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

Farhad Ghanipoor1, Carlos Murguia1, Peyman Mohajerin Esfahani2, Nathan van de
Wouw1

###### Abstract.

This paper proposes a nonlinear estimator for the robust reconstruction of
process and sensor faults for a class of uncertain nonlinear systems. The
proposed fault estimation method augments the system dynamics with an ultra-
local (in time) internal state-space representation (a finite chain of
integrators) of the fault vector. Next, a nonlinear state observer is designed
based on the known parts of the augmented dynamics. This nonlinear filter
(observer) reconstructs the fault signal as well as the states of the
augmented system. We provide sufficient conditions that guarantee stability of
the estimation error dynamics: firstly, asymptotic stability (i.e., perfect
fault estimation) in the absence of perturbations induced by fault model
mismatch (mismatch between internal, ultra-local model for the fault and the
actual fault characteristics), uncertainty, external disturbances, and
measurement noise and, secondly, Input-to-State Stability (ISS) of the
estimation error dynamics is guaranteed in the presence of these
perturbations. In addition, to support performance-based estimator design, we
provide Linear Matrix Inequality (LMI) conditions for $\mathcal{L}_{2}$-gain
and $\mathcal{L}_{2}-\mathcal{L}_{\infty}$ induced norm and cast the synthesis
of the estimator gains as a semi-definite program where the effect of model
mismatch and external disturbances on the fault estimation error is minimized
in the sense of $\mathcal{L}_{2}$-gain, for an acceptable
$\mathcal{L}_{2}-\mathcal{L}_{\infty}$ induced norm with respect to
measurement noise. The latter result facilitates a design that explicitly
addresses the performance trade-off between noise sensitivity and robustness
against model mismatch and external disturbances. Finally, numerical results
for a benchmark system illustrate the performance of the proposed
methodologies.

1\. Department of Mechanical Engineering, Eindhoven University of Technology,
The Netherlands

2\. Delft Center for Systems and Control, Delft University of Technology, The
Netherlands

##  1\. Introduction

Process reliability is essential in many engineering systems, such as high-
tech equipment, energy systems, automotive technology and health applications.
Predictive maintenance technology is a key enabler for improving process
reliability. A fundamental element for predictive maintenance is fault
estimation. That is, we do not only need to know the presence and source of
the fault (fault detection and isolation, see [1, 2, 3], and references
therein) but also its nature/severity (fault estimation). As an example,
suppose the fault is small and/or slowly increasing in magnitude (slow
compared to the system time scale). In this case, if accurate estimates of
fault-induced signals are available the fault severity can be quantified and
predictive maintenance can be scheduled accordingly. The latter is only
possible if we estimate fault signals (at least their magnitude) using
available information (inputs, measured outputs, and system models).
Therefore, this paper focuses on the problem of fault estimation for a class
of uncertain nonlinear systems.

_Existing Literature:_ Available methods for fault estimation can be divided
into three categories:

_1)_ Linear Systems: Numerous fault estimation methods have been developed for
linear dynamical systems (see, e.g., [4, 5] for results on linear stochastic
and switching systems). However, most practical systems, such as those in
robotics, transportation systems, power networks, manufacturing, and water
distribution, are nonlinear in nature.

_2)_ Nonlinear Systems/Linear Filters: Methods for nonlinear systems are still
under development, see, e.g., [6, 7, 8, 9]. For fault estimation in nonlinear
systems, linear or nonlinear filters can be developed. Some of the existing
literature on fault detection for nonlinear systems can be adapted to address
the fault estimation problem [10]. For instance, [11] provides a linear filter
for fault detection of nonlinear systems in which the linear filter is
designed by minimizing the nonlinearity effect on the filter output subject to
a bound on the effects of fault on the filter output. In this result, by
constraining the mapping from the fault to the filter output, the fault signal
can be estimated. Similar results can be found in [12] where instead of
minimizing the nonlinearity effect, the output mismatch of the actual and
simulation-based system is minimized to provide robustifcation against model
mismatch.

_3)_ Nonlinear Systems/Nonlinear Filters: For fault estimation in nonlinear
systems, also nonlinear filters can be used [13, 14, 15, 16]. Most of the
existing results construct nonlinear observers by incorporating nonlinear
dynamics of the system as nonlinear filters for fault estimation. Because
these results consider nonlinear dynamics in the filter structure, they can
capture the behavior of nonlinear systems accurately and as a consequence
provide better fault estimation. However, to provide such filters, due to
nonlinearities, some assumptions on the class of systems and faults are
required. Below we discuss some of these results.

As discussed in [13], the authors address the fault estimation 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 [17], 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.

In [6], the matching condition does not need to be satisfied. However, they do
not consider model uncertainty, external disturbances, and measurement noise.
The authors consider Lipschitz nonlinearities, simultaneous sensor and process
faults, and adopt a standard fault observability condition [18] on the linear
part of the dynamics. Therein, the problem is tackled 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 [19], simultaneous additive and multiplicative
process faults are considered in the scope of discrete-time system models.
They address the fault estimation problem by decoupling process nonlinearities
and perturbations 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 assumptions on the
system dynamics, which significantly limits the applicability of these
results.

We remark that all the above-mentioned results for uncertain nonlinear systems
guarantee approximate reconstruction of fault vector 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 state-space
representations of fault vector makes it challenging to enforce zero error
fault estimation. We propose a fault estimation scheme for process and sensor
faults that allows to guarantee zero error in the absence of uncertainty,
external disturbances, and measurement noise for some classes of faults and
ensures robust fault estimate in the presence of perturbations.

This scheme incorporates an internal representation of the fault vector, where
we use the notion of _ultra-local models_ [20], [21] phenomenological models
valid for short time intervals. We then extend the system dynamics with the
(internal) ultra-local state-space of the fault vector to construct an
augmented dynamics. Based on the known parts of the augmented dynamics, a
nonlinear observer is proposed to estimate the states of the original system
and the ultra-local fault (internal) system. We derive the error dynamics of
the observer in which the fault model mismatch (mismatch between actual
internal system and its model), uncertainty mismatch (mismatch between actual
uncertainty and its model), external disturbances, and measurement noise enter
as external perturbations. The fault estimation problem is reformulated as a
robust (against the mentioned perturbations) state estimation problem in the
error dynamics. The main contributions of this paper are as follows:

  1. (a)

We propose an optimization-based estimation scheme for time-varying process
and sensor faults that exploits the notion of ultra-local models, does not
assume the matching condition, and robustly provides fault estimates;

  2. (b)

We provide sufficient conditions that guarantee asymptotic stability of the
origin of the estimation error dynamics in the absence of perturbations
induced by the fault and uncertainty model mismatches, external disturbances,
and measurement noise; Input-to-State-Stability (ISS) guarantee in the
presence of these perturbations (Proposition 1);

  3. (c)

We present sufficient conditions that guarantee an upper bound on the
$\mathcal{L}_{2}$-gain from perturbations induced by fault and uncertainty
model mismatches and external disturbances to the fault estimation error
(Proposition 2);

  4. (d)

We put forward sufficient conditions that guarantee an upper bound on the
$\mathcal{L}_{2}-\mathcal{L}_{\infty}$ induced norm from perturbations induced
by measurement noise to the fault estimation error (Proposition 3);

  5. (e)

We come up with a synthesis algorithm for the estimator gains in terms of the
solution of semi-definite programs where we minimize the
$\mathcal{L}_{2}$-gains in (c), subject to stability conditions in (b) and
desired upper bounds on the $\mathcal{L}_{2}-\mathcal{L}_{\infty}$ induced
norms in (d) (Theorem 3.1). These computationally tractable design conditions
provide a means to performance trade-offs in terms of robustness with respect
to different disturbances/perturbations.

This paper is a generalized version of the preliminary result published in the
conference paper [22]. Compared to [22], here we address a more general
problem setting by considering state-dependent faults, external disturbances,
model uncertainties and measurement noise. Furthermore, the fault estimate is
robustified against perturbations induced by fault and uncertainty model
mismatches, external disturbances, and measurement noise in the
$\mathcal{L}_{2}$ sense for a desired worst-case noise amplification in the
$\mathcal{L}_{2}-\mathcal{L}_{\infty}$ sense.

The remainder of this paper is organized as follows. In Section 2, problem
formulation is presented, first in a high level sense and then with precise
mathematical details. The proposed method for the fault estimator design is
described in Section 3. In Section 4, the method is applied to a benchmark
example in simulation. Section 5 presents the conclusion and final remarks.

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 by
$I$ 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 by
$\boldsymbol{0}$ 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 $\ell_{2}$ vector norm (Euclidean norm) and the matrix
norm induced by the $\ell_{2}$ vector norm are both denoted as $||\cdot||$ and
the $\ell_{\infty}$ vector norm is showed by $||\cdot||_{\infty}$. We use
$\mathcal{L}_{2}(0,T)$ (or simply $\mathcal{L}_{2}$) to denote vector-valued
functions $z:[0,T]\to\mathbb{R}^{k}$ satisfying
$\|z(t)\|_{\mathcal{L}_{2}}:=\int_{0}^{T}\|z(t)\|^{2}dt<\infty$. For a vector-
valued signal $f$ defined for all $t\geq 0$,
$||f||_{\mathcal{L}_{\infty}}:=\sup_{t\geq 0}||f(t)||^{2}$ and $f^{(r)}$ shows
the entry-wise $r^{th}$-time total derivative. 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 (i.e., $\frac{\partial W}{\partial
e}\frac{de}{dt}$). We often omit time dependencies for notation simplicity.
The notation $(f,d)$ stands for the column vector composed of the (vector or
scalar) elements $f$ and $d$.

##  2\. Problem Formulation

Consider the nonlinear system

$\left\\{\begin{aligned}
\dot{x}&=Ax+B_{u}u+S_{g}g(V_{g}x,u,t)+S_{\eta}\eta(V_{\eta}x,u,t)+B_{f}f(x,u,t)\color[rgb]{0,0,0}+B_{\omega}\omega,\\\
y&=Cx+D_{f}f(x,u,t)\color[rgb]{0,0,0}+D_{\nu}\nu,\end{aligned}\right.$ (1a)
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, and function
$g:\mathbb{R}^{n_{v_{g}}}\times\mathbb{R}^{l}\times\mathbb{R}^{+}\to\mathbb{R}^{n_{g}}$
is a nonlinear known vector field. Function
$\eta:\mathbb{R}^{n_{v_{\eta}}}\times\mathbb{R}^{l}\times\mathbb{R}^{+}\to\mathbb{R}^{n_{\eta}}$
denotes unknown modeling uncertainty. Signals
$\omega:\mathbb{R}^{+}\to\mathbb{R}^{n_{\omega}}$ and
$\nu:\mathbb{R}^{+}\to\mathbb{R}^{m_{\nu}}$ are unknown bounded disturbances;
the former with unknown frequency range and the latter with high frequency
content (e.g., related to measurement noise). Function
$f:\mathbb{R}^{n}\times\mathbb{R}^{l}\times\mathbb{R}^{+}\to\mathbb{R}^{n_{f}}$
denotes the unknown fault vector, which contains both process and sensor
faults. Note that $f$ can represent any types of additive or multiplicative
faults. Matrices
$(A,B_{u},S_{g},V_{g},S_{\eta},V_{\eta},B_{f},B_{\omega},C,D_{f},D_{\nu})$ are
of appropriate dimensions,
${n},{m},l,n_{v_{g}},n_{g},n_{v_{\eta}},n_{\eta},n_{\omega},n_{f},m_{\nu}\in\mathbb{N}$.
Matrix $B_{f}$ denotes the process fault distribution matrix while matrix
$D_{f}$ represents the contribution of the fault signal to sensor
measurements. Matrices $S_{g}$ and $S_{\eta}$ are used to indicate in which
equation(s) the nonlinearity $g$ and the uncertainty $\eta$ appear explicitly,
and matrices $V_{g}$ and $V_{\eta}$ to indicate which states play a role in
the nonlinearity and uncertainty, respectively.

The objective of this paper is to estimate the fault vector $f$ using the real
time input, output data and the available known models. In the system (1a),
clearly the uncertainty $\eta(\cdot)$ is unknown, which challenges fault
estimation. We can either robustify the fault estimate error against the
complete uncertainty or use an approximated model of it (obtained, e.g., using
data-based methods) and robustify the fault estimate error agaisnt the
remaining (smaller) uncertainty model mismatch. Without loss of generality, we
assume that we can write $\eta(\cdot)$ in (1a) as:

$\displaystyle\eta(V_{\eta}x,u,t)=\eta_{l_{x}}(V_{\eta}x,u,t)+\delta\eta_{l_{x}}(V_{\eta}x,u,t),$
(1b)

where
$\delta\eta_{l_{x}}(V_{\eta}x,u,t):=\eta(V_{\eta}x,u,t)-\eta_{l_{x}}(V_{\eta}x,u,t)$
and $\eta_{l_{x}}(\cdot)$ denotes any prior (possibly inaccurate)
approximation, we may have of $\eta$. If we do not have such an approximated
model, we just take $\eta_{l_{x}}=\mathbf{0}$ an carry out further designs
considering the complete $\eta$.

In the following, we state the required assumptions on the system (1).

######  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 Vector)_ The fault vector $f(x(t),{u}(t),t)$ in (1)
is $r$ times differentiable, i.e., the total time derivatives
${f}^{(1)}(x(t),{u}(t),t)$, ${f}^{(2)}(x(t),{u}(t),t)$, …,
$f^{(r)}(x(t),{u}(t),t)$ exist and are continuous, and
$f^{(r)}(x(t),{u}(t),t)$ is bounded uniformly in $t$.

######  Assumption 3.

_(Bounded Disturbances)_ The disturbance vectors $\omega$ and $\nu$ in (1) are
bounded uniformly in $t$, and $\nu$ is differentiable, i.e., the total
derivative with respect to time $\dot{\nu}(t)$ exists, is continuous, and
bounded uniformly in $t$.

We aim to robustify the fault estimate error against unknown bounded external
disturbances (low or high frequency) and (fault and uncertainty) model
mismatches. For fault estimation, we consider nonlinear filters with the
following structure:

$\left\\{\begin{aligned} \dot{z}=&h(z,u,y;\theta),\\\
\hat{f}=&\phi(z,y;\theta),\end{aligned}\right.$ (2)

where $z\in{\mathbb{R}^{n_{z}}}$ is the internal state of the filter with
${n_{z}}\in\mathbb{N}$. Functions
$h:\mathbb{R}^{n_{z}}\times\mathbb{R}^{l}\times\mathbb{R}^{m}\to\mathbb{R}^{n_{z}}$
and $\phi:\mathbb{R}^{n_{z}}\times\mathbb{R}^{m}\to\mathbb{R}^{n_{f}}$
characterize the filer structure, $\theta$ denotes design parameters.

Define the fault estimate error as

$e_{f}:=\hat{f}-f.$

Now, we can state the problem we aim to solve at a high level.

######  Problem 1.

_(Fault Reconstruction - High Level)_ Consider the uncertain nonlinear system
(1) with known input and output signals, $u(t)$ and $y(t)$, and the nonlinear
fault estimator filter (2). For given $r$, design the filter parameters
$\theta$ such that:
_1) Stability:_ all trajectories of the filter (2) exist and are globally
uniformly ultimately bounded for $t\geq 0$_;_
_2) Disturbance Attenuation:_ for $\nu=\mathbf{0}$, the $\mathcal{L}_{2}$-gain
from $(\delta\eta_{l_{x}},\omega,f^{(r)})$ to $e_{f}$ is bounded by some known
$c_{1}>0$, for $t\geq 0$_;_
_3) Noise Rejection:_ for $(\delta\eta_{l_{x}},\omega,f^{(r)})=\mathbf{0}$,
the $\mathcal{L}_{2}-\mathcal{L}_{\infty}$ induced gain from $(\nu,\dot{\nu})$
to $e_{f}$ is bounded by some known $c_{2}>0$, for $t\geq 0$.

We can further state the above problem as optimal filter design in the sense
of disturbance attenuation or noise rejection. Now, we want to restate Problem
1, mathematically precisely. To this end, in what follows first, we discuss
the fault estimator filter architecture.

###  2.1. Ultra Local Fault Representation

Let us introduce some preliminaries which are required to present the fault
estimator filter architecture. Considering that the fault $f(x(t),{u}(t),t)$
in (1a) is an implicit function of time, for all $x(t)$ and $u(t)$, we can
write an entry-wise $r$-th order Taylor time-polynomial approximation at time
$t$ of $f$ as $\bar{f}=a_{0}+a_{1}t+\dots+a_{r-1}t^{r-1}$ with coefficients
$a_{i}\in{\mathbb{R}^{n_{f}}},i=0,\dots,r-1$. This model can be written in
state-space:

$\left\\{\begin{aligned} \dot{\bar{\zeta}}_{j}&=\bar{\zeta}_{j+1},\qquad
0<j<r,\\\ \dot{\bar{\zeta}}_{r}&=\mathbf{0},\\\
\bar{f}&=\bar{\zeta}_{1},\end{aligned}\right.$ (3)

where $\bar{\zeta}_{j}\in{\mathbb{R}^{n_{f}}}$. Clearly, in the above model we
have $\bar{f}^{(r)}=\mathbf{0}$, which might not be the case for actual fault
signal $f$. Under Assumption 2, the actual internal state-space representation
of the fault $f$ is as follows:

$\left\\{\begin{aligned} \dot{\zeta}_{j}&=\zeta_{j+1},\qquad 0<j<r,\\\
\dot{\zeta}_{r}&=f^{(r)},\\\ f&=\zeta_{1},\end{aligned}\right.$ (4)

where $\zeta_{j}\in{\mathbb{R}^{n_{f}}}$. As you see, the accuracy of the
approximated model (3) increases as $f^{(r)}$ goes to zero (entry-wise), and
it is exact for $f^{(r)}=\mathbf{0}$ (since we have
$\dot{\zeta}_{r}=\dot{\zeta}_{1}^{(r)}={f}^{(r)}=0$, see (4)). Model (3) is
used to construct a fault estimator filter that ultra-locally [20], [21] acts
as a self-updating polynomial spline approximating the actual value of the
fault. To design such a filter, in the following section, we extend the system
state, $x(t)$, with the states of the actual fault internal state
$\zeta_{j}(t),j\in\left\\{1,\ldots,r\right\\}$, and augment the system
dynamics in (1) with (4). We then design a nonlinear filter (observer) for the
augmented system to simultaneously estimate $x$ and $\zeta_{j}$ using model
(3). We remark that the number of the faults derivatives, $r$, added to the
approximated model (3) (and (4)) is problem-dependent, and an optimal
selection of $r$ would depend on the frequency characteristics of the fault.
Increasing $r$ results in higher-dimensional augmented dynamics and thus high-
dimensional observers as well. However, having larger observers also provides
more degrees of freedom for optimal synthesis.

###  2.2. Augmented Dynamics

Based on the fault model introduced above, define the augmented state vector
$x_{a}:=(x,\zeta_{1},\zeta_{2},\ldots,\zeta_{r})$ and write the augmented
dynamics using (1) and (4) as

$\displaystyle\left\\{\begin{aligned}
\dot{x}_{a}=&A_{a}x_{a}+B_{u_{a}}u_{a}+S_{g_{a}}g_{a}(V_{g_{a}}x_{a},u_{a},t)+B_{\omega_{a}}\omega_{a},\\\
y=&C_{a}x_{a}+D_{\nu}\nu,\end{aligned}\right.$ (5a) where $\displaystyle
A_{a}$
$\displaystyle:=\left[\begin{array}[]{ccccc}A&B_{f}&\boldsymbol{0}&\ldots&\boldsymbol{0}\\\
\boldsymbol{0}&\boldsymbol{0}&I_{n_{f}}&\ldots&\boldsymbol{0}\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\ldots&I_{n_{f}}\\\
\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}&\ldots&\boldsymbol{0}\end{array}\right],\quad
B_{u_{a}}:=\left[\begin{array}[]{cc}{B}_{u}\\\ \boldsymbol{0}\\\
\end{array}\right],\quad
S_{g_{a}}:=\left[\begin{array}[]{cc}{S}_{g}&{S}_{\eta}\\\
\boldsymbol{0}&\boldsymbol{0}\end{array}\right],$ (5b) $\displaystyle
V_{g_{a}}$ $\displaystyle:={\left[\begin{array}[]{cc}V_{g}&\boldsymbol{0}\\\
V_{\eta}&\boldsymbol{0}\end{array}\right]},\quad{g}_{a}(\cdot):={\left[\begin{array}[]{cc}g(\cdot)\\\
\eta_{l_{x}}(\cdot)\end{array}\right]},\quad
B_{\omega_{a}}:=\left[\begin{array}[]{ccc}{S}_{\eta}&B_{\omega}&\boldsymbol{0}\\\
\boldsymbol{0}&\boldsymbol{0}&\boldsymbol{0}\\\
\boldsymbol{0}&\boldsymbol{0}&I_{n_{f}}\\\
\end{array}\right],\quad\omega_{a}:=\left[\begin{array}[]{c}\delta\eta_{l_{x}}(\cdot)\\\
\omega\\\ f^{(r)}(\cdot)\\\ \end{array}\right],$ $\displaystyle C_{a}$
$\displaystyle:=\left[\begin{array}[]{lll}C&D_{f}&\boldsymbol{0}\end{array}\right],\qquad
u_{a}:=u.$

Note that we have stacked the uncertainty model $\eta_{l_{x}}(\cdot)$ with the
known nonlinearity $g(\cdot)$; however, if the uncertainty model is linear
(see, e.g., the example in Section 4), it has to be stacked with the linear
part of the augmented system $A_{a}$. Here we consider a more generic case
where the uncertainty model is a globally Lipschitz nonlinear function.

######  Assumption 4.

_(Globally Lipschitz Nonlinearity)_ The function
$g_{a}(V_{g_{a}}x_{a},u_{a},t)$ in (5a) is globally Lipschitz uniformly in
${u}_{a}(t)$ and $t$, i.e., there exists a known positive constant $\alpha$
satisfying

$\|g_{a}(V_{g_{a}}\hat{x}_{a},u_{a},t)-g_{a}(V_{g_{a}}x_{a},u_{a},t)\|\leq\alpha\|V_{g_{a}}(\hat{x}_{a}-x_{a})\|,$
(6)

for all $x_{a},\hat{x}_{a}\in{\mathbb{R}^{n+rn_{f}}}$,
${u}_{a}\in{\mathbb{R}^{l}}$, and $t\in{\mathbb{R}^{+}}$.

Note that in the above assumption, due to structure of $V_{g_{a}}$,
$g_{a}(\cdot)$ is function $x$ (not $x_{a}$) and therefore the assumption is
equivalent to $g(V_{g}x,u,t)$ and $\eta_{l_{x}}(V_{\eta}x,u,t)$ being globally
Lipschitz uniformly in ${u}(t)$ and $t$ for all
$x,\hat{x}\in{\mathbb{R}^{n}}$, ${u}\in{\mathbb{R}^{l}}$, and
$t\in{\mathbb{R}^{+}}$.

###  2.3. Fault Estimator

In this section, considering the fault estimator general structure in (2),
inspired from observer-based approaches, we propose $h(\cdot)$ and
$\phi(\cdot)$ as

$\left\\{\begin{aligned}
h(z,u,y;\theta)=&Nz+Gu_{a}+Ly+MS_{g_{a}}{g}_{a}(V_{g_{a}}\hat{x}_{a}+J\color[rgb]{0,0,0}(y-C_{a}\hat{x}_{a}),u_{a})\\\
\phi(z,y;\theta)=&\bar{C}(z-E\color[rgb]{0,0,0}y)\\\ \end{aligned}\right.$
(7a) with $\hat{x}_{a}=z-E\color[rgb]{0,0,0}y$,
$\bar{C}:={\left[\begin{array}[]{lll}\boldsymbol{0}_{n_{f}\times
n}&I_{n_{f}}&\boldsymbol{0}_{n_{f}\times n_{f}(r-1)}\end{array}\right]},$ (7b)
and matrices $\\{N,G,L,M\\}$ defined as $\displaystyle N$
$\displaystyle:=MA_{a}-K\color[rgb]{0,0,0}C_{a},$ $\displaystyle G$
$\displaystyle:=MB_{u_{a}},$ (7c) $\displaystyle L$
$\displaystyle:=K\color[rgb]{0,0,0}(I+C_{a}E\color[rgb]{0,0,0})-MA_{a}E\color[rgb]{0,0,0},$
$\displaystyle M$ $\displaystyle:=I+E\color[rgb]{0,0,0}C_{a}.$

The algebraic relations in (7c) are critical to nullify the effect of some
signals on the fault estimation error. (see (10)). Furthermore, filter
dimension $n_{z}$ equals to $n+rn_{f}$ and matrices $E$, $K$, and $J$ are
filter gains to be designed which can be collected as $\theta=\\{E,K,J\\}$. In
the following section, we analyze the fault estimator error dynamics.

###  2.4. Fault Estimator Error Dynamics

Consider the augmented state estimate $\hat{x}_{a}$ and define estimation
error as

$\displaystyle e:=$
$\displaystyle\hat{x}_{a}-x_{a}=z-x_{a}-Ey=z-Mx_{a}-ED_{\nu}\nu,$ (8)
$\displaystyle e_{f}=$ $\displaystyle\hat{f}-f=\bar{C}e.$

The related estimation error dynamics can then be written as follows:

$\displaystyle\dot{e}$
$\displaystyle=Ne+(NM+LC_{a}-MA_{a})x_{a}+(G-MB_{u_{a}})u_{a}$ (9)
$\displaystyle+MS_{g_{a}}\big{(}g_{a}(V_{g_{a}}\hat{x}_{a}+J(y-C_{a}\hat{x}_{a}),u_{a})-g_{a}(V_{g_{a}}{x}_{a},u_{a})\big{)}-MB_{\omega_{a}}\omega_{a}+(NE+L)D_{\nu}\nu-
ED_{\nu}\dot{\nu}.$

Given the algebraic relations in (7c), it can be verified that
$G-MB_{u_{a}}=\mathbf{0}$, $NM+LC_{a}-MA_{a}=\mathbf{0}$, and $NE+L=K$.
Therefore, (9) can be written as

$\displaystyle\dot{e}=$ $\displaystyle Ne+MS_{g_{a}}\delta
g_{a}-MB_{\omega_{a}}\omega_{a}+B_{\nu_{a}}\nu_{a}$ (10a) with
$\displaystyle\delta g_{a}$
$\displaystyle:=g_{a}(V_{g_{a}}\hat{x}_{a}+J(y-C_{a}\hat{x}_{a}),{u})-g_{a}(V_{g_{a}}{x}_{a},{u}),$
(10b) $\displaystyle B_{\nu_{a}}$
$\displaystyle:={\left[\begin{array}[]{cc}KD_{\nu},&-E{D}_{\nu}\end{array}\right]},\qquad\nu_{a}:={\left[\begin{array}[]{cc}\nu^{T},&\dot{\nu}^{T}\end{array}\right]^{T}}.$

By collecting all perturbations, error dynamics (10a) can be written as

$\displaystyle\dot{e}$ $\displaystyle=Ne+MS_{g_{a}}\delta
g_{a}+\bar{B}_{\omega_{a}}\bar{\omega}_{a},$ (11a) where
$\displaystyle\bar{B}_{\omega_{a}}$
$\displaystyle:={\left[\begin{array}[]{cc}-MB_{\omega_{a}}&B_{\nu_{a}}\end{array}\right]},\qquad\bar{\omega}_{a}$
$\displaystyle:={\left[\begin{array}[]{cccc}\omega_{a}^{T}&\nu_{a}^{T}\end{array}\right]^{T}}={\left[\begin{array}[]{ccccc}\delta\eta_{l_{x}}^{T}&\omega^{T}&f^{{(r)}^{T}}&\nu^{T}&\dot{\nu}^{T}\end{array}\right]^{T}}.$
(11b)

Now, we have all the machinery required to mathematically precisely restate
Problem 1 for the uncertain nonlinear system in (1).

######  Problem 2.

_(Fault Reconstruction)_ Consider the uncertain nonlinear system (1) with
known input and output signals, ${u}(t)$ and $y(t)$. Furthermore, consider the
internal fault dynamics (4), its Taylor approximation (3), the augmented
dynamics (5), the nonlinear fault estimator filter (2) with functions defined
in (7), and let Assumptions 1-4 be satisfied. Design the filter gain matrices
$\theta=\\{E,K,J\\}$ such that:
_1) Stability:_ all trajectories of the estimation error dynamics (11) exist
and are globally uniformly ultimately bounded for $t\geq 0$_;_
_2) Disturbance Attenuation:_

$J_{1}(\theta):=\sup_{(\delta\eta_{l_{x}},\omega,f)}\frac{\|e_{f}(\delta\eta_{l_{x}},\omega,f^{(r)},\mathbf{0})\|_{\mathcal{L}_{2}}}{\|(\delta\eta_{l_{x}},\omega,f^{(r)})\|_{\mathcal{L}_{2}}}$
(12)

is bounded with some known $c_{1}>0$_;_
_3) Noise Rejection:_

$J_{2}(\theta):=\sup_{\nu}\frac{\|e_{f}(\mathbf{0},\mathbf{0},\mathbf{0},\nu)\|_{\mathcal{L}_{2}}}{\|(\nu,\dot{\nu})\|_{\mathcal{L}_{\infty}}}$
(13)

is bounded with some known $c_{2}>0$_._

Under Assumptions 1-4, Problem 2 amounts to finding fault estimator gains that
guarantee a bounded estimation error $e(t)$ in (11); for
$\bar{\omega}_{a}=\mathbf{0}$, $e(t)$ goes to zero asymptotically (internal
stability); when $\nu=\mathbf{0}$ the $\mathcal{L}_{2}$-gain of the mapping
from $(\delta\eta_{l_{x}},\omega,f^{(r)})$ to $e_{f}$ (the fault estimation
error) is upper bounded by some $c_{1}>0$; and when
$(\delta\eta_{l_{x}},\omega,f^{(r)})=\mathbf{0}$, the
$\mathcal{L}_{2}-\mathcal{L}_{\infty}$ induced gain (or energy to peak norm
[23, pp. 78]) from $(\nu,\dot{\nu})$ to $e_{f}$ is upper bounded by some
$c_{2}>0$.

In what follows, we provide the solution to Problem 2.

##  3\. Fault Estimator Design

The solution to Problem 2 is given in the following three subsections with the
same order of the problem parts (i.e., 1. stability, 2. disturbance
attenuation, and 3. noise rejection).

###  3.1. ISS Estimation Error Dynamics

In this section, we derive Linear Matrix Inequality (LMI) conditions for
designing the matrices $\theta$ of the filter functions in (7). As a stepping
stone, we present a sufficient condition for asymptotic stability of the
origin of the estimation error dynamics (11) when the perturbation vector
$\bar{\omega}_{a}=(\omega_{a},\nu_{a})$ equals zero (internal stability).
Moreover, we prove the boundedness of the estimation error in the presence of
the perturbation using the input-to-state stability concept [24].

######  Definition 1.

_(Input-to-State Stability) The error dynamics (11) is said to be Input-to-
State Stable (ISS) if there exist a class $\mathcal{K}\mathcal{L}$ function
$\beta(\cdot)$ and a class $\mathcal{K}$ function $\mu(\cdot)$ such that for
any initial estimation error $e(t_{0})$ and any bounded input
$\bar{\omega}_{a}$, the solution $e(t)$ of (11) exists for all $t\geq t_{0}$
and satisfies_

$\|e(t)\|\leq\beta\left(\left\|e\left(t_{0}\right)\right\|,t-t_{0}\right)+\mu(\|\bar{\omega}_{a}\|_{\infty}).$
(14)

Note that ISS of the error dynamics (11) implies boundedness of the estimation
error for bounded $\bar{\omega}_{a}$. This follows directly from
$\eqref{ISS_def}$. Furthermore, we remark that $\bar{\omega}_{a}$ is a
function of $x(t)$, $u(t)$ and $t$, so it is an implicit function of time.
However, to ensure boundedness of $\bar{\omega}_{a}$, Assumptions 1 \- 3 are
required (see Remark 2 for further details).

The next proposition formalizes an LMI condition that guarantees an ISS
estimation error dynamics (11) with respect to input $\bar{\omega}_{a}$.

######  Proposition 1.

_(ISS Estimation Error Dynamics)_ Under Assumptions 1-4, consider the error
dynamics (11) and suppose there exist matrices $P\in{\mathbb{R}^{n_{z}\times
n_{z}}},$ with $P\succ 0,R\in{\mathbb{R}^{n_{z}\times
m}},Q\in{\mathbb{R}^{n_{z}\times m}}$, and $J\in{\mathbb{R}^{n_{v_{g}}\times
m}}$ satisfying the matrix inequality

$\left[\begin{array}[]{cc}X_{11}&X_{12}\\\ &-I\end{array}\right]\prec 0,$
(15a) where, matrices $X$ and $X_{12}$ defined as $\displaystyle X_{11}:=$
$\displaystyle
S_{11}+\alpha(V_{g_{a}}^{T}V_{g_{a}}-V_{g_{a}}^{T}JC_{a}-C_{a}^{T}J^{T}V_{g_{a}})$
(15b) $\displaystyle X_{12}:=$
$\displaystyle{\left[\begin{array}[]{c}\sqrt{2\alpha}\big{(}(P+RC_{a})S_{g_{a}}\big{)}^{T}\\\
\sqrt{\alpha}(JC_{a})^{T}\end{array}\right]^{T}}$ with $\displaystyle
S_{11}:=$ $\displaystyle
A_{a}^{\top}P+A_{a}^{\top}C_{a}^{\top}R^{\top}-C_{a}^{T}Q^{T}+PA_{a}+RC_{a}A_{a}-QC_{a},$
(15c) $\alpha$ from (6), and the remaining matrices in (5b). Then, the
estimation error dynamics in (11) is ISS with respect to input
$\bar{\omega}_{a}$. Moreover, when $\alpha=0$ (no nonlinearity) the condition
in (15a) transforms to $S_{11}\prec 0,$ (15d)

that is a necessary and sufficient condition for ISS with respect to input
$\bar{\omega}_{a}$.

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

######  Remark 1.

_(LMI Feasibility)_ If ${D}_{f}$ in (1a) is full row rank (i.e., there are as
many sensor faults, $f_{y}$, as sensors), the LMI in (15a) is always
infeasible (observability is lost) _[25, 26, 27]_. The standard practice to
circumvent this issue is to assume $\text{rank}[{D_{f}}]<m$ _[25, 26, 27]_.
Note that this is only a necessary condition and it does not guarantee the LMI
in (15a) to be feasible (this has to be checked on a case-by-case basis).

######  Remark 2.

_(Boundedness Assumptions)_ It is important to note that Assumption 1-3 are
required to conclude ISS in Proposition 1. That is, if either of the fault
derivatives, state, input, disturbances of the system are not bounded, the ISS
property cannot be exploited. More specifically, in the scope of using the ISS
property, the perturbation input in (11) (i.e., $\bar{\omega}_{a}(x,u,t)$)
should be bounded uniformly in $x(t),{u}(t)$ and $t$. We have
$\bar{\omega}_{a}=(\delta\eta_{l_{x}},\omega,f^{(r)},\nu,\dot{\nu})$, so it
consists of the bounded uncertainty model mismatch $\delta\eta_{l_{x}}$
(bounded by the extreme value theorem [28, Thm. 3.12] due to continuity and
bounded driving terms (Assumption 1)), the bounded exogenous disturbance
$\omega(t)$, the bounded high frequency disturbance $\nu(t)$, its time
derivative $\dot{\nu}(t)$ (Assumption 3), and the bounded fault model mismatch
$f^{{(r)}}(x,u,t)$ (Assumptions 1 and 2). It follows that $\bar{\omega}_{a}$
is bounded under the required assumptions.

In what follows, we shift our attention from the error dynamics (11) to its
equivalent (10), to characterize fault estimation robustness against
disturbances with unknown frequency range
$(\delta\eta_{l_{x}},\omega,f^{(r)})$ (external disturbances, and fault and
uncertainty model mismatches) and disturbances with high frequency content
$(\nu,\dot{\nu})$ (measurement noise and its derivative). Because
$(\delta\eta_{l_{x}},\omega,f^{(r)})$ and $(\nu,\dot{\nu})$ have different
frequency characteristics, robustness strategies for each of these
perturbations are different.

###  3.2. $\mathcal{L}_{2}$ Performance Criterion

To maximize the performance of the reconstruction scheme, we seek to minimize
the effect of $(\delta\eta_{l_{x}},\omega,f^{(r)})$ (treated as an arbitrary
energy bounded external disturbance) on the fault estimation error $e_{f}$. We
assume $\nu$ is zero (consequently $\nu_{a}=\boldsymbol{0}$) in the error
dynamics (10) since we apriori know that the measurement noise has high
frequency content and thus we consider the effect of $\nu_{a}$ in the next
section. To this end, we seek to minimize the $\mathcal{L}_{2}$-gain from
$(\delta\eta_{l_{x}},\omega,f^{(r)})$ to the fault estimation error $e_{f}$.
We could use the ISS formulation in Proposition 1 to cast an optimization
problem where we minimize the ISS gain and treat the LMI in (15a) as an
optimization constraint. By doing so, we would be reducing the effect of
$(\delta\eta_{l_{x}},\omega,f^{(r)})$ on the complete vector of estimation
errors $e$ (state, and fault and fault derivatives estimation errors). Note,
however, that the filter’s purpose is to reconstruct fault vector only, so the
performance in state estimation and the error of higher-order fault
derivatives is not relevant.

######  Definition 2.

_( $\mathcal{L}_{2}$-gain [29])_ We say that the estimation error dynamics
(10), assuming $\nu=\boldsymbol{0}$ with input
$\omega_{a}(t)=(\delta\eta_{l_{x}},\omega,f^{(r)})$ and output $e_{f}(t)$
(fault estimation error as in (8)) has a $\mathcal{L}_{2}$-gain less than or
equal to $\lambda$ if the following inequality is satisfied

$||e_{f}(t)||_{\mathcal{L}_{2}}:=\int_{0}^{T}\|e_{f}(t)\|^{2}dt\leq\lambda^{2}\int_{0}^{T}\|\omega_{a}(t)\|^{2}dt,$

for all $T\geq 0$ and $\omega_{a}(t)\in\mathcal{L}_{2}(0,T)$.

The following proposition formalizes an LMI-based condition guaranteeing that
(10) has the finite $\mathcal{L}_{2}$-gain property from
$(\delta\eta_{l_{x}},\omega,f^{(r)})$ to the fault estimation error $e_{f}$
(see Definition 2).

######  Proposition 2 ($\mathcal{L}_{2}$-gain LMI).

Under Assumptions 1-4, suppose there exist matrices $P\succ 0,R,$ $Q$, $J$,
and scalar $\rho\geq 0$ satisfying

$\displaystyle\left[\begin{array}[]{ccc}X_{11}+a\bar{C}^{T}\bar{C}&-(P+RC_{a})B_{\omega_{a}}&X_{12}\\\
&-{\rho}aI&\boldsymbol{0}\\\ &*&-I\end{array}\right]\preceq 0,$ (16d) for some
given $a>0$, $X_{11},X_{12}$ as defined in (15b), $\bar{C}$ in (7b), and the
remaining matrices in (5b). Then, $J_{1}(\cdot)$ in (12) is upper bounded by
$\sqrt{\rho}$, i.e., the $\mathcal{L}_{2}$-gain of (10) with $\nu=\mathbf{0}$
from $(\delta\eta_{l_{x}},\omega,f^{(r)})$ to the fault estimation error
$e_{f}$ is upper bounded by $\sqrt{\rho}$. Moreover, when $\alpha=0$ (no
nonlinearity) the condition in (16d) transforms to
$\displaystyle\left[\begin{array}[]{cc}S_{11}+\bar{C}^{T}\bar{C}&(P+RC_{a})B_{{d}_{1}}\\\
&-{\rho}I\\\ \end{array}\right]\preceq 0,$ (16e)

that is a necessary and sufficient condition for $\sqrt{\rho}$ to be the upper
bound for the $\mathcal{L}_{2}$-gain from
$(\delta\eta_{l_{x}},\omega,f^{(r)})$ to the fault estimation error $e_{f}$.

_Proof :_ The proof can be found in Appendix B. $\blacksquare$

###  3.3. $\mathcal{L}_{2}-\mathcal{L}_{\infty}$ Induced Norm Performance
Criterion

The other terms affecting the fault reconstruction are measurement noise and
its derivative with high frequency content $(\nu,\dot{\nu})$. Therefore, we
assume disturbances with unknown frequency range
$(\delta\eta_{l_{x}},\omega,f^{(r)})=\boldsymbol{0}$ in the error dynamics
(10) and seek to minimize the effect of disturbances with high frequency
content $(\nu,\dot{\nu})$ on the fault estimate error $e_{f}$. Here, we should
select an appropriate performance criterion to characterize the effect of
$(\nu,\dot{\nu})$ on the estimation error dynamics (10). Because
$(\nu,\dot{\nu})$ consists of disturbances with bounded energy, we consider
$\mathcal{L}_{2}$-norm for that and since it leads to abrupt changes in fault
estimates $e_{f}$, we use $\mathcal{L}_{\infty}$-norm for fault estimation
error $e_{f}$ to have a filtering effect. In other words, we minimize the
effect of abrupt changes and push the maximum amplitude in the fault estimate
error signal down to have a smooth fault estimate. Therefore, we seek to
minimize the $\mathcal{L}_{2}-\mathcal{L}_{\infty}$ induced norm from
$(\nu,\dot{\nu})$ to fault estimation error $e_{f}$. Moreover, one can note
that the $\mathcal{L}_{2}-\mathcal{L}_{\infty}$ induced norm is widely used in
the literature for filtering (See [30, 31, 32, 33, 34]).

######  Definition 3.

($\mathcal{L}_{2}-\mathcal{L}_{\infty}$ Induced Norm) We say that the
estimation error dynamics (10) assuming
$(\delta\eta_{l_{x}},\omega,f^{(r)})=\mathbf{0}$ with input
$\nu_{a}(t)=(\nu,\dot{\nu})$ and output $e_{f}(t)$ (fault estimation error as
in (8)) has a $\mathcal{L}_{2}-\mathcal{L}_{\infty}$ induced norm less than or
equal to $\gamma$ if the following inequality is satisfied

$||e_{f}||_{\mathcal{L}_{\infty}}:=\sup_{t\geq
0}\|e_{f}(t)\|^{2}\leq\gamma^{2}\int_{0}^{\infty}\|\nu_{a}(t)\|^{2}dt,$

for all $\nu_{a}(t)\in\mathcal{L}_{2}(0,\infty)$.

In the following proposition, we give a Lyapunov-based sufficient LMI
condition for having a bounded $\mathcal{L}_{2}-\mathcal{L}_{\infty}$ induced
norm of the mapping from $(\nu,\dot{\nu})$ to the fault estimation error
$e_{f}$.

######  Proposition 3 ($\mathcal{L}_{2}-\mathcal{L}_{\infty}$ Induced Norm
LMI).

Under Assumptions 3 and 4 , suppose there exist matrices $P\succ 0,R,$ $Q$,
$J$, and scalar $\sigma\geq 0$ satisfying

$\displaystyle\left[\begin{array}[]{cccc}X_{11}&H_{12}&\boldsymbol{0}&X_{12}\\\
&-b^{2}I&T_{\nu}^{T}J^{T}&\boldsymbol{0}\\\ &*&-I&\boldsymbol{0}\\\
&*&*&-I\end{array}\right]\preceq 0,$ (17e)
$\displaystyle\left[\begin{array}[]{cc}P&\bar{C}^{T}\\\ &\sigma
I\end{array}\right]\succeq 0,$ (17h) $\displaystyle
H_{12}:=[\begin{array}[]{ll}QD_{\nu},&-RD_{\nu}\end{array}],$ (17j)
$\displaystyle
T_{\nu}=[\begin{array}[]{ll}D_{\nu}&\boldsymbol{0}\end{array}],$ (17l) for
some given scalar $b$, $X_{11},X_{12}$ as defined in (15b), $\bar{C}$ in (7b),
and the remaining matrices in (5b). Then, $J_{2}(\cdot)$ in (13) is upper
bounded by $\sqrt{b\sigma}$, i.e., the $\mathcal{L}_{2}-\mathcal{L}_{\infty}$
induced norm of (10) with $\omega_{a}=\mathbf{0}$ from $(\nu,\dot{\nu})$ to
the fault estimation error $e_{f}$ is upper bounded by $\sqrt{b\sigma}$.
Moreover, when $\alpha=0$ (no nonlinearity) the condition in (17e) reduces to
$\displaystyle\left[\begin{array}[]{cc}S_{11}&H_{12}\\\ &-I\\\
\end{array}\right]\preceq 0,$ (17m)

that is a necessary and sufficient condition for $\sqrt{\sigma}$ to be the
upper bound for the $\mathcal{L}_{2}-\mathcal{L}_{\infty}$ induced gain from
$(\nu,\dot{\nu})$ to the fault estimation error $e_{f}$.

_Proof :_ The proof can be found in Appendix C. $\blacksquare$

######  Remark 3.

_(Extension)_ Although due to the nature of signals unknown frequency range
$(\delta\eta_{l_{x}},\omega,f^{(r)})$ and high frequency content
$(\nu,\dot{\nu})$, we have proposed $\mathcal{L}_{2}$-gain and
$\mathcal{L}_{2}-\mathcal{L}_{\infty}$ induced gain results, respectively, we
can apply $\mathcal{L}_{2}-\mathcal{L}_{\infty}$ induced norm norm to
$(\delta\eta_{l_{x}},\omega,f^{(r)})$, or the other way around. For example,
applying $\mathcal{L}_{2}-\mathcal{L}_{\infty}$ induced norm to
$(\delta\eta_{l_{x}},\omega,f^{(r)})$ can be useful when an uncertainty with
high-frequency content appears in $\delta\eta_{l_{x}}$. This case is, however,
not considered here.

######  Remark 4 (Perfect Estimation).

The developed methodology can guarantee zero estimation error for zero
$\bar{\omega}_{a}=(\delta\eta_{l_{x}},\omega,f^{(r)},\nu,\dot{\nu})$, i.e.,
when the $r^{th}$-time derivative of the fault vector vanishes (i.e., time
polynomial signals with a degree less than r), and the disturbances (the
uncertainty model mismatch $\delta\eta_{l_{x}}$, the external disturbance
$\omega$, and the measurement noise $\nu$) are zero. This follows directly
from the estimation error dynamics (11) since if
$\bar{\omega}_{a}=\boldsymbol{0}$, the error dynamics becomes independent of
external perturbations.

Propositions 1-3 provide sufficient conditions that we exploit to solve
Problem 2 (filter synthesis) in what follows. So far, we have presented
analysis tools to characterize the performance of a given filter of the form
(2) with functions in (7). In the following section, we provide a tool to
design the filter matrices $\theta=\\{E,K,J\\}$ in (7), in an optimal way in
the sense of achieving a desired trade-off between the $\mathcal{L}_{2}$-gain
and $\mathcal{L}_{2}-\mathcal{L}_{\infty}$ induced-gain introduced above.

###  3.4. Optimal Fault Estimator Design

Using Proposition 2, we can formulate a semi-definite program where we seek to
minimize the $\mathcal{L}_{2}$-gain from unknown frequency range disturbances
$\omega_{a}=(\delta\eta_{l_{x}},\omega,f^{(r)})$ to fault estimation error
$e_{f}$. Similarly, using Proposition 3, we can formulate another semi-
definite program where we seek to minimize the
$\mathcal{L}_{2}-\mathcal{L}_{\infty}$ induced norm from disturbances with
high frequency content $\nu_{a}=(\nu,\dot{\nu})$ to $e_{f}$. However, in the
presence of both unknown perturbations ($\omega_{a}$ and $\nu_{a}$), the
$\mathcal{L}_{2}$-gain and $\mathcal{L}_{2}-\mathcal{L}_{\infty}$ can not be
minimized simultaneously due to conflicting objectives. To attenuate the
effect of high-frequency disturbances on the estimation error, a relatively
slow (low-gain) filter is required, which does not react to every small and
fast change in the measured output. On the other hand, to reduce the effect of
$\omega_{a}$ on the estimation error, a high-gain filter 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 this trade-off, a convex program can be proposed where we seek to
minimize the $\mathcal{L}_{2}$-gain and constrain the
$\mathcal{L}_{2}-\mathcal{L}_{\infty}$ induced gain. The same tools allow
minimizing the $\mathcal{L}_{2}-\mathcal{L}_{\infty}$ norm for a constrained
$\mathcal{L}_{2}$-gain (as a dual problem). Moreover, we add the ISS LMI in
(15a) as a constraint to these programs to enforce that the resulting filter
also guarantees boundedness for bounded perturbations and asymptotic stability
for vanishing $\omega_{a}$ and $\nu_{a}$ (as having bounded signal norms does
not guarantee having bounded filter trajectories [35]).

######  Theorem 3.1.

_(Optimal Fault Estimator)_ Consider the augmented dynamics (5), the fault
estimator filter (2) with $h(\cdot)$ and $\phi(\cdot)$ as defined in (7), and
the corresponding estimation error dynamics (10)-(11). Let Assumptions 1-4 be
satisfied. To design the parameters of the fault estimator, solve the
following convex program

$\begin{array}[]{cl}\min\limits_{P,R,Q,J,\rho,\sigma}&\rho\\\
\text{s.t.}&\vspace{1 mm}\left[\begin{array}[]{cc}X_{11}&X_{12}\\\
&-I\end{array}\right]\prec 0,\\\
&\left[\begin{array}[]{ccc}X_{11}+a\bar{C}^{T}\bar{C}&-(P+RC_{a})B_{\omega_{a}}&X_{12}\\\
&-{\rho}aI&\boldsymbol{0}\\\ &*&-I\end{array}\right]\preceq 0,\\\\[14.22636pt]
&\left[\begin{array}[]{cccc}X_{11}&H_{12}&\boldsymbol{0}&X_{12}\\\
&-b^{2}I&T_{\nu}^{T}J^{T}&\boldsymbol{0}\\\\[14.22636pt]
*&*&-I&\boldsymbol{0}\\\ &*&*&-I\end{array}\right]\preceq 0,\\\\[14.22636pt]
&\left[\begin{array}[]{cc}P&\bar{C}^{T}\\\ &\sigma I\end{array}\right]\succeq
0,\\\ &P\succ 0,\quad\rho,\sigma\geq
0,\quad\sigma\leq\sigma_{x_{max}}\end{array}$ (18)

with given scalars $a,\sigma_{x_{max}}>0$ and $b$, $X_{11},X_{12}$ as defined
in (15b), $B_{\omega_{a}}$ and $C_{a}$ in (5b), $H_{12}$ in (17j), $T_{\nu}$
in (17l), and $\bar{C}$ in (7b). Denote the optimizers as $P^{\star}$,
$R^{\star}$, $Q^{\star}$, $J^{\star}$, $\rho^{\star}$ and $\sigma^{\star}$.
Then, the following parameters of (7),
$\theta=\theta^{\star}=\\{E^{\star}=P^{\star^{-1}}R^{\star},K^{\star}=P^{\star^{-1}}Q^{\star},J^{\star}\\}$
guarantees the following:

  1. (1)

The estimation error dynamics in (11) is ISS with respect to input
$\bar{\omega}_{a}=(\omega_{a},\nu_{a})$.

  2. (2)

$J_{1}(\cdot)$ in (12) is upper bounded by $\sqrt{\rho^{\star}}$, i.e., the
$\mathcal{L}_{2}$-gain of (10) with $\nu=\mathbf{0}$ from
$\omega_{a}=(\delta\eta_{l_{x}},\omega,f^{(r)})$ to the fault estimation error
$e_{f}$ is upper bounded by $\sqrt{\rho^{\star}}$.

  3. (3)

$J_{2}(\cdot)$ in (13) is upper bounded by $\sqrt{b\sigma^{\star}}$, i.e., the
$\mathcal{L}_{2}-\mathcal{L}_{\infty}$ induced norm of (10) with
$\omega_{a}=\mathbf{0}$ from $\nu_{a}=(\nu,\dot{\nu})$ to $e_{f}$ is upper
bounded by $\sqrt{b\sigma^{\star}}$.

_Proof :_ Theorem 3.1 follows from the above discussion and Propositions 1-3.
$\blacksquare$

Note that, for numerical tractability, we replace $X_{11}$ in the first strict
inequality and $P\succ 0$ in the last strict inequality with $X_{11}+\epsilon
I$ (it becomes non-strict inequality) and $P-\epsilon I\succeq 0$, with a
given $\epsilon>0$, respectively. In what follows, we discuss the uncertainty
model and how different types of models affect the provided solution of the
fault estimation problem.

###  3.5. Discussion on Uncertainty Models

To obtain an uncertainty model of the form $\eta_{l_{x}}$ in (1b), we can use
results in e.g., [36, 37, 38]. Available methods allow fitting parametric
static functions, which might be state or output dependent. To use these
results in the context of fault estimation, we have to assume that there is
some time window during the system operation in which no fault occurs and that
data is collected for this healthy mode. The collected data can be used to
learn uncertainty models to have a more accurate system description (valid at
least for trajectories close to the training data set) for fault estimation.

Based on (1b), we have assumed that we have the state dependent uncertainty
model $\eta_{l_{x}}(\cdot)$ to develop the results of this paper. In the case
of an output dependent uncertainty model $\eta_{l_{y}}(\cdot)$, (1b) will
change as follows:

$\displaystyle\eta(V_{\eta}x,u,t)=\eta_{l_{y}}(T_{\eta}y,u,t)+\delta\eta_{l_{y}}(x,u,t),$
(19)

where
$\delta\eta_{l_{y}}(x,u,t):=\eta(V_{\eta}x,u,t)-\eta_{l_{y}}(T_{\eta}y,u,t)$.
Matrix $T_{\eta}$ is a selection matrix that the user selects to specify what
particular outputs drive the model. If we use this $\eta_{l_{y}}(\cdot)$
instead of the state dependent $\eta_{l_{x}}(\cdot)$ in (1b), (5b) modifies as
follows:

$\displaystyle B_{u_{a}}$
$\displaystyle:=\left[\begin{array}[]{cc}{B}_{u}&S_{\eta}\\\
\boldsymbol{0}&\boldsymbol{0}\\\ \end{array}\right],\quad
u_{a}:=\left[\begin{array}[]{c}u\\\
\eta_{l_{y}}(T_{\eta}y,u,t)\end{array}\right],\quad
S_{g_{a}}:=\left[\begin{array}[]{c}{S}_{g}\\\
\boldsymbol{0}\end{array}\right],$ (20) $\displaystyle V_{g_{a}}$
$\displaystyle:={\left[\begin{array}[]{cc}V_{g}&\boldsymbol{0}\end{array}\right]},\quad{g}_{a}(\cdot):=g(\cdot),\quad\omega_{a}:=\left[\begin{array}[]{ccc}\delta\eta_{l_{y}}^{T}&\omega^{T}&f^{{(r)}^{T}}\end{array}\right]^{T}.$

As you can see above (see $g_{a}(\cdot)$ in (20)), $\eta_{l_{y}}(\cdot)$ does
not act as a nonlinearity for the augmented system (as $\eta_{l_{x}}$ does in
(5b)); instead, it acts as a known input signal. The latter affects the
required assumption for the application of this uncertainty model (See Remark
5).

######  Remark 5 (Uncertainty Model Selection).

In Assumption 4, the state-depended uncertainty model $\eta_{l_{x}}$ must be
globally Lipschitz to ensure the boundedness of the fault estimator. On the
other hand, the output-depended model $\eta_{l_{y}}$ relaxes this requirement
by only requiring $\eta_{l_{y}}$ to be continuous. The latter comes at the
price of lower model accuracy when the system uncertainty is not an explicit
function of the system output (i.e, $V_{\eta}x$ might contain more states than
those measured in the output).

##  4\. Simulation Results

In this section, we evaluate the proposed method using a benchmark example for
FDI [39, 40, 41, 22]. The system dynamics (a single-link robotic arm with a
revolute elastic joint, see Fig. 1 for a schematic) can be described as
follows:

$\displaystyle J_{b}\ddot{q}_{l}+F_{l}\dot{q}_{l}+(k_{s}+\Delta
k_{s})\left(q_{l}-q_{m}\right)+mg(c+\Delta c)\sin\left(q_{l}\right)=\omega,$
$\displaystyle J_{m}\ddot{q}_{m}+F_{m}\dot{q}_{m}-(k_{s}+\Delta
k_{s})\left(q_{l}-q_{m}\right)=k_{\tau}u,$

where ${q}_{l}$ and ${q}_{m}$ are the angular position of the link and the
angular position of the motor, respectively. Constants $J_{b}$ and $J_{m}$ are
the moments of inertia of the link and the motor, and $F_{l}$ and $F_{m}$ are
the viscous coefficients associated with friction acting at the link and the
motor, respectively. The flexibility in the joint is modeled by a spring with
a spring coefficient $k_{s}$, the inaccuracy in the spring coefficient is
denoted by $\Delta k_{s}$, $m$ is the link mass, $g$ is the gravity constant,
$c$ is the height of the link center of mass, $\Delta c$ is the inaccuracy in
the height of the link center of mass, $k_{\tau}$ is the amplifier gain, and
$u$ is the torque input delivered by the motor. Units are in SI, and the
parameters values are: $J_{b}=4.5$ $kgm^{2},J_{m}=1$ $kgm^{2},F_{l}=0.5$
$Nms/rad,F_{m}=1$ $Nms/rad,k_{s}=2$ $Nm/rad,\Delta k_{s}=-0.25k,m=4$
$kg,g=9.8$ $m/s^{2},c=0.5$ $m,\Delta c=0.25c$ and $k_{\tau}=1$. The torque
input is set to $u=2\text{sin}(0.25t)$, and $\omega=0.03\text{sin}(0.1t)$ is
an exogenous unknown (torque) disturbance affecting the link.

Figure 1. Benchmark System Schematic.

Let $x_{1}:=\dot{q}_{m},x_{2}:=q_{m},x_{3}:=\dot{q}_{l}$, $x_{4}:=q_{l}$, and
for the fault scenario, we consider the actuator fault; then, the system can
be written in the form of (1a)

$\left\\{\begin{aligned}
\dot{x}&=Ax+B_{u}u+S_{g}g\left(V_{g}x\right)+S_{\eta}\eta\left(V_{\eta}x\right)+B_{\omega}\omega+B_{f}f,\\\
y&=Cx+D_{f}f+D_{\nu}\nu,\end{aligned}\right.$ (21)

where $x:=[x_{1},x_{2},x_{3},x_{4}]^{T}$ is the state vector,

$\displaystyle A$
$\displaystyle=\left[\begin{array}[]{cccc}-\frac{F_{m}}{Jm}&-\frac{k_{s}}{J_{m}}&0&\frac{k_{s}}{J_{m}}\\\
1&0&0&0\\\ 0&\frac{k_{s}}{J_{l}}&-\frac{F_{l}}{J_{l}}&-\frac{k_{s}}{J_{l}}\\\
0&0&1&0\end{array}\right],\quad
B_{u}=\left[\begin{array}[]{cccc}\frac{k_{\tau}}{J_{m}}&0&0&0\\\
\end{array}\right]^{T},\quad C=\left[\begin{array}[]{cccc}0&1&0&0\\\
0&0&0&1\end{array}\right],$ $\displaystyle S_{g}$
$\displaystyle=\left[\begin{array}[]{cccc}0&0&-\frac{mgc}{J_{l}}&0\\\
\end{array}\right]^{T},\quad V_{g}=\left[\begin{array}[]{cccc}0&0&0&1\\\
\end{array}\right],\quad S_{\eta}=\left[\begin{array}[]{cccc}1&0&0&0\\\
0&0&1&0\\\ \end{array}\right]^{T},$ $\displaystyle V_{\eta}$
$\displaystyle=\left[\begin{array}[]{cccc}0&1&0&0\\\ 0&0&0&1\\\
\end{array}\right],\quad B_{\omega}=\left[\begin{array}[]{cccc}0&0&1&0\\\
\end{array}\right]^{T},\quad
B_{f}=\left[\begin{array}[]{llll}1&0&0&0\end{array}\right]^{T},$
$\displaystyle D_{f}$
$\displaystyle=\left[\begin{array}[]{ll}0&0\end{array}\right]^{T},\quad
D_{\nu}=I.$

The nonlinearity is given by $g\left(Vx\right)=\sin\left(x_{4}\right)$, which
is Lipschitz with constant $\alpha=1$. The uncertainty is
$\eta(V_{\eta}x)=[\frac{\Delta k_{s}}{J_{m}}(x_{4}-x_{2}),\frac{\Delta
k_{s}}{J_{l}}(x_{2}-x_{4})-\frac{mg\Delta c}{J_{l}}sin(x_{4})]^{T}$, which is
induced by the uncertainty on the stiffness and the location of the center of
mass of the link. We set initial conditions as
$x(0)=[0.01,0.01,0.01,0.01]^{T}$.

We have the following linear state- and output-dependent uncertainty models:

$\displaystyle\eta_{l_{x}}(V_{\eta}{x};\Theta_{x})=$
$\displaystyle\Theta_{x}V_{\eta}{x},$ (22)
$\displaystyle\eta_{l_{y}}(T_{\eta}{y};\Theta_{y})=$
$\displaystyle{\Theta_{y}}T_{\eta}y,$

where $T_{\eta}$ is the identity matrix (since the uncertainties are dependent
on both outputs). Matrices $\Theta_{x}$ and $\Theta_{y}$ are parameters of
state- and output-dependent models with appropriate dimensions, respectively.

Secondly, considering the system in (21) and either of state and output
dependent uncertainty models in (22), we construct augmented system (5a) with
$r=1$, using (5b) or its modified version (20) for state and output dependent
uncertainty models, respectively. Next, we design two actuator fault
estimators of the form (7), one for each of cases considering state- and
output-dependent uncertainty models, by solving the semi-definite problem in
Theorem 3.1. The initial condition of the filters in simulation is taken as
the zero vector. A sinusoidal actuator fault with the same frequency of input
is simulated (i.e., $f_{x}=0.1sin(0.25(t-25))$).

We evaluate two aspects of the proposed methods:

  1. (1)

Effect of the learning models for uncertainty in a noise-free case.

  2. (2)

Effect of robustification of fault estimate against noise.

###  4.1. Model Learning for Uncertainty

To indicate the performance of the proposed fault estimation approaches (two
cases considering two uncertainty models) in a noise-free situation, we
compare them with the case in which we neither use the uncertainty model in
the design of the fault estimator nor robustify fault estimation error against
perturbations induced by uncertainty (see the fault estimator given in [22]).
Figure 2 depicts the actual fault and its estimates for three different
approaches (two proposed methods plus one without any model for uncertainty).
It can be seen that the estimated actuator faults using both proposed
estimators follow the actual fault properly. Note that the fault estimate
accuracy for this system does not change that much using either of the
uncertainty models. However, this cannot be generalized since the used models
for uncertainty and what we robustify the fault estimate against is different
in these two proposed methods.

Figure 2. The actual actuator fault and its estimates.

###  4.2. Robustification against Noise

In this section, we aim to show that 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. Therefore, in
simulations of this section, the measurement noise $\nu$ is generated from a
uniform distribution with an amplitude of five percent of the output signals
for both sensors. To show the trade-off between fault estimation performance
and noise attenuation, we consider the filter of the form (7), which can be
designed in three different ways as follows:

  1. (1)

Only minimizing the $\mathcal{L}_{2}$-gain from $d_{a}$ to $\bar{C}e$ in (10)
(assuming $\nu_{a}=\boldsymbol{0}$), subject to ISS LMI in (15a).

  2. (2)

Only minimizing the $\mathcal{L}_{2}-\mathcal{L}_{\infty}$ induced gain from
$\nu_{a}$ to fault estimation error $\bar{C}e$ in (10) (assuming
$d_{a}=\boldsymbol{0}$), subject to ISS LMI in (15a).

  3. (3)

Minimizing the $\mathcal{L}_{2}$-gain, subject to an upper bound for
$\mathcal{L}_{2}-\mathcal{L}_{\infty}$ induced norm and ISS LMI in (15a)
(Theorem 3.1).

The first case of the above-mentioned scenarios is depicted in Fig. 3. One can
see that the effect of noise is dominant in fault estimate, and this is due to
the fact that by only minimizing $\mathcal{L}_{2}$-gain, we obtain a high-gain
filter, which amplifies the noise effect. In contrast, by only minimizing
$\mathcal{L}_{2}-\mathcal{L}_{\infty}$ induced norm, a low-gain filter is
found, which (almost) perfectly filters noise effect but sacrifices fault
estimation performance. The result for this filter is shown in Fig. 4. The
method proposed in this paper can provide a trade-off between the two previous
scenarios. Figure 4 depicts the result using the proposed method in Theorem
3.1. It can be seen that we have a decent trade-off between fault estimate
performance and noise attenuation. Note that by tuning the upper bound for
$\mathcal{L}_{2}-\mathcal{L}_{\infty}$ induced norm from $\nu_{a}$ to fault
estimate, you can increase noise filtering at the cost of losing fault
estimate accuracy.

Figure 3. The actual actuator fault and its estimates using the optimal
$\mathcal{L}_{2}$-gain estimator. Figure 4. The actual actuator fault and its
estimates using different optimal criteria.

##  5\. Conclusion

This paper proposes a method for the estimation of time-varying actuator and
sensor faults in uncertain nonlinear systems. The fault estimator exploits an
internal, ultra-local in time, model for the fault vector which allows us to
guarantee zero fault estimation error for a class of faults in the absence of
perturbations. The fault estimator can be designed by solving a semi-definite
program. Herein, the effect of (fault and uncertainty) model mismatches, and
external disturbances on the fault estimation error is minimized in the sense
of $\mathcal{L}_{2}$-gain, for an acceptable
$\mathcal{L}_{2}-\mathcal{L}_{\infty}$ induced norm with respect to
measurement noise. This allows to design fault estimators that exhibit a
favorable performance trade-off in the presence of these different
perturbations challenging fault estimation. Simulations for a benchmark system
illustrate the performance and potential of the proposed approach.

## Appendix

##  Appendix A Proof of Proposition 1

Let us first introduce the following lemma, which is used to ensure ISS using
an ISS Lyapunov function.

######  Lemma 1.

_(ISS Lyapunov Function[35, Thm. 4.19]) Consider the error dynamics (11) and
let $W(e)$ be a continuously differentiable function such that_

$\alpha_{1}(\|e\|)\leq W(e)\leq\alpha_{2}(\|e\|),$ $\dot{W}(e)\leq-
W_{3}(e),\quad\forall\hskip 2.84526pt\|e\|\geq\xi(\|\bar{\omega}_{a}(t)\|),$

_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 (11) is ISS with ISS gain
$\mu(\|\bar{\omega}_{a}\|)=\alpha_{1}^{-1}(\alpha_{2}(\xi(\|\bar{\omega}_{a}\|)))$._

Let $W(e):={e}^{T}P{e}$ be an ISS Lyapunov function candidate. Then, it
follows from (10) (equivalent to (11)) and the Lipschitz conditions for the
known nonlinearity in (6) that

$\displaystyle\dot{W}(e)\leq$ $\displaystyle e^{T}\Delta
e+\bar{\omega}_{a}^{T}\bar{T}^{T}M_{\nu}\bar{T}\bar{\omega}_{a}+2e^{T}P\bar{B}_{\omega_{a}}\bar{\omega}_{a}$
(23)

with $\bar{\omega}_{a},\bar{B}_{\omega_{a}}$ as defined in (11b),

$\displaystyle\Delta:=$ $\displaystyle N^{T}P+PN+2\alpha
PMS_{g_{a}}S_{g_{a}}^{T}M^{T}P+\alpha(V_{g_{a}}-JC_{a})^{T}(V_{g_{a}}-JC_{a}),$
(24)

and

$\displaystyle M_{\nu}:=T_{\nu}^{T}J^{T}JT_{\nu},\quad
T_{\nu}:=[\begin{array}[]{ll}D_{\nu}&\boldsymbol{0}\end{array}],\quad\bar{T}:=[\begin{array}[]{ll}\boldsymbol{0}&I_{2m}\end{array}].$
(25)

$\displaystyle\dot{W}=$ $\displaystyle
e^{T}\left(N^{T}P+PN\right)e+2e^{T}PMS_{g_{a}}\delta
g-2e^{T}PMB_{\omega_{a}}\omega_{a}+2e^{T}PB_{\nu_{a}}\nu_{a}$ (26)
$\displaystyle\leq$ $\displaystyle
e^{T}\left(N^{T}P+PN\right)e+2\left\|e^{T}PMS_{g_{a}}\right\|\|\delta
g\|-2e^{T}PMB_{\omega_{a}}\omega_{a}+2e^{T}PB_{\nu_{a}}\nu_{a}$
$\displaystyle\leq$ $\displaystyle
e^{T}\left(N^{T}P+PN\right)e+2\left\|e^{T}PMS_{g_{a}}\right\|\alpha\|(V_{g_{a}}-JC_{a}){e}+JD_{\nu}\nu\|-2e^{T}PMB_{\omega_{a}}\omega_{a}+2e^{T}PB_{\nu_{a}}\nu_{a}$
$\displaystyle\leq$ $\displaystyle
e^{T}\left(N^{T}P+PN\right)e+2\left\|e^{T}PMS_{g_{a}}\right\|\alpha(\|(V_{g_{a}}-JC_{a}){e}\|+\|JT_{\nu}\nu_{a}\|)-2e^{T}PMB_{\omega_{a}}\omega_{a}$
$\displaystyle+2e^{T}PB_{\nu_{a}}\nu_{a}$ $\displaystyle\leq$ $\displaystyle
e^{T}\left(N^{T}P+PN\right)e+\alpha\big{(}2\left\|e^{T}PMS_{g_{a}}\right\|^{2}+\|(V_{g_{a}}-JC_{a})e\|^{2}+\|JT_{\nu}\nu_{a}\|^{2}\big{)}-2e^{T}PMB_{\omega_{a}}\omega_{a}$
$\displaystyle+2e^{T}PB_{\nu_{a}}\nu_{a}$ $\displaystyle=$ $\displaystyle
e^{T}\big{(}N^{T}P+PN+2\alpha
PMS_{g_{a}}S_{g_{a}}^{T}M^{T}P+\alpha(V_{g_{a}}-JC_{a})^{T}(V_{g_{a}}-JC_{a})\big{)}e+\nu_{a}^{T}\big{(}T_{\nu}^{T}J^{T}JT_{\nu}\big{)}\nu_{a}$
$\displaystyle-2e^{T}PMB_{\omega_{a}}\omega_{a}+2e^{T}PB_{\nu_{a}}\nu_{a}$
$\displaystyle=$ $\displaystyle e^{T}\Delta
e+\nu_{a}^{T}M_{\nu}\nu_{a}+2e^{T}P{\left[\begin{array}[]{cc}-MB_{\omega_{a}}&B_{\nu_{a}}\end{array}\right]}{\left[\begin{array}[]{cc}\omega_{a}\\\
\nu_{a}\end{array}\right]}=e^{T}\Delta
e+\bar{\omega}_{a}^{T}\bar{T}^{T}M_{\nu}\bar{T}\bar{\omega}_{a}+2e^{T}P\bar{B}_{\omega_{a}}\bar{\omega}_{a}$

Now, inequality (23) (which is the same as (26)) by taking the norm of the
right-hand side of inequality implies the following inequality:

$\displaystyle\dot{W}(e)\leq$
$\displaystyle-\lambda_{\min}(-\Delta)\|e\|^{2}+\|\bar{T}^{T}M_{\nu}\bar{T}\|\|\bar{\omega}_{a}\|^{2}+2\|e\|\|P\bar{B}_{\omega_{a}}\|\|\bar{\omega}_{a}\|$
(27) $\displaystyle=$
$\displaystyle-(1-\chi)\lambda_{\min}(-\Delta)\|e\|^{2}-\chi\lambda_{\min}(-\Delta)\|e\|^{2}+\|\bar{T}^{T}M_{\nu}\bar{T}\|\|\bar{\omega}_{a}\|^{2}+2\|e\|\|P\bar{B}_{\omega_{a}}\|\|\bar{\omega}_{a}\|$

for any $\chi\in(0,1)$ and $\lambda_{\min}(\cdot)$ the minimum eigenvalue of
its symmetric argument. Now, a class $\mathcal{K}$ function
$\xi(\bar{\omega}_{a})$ exist such that we have

$\displaystyle\dot{W}(e)\leq$
$\displaystyle-(1-\chi)\lambda_{\min}(-\Delta)\|e\|^{2},\quad\forall\hskip
2.84526pt\|e\|\geq\xi(\|\bar{\omega}_{a}\|),$

$\xi(\bar{\omega}_{a})$ exists since the second quadratic term in $\|e\|$ can
dominate the third and fourth terms in (27) for large enough $\|e\|$.
$\xi(\bar{\omega}_{a})$ can be found by solving the following second-order
inequality for $\|e\|$

$\displaystyle\chi\lambda_{\min}(-\Delta)\|e\|^{2}\geq$
$\displaystyle\|\bar{T}^{T}M_{\nu}\bar{T}\|\|\bar{\omega}_{a}\|^{2}+2\|e\|\|P\bar{B}_{\omega_{a}}\|\|\bar{\omega}_{a}\|$

Since an explicit expression for the ISS gain is not needed in this paper, we
do not give a closed-form solution for $\xi(.)$.

Therefore, the conditions in Lemma 1 are satisfied if $\Delta$ is a negative
definite matrix. Hence, under such condition system (11) is ISS with input
$\bar{\omega}_{a}$. Based on the above analysis, the proposition formalizes an
LMI condition ($\Delta\prec 0$) that guarantees an ISS estimation error
dynamics (11).

As the final step, we want to prove that $\Delta\prec 0$ is equivalent to
(15a)-(15b). Using $\Delta$ defined in (24), (7c), and the Schur complements
on $\Delta\prec 0$, we can derive (15a) with $X$ and $X_{12}$ in terms of the
original observer gains $\\{E,K,J\\}$ as

$\displaystyle
X:=N^{T}P+PN+\alpha(V_{g_{a}}^{T}V_{g_{a}}-V_{g_{a}}^{T}JC_{a}-C_{a}^{T}J^{T}V_{g_{a}}),$
$\displaystyle
X_{12}:={\left[\begin{array}[]{ll}\sqrt{2\alpha}PMS_{g_{a}}&\sqrt{\alpha}C_{a}^{T}J^{T}\end{array}\right]},$

where $N^{T}P+PN$ expands as

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

and we have

$\displaystyle PMS_{g_{a}}$ $\displaystyle=PS_{g_{a}}+PEC_{a}S_{g_{a}}.$

Consider the following change of variables

$R:=PE,\qquad Q:=PK.$

Applying this change of variables on the above expanded $X$ and $X_{12}$, the
linear inequality (15a)-(15b) can be concluded.

When $\alpha$ is zero (no nonlinearity), it holds that $\Delta=N^{T}P+PN$ and,
therefore, we only require $N^{T}P+PN$ to be negative definite, which by
change of variables transforms to $S_{11}\prec 0$ is necessary and sufficient
condition for ISS of the linear error dynamics [35, Col. 5.1]. This conclude
the results of the proposition.

##  Appendix B Proof of Proposition 2

Let us first introduce the following Lemma, which is required to prove the
result of Proposition 2. In the following lemma, we state a Lyapunov-based
sufficient condition for having such a bounded $\mathcal{L}_{2}$-gain
property.

######  Lemma 2.

_( $\mathcal{L}_{2}$-gain Inequality)_ Consider (10) with $\nu(t)=\mathbf{0}$
and suppose there exists a continuously differentiable positive semi-definite
function $W(e)$ satisfying

$\displaystyle\dot{W}(e)$ $\displaystyle\leq
a(\lambda^{2}\omega_{a}^{T}\omega_{a}-e_{f}^{T}e_{f}),$ (28)

with $a,\lambda>0$ and the fault estimation error $e_{f}$ as in (8). Then, the
$\mathcal{L}_{2}$-gain from $\omega_{a}$ to $e_{f}$ in (10) is less than or
equal to $\lambda$.

_Proof : _ The proof is similar to the proof of Theorem 5.5 in [35]. Clearly,
by integrating (28) over a finite time $(0,\tau)$, we have

$W(e(\tau))-W(e(0))\leq
a\lambda^{2}\int_{0}^{\tau}\|\omega_{a}\|^{2}dt-a\int_{0}^{\tau}\|e_{f}\|^{2}dt.$

Using $W(e)\succeq 0$ and neglecting the initial condition, the above
inequality can be written as

$\int_{0}^{\tau}\|e_{f}\|^{2}dt\leq\lambda^{2}\int_{0}^{\tau}\|\omega_{a}\|^{2}dt$

Considering Definition 2, the above inequality concludes the result in the
lemma. $\blacksquare$

For the purpose of $\mathcal{L}_{2}$-gain analysis, we define
$W(e):={e}^{T}P{e}$ with positive definite matrix $P$. Then, we can guarantee
that the $\mathcal{L}_{2}$-gain inequality in (28) holds for the time-
derivative $\dot{W}(e)$ evaluated along solutions of the error-dynamics (10),
by using (26) (and assuming $\nu=\boldsymbol{0}$), as follows:

$\displaystyle\dot{W}(e)$ $\displaystyle\leq e^{T}\Delta
e-2e^{T}PMB_{\omega_{a}}\omega_{a}\leq
a(\lambda^{2}\omega_{a}^{T}\omega_{a}-e^{T}\bar{C}^{T}\bar{C}e).$

The above inequality can be written as

$e^{T}(\Delta+a\bar{C}^{T}\bar{C})e-2e^{T}PMB_{\omega_{a}}\omega_{a}-a\lambda^{2}\omega_{a}^{T}\omega_{a}\leq
0,$

for which we can give the following sufficient matrix inequality

$\left[\begin{array}[]{cc}\Delta+a\bar{C}^{T}\bar{C}&-PMB_{\omega_{a}}\\\
&-a\lambda^{2}I\end{array}\right]\preceq 0.$

If we follow the same procedure in the proof of Proposition 1, the equivalent
inequality can be found as

$\left[\begin{array}[]{ccc}X+a\bar{C}^{T}\bar{C}&-(P+RC_{a})B_{\omega_{a}}&X_{12}\\\
&-a\lambda^{2}I&\boldsymbol{0}\\\ &*&-I\end{array}\right]\preceq 0.$

Finally, by defining a change of variable as $\rho:=\lambda^{2}$, the LMI
condition (16d) implies the above inequality.

When $\alpha$ is zero (linear case), the LMI condition (16d) using the Schur
complement is equivalent to (16e). This LMI is necessary and sufficient
condition for the the $\mathcal{L}_{2}$-gain of (10) (with $\delta
g=\boldsymbol{0}$ for this linear case) from $\omega_{a}$ to the fault
estimation error $e_{f}$ to be upper bounded by $\sqrt{\rho}$, as given in
[23, Prop. 3.12].

##  Appendix C Proof of Proposition 3

For the purpose of $\mathcal{L}_{2}-\mathcal{L}_{\infty}$-gain analysis, we
define $W(e):={e}^{T}P{e}$ with positive definite matrix $P$. Then, we can
impose the quadratic performance inequality

$\displaystyle\dot{W}(e)$ $\displaystyle\leq b^{2}\nu_{a}^{T}\nu_{a}.$ (29)

Using (26) (and assuming $\omega_{a}=\boldsymbol{0}$), (29) can be written as
follows:

$\displaystyle\dot{W}(e)$ $\displaystyle\leq e^{T}\Delta
e+2e^{T}PB_{\nu_{a}}\nu_{a}+\nu_{a}^{T}M_{\nu}\nu_{a}\leq
b^{2}\nu_{a}^{T}\nu_{a}.$

The above inequality can be written as

$e^{T}\Delta
e+2e^{T}PB_{\nu_{a}}\nu_{a}+(M_{\nu}-b^{2}I)\nu_{a}^{T}\nu_{a}\leq 0,$

for which we can give the following sufficient matrix inequality

$\left[\begin{array}[]{cc}\Delta&PB_{\nu_{a}}\\\
&M_{\nu}-b^{2}I\end{array}\right]\preceq 0.$

Since $M_{\nu}=T_{\nu}^{T}J^{T}JT_{\nu}$ is not linear in the design variable
$J$, we use the Schur complement again to obtain the following equivalent
inequality:

$\left[\begin{array}[]{ccc}\Delta&PB_{\nu_{a}}&\boldsymbol{0}\\\
&-b^{2}I&T_{\nu}^{T}J^{T}\\\ &*&-I\end{array}\right]\preceq 0.$

If we follow the same procedure in the proof of Proposition 1, the equivalent
LMI can be found:

$\left[\begin{array}[]{cccc}X&H_{12}&\boldsymbol{0}&X_{12}\\\
&-b^{2}I&T_{\nu}^{T}J^{T}&\boldsymbol{0}\\\ &*&-I&\boldsymbol{0}\\\
&*&*&-I\end{array}\right]\preceq 0,$

where $H_{12}$ as defined in (17j). This completes the proof that the
condition (17e) in the proposition imply the satisfaction of (29). Besides, by
integrating (29) over a finite time $(0,\tau)$ and neglecting the initial
condition, we have

${e}(\tau)^{T}P{e}(\tau)\leq b^{2}\int_{0}^{\tau}\nu_{a}^{T}\nu_{a}dt.$

If we multiply both sides by $\gamma^{2}$ and impose a lower bound to enable
bounding the error on the fault estimate ($e_{f}=\bar{C}e$), we have

${e}^{T}(\tau){\bar{C}}^{T}\bar{C}e(\tau)\leq\gamma^{2}{e}^{T}(\tau)P{e}(\tau)\leq(\gamma
b)^{2}\int_{0}^{\tau}\nu_{a}^{T}\nu_{a}dt.$ (30)

Then, it follows from the first part of the above inequality that we need

${e}^{T}(\tau)(P-\frac{{\bar{C}}^{T}\bar{C}}{\gamma^{2}})e(\tau)\geq 0.$

Using the Schur complement, the following LMI implies the satisfaction of the
above inequality:

$\left[\begin{array}[]{cc}P&\bar{C}^{T}\\\ &\sigma I\end{array}\right]\succeq
0,$

where $\sigma:=\gamma^{2}$. Now, by imposing the above LMI, (30) indeed holds.
Therefore,

$\displaystyle\sup_{\tau\geq 0}{e}^{T}(\tau){\bar{C}}^{T}\bar{C}e(\tau)$
$\displaystyle\leq\sup_{\tau\geq 0}(\gamma
b)^{2}\int_{0}^{\infty}\nu_{a}^{T}\nu_{a}dt=(\gamma
b)^{2}\int_{0}^{\infty}\nu_{a}^{T}\nu_{a}dt$

which concludes the result in the proposition for nonlinear case.

When $\alpha$ is zero (linear case), the LMI condition (17e) using the Schur
complement is equivalent to (17m). Then, (17m) and (17h) are necessary and
sufficient conditions for the $\mathcal{L}_{2}-\mathcal{L}_{\infty}$ induced
norm of (10) (with $\delta g=\boldsymbol{0}$ for this linear case) from
$\nu_{a}$ to the fault estimation error $e_{f}$ to be upper bounded by
$\sqrt{\sigma}$, as given in [23, Prop. 3.15].

## Acknowledgements

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).

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