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# Sensor Fault Detection and Isolation in Autonomous Nonlinear Systems Using
Neural Network-Based Observers

John Cao∗ M. Umar B. Niazi† Matthieu Barreau∗ Karl Henrik Johansson∗ $*$
Division of Decision and Control Systems, Digital Futures, KTH Royal Institute
of Technology, SE-100 44 Stockholm, Sweden. Emails<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Laboratory for Information & Decision
Systems, Department of Electrical Engineering and Computer Science,
Massachusetts Institute of Technology, Cambridge, MA 02139, USA. Email:
<EMAIL_ADDRESS>work is supported by the Swedish Research Council and the
Knut and Alice Wallenberg Foundation, Sweden. It has also received funding
from the European Union’s Horizon Research and Innovation Programme under
Marie Skłodowska-Curie grant agreement No. 101062523.

###### Abstract

This paper presents a novel observer-based approach to detect and isolate
faulty sensors in nonlinear systems. The proposed sensor fault detection and
isolation (s-FDI) method applies to a general class of nonlinear systems. Our
focus is on s-FDI for two types of faults: complete failure and sensor
degradation. The key aspect of this approach lies in the utilization of a
neural network-based Kazantzis-Kravaris/Luenberger (KKL) observer. The neural
network is trained to learn the dynamics of the observer, enabling accurate
output predictions of the system. Sensor faults are detected by comparing the
actual output measurements with the predicted values. If the difference
surpasses a theoretical threshold, a sensor fault is detected. To identify and
isolate which sensor is faulty, we compare the numerical difference of each
sensor meassurement with an empirically derived threshold. We derive both
theoretical and empirical thresholds for detection and isolation,
respectively. Notably, the proposed approach is robust to measurement noise
and system uncertainties. Its effectiveness is demonstrated through numerical
simulations of sensor faults in a network of Kuramoto oscillators.

##  I Introduction

Sensor fault detection and isolation (s-FDI) plays a pivotal role in ensuring
the safe and efficient operation of numerous industrial processes. We address
two distinct types of sensor faults: complete failure and sensor degradation.
Complete failure occurs when a sensor becomes entirely nonfunctional, often
due to mechanical breakdown or a similar issue. In contrast, sensor
degradation results in a gradual decline in the sensor’s measurement accuracy.
When left undetected, these sensor faults can lead to disruptive consequences
for the system. Effective s-FDI methods serve as a proactive solution,
allowing system operators to detect sensor faults early, localize the faulty
sensor, and take corrective action before they escalate into issues that could
result in costly damage or downtime.

Fault detection and isolation methods can generally be categorized into two
main groups: hardware redundancy and analytical redundancy, as discussed in
[1]. Hardware redundancy methods entail using multiple sensors to acquire and
cross-reference data from processes. While effective, this approach does come
with the drawback of increased costs for acquiring and maintaining additional
hardware. Analytical redundancy methods offer an alternative solution. These
methods rely on the principle of residual generation, where the residual
represents the variance between a predicted system output and the actual
measurement. In the absence of faults, this residual remains close to zero.
However, as soon as faults occur, it significantly exceeds a predetermined
threshold, facilitating the detection of anomalies without the need for
additional hardware.

Historically, s-FDI methods based on analytical redundancy have primarily been
model-based, relying on explicit mathematical models of the system under
consideration, as noted in [2]. This model-based approach was initially
developed for linear systems in the 1970s, exemplified by pioneering work such
as [3], which demonstrated the feasibility of designing filters for detecting
and localizing faults within observable system dynamics. Subsequent
refinements and enhancements, as seen in [4], resulted in the renowned “Beard-
Jones Fault Detection filter.” Further extensions and practical applications
of these methods can be found in works such as [5, 6, 7, 8, 9].

Parallel to these developments, the framework of observer-based fault
detection schemes emerged for linear systems, with an early reference being
[10]. Over time, this approach has gained recognition as one of the most
successful methods for s-FDI, leading to diverse research directions. For
instance, the application of sliding-mode observers for s-FDI by [11, 12, 13]
allowed explicit reconstruction of the sensor faults by manipulating the
output injection error. Nonlinear unknown input observers have also been
prevalent for s-FDI [2]. However, in recent years, interval-based unknown
input observers have gained significant prominence. These observers offer
advantages because of relaxed assumptions on system inputs and nonlinearities
[14, 15, 16, 17].

Another principal approach to analytical redundancy-based s-FDI is by using
data-driven methods, which have gained popularity over the past decade, thanks
to significant advancements in deep learning. These methods do not require
explicit system models, relying instead on sensor data to approximate the
underlying dynamics and generate residuals, as highlighted in [18]. For
instance, [19] demonstrated s-FDI using long short-term memory neural networks
(LSTM), where residuals were formed by comparing network predictions based on
past time-series data with actual measurements. Similar neural network-based
approaches have been proposed for various domains, including industrial
manufacturing processes, power plants, and unmanned aerial vehicles, as
indicated in [18, 20, 21].

Despite the rich diversity of existing s-FDI methods, it’s important to
acknowledge that, to the best of the authors’ knowledge, these techniques are
often challenging to implement in practice. Observer-based methods are
constrained by their assumptions about specific system structures, while data-
driven approaches require substantial sensor data, which can be both difficult
and costly to obtain.

In this paper, we tackle these challenges by introducing a novel learning-
based approach to s-FDI using neural network-based nonlinear Luenberger
observers, specifically Kazantzis-Kravaris/Luenberger (KKL) observers [22,
23]. KKL observers lift the original system to a higher-dimensional state
space, where the system behaves linearly and is stable. The nonlinear
transformation required by the lift is obtained by solving a certain partial
differential equation. The observer’s design is based on this transformed
system. To estimate the system’s state, an inverse transformation is applied,
to obtain the estimate in the original state space. A notable advantage of KKL
observers is their flexibility; they do not rely on specific triangular
structure or normal form of the system but rather depend on relatively mild
observability conditions, rendering them applicable to a broad class of
nonlinear systems.

Our main contributions in this paper are

  1. 1.

We develop a novel s-FDI algorithm using a neural network-based KKL observer
with no assumptions on the structure of the system.

  2. 2.

We derive a theoretical threshold for sensor fault detection based on the
residual, and also devise an empirical method to obtain a threshold for sensor
fault isolation.

  3. 3.

Through simulations, we show that the proposed method is able to effectively
detect and isolate sensor faults under a variety of circumstances while
remaining robust and functional under the influence of model uncertainties and
measurement noise.

The outline of this paper is as follows. Section II formulates the s-FDI
problem under some mild assumptions. Section III presents the proposed s-FDI
methhod. Numerical results are provided in Section IV where the method is
tested in simulations on a variety of fault cases. Lastly, Section V concludes
the paper.

##  II Problem Formulation

We consider a nonlinear system

$\displaystyle\dot{x}(t)$ $\displaystyle=f(x(t))+w(t)$ (1a) $\displaystyle
y(t)$ $\displaystyle=\phi(t)\left[h(x(t))+v(t)+\zeta(t)\right]$ (1b)

where $x(t)\in\mathcal{X}\subset\mathbb{R}^{n_{x}}$ is the state,
$y(t)\in\mathbb{R}^{n_{y}}$ is the output,
$f:\mathbb{R}^{n_{x}}\rightarrow\mathbb{R}^{n_{x}}$ and
$h:\mathbb{R}^{n_{x}}\rightarrow\mathbb{R}^{n_{y}}$ are smooth maps, and
$w(t),v(t)$ are process and measurement noises, respectively. The measured
output (1b) might be affected by sensor faults $\phi(t)$ and $\zeta(t)$, where

$\phi(t)=\begin{bmatrix}\phi_{1}(t)&\dots&\phi_{n_{y}}(t)\end{bmatrix}^{\mathtt{T}}\in\\{0,1\\}^{n_{y}}$

models the complete failure of sensor $i$ when $\phi_{i}(t)=0$, and

$\zeta(t)=\begin{bmatrix}\zeta_{1}(t)&\dots&\zeta_{n_{y}}(t)\end{bmatrix}^{\mathtt{T}}\in\mathbb{R}^{n_{y}}$

models the degradation of sensor $i$, affecting its measurement accuracy, when
$\zeta(t)\neq 0$.

Note that a system with fault-free sensors at time $t$ will have
$\phi(t)=1_{n_{y}}$ and $\zeta(t)=0_{n_{y}}$. Any other value of these signals
represents a type of fault, such as sensor bias or degradation when
$\zeta_{i}(t)\neq 0$, for $t\geq T$, or sensor failure when $\phi_{i}(t)=0$,
for $t\geq T$, where $i\in\\{1,\dots,n_{y}\\}$ and $T\in\mathbb{R}_{>0}$ is a
time at which fault occurs.

######  Assumption 1.

We assume the following regarding the process and measurement noise:

  1. (i)

Essential boundedness: $\|w\|_{L^{\infty}}\leq\bar{w}$ and
$\|v\|_{L^{\infty}}\leq\bar{v}$, where $\bar{w},\bar{v}>0$ are known and
$\|\cdot\|_{L^{\infty}}$ denotes the essential supremum norm.

  2. (ii)

Bounded effect: There exists a class $\mathcal{K}_{\infty}$ function
$\psi:\mathbb{R}\rightarrow\mathbb{R}_{\geq 0}$ such that

$\|x(t;x_{0},w)-x(t;x_{0},0)\|\leq\psi(\bar{w})$

where $x(t;x_{0},w)$ denotes the state trajectory of (1a) initialized at
$x(0)=x_{0}$ and driven by the noise $w(t)$, and $x(t;x_{0},0)$ denotes the
state trajectory in the absence of noise.

######  Remark 1.

Assumption 1(i) is a standard assumption in the robust state estimation
literature. It says that the signals $w(t)$ and $v(t)$ almost always remain
bounded, and the instances at which $w(t)$ or $v(t)$ go unbounded are of zero
Lebesgue measure. On the other hand, Assumption 1(ii) requires that the model
$f$ without a process noise does a good job of describing the noisy system
(1a). Such guarantees are usually provided in the system identification
literature, and the reader is referred to [24, 25] for more details.
$\diamond$

Given a system (1), a neural network-based Kazantzis-Kravaris/Luenberger (NN-
KKL) observer [26] is given by

$\displaystyle\dot{\hat{z}}(t)=A\hat{z}(t)+By(t)$ (2a)
$\displaystyle\hat{x}(t)=\hat{\mathcal{T}}_{\eta}^{*}(\hat{z}(t))$ (2b)
$\displaystyle\hat{y}(t)=h(\hat{x}(t))$ (2c)

which takes output measurements $y(t)$ as input and gives the estimated state
$\hat{x}(t)$ and predicted output $\hat{y}(t)$ as outputs. In (2a),
$A\in\mathbb{R}^{n_{z}\times n_{z}}$ is a Hurwitz matrix and
$B\in\mathbb{R}^{n_{z}\times n_{y}}$ is such that the pair $(A,B)$ is
controllable. The observer state $\hat{z}(t)\in\mathbb{R}^{n_{z}}$ follows a
nonlinear transformation $\hat{z}=\hat{\mathcal{T}}_{\theta}(\hat{x})$, where
$\hat{\mathcal{T}}_{\theta}$ is a neural network approximation of
$\mathcal{T}:\mathcal{X}\rightarrow\mathbb{R}^{n_{z}}$, which is an injective
transformation of the state space to a higher dimension
$n_{z}=n_{y}(2n_{x}+1)$. On the other hand, in (2b),
$\hat{\mathcal{T}}^{*}_{\eta}$ is a neural network approximation of
$\mathcal{T}^{*}:\mathbb{R}^{n_{z}}\rightarrow\mathcal{X}$, which is the left
inverse of $\mathcal{T}$. The maps $\hat{\mathcal{T}}_{\theta}$ and
$\hat{\mathcal{T}}^{*}_{\eta}$ are neural networks with parameters $\theta$
and $\eta$, respectively. Note that the observer is trained by considering
fault-free sensors. We refer to [26] for details on the design of NN-KKL
observers.

######  Assumption 2.

We assume the following regarding nonlinear transformations
$\mathcal{T},\mathcal{T}^{*}$ and their corresponding neural network
approximations $\hat{\mathcal{T}}_{\theta},\hat{\mathcal{T}}_{\eta}^{*}$:

  1. (i)

Lipschitzness: $\hat{\mathcal{T}}_{\theta}$ and $\hat{\mathcal{T}}_{\eta}^{*}$
are Lipschitz continuous over $\mathcal{X}$ and $\mathcal{Z}$, respectively,
where $\mathcal{Z}\supseteq\mathcal{T}(\mathcal{X})$. In particular, for every
$x,\hat{x}\in\mathcal{X}$, there exists $\ell_{\theta}\in\mathbb{R}_{>0}$ such
that

$\|\hat{\mathcal{T}}_{\theta}(x)-\hat{\mathcal{T}}_{\theta}(\hat{x})\|\leq\ell_{\theta}\|x-\hat{x}\|$

and, for every $z,\hat{z}\in\mathcal{Z}$, there exists
$\ell_{\eta}\in\mathbb{R}_{>0}$ such that

$\|\hat{\mathcal{T}}_{\eta}^{*}(z)-\hat{\mathcal{T}}_{\eta}^{*}(\hat{z})\|\leq\ell_{\eta}\|z-\hat{z}\|.$

  2. (ii)

Uniform injectivity: There exists a class $\mathcal{K}$ functions
$\rho,\rho^{*}$ such that, for every $x,\hat{x}\in\mathcal{X}$,

$\|x-\hat{x}\|\leq\rho(\|\mathcal{T}(x)-\mathcal{T}(\hat{x})\|)$

and, for every $z,\hat{z}\in\mathcal{Z}$,

$\|z-\hat{z}\|\leq\rho^{*}(\|\mathcal{T}^{*}(z)-\mathcal{T}^{*}(\hat{z})\|).$

######  Remark 2.

Assumption 2(i) is satisfied if the activation function of neural networks is
chosen to be Lipschitz continuous. Among others, one example is a neural
network with ReLU activation function, which is Lipschitz with the Lipschitz
constant equal to 1. On the other hand, Assumption 2(ii) is needed for the
existence of a KKL observer [23]. Notice that uniformly injective
$\mathcal{T}$ implies the uniform injectivity of its inverse
$\mathcal{T}^{*}$. It is important to remark that Assumption 2(i) and (ii)
imply boundedness of approximation errors, i.e.,

$\|\mathcal{T}(x)-\hat{\mathcal{T}}_{\theta}(x)\|<\infty,\;\forall
x\in\mathcal{X}$

and

$\|\mathcal{T}^{*}(z)-\hat{\mathcal{T}}_{\eta}^{*}(z)\|<\infty,\;\forall
z\in\mathcal{Z}$

where $\mathbb{R}^{n_{z}}\supset\mathcal{Z}\supseteq\mathcal{T}(\mathcal{X})$.
This holds because

$\|\mathcal{T}(x)-\hat{\mathcal{T}}_{\theta}(x)\|\leq\|\mathcal{T}(x)\|+\|\hat{\mathcal{T}}_{\theta}(x)\|\leq\rho^{*}(\|x\|)+\ell_{\theta}\|x\|$

and $\mathcal{X}\subset\mathbb{R}^{n_{x}}$ is a compact set. Further
improvement on this bound can be achieved using the universal approximation
property of neural networks given that an ample amount of data is generated
and an appropriate network architecture is chosen. $\diamond$

Observer-based s-FDI of nonlinear systems relies on the design of observers
that are accurate in state estimation and output prediction. However, for
general nonlinear systems, designing observers is a challenging task.
Therefore, we consider an NN-KKL observer (2) proposed in [26], which is
designed using a physics-informed learning approach. The focus of the present
paper is on developing an s-FDI method for general nonlinear systems using an
NN-KKL observer (2). Specifically, we address the following problems:

  1. (P1)

Detection: How to detect whenever a fault occurs in one or more sensors?

  2. (P2)

Isolation: How to identify a sensor whenever it becomes faulty?

To solve the problems stated above, we define residuals to compare a measured
output from the system with a predicted output from the NN-KKL observer. The
residual corresponding to $i$th sensor is defined as

$\begin{array}[]{ccl}r_{i}(t)&\doteq&|y_{i}(t)-\hat{y}_{i}(t)|\\\
&=&|\phi_{i}(t)[h_{i}(x(t))+v_{i}(t)+\zeta_{i}(t)]-h_{i}(\hat{x}(t))|.\end{array}$
(3)

Consider equally-distant discrete time samples $t_{0},t_{1},t_{2},\dots$ with
$\delta=t_{k}-t_{k-1}$ for $k\in\mathbb{N}$, then we define a differentiated
residual of $i$th sensor as the absolute value of numerical differentiation of
the residual, i.e.,

$\tilde{r}_{i}(t_{k})\doteq\left|\frac{r_{i}(t_{k})-r_{i}(t_{k-1})}{\delta}\right|$
(4)

Let $\tau_{i}\in\mathbb{R}_{>0}$ denote a threshold for residual $i$ such
that, in the steady state, $r_{i}(t)\leq\tau_{i}$ when there are no faults. In
addition, let $r_{\Delta}\in\mathbb{R}_{>0}$ denote a threshold for
differentiated residuals such that, in the steady state, $\tilde{r}_{i}(t)\leq
r_{\Delta}$ when sensor $i$ is not faulty.

##  III s-FDI using NN-KKL Observers

In this section, we first derive theoretical bounds for the sensor residuals,
which allow us to choose their thresholds. Then, we devise an empirical method
to compute the threshold for differentiated residuals. Finally, we present the
proposed s-FDI method.

###  III-A Upper bounds of the residuals

We derive two upper bounds on the residuals. The first bound is
straightforward but requires an additional stricter assumption that both
$\hat{\mathcal{T}}_{\theta}$ and $\hat{\mathcal{T}}_{\eta}^{*}$ are
contractions, i.e., their Lipschitz constants are less than 1. The second
bound relies on Assumption 1 and 2 and provides a bound that is more general.

######  Proposition 1.

Suppose the neural networks $\hat{\mathcal{T}}_{\theta}$ and
$\hat{\mathcal{T}}_{\eta}^{*}$ are Lipschitz continuous with
$\ell_{\theta}\in[0,1)$ and $\ell_{\eta}\in[0,1)$. Then, in a fault-less case,

$r_{i}(t)\leq\ell_{h_{i}}\frac{\ell_{\eta}\xi(x)+\xi^{*}(z)}{1-\ell_{\theta}\ell_{\eta}}+\bar{v}$
(5)

where $\ell_{h_{i}}$ is the Lipschitz constant of $h_{i}(x)$ for
$x\in\mathcal{X}$, $\bar{v}$ is given in Assumption 1(i), and

$\displaystyle\xi(x)$
$\displaystyle\doteq\mathcal{T}(x)-\hat{\mathcal{T}}_{\theta}(x)$ (6)
$\displaystyle\xi^{*}(z)$
$\displaystyle\doteq\mathcal{T}^{*}(z)-\hat{\mathcal{T}}_{\eta}^{*}(z)$ (7)

which are bounded for $x\in\mathcal{X}$ and $z\in\mathcal{Z}$ by Assumption
2(ii).

###### Proof.

See Appendix -A. ∎

######  Remark 3.

The assumption of Proposition 1 that both Lipschitz constants are less than
one can be satisfied by regularizing the weights of neural networks during
training. For instance, an $l$-layer ReLU network with
$W^{(1)},\dots,W^{(l)}$, which are the weight matrices of the neural network
layers, has a Lipschitz constant $\ell=\|W^{(1)}\|\dots\|W^{(l)}\|$. By
regularizing weights such that the maximum singular value of each weight
matrix is less than one, it can be ensured that $\ell<1$. Nevertheless, we
acknowledge that the upper bound (5) is quite conservative because if
$\ell_{\theta}\ell_{\eta}$ is close to one, the bound could become very large.
In this case, such a bound may not be appropriate for choosing a threshold.
$\diamond$

Recall $z=\mathcal{T}(x)$ and $\hat{z}=\hat{\mathcal{T}}_{\theta}(x)$, where

$\dot{z}(t)=Az(t)+Bh(\bar{x}(t))$

with $\bar{x}(t)\doteq x(t;x_{0},0)$ the noise-free state trajectory, and the
dynamics of $\hat{z}(t)$ is given in (2a).

######  Lemma 1.

Suppose the matrix $A$ is Hurwitz and diagonalizable with eigenvalue
decomposition $A=V\Lambda V^{-1}$. Then, the error
$\tilde{z}(t):=z(t)-\hat{z}(t)$ satisfies

$\begin{array}[]{l}\|\tilde{z}(t)\|\leq\|\tilde{z}_{0}\|\kappa(V)e^{-ct}\\\\[10.00002pt]
\qquad\qquad\displaystyle+\frac{\kappa(V)}{c}\|B\|(1-e^{-ct})\big{[}\ell_{h}\psi(\bar{w})+\sqrt{n_{y}}\bar{v}\big{]}\end{array}$

where $c$ is given in (16), $\psi$ in Assumption 1(ii), and $\ell_{h}$ is the
Lipschitz constant of $h(x)$ for $x\in\mathcal{X}$.

###### Proof.

See Appendix -B. ∎

Using the above lemmas, we can obtain a general upper bound on the residuals
under Assumption 1 and 2.

######  Proposition 2.

Suppose the matrix $A$ is diagonalizable with eigenvalue decomposition
$A=V\Lambda V^{-1}$. Then, in a fault-less case,

$\begin{array}[]{l}r_{i}(t)\leq\bar{v}+\ell_{h_{i}}\bigg{[}\xi^{*}(z)+\ell_{\eta}\kappa(V)e^{-ct}\|\tilde{z}_{0}\|\\\
\hskip
28.45274pt+\frac{\kappa(V)}{c}\|B\|(1-e^{-ct})(\ell_{h}\psi(\bar{w})+\sqrt{n_{y}}\bar{v})\bigg{]}\end{array}$
(8)

where $\bar{v}$ is from Assumption 1(i), $\ell_{h_{i}}$ is the Lipschitz
constant of $h_{i}(x)$ for $x\in\mathcal{X}$, $\xi^{*}(z)$ is defined in (7),
$\kappa(V)$ is the condition number of $V$, $c$ is given in (16),
$\tilde{z}_{0}=\tilde{z}(0)$, $\ell_{h}$ is the Lipschitz constant of $h(x)$
for $x\in\mathcal{X}$, and $\psi(\bar{w})$ is from Assumption 1(ii).

###### Proof.

From (12) and $\eqref{eq:prop_proof2}$, we have

$r_{i}(t)\leq\bar{v}+\ell_{h_{i}}\left[\xi^{*}(z)+\ell_{\eta}\|z-\hat{z}\|\right].$

Therefore, by applying Lemma 1, we obtain (8). ∎

###  III-B Fault detection by computing a theoretical threshold for the
residuals

After learning the NN-KKL observer on the training datasets
$\mathcal{X}_{\text{train}}\subset\mathcal{X}$ and
$\mathcal{Z}_{\text{train}}\subset\mathcal{Z}$, generate a test dataset
$\mathcal{X}_{\text{test}}\subset\mathcal{X}$ by simulating a noise-less
($w\equiv 0,v\equiv 0$) and fault-free ($\phi\equiv 1,\zeta\equiv 0$) system
(1a). Then, following the methodology presented in [26], we can generate
$\mathcal{Z}_{\text{test}}\approx\mathcal{T}(\mathcal{X}_{\text{test}})$,
where $\mathcal{Z}_{\text{test}}\subset\mathcal{Z}$. The datasets
$\mathcal{X}_{\text{test}}$ and $\mathcal{Z}_{\text{test}}$ are then used to
estimate the approximation errors incurred by $\hat{\mathcal{T}}_{\theta}$ and
$\hat{\mathcal{T}}_{\eta}^{*}$. Define

$\hat{\epsilon}\doteq\max_{x\in\mathcal{X}_{\text{test}},z\in\mathcal{Z}_{\text{test}}}\left\|z-\hat{\mathcal{T}}_{\theta}(x)\right\|$

and

$\hat{\epsilon}^{*}\doteq\max_{x\in\mathcal{X}_{\text{test}},z\in\mathcal{Z}_{\text{test}}}\left\|x-\hat{\mathcal{T}}_{\eta}^{*}(z)\right\|.$

Then, by relying on Proposition 2 for threshold computation, we obtain

$\displaystyle\limsup_{t\rightarrow\infty}r_{i}(t)$
$\displaystyle\qquad\leq\bar{v}+\ell_{h_{i}}\bigg{[}\xi^{*}(z)+\frac{\kappa(V)}{c}\|B\|(\ell_{h}\psi(\bar{w})+\sqrt{n_{y}}\bar{v})\bigg{]}.$

Therefore, in this case, we can choose the threshold to be

$\boxed{\tau_{i}=\bar{v}+\ell_{h_{i}}\left[\hat{\epsilon}^{*}+\frac{\kappa(V)}{c}\|B\|(\ell_{h}\psi(\bar{w})+\sqrt{n_{y}}\bar{v})\right].}$
(9)

Fault detection: At a steady state when $t$ becomes large, it holds that
$r_{i}(t)\leq\tau_{i}$ when there are no sensor faults. When the residual
$r_{i}(t)$ surpasses $\tau_{i}$, it signifies that the measured output is
significantly different from the predicted output. Since $\tau_{i}$ has been
computed using tight inequalities, sensor faults can be detected by measuring
the residuals $r_{i}(t)$ and raising an alert whenever $r_{i}(t)>\tau_{i}$ for
any $i$ and any $t\geq 0$.

###  III-C Fault isolation by computing an empirical threshold for
differentiated residuals

As described in the previous subsection, a sensor fault is detected whenever
the residual $r_{i}(t)$ surpasses the threshold $\tau_{i}$. However,
$r_{i}(t)$ surpassing $\tau_{i}$ and having the largest value among other
residuals do not mean that sensor $i$ is faulty111Assumptions about the
triangular structure of the system made in [12, 13, 27, 14, 15, 16, 17, 28]
would actually enable a simpler treatment because of easy decoupling between
the outputs. However, such an assumption holds for a restrictive class of
systems.. This is because a fault in sensor $j$ distorts the output
$y_{j}(t)$, which is filtered through the observer dynamics and then
transformed by $(h_{j}\circ\hat{\mathcal{T}}_{\eta}^{*})(\cdot)$. Since the
neural network $\hat{\mathcal{T}}_{\eta}^{*}$ is not diagonal, the fault in
sensor $j$ could induce a large residual $r_{i}(t)$ even when sensor $i$ is
not faulty. This phenomenon can also be observed in the third row of Fig. 2 in
Section IV. Because of the inter-dependencies due to non-diagonal
$\hat{\mathcal{T}}_{\eta}^{*}$, the transients after the occurrence of a fault
may persist above the threshold, leading to an inability to exactly isolate
the sensor faults from the residuals. Therefore, we propose fault isolation by
evaluating differentiated residuals $\tilde{r}_{i}(t_{k})$.

Based on our empirical observations, a spike is generated in the
differentiated residual $\tilde{r}_{i}(t_{k})$ whenever an abrupt fault
($\phi_{i}(t)=0$ or $\zeta_{i}(t)\neq 0$) is introduced in sensor $i$ at time
$t_{k}$. The explanation for this spike is straightforward because in the
differentiated residual, the fault in the output also gets numerically
differentiated. Therefore, an abrupt fault results in a spike in the
differentiated residual. However, when the fault signal $\zeta_{i}(t)$ is very
smooth and of low magnitude, it is difficult to isolate it. Nevertheless, it
is reasonable to assume faults to be irregular, non-smooth signals as compared
to a smooth, analytical signal.

Consider the test datasets $\mathcal{X}_{\text{test}}$ and
$\mathcal{Z}_{\text{test}}$, where we denote the state trajectories
$x^{j}(t_{k})\doteq x(t_{k};x_{0}^{j},0)$ that are initialized at $p$ initial
points $x(0)=x_{0}^{j}$, for $j=1,\dots,p$, and the corresponding observer
estimates as
$\hat{x}^{j}(t_{k})\doteq\hat{x}(t_{k};\hat{\mathcal{T}}_{\eta}^{*}(z_{0}^{j}))$.
We simulate the NN-KKL observer and compute the residuals
$r_{1}^{j}(t_{k}),\dots,r_{n_{y}}^{j}(t_{k})$, for $k=0,\dots,T$, where
$T\in\mathbb{N}$ and

$r_{i}^{j}(t_{k})=|y_{i}(t_{k};x^{j})-\hat{y}_{i}(t_{k};\hat{x}^{j})|.$

Here, $t_{T}>0$ is chosen large enough to allow the observer to converge to a
neighborhood of the original state. Then, the corresponding differentiated
residuals $\tilde{r}_{1}^{j}(t_{k}),\dots,\tilde{r}_{n_{y}}^{j}(t_{k})$, for
$k=1,\dots,T$, are obtained using (4). Let $t_{c}$, for $c<T$, denote the
minimum time at which the observer is in the steady state. Then an empirical
threshold for differentiated residuals is computed as follows:

$\boxed{r_{\Delta}=\max_{\begin{subarray}{c}k\in\\{c,\dots,T\\}\\\
j\in\\{1,\dots,p\\}\\\
i\in\\{1,\dots,n_{y}\\}\end{subarray}}\tilde{r}_{i}^{j}(t_{k}).}$ (10)

Fault isolation: Given that the learned NN-KKL observer estimates the system
state accurately, it holds that $\tilde{r}_{i}(t)\leq r_{\Delta}$ at a steady
state when $t_{k}$ becomes sufficiently large. Therefore, whenever the
differentiated residual $\tilde{r}_{i}(t)$ surpasses the empirical threshold
$r_{\Delta}$ given by (10), an alert can be raised that sensor $i$ is probably
faulty.

##  IV Simulation Results

Our approach is demonstrated in this section using numerical simulations. We
show that s-FDI of a highly nonlinear system can be achieved using NN-KKL
observers, even in the presence of additive process and measurement noises.
Both types of faults, sensor degradation represented by the fault signal
$\zeta_{i}(t)\neq 0$ and sensor failure represented by $\phi_{i}(t)=0$ in the
system output (1b), are demonstrated.

Figure 1: Estimated and true trajectories of states $\theta_{7}$ and
$\theta_{8}$ under the influence of process and sensor noise.

(a)

(b)

(c)

(d)

(e)

Figure 2: (a) Complete failure of sensor 4. (b) Degradation of sensor 1 and 5
at different time points with abrupt faults. (c) Smooth sigmoidal fault in
sensor 2. (d) Gradually increasing white noise in sensor 3. (e) Gradually
increasing sinusoidal signal in sensor 3.

###  IV-A Kuramoto Model

We consider the Kuramoto model for demonstrating our s-FDI method. The
Kuramoto model describes the phenomena of synchronization in a multitude of
systems, including electric power networks, multi-agent coordination, and
distributed software networks [29]. The dynamics of a network with $n$ nodes
are given as

$\dot{\theta_{i}}(t)=\omega_{i}+\sum_{j=1}^{n}a_{ij}\sin(\theta_{i}(t)-\theta_{j}(t))$
(11)

where $\theta_{i}(t)\in\mathbb{S}^{1}$ is the phase angle of node $i=1,..,n$,
$\omega_{i}\in\mathbb{R}$ is the natural frequency of node $i$, and
$a_{ij}\geq 0$ denotes the coupling between node $i$ and $j$. In the
literature, the state trajectories of (11) are often represented graphically
as $\sin(\theta_{i})$ in order to better illustrate their synchronization. We
follow the same convention in our simulations.

###  IV-B Experimental Setup

For (11), we consider a network of 10 nodes with randomly generated natural
frequencies $\omega_{i}$ and couplings $a_{ij}$. The measurements are chosen
as
$y=\begin{bmatrix}\theta_{1}&\theta_{2}&\theta_{3}&\theta_{4}&\theta_{5}\end{bmatrix}^{\mathtt{T}}$.
A set of 50 initial conditions is generated using Latin hypercube sampling,
with $\mathcal{X}_{\text{train}}\subset[-2,2]^{10}$. We choose Runge-Kutta-4
as our numerical ODE solver to simulate (11) and (2a) over a time interval of
$[0,30]$, partitioned into 4000 sample points for each trajectory. The neural
network $\hat{\mathcal{T}}^{*}_{\eta}$ is chosen as a fully connected feed-
forward network, consisting of 3 hidden layers of 250 neurons with ReLU
activation function. Model training is facilitated by data standardization and
learning rate scheduling. Following [30], the matrices of (2a) are chosen as

$A=\Lambda\otimes I_{n_{y}},\qquad B=\Gamma\otimes I_{n_{y}}$

where $\Lambda\in\mathbb{R}^{(2n_{x}+1)\times(2n_{x}+1)}$ is a diagonal matrix
with diagonal elements linearly distributed in $[-15,-21]$,
$\Gamma\in\mathbb{R}^{2n_{x}+1}$ is a column vector of ones, and $I_{n_{y}}$
is the identity matrix of size $n_{y}\times n_{y}$. Here, $n_{x}$ and $n_{y}$
are 10 and 5, respectively, and $n_{z}=n_{y}(2n_{x}+1)=105$.

The estimation capabilities of the observer under fault-free conditions are
demonstrated in Fig. 1, which shows the estimated and true trajectories of two
(randomly chosen) unmeasured states $\theta_{7}$ and $\theta_{8}$ over a time
interval of $[0,20]$, with noise terms $w(t),v(t)\sim\mathcal{N}(0,0.02)$. The
figure demonstrates that the estimation error is stable under noise and neural
network approximation error.

###  IV-C Numerical Results

We now apply the learned neural network-based KKL observer to perform s-FDI.
The theoretical thresholds $\tau_{i}$ of the residuals $r_{i}(t)$ computed by
(9) are shown in the third row of Fig. 2. On the other hand, the empirical
threshold $r_{\Delta}=4.74$ is computed using (10) according to the method
described in Section III-C. We choose $N=100$ initial conditions to create
$\mathcal{X}_{\text{test}}\in[-2,2]^{10}$, again using Latin hypercube
sampling to compute $r_{\Delta}$ using (10). Fig. 2(c)–2(e) demonstrate the
detection and isolation capabilities of our method under a variety of faults.
In the figure, the first, second, third and fourth rows correspond to the
measured and estimated state trajectories, the fault signal, the residuals
$r_{i}=|y_{i}-\hat{y}_{i}|$ and the finite difference approximation (4)
respectively.

In Fig. 2(a), we show that our method is capable of detecting sensor shutdowns
due to complete failure, which we demonstrate by modeling the fault in sensor
${4}$ with $\phi_{4}(t)=0$. Fig. 2(b) illustrates the situation when more than
one fault is present. Sensors ${1}$ and ${5}$ are disturbed by
$\zeta_{1}(t)=1$, at $t\geq 5$, and $\zeta_{5}(t)=1$, at $t\geq 15$. Each
fault is distinctly detectable at the moment of occurrence.

In Fig. 2(c), sensor ${2}$ is disturbed by a smooth sigmoidal fault term
$\zeta_{2}(t)$ introduced at $t=5$. Because of the smooth sigmoidal signal as
fault, the observer does not detect any anamoly and continues to follow the
measured state trajectories, thus generating a small residual. Although
introducing an abrupt fault as in Fig. 2(a) and 2(b) induces a transient in
the observer, causing a large residual to be generated, introducing a smooth
fault in Fig. 2(c) resulted in only detection but not isolation. Due to the
stability of the observer, it attempts to track the faulty trajectories after
the occurrence of the fault, leading to a decrease in the residual magnitudes
subsequently.

Fig. 2(d) and 2(e) illustrates the case when the fault signal on sensor $3$ is
introduced gradually. In Fig. 2(d), we simulate a fault in sensor $y_{3}$
which is a gradually increasing white noise. In Fig. 2(e), the fault is a
sinusoidal signal that gradually increases its magnitude from 0 to 5. Here,
due to a gradual increase in the fault signals’ magnitudes, it can be observed
that both detection and isolation are successfully performed but with a small
delay.

##  V Conclusion and Future Work

We have introduced a novel sensor fault detection and isolation method
tailored for a general class of nonlinear systems. For s-FDI, we leverage a
neural network-based Kazantzis-Kravaris/Luenberger (KKL) observer for residual
generation. We derived theoretical upper bounds for the residuals that are
obtained by comparing measured outputs with those predicted by the observer.
These upper bounds serve as the foundation for analytically computing
thresholds. When a residual crosses its corresponding threshold, it indicates
the detection of sensor faults. However, when it comes to the critical task of
fault isolation, specifically pinpointing the malfunctioning sensor, we
observed the importance of examining numerically differentiated residuals
rather than the usual residuals. To this end, we introduced an empirical
methodology, involving experimentation with the learned KKL observer, to
calculate the threshold for the differentiated residuals.

We demonstrated the efficacy of our approach on a network of Kuramoto
oscillators by evaluating various fault scenarios, including sensor
degradation and complete failure. The comprehensive set of simulations
provides compelling evidence of the method’s robustness and effectiveness in
both fault detection and isolation within the context of autonomous nonlinear
systems.

The theory of KKL observers extends to non-autonomous systems, and adapting
our method to those systems remains an open research topic. It is also of
interest to study the performance of the method in real applications,
especially in systems where conventional solutions are known to fail.

###  -A Proof of Proposition 1

In a fault-less case, $\phi(t)=1$ and $\zeta(t)=0$. Thus, from (3),

$\displaystyle|r_{i}|$ $\displaystyle=|h_{i}(x)-h_{i}(\hat{x})+v_{i}|$
$\displaystyle\leq\ell_{h_{i}}\|x-\hat{x}\|+|v_{i}|.$ (12)

We have

$\displaystyle\|x-\hat{x}\|$
$\displaystyle=\|\mathcal{T}^{*}(z)-\hat{\mathcal{T}}_{\eta}^{*}(\hat{z})\|$
$\displaystyle=\|\mathcal{T}^{*}(z)-\hat{\mathcal{T}}_{\eta}^{*}(z)+\hat{\mathcal{T}}_{\eta}^{*}(z)-\hat{\mathcal{T}}_{\eta}^{*}(\hat{z})\|$
$\displaystyle\leq\|\mathcal{T}^{*}(z)-\hat{\mathcal{T}}_{\eta}^{*}(z)\|+\|\hat{\mathcal{T}}_{\eta}^{*}(z)-\hat{\mathcal{T}}_{\eta}^{*}(\hat{z})\|$
$\displaystyle\leq\xi^{*}(z)+\ell_{\eta}\|z-\hat{z}\|.$ (13)

Similarly, we have

$\displaystyle\|z-\hat{z}\|$
$\displaystyle=\|\mathcal{T}(x)-\hat{\mathcal{T}}_{\theta}(\hat{x})\pm\hat{\mathcal{T}}_{\theta}(\hat{x})\|$
$\displaystyle\leq\|\mathcal{T}(x)-\hat{\mathcal{T}}_{\theta}(x)\|+\|\hat{\mathcal{T}}_{\theta}(x)-\hat{\mathcal{T}}_{\theta}(\hat{x})\|$
$\displaystyle\leq\xi(x)+\ell_{\theta}\|x-\hat{x}\|.$ (14)

By substituting (14) into (13) and rearranging, we obtain

$\|x-\hat{x}\|\leq\frac{\xi^{*}(z)+\ell_{\eta}\xi(x)}{1-\ell_{\theta}\ell_{\eta}}.$

Substituting this into (12) and using Assumption 1(i) proves the result.

###  -B Proof of Lemma 1

Define $\sigma(t)\doteq h(\bar{x}(t))-h(x(t))-v(t)$. Then, from the dynamics
$\dot{\tilde{z}}(t)$, we have that the solution
$\tilde{z}(t;\tilde{z}_{0},\sigma)$ satisfies

$\|\tilde{z}(t)\|\leq\|\exp(At)\tilde{z}_{0}\|+\int_{0}^{t}\|\exp(A\tau)B\sigma(t-\tau)\|\text{d}\tau.$
(15)

######  Lemma 2.

Suppose the matrix $A$ is Hurwitz and diagonalizable with eigenvalue
decomposition $A=V\Lambda V^{-1}$. Then,

$\|\exp(At)\|\leq\kappa(V)e^{-ct}$

and

$\int_{0}^{t}\|\exp(A\tau)B\|\text{d}\tau\leq\frac{\kappa(V)}{c}\|B\|(1-e^{-ct})$

where $\kappa(V)=\|V\|\|V^{-1}\|$ is the condition number of $V$ and

$c=\min_{\lambda\in\mathrm{eig}(A)}|\text{Re}(\lambda)|.$ (16)

The proof of this lemma is straightforward. A similar result when $A$ is not
diagonalizable can be obtained via the Jordan form of $A$; see [31, Appendix
C.5]. However, since diagonalizable matrices are dense in the space of square
matrices, the diagonalizability assumption is very mild.

The second term on the right-hand side of (15) satisfies

$\displaystyle\int_{0}^{t}\|\exp(A\tau)B\sigma(t-\tau)\|\text{d}\tau\hskip
56.9055pt$ $\displaystyle\hskip
56.9055pt\leq\int_{0}^{t}\|\exp(A\tau)B\|\|\sigma(t-\tau)\|\text{d}\tau$
$\displaystyle\hskip
56.9055pt\leq\int_{0}^{t}\|\exp(A\tau)B\|\text{d}\tau\,\max_{\tau\geq
0}\|\sigma(\tau)\|$ (17)

From the definition of $\sigma(t)$, we have

$\begin{array}[]{ccl}\|\sigma(t)\|&\leq&\|h(\bar{x}(t))-h(x(t))\|+\|v(t)\|\\\
&\leq&\ell_{h}\|\bar{x}(t)-x(t)\|+\|v(t)\|\\\
&\leq&\ell_{h}\psi(\|w_{[0,t]}\|_{L^{\infty}})+\sqrt{n_{y}}\|v(t)\|_{\infty}.\end{array}$

The first step is due to the triangle inequality, the second step is due to
the smoothness of $h(\cdot)$, and the last step is due to Assumption 1(ii) and
properties of vector norms. Therefore,

$\max_{\tau\geq
0}\|\sigma(\tau)\|\leq\ell_{h}\psi(\bar{w})+\sqrt{n_{y}}\bar{v}.$ (18)

The result then follows by substituting (17) and (18) into (15), and then
using the Lemma 2.

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