# Electromagnetic Transient Model of Cryptocurrency Mining Loads for Low-
Voltage Ride Through Assessment in Transmission Grids

Anindita Samanta, Subir Majumder, Hasan Ibrahim, Prasad Enjeti, Le Xie
Anindita Samanta, Subir Majumder, Hasan Ibrahim, Prasad Enjeti, and Le Xie are
with the Department of Electrical Engineering and Computer Sciences, Texas A&M
University, USA. (Corresponding author: le.xie@tamu.edu)This work is supported
in part by Texas A&M Energy Institute and in part by the Blockchain and Energy
Research Consortium.

###### Abstract

In this paper, we developed an Electromagnetic Transient (EMT) model tailored
for large cryptocurrency mining loads to understand the cross-interaction of
these loads with the electric grid. The load model has been built using
Electromagnetic Transients Program (EMTP) software. We have cross-validated
the performance of the EMT model of the load with commercial application-
specific integrated circuit miners, typically used by large-scale mining
facilities, by comparing their low-voltage ride-through (LVRT) capabilities.
Subsequently, LVRT capabilities of the large-scale miners have been tested
against various fault scenarios both within the miner’s remote facility as
well as at one of the distant buses of the interconnected grid. The
significance of this model lies in its scalability to accommodate larger
blocks of mining loads and its seamless integration into a larger electric
grid.

###### Index Terms:

Cryptocurrency mining loads, Electromagnetic model, EMTP, Low Voltage Ride
Through, Transient studies

##  I Introduction

Over the past decade, blessed with abundant wind and solar resources, the
Texas power grid has witnessed a significant increase in renewable energy
penetration, accounting for almost 40% of the state’s total generation
capacity [1]. This was partly the reason that drew attention from many
emerging electric loads, such as large cryptocurrency mining operations [2].
These crypto-mining industrial facilities with their large sitewide energy
demand (a typical facility has a load of $\sim$ 75MW, see Fig. 1 that shows a
typical mining firm), integrated with power-electronic converters, play a
crucial role in providing frequency support to the electricity grid based on
price signals. However, these resources are similar to wind and solar
resources, and operate on inverter-based (IBR) systems with distinct dynamics
in contrast with conventional synchronous generators and loads. Notably,
recent outage events at wind and solar farms integrated with IBR-based
resources in the Panhandle and Odessa regions of Texas have raised significant
concerns [3],[4]. Such incidents were considered to be associated with a lack
of proper electromagnetic transient responses from IBR-based resources [5],
which would likely exacerbate with increasing penetration of crypto-mining
loads.

Figure 1: Mining firm Riot Platforms, Inc. with own substation of 750 MW
located in Rockdale, Texas [6].

Owing to ERCOT’s (major power grid operator operating within the state of
Texas, USA) interconnection guidelines [7], IBR-based generating resources
should maintain connectivity even during periods of low grid voltage – a
requirement known as Low Voltage Ride Through (LVRT) capability. While ERCOT’s
existing framework does not impose specific mandates on these IBR-based large
loads, recent incidents in West Texas on October 12, 2022 [4] vividly
demonstrated the repercussions of a low voltage event, triggering four
cascading faults and ultimately leading to an outage exceeding 400MW, which
included large-flexible loads, such as crypto-mining facilities. The recorded
minimum voltage during this episode plummeted to 0.36 pu. With a surge in
requests for interconnection from these cryptocurrency loads [8], ERCOT has
been scrambling to conduct power grid dynamic performance analysis for
developing newer LVRT standards for these loads.

In regards to the modeling efforts for crypto-mining facilities, efforts have
been made toward the use of power quality analyzers within a farm. In this
context, Wheeler et al. have conducted power factor and harmonic analysis for
a physical site with S9 AntMiners as processing units [9]. As demonstrated in
this study, which is also corroborated by our lab tests (see Fig. 2) available
in [10], the voltage waveform does not suffer from distortions owing to being
connected to an ideal power source, but, the current waveform shows distortion
during startup. In [10], we also conducted power quality analysis at two
industrial facilities where we observed similar performance. During the steady
state, the power factor remains between 0.994 and 0.995 leading. However,
these loads show non-linear characteristics during power system transients.
While data center loads may not be as flexible as crypto-mining loads, being
majorly computing demand, they are expected to behave as a crypto-mining load
during transients. Regarding this, [11] analyses the utilization of automatic
power factor corrector to compensate for the required reactive power demand
for IT loads and UPSs emulating datacenter. Ref. [12] also showed high
harmonics content in the power supply measurements, significantly distinct
from cryptocurrency mining loads.

The need for accurate modeling of crypto-mining facilities through dynamic
load modeling, frequency scan, and eigenvalue analysis has already been
discussed in the existing literature [13]. However, as discussed in [13], the
electromagnetic transient (EMT) model of these loads for their transient
performance analysis is largely missing. In a recent study, the LVRT
capabilities of a S19 miner have been characterized in laboratory setup
utilizing actual crypto-mining power supply [10]. However, in order to set
standards for grid interconnection in Texas, the development of an EMT model
for these power supplies, appropriate analysis of their transient performance,
and their interaction with other resources in the power grid are extremely
important. The contribution of this proposed work is therefore twofold:

  * i.

This paper aims to develop an EMT model of a cryptocurrency mining facility.
Both, the time-domain response and voltage sag magnitude-duration capability
characteristics of the developed model have been validated against a
laboratory-scale crypto-mining load. The EMT model has been developed in the
Electromagnetic Transients Program (EMTP) software. It has been made open
source111Repository is available at: https://github.com/tamu-engineering-
research/COVID-EMDA.

  * ii.

The performance of the developed EMT model has been demonstrated utilizing a
120 kV 6-bus test system considering faults at two different locations: (i) at
the crypto-miner’s premises, and (ii) at bus 3 of the test system.

##  II Development of EMT Model of a Cryptocurrency Mining Load

Contrary to general-purpose computing resources, an industrial-scale
cryptocurrency mining facility utilizes application-specific integrated
circuit (ASIC) chips embedded in a hashing board. These hash boards are
powered by the mining power supply. Typically these mining power supplies are
driven by single-phase 240 V source, however, in recent times, the use of
three-phase power supplies are also notable. In this analysis, we’ll focus on
the widely-used S19 Antminer with a 240 V single-phase power supply. Based on
Fig. 2, and the power supply repair guide for the Antminer [14], we model the
crypto-mining power supply as active power factor correction (PFC)-boost
converters. Although diverse PFC converters, each with their unique dynamic
performances, exist [15], we have opted for conventional boost PFC converters
for their versatile uses.

Figure 2: IV Characteristics of mining power supply: (a) Based on power supply
available in the lab, (b) Startup time-domain voltage and current waveforms
reproduced from [9].

Electromagnetic Transient Program (EMTP) software is widely used for transient
performance analysis [16] of the power system. Here, we have developed
switching-based models of an active PFC-boost converter in the EMTP to model
the crypto-mining facility. A block diagram of this converter is provided in
Fig. 3. The converter consists of a bridge rectifier at the input side and a
boost circuit aiming to maintain a constant DC-bus voltage $v_{dc}$. As shown
in the figure, the profile of rectified loads and the deviation in DC-bus
voltage passed through the PI controller provide the references for the
inductor current in the boost converter. The second PI controller with a
limiter generates PWM signals for the IGBT within the converter to track the
inductor current with the determined reference. The inductor and capacitors
are designed to allow a certain amount of ripples in the voltage and current
waveform. The gains of the PI controllers could be tuned for desired fastness
in response from the converters. As described in (1), these loads behave as
constant power loads (CPLs). Here, $V_{rms}(t)$, $I_{rms}(t)$ are the rms
voltage and current magnitudes and $P_{av}$ is the average power drawn by the
crypto-mining power supply.

$V_{rms}(t)I_{rms}(t)=P_{av}\implies\frac{\partial V_{rms}}{\partial
t}I_{rms}+V_{rms}\frac{\partial I_{rms}}{\partial t}=0$ (1)

Figure 3: Block diagram of active PFC-Boost converter representing mining
power supply. Figure 4: Start-up waveform for the EMT model of cryptocurrency
miner power supply.

Being operated as a CPL implies that during the transients the product of the
rms voltage and the current remains constant. The implications of this
behavior are imminent across all the time-duration characteristics shown in
this paper. During start-up, depending upon the DC-bus voltage, the converter
would show inrush, which can be found in both Figs. 2(b) and 4. Both of these
figures also show that after the initial transient, the voltage and current
become in phase. During sags, the converter current drops suddenly but grows
as time progresses, as shown in Fig. 5. As the voltage recovers, the inrush
kicks in as shown in Figs. 3(a), and 5, which may lead to tripping of mining
power supply. Fig. 3 demonstrates that during the zero-voltage condition, the
current reference would be zero, and therefore, as shown in Fig. 2(a), the
current drawn by the mining power supply would also be zero. Ideal voltage and
current waveforms of the crypto-mining power supply could be given as follows:

$i(t)=\frac{P_{av}}{V_{rms}(t){}^{2}+\epsilon}v(t)$ (2)

$\epsilon$ is a small positive real number prohibiting division by zero.
$v(t)$ and $i(t)$ are instantaneous voltage and current drawn by the power
supply.

The converters are designed to operate below certain loading capabilities,
determining their overcurrent limits. They are also equipped with both over
and under-voltage protection. During the inrush condition, if the overcurrent
circuitry gets triggered, or the DC-bus voltage drops significantly, it may
eventually cause the mining loads to trip. In regards to the protection
circuitry of the model, we have used a simplified overcurrent protection,
where, the crypto-mining power supply trips if the rms value of the current
through the inductor exceeds a certain threshold. We have performed extensive
tests with the developed model by connecting the developed model to an ideal
power supply and subjecting it to voltage sags of various magnitudes and
durations. For further model validation, we compared this sag magnitude-
duration capability characteristics to a S19 ASIC miner available in our lab.
The results are presented in Fig. 6. Individual measurements obtained out of
the laboratory testing are provided in [10], and for representational
simplicity, we have denoted it by a continuous line.

(a) Simulated fault of 50% pre-fault voltage for 45 ms duration.

(b) Simulated fault of 50% pre-fault voltage for 16 ms duration.

Figure 5: Time-domain response of the EMT model of the crypto-mining power
supply. Figure 6: Validation with laboratory crypto-mining power supply. The
solid black line corresponds to the laboratory results from the mining power
supply.  show experiments where the power supply was tripped.  show
experiments where the power supply did not trip.

During model testing and simulation, it became evident that the severity of
voltage sags and the current drawn during these events are influenced not only
by the magnitude and duration of the sag but also by the specific point on the
wave[17], (see case study section, where tripping of a partial set of
converters is notable). Nevertheless, this additional information is absent in
the voltage sag magnitude-duration characteristics. Hence, modeling the
converter to align exclusively with magnitude-duration characteristics, while
neglecting the broader converter protection system could lead to deviations
from laboratory experiments. (see one of the test results deviates from the
ideal scenario).

##  III Case Studies

The performance of the developed EMT model has been tested in a 120 kV, 60 Hz,
6 bus transmission network in EMTP. A detailed description of test system
containing the solar PV park is provided in [18]. A moderately sized crypto-
mining facility of $\sim$ 1 MW with loads equally distributed across all three
phases was considered. As shown in Fig. 7, the mining facility is connected
with the rest of the transmission network through a $\Delta$ \- Yg 25 kV/415 V
transformer with a percentage impedance of 6 % at Bus 6. Therefore, the miners
are supplied with an input voltage at 240 V with the DC-bus voltage to be
maintained at 400 V.

Figure 7: Cryptocurrency farm interconnected with 120 kV Test System. The
performance of mining converters is compared for two distinct scenarios
considering three-phase line-to-ground (LLLG) faults: (i) FAULT 1: fault
within the cryptocurrency miner’s premises, and (ii) FAULT 2: fault at bus 3.

The non-mining loads in the test system are modeled as constant impedance
loads and the transmission lines are modeled using constant parameter models.
Both loads are 30 MW each and the solar PV park has a capacity of 75MVA. We
consider the constant power output from the solar park across all scenarios
and constant load demand from the crypto-mining facility at their respective
rated capacity. The projected sites in Fig. 7 represent the scalability of our
model. The faults were initiated after the cryptocurrency miners reached a
steady state. We explored multiple cases under each of these scenarios based
on fault duration and fault impedance dictating the sag voltage magnitude.

###  III-A Fault applied at Load End

Compared to Fig. 5, limited short-circuit capability implies that the post-
fault network voltage may not remain as ideal sinusoid as depicted in Figs.
8(b) and 9(b). Here, we have compared the LVRT capability of the crypto-
miners’ power supply with a 3-$\phi$ bolted fault and a fault with 50% pre-
fault voltage, each for a duration of 15 ms. While the miners in all three
phases were able to ride through with a fault of 50% pre-fault voltage, none
of the miners were able to ride through with the bolted fault, which is in
line with the ride-through characteristics shown in Fig. 6. Also, it can be
seen that after the miners were tripped, in Fig. 9(b), the voltage at the
miners’ premises contains a large amount of harmonics that tend to die out
with time. It could safely be attributed to the PV farm.

(a) Current Waveforms of Miners (Aggregated) Facility.

(b) Fault Voltage at Miner Premises.

Figure 8: Voltage and current at miner’s premises with a 3-$\phi$ fault of 50%
pre-fault voltage of 15 ms duration applied at load end.

(a) Current Waveforms of Miners (Aggregated) Facility.

(b) Fault Voltage at Miner Premises.

Figure 9: Voltage and current at miner’s premises with a bolted 3-$\phi$ fault
of 15 ms duration applied at load end. TABLE I: LVRT Capability with Fault
within Miner’s Premises ($*\rightarrow$ all miners in one out of three phases
trip).

Sag Voltage | Duration of fault
---|---
(% of Prefault) | 09ms | 1 cycle (15ms) | 3 cycles (45ms) | 100ms
75% | NO | NO | NO | NO
50% | NO | NO | 1 / 3 TRIP∗ | YES
25% | YES | YES | YES | YES
0% | YES | YES | YES | YES

We have conducted additional experiments to understand the LVRT capability of
the mining facility if the faults are within the miner’s premises, and the
results are tabulated in Table I. As highlighted in Fig. 6, demonstrating the
LVRT capability of crypto miner’s power supply, with sag voltage of 75% pre-
fault voltage magnitude none of the miners should trip, and it has been
validated in Table I. As with 50 % pre-fault voltage and fault duration of 45
ms, all the miners in one of the phases out of all the three phases trips.
This can be attributed to the large post-fault inrush current drawn from the
PCC which is a function of the point-on-wave. For a fault duration of less
than 100 ms and with a sag voltage of less than 25 % pre-fault voltage, none
of the mining power supplies are able to ride through.

(a) Current Waveforms of Miners (Aggregated) Facility.

(b) Voltage Waveforms at Miner Premises.

(c) Fault Voltage at Bus 3.

Figure 10: Voltage and current at miner’s premises and voltage at the faulted
bus with a 3-$\phi$ fault of 75% pre-fault voltage for a period of 100 ms
duration applied at bus 3.

###  III-B Fault at bus 3

Owing to the transmission line’s impedance, the sag voltage observed at the
crypto-miners’s premises would be less severe than in the previous scenario.
However, as demonstrated in Fig. 10, the occurrence of faults near the PV park
appears to inject a significant amount of harmonics. This interacts with the
controller of the mining power supply, and depending upon the phase it may
lead to an outage of all the mining power supplies. We have further conducted
an extensive number of experiments to understand the mining power supplies’
LVRT capability. It can be seen that even with 75% pre-fault voltage at the
faulted node, or, a better voltage within the miner’s premises, all the miners
are not able to ride through. While not shown for brevity, it has also been
observed that for a larger sag duration, more and more miners are unable to
ride through even with 75% pre-fault voltage.

TABLE II: Miner’s LVRT Capability with Fault at bus 3 ($*\rightarrow$ all
miners in one out of three phases trip, $\dagger\rightarrow$ all miners in two
out of three phases trip).

Sag Voltage | Time duration of fault
---|---
(% of Prefault) | 09ms | 1 cycle (15ms) | 3 cycles (45ms) | 100ms
75% | 1 /3 TRIP∗ | 1 /3 TRIP∗ | 1 /3 TRIP∗ | 1 /3 TRIP ∗
50% | 1 /3 TRIP∗ | 2 /3 TRIP† | 2 /3 TRIP† | YES
25% | 1 /3 TRIP∗ | YES | YES | YES
0% | 2 /3 TRIP† | YES | YES | YES

##  IV Conclusion

This paper provides a detailed switching-based EMT model of a typical
cryptocurrency mining facility based on an active power factor correction
(PFC) boost converter first by validating with a laboratory-based crypto-
mining power supply, and then by integrating and testing it with a small-scale
transmission system with a PV farm. The efficacy of the mining facility in
terms of its LVRT capability was compared considering two fault scenarios: (i)
fault within the miner’s premises, and (ii) fault near the PV farm. The
resulting harmonics from the PV farm in the aftermath of the fault can
interact with the controllers of the cryptocurrency miner’s power supply
resulting in an outage of these cryptominers, which would be challenging,
especially in a network with a low short circuit ratio. While the developed
model of the cryptocurrency miners is highly scalable, the use of a detailed
switching model hinders large-scale performance analysis of multiple
cryptocurrency mining facilities in a larger system. Therefore, in the future,
we would like to develop an average value model and utilize it for larger-
scale performance analysis.

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# Nonlinear Control Synthesis
for a Self-energizing Electro-Hydraulic Brake

Alexey Starykh A. Starykh is with the Department for Electric Drive and
Electrical Equipment of Tomsk Polytechnic University, Lenin Ave. 30, 634050
Tomsk, Russia. E-mail<EMAIL_ADDRESS>
###### Abstract

Nonlinear control algorithm for a self-energizing electro-hydraulic brake is
analytically designed. The desired closed-loop system behavior is reached via
a synthesized nonlinear controller.

###### Index Terms:

nonlinear system, nonlinear control, algorithm, electro-hydraulic brake.

##  I Introduction

Self-reinforcing brakes are a subject of intensive investigation during last
years. Working principle of such brakes is to use the wheelset’s inertia
momentum of a vehicle as the source of power for braking. One major advantage
of self-reinforcing brakes is the energy consumption decrease which makes this
research direction perspective.

At present the development of a new brake concept of a self-energizing
electro-hydraulic brake for a railway application is being carried out at the
Institute for Fluid Power Drives and Control (IFAS, RWTH Aachen University).
Working principle of the braking system can be found in [6–9].

##  II Problem Statement

The control task of the self-energizing electro-hydraulic brake is to track
the reference signal of the pressure $p_{sup}$ (the output variable) in the
supply line of the brake system. The control system with a pure proportional
controller allows reaching the goal, as shown in Fig. 1, where the reference
signal represents a step function, whose values are the sequence {27, 59, 91,
59, 27 bar}, which corresponds to brake forces of {5, 10, 15, 10, 5 kN}.
However, as seen from the figure, the oscillations of the supply line pressure
appear in the system, which cause the oscillations of the brake force.

Figure 1: The supply line pressure in the control system with a pure
proportional controller

Such a behavior of braking, of course, is inadmissible for passenger trains.
Therefore, the main requirement for the closed-loop system behavior is to get
an aperiodic process of supply line pressure changes independent of the
desired brake force level and friction coefficient variations of brake pads.

##  III Coordinates Transformation and Input-Output Linearization of a Plant
Model

To reach the required behavior of the controlled variable a controller of
nonlinear structure is proposed to synthesize on basis of the exact feedback
linearization method [3], [5].

Mathematical description of the electro-hydraulic system can be represented by
a nonlinear model of 10th order [6]. High order of the model result in the
analytically complicated controller structure, which is a problem for
simulation and experiments. Due to simplifications based on the results of the
brake system analysis [6] as well as the approach used in [1], the basic model
can be reduced to a 4th order nonlinear model. For positive (Case A) and
negative (Case B) valve opening this model has the form

Case A:

$\begin{pmatrix}\dot{p}_{L}\\\ \dot{p}_{sup}\\\ \dot{v}_{\textsf{v}}\\\
\dot{x}_{\textsf{v}}\\\
\end{pmatrix}\\!=\\!\begin{pmatrix}T^{A}_{L}x_{\textsf{v}}\sqrt{p_{sup}-p_{L}-\alpha
p_{lp}}\\\ T^{A}_{sup}x_{\textsf{v}}\sqrt{p_{sup}-p_{L}-\alpha p_{lp}}\\\
-2D_{\textsf{v}}\omega_{\textsf{v}}v_{\textsf{v}}-\omega^{2}_{\textsf{v}}x_{\textsf{v}}\\\
v_{\textsf{v}}\\\ \end{pmatrix}\\!+\\!\begin{pmatrix}0\\\ 0\\\
\omega^{2}_{\textsf{v}}K_{\textsf{v}}\\\ 0\\\ \end{pmatrix}\\!u$ (1)

Case B:

$\begin{pmatrix}\dot{p}_{L}\\\ \dot{p}_{sup}\\\ \dot{v}_{\textsf{v}}\\\
\dot{x}_{\textsf{v}}\\\
\end{pmatrix}\\!=\\!\begin{pmatrix}T^{B}_{L}x_{\textsf{v}}\sqrt{p_{L}+\alpha
p_{sup}-p_{lp}}\\\ T^{B}_{sup}x_{\textsf{v}}\sqrt{p_{L}+\alpha
p_{sup}-p_{lp}}\\\
-2D_{\textsf{v}}\omega_{\textsf{v}}v_{\textsf{v}}-\omega^{2}_{\textsf{v}}x_{\textsf{v}}\\\
v_{\textsf{v}}\\\ \end{pmatrix}\\!+\\!\begin{pmatrix}0\\\ 0\\\
\omega^{2}_{\textsf{v}}K_{\textsf{v}}\\\ 0\\\ \end{pmatrix}\\!u,$ (2)

where $p_{L},p_{sup},v_{\textsf{v}},x_{\textsf{v}}$ – the load pressure of the
brake actuator, the pressure in the supply line, the velocity and the control
valve spool movement, respectively; $\alpha$ – the ratio between piston areas
of the brake actuator; $p_{lp}$ – the pressure in the low pressure line;
$K_{\textsf{v}},D_{\textsf{v}},\omega_{\textsf{v}}$ – the control valve
parameters; $T^{A}_{L},T^{B}_{L}$ – the known constant parameters;
$T^{A}_{sup},T^{B}_{sup}$ – the parameters dependent of the brake pads
friction coefficient; $u$ – the control valve input signal. The output of the
brake system is

$y=p_{sup}$ (3)

This model will be used for the controller synthesis in the following.

The nonlinear model (1) – (3) can be represented as

$\displaystyle\dot{x}$ $\displaystyle=f(x)+g(x)u$ (4) $\displaystyle y$
$\displaystyle=h(x)$ (5)

The idea of the exact feedback linearization method is to find a nonlinear
transformation (linearization algorithm) of a control signal, for which the
model (4), (5) is linear or equivalent to a linear model in new coordinates
$z=H(x)$, where $H(x)$ is the coordinates transformation.

An important notion of an input-output model is its relative degree. For
linear systems, as known, the relative degree is the difference between the
number of poles and zeros of the transfer function of a system. This is also
the number of times an output needs to be differentiated in order that an
input appears in the equation. For nonlinear systems it is defined in the
similar manner. By differentiating the output $y$ and substituting (4) we get
(see [2])

$\begin{split}y^{(1)}=\frac{\partial h}{\partial x}\frac{\partial x}{\partial
t}&=\frac{\partial h}{\partial x}(f(x)+g(x)u)={}\\\
&=L_{f}h(x)+L_{g}h(x)u,\end{split}$ (6)

where $L_{f}h$ and $L_{g}h$ are Lie derivatives of the function $h$ along the
vector field $f$ and $g$, respectively.

For the electro-hydraulic brake, for positive and negative valve opening,
$\frac{\partial h}{\partial x}g(x)=L_{g}h(x)=0$ for all $x$ in the range of
operating points. Therefore

$y^{(1)}=L_{f}h(x)$

Continuing in this way, we get

$\begin{split}y^{(2)}=\frac{\partial L_{f}h}{\partial x}\frac{\partial
x}{\partial t}&=\frac{\partial L_{f}h}{\partial x}(f(x)+g(x)u)={}\\\
&=L^{2}_{f}h(x)+L_{g}L_{f}h(x)u,\end{split}$ (7)

where $L_{g}L_{f}h(x)=0$. Consequently,

$y^{(2)}=L^{2}_{f}h(x)$

The time derivative $y^{(3)}$ yields:

$y^{(3)}=L^{3}_{f}h(x)+L_{g}L^{2}_{f}h(x)u,$ (8)

where the Lie derivative $L_{g}L^{2}_{f}h(x)\neq 0$ for all $x$ in the range
of operating points and for positive and negative valve opening has the form

Case A:

$L_{g}L^{2}_{f}h(x)=T^{A}_{sup}\sqrt{p_{sup}-p_{L}-\alpha
p_{lp}}\>\omega^{2}_{\textsf{v}}K_{\textsf{v}}$ (9)

Case B:

$L_{g}L^{2}_{f}h(x)=T^{B}_{sup}\sqrt{p_{L}+\alpha
p_{sup}-p_{lp}}\>\omega^{2}_{\textsf{v}}K_{\textsf{v}}$ (10)

Thus, the equation (8) with the nonzero factor for $u$ describes the relation
between the input $u$ and the output $y$. Here, according to the definition in
[3], the relative degree of the 4th order nonlinear model of the electro-
hydraulic brake is 3.

For linear state-space systems, derivatives of an output are chosen as state
variables of a plant. Using the similar approach, we determine new coordinates
as

$z=\begin{pmatrix}z_{1}\\\ z_{2}\\\ z_{3}\\\ \end{pmatrix}=\begin{pmatrix}y\\\
y^{(1)}\\\ y^{(2)}\\\ \end{pmatrix}=\begin{pmatrix}h(x)\\\ L_{f}h(x)\\\
L^{2}_{f}h(x)\\\ \end{pmatrix}\doteq H(x)$ (11)

For positive and negative valve spool movement, the new coordinates $z$ can be
expressed by the coordinates $x$ of the simplified brake model (1) – (3), i.e.

Case A:

$\displaystyle z_{1}$ $\displaystyle=p_{sup}$ $\displaystyle z_{2}$
$\displaystyle=T^{A}_{sup}x_{\textsf{v}}\sqrt{p_{sup}-p_{L}-\alpha p_{lp}}$
$\displaystyle z_{3}$
$\displaystyle=\frac{1}{2}T^{A}_{sup}(T^{A}_{sup}-T^{A}_{L})x^{2}_{\textsf{v}}+T^{A}_{sup}v_{\textsf{v}}\sqrt{p_{sup}-p_{L}-\alpha
p_{lp}}$

Case B:

$\displaystyle z_{1}$ $\displaystyle=p_{sup}$ $\displaystyle z_{2}$
$\displaystyle=T^{B}_{sup}x_{\textsf{v}}\sqrt{p_{L}+\alpha p_{sup}-p_{lp}}$
$\displaystyle z_{3}$ $\displaystyle=\frac{1}{2}T^{B}_{sup}(\alpha
T^{B}_{sup}-T^{B}_{L})x^{2}_{\textsf{v}}+T^{A}_{sup}v_{\textsf{v}}\sqrt{p_{L}+\alpha
p_{sup}-p_{lp}}$

In accordance with (11), equations (5) – (7) can be rewritten in the form

$\dot{z}_{1}=z_{2},\quad\dot{z}_{2}=z_{3},\quad\dot{z}_{3}=a(x)+b(x)u,\quad
y=z_{1},$ (12)

where

$a(x)=L^{3}_{f}h(x),\quad b(x)=L_{g}L^{2}_{f}h(x)$

The function $b(x)$, for positive and negative valve spool movement, is
determined by equations (9), (10), the function $a(x)$ is expressed as

Case A:

$\begin{split}a(x)&=\frac{3}{2}T^{A}_{sup}(T^{A}_{sup}-T^{A}_{L})v_{\textsf{v}}x_{\textsf{v}}-{}\\\
&-T^{A}_{sup}(2D_{\textsf{v}}\omega_{\textsf{v}}v_{\textsf{v}}+\omega^{2}_{\textsf{v}}x_{\textsf{v}})\sqrt{p_{sup}-p_{L}-\alpha
p_{lp}}\end{split}$

Case B:

$\begin{split}a(x)&=\frac{3}{2}T^{B}_{sup}(\alpha
T^{B}_{sup}+T^{B}_{L})v_{\textsf{v}}x_{\textsf{v}}-{}\\\
&-T^{B}_{sup}(2D_{\textsf{v}}\omega_{\textsf{v}}v_{\textsf{v}}+\omega^{2}_{\textsf{v}}x_{\textsf{v}})\sqrt{p_{L}+\alpha
p_{sup}-p_{lp}}\end{split}$

If models (1) – (3), and (12) where equivalent, than the exact feedback
linearization problem would be solvable. This, in its turn, would mean that a
control signal of the form

$u=\frac{1}{b(x)}(-a(x)+\nu),$ (13)

where $\nu$ is a new control signal, due to overall compensation of the
nonlinear functions $a(x)$ and $b(x)$, would lead the plant model (1) – (3),
on the assumption of parametric certainty of the model and measurability of
the state variables $x$, to a system, whose behaviour is exactly identical
with the behaviour of the linear model

$\displaystyle\dot{z}=\begin{pmatrix}0&1&0\\\ 0&0&1\\\ 0&0&0\\\
\end{pmatrix}\begin{pmatrix}z_{1}\\\ z_{2}\\\ z_{3}\\\ \end{pmatrix}$
$\displaystyle+\begin{pmatrix}0\\\ 0\\\ 1\\\ \end{pmatrix}\\!\nu,$ (14)
$\displaystyle y=\begin{pmatrix}1&0&0\\\ \end{pmatrix}$
$\displaystyle\begin{pmatrix}z_{1}\\\ z_{2}\\\ z_{3}\\\ \end{pmatrix}\\!,$
(15)

whose output $y=z_{1}$ would coincide with the output $y=p_{sup}$ of the
simplified nonlinear model.

However, since rank $H(x)=3$ for all $x$ in the range of operating points,
which is less then the dimension $n=4$ of the model (1) – (3), the mapping
$H(x)$ is not a diffeomorphism. Let us note that a mapping $\varphi\colon X\to
Y$, where $X$ and $Y$ are smooth manifolds of dimension $n$, is a
diffeomorphism if $\varphi$ is bijective and both $\varphi$ and $\varphi^{-1}$
are smooth mapping (see Appendix A in [3]).

Considering aforesaid, the model (12) is not equivalent to the simplified
nonlinear model (1) – (3). This is caused by the internal dynamics, i.e.
behaviour of $(n-3)$-dimensional part of the nonlinear model, which, however,
has no influence on the output (15) of the linear model (14).

In that case the problem of input-output linearization can be solved [4]. This
means that the control (13) transforms the nonlinear model (1) – (3) of the
electro-hydraulic brake into a model whose input-output behaviour can be
represented in the form (14), (15). Additionally, one more variable
$\eta=T(x)$ has to be supplemented with the new coordinates $z$. According to
Proposition 4.1.3 in [3] it is always possible to find such a function that
the jacobian matrix of the mapping

$M(x)=\begin{pmatrix}T(x)\\\ H(x)\\\ \end{pmatrix}$

is nonsingular at some point $x=x^{*}$.

For the plant in question the load pressure of the brake actuator $p_{L}$ was
chosen as the additional variable, i.e. $\eta=T(x)=p_{L}$. The jacobian matrix
of the mapping $M(x)$ is nonsingular at each operating point of the brake. The
time derivative of $\eta$ for positive and negative valve opening has the form

Case A:

$\dot{\eta}\doteq q(z)=\frac{T^{A}_{L}}{T^{A}_{sup}}z_{2}$ (16)

Case B:

$\dot{\eta}\doteq q(z)=\frac{T^{B}_{L}}{T^{B}_{sup}}z_{2}$ (17)

Thus, the equivalent model in the new coordinates can be written as

$\displaystyle\dot{\eta}$ $\displaystyle=q(z),$ (18)
$\displaystyle\dot{z}=\begin{pmatrix}0&1&0\\\ 0&0&1\\\ 0&0&0\\\
\end{pmatrix}\begin{pmatrix}z_{1}\\\ z_{2}\\\ z_{3}\\\ \end{pmatrix}$
$\displaystyle+\begin{pmatrix}0\\\ 0\\\ 1\\\
\end{pmatrix}(\overline{a}(\eta,z)+\overline{b}(\eta,z)u),$ $\displaystyle y$
$\displaystyle=z_{1},$

where the equation (18) for $z\equiv 0$ describes the zero-dynamics of the
model (1) – (3). Taking into consideration that $x=M^{-1}(\eta,z)$, the
functions $\overline{a}(\eta,z)$ and $\overline{b}(\eta,z)$ are expressed as

$\overline{a}(\eta,z)=a\\{M^{-1}(\eta,z)\\},\quad\overline{b}(\eta,z)=b\\{M^{-1}(\eta,z)\\}$

The block diagram of the equivalent model is depicted in Fig. 2.

Figure 2: The normal form of the electro-hydraulic brake model (1) – (3)

Thus we can conclude that for the self-energizing electro-hydraulic brake the
controller synthesis based on the exact feedback linearization method is
conventionally divided into the following steps. Firstly, the control signal
$\nu$ is determined for the linear model (14), (15), which describes the
relationship between input and output variables of the nonlinear model (1) –
(3). Secondly, we take into consideration the internal dynamics of the plant,
described by the equation (18), which has no influence on the output of the
linear model, see Fig. 3.

Figure 3: The linearized model and the internal dynamics

Finally, we generate the control signal $u$ of the 4th order nonlinear model
using the nonlinear transformation for the signal $\nu$. The control $u$ is
expressed as

$u=\frac{1}{\overline{b}(\eta,z)}(-\overline{a}(\eta,z)+\nu)$

The obtained signal $u$ is the input signal for the real brake system,
described by the nonlinear model of 10th order.

##  IV Control Synthesis for the Linear Model of the Electro-Hydraulic Brake

The linear model (14) gives the opportunity to use classical methods of linear
systems theory to achieve the required control performance, i.e. to get an
aperiodic behaviour of the output variable (15) of the model (14). One such
approach is the pole placement method which allows reaching any prescribed
placement of closed loop system poles [10].

According to the method the control input signal $\nu$ for the linear model
(14) is chosen in the form

$\nu=K_{z}e,$

where $e=z^{*}-z$ is the error, i.e. the difference between the reference
state vector and the real state vector. Since the reference signal
$p^{*}_{sup}$ is assumed to be a piecewise constant function, the reference
signal $z^{*}$ of the model (14) during each time period, where $p^{*}_{sup}$
is constant, can be represented as

$z^{*}=\begin{pmatrix}p^{*}_{sup}\\\ \dot{p}^{*}_{sup}\\\
\ddot{p}^{*}_{sup}\\\ \end{pmatrix}=\begin{pmatrix}p^{*}_{sup}\\\ 0\\\ 0\\\
\end{pmatrix}$

The row-vector $K_{z}$ provides prescribed poles placement of the following
closed loop system

$\dot{e}=(A-bK_{z})e,$

where

$A=\begin{pmatrix}0&1&0\\\ 0&0&1\\\ 0&0&0\\\ \end{pmatrix};\quad
b=\begin{pmatrix}0\\\ 0\\\ 1\\\ \end{pmatrix}$

Since the electro-hydraulic brake system in question is a SISO system, i.e. a
single-input single-output system, only the output $p_{sup}$ is measured.
Consequently, not all state vector $x$ is known. It means that both the state
vector $z$ of the model (14), (15) and the variable $\eta$ of the model (18)
are unmeasured. Instead of these variables their estimates have to be used.
Therefore, the control system is supplemented with a full-order state observer
for (14), (15). By means of the estimate $\hat{z}$ of the vector $z$ the
estimate $\hat{\eta}$ of the variable $\eta$ can be calculated (see (16) and
(17)). The full-order state observer model is determined on the basis of the
model (14), (15) and has the form:

$\displaystyle\dot{\hat{z}}$ $\displaystyle=A\hat{z}+b\nu+K_{obs}(y-\hat{y}),$
$\displaystyle\hat{y}$ $\displaystyle=c^{T}\hat{z},$

where $K_{obs}$ is a column-vector of preset coefficients and
$c^{T}=(1\;0\;0)$. The vector $K_{obs}$ is chosen such that the matrix of
closed loop estimation system $(A-K_{obs}c^{T})$ be Hurwitz stable [10].

Simulation results of the electro-hydraulic brake system of 10th order with
the controller of described nonlinear structure are shown in Fig. 4.

Figure 4: The supply line pressure in the control system with the nonlinear
controller

One can see from the figure, that the nonlinear control algorithm synthesized
on the basis of the exact feedback linearization method allows to get the
desired closed-loop system behavior. The results of accomplished simulation
demonstrated as well, that the control system is robust with respect to small
variations of the friction coefficient. Expansion of the robust stability
ranges stimulates further development of the controller for the self-
energizing electro-hydraulic brake.

##  V Conclusion

Nonlinear control algorithm synthesis of the self-energizing electro-hydraulic
brake on the basis of the exact feedback linearization method has been
fulfilled. With the help of the obtained algorithm the problem of the desired
closed-loop system behavior has been solved.

## Acknowledgment

The author would like to thank German Academic Exchange Service (DAAD) and
Ministry of Education and Science of Russian Federation for supporting the
research stay at the Institute for Fluid Power Drives and Control in Aachen
(Project No. 7.375.C2007).

## References

  * [1] H.J. Feigel. Nichtlineare Effekte am servoventilgesteuerten Differentialzylinder [Nonlinear effects of a servo controlled differential cylinder]. _Ölhydraulik und Pneumatik, O+P_ , 31(1):42–48, 1987\.
  * [2] A.L. Fradkov, I.V. Miroshnik and V.O. Nikiforov. _Nonlinear and Adaptive Control of Complex Systems_. Dordrecht: Kluwer Academic Publishers, 1999.
  * [3] A. Isidori. _Nonlinear Control Systems_. 3rd ed. London: Springer–Verlag, 1995.
  * [4] H.K. Khalil. _Nonlinear Systems_. 2nd ed. N.–Y.: Prentice–Hall, 1996.
  * [5] M. Krstić, I. Kanellakokoulos and P.V. Kokotović. _Nonlinear and Adaptive Control Design_. N.–Y.: John Willey and Sons, 1995.
  * [6] M. Liermann. _Self-energizing Electro-hydraulic Brake_ , Dissertation RWTH Aachen University, Aachen: Shaker, 2008.
  * [7] M. Liermann and C. Stammen. Development of a self-energizing electro-hydraulic brake (SEHB) for rail vehicles. In: _The Tenth Scandinavian International Conference on Fluid Power: SICFP’07_ , Tampere: University of Technology, Institute of Hydraulics and Automation, 2007.
  * [8] M. Liermann, C. Stammen and H. Murrenhoff. Development of A Self-energizing Electro-Hydraulic Brake (SEHB). _Paper No. 2007-01-4236, SAE Commercial Vehicle Congress and Exhibition 2007_ , Chicago, USA, 2007.
  * [9] M. Liermann, C. Stammen and H. Murrenhoff. Pressure tracking control for a self-energizing hydraulic brake. In _The 20th Bath Symposium on Power Transmission and Motion Control_ , Bath: Hadleys Ltd., p. 315–330, 2007.
  * [10] I.V. Miroshnik. _Teoriya avtomaticheskogo upravleniya. Lineĭnye sistemy_ _[Automatic Control Theory. Linear Systems]_. SPb.:Piter, 2005.


# Chow and augmented Chow polynomials as evaluations of Poincaré-extended
$\mathbf{a}\mathbf{b}$-indices

Christian Stump  Ruhr-Universität Bochum, Germany<EMAIL_ADDRESS>
###### Abstract.

We show that Chow polynomials and augmented Chow polynomials of matroids, and
more generally of finite graded posets admitting $R$-labellings, are obtained
as evaluations of their Poincaré-extended $\mathbf{a}\mathbf{b}$-indices. This
implies in particular explicit combinatorial $\gamma$-positive expansions for
both.

###### 2020 Mathematics Subject Classification:

Primary 06A07; Secondary 05B35, 52C40

##  1\. Main result

The Chow ring $\underline{\textsf{CH}}(M)$ and the augmented Chow ring
$\textsf{CH}(M)$ are important geometric-algebraic object attached to a
loopless matroid $M$ that have been introduced in [FY04] and in [BHM+22],
respectively. They have attracted a lot of attention in recent years, maybe
most notable in their celebrated use in the proof of the log-concavity of the
Whitney numbers of the second kind in [AHK18]. We follow [FMSV24] in our
notations and call their Hilbert-Poincaré series
$\underline{\textsf{H}}_{M}(x)$ and $\textsf{H}_{M}(x)$ the _Chow polynomial_
and the _augmented Chow polynomial_.

To state the two main results, recall that the _lattice of flats_
$\mathcal{L}(M)$ of the matroid $M$ is geometric and thus admits an _$R$
-labeling_ $\lambda$, i.e., an edge-labeling $\lambda$ such that each interval
in $\mathcal{L}(M)$ contains a unique maximal chain along which $\lambda$ is
increasing [Sta74]. To a maximal chain
$\mathcal{F}=\\{\mathcal{F}_{0}\prec\dots\prec\mathcal{F}_{n}\\}$ in
$\mathcal{L}(M)$ associate the sequence
$\lambda_{\mathcal{F}}=(\lambda_{1},\dots,\lambda_{n})$ of edge labels
$\lambda_{i}=\lambda(\mathcal{F}_{i-1}\prec\mathcal{F}_{i})$ for $n={\sf
rank}(M)$ being the rank of $M$.

######  Theorem 1.1.

The Chow polynomial $\underline{\textsf{H}}_{M}(x)$ has the expansion

$\displaystyle\underline{\textsf{H}}_{M}(x)$
$\displaystyle=\sum_{\mathcal{F}}x^{\mathsf{des}(\lambda_{\mathcal{F}})}(x+1)^{n-1-2\
\\!\\!\mathsf{des}(\lambda_{\mathcal{F}})}\,,$

where the sum ranges over all maximal chains $\mathcal{F}$ in $\mathcal{L}(M)$
such that for $\lambda_{\mathcal{F}}=(\lambda_{1},\dots,\lambda_{n})$ it holds
that $\lambda_{1}<\lambda_{2}$ and also that $\lambda_{i}>\lambda_{i+1}$
implies $\lambda_{i-1}<\lambda´_{i}$ for $i\in\\{2,\dots,n-1\\}$.

######  Theorem 1.2.

The augmented Chow polynomial $\textsf{H}_{M}(x)$ has the expansion

$\displaystyle\textsf{H}_{M}(x)$
$\displaystyle=\sum_{\mathcal{F}}x^{\mathsf{des}(\lambda_{\mathcal{F}})}(x+1)^{n-2\
\\!\\!\mathsf{des}(\lambda_{\mathcal{F}})}\,,$

where the sum ranges over all maximal chains $\mathcal{F}$ in $\mathcal{L}(M)$
such that for $\lambda_{\mathcal{F}}=(\lambda_{1},\dots,\lambda_{n})$ it holds
that $\lambda_{i}>\lambda_{i+1}$ implies $\lambda_{i-1}<\lambda_{i}$ for
$i\in\\{2,\dots,n-1\\}$.

######  Corollary 1.3.

The Chow polynomial $\underline{\textsf{H}}_{M}(x)$ and the augmented Chow
polynomial $\textsf{H}_{M}(x)$ are $\gamma$-positive. They are in particular
nonnegative, palindromic and unimodal.

To the best of our knowledge, the only known proof of the $\gamma$-positivity
appears in [FMSV24] and relies on semi-small decompositions for the Chow and
the augmented Chow rings [BHM+22] or, equivalently, on the Kähler package for
these rings [AHK18]. Hence, Theorems 1.1, LABEL: and 1.2 may be viewed as a
Hodge-theory free proof to the $\gamma$-positivity and also of unimodality of
Chow and augmented Chow polynomials of geometric lattices.

######  Remark 1.4 (Generalization to finite graded posets admitting
$R$-labelings).

In forthcoming work by Ferroni–Matherne–Vecchi, the definitions of the Chow
and augmented Chow polynomials are generalized to finite graded posets with
$\hat{0}$ and $\hat{1}$ [FMV24], see Definition 2.1. Using this definition for
general finite graded posets, our proofs hold for all such posets admitting
$R$-labelings. We in particular obtain non-negativity, palindromicity,
unimodality, and $\gamma$-positivity in this generality.

To prove the main results, we show for finite graded posets admitting
$R$-labelings that the Chow and the augmented Chow polynomials are obtained as
evaluations of the Poincaré-extended $\mathbf{a}\mathbf{b}$-index defined and
studied by the author and Dorpalen-Barry–Maglione in [DMS23]. It turns out
that the Chow polynomial, but not the augmented Chow polynomial, is an
evaluation of the coarse flag Hilbert-Poincaré series as defined and studied
by Maglione–Voll in [MV23], which itself is also an evaluation of the extended
$\mathbf{a}\mathbf{b}$-index. As the Poincaré-extended
$\mathbf{a}\mathbf{b}$-indices of finite graded poset admitting $R$-labelings
have explicit descriptions using the labeling along maximal chains, we then
deduce Theorems 1.1, LABEL: and 1.2.

######  Remark 1.5 (Three existing real-rootedness conjectures).

We show in Theorem 2.6 that the Chow polynomial is an evaluation of the
extended $\mathbf{a}\mathbf{b}$-index. This evaluation is conjectured to be
real-rooted [FS23, Conjecture 8.18]. It is shown in [DMS23, Corollary 2.23]
that the numerator polynomial of the coarse flag Hilbert-Poincaré series is an
evaluation of the extended $\mathbf{a}\mathbf{b}$-index and asked in [KM23,
Question 1.6] for which matroids a further evaluation is real-rooted. Also the
chain polynomial of the lattice of flats is yet another evaluation of the
extended $\mathbf{a}\mathbf{b}$-index which is conjectured in [AKE23,
Conjecture 1.2] to be real-rooted. This suggests the following question.

######  Question 1.6.

Which evaluations of the extended $\mathbf{a}\mathbf{b}$-index of a matroid,
or of a more general finite graded poset admitting an $R$-labeling, are real-
rooted?

### Acknowledgment

This work was conducted at the _Mathematisches Forschungsinstitut Oberwolfach_
in June 2024 during the workshop “Arrangements, Matroids and Logarithmic
Vector Fields”. We very much thank the institute for their hospitality and the
organizers Takuro Abe, Graham Denham, Eva-Maria Feichtner, and Gerhard Röhrle
for setting up this wonderful and inspiring meeting. The results in this paper
were obtained after attending the talk by Luis Ferroni entitled “Chow
polynomials of posets”. The author would very much like to thank Luis and his
collaborators Chris Eur, Jacob Matherne, and Lorenzo Vecchi, and as well Elena
Hoster and Lukas Kühne for inspiring discussions concerning Chow polynomials
and their combinatorial interpretations, and for comments on a preliminary
version of this paper.

##  2\. The Poincaré-extended $\mathbf{a}\mathbf{b}$-index and its evaluations

Let $P$ be a finite _graded poset_ of rank $n$. That is, $P$ is a finite poset
with unique minimum element $\hat{0}$ of rank $0$ and unique maximum element
$\hat{1}$ of rank $n$ such that ${\sf rank}(F)$ is equal to the length of any
maximal chain from $\hat{0}$ to $F$. Its _Möbius function_ $\mu$ is given by
$\mu(F,F)=1$ for all $F\in P$ and $\mu(F,G)=-\sum_{F\leq H<G}\mu(F,H)$ for all
$F<G$. Its _Poincaré polynomial_ is

$\displaystyle\mathcal{\sf Poin}_{P}(q)$ $\displaystyle=\sum_{F\in
P}\mu(\hat{0},F)\cdot(-q)^{{\sf rank}(F)}$ its _characteristic polynomial_ is
$\displaystyle\chi_{P}(q)$ $\displaystyle=\sum_{F\in P}\mu(\hat{0},F)\cdot
q^{n-{\sf rank}(F)}$ and its _reduced characteristic polynomial_ is
$\displaystyle\overline{\chi}_{P}(q)$
$\displaystyle=\chi_{P}(q)/(q-1)=q^{n}\cdot\mathcal{\sf
Poin}_{P}(-q^{-1})/(q-1)\,.$

A (not necessary maximal) chain
$\mathcal{C}=\\{\mathcal{C}_{1}<\dots<\mathcal{C}_{k}<\mathcal{C}_{k+1}\\}$ in
$P$ is an ordered set of pairwise comparable elements. We assume throughout
this paper that all chains end in the maximum element
$\mathcal{C}_{k+1}=\hat{1}$. A chain is maximal if $k=n$ and we usually write
$\\{\mathcal{F}_{0}\prec\dots\prec\mathcal{F}_{n}\\}$ in this case in order to
clearly distinguish between general chains ending in $\hat{1}$ and maximal
chains. The _chain Poincaré polynomial_ and the _reduced chain characteristic
polynomial_ of such a chain $\mathcal{C}$ are

$\mathcal{\sf Poin}_{P,\mathcal{C}}(q)=\prod_{i=1}^{k}\mathcal{\sf
Poin}_{[\mathcal{C}_{i},\mathcal{C}_{i+1}]}(q)\quad\text{ and
}\quad\overline{\chi}_{P,\mathcal{C}}(q)=\prod_{i=1}^{k}\overline{\chi}_{[\mathcal{C}_{i},\mathcal{C}_{i+1}]}(q)\,,$

where $[\mathcal{C}_{i},\mathcal{C}_{i+1}]\subseteq P$ denotes the interval
between two consecutive elements in the chain. In particular,

$\mathcal{\sf Poin}_{P,\\{\hat{0}<\hat{1}\\}}(q)=\mathcal{\sf
Poin}_{P}(q),\quad\overline{\chi}_{P,\\{\hat{0}<\hat{1}\\}}(q)=\overline{\chi}_{P}(q)\,.$

###  2.1. The Chow and the augmented Chow polynomials

The following description of the Chow and augmented Chow polynomials of a
matroid is taken from [FMSV24, Theorem 1.3 & 1.4]. We present it in the
generality of general finite graded posets as introduced by [FMV24].

######  Definition 2.1 (Ferroni–Matherne–Vecchi).

Let $P$ be a finite graded poset. The _Chow polynomial_ is defined to be $1$
if ${\sf rank}(P)=0$ and otherwise

$\underline{\textsf{H}}_{P}(x)=\sum_{\mathcal{C}}\overline{\chi}_{P,\mathcal{C}}(x)\
\in\mathbb{Z}[x]\,,$

where the sum ranges over all chains
$\mathcal{C}=\\{\mathcal{C}_{1}<\dots<\mathcal{C}_{k+1}\\}$ in $P$ starting at
$\mathcal{C}_{1}=\hat{0}$ and ending in $\mathcal{C}_{k+1}=\hat{1}$. The
_augmented Chow polynomial_ is defined to be $1$ if ${\sf rank}(P)=0$ and
otherwise

$\textsf{H}_{P}(x)=\sum_{\mathcal{C}}x^{{\sf
rank}(\mathcal{C}_{1})}\cdot\overline{\chi}_{P,\mathcal{C}}(x)\
\in\mathbb{Z}[x]\,,$

where the sum ranges over all chains
$\mathcal{C}=\\{\mathcal{C}_{1}<\dots<\mathcal{C}_{k+1}\\}$ in $P$ ending in
$\mathcal{C}_{k+1}=\hat{1}$.

It is shown in [FMSV24, Theorems 1.3 & 1.4] that for the lattice of flats
$P=\mathcal{L}(M)$, this definition agrees with the Hilbert-Poincaré series of
the Chow and the augmented Chow rings for a loopless matroid $M$ as presented
in Section 1.

###  2.2. The Poincaré-extended $\mathbf{a}\mathbf{b}$-index

The following description and properties of the extended
$\mathbf{a}\mathbf{b}$-index are taken from [DMS23]. Throughout this section,
let $P$ be a finite graded poset admitting an _$R$ -labeling_, i.e., an edge-
labeling $\lambda:\mathcal{E}(P)\rightarrow\mathbb{N}_{+}$ such that each
interval in $P$ contains a unique maximal chain along which $\lambda$ is
(weakly) increasing.

Let $\mathbb{Z}[y]\langle\mathbf{a},\mathbf{b}\rangle$ be the polynomial ring
in two noncommuting variables $\mathbf{a},\mathbf{b}$ with coefficients being
polynomials in the variable $y$. For a subset $S\subseteq\\{0,\dots,n-1\\}$,
we set ${\sf wt}_{S}(\mathbf{a},\mathbf{b})=w_{0}\cdots w_{n-1}$ for the
polynomial with

(1) $w_{k}=\begin{cases}\mathbf{b}&\text{if }k\in S,\\\
\mathbf{a}-\mathbf{b}&\text{if }k\notin S\,.\end{cases}$

For a chain
$\mathcal{C}=\\{\mathcal{C}_{1}<\dots<\mathcal{C}_{k}<\mathcal{C}_{k+1}\\}$ in
$P$, ending in $\mathcal{C}_{k+1}=\hat{1}$, we moreover set

${\sf wt}_{\mathcal{C}}={\sf wt}_{\\{{\sf rank}(\mathcal{C}_{1}),\dots,{\sf
rank}(\mathcal{C}_{k})\\}}\,.$

######  Definition 2.2.

The (_Poincaré-_)_extended $\mathbf{a}\mathbf{b}$-index_ of $P$ is

${\text{\scriptsize\sf
ex}}\Psi_{P}(y,\mathbf{a},\mathbf{b})=\sum_{\mathcal{C}}\mathcal{\sf
Poin}_{P,\mathcal{C}}(y)\cdot{\sf wt}_{\mathcal{C}}(\mathbf{a},\mathbf{b})\
\in\mathbb{Z}[y]\langle\mathbf{a},\mathbf{b}\rangle\,,$

where the sum ranges over all chains
$\mathcal{C}=\\{\mathcal{C}_{1}<\dots<\mathcal{C}_{k+1}\\}$ in $P$ ending in
$\mathcal{C}_{k+1}=\hat{1}$.

Since $P$ has a unique minimum, we always have $\mathcal{\sf Poin}_{P}(0)=1$,
implying

(2) ${\text{\scriptsize\sf
ex}}\Psi_{P}(0,\mathbf{a},\mathbf{b})=\Psi_{P}(\mathbf{a},\mathbf{b})=\sum_{\mathcal{C}}{\sf
wt}_{\mathcal{C}}(\mathbf{a},\mathbf{b})$

for (a mild variation of) the _$\mathbf{a}\mathbf{b}$ -index_
$\Psi_{P}(\mathbf{a},\mathbf{b})$ as given, for example, in [Bay21, Section
2].

######  Proposition 2.3 ([DMS23, Corollary 2.22]).

We have

$\mathbf{b}\cdot\iota\big{(}{\text{\scriptsize\sf
ex}}\Psi_{P}(y,\mathbf{a},\mathbf{b})\big{)}=\sum_{\mathcal{C}}\mathcal{\sf
Poin}_{P,\mathcal{C}}(y)\cdot{\sf wt}_{\mathcal{C}}(\mathbf{a},\mathbf{b})\
\in\mathbb{N}[y]\langle\mathbf{a},\mathbf{b}\rangle\,,$

where the sum ranges over all chains
$\mathcal{C}=\\{\mathcal{C}_{1}<\dots<\mathcal{C}_{k+1}\\}$ starting in
$\mathcal{C}_{1}=\hat{0}$ and ending in $\mathcal{C}_{k+1}=\hat{1}$ and where
$\iota$ deletes the first variable from every $\mathbf{a}\mathbf{b}$-monomial.

Observe that the factor $\mathbf{b}$ on the left-hand side comes from the
assumption that $\mathcal{C}_{1}=\hat{0}$ and thus, every
$\mathbf{a},\mathbf{b}$-polynomial ${\sf
wt}_{\mathcal{C}}(\mathbf{a},\mathbf{b})$ starts with the variable
$\mathbf{b}$ as $0={\sf rank}(\mathcal{C}_{1})$. For later reference, we set

(3) ${\text{\scriptsize\sf
ex}}\widetilde{\Psi}_{P}(y,\mathbf{a},\mathbf{b})=\iota\big{(}{\text{\scriptsize\sf
ex}}\Psi_{P}(y,\mathbf{a},\mathbf{b})\big{)}\,.$

We next present two of the main result of [DMS23], from which we then deduce
the proposed evaluation to obtain the Chow and the augmented Chow polynomials.
As in Section 1, we write
$\lambda_{\mathcal{F}}=(\lambda_{1},\dots,\lambda_{n})$ for the labeling along
a maximal chain
$\mathcal{F}=\\{\mathcal{F}_{0}\prec\dots\prec\mathcal{F}_{n}\\}$ with
$\lambda_{i}=\lambda(\mathcal{F}_{i-1}\prec\mathcal{F}_{i})$. For a subset
$E\subseteq\\{1,\dots,n\\}$, we moreover set
$\lambda_{\mathcal{F},E}=(\lambda^{\prime}_{0},\lambda^{\prime}_{1},\dots,\lambda^{\prime}_{n})$
with $\lambda^{\prime}_{0}=0$ and $\lambda^{\prime}_{i}=-\lambda_{i}$ if $i\in
E$ and $\lambda^{\prime}_{i}=\lambda_{i}$ if $i\notin E$. We in addition
define ${\sf u}_{\mathcal{F},E}(\mathbf{a},\mathbf{b})=u_{0}\cdots u_{n-1}$ to
be the monomial in $\mathbf{a},\mathbf{b}$ with

$u_{i}=\begin{cases}\mathbf{a}&\text{if
}\lambda^{\prime}_{i}\leq\lambda^{\prime}_{i+1}\,,\\\ \mathbf{b}&\text{if
}\lambda^{\prime}_{i}>\lambda^{\prime}_{i+1}\,,\\\ \end{cases}$

so the monomial ${\sf u}_{\mathcal{F},E}(\mathbf{a},\mathbf{b})$ detects the
ascent-descent pattern of $\lambda_{\mathcal{F},E}$.

######  Theorem 2.4 ([DMS23, Theorem 2.7]).

Let $P$ be a finite graded poset of rank $n$, admitting an $R$-labeling
$\lambda$. Then

${\text{\scriptsize\sf
ex}}\Psi_{P}(y,\mathbf{a},\mathbf{b})=\sum_{\mathcal{F},E}y^{\\#E}\cdot{\sf
u}_{\mathcal{F},E}(\mathbf{a},\mathbf{b})$

where the sum ranges over all maximal chains $\mathcal{F}$ in $P$ and all
subsets $E\subseteq\\{1,\dots,n\\}$.

######  Corollary 2.5 ([DMS23, Corollary 2.9]).

Let $P$ be a finite graded poset of rank $n$, admitting an $R$-labeling
$\lambda$. Then

${\text{\scriptsize\sf
ex}}\Psi_{P}(y,\mathbf{a},\mathbf{b})=\omega\big{(}\Psi_{P}(\mathbf{a},\mathbf{b})\big{)}$

where the substitution $\omega$ first replaces all occurrences of
$\mathbf{a}\mathbf{b}$ with
$\mathbf{a}\mathbf{b}+y\mathbf{b}\mathbf{a}+y\mathbf{a}\mathbf{b}+y^{2}\mathbf{b}\mathbf{a}$
and then simultaneously replaces all remaining occurrences of $\mathbf{a}$
with $\mathbf{a}+y\mathbf{b}$ and $\mathbf{b}$ with $\mathbf{b}+y\mathbf{a}$.

It is immediate that the $\mathbf{a}\mathbf{b}$-index in (2) has the expansion

(4) $\Psi_{P}(\mathbf{a},\mathbf{b})=\sum_{\mathcal{F}}{\sf
u}_{\mathcal{F},\emptyset}(\mathbf{a},\mathbf{b})$

and that every monomial ${\sf
u}_{\mathcal{F},\emptyset}(\mathbf{a},\mathbf{b})=u_{0}\cdots u_{n-1}$ starts
with the variable $u_{0}=\mathbf{a}$.

###  2.3. Evaluating the Poincaré-extended $\mathbf{a}\mathbf{b}$-index

The main results of this section are the following evaluations.

######  Theorem 2.6.

Let $P$ be a finite graded poset of rank $n$ admitting an $R$-labeling. Then

$\displaystyle{\text{\scriptsize\sf ex}}\Psi_{P}(-x,1,x)$
$\displaystyle=(1-x)^{n}\cdot\textsf{H}_{P}(x)\,,$
$\displaystyle{\text{\scriptsize\sf ex}}\widetilde{\Psi}_{P}(-x,1,x)$
$\displaystyle=(1-x)^{n}\cdot\underline{\textsf{H}}_{P}(x)\,.$

Before proving this evaluation, we deduce Theorems 1.1, LABEL: and 1.2.

###### Proof of Theorems 1.1, LABEL: and 1.2.

Using the expansion (4), an immediate consequence of Theorem 2.4 and Corollary
2.5 is that

${\text{\scriptsize\sf
ex}}\Psi_{P}(-x,1,x)=\sum_{\mathcal{F}}\omega\big{(}{\sf
u}_{\mathcal{F},\emptyset}(\mathbf{a},\mathbf{b})\big{)}\big{|}_{y=-x,\
\mathbf{a}=1,\ \mathbf{b}=x}$

where the sum ranges over all maximal chains $\mathcal{F}$ in $P$. Applying
the evaluation $y=-x,\ \mathbf{a}=1,\ \mathbf{b}=x$ to the substitution
$\omega$ gives that every occurrence in ${\sf u}_{\mathcal{F},\emptyset}$ of
the pattern $\mathbf{a}\mathbf{b}$ becomes a factor
$x-x^{2}-x^{2}+x^{3}=x(1-x)^{2}$, every remaining $\mathbf{a}$ yields a factor
$1-x^{2}=(1+x)(1-x)$, and every remaining $\mathbf{b}$ yields a factor $0$. We
therefore obtain

$\displaystyle\omega\big{(}{\sf
u}_{\mathcal{F},\emptyset}(\mathbf{a},\mathbf{b})\big{)}\big{|}_{y=-x,\
\mathbf{a}=1,\ \mathbf{b}=x}$ $\displaystyle=0$ if $\lambda_{\mathcal{F}}$
contains a descent that does not immediately follow an ascent, and otherwise
$\displaystyle\omega\big{(}{\sf
u}_{\mathcal{F},\emptyset}(\mathbf{a},\mathbf{b})\big{)}\big{|}_{y=-x,\
\mathbf{a}=1,\ \mathbf{b}=x}$ $\displaystyle=(1-x)^{n}\cdot
x^{\mathsf{des}(\lambda_{\mathcal{F}})}(1+x)^{n-2\
\\!\\!\mathsf{des}(\lambda_{\mathcal{F}})}\,.$

Summing over all maximal chains $\mathcal{F}$ in $P$ such that for
$\lambda_{\mathcal{F}}=(\lambda_{1},\dots,\lambda_{n})$ it holds that
$\lambda_{i}>\lambda_{i+1}$ implies $\lambda_{i-1}<\lambda_{i}$ for
$i\in\\{2,\dots,n-1\\}$ then gives

${\text{\scriptsize\sf
ex}}\Psi_{P}(-x,1,x)=(1-x)^{n}\sum_{\mathcal{F}}x^{\mathsf{des}(\lambda_{\mathcal{F}})}(x+1)^{n-2\
\\!\\!\mathsf{des}(\lambda_{\mathcal{F}})}\,.$

This proves Theorem 1.2.

The argument to deduce Theorem 1.1 is almost the same. The only difference is
that now every initial variable is removed from every
$\mathbf{a}\mathbf{b}$-monomial to obtain ${\text{\scriptsize\sf
ex}}\widetilde{\Psi}_{P}(y,\mathbf{a},\mathbf{b})$ from ${\text{\scriptsize\sf
ex}}\Psi_{P}(x,\mathbf{a},\mathbf{b})$. This is, we have

${\text{\scriptsize\sf
ex}}\widetilde{\Psi}_{P}(y,\mathbf{a},\mathbf{b})=\iota\big{(}{\text{\scriptsize\sf
ex}}\Psi_{P}(y,\mathbf{a},\mathbf{b})\big{)}=\sum_{\mathcal{F}}\iota\big{(}\omega\big{(}{\sf
u}_{\mathcal{F},\emptyset}(\mathbf{a},\mathbf{b})\big{)}\big{)}\,.$

We consider the two cases whether ${\sf
u}_{\mathcal{F},\emptyset}(\mathbf{a},\mathbf{b})$ starts with
$\mathbf{a}\mathbf{a}$ or with $\mathbf{a}\mathbf{b}$. If it starts with
$\mathbf{a}\mathbf{a}$, we obtain

$\iota\big{(}\omega\big{(}{\sf
u}_{\mathcal{F},\emptyset}(\mathbf{a},\mathbf{b})\big{)}\big{)}=(1+y)\omega\big{(}\iota\big{(}{\sf
u}_{\mathcal{F},\emptyset}(\mathbf{a},\mathbf{b})\big{)}\big{)}$

as the first variable $\mathbf{a}$ is evaluated to
$\iota\omega(\mathbf{a})=\iota(\mathbf{a}+y\mathbf{b})=1+y$. If it starts with
$\mathbf{a}\mathbf{b}$, we obtain

$\iota\omega\big{(}{\sf
u}_{\mathcal{F},\emptyset}(\mathbf{a},\mathbf{b})\big{)}\big{|}_{y=-x,\
\mathbf{a}=1,\ \mathbf{b}=x}=0$

as the initial two variables $\mathbf{a}\mathbf{b}$ are evaluated to

$\iota\omega(\mathbf{a}\mathbf{b})=\iota((1+y)\mathbf{a}\mathbf{b}+y(1+y)\mathbf{b}\mathbf{a})=(1+y)\mathbf{b}+y(1+y)\mathbf{a}$

and

$(1+y)\mathbf{b}+y(1+y)\mathbf{a}\big{|}_{y=-x,\ \mathbf{a}=1,\
\mathbf{b}=x}=0$

Summing now over all maximal chains $\mathcal{F}$ in $P$ such that for
$\lambda_{\mathcal{F}}=(\lambda_{1},\dots,\lambda_{n})$ it holds that
$\lambda_{1}<\lambda_{2}$ and also that $\lambda_{i}>\lambda_{i+1}$ implies
$\lambda_{i-1}<\lambda_{i}$ for $i\in\\{2,\dots,n-1\\}$ then gives

${\text{\scriptsize\sf
ex}}\widetilde{\Psi}_{P}(-x,1,x)=(1-x)^{n}\sum_{\mathcal{F}}x^{\mathsf{des}(\lambda_{\mathcal{F}})}(x+1)^{n-1-2\
\\!\\!\mathsf{des}(\lambda_{\mathcal{F}})}\,.$

This proves Theorem 1.1. ∎

###### Proof of Theorem 2.6.

We evaluate the Chow polynomial and the extended $\mathbf{a}\mathbf{b}$-index
as follows, where the sums range over all chains
$\mathcal{C}=\\{\mathcal{C}_{1}<\dots<\mathcal{C}_{k}<\mathcal{C}_{k+1}\\}$
starting in $\mathcal{C}_{1}=\hat{0}$ and ending in
$\mathcal{C}_{k+1}=\hat{1}$. We write $k_{\mathcal{C}}=k$ to denote its
length. Then

$\displaystyle\underline{\textsf{H}}_{P}(x)$
$\displaystyle=\sum_{\mathcal{C}}\overline{\chi}_{P,\mathcal{C}}(x)=\sum_{\mathcal{C}}\prod_{i=1}^{k_{\mathcal{C}}}\overline{\chi}_{[\mathcal{C}_{i},\mathcal{C}_{i+1}]}(x)$
$\displaystyle=\sum_{\mathcal{C}}\prod_{i=1}^{k_{\mathcal{C}}}x^{{\sf
rank}(\mathcal{C}_{i+1})-{\sf
rank}(\mathcal{C}_{i})}(x-1)^{-1}\cdot\mathcal{\sf
Poin}_{[\mathcal{C}_{i},\mathcal{C}_{i+1}]}(-x^{-1})$
$\displaystyle=x^{n}\sum_{\mathcal{C}}\prod_{i=1}^{k_{\mathcal{C}}}(x-1)^{-1}\cdot\mathcal{\sf
Poin}_{[\mathcal{C}_{i},\mathcal{C}_{i+1}]}(-x^{-1})$
$\displaystyle=x^{n}\sum_{\mathcal{C}}(x-1)^{-k_{\mathcal{C}}}\mathcal{\sf
Poin}_{P,\mathcal{C}}(-x^{-1})$
$\displaystyle=x^{n}(x-1)^{-1}{\text{\scriptsize\sf
ex}}\widetilde{\Psi}_{P}(-x^{-1},\tfrac{x}{x-1},\tfrac{1}{x-1})$

where we used in the last equation that ${\sf
wt}_{\mathcal{C}}(\mathbf{a},\mathbf{b})$ has exactly $n-1$ factors, either
being $\mathbf{b}$ for ranks in $\mathcal{C}\setminus\\{\mathcal{C}_{k+1}\\}$
or being $\mathbf{a}-\mathbf{b}$ for ranks not in $\mathcal{C}$, and moreover
that evaluating $\mathbf{b}\mapsto\frac{1}{x-1}$ and
$\mathbf{a}\mapsto\frac{x}{x-1}$ evaluates $\mathbf{a}-\mathbf{b}\mapsto 1$
and at the same time yields a factor $(x-1)^{1-k_{\mathcal{C}}}$. Using the
symmetry
$\underline{\textsf{H}}_{P}(x)=x^{n-1}\underline{\textsf{H}}_{P}(x^{-1})$, we
obtain

$\displaystyle\underline{\textsf{H}}_{P}(x)$
$\displaystyle=x^{-1}(x^{-1}-1)^{-1}{\text{\scriptsize\sf
ex}}\widetilde{\Psi}_{P}(-x,\tfrac{1}{1-x},\tfrac{x}{1-x})$
$\displaystyle=(1-x)^{-n}{\text{\scriptsize\sf
ex}}\widetilde{\Psi}_{P}(-x,1,x)$

where we used that ${\text{\scriptsize\sf
ex}}\widetilde{\Psi}_{P}(y,\mathbf{a},\mathbf{b})$ is homogeneous of
$\mathbf{a}\mathbf{b}$-degree $n-1$ and thus

${\text{\scriptsize\sf
ex}}\widetilde{\Psi}_{P}(-x,\tfrac{1}{1-x},\tfrac{x}{1-x})=(1-x)^{n-1}{\text{\scriptsize\sf
ex}}\widetilde{\Psi}_{P}(-x,1,x)\,.$

The argument for

$\textsf{H}_{P}(x)=(1-x)^{-n}{\text{\scriptsize\sf ex}}\Psi_{P}(-x,1,x)$

is word for word the same with the only difference that chains
$\mathcal{C}=\\{\mathcal{C}_{1}<\dots<\mathcal{C}_{k}<\mathcal{C}_{k+1}\\}$ do
not necessarily need to start in $\mathcal{C}_{1}=\hat{0}$, and thus the
factor $x^{{\sf rank}(\mathcal{C}_{1})}$ in the definition of the augmented
Chow polynomial ensures that $x^{n}=x^{{\sf
rank}(\mathcal{C}_{1})}\cdot\prod_{i=1}^{k_{\mathcal{C}}}x^{{\sf
rank}(\mathcal{C}_{i+1})-{\sf rank}(\mathcal{C}_{i})}$. ∎

######  Remark 2.7.

We used in the proof the symmetry
$\underline{\textsf{H}}_{P}(x)=x^{n-1}\underline{\textsf{H}}_{P}(x^{-1})$. As
the symmetry of the evaluated extended $\mathbf{a}\mathbf{b}$-index is clear
from its combinatorial description, this is not a prerequired property of the
Chow polynomial and we only used it here for the clarity of the argument.

###  2.4. Evaluating the coarse flag Hilbert-Poincaré series

The coarse flag Hilbert-Poincaré series ${\sf cfHP}_{P}(y,t)$ is originally
defined and studied in [MV23] as an algebraic invariant in geometric group
theory. It is shown in [DMS23, Corollary 2.22] that this rational function can
be expressed as

${\sf cfHP}_{P}(y,t)=(1-t)^{-n}{\text{\scriptsize\sf
ex}}\widetilde{\Psi}_{P}(y,1,t)\,.$

We thus immediately obtain the following specialization from Theorem 2.6.

######  Corollary 2.8.

Let $P$ be a finite graded poset of rank $n$ admitting an $R$-labeling. The
Chow polynomial is then obtained from the coarse flag Hilbert-Poincaré series
by the specialization

$\displaystyle\underline{\textsf{H}}_{P}(x)={\sf cfHP}_{P}(x,-x)\,.$

Applying $\leavevmode\nobreak\ \iota$ to ${\text{\scriptsize\sf
ex}}\Psi_{P}(y,\mathbf{a},\mathbf{b})$ restricts the situation to chains
starting in $\hat{0}$ as in Proposition 2.3. It is thus not immediately
possible to also obtain the augmented Chow polynomial from the coarse flag
Hilbert-Poincaré series.

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