# Wizard of Errors: Introducing and Evaluating Machine Learning Errors in
Wizard of Oz Studies

Anniek Jansen<EMAIL_ADDRESS>Department of Industrial Design,
Eindhoven University of TechnologyEindhovenThe Netherlands and  Sara Colombo
<EMAIL_ADDRESS>Department of Industrial Design, Eindhoven University of
TechnologyEindhovenThe Netherlands

(2022)

###### Abstract.

When designing Machine Learning (ML) enabled solutions, designers often need
to simulate ML behavior through the Wizard of Oz (WoZ) approach to test the
user experience before the ML model is available. Although reproducing ML
errors is essential for having a good representation, they are rarely
considered. We introduce Wizard of Errors (WoE), a tool for conducting WoZ
studies on ML-enabled solutions that allows simulating ML errors during user
experience assessment. We explored how this system can be used to simulate the
behavior of a computer vision model. We tested WoE with design students to
determine the importance of considering ML errors in design, the relevance of
using descriptive error types instead of confusion matrix, and the suitability
of manual error control in WoZ studies. Our work identifies several
challenges, which prevent realistic error representation by designers in such
studies. We discuss the implications of these findings for design.

Wizard of Oz, Machine Learning, Machine Learning Errors, User Experience
Design, User Experience Analysis, Interaction Design, Computer Vision,
Prototyping Methods

††journalyear: 2022††copyright: rightsretained††conference: CHI Conference on
Human Factors in Computing Systems Extended Abstracts; April 29-May 5, 2022;
New Orleans, LA, USA††booktitle: CHI Conference on Human Factors in Computing
Systems Extended Abstracts (CHI ’22 Extended Abstracts), April 29-May 5, 2022,
New Orleans, LA, USA††doi: 10.1145/3491101.3519684††isbn:
978-1-4503-9156-6/22/04††ccs: Human-centered computing User studies††ccs:
Human-centered computing Interactive systems and tools††ccs: Human-centered
computing Systems and tools for interaction design

##  1\. Introduction

Machine Learning (ML) is becoming an increasingly important asset for
designers as it provides, among others, new interaction possibilities (e.g.
(Laput et al., 2016, 2017)) and new personalization techniques (e.g. (Yang et
al., 2016, 2018a; Gillies et al., 2016)). ML provides the functional backbone
of applications such as voice assistants, recommender systems, and face
recognition. It can be embedded into consumer products in various ways, from
narrow backend functionalities such as email spam filtering, to complex
autonomous systems like robots or self-driving cars.

Despite its potential, research has highlighted that designers find it
challenging to use ML as a design material and to prototype with ML (Yang et
al., 2020; Yang et al., 2018b; Dove et al., 2017), especially in the early
phases of a design process. Training a custom ML model is a resource-intensive
process and does not fit with the ’fail fast, fail often’ approach typical of
the design process early stages (Dove et al., 2017). To overcome the
difficulty of embedding working ML models in early prototypes to assess ML-
enabled solutions, the Wizard of Oz (WoZ) method can be used (Maulsby et al.,
1993). In this method, the wizard - a designer behind the screen who mimics
the technology, can act like an ML model. The goal is to provide the user with
the impression of interacting with a working system, and to test their
reactions to the experience it generates (Browne, 2019). This method is often
used in the fields of design and HCI to prototype non-existing (or not yet
realized) technologies and therefore represents a valid alternative to
simulate the behavior of ML even when a model has not been trained yet. WoZ
has already been applied to evaluate ML models in previous studies (e.g.
(Müller et al., 2019; Finke, 2019; Viswanathan et al., 2020)). However, both
these studies and additional research (van Allen, 2018; Browne, 2019) show
that simulating ML models behaviors in WoZ is no easy task. Among the
challenges identified is the difficulty to realistically reproduce ML errors,
because they are unlike human errors (Begel et al., 2020; Yang et al., 2019).
For instance, Riek (Riek, 2012) shows that only 3.7% of the WoZ studies in the
Human-Robot Interaction domain include deliberate errors. ML errors are an
intrinsic feature of ML models, and omitting them in a WoZ can lead to
findings that are not representative of the user experience that is being
simulated (Begel et al., 2020).

Figure 1. Prediction interface with explanations at the left-hand side

The live predict of the Wizard of Errors webpage. The page consists of a small
header with the title. The elements on the site are annotated to explain there
function. First there is a list of ingredients which are the ground truths
with the annotation: All correct labels from the error repository. The wizard
has to select the correct label (ground truth) as a first step. Next there is
an overview of text witht the annotation: The selected ground truth is
displayed together with the entered target accuracy and the current accuracy
(live updated). Underneath there is a slider with the annotation: The
confidence score for the prediction that will be send, can be set between
0-100%. Then there are five buttons with different types of errors and labels
inside of them with the annotation: The wizard can select here one of the
prediction options which include the error type and the corresponding label
that will be send to the user. Finally there is a last button with download
log with annotation: Possibility to download a logbook with all actions done
in the WoZ study.

#### Advantages of considering ML errors early on

Being able to successfully include ML errors in the early evaluation phases of
a design process would allow designers to assess users’ reactions to different
performance levels and errors of an ML model. This would have two main
advantages. Firstly, it would allow designers to test the overall
acceptability of a solution well in advance. Although the accuracy of an ML
model is unknown before a model is trained, it can greatly influence the user
experience (UX). For instance, in some applications like movie recommender
systems, an 80% accuracy score may be acceptable, while in others, such as in
diagnostics systems based on image recognition (e.g. to detect skin conditions
(Liu et al., 2020)) the same accuracy might be deemed unacceptable. Testing
the impact of different accuracy levels on the UX early on would prevent the
development of systems that are unsuccessful from a UX viewpoint.

Secondly, testing different ML errors in advance would allow designers to
create user interfaces that respond or adapt to ML errors, making the overall
UX more pleasant or understandable. Previous studies have shown that ML errors
can greatly impact the UX (Agarwal and Regalado, nd; Kocielnik et al., 2019;
Hong et al., 2021), the mental models users form to interact with the system,
and their trust in the ML model (Nourani et al., 2021). Different error types
could differently influence the UX, e.g. categorizing a Golden Retriever as a
Labrador might be perceived as an understandable ML error by users, while
mistaking a dog for a bird might cause more frustration and distrust in the
ML-based system. Considering different error types and testing their effects
on the UX early on would inform design decisions to ensure that the system can
’fail gracefully’ (PAIR, 2019), for instance giving users the possibility to
report an error, or providing information on the types of errors that might
occur, and their causes.

In this paper, we introduce Wizard of Errors (WoE), a web interface that
facilitates the inclusion of ML errors in WoZ studies. This interface can be
used in WoZ studies where the wizard can recognize the ground truth, i.e. the
correct outcome of the ML prediction, from the input data. This happens, for
instance, in object recognition, where the designer is able to recognize an
object correctly, the same way an ML model is expected to do. We tested the
WoE prototype with 10 design students to investigate if designers can
successfully mirror ML errors during a WoZ study, in order to have a more
representative simulation of the UX in the early phases of a design process.
This study aims to contribute to the current discussion on designing with ML,
by showing how ML errors can be included and tested in WoZ studies, and by
uncovering designers’ challenges and misconceptions in mirroring an ML model
behavior.

##  2\. Machine Learning failures and errors

ML errors can occur in many forms and can have different origins. In this
paper, we focus on errors that may occur during the user’s interaction with
ML-enabled solutions due to an incorrect ML prediction. During the remainder
of this paper, the term ML error will be used to describe such types of
errors.

A standard method to report ML errors is a confusion matrix. Here, errors are
described as False Positives (FPs) and False Negatives (FNs). However, the
confusion matrix, especially for a multi-class classification model (where the
prediction is not a binary yes/no, but is based on three or more labels)
(Tharwat, 2020) is hard to interpret for ML non-experts since the terminology
is confusing and the reading direction and structure are unintuitive (Shen et
al., 2020; Beauxis-Aussalet et al., 2018). To overcome these issues and make
errors more understandable to designers, we adopted the terminology introduced
by Bott and Laviola (Bott and Laviola Jr., 2015) where they differentiate
between four types of errors. The definition of the errors is tailored to
their use case - classifying handwritten mathematical equations, but it can be
generalized into the following definitions:

  * •

_Segmentation error_ : incorrect segmentation of the data that results in an
incorrect prediction (e.g. recognizing a face in a photo in the background of
a photo instead of the person in the foreground)

  * •

_Similarity error_ : incorrect prediction that is somewhat related to the
correct answer (e.g. recognizing sugar in a photo as salt)

  * •

_Wild errors_ : incorrect prediction that appears to have no relationship with
the correct answer (e.g. recognizing sugar in a photo as a carrot)

  * •

_No-recognition error_ : failing to give a prediction (e.g. a face recognition
system not responding when a face appears in front of the camera)

These descriptive error types can be used to create an error repository for an
ML model, which contains all possible error options for each correct label.
Such a repository can then be used in a WoZ study. Compared to the confusion
matrix, these four error types are expected to be easier to understand and to
better support designers in testing the users’ response to different types of
errors.

##  3\. The Wizard of Errors Interface

To enable the inclusion of the four ML error types in WoZ studies, we
developed _Wizard of Errors_ (WoE). WoE is a web interface, which aims to help
designers test the effects of different ML errors on users’ experience while
interacting with an ML-enabled system. The WoE application can be connected to
different prototyping platforms (e.g. Processing, JavaScript, Python, Arduino)
to perform WoZ studies. During the study, the WoE enables designers to
simulate the predictions of supervised ML models by selecting either correct
or incorrect labels (i.e. ML errors). For each correct label, four incorrect
labels can be selected, based on the four different types of errors. Designers
can also set an ML model target accuracy, so they can adjust the number of
errors dynamically throughout the test, in order to reach the desired accuracy
they intend to assess.

The WoE application consists of two elements: (i) _the error repository_ ;
(ii) _The WoE interface_. The error repository needs to be created by the
designer in advance, as part of the WoZ setup, and it consists of a table that
includes all the possible correct labels, and the corresponding incorrect
labels - one for each of the four error types (see Table 1). Once the error
repository is uploaded to the WoE interface, the wizard-designer can decide to
select a target accuracy, which will guide them in selecting the right amount
of ML errors (see Figure 2(a)).

The WoE interface is used during the WoZ study to simulate different errors
during the user’s interaction with the ML-enabled system (see Figure 1 for an
overview of the system). The WoE interface is controlled by the wizard-
designer, who can insert ML errors during the WoZ test, in order to determine
how users react to potential ML errors. The interface allows designers to test
three elements that potentially affect the overall UX: (i) the accuracy of the
ML model; (ii) the types of ML errors that can occur; (iii) the confidence
score for the predictions (a number between 0-1 or 0-100% that indicates the
probability associated to that label. A high confidence score does not mean
that the label is correct, just that the machine associates that label to the
input with a high probability).

Table 1. A selection from the error repository ID | correctAnswer | segmentationError | similarityError | wildError | noRecognitionError
---|---|---|---|---|---
0 | oats | cinnamon | flour | carrots | null
1 | flour | salt | oats | maple syrup | null

(a) Setup page

The setup page of the Wizard of Errors webpage. The page consists of a small
header with the tile. Next a textbox where the table name of an uploaded error
repository can be entered and a textbox for entering the target accuracy
score. At the bottom of the page is a button to start the study.

(b) A screenshot of the video with ingredient list that was added to the
prediction interface in Figure 1

A screenshot of the video that was added to the prediction figure during the
research. It shows a list of ingredients on the left side with a check mark, a
crossmark or nothing in front of it and a glass with maple syrup with
underneath 80 ml written on the screen.

Figure 2. Elements of the WoE interface

##  4\. Method

To assess if designers could mirror the behavior of an ML model in a WoZ
study, we tested the WoE interface with 10 design students. We aimed to
investigate if designers would be able to simulate the ML behavior, including
its errors, through the WoE interface, and what issues they would encounter.
As a design case for our test, we used a smart kitchen countertop inspired by
the IKEA concept kitchen (Sly Golovanov, 2015), which adopts ML image
recognition to identify ingredients placed on the countertop. We asked
participants to simulate the image recognition (ML) system performance through
WoE, as if they were testing the user experience of the smart countertop in a
WoZ study. Because in-person interactions were not possible, due to COVID-19
restrictions, the tests were performed online. In order to create the setting
of a WoZ test, a video was generated, which simulated a sequence of user’s
actions, i.e. a series of ingredients being added to different bowls on the
countertop. The video also showed an ingredient list on the screen. This video
was shown next to the WoE interface (see Figure 2(b)) and participants were
asked to use it as an input for simulating the ML system behavior in correctly
or incorrectly classifying the ingredient being added to the bowl. We chose to
simulate a user’s interactions through a video because it allowed for
consistency between the different tests - by providing all participants with
the same inputs, and because in-person studies were not allowed. For this
study, an error repository was created by the researchers and uploaded to WoE.
An extract of the error repository can be found in Table 1.

###  4.1. Participants

Participants were selected through snowball sampling (Goodman, 1961). Students
who were known to have worked with ML in a design project were approached, and
they referred other students who also had experience with using ML in a design
project. 10 participants were recruited (n=5 female; age: 22-27, M:23.5, SD:
1.71). All participants were Bachelor (n=1) or Master (n=9) Industrial Design
students from a technical university. All of them had experience with using ML
in at least one design project, up to six projects.

###  4.2. Procedure

The study consisted of three parts and was conducted entirely online through a
video-conference platform. The study protocol complied with the university
Ethical Review Board procedures and all data were managed in accordance with
GDPR regulations.

#### Part I - pre-experiment interview

For each subject, a pre-experiment semi-structured interview and a short
survey were used to get insights into the participant’s experience with and
knowledge of ML errors.

#### Part II - the experiment

The design case of the smart countertop was introduced to the participant,
together with the aim of the WoZ study, i.e. simulating the behavior of an ML
system for image recognition, including its errors. After describing the WoE
interface, participants were asked to set the target accuracy score to 50% and
to start the video. The video showed an ingredient being poured into a bowl,
and the task of the participant was to classify that ingredient, to simulate
the ML prediction. After the ingredient was added, the video paused. The
participant was then asked to (i) select the ground truth from the upper list
(see Figure 1), (ii) choose the confidence score for their prediction, and
(iii) simulate the ML prediction by selecting either the correct label or one
of the ML errors. If the correct label was chosen, a check mark would appear
next to the ingredient in the video. In case of an error, a cross mark would
be shown. This was meant to help the participant keep track of the number of
correct predictions and errors they had simulated during the study. After
recording the prediction, the video would continue by showing the next
ingredient. The same procedure was repeated for 12 ingredients in total.

The WoE interface also showed the real-time accuracy score, based on the
number of (in)correct predictions made up to that point. This allowed the
participant to adjust their behavior over time, in order to reach the pre-
defined target accuracy, i.e. make more correct predictions if the accuracy
score was below the target accuracy score, or more incorrect predictions if
the accuracy score was above the target. Once the video ended, the participant
was asked to follow the same procedure, watching the video again, this time
with a target accuracy score set to 70%. Participants were asked to think
aloud during the whole study, to explain the reasoning behind their choices
and actions. All participants’ interactions with WoE (i.e. logs) were recorded
for subsequent analysis.

#### Part III - post-experiment interview

At the end of the experiment, a semi-structured interview was conducted and
participants filled out a post-experiment survey. The interview and the survey
were aimed to assess the participant’s understanding of the error types, their
view on how different error types could influence the UX, and what error they
found most difficult to make.

##  5\. Preliminary results

Results stem from a thematic analysis of qualitative data (i.e. audio
recordings and interviews), as well as from the analysis of quantitative data
(i.e. Likert scales from the survey and logs of the WoE system). Since the
sample was small and homogeneous, we see these results as interesting initial
findings that need to be validated through a larger study.

###  5.1. Machine Learning errors

Participants indicated that before working with ML, they were unaware of
possible ML errors, but they became familiar with them during their first
project. They assessed errors as increasingly important throughout the design
process (Ideation phase: M=2.9, SD= 1.5; Realization phase: M=6.4, SD= 0.5).
This assessment barely changed after the experiment (Ideation phase: M=3.0,
SD= 1.7; Realization phase: M=6.5, SD= 0.5).

The descriptive types of ML errors used in this experiment were new to all
participants. While most ML errors were clear, mainly due to their self-
explanatory names, the segmentation error was more difficult to understand and
only two participants could correctly define it afterwards. During the study,
the participants could read the explanation by hovering over the buttons, but
this option was hardly used.

Moreover, participants expected the different error types to have different
impacts on the UX. While the similarity error was seen by most as less harmful
because it was expected to be understandable by humans, the other errors were
considered to have a more negative impact on the UX. Participants expected
these errors to decrease the trust in the model or elicit frustration. _”But
if I put a carrot on the table and it is like “Oh strawberry” then I would be
like, this is a stupid system it cannot even recognize such a simple thing
then I must just work horribly”_ (P2).

###  5.2. Mirroring Machine Learning

The main focus of the study was to evaluate if participants could mirror the
behavior of an ML model by simulating ML errors. While all participants had
worked with ML before and trained models themselves, their behavior did not
match with that of an ML model. Several misconceptions surfaced during the
experiment, as explained below. Nevertheless, the WoE interface was successful
in encouraging wizards to make ML errors and to reach the target accuracy
score, although they were hesitant to make ML errors in the beginning (for
$accuracy=50(\%):M=59.10,SD=10.47;$ for $70(\%):M=68.72,SD=9.67$).
Participants claimed that WoE was a useful tool to test the interactions in an
early stage: _”I think it would be helpful to explore the opportunities and
limitations of ML in combination with a specific project. Normally this is
difficult to do because in an early stage of a project a full system is often
not working already.”_ (P6). However, they also indicated that it was hard to
assess a realistic number of errors to include (P9) and that they would like
to have something that could be applicable even earlier in the design process,
to evaluate the potential impact of errors (P3, P4 and P5).

#### Logical errors are easier to make

The ML errors made during the experiment showed that participants mainly
selected similarity errors and hardly any wild errors (see Figure 3). From the
think-aloud transcripts, it became clear that participants based their choices
mainly on the label associated to each error type and not the error type
itself (e.g. choosing the ’sugar’ label for salt, instead of choosing a
similarity error). By using labels, participants often selected an
understandable or logical error from a human perspective. They also expressed
it felt unnatural to send an unrelated error. As participant 9 stated: _”For
me as a wizard, a no-recognition error would be weird to make because I know
what it is and rationalizing why it would not recognize it at all would be a
bit weird, in my opinion.”_

(a) First cycle with target accuracy score 50%

A stacked bar chart for the type of errors made for each ingrediens in case of
a target accuracy score of 50%. The majority of the predictions was correct,
next a similarity error, a segmentation error, a no-recognition error and only
three times a wild error. Both the oats, eggs, and carrots were predicted as
correct by all participants. The coconut oil was only predicted correct by one
participant.

(b) Second cycle with target accuracy score 70%

A stacked bar chart for the type of errors made for each ingredients in case
of a target accuracy score of 70%. The majority of the predictions was
correct, and more compared to the first cycle with target accuracy 50%. The
similarity error was made most often of the errors, next a segmentation error
and finally a no-recognition error. There are no wild errors made. Both the
oats and the eggs were predicted as correct by all participants. The coconut
oil was only predicted correct by two participants.

Figure 3. Overview of the predictions for each ingredient

#### Projecting personal knowledge

Similarly to the tendency to select logical errors, participants based the
behavior of the ML model on their own knowledge. This was mainly visible when
coconut oil was shown in the video, and 85% of the predictions were an ML
error - while the mean for all other ingredients, excluding coconut oil, was
31.8%. As participant 7 put it: _”I myself have never used coconut oil before,
so I am not really sure if it is an oil or if it looks like this.”_

The tendency to rely on their own knowledge could also be seen in
participants’ assumption that common ingredients (in their food culture) are
easier to recognize: _”If there are ingredients that are exotic, then it might
be easier to make [a no-recogntion error]”_ (P1). Moreover, they did not
consider that ingredients can exist in different shapes and forms. Eggs and
carrots were seen as very easy to classify for the machine because they had a
distinctive shape, but participants did not take into account that they could
exist in many forms and shapes, which would make it harder for the ML model to
recognize them correctly.

#### Higher confidence score for correct answers

To check if participants gave higher confidence scores to correct predictions,
a linear regression was calculated. A significant regression was found
($F(1,238)=35.618,p<.001$), with an ${R}^{2}$ of $.130$ and an
${R}^{2}_{Adjusted}$ of $.127$, indicating that the error type (correct vs
incorrect) is a significant predictor for the confidence score
($\beta=.361,t(238)=5.968,p<.001$) and that the confidence score increased
when a correct prediction was selected. This differs from a real ML model,
where a confidence score can be equally high for incorrect predictions.

#### ML model self-awareness

Finally, participants expected the model to be aware that it was making a
mistake and mentioned that it would be useful if it would indicate to the
final user what type of error it was making: _”I think for the user it is very
helpful to know that there is an error, and they know what kind of error it
is.”_ (P3). A possible explanation for this misconception might be that the
participants had only trained and tested an ML model and not deployed it live,
since in training and testing supervised classification models the correct
label is available and makes it possible to detect errors - although not the
error types.

##  6\. Discussion

The preliminary results presented in the previous section provide insights
into what aspects of mirroring an ML model are difficult for wizard-designers.
These findings have implications both for designing WoZ studies for ML
applications, and for designing with ML in general.

When designing WoZ studies to test user experiences and interactions based on
ML, one needs to take into account the fact that wizards will not be able to
ignore their human rationale, therefore their behavior will not be a good
representation of the ML behavior. Our study showed that the WoE interface and
the ML errors can already contribute to a better representation of ML models
in WoZ studies and can trigger designers to reflect on the importance of
testing different types of ML errors. However, it also is clear that this is
not sufficient, as designers’ misconceptions prevent them from properly
mirroring ML models. One way to overcome this limitation is to recommend
wizards certain error types during the use of the WoE interface, for instance
in case certain error types are more rarely selected, compared to others.

Another option is to allow designers to pre-define how many errors for each
type will be sent by the WoE interface, and let the interface randomly assign
those errors, while asking designers only to select the ground truth. While
limiting designers’ ability to adjust the number and type of errors in real-
time, this would make the study more realistic.

Next to that, designers not only need to test different error types, but also
vary on the number and the moment of occurrence to explore the effect of all
these factors on the user’s experience and their trust in the ML predictions.
While this study only focused on using WoE in the early phase of a design
process, ML errors can also be introduced during other phases. For instance,
in the ideation and conceptualization phase, cards with the error types,
potential consequences and options for adjusting the design can help to
consider errors from the beginning. When designing a UX wireframe, the
designer can go over each step and consider what would happen if each type of
error occurred, to check if their interface is able to fail gracefully (PAIR,
2019) or if it needs to be adjusted so that it can fail gracefully.

##  7\. Limitations and Future Work

A limitation of the WoE interface, and WoZ testing with ML in general, is that
it is only applicable in cases where the humans are able to make the correct
predictions - i.e. recognize the ground truth, during the study. However, this
still leaves out a considerable number of ML models within supervised
learning. This study only covered object recognition, but we expect the error
types to be also transferable to other subsections of computer vision and
potentially also to other types of supervised classification. Further research
should explore and validate the use of ML in these applications.

Another limitation of this study is that it was conducted with students from
one Industrial Design faculty, which could give a biased view on how design
students, and designers in general, approach and use ML. Therefore, we
consider these findings as preliminary results and future studies should be
conducted with a larger and more diverse sample to confirm their validity.
Furthermore, the WoE interface should be tested in a WoZ study with actual
users, to gather more insights on designers’ behaviors in a real context.

##  8\. Conclusion

Using ML in prototypes during the early phases of a design process is
challenging. In this study, we introduced Wizard of Errors (WoE), a
prototyping tool for conducting WoZ studies on ML-enabled interactions. The
WoE interface facilitates the inclusion of ML errors into WoZ studies, since
these are essential for UX assessment but are currently rarely included. In
WoE we used four descriptive error types instead of the confusion matrix and
evaluated with design students if ML errors are important to consider and how
relevant the four error types are in comparison to the confusion matrix.
Moreover, during a WoZ simulation, we evaluated if it is possible for a
designer to simulate an ML model in terms of ML error behavior. The error
types showed to have potential to be used during WoZ studies, but we also
identified several challenges that still prevent the designer from realistic
error representation in WoZ. In this study, we designed and tested the WoE
interface to embed ML errors in WoZ studies, and provide preliminary knowledge
that can help both design researchers and practitioners in this field to
consider ML errors as a regular component of the design process for ML-enabled
solutions.

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# Linear Fault Estimators for Nonlinear Systems: An Ultra-Local Model Design

Farhad Ghanipoor  Carlos Murguia  Peyman Mohajerin Esfahani  Nathan van de
Wouw  Mechanical Engineering Department, Eindhoven University of Technology,
Eindhoven, The Netherlands (e-mails<EMAIL_ADDRESS><EMAIL_ADDRESS>and N.v.d.Wouw@tue.nl).  Delft Center for Systems and Control, Delft
University of Technology, Delft, The Netherlands (e-mail:
P.MohajerinEsfahani@tudelft.nl).

###### Abstract

This paper addresses the problem of robust process and sensor fault
reconstruction for nonlinear systems. The proposed method augments the system
dynamics with an approximated internal linear model of the combined
contribution of _known_ nonlinearities and _unknown_ faults – leading to an
approximated linear model in the augmented state. We exploit the broad
modeling power of ultra-local models to characterize this internal dynamics.
We use a linear filter to reconstruct the augmented state (simultaneously
estimating the state of the original system and the sum of nonlinearities and
faults). Having this combined estimate, we can simply subtract the analytic
expression of nonlinearities from that of the corresponding estimate to
reconstruct the fault vector. Because the nonlinearity does not play a role in
the filter dynamics (it is only used as a static nonlinear output to estimate
the fault), we can avoid standard restrictive assumptions like globally (one-
sided) Lipschitz nonlinearities and/or the need for Lipschitz constants to
carry out the filter design. The filter synthesis is posed as a mixed
$H_{2}/H_{\infty}$ optimization problem where the effect of disturbances and
model mismatches is minimized in the $H_{\infty}$ sense, for an acceptable
$H_{2}$ performance with respect to measurement noise.

###### keywords:

Fault estimation for nonlinear systems; Observer-based fault estimation; Mixed
$H_{2}/H_{\infty}$; Ultra local model;

††thanks: This publication is part of the project Digital Twin project 4.3
with project number P18-03 of the research programme Perspectief which is
(mainly) financed by the Dutch Research Council (NWO).

##  1 Introduction

Predictive maintenance is a vital technology that guarantees the reliable
operation of high-tech industrial systems. A fundamental element for
predictive maintenance is fault estimation (for severity assessment). If we
have an accurate estimate of the fault effect in the system, predictions of
its severity can be performed and predictive maintenance can be scheduled
accordingly. Many fault estimation methods have been developed for linear
dynamical systems (see, e.g., Liu and Shi (2013); Liu et al. (2012) for
results on linear stochastic and switching systems). However, most engineered
systems (e.g., robotics, power/water networks, transportation, and
manufacturing) are highly nonlinear in nature. Methods for nonlinear systems
are still under development, see, e.g., Mohajerin Esfahani and Lygeros (2015),
Valibeygi et al. (2016), Pan et al. (2021), Lan and Patton (2020), and Jiang
et al. (2006). In Jiang et al. (2006), the authors address the problem for
nonlinear systems with uniformly Lipschitz nonlinearities, process faults only
(i.e., no sensor faults), and assume the so-called matching condition (the
rank of the fault distribution matrix is invariant under left multiplication
by the output matrix) is satisfied. An adaptive filter is provided that
approximately reconstructs the actuator fault vector in this configuration.
Although the matching condition makes the problem tractable, it significantly
reduces the class of systems that can benefit from the results. In Vo et al.
(2021), a fault estimation scheme is introduced for both sensor and process
faults using Nonlinear Unknown Input Observers (NUIO), adaptive Radial Basis
Function Neural Networks (RBFNN), and assuming the matching condition is
satisfied. The authors prove that their scheme provides boundedness of fault
estimation errors.

The authors in Zhu et al. (2015) do not require the matching condition to be
satisfied. They consider Lipschitz nonlinearities, simultaneous sensor and
process faults, and adopt a standard fault observability condition on the
linear part of the dynamics, Hou and Patton (1998). They tackle the problem
using the notion of intermediate observers, consisting of two dedicated
observers, one that estimates the fault and the other the state. Their scheme
guarantees bounded fault estimation errors. In Van der Ploeg et al. (2022),
simultaneous additive and multiplicative process faults are considered in the
scope of discrete-time system models. Herein, the fault estimation problem is
addressed by decoupling process nonlinearities and disturbances from the
estimation filter dynamics and using regression techniques to approximately
estimate fault signals. Decoupling nonlinearities leads to linear filters for
which linear methods can be used to reconstruct fault signals. However,
decoupling conditions impose strong constraints on the system dynamics, which
significantly limits the applicability of these results. Furthermore, in Van
der Ploeg et al. (2022), the considered system is linear during the healthy
operational mode (i.e., in the absence of the fault signals).

We remark that the above-mentioned results for nonlinear systems guarantee
approximate reconstruction of fault vectors only, i.e., they ensure bounded
estimation errors, which, if small enough, still lead to a potentially good
estimate of the true fault. Not having internal models of fault signals makes
it challenging to enforce zero error fault estimation. In Ghanipoor et al.
(2022), an internal model for the fault vector is used, and this allows to
guarantee zero error in the absence of external disturbances and noise for
some classes of faults. However, this solution only applies to systems with
globally Lipschitz nonlinearities and the effect of measurement noise and
exogenous disturbances is not addressed.

Considering the discussed literature, the main contribution of this manuscript
is a fault estimation scheme for process and sensor faults that _does not
require_ globally Lipschitz nonlinearities, known Lipschitz constants, and the
matching condition. This scheme augments the system dynamics with an
approximated internal linear model of the combined contribution of _known_
nonlinearities and _unknown_ faults – leading to an approximated linear model
in the augmented state. We exploit the broad modeling power of _ultra-local
models_ , see Fliess and Join (2013) and Sira-Ramírez et al. (2018) – which
refers to a class of phenomenological models that are only valid for very
short time intervals – to characterize this internal dynamics. We use a linear
filter to reconstruct the augmented state (simultaneously estimating the state
of the original system and the sum of nonlinearities and faults). If this
estimate is sufficiently accurate, we can simply subtract the analytic
expression of nonlinearities from that of the corresponding estimate to
reconstruct the fault vector. Because the nonlinearity does not play a role in
the filter dynamics (it is only used as a static nonlinear output to estimate
the fault), we can avoid standard restrictive assumptions like globally (one-
sided) Lipschitz nonlinearities and/or the need of Lipschitz constants to
carry out the filter design. The filter synthesis is posed as a mixed
$H_{2}/H_{\infty}$ semi-definite program where the effect of exogenous
disturbances and the mismatch between ultra-local models and true internal
dynamics is minimized in the $H_{\infty}$ sense, for an acceptable $H_{2}$
performance with respect to measurement noise. We provide sufficient
conditions that guarantee the existence of the filter and ISS of the
estimation error dynamics with respect to exogenous disturbances, model
mismatches, and measurement noise.

Notation: The symbol $\mathbb{R}^{+}$ denotes the set of nonnegative real
numbers. The $n\times n$ identity matrix is denoted by $I_{n}$ or simply $I$
if $n$ is clear from the context. Similarly, $n\times m$ matrices composed of
only zeros are denoted by $\boldsymbol{0}_{n\times m}$ or simply
$\boldsymbol{0}$ when their dimensions are clear. For positive definite (semi-
definite) matrices, we use the notation $P\succ 0$ $(P\succeq 0)$. For
negative definite (semi-definite) matrices, we use the notation $P\prec 0$
$(P\preceq 0)$. The notation $\text{col}[x_{1},\ldots,x_{n}]$ stands for the
column vector composed of the elements $x_{1},\ldots,x_{n}$. This notation is
also used when the components $x_{i}$ are vectors. The $\ell_{2}$ vector norm
(Euclidean norm) and the matrix norm induced by the $\ell_{2}$ vector norm are
both denoted as $||\cdot||$. For a vector-valued signal $f(t)$ defined for all
$t\geq 0$, $||f(t)||_{\infty}:=\sup_{t\geq 0}||f(t)||$. For a differentiable
function $W:\mathbb{R}^{n}\to\mathbb{R}$ we denote by $\frac{\partial
W}{\partial e}$ the row-vector of partial derivatives and by $\dot{W}(e)$ the
total derivative of $W(e)$ with respect to time. For the transfer function
$T(s)$, $\sigma_{\max}(T(s))$ shows the maximum singular value, and $T^{H}(s)$
the Hermitian transpose.

##  2 Problem Formulation

Consider the nonlinear system

$\left\\{\begin{aligned}
\dot{x}(t)=&Ax(t)+Bu(t)+Sg(Vx(t),u(t),t)+D\omega(t)\\\\[2.84526pt]
&+F_{x}f_{x}(x(t),u(t),t),\\\\[4.2679pt]
y(t)=&Cx(t)+F_{y}f_{y}(x(t),t)+\nu(t),\end{aligned}\right.$ (1)

where $t\in\mathbb{R}^{+}$, $x\in{\mathbb{R}^{n}}$, $y\in{\mathbb{R}^{{m}}}$,
and $u\in{\mathbb{R}^{{l}}}$ are time, state, measured output, and known input
vectors, respectively. Function
$g:\mathbb{R}^{n_{v}}\times\mathbb{R}^{l}\times\mathbb{R}^{+}\to\mathbb{R}^{n_{g}}$
is a nonlinear _known_ vector field. Vectors
$\omega:\mathbb{R}^{+}\to\mathbb{R}^{n_{d}}$ and
$\nu:\mathbb{R}^{+}\to\mathbb{R}^{m}$ are unknown bounded external
disturbances and measurement noise, respectively. Functions
$f_{x}:\mathbb{R}^{n}\times\mathbb{R}^{l}\times\mathbb{R}^{+}\to\mathbb{R}^{n_{f_{x}}}$
and $f_{y}:\mathbb{R}^{n}\times\mathbb{R}^{+}\to\mathbb{R}^{n_{f_{y}}}$ denote
_unknown_ process and sensor fault vectors, respectively. Matrices
$(A,B,S,V,C,F_{x},F_{y})$ are of appropriate dimensions and
${n},{m},l,n_{g},n_{v},n_{d},n_{f_{x}},n_{f_{y}}\in\mathbb{N}$. We use matrix
$S$ to indicate in what equations the nonlinearity $g(Vx,u,t)$ appears
explicitly, and $V$ is used to specify which states play a role in the known
nonlinearity. Matrices $F_{x}$ and $F_{y}$ denote process and sensor fault
distribution matrices, respectively. We often omit implicit time dependencies
for notational simplicity.

Let $S^{\dagger}$ denote the Moore-Penrose pseudo-inverse of matrix $S$, and
define $Q_{1}R_{1}:=S^{\dagger}F_{x}$ and $Q_{2}R_{2}:=(I-SS^{\dagger})F_{x}$,
where $Q_{i}$ ($R_{i}$) denotes a full column (row) rank matrix of appropriate
dimensions, $i=1,2$. Note that this factorization always exists (rank
decomposition). Then, we can rewrite the nonlinear system in (1) as follows:

$\left\\{\begin{aligned}
\dot{x}=&Ax+Bu+S\big{(}g(Vx,u,t)+Q_{1}f_{n}(x,u,t)\big{)}\\\\[2.84526pt]
&+Q_{2}f_{l}(x,u,t)+D\omega,\\\\[4.2679pt]
y=&Cx+F_{y}f_{y}(x,t)+\nu,\end{aligned}\right.\\\ $ (2a)
$\Bigg{\\{}\begin{array}[]{r@{}l}f_{n}&{}:=R_{1}f_{x},\\\\[2.84526pt]
f_{l}&{}:=R_{2}f_{x}.\end{array}\hskip 140.84114pt$ (2b)

This transformation does not lead to loss of generality and will play an
instrumental role to the fault estimation algorithm that we propose.

######  Assumption 1.

_(State and Input Boundedness) The state variable $x$ and the input $u$ remain
bounded in some compact region of interest
$\mathcal{D}=\mathcal{D}_{x}\times\mathcal{D}_{u}\subset\mathbb{R}^{n}\times\mathbb{R}^{l}$,
before and after the occurrence of a fault._

######  Assumption 2.

_( $\mathcal{C}^{r}$ Fault and Nonlinearity) Vectors
$g(Vx,u,t)+Q_{1}f_{n}(x,u,t)$, $f_{l}(x,u,t)$, and $f_{y}(x,t)$ are $r$ times
differentiable with respect to $t$, and their $r$-th derivative is uniformly
bounded._

###  2.1 Ultra Local Model

Define the functions $\beta_{1}(x,u,t):=g(Vx,u,t)+Q_{1}f_{n}(x,u,t)$,
$\beta_{2}(x,u,t):=f_{l}(x,u,t)$, and $\beta_{3}(x,t):=f_{y}(x,t)$. We refer
to these $\beta_{k}(t)$ as the (fault-induced) lumped disturbances. Under
Assumption 2, using the notion of ultra-local models, Fliess and Join (2013),
we can write linear approximated internal models of $\beta_{k}(t)$. Consider
the approximated model:

$\left\\{\begin{aligned}
\dot{\zeta}_{j_{k}}(t)&=\zeta_{{(j+1)}_{k}}(t),\qquad&&j_{k}\in\\{1,\dots,r_{k}-1\\},\\\\[0.7113pt]
\dot{\zeta}_{r_{k}}(t)&=\mathbf{0},\qquad&&k\in\\{1,2,3\\},\\\\[0.7113pt]
\bar{\beta}_{k}(t)&=\zeta_{1_{k}}(t),\end{aligned}\right.$ (3)

where $\bar{\beta}_{k}(t),k\in\\{1,2,3\\}$, is an approximation of the actual
$\beta_{k}(t)$ introduced above. Note that each of these models corresponds to
an entry-wise $r_{k}$-th order Taylor polynomial-in-time approximation at time
$t$ of $\beta_{k}(t)$. The accuracy of the approximated model increases as
$d\beta_{k}(t)/dt^{r_{k}}$ goes to zero (entry-wise) and it is exact for
$d\beta_{k}(t)/dt^{r_{k}}=\mathbf{0}$. Model (3) is used to construct an
observer that ultra-locally, see Fliess and Join (2013) and Sira-Ramírez et
al. (2018), acts as a self-updating polynomial spline approximating the actual
value of $\beta_{k}(t)$. To design such an observer, we extend the system
state, $x(t)$, with the states of the internal model for each of the lumped
disturbances, $\zeta_{j}(t)$, $j\in\\{1,\dots,r_{k}-1\\}$, $k\in\\{1,2,3\\}$,
and augment the system dynamics in (2) with (3). We then design a linear
observer for the augmented linearized system to simultaneously estimate $x(t)$
and $\beta_{k}(\cdot)$. We remark that the number of the derivatives, $r_{k}$,
for each of the above-mentioned models in (3) is problem-dependent. An optimal
selection depends on the (usually unknown) frequency characteristics of the
functions being approximated. Increasing $r_{k}$ results in higher-dimensional
augmented dynamics and, thus, high-dimensional observers. Such higher
dimension of the observers provide more degrees of freedom for optimal
synthesis, see Ghanipoor et al. (2022).

###  2.2 Augmented Dynamics

Define $\rho_{k}:=\text{col}[\dot{\beta_{k}},\ldots,\beta_{k}^{(r_{k}-1)}]$,
$k\in\\{1,2,3\\}$ and the augmented state
$x_{a}:=\text{col}[x,\beta_{1},\rho_{1},\beta_{2},\rho_{2},\beta_{3},\rho_{3}]$,
and write the augmented linear dynamics using (2)-(3) as

$\displaystyle\left\\{\begin{aligned}
\dot{x}_{a}&=A_{a}x_{a}+B_{a}u+D_{a}\omega_{a},\\\
y&=C_{a}x_{a}+\nu,\end{aligned}\right.$ (4a) $\displaystyle A_{a}$
$\displaystyle:=\left[\begin{array}[]{ccccccccccc}A&S&\boldsymbol{0}&Q_{2}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}\\\
\boldsymbol{0}&\boldsymbol{0}&I_{d_{1}}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}\\\
\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}\\\
\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&I_{d_{2}}&\boldsymbol{0}&\boldsymbol{0}\\\
\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}\\\
\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&I_{d_{3}}\\\
\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}\\\
\end{array}\right],$ (4b) $\displaystyle B_{a}$
$\displaystyle:=\left[\begin{array}[]{cc}B\\\
\boldsymbol{0}\end{array}\right],\quad
C_{a}:=\left[\begin{array}[]{lllllll}C&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&F_{y}&\boldsymbol{0}\end{array}\right],$
$\displaystyle D_{a}$
$\displaystyle:=\left[\begin{array}[]{cccc}D&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}\\\
\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}\\\
\boldsymbol{0}&I_{d_{1}}&\boldsymbol{0}&\boldsymbol{0}\\\
\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}\\\
\boldsymbol{0}&\boldsymbol{0}&I_{d_{2}}&\boldsymbol{0}\\\
\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}\\\
\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&I_{d_{3}}\\\
\end{array}\right],\quad\omega_{a}:=\left[\begin{array}[]{c}\omega(t)\\\
\beta_{1}^{{(r_{1})}}(x,u,t)\\\\[2.84526pt]
\beta_{2}^{{(r_{2})}}(x,u,t)\\\\[2.84526pt]
\beta_{3}^{{(r_{3})}}(x,t)\\\\[2.84526pt] \end{array}\right],$

with $d_{1}=(r_{1}-1)n_{g},d_{2}=(r_{2}-1)n_{f_{l}}$, $n_{f_{l}}$ being the
number of rows of $R_{2}$ in (2b) and $d_{3}=(r_{3}-1)n_{f_{y}}$.

###  2.3 Linear Joint State-Fault Estimator

We propose the following linear observer to estimate the fault vector:

$\displaystyle\dot{z}$ $\displaystyle=Nz+Gu+Ly,$ (5a)
$\displaystyle\hat{x}_{a}$ $\displaystyle=z-Ey,$ with observer state
$z\in{\mathbb{R}^{n_{z}}}$,
$n_{z}=n+r_{1}n_{g}+r_{2}n_{f_{l}}+r_{3}n_{f_{y}}$, estimate $\hat{x}_{a}$,
and matrices $(N,G,L)$ defined as $\displaystyle N$
$\displaystyle:=MA_{a}-KC_{a},$ $\displaystyle M$ $\displaystyle:=I+EC_{a},$
(5b) $\displaystyle G$ $\displaystyle:=MB_{a},$ $\displaystyle L$
$\displaystyle:=K(I+C_{a}E)-MA_{a}E.$

Matrices $E$ and $K$ are observer gains to be designed. Note that, given
$\hat{x}_{a}$, fault estimates, $\hat{f}_{x}(\cdot)$ and $\hat{f}_{y}(\cdot)$,
can be constructed as follows:

$\displaystyle\hat{f}_{x}$ $\displaystyle=\left[\begin{array}[]{c}R_{1}\\\
R_{2}\end{array}\right]^{L}\left[\begin{array}[]{c}Q_{1}^{L}\big{(}\bar{C}_{1}\hat{x}_{a}-g\left(V_{a}\hat{x}_{a},u,t\right)\big{)}\\\
\bar{C}_{2}\hat{x}_{a}\end{array}\right],$ (6) $\displaystyle\hat{f}_{y}$
$\displaystyle=\bar{C}_{3}\hat{x}_{a},$

where

$\displaystyle\bar{C}_{1}$
$\displaystyle:={\left[\begin{array}[]{ccccccc}\boldsymbol{0}&I_{n_{g}}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}\end{array}\right]},\quad\bar{C}_{2}:={\left[\begin{array}[]{ccccccc}\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&I_{n_{f_{l}}}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}\end{array}\right]},$
$\displaystyle\bar{C}_{3}$
$\displaystyle:={\left[\begin{array}[]{ccccccc}\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&I_{n_{f_{y}}}&\boldsymbol{0}\end{array}\right]},\quad
V_{a}:={\left[\begin{array}[]{cc}V&\boldsymbol{0}\end{array}\right]},$

and $[\cdot]^{L}$ denotes left inverse. Note that, by construction
$[R_{1}^{\top},R_{2}^{\top}]^{\top}$ and $Q_{1}^{L}$, are full column rank;
hence, they are always left invertible. Note that we assume that the fault-
free system dynamics is known. Hence, the nonlinear function, $g(\cdot)$, can
be used to extract faults signals algebraically as in (6).

Note that, according to (6), the part of the extended state, $x_{a}$, that we
use to reconstruct faults is $\bar{C}_{a}x_{a}$ with

$\bar{C}_{a}:={\left[\begin{array}[]{cccc}V_{a}^{\top}&\bar{C}_{1}^{\top}&\bar{C}_{2}^{\top}&\bar{C}_{3}^{\top}\end{array}\right]^{\top}}.$
(7)

That is, we mainly require the observer to provide accurate estimates
$\bar{C}_{a}\hat{x}_{a}$ of $\bar{C}_{a}x_{a}$ and not of the complete state
$x_{a}$. Let us now define the estimation error

$e:=\hat{x}_{a}-x_{a}=z-x_{a}-Ey=z-Mx_{a}-E\nu.$

Then, the estimation error dynamics is given by

$\displaystyle\dot{e}=$ $\displaystyle Ne+(NM+LC_{a}-MA_{a})x_{a}+(G-MB_{a})u$
(8) $\displaystyle-MD_{a}\omega_{a}+(NE+L)\nu-E\dot{\nu}.$

Given the algebraic relations in (5b), it can be verified that
$G-MB_{a}=\mathbf{0}$, $NM+LC_{a}-MA_{a}=\mathbf{0}$, and $NE+L=K$. Therefore,
(8) amounts to

$\displaystyle\dot{e}$ $\displaystyle=Ne-
MD_{a}\omega_{a}+{\left[\begin{array}[]{cc}K&-E\end{array}\right]}{\left[\begin{array}[]{cc}\nu\\\
\dot{\nu}\end{array}\right]}.$ (9)

Define $\nu_{a}:={\left[\begin{array}[]{cc}\nu\\\
\dot{\nu}\end{array}\right]}$, $e_{d}:=\bar{C}_{a}e$ with $\bar{C}_{a}$ as in
(7), and

$\bar{B}:={\left[\begin{array}[]{cc}K&-E\end{array}\right]}.$ (10)

Then, the estimation error dynamics is given by

$\left\\{\begin{aligned} \dot{e}&=Ne-MD_{a}\omega_{a}+\bar{B}\nu_{a},\\\
e_{d}&=\bar{C}_{a}e.\end{aligned}\right.$ (11)

Define the transfer matrices

$\displaystyle T_{e_{d}\omega_{a}}(s)$
$\displaystyle:=-\bar{C}_{a}(sI-N)^{-1}MD_{a},$ (12) $\displaystyle
T_{e_{d}\nu_{a}}(s)$ $\displaystyle:=\bar{C}_{a}(sI-N)^{-1}\bar{B},$

where $T_{e_{d}\omega_{a}}(s)$ and $T_{e_{d}\nu_{a}}(s)$ denote the
corresponding transfer matrices from $\omega_{a}$ and $\nu_{a}$, both to
$e_{d}$, respectively. Now, we can state the fault reconstruction problem that
we seek to address.

######  Problem 3.

_(Fault Reconstruction) Consider the nonlinear system (2) with known input and
output signals, $u(t)$ and $y(t)$, and let Assumptions 1 and 2 be satisfied.
Further, consider the approximated internal model (3), the augmented dynamics
(4), and the observer (5). Design the observer gain matrices $(E,K)$ such
that:
1) All trajectories of the error dynamics (11) exist and are globally
uniformly ultimately bounded in $t\geq 0$;
2)
$\sup_{\mu\in\mathbb{R}^{+}}\sigma_{\max}(T_{e_{d}\omega_{a}}(i\mu))<\lambda$,
for some finite $\lambda>0$, transfer matrix $T_{e_{d}\omega_{a}}(i\mu)$ in
(12), and frequencies $\mu\in\mathbb{R}^{+}$;
3)
$\sqrt{\frac{1}{2\pi}\operatorname{trace}\int_{-\infty}^{\infty}T_{e_{d}\nu_{a}}(i\mu)T_{e_{d}\nu_{a}}^{H}(i\mu)\mathrm{~{}d}\mu}<\gamma$,
for some finite $\gamma>0$, transfer matrix $T_{e_{d}\nu_{a}}(i\mu)$ in (12);
and
4) for $\omega_{a}(x,u,t)=\mathbf{0}$, $\nu_{a}(t)=\mathbf{0}$,
$x\in\mathcal{D}_{x}$, $u\in\mathcal{D}_{u}$, and $t\geq 0$, $e=\mathbf{0}$ is
an asymptotically stable equilibrium of the estimation error dynamics (11)._

Under Assumptions 1 and 2, Problem 1 amounts to finding a fault estimator
that: guarantees a bounded estimation error, $e(t)$; the $H_{\infty}$-norm of
$T_{e_{d}\omega_{a}}(s)$, the transfer matrix from external disturbances and
model mismatches to the performance output $\bar{C}_{a}e$ (fault and some
state estimation errors) is upper bounded by $\lambda$; the $H_{2}$-norm of
$T_{e_{d}\nu_{a}}(s)$, the transfer matrix from measurement noise to the
performance output, is upper bounded by $\gamma$; and, for
$\omega_{a}(x,u,t)=\mathbf{0}$ and $\nu_{a}(t)=\mathbf{0}$, $e(t)$ goes to
zero asymptotically (internal stability).

######  Remark 4.

_( $H_{2}$/$H_{\infty}$ Interpretation) In the SISO case, the
$H_{\infty}$-norm of a transfer function $T(s)$ can be interpreted as the
highest peak value in its frequency response function. In other words, it is
the largest amplification over all frequencies of input signals passed through
$T(s)$ (Scherer and Weiland, 2000, pp. 73). Then, minimizing the
$H_{\infty}$-norm would decrease the worse-case amplification of inputs
(disturbances) in the frequency domain. The $H_{2}$-norm of $T(s)$ has a
stochastic interpretation – it equals the asymptotic output variance of the
system when driven by white noise signals (Scherer and Weiland, 2000, pp. 74).
Then, minimizing the $H_{2}$-norm would decrease the effect (in terms of
variance) of stochastic inputs on the system output._

##  3 Fault estimator design

###  3.1 ISS Estimation Error Dynamics

In this section, we derive Linear Matrix Inequality (LMI) conditions for
designing the matrices $E$ and $K$ of the augmented system observer (5). As a
stepping stone, we present a sufficient condition for asymptotic stability of
the origin of the estimation error dynamics when the perturbations,
$\omega_{a}$ and $\nu_{a}$, equal zero (internal stability). Furthermore, we
prove the boundedness of the estimation error in the presence of perturbations
using the concept of input-to-state stability, see Sontag (2008).

######  Definition 5.

_(Input-to-State Stability) The error dynamics (9) is said to be Input-to-
State Stable (ISS) if there exist a class $\mathcal{K}\mathcal{L}$ function
$\psi(\cdot)$ and a class $\mathcal{K}$ function $\bar{\zeta}(\cdot)$ such
that for any initial estimation error $e(t_{0})$ and any bounded inputs
$\omega_{a}(x,u,t)$ and $\nu_{a}(t)$, the solution $e(t)$ of (9) exists for
all $t\geq t_{0}$ and satisfies_

$\|e(t)\|\leq\psi\left(\left\|e\left(t_{0}\right)\right\|,t-t_{0}\right)+\bar{\zeta}\big{(}\|{\left[\begin{array}[]{cc}\omega_{a}(x,u,t)\\\
\nu_{a}(t)\end{array}\right]}\|_{\infty}\big{)}.$ (13)

Note that ISS of the error dynamics (9) implies boundedness of the estimation
error for bounded $\omega_{a}$ and $\nu_{a}$. This follows directly from
$\eqref{ISS_def}$.

######  Lemma 6.

_(ISS Lyapunov Function(Khalil, 2002, Thm. 4.19)) Consider the error dynamics
(9) and let $W(e)$ be a continuously differentiable function such that_

$\alpha_{1}(\|e\|)\leq W(e)\leq\alpha_{2}(\|e\|),$ $\dot{W}(e)\leq-
W_{3}(e),\quad\forall\hskip
2.84526pt\|e\|\geq\xi(\|{\left[\begin{array}[]{cc}\omega_{a}(x,u,t)\\\
\nu_{a}(t)\end{array}\right]}\|),$

_where $\alpha_{1}(\cdot)$ and $\alpha_{2}(\cdot)$ are class
$\mathcal{K}_{\infty}$ functions, $\xi(\cdot)$ is a class $\mathcal{K}$
function, and $W_{3}$ is a continuous positive definite function. Then, the
estimation error dynamics (9) is ISS with gain
$\bar{\zeta}=\alpha_{1}^{-1}(\alpha_{2}(\xi))$._

Let $W(e):={e}^{T}P{e}$ be an ISS Lyapunov function candidate. Then, it
follows from (9) that

$\displaystyle\dot{W}(e)$ $\displaystyle\leq e^{T}\Delta
e-2e^{T}P\bar{B}_{a}{\left[\begin{array}[]{cc}\omega_{a}(x,u,t)\\\
\nu_{a}(t)\end{array}\right]},$ (14)

where

$\displaystyle\Delta:=$ $\displaystyle N^{T}P+PN,\qquad\bar{B}_{a}:=$
$\displaystyle{\left[\begin{array}[]{cc}MD_{a}&-\bar{B}\end{array}\right]}.$
(15)

Inequality (14) implies the following

$\displaystyle\dot{W}(e)\leq$
$\displaystyle-\lambda_{\min}(-\Delta)\|e\|^{2}+2\|e\|\|P\bar{B}_{a}\|\|{\left[\begin{array}[]{cc}\omega_{a}\\\
\nu_{a}\end{array}\right]}\|$ (16) $\displaystyle=$
$\displaystyle-(1-\theta)\lambda_{\min}(-\Delta)\|e\|^{2}-\theta\lambda_{\min}(-\Delta)\|e\|^{2}$
$\displaystyle+2\|e\|\|P\bar{B}_{a}\|\|{\left[\begin{array}[]{cc}\omega_{a}\\\
\nu_{a}\end{array}\right]}\|,$

for any $\theta\in(0,1)$. Therefore, by (16) and Lemma 6, if $\Delta$ is
negative definite, error dynamics (9) is ISS with inputs $\omega_{a}$ and
$\nu_{a}$ and linear ISS-gain

$\bar{\zeta}\biggl{(}\biggl{\|}{\left[\begin{array}[]{cc}\omega_{a}\\\
\nu_{a}\end{array}\right]}\biggl{\|}\biggl{)}=\frac{2\|P\bar{B}_{a}\|}{\theta\lambda_{\min}(-\Delta)}\biggl{\|}{\left[\begin{array}[]{cc}\omega_{a}\\\
\nu_{a}\end{array}\right]}\biggl{\|}.$ (17)

Based on the above discussion, the next proposition formalizes an LMI
condition ($\Delta\prec 0$) that guarantees ISS estimation error dynamics (9).
Without loss of generality, for numerical tractability, we enforce
$\Delta+\epsilon I\preceq 0$ for some arbitrarily small given $\epsilon>0$
instead of $\Delta\prec 0$.

######  Proposition 7.

_(ISS Estimation Error Dynamics) Consider the error dynamics (9) and suppose
there exist matrices $P\in{\mathbb{R}^{n_{z}\times n_{z}}},P\succ
0,R\in{\mathbb{R}^{n_{z}\times m}}$, and $Q\in{\mathbb{R}^{n_{z}\times m}}$
satisfying the inequality_

$X+\epsilon I\preceq 0,$ (18)

_for some given $\epsilon>0$ and matrix $X$ defined as_

$\displaystyle X:=$ $\displaystyle
A_{a}^{\top}P+A_{a}^{\top}C_{a}^{\top}R^{\top}-C_{a}^{T}Q^{T}+PA_{a}+RC_{a}A_{a}$
(19) $\displaystyle-QC_{a},$

_with $A_{a}$ and $C_{a}$ in (4b). Then, the estimation error dynamics in (9)
is ISS with input $[\omega_{a}^{\top},\nu_{a}^{\top}]^{\top}$. Moreover, the
ISS-gain from $[\omega_{a}^{\top},\nu_{a}^{\top}]^{\top}$ to the estimation
error $e$ in (9) is upper bounded by
$2\|P{\left[\begin{array}[]{ccc}(I+EC_{a})D_{a}&-K&E\end{array}\right]}\|\epsilon^{-1}$
with $E=P^{{}^{-1}}R$ and $K=P^{{}^{-1}}Q$._

_Proof_ : The proof can be found in Appendix A. $\blacksquare$

######  Remark 8.

_(LMI Feasibility) If $F_{y}$ in (1) is full row rank (i.e., there are as many
sensor faults as sensors), the LMI in (18) is always infeasible (observability
of $(A_{a},C_{a})$ is lost) Edwards and Tan (2006); Edwards and Spurgeon
(2000); Frank and Wunnenberg (1989). The standard practice to circumvent this
issue is to assume $\text{rank}[F_{y}]<m$, Edwards and Tan (2006); Edwards and
Spurgeon (2000); Frank and Wunnenberg (1989). Note that this is only a
necessary condition for the estimator to exist, but it does not guarantee the
LMI in (18) to be feasible (this has to be checked on a case-by-case basis)._

Proposition 1 is used to enforce that all trajectories of the estimation error
dynamics (9) exist and are bounded for all $t\geq 0$. If the observer gains
$(E,K)$ satisfy the ISS LMI in (18), boundedness and asymptotic stability for
vanishing $\omega_{a}$ and $\nu_{a}$ is guaranteed. This follows directly from
Assumption 1, Assumption 2, and Definition 13.

###  3.2 $H_{\infty}$ Performance Criteria

To maximize the fault reconstruction performance, we seek to minimize the
effect of $\omega_{a}$ (treated as an external disturbance) on the estimation
error dynamics (9). We could use the ISS formulation in Proposition 7 to cast
an optimization problem where we minimize the ISS gain and treat the LMI in
(18) as an optimization constraint. This would indeed reduce the effect of
$\omega_{a}$ and $\nu_{a}$ on the complete vector of estimation errors $e$.
Note, however, that the purpose of the observer is to reconstruct faults only,
so the performance of unrelated estimation errors is not directly relevant. To
this end, we seek to minimize the $H_{\infty}$-norm of the transfer function
from $\omega_{a}$ to the relevant (according to (6)) estimation errors $e_{d}$
as defined in (12).

######  Definition 9.

_( $H_{\infty}$-norm) The $H_{\infty}$-norm $\|T_{e_{d}\omega_{a}}\|_{\infty}$
of the transfer matrix $T_{e_{d}\omega_{a}}(s)$ is given by_

$\|T_{e_{d}\omega_{a}}\|_{\infty}:=\sup_{\mu\in\mathbb{R}^{+}}\sigma_{\max}(T_{e_{d}\omega_{a}}(i\mu)).$

In the following lemma, we state a necessary and sufficient condition for
having a bounded $H_{\infty}$-norm of $T_{e_{d}\omega_{a}}(s)$.

######  Lemma 10.

_(Finite $H_{\infty}$-norm (Scherer and Weiland, 2000, Prop. 3.12)) Consider
the estimation error dynamics (11) with transfer matrix
$T_{e_{d}\omega_{a}}(s)$ in (12). Assume $N$ is Hurwitz and consider a finite
$\lambda>0$. Then, the following statements are equivalent_

  1. 1.

$\|T_{e_{d}\omega_{a}}\|_{\infty}<\lambda$.

  2. 2.

There exists a $P\in{\mathbb{R}^{n_{z}\times n_{z}}},P\succ 0$ satisfying

$\left[\begin{array}[]{ccc}N^{\top}P+PN&-PMD_{a}&\bar{C}_{a}^{\top}\\\
&-\lambda I&\boldsymbol{0}\\\ &*&-\lambda I\end{array}\right]\prec 0,$ (20)

with $(N,M)$ in (5b), $\bar{C}_{a}$ in (7), and $D_{a}$ in (4b).

Based on Lemma 10, the following proposition formalizes an LMI-based condition
guaranteeing $\|T_{e_{d}\omega_{a}}\|_{\infty}<\lambda$, i.e., a finite
$H_{\infty}$-norm of the transfer function $T_{e_{d}\omega_{a}}(s)$.

######  Proposition 11.

_(Finite $H_{\infty}$-norm) Consider the estimation error dynamics (11) with
transfer matrix $T_{e_{d}\omega_{a}}(s)$ in (12). Assume there exist matrices
$P\succ 0,R,$ $Q$, and scalar $\lambda>0$ satisfying_

$\displaystyle\left[\begin{array}[]{ccc}X&-(P+RC_{a})D_{a}&\bar{C}_{a}^{\top}\\\
&-\lambda I&\boldsymbol{0}\\\ &*&-\lambda I\end{array}\right]\prec 0,$ (21)

_with $X$ defined in (19), $\bar{C}_{a}$ in (7), and the remaining matrices in
(4b). Then, $\|T_{e_{d}\omega_{a}}\|_{\infty}<\lambda$._

_Proof_ : Clearly, using the change of variables in (26), and discussion in
the Appendix A, (21) is equivalent to (20), which concludes the result of the
proposition. $\blacksquare$

###  3.3 $H_{2}$ Performance Criteria

The other terms affecting the performance of the fault reconstruction are
measurement noise and its derivative, collected in the vector $\nu_{a}$. We
seek to minimize the effect of $\nu_{a}$ on the estimation error dynamics (9).
To this end, we seek to minimize the $H_{2}$-norm of the transfer function
from $\nu_{a}$ to the performance output $e_{d}$ defined in (12).

######  Definition 12.

_( $H_{2}$-norm) The $H_{2}$-norm, $\|T_{e_{d}\nu_{a}}\|_{H_{2}}$, of the
transfer matrix $T_{e_{d}\nu_{a}}(s)$ is given by_

$\|T_{e_{d}\nu_{a}}\|_{H_{2}}:=\sqrt{\frac{1}{2\pi}\operatorname{trace}\int_{-\infty}^{\infty}T_{e_{d}\nu_{a}}(i\mu)T_{e_{d}\nu_{a}}^{H}(i\mu)\mathrm{~{}d}\mu}.$

In the following lemma, we state a necessary and sufficient condition for
having a bounded $H_{2}$-norm of $T_{e_{d}\nu_{a}}(s)$.

######  Lemma 13.

_(Finite $H_{2}$-norm (Scherer and Weiland, 2000, Prop. 3.13)) Consider the
estimation error dynamics (11) with transfer matrix $T_{e_{d}\nu_{a}}(s)$ in
(12). Assume $N$ is Hurwitz and consider a finite $\gamma>0$. Then, the
following statements are equivalent_

  1. 1.

$\|T_{e_{d}\nu_{a}}\|_{H_{2}}<\gamma$.

  2. 2.

There exists a $P\succ 0$ and $Z$ satisfying

$\displaystyle\left[\begin{array}[]{cc}N^{\top}P+PN&P\bar{B}\\\ &-\gamma
I\end{array}\right]\prec
0,\quad\left[\begin{array}[]{cc}P&\bar{C}_{a}^{\top}\\\
&Z\end{array}\right]\succ 0,$ (22)
$\displaystyle\operatorname{trace}(Z)<\gamma,$

with $N$ in (5b), $\bar{B}$ in (10), and $\bar{C}_{a}$ in (7).

Based on Lemma 13, the following proposition formalizes an LMI-based condition
guaranteeing $\|T_{e_{d}\nu_{a}}\|_{H_{2}}<\gamma$, i.e., a finite
$H_{2}$-norm of $T_{e_{d}\nu_{a}}(s)$.

######  Proposition 14.

_(Finite $H_{2}$-norm) Consider the estimation error dynamics (11) with
transfer matrix $T_{e_{d}\nu_{a}}(s)$ in (12). Assume there exist matrices
$P\succ 0,R,$ $Q$, and scalar $\gamma>0$ satisfying_

$\displaystyle\left[\begin{array}[]{cc}X&{\left[\begin{array}[]{cc}Q&-R\end{array}\right]}\\\
&-\gamma I\end{array}\right]\prec
0,\quad\left[\begin{array}[]{cc}P&\bar{C}_{a}^{\top}\\\
&Z\end{array}\right]\succ 0,$ (23)
$\displaystyle\operatorname{trace}(Z)<\gamma,$

_with $X$ defined in (19) and $\bar{C}_{a}$ in (7). Then,
$\|T_{e_{d}\nu_{a}}\|_{H_{2}}<\gamma$._

_Proof_ : Similar to Proposition 11, it can be shown that (23) is equivalent
to (22). $\blacksquare$

Using Proposition 11, we can formulate a semi-definite program where we seek
to minimize the $H_{\infty}$-norm of $T_{e_{d}\omega_{a}}(s)$. Similarly,
using Proposition 14, we can formulate another semi-definite program where we
seek to minimize the $H_{2}$-norm of $T_{e_{d}\nu_{a}}(s)$. However, in the
presence of both unknown perturbations ($\omega_{a}$ and $\nu_{a}$), the
$H_{\infty}$-norm and $H_{2}$-norm can not be minimized simultaneously due to
conflicting objectives. To attenuate the average effect of the measurement
noise on the estimations errors, a relatively slow observer is required, which
does not react to every small change in the measured output. On the other
hand, to reduce the effect of $\omega_{a}$, a high-gain observer is preferred,
which tries to estimate the fault as accurately as possible. It follows that
there is a trade-off between estimation performance the noise sensitivity. To
address

To address this trade-off, a convex program where we seek to minimize the
$H_{\infty}$-norm and constrain the $H_{2}$-norm, for the same performance
output $e_{d}$, is proposed. The same tools allow minimizing the $H_{2}$-norm
for a constrained $H_{\infty}$-norm (as a dual problem), for which we present
simulation results in Section 4. Moreover, we add the ISS LMI in (18) as a
constraint to these programs to enforce that the resulting observer also
guarantees boundedness for bounded perturbations and asymptotic stability for
vanishing $\omega_{a}$ and $\nu_{a}$. The latter is essential to avoid
observer divergence.

######  Theorem 15.

_(Fault Estimator Design) Consider the augmented dynamics (4), the estimator
(5), the corresponding estimation error dynamics (11), and the transfer
functions (12). To design optimal mixed $H_{2}/H_{\infty}$ fault estimators
(5)-(6), solve the following convex program_

$\begin{array}[]{cl}\min\limits_{P,R,Q,Z,\lambda,\gamma}&\lambda\\\
\text{s.t.}&\vspace{1 mm}X+\epsilon I\preceq 0,\\\
&\left[\begin{array}[]{ccc}X&-(P+RC_{a})D_{a}&\bar{C}_{a}^{\top}\\\ &-\lambda
I&\boldsymbol{0}\\\ &*&-\lambda I\end{array}\right]\prec 0,\\\\[14.22636pt]
&\left[\begin{array}[]{cc}X&{\left[\begin{array}[]{cc}Q&-R\end{array}\right]}\\\
&-\gamma I\end{array}\right]\prec
0,\quad\left[\begin{array}[]{cc}P&\bar{C}_{a}^{\top}\\\
&Z\end{array}\right]\succ 0,\\\\[8.53581pt]
&\operatorname{trace}(Z)<\gamma,\qquad\gamma\leq\gamma_{max},\\\\[2.84526pt]
&P\succ 0,\qquad\lambda,\gamma>0,\end{array}$ (24)

_with given $\epsilon,\gamma_{max}>0$, $\bar{C}_{a}$ in (7), $X$ as defined in
(19), and the remaining matrices as defined in (4b). Denote the optimizers as
$P^{\star},R^{\star},Q^{\star},Z^{\star},\lambda^{\star},\gamma^{\star}$ and
define the matrices $E^{\star}:=P^{\star^{-1}}R^{\star}$ and
$K^{\star}:=P^{\star^{-1}}Q^{\star}$. Then, we have the following:_

  1. 1.

The ISS-gain from inputs $\omega_{a}$ and $\nu_{a}$ to the estimation error
$e$ in (11) is upper bounded by
$2\|P^{\star}{\left[\begin{array}[]{ccc}(I+E^{\star}C_{a})D_{a}&-K^{\star}&E^{\star}\end{array}\right]}\|\epsilon^{-1}$.

  2. 2.

The $H_{\infty}$-norm of transfer function $T_{e_{d}\omega_{a}}(s)$ in (12) of
the error dynamics (11) is upper bounded by $\lambda^{\star}$ (i.e.,
$\|T_{e_{d}\omega_{a}}\|_{\infty}<\lambda^{\star}$).

  3. 3.

The $H_{2}$-norm of transfer function $T_{e_{d}\nu_{a}}(s)$ in (12) of the
error dynamics (11) is upper bounded by $\gamma^{\star}$ (i.e.,
$\|T_{e_{d}\nu_{a}}\|_{H_{2}}<\gamma^{\star}$).

_Proof :_ Theorem 15 follows from the above discussion and Propositions 7-14.
$\blacksquare$

##  4 Simulation Results

In this section, we evaluate the proposed method by a two-link robotic
manipulator with revolute joints, as shown in Figure 1. This system is
employed as a case study due to its highly nonlinear behavior. The Euler-
Lagrange equations of motions of this system are described by two second-order
coupled differential equations (one for each link), which are given as (Sira-
Ramírez et al., 2018, pp. 86-88)

Figure 1: Schematic of the two-link robot manipulator.

$\underline{M}(q)\ddot{q}+\underline{C}(q,\dot{q})\dot{q}+\underline{G}(q)=\tau+\tau_{f}-\underline{D}\dot{q},$

with

$\begin{split}&\underline{M}(q)=\begin{bmatrix}\underline{M}_{11}&\underline{M}_{12}\\\
*&\underline{M}_{22}\end{bmatrix},\quad\underline{D}=\begin{bmatrix}d_{1}&0\\\
0&d_{2}\end{bmatrix},\quad\underline{G}(q)=\begin{bmatrix}\underline{G}_{1}\\\
\underline{G}_{2}\end{bmatrix},\\\
&\underline{C}(q,\dot{q})=\begin{bmatrix}-2m_{2}l_{1}l_{c_{2}}\sin(\phi)\dot{\phi}&-m_{2}l_{1}l_{c_{2}}\sin(\phi)\dot{\phi}\\\
m_{2}l_{1}l_{c_{2}}\sin(\phi)\dot{\theta}&0\end{bmatrix},\\\ \end{split}$

where

$\displaystyle\underline{M}_{11}=$ $\displaystyle
m_{1}l_{c_{1}}^{2}+m_{2}l_{1}^{2}+m_{2}l_{c_{2}}^{2}+2m_{2}l_{1}l_{c_{2}}\cos(\phi)+I_{1}+I_{2},$
$\displaystyle\underline{M}_{12}=$ $\displaystyle
m_{2}l_{c_{2}}^{2}+m_{2}l_{1}l_{c_{2}}\cos(\phi)+I_{2},\quad\underline{M}_{22}=m_{2}l_{c_{2}}^{2}+I_{2},$
$\displaystyle\underline{G}_{1}=$
$\displaystyle\left(m_{1}l_{c_{1}}+m_{2}l_{1}\right)g\sin(\theta)+m_{2}l_{c_{2}}g\sin(\theta+\phi),$
$\displaystyle\underline{G}_{2}=$ $\displaystyle
m_{2}l_{c_{2}}g\sin(\theta+\phi),$

and $q=[\theta\,\,\phi]^{T}$ is the generalized coordinate,
$\tau=[\tau_{\theta}\,\,\tau_{\phi}]^{T}=[0.5\sin\left(\frac{2\pi}{40}t\right)\,\,0]^{T}$
the input, and $\tau_{f}=[\tau_{f_{\theta}}\,\,\tau_{f_{\phi}}]^{T}$ the
actuator fault. In addition, the measurement is $q$, which contains the
angular positions of both links. The parameters are as follows (Sira-Ramírez
et al., 2018, p. 92):

$\begin{array}[]{lll}m_{1}=0.263\hskip 2.84526ptkg,&m_{2}=0.1306\hskip
2.84526ptkg,\\\ I_{1}=0.002\hskip 2.84526ptNm^{2},&I_{2}=0.00098\hskip
2.84526ptNm^{2},\\\ l_{1}=0.3\hskip 2.84526ptm,&l_{2}=0.3\hskip 2.84526ptm,\\\
l_{c_{1}}=0.15\hskip 2.84526ptm,&l_{c_{2}}=0.15\hskip 2.84526ptm,\\\
d_{1}=0.03\hskip 2.84526ptNms/rad,&d_{2}=0.005\hskip 2.84526ptNms/rad,\\\
g=9.81\hskip 2.84526ptm/s^{2}.\end{array}$

By selecting $x_{1}:=q$ and $x_{2}:=\dot{q}$, the system can be written
directly in the form of (2) (no transformation is needed for this example) as
follows

$\left\\{\begin{aligned} \dot{x}=&Ax+S\big{(}g(Vx,u)+f_{n}(x)\big{)},\\\
y=&Cx+\nu,\end{aligned}\right.$ (25)

where $x:=[x_{1},x_{2}]^{T}=[q,\dot{q}]^{T}$ is the state vector,

$\displaystyle A$
$\displaystyle=\left[\begin{array}[]{cc}\boldsymbol{0}_{2\times 2}&I_{2}\\\
\boldsymbol{0}_{2\times 2}&-\underline{M}_{l}^{-1}\underline{D}\\\
\end{array}\right],$ $\displaystyle S$
$\displaystyle=\left[\begin{array}[]{l}\boldsymbol{0}_{2\times 2}\\\
I_{2}\end{array}\right],$ $\displaystyle C$
$\displaystyle=\left[\begin{array}[]{cc}I_{2}&\boldsymbol{0}_{2\times
2}\end{array}\right],$ $\displaystyle V$ $\displaystyle=I_{4},$ $\displaystyle
g_{nf}(Vx,u)=$
$\displaystyle\underline{M}^{-1}(x_{1})\big{(}u-\underline{D}x_{2}-\underline{C}(x_{1},x_{2})x_{2}-\underline{G}(x_{1})\big{)}$
$\displaystyle+\underline{M}_{l}^{-1}\underline{D}x_{2},\qquad u=\tau,$
$\displaystyle f_{n}(x)=$ $\displaystyle\underline{M}^{-1}(x_{1})\tau_{f},$

and $\underline{M}_{l}=\underline{M}(\boldsymbol{0})$. The measurement noise
$\nu$ is generated from a uniform distribution with the maximum of $0.1$ $rad$
and the minimum of $-0.1$ $rad$, sample time of $0.1$ $sec$, for both sensors,
which is a large noise level. We have used such noise levels here to compare
different above-mentioned approaches. We set the initial conditions of the
system to zero. The actuator fault signal is given as

${}\tau_{f}(t)=\begin{cases}\tau_{f_{\theta}}=0,&\quad\text{if }t<50,\\\
\tau_{f_{\phi}}=0,&\quad\text{if }t<50,\\\
\tau_{f_{\theta}}=0.2\sin\left(\frac{2\pi}{10}(t-50)\right),&\quad\text{if
}t\geq 50,\\\ \tau_{f_{\phi}}=-0.05,&\quad\text{if }t\geq 50.\\\ \end{cases}$

Note that the nonlinear fault $f_{n}(x,u)$ has two fault entries; one affects
the third entry and the other, the fourth entry of the manipulator dynamics.
Although each actuator fault affects just one actuator, since we are
estimating $f_{n}(x)$ and not $\tau_{f}$, the occurrence of only one actuator
fault causes non-zero values for both entries of $f_{n}(x)$.

Figure 2: The actual third entry fault (blue) and its estimates using the
optimal $H_{\infty}$ estimator (red), the optimal $H_{2}$ estimator with
constrained $H_{\infty}$-norm (magenta), and the optimal $H_{\infty}$
estimator with constrained $H_{2}$-norm (black). Figure 3: The actual third
entry fault (blue) and its estimates using the optimal $H_{2}$ estimator
(red). Figure 4: The actual fourth entry fault (blue) and its estimates using
the optimal $H_{\infty}$ estimator (red), the optimal $H_{2}$ estimator with
constrained $H_{\infty}$-norm (magenta), and the optimal $H_{\infty}$
estimator with constrained $H_{2}$-norm (black). Figure 5: The actual fourth
entry fault (blue) and its estimates using the optimal $H_{2}$ estimator
(red).

Next, we augment the system (25) using (4) with $r=4$. For the augmented
system, the observer of the form (5) can be designed in four different ways as
follows:

  1. 1.

Only minimizing the $H_{\infty}$-norm of $T_{e_{d}\omega_{a}}(s)$ in (12) of
the error system (11), subject to ISS LMI in (18).

  2. 2.

Only minimizing the $H_{2}$-norm of $T_{e_{d}\nu_{a}}(s)$ in (12) of the error
system (11), subject to ISS LMI in (18).

  3. 3.

Minimizing the $H_{\infty}$-norm of $T_{e_{d}\omega_{a}}(s)$ in (12) of the
error system (11), subject to ISS LMI in (18) and an upper bound for the
$H_{2}$-norm of transfer function $T_{e_{d}\nu_{a}}(s)$ in (12) of the error
system (11) with an upper bound (Theorem 15).

  4. 4.

Minimizing the $H_{2}$-norm of $T_{e_{d}\nu_{a}}(s)$ in (12) of the error
system (11), subject to ISS LMI in (18) and an upper bound for the
$H_{\infty}$-norm of transfer function $T_{e_{d}\omega_{a}}(s)$ in (12) of the
error system (11) with an upper bound.

To solve the associated convex programs, we use the YALMIP toolbox and MOSEK
solver in MATLAB. Since the designed observer is estimating the lumped
disturbance, the nonlinear actuator fault estimates are commutable using (6).

Finally, we apply all of the designed observers to the manipulator system in
simulation. The initial conditions of the observers in the simulation are
taken as a vector with all values equal to $0.01$. Figure 2 and Figure 4
depict the estimated nonlinear faults and their actual values, using all the
observers described above but the second one (i.e., only minimizing the
$H_{2}$-norm, which is given in Fig. 3 and Fig. 5) for the faults affecting
the third and the fourth entries of the manipulator dynamics, respectively. It
can be seen that only for the $H_{\infty}$-norm is minimized, although the
fault estimation performance is decent, the results are too noisy. On the
other hand, when we only minimize for the $H_{2}$-norm, the fault estimation
performance is too poor. Therefore, minimizing one of the $H_{\infty}$ or
$H_{2}$ norms and constraining the other one, known as a mixed
$H_{2}/H_{\infty}$ approach, gives the best solution. Herein, the fault
estimation performance and the noise attenuation are always a trade-off, and
the proposed synthesis approach allows us to make this trade-off in a
constructive manner.

##  5 Conclusion

This paper presents a method for the robust reconstruction of time-varying
faults in nonlinear systems, using ultra-local observers for modeling faults
and nonlinearities. The design approach proposes semi-definite programs that
allow for a trade-off between fault estimation performance and noise
sensitivity. The numerical simulations for a two-link manipulator illustrate
the performance of the proposed approach and the potential for highly
nonlinear systems. Future work includes incorporating model uncertainty.

##  Appendix A Proof of Proposition 1

First, we need to show that $\Delta$ is equivalent to (19). Using $\Delta$
defined in (15) and (5b), we can write $\Delta$ in terms of the original
observer gains $(E,K)$ as

$\displaystyle\Delta=$ $\displaystyle
A_{a}^{T}\left(I+EC_{a}\right)^{T}P-C_{a}^{T}KP+P\left(I+EC_{a}\right)A_{a}$
$\displaystyle-PKC_{a},$

Consider the following change of variables

$R:=PE,\qquad Q:=PK.$ (26)

Applying (26) on the above expanded $\Delta$, the linear inequality (19) can
be concluded. Clearly, $\Delta+\epsilon I\preceq 0$ implies
$\lambda_{\min}(-\Delta)\geq\epsilon$. Then, using (17), we can conclude the
bound on the ISS-gain.

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