say that an exotic torus $L\subset\mathbb{C}^{n}$ is
_unstable_ if an open dense subset of tori in its versal deformation are not
exotic, meaning they are equivalent to product tori. In a general symplectic
manifold $X$, we can ask if there is a pair of Lagrangians
$L,L^{\prime}\subset X$ which are _stably distinct_. By this we mean that
$L\ncong L^{\prime}$ and $L\sim L^{\prime}$ such that their versal
deformations $v_{L}\colon U\rightarrow{\mathcal{L}}$ and $v_{L^{\prime}}\colon
U^{\prime}\rightarrow{\mathcal{L}}$ are distinct in the sense that
$v_{L}(b)\ncong v_{L^{\prime}}(b^{\prime})$ for all $b\in V,b^{\prime}\in
V^{\prime}$, where $V\subset U,V^{\prime}\subset U^{\prime}$ are
neighbourhoods of the origin.

All exotic tori known to the author are unstable. We heavily use the
instability of the $\Upsilon_{k}$\- and $\Theta_{{\mathfrak{m}}}$-tori to
compute their displacement energy germs. Every exotic torus known to the
author extends to a codimension one family of exotic tori, see for example
Remarks 1.7 and 4.12 and §1.3. We can refine Question 5.1.

######  Question 5.2.

Does every exotic torus come in a codimension one family of exotic tori?

Note that the Vianna tori (and their lifts) lie at the intersection of three
codimension one families of exotic tori. This suggests a stratified structure
of the set of exotic tori in the space of all tori.

######  Question 5.3.

Is the displacement energy germ of every Lagrangian torus of the form (43)?

Again, this is the case in all examples known to the author. One way of
answering Question 5.3 may be to link the displacement energy germ to an
invariant coming from Maslov two $J$-holomorphic disks of minimal area as
suggested in Remark 2.18. Let $L\subset X$ be a (not necessarily monotone)
Lagrangian torus.

######  Question 5.4.

Is there a symplectic invariant of $L$ based on counting Maslov two
$J$-holomorphic disks (of minimal area) which has the locality property? Can
${\mathcal{E}}_{L}$ be recovered from this invariant?

In case $X$ is monotone and (almost) toric, the displacement energy germ
${\mathcal{E}}_{L}$ of the monotone fibre $L$ is of the form (43) and it
recovers the geometry of the toric base or of the ATF base diagram,
respectively, in the sense that the base polytope is given by

$\Delta_{{\mathcal{E}}_{L}}=\\{x\in\mathbb{R}^{n}\,|\,\langle
x,v_{i}\rangle+a\geqslant 0\\},$ (111)

where $v_{1},\ldots,v_{n}\in\mathbb{Z}^{n}$ and $a>0$ are given by (43).

######  Question 5.5.

Let $L\subset X$ be a monotone Lagrangian torus with displacement energy germ
${\mathcal{E}}_{L}$ given by (43). Is there a Lagrangian torus fibration with
base $\Delta_{{\mathcal{E}}_{L}}$ as defined in (111)?

Note that in that case ${\mathcal{E}}_{L}$ is the germ of the function $d_{\rm
IA}(\cdot,\partial\Delta_{{\mathcal{E}}_{L}})$. In that sense, one might
expect a correspondence between the set of monotone Lagrangian tori in $X$ and
certain Lagrangian torus fibrations (up to a reasonable equivalence relation,
e.g. nodal slides in the almost toric case) on $X$. In Remark 4.10 and §5.3 we
discuss some evidence for a positive answer to Question 5.5 in the case of the
$\Theta_{{\mathfrak{m}}}$\- and $\Upsilon_{k}$-tori constructed in this paper.
In a second step, one can ask which singularities appear in the Lagrangian
torus fibration and how these are related to properties of $L\subset X$.

###  5.2. Dimension four

In dimension four there is, as of now, no analogue of the local exotic tori
constructed in this paper. It is still open whether every Lagrangian torus
$L\subset\mathbb{C}^{2}$ is equivalent to either a product or a Chekanov
torus, see [18] for evidence in that direction. In case this classification
holds, there can be no such analogue due to a lack of exotic tori, even in
$\mathbb{C}^{2}$. Furthermore, embeddings of the Chekanov torus by Darboux
charts are not always exotic in a robust sense. The following example makes
use of symmetric probes, see [8], and will appear as an application of the
classification of toric fibres in non-monotone $S^{2}\times S^{2}$ in [11].

######  Example 5.6.

Let $X_{\lambda}$ be the symplectic manifold obtained by equipping
$S^{2}\times S^{2}$ with the symplectic form
$(\lambda\omega_{S^{2}})\oplus\omega_{S^{2}}$. Recall that $T_{\rm
Ch}(a)\subset\mathbb{C}^{2}$ is defined as the lift of a curve enclosing area
$a>0$ in the reduced space fibering over the diagonal in the toric moment map
image of $\mathbb{C}^{2}$, see §1.4. For every irrational (and some, but not
all, rational) $a\in(0,1)$, there is a Darboux embedding $\varphi\colon
B^{4}(2)\rightarrow X_{\lambda}$ such that
$\varphi(T_{\rm{Ch}}(a))\cong\varphi(T(x_{1},x_{2}))$, where
$T(x_{1},x_{2})\subset\mathbb{C}^{2}$ is a non-monotone product torus. Note
that although $T_{\rm Ch}(a)\subset\mathbb{C}^{2}$ is monotone,
$\varphi(T_{\rm Ch}(a))\subset X_{\lambda}$ is not. By choosing $a>0$ small
enough, one can arrange for $\varphi(T(x_{1},x_{2}))$ to have property (CS).
Thus this example stands in contrast to the local exotic tori in dimension six
and higher constructed in this paper, see Remark 1.20. Furthermore, the
Darboux embedding $\varphi$ is equivariant with respect to the natural toric
structures on $B^{4}(2)$ and $X_{\lambda}$, respectively, meaning that
$\varphi(T(x_{1},x_{2}))$ is a toric fibre. Thus this also contrasts with
Corollary 2.20.

However, in work in progress joint with Johannes Hauber and Joel Schmitz [10],
we treat a _semi-local_ analogue of the ideas developed in this paper yielding
tori which are exotic in a more robust way. Instead of embedding Lagrangian
tori contained in symplectic balls, we treat Lagrangian tori contained in more
complicated symplectic domains. For example, let $L\subset X^{4}$ be a
Lagrangian sphere in a symplectic four-manifold. Then a neighbourhood of $L$
is symplectomorphic to a neighbourhood of the zero-section in $T^{*}S^{2}$
which serves as our domain. Any neighbourhood of the zero-section in
$T^{*}S^{2}$ contains three types of tori, namely product, Chekanov and
Polterovich tori. For a discussion of the Polterovich torus, we refer to [2].
Under similar smallness assumptions on the tori as in Theorems B and C, see
also Remark 1.16, one can prove that these types of tori are not equivalent in
$X^{4}$. More generally, we consider embeddings of certain Liouville domains
$B_{d,p,q}$ into $X^{4}$, yielding $d+1$ distinct tori. For example
$B_{2,1,1}\cong T^{*}S^{2}$ and $B_{1,2,1}\cong T^{*}\mathbb{R}P^{2}$, see
[24] for a discussion of the spaces $B_{d,p,q}$. These spaces appear as Milnor
fibres of cyclic quotient singularities, or equivalently, neighbourhoods of
_Lagrangian pinwheels_ , see [25] for more details. The space $B_{d,p,q}$
admits an almost toric fibration with $d$ nodes lying on one branch cut line
and the tori studied in [10] appear as regular fibres over the branch cut line
of these ATFs. For a fixed area class $a>0$, the corresponding torus changes
equivalence class whenever a node is moved across it. As in the present paper,
in [10] we prove a locality property which implies that the tori remain
distinct after embedding them into a geometrically bounded $X^{4}$.

The main difference with the purely local case of Darboux embeddings is that
there are constraints from symplectic topology for symplectic embeddings of
the type $B_{d,p,q}\supset U\rightarrow X^{4}$ to exist. A full discussion of
this can be found in [25] for the case $d=1$ for $X=\mathbb{C}P^{2}$.

###  5.3. Almost toric fibrations in higher dimensions

In the same vein as Question 5.5, one can try to construct a Lagrangian torus
fibration on $\mathbb{C}^{3}$ having the $\Upsilon_{k}$-tori as fibres. Note
that for the Chekanov torus $T_{\rm Ch}\subset\mathbb{C}^{2}$, this works and
yields an almost toric fibration on $\mathbb{C}^{2}$. We suspect that this
leads to a notion of almost toric fibrations in arbitrary dimension. Let us
give a rough outline of this.

One can construct a Lagrangian torus fibration by hand, using a global version
of the arguments we have used to deal with the versal deformation of the
$\Upsilon_{k}$. As projection map, consider

$\pi_{k,G}\colon\mathbb{C}^{3}\rightarrow\Delta_{k}\times\mathbb{R}_{\geqslant
0},\quad\pi_{k,G}(z)=(\nu_{k}(z),\pi_{k,\nu_{k}(z)}^{*}G_{k,\nu_{k}(z)}(z)).$
(112)

Since $\nu_{k}$ generates the $T_{k}$-action considered in §4.1, we can
interpret $\pi_{k,G}$ as a $T_{k}$-equivariant perturbation of the toric
system on $\mathbb{C}^{3}$. Indeed, the last component of $\pi_{k,G}$ is a
lift of a Hamiltonian $G_{k,c}\colon\Sigma_{k,c}\rightarrow\mathbb{R}$ from
the reduced space $\Sigma_{k,c}$ of the $T_{k}$-action. Note that if we lifted
$H_{k,c}$ instead, then we would end up with the standard toric system on
$\mathbb{C}^{3}$, since the fibres of $\pi_{k,H}$ are product tori by
Proposition 4.1. However, by lifting perturbations of $H_{k,c}$ having the
circles $C(\cdot)\subset\Sigma_{k,c}$ appearing in the definition of the
$\Upsilon_{k}$-tori as level sets, we end up with an Lagrangian torus
fibration having $\Upsilon_{k}$-tori as fibres. A good candidate for a family
$\\{G_{k,c}\\}_{c\in\Delta_{k}}$ of functions to define $\pi_{k,G}$ seems to
be: start with $H_{k,c}$ and perturb it on a small neighbourhood of the ray
$\\{(c_{1},0)\\}\subset\Delta_{k}$. Just as a similar $S^{1}$-equivariant
perturbation of the toric system on $\mathbb{C}^{2}$ produces an almost toric
nodal trade at the vertex of $\mathbb{R}_{\geqslant 0}^{2}$, this
$T_{k}$-equivariant procedure produces an analoguous _nodal trade at the edge_
$\\{x_{2}=x_{3}=0\\}\subset\mathbb{R}^{3}_{\geqslant 0}$. Singular fibres
appear where the level sets of $G_{k,(c_{1},0)}$ intersect the punctures of
$\Sigma_{k,(c_{1},0)}$ and they are of focus-focus-regular type, since locally
the nodal trade at the vertex is equivalent to starting with the standard
toric system on $\mathbb{C}^{2}\times T^{*}S^{1}$ and performing a nodal trade
in the $\mathbb{C}^{2}$-component. Nodal slides correspond, in this picture,
to varying the level of $G_{k,(c_{1},0)}$ at which the puncture is
intersected. The corresponding ATF base diagrams are computed by removing two-
dimensional branch sets and computing action coordinates on their complement.

In $\mathbb{R}_{\geqslant 0}^{3}$, there are three edges at which one can
perform nodal trades and for each of those, one can choose
$k\in\mathbb{Z}_{\geqslant 0}$. Therefore, for each triple
$(k_{1},k_{2},k_{3})\in\mathbb{Z}_{\geqslant 2}^{3}$, we obtain a series of
ATFs and mutations of $\mathbb{C}^{3}$. We call $(k_{1},k_{2},k_{3})$ the
_nodal trade vector_. Note that for the nodal trade vector
$(k_{1},k_{2},k_{3})=(2,2,2)$, the $T^{2}$-actions at chosen at each edge
intersect in a common circle, namely the diagonal circle $S^{1}\subset T^{3}$.
Therefore, the corresponding mutation in $\mathbb{C}^{3}$ are compatible with
symplectic reduction by the diagonal circle and project to mutations of
$\mathbb{C}P^{2}$. This reproduces the Markov mutations in $\mathbb{C}P^{2}$,
see §3.2. In other words, our construction for $k_{i}=2$ reproduces the cone
construction discussed in Remark 4.10 and thus fibres of the higher mutations
correspond, in this case, to the tori $\Theta_{{\mathfrak{m}}}$. On the other
hand, simple mutations at only one edge correspond to the tori $\Upsilon_{k}$
as we have seen. Therefore these conjectural ATF-mutations in higher
dimensions have fibres which generalize both the $\Upsilon_{k}$\- and
$\Theta_{{\mathfrak{m}}}$-tori, with $\Theta_{{\mathfrak{m}}}$-tori appearing
as fibres of ATFs obtained by _higher mutations for the fixed nodal trade
vector_ $(k_{1},k_{2},k_{3})=(2,2,2)$ and $\Upsilon_{k}$-tori appearing as
fibres of ATFs obtained by _first order mutations for the nodal trade vectors_
$(k_{1},k_{2},k_{3})=(k,\varnothing,\varnothing)$, where $\varnothing$ stands
for not making a nodal trade. For every
$(k_{1},k_{2},k_{3})\in\mathbb{Z}_{\geqslant 2}^{3}$, we obtain a Markov-like
tree of Lagrangian tori. It would be highly interesting to give a proper
construction of these families tori and distinguish them. As in the four-
dimensional analogue in §3.3, this should be possible by the use of versal
deformations. In particular, it should be possible to find out how the
displacement energy germ behaves under mutations and that it can be read off
from the base of the corresponding ATF.

######  Remark 5.7.

Note that this could lead to a theory of ATFs for compact toric manifolds of
arbitrary dimension. Indeed, every compact toric manifold $X$ is, via the
Delzant construction, see [34], an equivariant symplectic quotient of some
$\mathbb{C}^{N}$ by a subtorus $K\subset T^{N}$ acting on $\mathbb{C}^{N}$. We
can restrict our attention to nodal trades at codimension $(k\geqslant
2)$-faces of $\mathbb{R}^{N}_{\geqslant 0}$ such that their corresponding tori
contain $K$. Just as the $(2,2,2)$-ATFs on $\mathbb{C}^{3}$ induce ATFs on
$\mathbb{C}P^{2}$, this general recipe should induce ATFs on $X$.

######  Remark 5.8.

Note that the Lagrangian torus fibrations in (112) are perturbations of the
toric moment map which are somewhat _mild_ in the sense that, locally, a
$T^{2}$-action is preserved. What happens if one decides to only preserve an
$S^{1}\subset T^{3}$? This line of inquiry should be started by analzing what
perturbations of the toric system on $\mathbb{C}^{2}$ make sense in this
context, since the reduced spaces are two dimensional in this case.

## References

  * [1] M. Abreu and L. Macarini.  Remarks on Lagrangian intersections in toric manifolds.  Trans. Amer. Math. Soc., 365(7):3851–3875, 2013.
  * [2] P. Albers and U. Frauenfelder.  A nondisplaceable Lagrangian torus in $T^{*}S^{2}$.  Comm. Pure Appl. Math., 61(8):1046–1051, 2008.
  * [3] D. Auroux.  Mirror symmetry and $T$-duality in the complement of an anticanonical divisor.  J. Gökova Geom. Topol. GGT, 1:51–91, 2007.
  * [4] D. Auroux.  Special Lagrangian fibrations, wall-crossing, and mirror symmetry.  In Surveys in differential geometry. Vol. XIII. Geometry, analysis, and algebraic geometry: forty years of the Journal of Differential Geometry, volume 13 of Surv. Differ. Geom., pages 1–47. Int. Press, Somerville, MA, 2009.
  * [5] D. Auroux.  Infinitely many monotone Lagrangian tori in $\mathbb{R}^{6}$.  Invent. Math., 201(3):909–924, 2015.
  * [6] P. Biran and O. Cornea.  Rigidity and uniruling for Lagrangian submanifolds.  Geom. Topol., 13(5):2881–2989, 2009.
  * [7] J. Brendel.  An introduction to the Chekanov torus.  In Proceedings of the Gökova Geometry-Topology Conference 2021, pages 1–56. Int. Press, Somerville, MA, 2022.
  * [8] J. Brendel.  Hamiltonian classification of toric fibres and symmetric probes.  arXiv:2302.00334, 2023.
  * [9] J. Brendel.  Real Lagrangian tori and versal deformations.  to appear in Journal of Symplectic Geometry, 2023.
  * [10] J. Brendel, J. Hauber, and J. Schmitz.  Semi-local exotic tori in dimension four.  in preparation, 2023.
  * [11] J. Brendel and J. Kim.  On Lagrangian tori in $S^{2}\times S^{2}$.  in preparation, 2023.
  * [12] A. Cannas da Silva.  Lectures on symplectic geometry, volume 1764 of Lecture Notes in Mathematics.  Springer-Verlag, Berlin, 2001.
  * [13] Y. Chekanov.  Lagrangian tori in a symplectic vector space and global symplectomorphisms.  Math. Z., 223(4):547–559, 1996.
  * [14] Y. Chekanov.  Lagrangian intersections, symplectic energy, and areas of holomorphic curves.  Duke Math. J., 95(1):213–226, 1998.
  * [15] Y. Chekanov and F. Schlenk.  Notes on monotone Lagrangian twist tori.  Electron. Res. Announc. Math. Sci., 17:104–121, 2010.
  * [16] Y. Chekanov and F. Schlenk.  Lagrangian product tori in symplectic manifolds.  Comment. Math. Helv., 91(3):445–475, 2016.
  * [17] T. Delzant.  Hamiltoniens périodiques et images convexes de l’application moment.  Bull. Soc. Math. France, 116(3):315–339, 1988.
  * [18] G. Dimitroglou Rizell.  The classification of Lagrangians nearby the Whitney immersion.  Geom. Topol., 23(7):3367–3458, 2019.
  * [19] J. J. Duistermaat.  On global action-angle coordinates.  Comm. Pure Appl. Math., 33(6):687–706, 1980.
  * [20] Y. Eliashberg and L. Polterovich.  The problem of Lagrangian knots in four-manifolds.  In Geometric topology (Athens, GA, 1993), volume 2 of AMS/IP Stud. Adv. Math., pages 313–327. Amer. Math. Soc., Providence, RI, 1997\.
  * [21] M. Entov and L. Polterovich.  Quasi-states and symplectic intersections.  Comment. Math. Helv., 81(1):75–99, 2006.
  * [22] M. Entov and L. Polterovich.  Rigid subsets of symplectic manifolds.  Compos. Math., 145(3):773–826, 2009.
  * [23] J. Evans.  Lectures on Lagrangian torus fibrations, volume 105 of London Mathematical Society Student Texts.  Cambridge University Press, Cambridge, 2023.
  * [24] J. D. Evans.  KIAS Lectures on symplectic aspects of degenerations, 2019.
  * [25] J. D. Evans and I. Smith.  Markov numbers and Lagrangian cell complexes in the complex projective plane.  Geom. Topol., 22(2):1143–1180, 2018.
  * [26] K. Fukaya, Y.-G. Oh, H. Ohta, and K. Ono.  Lagrangian Floer theory on compact toric manifolds. I.  Duke Math. J., 151(1):23–174, 2010.
  * [27] K. Fukaya, Y.-G. Oh, H. Ohta, and K. Ono.  Lagrangian Floer theory on compact toric manifolds II: bulk deformations.  Selecta Math. (N.S.), 17(3):609–711, 2011.
  * [28] K. Fukaya, Y.-G. Oh, H. Ohta, and K. Ono.  Lagrangian Floer theory on compact toric manifolds: survey.  In Surveys in differential geometry. Vol. XVII, volume 17 of Surv. Differ. Geom., pages 229–298. Int. Press, Boston, MA, 2012.
  * [29] K. Fukaya, Y.-G. Oh, H. Ohta, and K. Ono.  Toric degeneration and nondisplaceable Lagrangian tori in $S^{2}\times S^{2}$.  Int. Math. Res. Not. IMRN, (13):2942–2993, 2012.
  * [30] A. Gadbled.  On exotic monotone Lagrangian tori in $\mathbb{CP}^{2}$ and $\mathbb{S}^{2}\times\mathbb{S}^{2}$.  J. Symplectic Geom., 11(3):343–361, 2013.
  * [31] S. Galkin and G. Mikhalkin.  Singular symplectic spaces and holomorphic membranes.  Eur. J. Math., 8(3):932–951, 2022.
  * [32] S. Galkin and A. Usnich.  Laurent phenomenon for Landau–Ginzburg potential, 2010.
  * [33] Y. Groman and U. Varolgunes.  Locality of relative symplectic cohomology for complete embeddings.  arXiv:2110.08891, 2021.
  * [34] V. Guillemin.  Moment maps and combinatorial invariants of Hamiltonian $T^{n}$-spaces, volume 122 of Progress in Mathematics.  Birkhäuser Boston, Inc., Boston, MA, 1994.
  * [35] V. Guillemin and S. Sternberg.  Symplectic techniques in physics.  Cambridge University Press, Cambridge, second edition, 1990.
  * [36] P. Hacking and Y. Prokhorov.  Smoothable del Pezzo surfaces with quotient singularities.  Compos. Math., 146(1):169–192, 2010.
  * [37] O. N. Karpenkov.  Geometry of continued fractions, volume 26 of Algorithms and Computation in Mathematics.  Springer, Berlin, 2022.
  * [38] J. Lafontaine and M. Audin.  Introduction: applications of pseudo-holomorphic curves to symplectic topology.  In Holomorphic curves in symplectic geometry, volume 117 of Progr. Math., pages 1–14. Birkhäuser, Basel, 1994.
  * [39] E. Lerman and S. Tolman.  Hamiltonian torus actions on symplectic orbifolds and toric varieties.  Trans. Amer. Math. Soc., 349(10):4201–4230, 1997.
  * [40] D. McDuff.  Displacing Lagrangian toric fibers via probes.  In Low-dimensional and symplectic topology, volume 82 of Proc. Sympos. Pure Math., pages 131–160. Amer. Math. Soc., Providence, RI, 2011\.
  * [41] D. McDuff and D. Salamon.  Introduction to symplectic topology.  Oxford Graduate Texts in Mathematics. Oxford University Press, Oxford, third edition, 2017.
  * [42] E. Meinrenken.  _Symplectic Geometry, Lecture Notes_.
  * [43] J. Oakley and M. Usher.  On certain Lagrangian submanifolds of $S^{2}\times S^{2}$ and $\mathbb{C}{\rm P}^{n}$.  Algebr. Geom. Topol., 16(1):149–209, 2016.
  * [44] Y.-G. Oh.  Gromov-Floer theory and disjunction energy of compact Lagrangian embeddings.  Math. Res. Lett., 4(6):895–905, 1997.
  * [45] K. Ono.  A question analogous to the flux conjecture concerning Lagrangian submanifolds.  In Proceedings of Gökova Geometry-Topology Conference 2007, pages 1–14. Gökova Geometry/Topology Conference (GGT), Gökova, 2008.
  * [46] L. Polterovich.  The geometry of the group of symplectic diffeomorphisms.  Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2001\.
  * [47] E. Shelukhin, D. Tonkonog, and R. Vianna.  Geometry of symplectic flux and lagrangian torus fibrations, 2019.
  * [48] J.-C. Sikorav.  Systèmes hamiltoniens et topologie symplectique.  ETS Editrice Pisa, 1990.
  * [49] M. Symington.  Four dimensions from two in symplectic topology.  In Topology and geometry of manifolds (Athens, GA, 2001), volume 71 of Proc. Sympos. Pure Math., pages 153–208. Amer. Math. Soc., Providence, RI, 2003.
  * [50] R. Vianna.  Infinitely many exotic monotone Lagrangian tori in $\mathbb{CP}^{2}$.  J. Topol., 9(2):535–551, 2016.
  * [51] R. Vianna.  Infinitely many monotone Lagrangian tori in del Pezzo surfaces.  Selecta Math. (N.S.), 23(3):1955–1996, 2017.
  * [52] W. Wu.  On an exotic Lagrangian torus in $\mathbb{C}P^{2}$.  Compos. Math., 151(7):1372–1394, 2015.
  * [53] N. T. Zung.  Symplectic topology of integrable Hamiltonian systems. I. Arnold-Liouville with singularities.  Compositio Math., 101(2):179–215, 1996.
  * [54] N. T. Zung.  A note on focus-focus singularities.  Differential Geom. Appl., 7(2):123–130, 1997.
  * [55] N. T. Zung.  Symplectic topology of integrable Hamiltonian systems. II. Topological classification.  Compositio Math., 138(2):125–156, 2003.