# Fault Detection for Grid-Forming Inverters in Islanded Droop-Controlled AC
Microgrids

Gabriel Intriago, , Andres Intriago, ,
Charalambos Konstantinou, and Yu Zhang G. Intriago is with the Department of
Electrical and Computer Engineering at the University of California Santa
Cruz, USA (e-mail: gintriag@ucsc.edu). Also, he is with the Math Department at
the Escuela Politécnica Nacional, Ecuador (e-mail:
gabriel.intriago01@epn.edu.ec).A. Intriago and C. Konstantinou are with the
Computer, Electrical, and Mathematical Sciences and Engineering Division, King
Abdullah University of Science and Technology, Saudi Arabia (e-mail:
{andres.intriagovelasquez,charalambos.konstantinou}@kaust.edu.sa).Y. Zhang is
with the Department of Electrical and Computer Engineering at the University
of California, Santa Cruz, USA (e-mail: zhangy@ucsc.edu).

###### Abstract

In this paper, we develop an observer-based fault detection mechanism for
grid-forming inverters operating in islanded droop-controlled AC microgrids.
The detection scheme uses linear matrix inequalities as constraints with
$\mathcal{H}_{-}/\mathcal{H}_{\infty}$ optimization to achieve sensitivity to
faults and robustness against disturbances or parametric uncertainties. We
explore a nonlinear inverter model formulation based on the less-restrictive
one-sided Lipschitz and quadratic inner-boundedness conditions instead of the
state-of-the-art formulation based on the Lipschitz condition. In this sense,
we aim to overcome the sensitivity of observer-based schemes to the Lipschitz
constant. The relation between these two formulations for fault detection is
analyzed theoretically. We find the deterministic matrix expressions of
different internal faults, including busbar, actuator, and inverter bridge
faults. The performance of the proposed detection method is tested on an
islanded AC microgrid with four grid-forming inverters and compared against
the state-of-the-art nonlinear detection based on the Lipschitz condition.
Most importantly, this method requires no additional sensors, a crucial
advantage over many proposed solutions in the literature.

###### Index Terms:

AC microgrids, grid-forming inverter, fault detection, one-sided Lipschitz,
quadratic inner-boundedness.

##  I Introduction

The development of converter-interfaced power generation has facilitated the
successful integration of renewable generation into small-scale power systems,
also known as microgrids [1] [2]. Microgrids have two modes of operation,
islanded or grid-connected. The grid-connected mode forces the voltage and
frequency of the microgrid to be dominated by the main grid. In this mode, the
microgrid inherits the upstream protection of the main grid. On the other
hand, an islanded microgrid is an independent system that must provide a
reference voltage magnitude and frequency to the active and passive elements
of the microgrid. In this regard, maintaining the stability of the microgrid
is more challenging for the islanded mode than for the grid-connected mode
[3]. In addition, the stability of independent microgrids is at risk when
abnormal events appear, for instance, internal faults. Such events may incur a
significant imbalance between energy demand and supply, leading to unwanted
scenarios, such as partial power outages or blackouts. Moreover, the
disconnection from the bulk power system reduces the fault current strength in
the islanded microgrid, which helps halt the decline of voltage magnitude and
frequency following a severe event [4]. The situation is further aggravated
when the power generators are grid-forming inverters reducing the available
electrical inertia in the system [5]. In this regard, a fault detection (FD)
scheme that is robust against disturbances and sensitive to faults becomes
vital to promote a stable microgrid performance and avoid energy supply
interruption.

Several fault detection techniques have been developed in the last decade to
guarantee an accurate and reliable operation of microgrids [6]. Despite data-
driven techniques demanding vast training data, they have gained popularity
for fault detection due to the advances in microgrids software and hardware
assets [7, 8, 9]. Methods based on signal processing have proven to be an
effective policy for fault detection in microgrids; however, their solutions
are difficult to generalize because they depend on a case-by-case basis [10,
11, 12]. Other strategies rely on signal patterns and local measurements close
to the generation units to build a reliable fault detection module [13, 14,
15, 16]. However, such strategies demand additional hardware and can be
costly. Besides these techniques, model-based methods are considered an option
for FD when a system model is available. Recently, these methods have become a
popular technique for fault detection because they are built upon the physical
relationships that govern the system dynamics. Among the model-based methods,
the observer-based methods have received considerable interest due to their
fast detection capabilities, low-cost implementation, and the availability of
powerful tools for observer design [17, 18, 19, 20, 21].

A model-based nonlinear observer method for induction motor drives that
detects and isolates faults is studied in [17]. In [18], the authors propose a
nonlinear observer for a grid-connected PV circuit that supervises the values
of unmodeled fault signatures for measuring output deviations that might
suggest the existence of a fault. However, the effect of disturbances and
parametric uncertainties was not considered. Fault detection and
identification method enhanced with $\mathcal{H}_{-}/\mathcal{H}_{\infty}$
optimization that is able to respond to different component faults is proposed
in [19]. Nevertheless, the study is carried out for DC microgrids, and the
state-space model is linear. A time-varying observer for predicting the states
of synchronous machines is developed in [20]. The study relies on a linearized
model computed at each time step, increasing the proposed approach’s
computational complexity. In addition, model uncertainties and disturbances
have not been considered. The authors in [21] consider a Lipschitz equivalent
nonlinear model to pose an optimization problem with matrix inequalities
constraints and $\mathcal{L}_{-}/\mathcal{L}_{\infty}$ performance indexes for
detecting faults in a microgrid composed by synchronous machines. However, the
study does not consider parametric uncertainties and fault influence in the
nonlinear function that arises in the states transition model. Moreover, the
study does not address how the Lipschitz condition’s restrictiveness may
affect the constraints’ feasibility expressed as matrix inequalities.

###  I-A Contribution

A main drawback of observer-based fault detection schemes for nonlinear
systems is their sensitivity to the constant of the Lipschitz condition,
leading to conservative observer designs [22, 23]. In [22], the authors
introduce observer design with two less conservative conditions, the one-sided
Lipschitz and quadratic inner-boundedness conditions. A full-order and
reduced-order design problem for one-sided Lipschitz nonlinear systems using a
Ricatti equation approach is addressed in [23]. This work proposes a novel
fault detection scheme for GFMs based on the above observers. The
contributions of our work are listed as follows:

  1. 1.

We show that LMI-based techniques combined with
$\mathcal{H}_{-}/\mathcal{H}_{\infty}$ optimization can be successfully
applied for fault detection in GFMs connected to islanded droop-controlled AC
microgrids.

  2. 2.

Our proposed observer-based fault detection scheme is robust against
disturbances while sensitive to faults.

  3. 3.

Leveraging the conditions of one-sided one-sided Lipschitz continuity and
quadratic inner-boundedness, we propose an GFM nonlinear model instead of the
Lipschitz equivalent formulation. Furthermore, we investigate the relation
between the two formulations based on one-sided Lipschitz continuity and
Lipschitz continuity.

  4. 4.

We derive the deterministic matrix representation for different types of
faults acting on GFMs.

  5. 5.

We consider the effect of disturbances and parametric uncertainties on the
system model, including the nonlinear function in the state transition
equation.

###  I-B Notation and Organization

Symbol $\times$ denotes the Cartesian product. Bold letters represent vectors.
$\left\lVert\mathbf{a}\right\rVert$ is the $\ell_{2}$-norm of the vector while
$\|\mathbf{a}\|_{[0,t]}:=\sqrt{\int_{0}^{t}\|\mathbf{a}\|_{2}^{2}\>dt}$ is the
$\ell_{2}$-norm up to time $t$. $A\succ 0$ indicates that matrix $A$ is
positive definite. $\langle\mathbf{a},\mathbf{b}\rangle$ represents the dot
product between vectors $\mathbf{a}$ and $\mathbf{b}$. $\mathbb{R}$ denotes
the set of real numbers. $\mathbf{a}^{\top}$ is the transpose of vector
$\mathbf{a}$.

This paper is organized as follows. Section II presents the background for
observers design in nonlinear systems. Section III presents the state-space
model for grid-forming inverters. A set of fault and disturbance with their
matrix representation are shown in section IV. The proposed observer design is
in Section V. The numerical tests and concluding remarks are given in Sections
VI and VIII, respectively.

##  II Preliminaries

###  II-A Observers for Nonlinear Dynamic Systems

The state-space representation of a nonlinear system can be typically
expressed as:

$\displaystyle\left\\{\begin{matrix}\dot{\mathbf{x}}=A\mathbf{x}+B\mathbf{u}+\bm{\phi}(\mathbf{x},\mathbf{u})\\\
\mathbf{y}=C\mathbf{x}+D\mathbf{u}\qquad\qquad\qquad\end{matrix}\right.\,,$
(1)

where $\bm{x}$ is the state vector, $\bm{y}$ represents the measurement
vector, $\bm{u}$ the inputs vector, and function
$\bm{\phi}(\mathbf{x},\mathbf{u})$ captures nonlinearity of the system. The
parametric matrices $A$, $B$, $C$ and $D$ are formed according to the
equations that govern the system. We further define two convex sets
$\mathcal{D}$ and $\mathcal{U}$ that make up the feasible operating region of
the system. They are the Cartesian products of all intervals formed by the
upper and lower limits of all states and control inputs, respectively [24]:

$\displaystyle\mathcal{D}$
$\displaystyle=[x_{1}^{\min},x_{1}^{\max}]\times\dots\times[x_{n}^{\min},x_{n}^{\max}],$
$\displaystyle\mathcal{U}$
$\displaystyle=[u_{1}^{\min},u_{1}^{\max}]\times\dots\times[u_{p}^{\min},u_{p}^{\max}].$

The nonlinear model (1) can be modified to incorporate both disturbances and
faults as follows:

$\begin{cases}\dot{\mathbf{x}}&=A\mathbf{x}+B\mathbf{u}+\eta_{1}(\mathbf{x},\mathbf{u})+\bm{\phi}(\mathbf{x},\mathbf{u})+E_{w}\mathbf{w}+E_{f}\mathbf{f}\\\
\mathbf{y}&=C\mathbf{x}+D\mathbf{u}+\eta_{2}(\mathbf{x},\mathbf{u})+F_{w}\mathbf{w}+F_{f}\mathbf{f}\end{cases}\,,$
(2)

where $\mathbf{w}$ is a disturbance input vector, $\mathbf{f}$ is the fault
vector, $E_{w}$ and $F_{w}$ are constant disturbance matrices, $E_{f}$ and
$F_{f}$ are deterministic fault matrices. The unknown functions $\eta_{1}$ and
$\eta_{2}$ represent the model uncertainties which are assumed to be bounded,
i.e.,

$\|\eta_{i}(\mathbf{x},\mathbf{u})\|<\infty,\quad\forall\>\mathbf{x}\in\mathcal{D}~{}\text{and}~{}\forall\>\mathbf{u}\>\in\mathcal{U},~{}i=1,2.$

A Luenberger observer of the nonlinear system is given as:

$\begin{cases}\dot{\hat{\mathbf{x}}}&=A\mathbf{\hat{x}}+B\mathbf{u}+\phi(\mathbf{\hat{x}},\mathbf{u})+L(\mathbf{y}-\hat{\mathbf{y}})\\\
\hat{\mathbf{y}}&=C\mathbf{\hat{x}}+D\mathbf{{u}}\end{cases}\,,$ (3)

where $\mathbf{\hat{x}}$ is the vector of estimated states, $\hat{\mathbf{y}}$
is the vector of estimated measurements, and $L$ is the observer gain matrix
to be designed. We have two assumptions of model (2):

  1. 1.

The pair $(A,C)$ is observable.

  2. 2.

The signal vectors $\mathbf{w}$ and $\mathbf{f}$are $\mathcal{L}_{2}$ square-
integrable that satisfy: $\|\mathbf{w}\|_{[0,t]}<\infty$ and
$\|\mathbf{f}\|_{[0,t]}<\infty$.

###  II-B Residual Generation

Define observation error $\mathbf{e}={{\mathbf{x}}}-{\hat{\mathbf{x}}}$, and
measurement residual $\mathbf{r}=W(\mathbf{y}-\hat{\mathbf{y}})$. Then, the
error dynamics can be derived as follows:

$\begin{cases}\dot{\mathbf{e}}&=(A-LC)\mathbf{e}+\phi(\mathbf{x},\mathbf{u})-\phi(\hat{\mathbf{x}},\mathbf{u})\\\
&\quad+(E_{w}-LF_{w})\mathbf{w}+(E_{f}-LF_{f})\mathbf{f}\\\
\mathbf{r}&=W(\mathbf{y}-C\mathbf{\hat{x}}-D\mathbf{{u}})\end{cases}\,.$ (4)

We can compactly rewrite (4) as follows:

$\begin{cases}\dot{\mathbf{e}}&=\bar{A}\mathbf{e}+\Phi+\bar{E}_{w}\mathbf{w}+\bar{E}_{f}\mathbf{f}\\\
\mathbf{r}&=\bar{C}\mathbf{e}+\bar{F}_{w}\mathbf{w}+\bar{F}_{f}\mathbf{f}\end{cases}$
(5)

with
$\Phi\triangleq\phi(\mathbf{x},\mathbf{u})-\phi(\hat{\mathbf{x}},\mathbf{u})$
and

$\displaystyle\left\\{\begin{matrix}\bar{A}=A-LC,&\bar{E}_{w}=E_{w}-LF_{w},&\bar{E}_{f}=E_{f}-LF_{f}\\\
\bar{C}=WC,&\bar{F}_{w}=WF_{w},&\bar{F}_{f}=WF_{f}\end{matrix}\right.$ (6)

We can assume $W=I$ as suggested in [25]. The residual $\mathbf{r}$ in (5)
will be used for fault detection [21].

###  II-C Mixed $\mathcal{H}_{-}/\mathcal{H}_{\infty}$ Optimization for
Observer Design

The fault detection problem becomes challenging when there is no clear
distinction between disturbances and faults. Such a situation may mislead the
fault detection filter triggering false alarms. In order to deal with this
issue, we considered the mixed $\mathcal{H}_{-}/\mathcal{H}_{\infty}$
optimization framework for observer design [21]. This framework aims to
simultaneously make the fault detector filter robust against disturbances and
sensitive to faults by satisfying the following criteria:

  1. 1.

Robustness against disturbances $\mathbf{w}$:

$\displaystyle\|\mathbf{r}_{w}\|_{[0,t]}\leq\alpha\|\mathbf{w}\|_{[0,t]}$ (7)

  2. 2.

Sensitivity to faults $\mathbf{f}$:

$\displaystyle\|\mathbf{r}_{f}\|_{[0,t]}\geq\beta\|\mathbf{f}\|_{[0,t]}$ (8)

where

$\displaystyle\mathbf{r}_{w}$
$\displaystyle=\bar{C}\mathbf{e}+\bar{F}_{w}\mathbf{w}$ (9)
$\displaystyle\mathbf{r}_{f}$
$\displaystyle=\bar{C}\mathbf{e}+\bar{F}_{f}\mathbf{f}.$ (10)

In the next section, we pose the task of designing a fault detection filter as
a convex optimization problem. We will derive sufficient conditions in the
form of linear matrix inequalities to guarantee the existence of such a
filter. In this sense, we transform conditions (7) and (8) into linear matrix
inequalities such that we can effectively incorporate them in the filter
design.

###  II-D Residual Evaluation Function and Threshold Computation

We follow the procedure presented in [26] to decide the occurrence of faults.
The residual evaluation function is defined as $J=\|\mathbf{r}\|_{2}$.
Ideally, we might be able to detect a fault when $J$ is non-zero. However,
disturbances make $J$ non-zero even in the absence of faults. To cope with
this issue, we compute a threshold for $J$ considering the system is in a
fault-free condition and subjected to disturbances:

$\displaystyle
J_{\mathrm{th}}=\underset{\mathbf{w}\in\mathcal{L}_{2}}{\sup}\>\|\mathbf{r}_{\mathbf{w}}\|$
(11)

Hence, the logic for fault detection is defined as:

$\begin{cases}J\leq J_{\mathrm{th}}&\longrightarrow\>\text{No faults}\\\
J>J_{\mathrm{th}}&\longrightarrow\>\text{Fault detected}\end{cases}$ (12)

The threshold is a vital component of a residual-based fault detection
strategy. A reliable threshold minimizes the false alarm and missed detection
rates, improving the fault detection ability of our proposed approach.

##  III Dynamic Model of the Grid-forming Inverter

We use the $dq$ reference frame to represent the dynamics of the GFMs
according to [27] where the $i$-th GFM is indexed with the subscript $i$. The
state space representation of the disturbance- and fault-free dynamical model
of each GFM can be expressed as follows:

$\displaystyle\left\\{\begin{matrix}\dot{\mathbf{x}}=A\mathbf{x}+\bm{\phi}(\mathbf{x},\mathbf{u})+B\mathbf{u}\\\
\mathbf{y}=C\mathbf{x}+D\mathbf{u}\qquad\qquad\qquad\end{matrix}\right.\,,$
(13)

where

$\displaystyle\mathbf{x}$
$\displaystyle=\left[\alpha_{i}\>\>P_{i}\>\>Q_{i}\>\>\phi_{di}\>\>\phi_{qi}\>\>\gamma_{di}\>\>\gamma_{qi}\>\>i_{ldi}\>\>i_{lqi}\>\>v_{odi}\>\>v_{oqi}\>\>i_{odi}\>\>i_{oqi}\right]^{\top}$
$\displaystyle\mathbf{u}$
$\displaystyle=\left[\omega_{com}\>\>\omega_{ni}\>\>V_{ni}\>\>v_{bdi}\>\>v_{bqi}\right]^{\top}$
$\displaystyle\mathbf{y}$
$\displaystyle=\left[\alpha_{i}\>\>\omega_{i}\>\>v_{odi}^{*}\>\>i_{ldi}^{*}\>\>i_{lqi}^{*}\>\>v_{idi}^{*}\>\>v_{iqi}^{*}\right]^{\top}.$

The parametric matrices $A$, $B$, $C$, $D$, and the nonlinear function
$\bm{\phi}(\mathbf{x},\mathbf{u})$ are formed according to the differential
equations governing the GFMs as shown in [28]. The nonlinear function
$\bm{\phi}(\mathbf{x},\mathbf{u})$ in (13) arises from the low-pass filtered
active and reactive power, and the Kirchoff voltage and current laws.

##  IV Fault Model

This section introduces five faults that are likely to affect the operating
point of a GFM. Also, we express the faults and disturbances as a
deterministic matrix representation using matrices $E_{f}$, $F_{f}$, $E_{w}$
and $F_{w}$.

###  IV-A Busbar Fault

A busbar fault affects the voltage phasor at the end of the output connector
or point of common coupling (PCC) with the microgrid; such voltage is
represented by the inputs $v_{bdi}$ and $v_{bqi}$. The busbar fault is modeled
as follows:

$\displaystyle v_{bdi}^{\prime}$ $\displaystyle=v_{bdi}+\Delta v_{bdi}$ (14)
$\displaystyle v_{bqi}^{\prime}$ $\displaystyle=v_{bqi}+\Delta v_{bqi}$ (15)
$\displaystyle f$ $\displaystyle=\left[\begin{matrix}\Delta v_{bdi}&\Delta
v_{bqi}\end{matrix}\right]^{\top}$

Hence, by replacing (14) & (15) in (13), we find the fault matrices $E_{f}$
and $F_{f}$ as:

$\displaystyle E_{f}$ $\displaystyle=\left[\begin{matrix}0_{1\times
11}&-\frac{1}{L_{ci}}&0\\\ 0_{1\times
11}&0&-\frac{1}{L_{ci}}\end{matrix}\right]^{\top}$ $\displaystyle F_{f}$
$\displaystyle=\left[\begin{matrix}0_{7\times 2}\end{matrix}\right]$

###  IV-B Actuator Faults

The actuator signals consist of the inputs $\omega_{ni}$ and $V_{ni}$, which
set the GFMs’ desired frequency and voltage magnitude, respectively. The
actuator faults are modeled as follows:

$\displaystyle\omega_{ni}^{\prime}$
$\displaystyle=\omega_{ni}+\Delta\omega_{ni}$ (16) $\displaystyle
V_{ni}^{\prime}$ $\displaystyle=V_{ni}+\Delta V_{ni}$ (17)

The actuator fault on $\omega_{ni}$ affects the linear term $B\mathbf{u}$ and
the nonlinear function $\bm{\phi}(\mathbf{x},\mathbf{u})$ of (13). Hence, by
replacing (16) in (13), and after some mathematical manipulations, we have:

$\displaystyle f_{\omega_{ni}}$
$\displaystyle=\left[\begin{matrix}\Delta\omega_{ni}&\Delta\omega_{ni}i_{lqi}&\Delta\omega_{ni}i_{ldi}&\Delta\omega_{ni}v_{oqi}\end{matrix}\right.$
$\displaystyle\qquad\left.\begin{matrix}\Delta\omega_{ni}v_{odi}&\Delta\omega_{ni}i_{oqi}&\Delta\omega_{ni}i_{odi}\end{matrix}\right]^{\top}$
$\displaystyle E_{f_{\omega_{ni}}}$
$\displaystyle=\left[\begin{matrix}1&0_{1\times 6}\\\ 0_{6\times 1}&0_{6\times
6}\\\ 0_{6\times 1}&A_{\omega_{ni}}\end{matrix}\right]$ $\displaystyle
A_{\omega_{ni}}$
$\displaystyle=\text{diag}\left([\begin{matrix}1&-1&1&-1&1&-1\end{matrix}]\right)$
$\displaystyle F_{f_{\omega_{ni}}}$
$\displaystyle=\left[\begin{matrix}0&1&0_{1\times 5}\\\ 0_{6\times
1}&0_{6\times 1}&0_{6\times 5}\end{matrix}\right]^{\top}$

The actuator fault on $V_{ni}$ affects the linear term $B\mathbf{u}$ of (13)
exclusively. Similarly, by replacing (17) in (13), and after mathematical
manipulations, we find the following fault vector and matrices:

$\displaystyle f_{V_{ni}}$ $\displaystyle=\left[\begin{matrix}\Delta
V_{ni}\end{matrix}\right]$ $\displaystyle E_{f_{V_{ni}}}$
$\displaystyle=\left[\begin{matrix}0_{1\times
3}&1&0&K_{PV_{i}}&0&\frac{1}{L_{fi}}K_{PC_{i}}K_{PV_{i}}&0_{1\times
5}\end{matrix}\right]^{\top}$ $\displaystyle F_{f_{V_{ni}}}$
$\displaystyle=\left[\begin{matrix}0&0&1&K_{PV_{i}}&0&K_{PC_{i}}K_{PV_{i}}&0\end{matrix}\right]^{\top}$

###  IV-C Inverter Bridge Faults

Ideally, the DQ output voltages of the current controller $v_{idi}^{*}$ and
$v_{iqi}^{*}$ are equal to the inverter bridge’s output voltage $v_{idi}$ and
$v_{iqi}$:

$\displaystyle v_{idi}$
$\displaystyle=\eta_{v_{idi}}v_{idi}^{*},\quad\eta_{v_{idi}}=1$ $\displaystyle
v_{iqi}$ $\displaystyle=\eta_{v_{iqi}}v_{iqi}^{*},\quad\eta_{v_{iqi}}=1$

In this sense, we model inverter bridge’s faults as a parametric fault
representing a reduction in the bridge’s efficiency ($\Delta\eta_{v_{idi}}$,
$\Delta\eta_{v_{iqi}}$):

$\displaystyle v_{idi}$
$\displaystyle=(1-\Delta\eta_{v_{idi}})v_{idi}^{*},\quad\Delta\eta_{v_{idi}}\in(0,1]$
$\displaystyle v_{iqi}$
$\displaystyle=(1-\Delta\eta_{v_{iqi}})v_{iqi}^{*},\quad\Delta\eta_{v_{iqi}}\in(0,1]$

We find the fault vector $f$, fault matrices $E_{f}$, and $F_{f}$ by modifying
the corresponding dynamical equations that depend on $v_{idi}^{*}$ and
$v_{iqi}^{*}$. Consequently, we split the analysis into two parts.

####  IV-C1 Fault vector and fault matrices for $v_{idi}$

$\displaystyle\mathbf{f}_{v_{idi}}$
$\displaystyle=\Delta\eta_{v_{idi}}\left[\begin{matrix}Q_{i}&\phi_{di}&\gamma_{di}&i_{ldi}&i_{lqi}&v_{odi}\end{matrix}\right.$
$\displaystyle\qquad\qquad\qquad\left.\begin{matrix}v_{oqi}&i_{odi}&V_{ni}\end{matrix}\right]^{\top}$
$\displaystyle E_{f_{v_{idi}}}$ $\displaystyle=\left[\begin{matrix}0_{9\times
7}&\bm{\xi}_{v_{idi}}&0_{9\times 5}\end{matrix}\right]^{\top}$ $\displaystyle
F_{f_{v_{idi}}}$ $\displaystyle=\left[\begin{matrix}0_{9\times
5}&\bm{\tau}_{v_{idi}}&0_{9\times 1}\end{matrix}\right]^{\top}$
$\displaystyle\bm{\xi}_{v_{idi}}$
$\displaystyle=\left[\begin{matrix}\frac{K_{PC_{i}}K_{PV_{i}}n_{Q_{i}}}{L_{fi}}&-\frac{K_{PC_{i}}K_{IV_{i}}}{L_{fi}}&-\frac{K_{IC_{i}}}{L_{fi}}&\frac{K_{PC_{i}}}{L_{fi}}&\omega_{b}\end{matrix}\right.$
$\displaystyle\quad\left.\begin{matrix}\frac{K_{PC_{i}}K_{PV_{i}}}{L_{fi}}&\frac{K_{PC_{i}}\omega_{b}C_{fi}}{L_{fi}}&-\frac{K_{PC_{i}}F_{i}}{L_{fi}}&-\frac{K_{PC_{i}}K_{PV_{i}}}{L_{fi}}\end{matrix}\right]^{\top}$
$\displaystyle\bm{\tau}_{v_{idi}}$
$\displaystyle=\left[\scalebox{1.0}{\mbox{$\displaystyle\begin{matrix}K_{PC_{i}}K_{PV_{i}}n_{Q_{i}}&-K_{PC_{i}}K_{IV_{i}}&-K_{IC_{i}}&K_{PC_{i}}\end{matrix}$}}\right.$
$\displaystyle\quad\quad\left.\begin{matrix}\omega_{b}L_{fi}&K_{PC_{i}}K_{PV_{i}}&K_{PC_{i}}\omega_{b}C_{fi}&-K_{PC_{i}}F_{i}\end{matrix}\right.$
$\displaystyle\quad\quad\left.\begin{matrix}-K_{PC_{i}}K_{PV_{i}}\end{matrix}\right]^{\top}$

####  IV-C2 Fault vector and fault matrices for $v_{iqi}$

$\displaystyle\mathbf{f}_{v_{iqi}}$
$\displaystyle=\Delta\eta_{v_{iqi}}\left[\begin{matrix}\phi_{qi}&\gamma_{qi}&i_{ldi}&i_{lqi}&v_{odi}&v_{oqi}&i_{oqi}\end{matrix}\right]^{\top}$
$\displaystyle E_{f_{v_{iqi}}}$ $\displaystyle=\left[\begin{matrix}0_{7\times
8}&\bm{\xi}&0_{7\times 4}\end{matrix}\right]^{\top}$ $\displaystyle
F_{f_{v_{iqi}}}$ $\displaystyle=\left[\begin{matrix}0_{7\times
6}&\bm{\tau}_{v_{iqi}}\end{matrix}\right]^{\top}$
$\displaystyle\bm{\xi}_{v_{iqi}}$
$\displaystyle=\left[\begin{matrix}-\frac{K_{PC_{i}}K_{IV_{i}}}{L_{fi}}&-\frac{K_{IC_{i}}}{L_{fi}}&-\omega_{b}&\frac{K_{PC_{i}}}{L_{fi}}\end{matrix}\right.$
$\displaystyle\quad\quad\left.\begin{matrix}-\frac{K_{PC_{i}}\omega_{b}C_{fi}}{L_{fi}}&\frac{K_{PC_{i}}K_{PV_{i}}}{L_{fi}}&-\frac{K_{PC_{i}}F_{i}}{L_{fi}}\end{matrix}\right]$
$\displaystyle\bm{\tau}_{v_{iqi}}$
$\displaystyle=\left[\scalebox{1.0}{\mbox{$\displaystyle\begin{matrix}-K_{PC_{i}}K_{IV_{i}}&-K_{IC_{i}}&-\omega_{b}L_{fi}&K_{PC_{i}}\end{matrix}$}}\right.$
$\displaystyle\quad\quad\left.\begin{matrix}-K_{PC_{i}}\omega_{b}C_{fi}&K_{PC_{i}}K_{PV_{i}}&-K_{PC_{i}}F_{i}\end{matrix}\right]^{\top}$

The inverter bridge fault affects both efficiencies simultaneously, so we
merge previous fault vectors and matrices as follows:

$\displaystyle\mathbf{f}$
$\displaystyle=\left[\begin{matrix}\mathbf{f}_{v_{idi}}&\mathbf{f}_{v_{iqi}}\end{matrix}\right]^{\top}$
$\displaystyle E_{f}$
$\displaystyle=\left[\begin{matrix}E_{f_{v_{idi}}}&E_{f_{v_{iqi}}}\end{matrix}\right]$
$\displaystyle F_{f}$
$\displaystyle=\left[\begin{matrix}F_{f_{v_{idi}}}&F_{f_{v_{iqi}}}\end{matrix}\right]$

##  V Observer Design for Nonlinear Systems

###  V-A Lipschitz Nonlinear Systems

Lipschitz systems include a broad range of physical systems providing
flexibility for the design of observers. Let us assume the nonlinear function
$\bm{\phi}(\cdot)$ satisfies the Lipschitz condition given by the following
definition:

######  Definition 1.

The function $\bm{\phi}(\cdot)$ is said to be Lipschitz continuous if there
exists a Lipschitz constant $\gamma\in\mathbb{R}^{+}$ such that
$\forall\>\mathbf{x},\hat{\mathbf{x}}\in\mathcal{D}$ and
$\forall\>\mathbf{u}\in\mathcal{U}$:

$\displaystyle\|\bm{\phi}(\mathbf{x},\mathbf{u})-\bm{\phi}(\hat{\mathbf{x}},\mathbf{u})\|\leq\gamma\|\mathbf{x}-\hat{\mathbf{x}}\|$
(18)

The following lemma provides sufficient conditions for the existence of a
filter gain matrix $L$ based on the mixed
$\mathcal{H}_{-}/\mathcal{H}_{\infty}$ optimization framework for Lipschitz
nonlinear systems:

######  Lemma 1 (see [21]).

Consider the system (2) with a Lipschitz continuous observer defined in (3)
and (18). If there exist identical positive definite matrices $P$ and $Q$,
gain matrix $L$, and strictly positive scalars $\alpha$, $\beta$,
$\varepsilon_{1}$, $\varepsilon_{2}$ such that the following two LMIs hold

$\displaystyle\begin{split}\left[\begin{matrix}\Omega_{1}&P\bar{E}_{w}+\bar{C}^{\top}\bar{F}_{w}&P\\\
*&-\alpha^{2}I+\bar{F}_{w}^{\top}\bar{F}_{w}&0\\\
*&*&-\varepsilon_{1}I\end{matrix}\right]\prec 0\end{split}$ (19)
$\displaystyle\begin{split}\left[\begin{matrix}\Omega_{2}&Q\bar{E}_{f}-\bar{C}^{\top}\bar{F}_{f}&Q\\\
*&-\beta^{2}I+\bar{F}_{f}^{\top}\bar{F}_{f}&0\\\
*&*&-\varepsilon_{2}I\end{matrix}\right]\prec 0,\end{split}$ (20)

where
$\Omega_{1}=\bar{A}^{\top}P+P\bar{A}+\bar{C}^{\top}\bar{C}+\varepsilon_{1}\gamma^{2}I$,
$\Omega_{2}=\bar{A}^{\top}Q+Q\bar{A}-\bar{C}^{\top}\bar{C}+\varepsilon_{2}\gamma^{2}I$,
and $\gamma$ is the given Lipschitz constant. Then, i) the residual generator
in (4) is stable; and ii) the resulting observer satisfies the robustness and
sensitivity constraints (7) and (8).

The proof of Lemma 1 can be found in [21], where an LMI solution is proposed
for the observer design problem by considering a Lipschitz nonlinear system
under the $\mathcal{H}_{-}/\mathcal{H}_{\infty}$ framework.

###  V-B Observer Design for One-sided Lipschitz Nonlinear Systems

As discussed, the Lipschitz constant is sensitive to the system’s operating
region $\mathcal{D}\times\mathcal{U}$ and the parameters defining the
nonlinear function $\bm{\phi}(\cdot)$. In this situation, Lemma 1 may likely
fail to stabilize the dynamics of the residual generator and find the observer
matrix gain $L$. An alternative is to consider the less restrictive one-sided
Lipschitz and quadratic inner-boundedness conditions, which generalize the
Lipschitz condition [22] [23].

######  Definition 2.

The function $\bm{\phi}(\cdot)$ is said to be one-sided Lipschitz continuous
if there exists $\rho\in\mathbb{R}$ such that
$\forall\>\mathbf{x},\hat{\mathbf{x}}\in\mathcal{D}$ and
$\forall\>u\in\mathcal{U}$ we have

$\displaystyle\langle\bm{\phi}(\mathbf{x},\mathbf{u})-\bm{\phi}(\hat{\mathbf{x}},\mathbf{u}),\mathbf{x}-\hat{\mathbf{x}}\rangle\leq\rho\|\mathbf{x}-\hat{\mathbf{x}}\|^{2}.$
(21)

######  Definition 3.

The function $\bm{\phi}(\cdot)$ is said to be quadratic inner-bounded if there
exist $\delta,\varphi\in\mathbb{R}$ such that
$\forall\>\mathbf{x},\hat{\mathbf{x}}\in\mathcal{D}$ and
$\forall\>u\in\mathcal{U}$ we have

$\displaystyle\|\bm{\phi}(\mathbf{x},\mathbf{u})-\bm{\phi}(\hat{\mathbf{x}},\mathbf{u})\|^{2}$
$\displaystyle\leq\delta\|\mathbf{x}-\hat{\mathbf{x}}\|^{2}+$
$\displaystyle\quad\varphi\langle\mathbf{x}-\hat{\mathbf{x}},\bm{\phi}(\mathbf{x},\mathbf{u})-\bm{\phi}(\hat{\mathbf{x}},\mathbf{u})\rangle.$
(22)

######  Remark.

Different from the Lipschitz constant $\gamma$, the constants $\rho$,
$\delta$, and $\varphi$ are not required to be positive. Hence, it is
beneficial for a less conservative design of observers.

In the following theorem, we propose sufficient conditions that incorporate
the $\mathcal{H}_{-}/\mathcal{H}_{\infty}$ framework, and conditions (21) and
(22) to demonstrate the existence of the matrix gain $L$.

######  Theorem 2.

Consider the constants $\rho$, $\delta$, and $\varphi$, the system in (2)
satisfying the conditions (21) and (22), the observer defined in (3) and the
residual generator in (4). If there exist identical positive definite matrices
$P$ and $Q$, filter gain matrix $L$, strictly positive scalars $\alpha$,
$\beta$, and $\\{\epsilon_{i}\\}_{i=1}^{4}$ such that the following two LMIs
hold:

$\displaystyle\begin{split}\left[\begin{matrix}\Omega_{1}&P\bar{E}_{w}+\bar{C}^{\top}\bar{F}_{w}&P+\frac{1}{2}(\epsilon_{2}\varphi-\epsilon_{1})I\\\
*&-\alpha^{2}I+\bar{F}_{w}^{\top}\bar{F}_{w}&0\\\
*&*&-\epsilon_{2}I\end{matrix}\right]\prec 0\end{split}$ (23)
$\displaystyle\begin{split}\left[\begin{matrix}\Omega_{2}&Q\bar{E}_{f}-\bar{C}^{\top}\bar{F}_{f}&Q+\frac{1}{2}(\epsilon_{4}\varphi-\epsilon_{3})I\\\
*&-\beta^{2}I+\bar{F}_{f}^{\top}\bar{F}_{f}&0\\\
*&*&-\epsilon_{4}I\end{matrix}\right]\prec 0\end{split}$ (24)

where
$\Omega_{1}=\bar{A}^{\top}P+P\bar{A}+\bar{C}^{\top}\bar{C}+(\epsilon_{1}\rho+\epsilon_{2}\delta)I$,
$\Omega_{2}=\bar{A}^{\top}Q+Q\bar{A}-\bar{C}^{\top}\bar{C}+(\epsilon_{3}\rho+\epsilon_{4}\delta)I$.
Then, we have i) the residual generator is stable, and ii) the observer
satisfies the robustness and sensitivity constraints (7) and (8).

The proof of Theorem 2 is presented in the Appendix. The matrices in (23) and
(24) are not linear in terms of $P$, $Q$ and $L$. However, if we set $Y=PL$,
then (23) and (24) become LMIs in $Y$. Once the problem is solved, we can
recover the matrix gain using $L=P^{-1}Y$.

###  V-C Relation between Lipschitz and one-sided Lipschitz Observer Design
for Nonlinear Systems

The relation between Lemma 1 and Theorem 2 is established in Theorem 3, which
indicates that the latter is less conservative than the former.

######  Theorem 3.

Assume the nonlinear function $\bm{\phi}(\mathbf{x},\mathbf{u})$ is Lipschitz
continuous with Lipschitz constant $\gamma$, and there exists the gain
matrices $L$, $P$, $Q$, non-negative scalars $\varepsilon_{1}$,
$\varepsilon_{2}$ such that the inequalities (19) and (20) hold. Then, there
exist real scalars $\rho$, $\delta$, $\varphi$, non-negative scalars
$\epsilon_{1}$, $\epsilon_{2}$, $\epsilon_{3}$, $\epsilon_{4}$, together with
the matrices $L$, $P$, and $Q$ such that the inequalities in (23) and (24) are
satisfied.

The proof of Theorem 3 can be found in the Appendix.

######  Remark.

One-sided Lipschitz continuity does not imply Lipschitz continuity, because
the Lipschitz condition is a two-sided inequality [22]. Hence, the converse of
Theorem 3 does not hold, which implies that Theorem 2 is less conservative
than Lemma 1.

Figure 1: Islanded droop-controlled AC microgrid test system [27].

Figure 2: Residual norms under a busbar fault at the PCC of each GFM using a
one-sided Lipschitz observer: (a) GFM $\\#1$ at time $t=4$, (b) GFM $\\#2$ at
$t=5$, (c) GFM $\\#3$ at $t=6$, and (d) GFM $\\#4$ at $t=7$. The faults are
cleared after $0.2$ seconds.

Figure 3: Residual norms under an actuator fault corresponding to input
$\omega_{ni}$ using a one-sided Lipschitz observer (left column) and Lipschitz
observer (right column): (a) $\&$ (b) GFM $\\#2$ at time $t=5$; (c) $\&$ (d)
GFM $\\#4$ at time $t=7$. The fault is cleared after $0.2$ seconds.

Figure 4: Residual norms under an actuator fault corresponding to input
$V_{ni}$ using a one-sided Lipschitz observer (left column) and Lipschitz
observer (right column): (a) $\&$ (b) GFM $\\#1$ at time $t=4$; (c) $\&$ (d)
GFM $\\#3$ at time $t=6$. The fault is cleared after $0.2$ seconds.

Figure 5: Residual norms under an inverter bridge fault (sudden efficiency
reduction) using a one-sided Lipschitz observer (left column) and Lipschitz
observer (right column): (a) $\&$ (b) GFM $\\#1$ at time $t=4$; (c) $\&$ (d)
GFM $\\#4$ at time $t=7$. The fault is cleared after $0.2$ seconds.

##  VI Numerical Tests

We evaluate our proposed observer- and residual-based strategy using the
islanded AC microgrid shown in Fig. 1. The microgrid has four GFMs, three
inductive branches, and four RL loads. The parameters of the microgrid, GFMs,
branches, and loads can be found in [27]. Droop control accommodates the
power-sharing interaction between the GFMs. We run the simulations using a
MacBook Pro 2019, 16 GB 2133 MHz LPDDR3 RAM, 1 TB hard disk drive, and 2.8 GHz
Intel Core i7 processor. The experiments were performed using MATLAB,
Simulink, and the YALMIP toolbox with the semidefinite programming solvers
LMILAB, MOSEK, SeDuMi, and SDPT3 to solve optimization problem described by
Lemma 1 and Theorem 2. The Lipschitz, one-sided Lipschitz, and quadratic
inner-boundedness constants are presented in Table I and computed according to
[24]. For a single type of fault, the fault simulations happen in this order:
at time $t=4$ for GFM #1, at time $t=5$ for GFM #2, at time $t=6$ for GFM #3,
and at time $t=7$ for GFM #4. All the faults are cleared after $0.2$ seconds.
The disturbances are modeled as uncorrelated white Gaussian noise on the
process and measurements equations [21]. The disturbance matrices are
$E_{w}=B$, and $F_{w}=D$ [21]. And the faults of other GFMs are also
considered disturbances or unknown inputs. The following experiments compare
our proposed approach based on the one-sided Lipschitz condition against the
state-of-the-art Lipschitz design.

Fig. 2 shows the response of the residual norm using a one-sided Lipschitz
observer when a busbar fault occurs at the PCC of all GFMs. The impedance of
the fault is 10% the nominal voltage magnitude of the busbar. This figure
shows that the four residuals are sensitive to the busbar fault with residuals
less than two units. A mentionable observation is that the residual for a
single GFM is robust against disturbances and the faults occurring at the PCC
of other GFMs. The fault location can be clearly identified because each
residual is sensitive to its corresponding busbar fault. The observer
accommodates the faults at other busbars as disturbances. Table II shows the
computational time of the one-sided Lipschitz observer only. The state-of-the-
art Lipschitz design was reported as infeasible by the optimization solvers.

Actuator faults considering the input signal $\omega_{ni}$ for GFMs #2 and #4,
and the input signal $V_{ni}$ for GFMs #1 and #3 are shown in Figs. 3 and 4
respectively. We notice that the behavior of the residual norms is very
similar for both the one-sided Lipschitz and Lipschitz observers. The residual
thresholds are a little less for the one-sided Lipschitz observer. Notice that
the residual norm of each GFM remains unaffected by the occurrence of the
faults in the other GFMs. In other words, the actuator faults are correctly
identified among the GFMs.

A similar situation occurs with the inverter bridge fault, as shown in Fig. 5.
The inverter bridge fault corresponds to an efficiency reduction of the
bridge. The residual norm is below the threshold during the absence of the
fault, and it stays above the threshold under the presence of the fault. The
location of the bridge faults is properly identified because the residual
norms are sensitive to the corresponding GFM. Table II shows that the
computational time required by the one-sided Lipschitz observer is almost
$12\%$ less than that of the Lipschitz observer.

TABLE I: Lipschitz, one-sided Lipschitz and quadratic inner-boundedness constants. GFM | Constants
---|---
# | $\gamma$ | $\rho$ | $\delta$ | $\varphi$
1 & 2 | 44.7488 | 22.3688 | -0.7493 | 2.3599
3 & 4 | 44.7488 | 22.3688 | -0.7535 | 2.3679
TABLE II: 5-fold average $\mu$, and standard deviation $\sigma$ of the required computational time of one-sided Lipschitz and Lipschitz observers. The Lipschitz observer can not be obtained under a three-phase fault, refer to the remark at the end of section VI. | One-sided Lipschitz | Lipschitz
---|---|---
Fault type | $\mu$ | $\sigma$ | $\mu$ | $\sigma$
| (seconds) | (seconds) | (seconds) | (seconds)
Three-phase | 1144 | 7.77 | – | –
$\omega_{ni}$ | 1736 | 3.85 | 4076 | 14.89
$V_{ni}$ | 1472 | 3.11 | 3784 | 12.76
Inverter bridge | 1686 | 6.91 | 1916 | 6.02

Figure 6: Residual norms of GFM #1 using a one-sided Lipschitz observer in an
AC microgrid with technology mix. (a) busbar fault, (b) $\omega_{ni}$ actuator
fault, (c) $V_{ni}$ actuator fault, and (d) inverter bridge fault. The faults
last $0.1$ seconds.

We incorporate an experiment to extend the practical contributions of this
work, and to evaluate the feasibility of the proposed method for real-world
implementation. First, we modify the technology mix of the microgrid by
replacing GFM #2 and GFM #4 with a grid-following inverter (GFL) and a fourth-
order synchronous machine. The dynamic equations of the GFL and parameters can
be found in [29]. Similarly, the synchronous machine (SM) parameters and model
with turbine-governor and exciter control are found in [30, 31, 32]. Second,
the droop control of the remaining GFMs #1 and #3 is transformed into its
equivalent virtual synchronous machine control, as described in the
supplementary material of [33]. Third, the GFMs incorporate a distributed
secondary layer control for the voltage magnitude ($V_{ni}$) and frequency
($\omega_{ni}$), as described in [29]. Fig. 6 presents the response of the
residual norm of GFM #1 under four types of faults, which are cleared after
$0.1$ seconds. Although the maximum values of some residual norms and
thresholds have changed, all the faults can be clearly detected.

##  VII Discussion

We identify a possible limitation of the state-of-the-art Lipschitz observer
design. For this particular design, notice that the squared Lipschitz constant
in the block matrices $\Omega_{1}$ and $\Omega_{2}$ forces the positive
definiteness of the LMIs in (19) and (20). Hence, the feasible set for the
matrix gain $L$ becomes more restricted than the counterpart of the one-sided
Lipschitz observer. The situation is further aggravated by the magnitude of
the Lipschitz constant, which is approximately 44. The feasibility issue
becomes noticeable for the busbar faults. The four solvers report the
optimization problem described by Lemma 1 as infeasible. On the other hand,
the one-sided Lipschitz and quadratic inner-boundedness constants are not
squared in their corresponding constraints and are at least half of the
Lipschitz constant, as seen in Table I.

We acknowledge that a drawback of either the Lipschitz or the one-sided
Lipschitz design is that a single fault may perturb the residual norms of
different observers. In such a situation, we suggest using a bank of observers
to find triggering patterns among the observers to avoid false alarms. Another
solution is to include the matrix expressions $E_{f}$, $F_{f}$ of other faults
into the disturbance matrix expressions $E_{w}$, $F_{w}$ of the fault under
analysis. These solutions are beyond the scope of this work and are left for
future work. Despite the drawbacks, we clearly demonstrate that our proposed
approach performs better than the state-of-the-art Lipschitz design regarding
trustworthy detection and computational time. Moreover, our proposed approach
uses the available signals in a typical grid-forming inverter design. Hence,
our approach does not require additional sensors yielding a cost-effective
solution.

##  VIII Conclusion

This work introduces an observer-based fault detection scheme for grid-forming
inverters operating in islanded AC microgrids. We derive the matrix
expressions of faults and disturbances affecting GFMs and consider them in the
observer design process. The nonlinear model of the GFM is expressed as a one-
sided Lipschitz formulation. We pose a set of LMI constraints and
$\mathcal{H}_{-}/\mathcal{H}_{\infty}$ optimization to design an observer that
achieves sensitivity to faults and robustness against disturbances. The
proposed observer design is compared with the state-of-the-art design based on
the Lipschitz condition. The relation between the Lipschitz and the one-sided
Lipschitz observer is studied theoretically and experimentally, showing that
the latter allows for a less-restrictive observer design with less
computational time. We showcase that the state-of-the-art design leads to an
infeasible optimization problem under a three-phase fault. Our proposed
approach is evaluated using an islanded droop-controlled microgrid with four
GFMs. The numerical tests corroborate our proposed approach’s effectiveness
and feasibility of real-world implementation.

###  -A Proof of Theorem 2

###### Proof.

Let us choose the following candidate Lyapunov function
${V}=\mathbf{e}^{\top}P\mathbf{e}$ with $P>0$ to prove the internal stability
of the residual generator (4) in the fault-free case. The time derivative of
$V$ is:

$\displaystyle\dot{V}$
$\displaystyle=\mathbf{e}^{\top}(\bar{A}^{\top}P+P\bar{A})\mathbf{e}+\mathbf{e}^{\top}P\bar{E}_{w}\mathbf{w}+\mathbf{w}^{\top}\bar{E}_{w}^{\top}P\mathbf{e}$
$\displaystyle\quad+\Phi^{\top}P\mathbf{e}+\mathbf{e}^{\top}P\Phi$ (25)

For any strictly positive scalars $\epsilon_{1}$ and $\epsilon_{2}$, the
inequalities in (21) and (22) can be rewritten as:

$\displaystyle\epsilon_{1}\left(\rho\mathbf{e}^{\top}\mathbf{e}-\Phi^{\top}\mathbf{e}\right)$
$\displaystyle\geq 0$ (26)
$\displaystyle\epsilon_{2}\left(\delta\>\mathbf{e}^{\top}\mathbf{e}+\varphi\>\mathbf{e}^{\top}\Phi-\Phi^{\top}\Phi\right)$
$\displaystyle\geq 0$ (27)

Adding (26), and (27) to the right-hand side of (25), we have:

$\displaystyle\begin{split}\dot{V}\leq\mathbf{e}^{\top}(\bar{A}^{\top}P+P\bar{A})\mathbf{e}+\mathbf{e}^{\top}P\bar{E}_{w}\mathbf{w}+\mathbf{w}^{\top}\bar{E}_{w}^{\top}P\mathbf{e}\\\
+\Phi^{\top}P\mathbf{e}+\mathbf{e}^{\top}P\Phi+\epsilon_{1}\rho\mathbf{e}^{\top}\mathbf{e}-\epsilon_{1}\Phi^{\top}\mathbf{e}\\\
+\epsilon_{2}\delta\>\mathbf{e}^{\top}\mathbf{e}+\epsilon_{2}\varphi\>\mathbf{e}^{\top}\Phi-\epsilon_{2}\Phi^{\top}\Phi\end{split}$
(28)

Rewriting (28) as in a matrix inequality:

$\displaystyle\dot{V}\leq\left[\begin{matrix}\mathbf{e}\\\ \mathbf{w}\\\
\mathbf{\Phi}\end{matrix}\right]^{\top}\underbrace{\scalebox{0.6}{\mbox{$\displaystyle\left[\begin{matrix}(\bar{A}^{\top}P+P\bar{A})&P\bar{E}_{w}&P+\frac{1}{2}(\epsilon_{2}\varphi-\epsilon_{1})I\\\
(\epsilon_{1}\rho+\epsilon_{2}\delta)I&&\\\ &&\\\ \bar{E}_{w}^{\top}P&0&0\\\
&&\\\
P+\frac{1}{2}(\epsilon_{2}\varphi-\epsilon_{1})I&0&-\epsilon_{2}I\end{matrix}\right]$}}}_{\text{${\mathbf{M}}$}}\left[\begin{matrix}\mathbf{e}\\\
\mathbf{w}\\\ \mathbf{\Phi}\end{matrix}\right]$ (29)

Hence, $\dot{V}<0$ if $M<0$. Considering that $V>0$ and $\mathbf{r}_{w}$ as in
(9), we can satisfy constraint (7) rewriting it as:

$\displaystyle\int_{0}^{t}\|\mathbf{r}_{w}\|_{2}^{2}\>dt-\int_{0}^{t}\alpha^{2}\>\|\mathbf{w}\|_{2}^{2}\>dt+V$
$\displaystyle\leq 0$
$\displaystyle\int_{0}^{t}\|\mathbf{r}_{w}\|_{2}^{2}\>dt-\int_{0}^{t}\alpha^{2}\>\|\mathbf{w}\|_{2}^{2}\>dt+\int_{0}^{t}\dot{V}\>dt\leq
0$
$\displaystyle\int_{0}^{t}\left(\mathbf{r}_{w}^{\top}\mathbf{r}_{w}-\alpha^{2}\>\mathbf{w}^{\top}\mathbf{w}+\dot{V}\right)\>dt$
$\displaystyle\leq 0$
$\displaystyle\mathbf{r}_{w}^{\top}\mathbf{r}_{w}-\alpha^{2}\>\mathbf{w}^{\top}\mathbf{w}+\dot{V}$
$\displaystyle\leq 0$
$\displaystyle-\alpha^{2}\>\mathbf{w}^{\top}\mathbf{w}+\mathbf{e}^{\top}\bar{C}^{\top}\bar{C}\mathbf{e}+\mathbf{e}^{\top}\bar{C}^{\top}\bar{F}_{w}\mathbf{w}$
$\displaystyle+\mathbf{w}^{\top}\bar{F}_{w}\bar{C}\mathbf{e}+\mathbf{w}^{\top}\bar{F}_{w}^{\top}\bar{F}_{w}\mathbf{w}+\dot{V}$
$\displaystyle\leq 0$ (30)

Assuming zero initial conditions, combining (30) with (29) we have:

$\displaystyle\dot{V}\leq\left[\begin{matrix}\mathbf{e}\\\ \mathbf{w}\\\
\mathbf{\Phi}\end{matrix}\right]^{\top}\underbrace{\scalebox{0.6}{\mbox{$\displaystyle\left[\begin{matrix}(\bar{A}^{\top}P+P\bar{A})+\bar{C}^{\top}\bar{C}&P\bar{E}_{w}+\bar{C}^{\top}\bar{F}_{w}&P+\frac{1}{2}(\epsilon_{2}\varphi-\epsilon_{1})I\\\
(\epsilon_{1}\rho+\epsilon_{2}\delta)I&&\\\ &&\\\
\bar{E}_{w}^{\top}P+\bar{F}_{w}\bar{C}&-\alpha^{2}I+\bar{F}_{w}^{\top}\bar{F}_{w}&0\\\
&&\\\
P+\frac{1}{2}(\epsilon_{2}\varphi-\epsilon_{1})I&0&-\epsilon_{2}I\end{matrix}\right]$}}}_{\text{${\mathbf{M}^{*}}$}}\left[\begin{matrix}\mathbf{e}\\\
\mathbf{w}\\\ \mathbf{\Phi}\end{matrix}\right]$ (31)

If ${\mathbf{M}^{*}}<0$, then $\dot{V}<0$ which means that the residual
generator is asymptotically stable in the Lyapunov sense. Therefore, the
inequality ${\mathbf{M}^{*}}<0$ is equivalent to (23).

Similarly, defining the Lyapunov function $Y=\mathbf{e}^{\top}Q\mathbf{e}$
with $Q>0$ to demonstrate the internal stability of (4) in the disturbance-
free case:

$\displaystyle\begin{split}\dot{Y}=\mathbf{e}^{\top}(\bar{A}^{\top}P+P\bar{A})\mathbf{e}+\mathbf{e}^{\top}P\bar{E}_{f}\mathbf{f}+\mathbf{f}^{\top}\bar{E}_{f}^{\top}P\mathbf{e}\\\
+\Phi^{\top}P\mathbf{e}+\mathbf{e}^{\top}P\Phi\end{split}$ (32)

Adding (26), and (27) with strictly positive scalars $\epsilon_{3}$ and
$\epsilon_{4}$ to the right-hand side of (32):

$\displaystyle\begin{split}\dot{Y}\leq\mathbf{e}^{\top}(\bar{A}^{\top}P+P\bar{A})\mathbf{e}+\mathbf{e}^{\top}P\bar{E}_{f}\mathbf{f}+\mathbf{f}^{\top}\bar{E}_{f}^{\top}P\mathbf{e}\\\
+\Phi^{\top}P\mathbf{e}+\mathbf{e}^{\top}P\Phi+\epsilon_{3}\rho\mathbf{e}^{\top}\mathbf{e}-\epsilon_{3}\Phi^{\top}\mathbf{e}\\\
+\epsilon_{4}\delta\>\mathbf{e}^{\top}\mathbf{e}+\epsilon_{4}\varphi\>\mathbf{e}^{\top}\Phi-\epsilon_{4}\Phi^{\top}\Phi\end{split}$
(33)

And proceeding in a similar way as above for constraint (8) with Schur
complements and mathematical manipulations yield the matrix inequality in
(24). ∎

###  -B Proof of Theorem 3

###### Proof.

The Lipschitz condition implies the one-sided Lipschitz, and the quadratic
inner-boundedness conditions [22]. Using the Cauchy-Schwarz inequality and the
assumption that $\bm{\phi}(\mathbf{x},\mathbf{u})$ is Lipschitz continuous in
$\mathcal{D}\times\mathcal{U}$, we have:

$\displaystyle\lvert\langle\bm{\phi}(\mathbf{x},\mathbf{u})-\bm{\phi}(\hat{\mathbf{x}},\mathbf{u}),\mathbf{x}-\hat{\mathbf{x}}\rangle\rvert$
$\displaystyle\leq\|\bm{\phi}(\mathbf{x},\mathbf{u})-\bm{\phi}(\hat{\mathbf{x}},\mathbf{u})\|\|\mathbf{x}-\hat{\mathbf{x}}\|$
$\displaystyle\leq\gamma\|\mathbf{x}-\hat{\mathbf{x}}\|^{2}$

which is equivalent to:

$\displaystyle-\gamma\|\mathbf{x}-\hat{\mathbf{x}}\|^{2}\leq\langle\bm{\phi}(\mathbf{x},\mathbf{u})-\bm{\phi}(\hat{\mathbf{x}},\mathbf{u}),\mathbf{x}-\hat{\mathbf{x}}\rangle\leq\gamma\|\mathbf{x}-\hat{\mathbf{x}}\|^{2}$

According to Definition 2, the one-sided Lipschitz constant $\rho$ can be set
equal to $\pm\gamma$. Similarly, using Definition 3 we have that:

$\displaystyle\|\bm{\phi}(\mathbf{x},\mathbf{u})-\bm{\phi}(\hat{\mathbf{x}},\mathbf{u})\|^{2}$
$\displaystyle\leq\delta\|\mathbf{x}-\hat{\mathbf{x}}\|^{2}$
$\displaystyle+\varphi\langle\mathbf{x}-\hat{\mathbf{x}},\bm{\phi}(\mathbf{x},\mathbf{u})-\bm{\phi}(\hat{\mathbf{x}},\mathbf{u})\rangle$
$\displaystyle\leq\delta\|\mathbf{x}-\hat{\mathbf{x}}\|^{2}+\varphi\rho\|\mathbf{x}-\hat{\mathbf{x}}\|^{2}$
$\displaystyle\leq\left(\delta+\varphi\rho\right)\|\mathbf{x}-\hat{\mathbf{x}}\|^{2}\,.$

Based on Definition 1, let $\gamma^{2}=\left(\delta+\varphi\rho\right)$. Then,
we can obtain the value of $\delta$. Notice that the previous inequality is
valid if $\varphi>0$. By comparing (23) and (19), we establish the following
relations:

$\displaystyle\epsilon_{1}$ $\displaystyle=\varepsilon_{1}\varphi$
$\displaystyle\epsilon_{2}$ $\displaystyle=\varepsilon_{1}$
$\displaystyle\gamma^{2}$ $\displaystyle=\delta+\varphi\rho.$

Given that $\varphi>0$, then $\epsilon_{1}>0$. A similar analysis can be done
by using the entries of inequalities (24) and (20). ∎

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# Fundamental Limit on Angular Measurements and Rotations from Quantum
Mechanics and General Relativity

Xavier Calmet<EMAIL_ADDRESS>Department of Physics and Astronomy,
University of Sussex, Brighton, BN1 9QH, United Kingdom  Stephen D. H. Hsu
<EMAIL_ADDRESS>Department of Physics and Astronomy
Michigan State University

###### Abstract

We show that the precision of an angular measurement or rotation (e.g., on the
orientation of a qubit or spin state) is limited by fundamental constraints
arising from quantum mechanics and general relativity (gravitational
collapse). The limiting precision is $r^{-1}$ in Planck units, where $r$ is
the physical extent of the (possibly macroscopic) device used to manipulate
the spin state. This fundamental limitation means that spin states $S_{1}$ and
$S_{2}$ cannot be experimentally distinguished from each other if they differ
by a sufficiently small rotation. Experiments cannot exclude the possibility
that the space of quantum state vectors (i.e., Hilbert space) is fundamentally
discrete, rather than continuous. We discuss the implications for finitism:
does physics require infinity or a continuum?

today

##  I Introduction: Fundamental limits on measurement

Gedanken experiments can reveal fundamental limitations on measurements or
other experimental procedures that arise from the laws of physics, see e.g.
Garay:1994en ; Mead:1964zz ; Padmanabhan:1987au ; Baez:2002ra ; Ng:1999hm ;
Hossenfelder:2012jw ; Gibbons:2002iv . The best known example of this is the
Heisenberg uncertainty principle, which is motivated by elementary
considerations of particle scattering. Heuristic arguments suggested that
localization of a particle in position space would inevitable contribute to
uncertainty in its momentum, and vice versa. It was only later that the formal
theory of quantum mechanics incorporated this uncertainty in the form of
operator commutation relations $[x,p]=i\hbar$.

More recently, it has been shown that discreteness of space-time on length
scales smaller than the Planck length cannot be detected due to limitations on
measuring devices which arise from quantum mechanics and general relativity
Calmet:2004mp ; Calmet:2005mh ; Calmet:2007vb . This suggests, but does not
prove, that models of quantum gravity that are consistent with what is
currently known about low energy (long distance) physics will incorporate
minimal length in some fundamental way. Examples are models which incorporate
a minimal length via a generalized uncertainty principle Maggiore:1993kv ;
Kempf:1994su ; Scardigli:1999jh ; Scardigli:2003kr  or even string theory.

In this letter we deduce the existence of minimal rotations which are
analogous to minimal length. Specifically, we deduce limits on the precision
with which a rotation can be applied (e.g., to a microscopic state), or a
device can measure spin orientation. These results suggest a fundamental
discreteness in the structure of Hilbert space itself BHZ1 ; BHZ2 .

The result described here was first obtained in BHZ1  using minimal length
(the Planck length) as an input assumption. Consider the rotation of a
macroscopic device of size $r$. If the angle of rotation is sufficiently small
no part of the device is displaced by more than the minimal length, and the
device is not distinguishable from its unrotated configuration. Thus rotations
smaller than $r^{-1}$ in Planck units cannot be realized and measurements with
better than this precision cannot be performed.

Below we give a more complete derivation of the result: we consider the
angular displacement operator $\varphi(t)-\varphi(0)$ and examine limits on
related experimental procedures. This is analogous to the approach used in
Calmet:2004mp  to deduce minimal length.

##  II Angular measurements and rotations

In this section, we show that quantum mechanics and general relativity
considered simultaneously imply the existence of a bound on the precision of
the measurement of an angle: i.e., no operational procedure exists which can
measure an angle less than this fundamental angle. The key ingredients used to
reach this conclusion are the uncertainty principle from quantum mechanics,
and gravitational collapse from general relativity in the form of the hoop
conjecture.

A dynamical condition for gravitational collapse is given by the Hoop
Conjecture hoop : if an amount of energy $E$ is confined at any instant to a
ball of size $R$, where $R<G_{N}E$ ($G_{N}$ is Newton’s constant), then that
region will eventually evolve into a black hole. Recent theoretical results
Penrose74 ; D'Eath:1992hb ; D'Eath:1992hd ; D'Eath:1992qu ; Eardley:2002re ;
Hsu:2002bd ; Calmet:2014dea  support the Hoop Conjecture : even the scattering
of extremely high energy particles cannot avoid black hole formation if their
impact parameter is less than of order $R$ given above. Henceforth, we use
natural units where $\hbar,c$ and Newton’s constant (or $l_{P}$) are unity. We
also sometimes neglect numerical factors of order one.

Let us consider the Hamiltonian for a free particle on a circle of radius $r$.
The Hamiltonian is given by

$\displaystyle H=\frac{p_{\varphi}^{2}}{2mr^{2}},$ (1)

where $p_{\varphi}$ is the angular momentum. The angular position of the
particle is given by the classical equation of motion

$\displaystyle\dot{\varphi}=\frac{\partial H}{\partial p_{\varphi}}=\omega,$
(2)

which admits the solution

$\displaystyle\varphi(t)=\omega t+\varphi(0).$ (3)

Passing to quantum mechanics, we can calculate the commutator of $\varphi(t)$
and $\varphi(0)$ and obtain

$\displaystyle[\varphi(0),\varphi(t)]=i\frac{t}{mr^{2}},$ (4)

where $t$ is the time between the two measurements. From this we obtain

$\displaystyle|\Delta\varphi(0)||\Delta\varphi(t)|\geq\frac{t}{2mr^{2}}.$ (5)

As measuring an angle requires two measurements, it is limited by the greater
of $\Delta\varphi(0)$ and $\Delta\varphi(t)$, which is at least
$(t/2mr^{2})^{1/2}$. At first sight, it appears that one could make this
difference arbitrarily small by making $m$ very large. However, in order to
avoid gravitational collapse, the size $r$ of the object must scale
proportionally to $m$ so that $r>m$ (the inequality holds up to factors of
order one). By causality $r$ cannot exceed $t$. We thus find

$\displaystyle|\Delta\varphi(t)|\geq\frac{l_{P}}{\sqrt{2}r},$ (6)

where, leaving natural units momentarily, $l_{P}$ is the Planck length given
by $\sqrt{\hbar G_{N}/c^{3}}\sim 1.6\times 10^{-35}$ m. The uncertainty in the
measurement of $\varphi$ can be reduced by making $r$ large, but it cannot
approach zero without taking the rotational inertia of the device to infinity.
Of course, more practical limitations resulting from material properties
(e.g., causal bounds on rigidity) will intervene before this limit can be
taken.

We neglected interactions between the experimental apparatus and the region of
the universe outside the causal region of the measurement. The possibility of
interactions, and a more complicated Hamiltonian, were considered in
Calmet:2004mp  and shown not to alter the conclusions.

Our result can be interpreted as a proof of the existence of a minimal angle
in full analogy to the minimal length related to the Planck mass. This minimal
angle could be called the Planck angle. Basically, it implies that no
operational procedure can exclude the quantization of space-time for distances
or angles less than the Planck length or the Planck angle. We emphasize that
our result does not rely on a discretization of space-time; the angular
evolution of the object on the circle is given by standard quantum mechanics
on a continuous space-time. However, given the existence of a minimal length,
one could speculate that space-time has a short distance (high momentum)
cutoff, and thus physics in a bounded region of space-time is described by a
finite dimensional Hilbert space. We have shown that no experiment could rule
this out.

We note that canonical commutation relations such as $[x,p]=i\hbar$ cannot be
realized in a finite dimensional Hilbert space. This is easy to verify by
taking the trace of both sides of the equation: the left hand side is
traceless but the right hand side has trace proportional to the dimensionality
finite . However, this is primarily a technical issue because finite
dimensional quantum systems continue to obey the usual uncertainty relations.
For example, a wavepacket state realized in a discrete and finite setting
(e.g., a space-time lattice with finite volume) cannot reduce its position
uncertainty $\Delta x$ arbitrarily without increasing the corresponding
momentum uncertainty $\Delta p$, and vice versa. It is sometimes argued
(erroneously) that quantum mechanics must have an infinite dimensional Hilbert
space because of the position-momentum commutation relation. However in fact
what is really known about quantum physics from direct observation is not the
commutation relation itself but the uncertainty relation that it encodes. The
uncertainty relation persists in a finite dimensional version of quantum
mechanics (e.g., the Schrödinger evolution of states on a discrete space-time
lattice).

##  III Superpositions and Spin States

The results of the previous section limit the precision with which we can
measure or manipulate the state of a single qubit – the orientation of a spin:

$|\psi\rangle=\cos\theta\,|+\rangle~{}+~{}e^{i\phi}\sin\theta\,|-\rangle~{}~{}~{}.$
(7)

Limits on the precision of $(\theta,\phi)$ are actually limits on the
precision of Hilbert space (or state vector) coefficients. These limits,
together with the Planck length short distance cutoff, imply that no
experiment can exclude the possibility that Hilbert space is discrete and
finite dimensional, i.e.,

$|\psi\rangle=\sum c_{n}\,|n\rangle~{}~{}~{}$ (8)

where 1. the values of the coefficients $c_{n}$ are only defined to some
finite accuracy – they are not continuous complex parameters, and 2. the sum
itself is finite.

Physicists who simulate quantum phenomena on classical computers are already
familiar with properties 1 and 2. What we describe above as fundamental
consequences for quantum mechanics arising from gravity are approximations
made out of necessity in everyday computation. Any quantum phenomenon that
can, in principle, be reproduced to satisfactory approximation in computer
simulation is entirely consistent with properties 1 and 2.

We can also pursue a quantum information approach to these questions. Consider
an experiment which takes place in a space-time region of extent $r$. Given
the short distance cutoff at the Planck length, $l_{p}$, the number of degrees
of freedom relevant to the experiment is itself finite. For a given $r$, the
number of distinct configurations of the experimental apparatus (i.e., the
number of distinct quantum operators represented by the possible measurements)
is bounded above. Thus the number of distinct spin orientations (qubit states
which are eigenstates of the measurement operator) that can be resolved is
also bounded above. Physics can therefore be described by a discretized
Hilbert space in which the angles $(\theta,\phi)$ are discrete and take on
only a finite (but presumably very large) number of values.

Holography (another aspect of quantum gravity) provides a stronger bound on
the scale of discreteness: the total entropy of the measurement apparatus is
bounded above (i.e., limiting its configurations and accuracy of read out) by
the boundary area rather than the volume of the region in Planck units.

We can illustrate the connection to holography BHZ1  using a composite state
built from many qubits:

$\Psi=\psi_{1}\otimes\psi_{2}\otimes\psi_{3}\otimes\cdots\otimes\psi_{n}~{}.$
(9)

Assume a fundamental uncertainty $\epsilon$ in the state of each of the
qubits, so that $\psi$ and $\psi^{\prime}$ are indistinguishable when
$|\psi-\psi^{\prime}|<\epsilon$. Now consider the composite state

$\Psi^{\prime}=\psi^{\prime}_{1}\otimes\psi^{\prime}_{2}\otimes\psi^{\prime}_{3}\otimes\cdots\otimes\psi^{\prime}_{n}~{}~{}~{}.$
(10)

If uncertainties $\epsilon$ for each qubit are independent, the resulting
uncertainty in the composite state $\Psi$ is

$|\Psi-\Psi^{\prime}|^{2}\sim n\epsilon^{2}$ (11)

and requiring that this be less than unity implies the holographic bound

$n~{}<~{}\epsilon^{-2}~{}\sim~{}r^{2}$ (12)

where $r$ is the size of the system. In other words, the requirement of a
small aggregate uncertainty in the composite state $\Psi$ due to the
individual qubit uncertainties $\epsilon$ is equivalent to the holographic
upper bound on the entropy or number of degrees of freedom $n$ in a finite
region of space of size $r$.

To summarize, the observation that only a finite amount of quantum information
can be encoded in a finite region provides an upper limit on the precision of
a measurement conducted in that region. However, we can go further: the
measured state of a qubit can only be determined to some limited precision,
and this is consistent with a finite (rather than infinite, as is usually
assumed) set of possible orientations $(\theta,\phi)$ for each qubit in the
universe.

##  IV Finitism: Does Physics require a Continuum?

Our intuitions about the existence and nature of a continuum arise from
perceptions of space and time Weyl . But the existence of a fundamental Planck
length suggests that space-time may not be a continuum. In that case, our
intuitions originate from something (an idealization) that is not actually
realized in Nature.

Quantum mechanics is formulated using continuous structures such as Hilbert
space and a smoothly varying wavefunction, incorporating complex numbers of
arbitrary precision. However beautiful these structures may be, it is possible
that they are idealizations that do not exist in the physical world. The
introduction of gravity limits the precision necessary to formulate a model of
fundamental quantum physics. Indeed, any potential structure smaller than the
Planck length or the minimal angle considered here cannot be observed by any
device subject to quantum mechanics, general relativity, and causality. Our
results suggest that quantum mechanics combined with gravity does not require
a continuum, nor any concept of infinity.

It may come as a surprise to physicists that infinity and the continuum are
even today the subject of debate in mathematics and the philosophy of
mathematics. Some mathematicians, called finitists, accept only finite
mathematical objects and procedures Ye . The fact that physics does not
require infinity or a continuum is an important empirical input to the debate
over finitism. For example, a finitist might assert (contra the Platonist
perspective adopted by many mathematicians) that human brains built from
finite arrangements of atoms, and operating under natural laws (physics) that
are finitistic, are unlikely to have trustworthy intuitions concerning
abstract concepts such as the continuum. These facts about the brain and about
physical laws stand in contrast to intuitive assumptions adopted by many
mathematicians. For example, Weyl (Das Kontinuum Weyl ; Feferman ) argues that
our intuitions concerning the continuum originate in the mind’s perception of
the continuity of space-time.

There was a concerted effort beginning in the 20th century to place infinity
and the continuum on a rigorous foundation using logic and set theory. As
demonstrated by Gödel, Hilbert’s program of axiomatization using finitary
methods (originally motivated, in part, by the continuum in analysis) could
not succeed. Opinions are divided on modern approaches which are non-finitary.
For example, the standard axioms of Zermelo-Fraenkel (ZFC) set theory applied
to infinite sets lead to many counterintuitive results such as the Banach-
Tarski Paradox: given any two solid objects, the cut pieces of either one can
be reassembled into the other BT . When examined closely all of the axioms of
ZFC (e.g., Axiom of Choice) are intuitively obvious if applied to finite sets,
with the exception of the Axiom of Infinity, which admits infinite sets.
(Infinite sets are inexhaustible, so application of the Axiom of Choice leads
to pathological results.) The Continuum Hypothesis, which proposes that there
is no cardinality strictly between that of the integers and reals, has been
shown to be independent (neither provable nor disprovable) in ZFC cohen .
Finitists assert that this illustrates how little control rigorous mathematics
has on even the most fundamental properties of the continuum.

David Deutsch Deutsch :

> The reason why we find it possible to construct, say, electronic
> calculators, and indeed why we can perform mental arithmetic, cannot be
> found in mathematics or logic. The reason is that the laws of physics
> “happen to” permit the existence of physical models for the operations of
> arithmetic such as addition, subtraction and multiplication.

This suggests the primacy of physical reality over mathematics, whereas
usually the opposite assumption is made. From this perspective, the parts of
mathematics which are simply models or abstractions of “real” physical things
are most likely to be free of contradiction or misleading intuition. Aspects
of mathematics which have no physical analog (e.g., infinite sets, the
continuum) are prone to problems in formalization or mechanization. Physics –
i.e., models which can be compared to experimental observation, actual
“effective procedures” – does not ever require infinity, although it may be of
some conceptual convenience. Hence it seems possible, and the finitists
believe, that the Axiom of Infinity and its equivalents do not provide a sound
foundation for mathematics.

Acknowledgments: The work of X.C. is supported in part by the Science and
Technology Facilities Council (grants numbers ST/T00102X/1, ST/T006048/1 and
ST/S002227/1).

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