Nicolas Langlois

Mrs Laura Guillet

Mr Antoine Ferreira

Tingting Zhang

Keywords: Observer design, Nonlinear dynamical systems, Fault diagnosis, Unknown input, Time-varying, Finite memory observer

en Ingénierie des Systèmes, Mécanique et Énergétique) at INSA Centre Val de Loire. There are many people whom I would like to thank for their contributions and support, both directly and indirectly, to this PhD thesis. First and foremost, I wish to thank all my committee members who were more than generous with their expertise and precious time. A special thanks to Mr. Youssoufi TOURE for accepting being the committee chairman in spite of the pandemic

General Introduction

Context

Most geometric control design methods for nonlinear systems assume that all state variables are available for feedback in real time [START_REF] Rajamani | Observers for nonlinear systems: Introduction to part 1 of the special issue[END_REF]]. Unfortunately, the entire state is usually too expensive or impossible to measure, and certain form of estimation is necessary.

A common way to solve this problem is to set several sensors in the real physical system and design an algorithm, so-called observer, whose role is to perform a reliable estimate of the whole system state through the information provided by the sensors. It is worth to mention that such an algorithm can exist only if the measurements from the sensors contain enough amount of information to uniquely determine the system states, which is what we 1 said "the system is observable". Observer is commonly seen as a "soft sensor". Meanwhile, it can be noticed that observer-based estimation provides the basis of implementing many important engineering applications over past decades, especially fault diagnosis of different fields. As the real physical systems become more and more complex, more and more system components are involved. The requirements of fault diagnosis are therefore much more important since they are crucial means to maintain system safety and reliability, namely to ensure that the system is well functioning. Under such circumstances, the corresponding growing demand of nonlinear observer design have gained increasing consideration as well. However, unlike linear time-invariant systems, the design of a stable observer for nonlinear systems still suffers from a significant lack of general systematic method despite the existing developed techniques, including Luenberger-like nonlinear observer [START_REF] Ciccarella | A luenberger-like observer for nonlinear systems[END_REF][START_REF] Rajamani | Observers for lipschitz nonlinear systems[END_REF]], linear matrix inequalities (LMI) [START_REF] Zemouche | On LMI conditions to design observers for lipschitz nonlinear systems[END_REF], high-gain observer [START_REF] Ahrens | High-gain observers in the presence of measurement noise: A switched-gain approach[END_REF][START_REF] Sanfelice | On the performance of high-gain observers with sign-indefinite gain adaptation under measurement noise[END_REF][START_REF] Prasov | A nonlinear high-gain observer for systems with measurement noise in a feedback control framework[END_REF]]. 1 In this thesis, the word we is often used to refer to my colleagues and myself. Sometimes, we also indicates that the control and estimation community. The distinction should be clear from the context. However, we encourage the reader not to read too much into my use of the word we, it is more a matter of personal preference and style rather than a claim to authority.

Motivations and Objective

The above context reveals the needs of observer design techniques as well as observer-based fault diagnosis tool for nonlinear dynamical systems. This motivates me to look for some contributions to this subject. Meanwhile, when I started my thesis, some preliminary results about finite memory observer design had already been obtained in the case of time-invariant linear systems, where the nonlinear and time-varying cases remained to be developed. It then brought the beginning of my work in this thesis.

In the literature or textbooks of control theory, the general continuous-time dynamical models for nonlinear systems is under the form ẋ(t) = f (x(t), u(u)) y(t) = h(x(t), u(t))

where x(t) is the state, u(t) is the control input and y(t) is the measurement at time instant t. However, it may be worth noting that: If either the state equation or the output equation has nonlinear terms, then the system is called a nonlinear system. This means that the measurement equation h(•) of a nonlinear system does not have to be always nonlinear.

As the dynamics of most engineering systems are naturally continuous, e.g., trajectories of vehicles and flow of electric current, it is therefore more convenient and accurate to model the physical processes in continuous time with nonlinear differential equations. However, observations or measurements are usually taken at discrete time instants using digital sensors, which are massively used in today's "digital era". For this reason, the observer designed in this thesis are investigated based on a discrete-time measurement model.

Note that the purpose of the observer design in this work is to support the fault diagnosis for practical systems, the considered nonlinear systems should be as close as possible to the practical one, in which the disturbance and uncertainty is inevitable. The measurement noise, system uncertainties including process noise and modeling uncertainties are therefore taken into consideration in this thesis.

In general, most practical systems can be described by time-invariant model, however, there are some systems that still cannot be modeled by assuming that they are time invariant, such as aircraft [START_REF] Cook | Flight dynamics principles[END_REF]] and the human vocal tract [START_REF] Strube | Time-varying wave digital filters and vocal-tract models[END_REF]]. In the meantime, it should be noted that the linearization of a time-invariant nonlinear system also (when the nominal solution is not a constant) gives rise to time-varying systems, and this is actually one of the chief ways time-varying systems are encountered in system analysis [START_REF] Kailath | Linear systems[END_REF]]. This is why we also devote part of our efforts to study time-varying systems (linear case and nonlinear case). The overall scheme is shown in Figure 1.1.

Chapter 1 General Introduction

Based on the context introduced in section 1.1 and the above presented motivations, our objective in this thesis is to design a nonlinear observer for a class of continuous-time nonlinear systems with discrete-time measurements in the presence of measurement noise and process noise in order to perform fault diagnosis.

Outline of Manuscript

This manuscript is organized as follows:

Chapter 1 The thesis begins with this chapter by a brief introduction, which presents the context of this work, the problem statements and the organization of the manuscript.

Chapter 2 A review of the relevant definitions and notations of observability is firstly recalled in this chapter. Then there follows three typical optimization-based moving horizon nonlinear observer design techniques. Observer-based fault diagnosis (fault detection and isolation) is performed at the end of this chapter as well.

Chapter 3 This chapter is devoted to develop a continuous-discrete finite memory observer for a class of nonlinear systems under the presence of both process noise and measurement noise. The properties of estimation performance have been theoretically proved and practically validated by simulations. For the diagnosis purpose, the designed nonlinear observer is used to realize fault detection and isolation of different sensor and actuator faults of a nonlinear robotic arm system.

Chapter 4 Based on chapter 3, a nonlinear observer with unknown input is proposed in this chapter by an augmented model. It is then applied to an actuator fault detection and estimation, where the actuator fault is modeled as unknown input. Moreover, multiple simultaneous sensor and actuator faults are successfully detected and isolated thanks to the unknown input estimation.

Chapter 5 A time-varying observer is designed for both linear and nonlinear time-varying systems in this chapter. The detailed illustration and proofs of the proposed observer are firstly provided for linear case, then further extended to nonlinear case.

Chapter 6 Conclusions and perspectives are summarized in this final chapter.

A Note on Symbols: In this thesis, we may sometimes use the same alphabet letter, Greek or Romain symbols representing different variables in different chapters. The specific meaning needs to be inferred from the context.

Chapter 2

On Some Nonlinear Observers (NLOs) and Fault Diagnosis "If I have seen further, it is by standing on the shoulders of Giants."

-Isaac Newton,

Contents of chapter

Introduction

Résumé en français :

Suite aux motivations et contextes introduits dans le chapitre précédent, nous allons présenter dans ce chapitre quelques observateurs non linéaires et méthodes de diagnostic de défauts basées sur observateurs. Comme nous l'avons mentionné précédemment, la synthèse d'un observateur stable pour les systèmes non linéaires est un défi très important depuis des décennies, et ce défi a été abordé dans la littérature grâce au développement de diverses techniques de synthèse dans le cadre non linéaire. Il convient de noter que notre objectif dans ce chapitre n'est pas de fournir un aperçu exhaustif des observateurs non linéaires, mais plutôt de situer notre contribution et d'introduire une synthèse basée sur l'optimisation, en particulier des techniques à horizon glissant. En plus, quelques méthodes de diagnostic à base d'observateur, des définitions/concepts de base mais importants et des étapes de diagnostic, qui sont nécessaires dans le reste de cette thèse, seront également établis à la fin de ce chapitre.

Introduction

Based on the motivations and backgrounds introduced in the previous chapter, we are going to review in this chapter some nonlinear observers and observer-based fault diagnostic methods.

As we mentioned before, the design of a stable observer for nonlinear systems has been a significant challenge over the past decades, and this challenge has been addressed in the literature through the development of various observer design techniques specific to nonlinear systems. Our intension in this chapter is not to provide an exhaustive overview on nonlinear observer design, but rather to situate our contribution and introduce some optimization-based methods, specifically moving horizon techniques. Moreover, some observer-based diagnostic methods, basic but important definitions/concepts and diagnostic steps, which are needed in the rest of this thesis, will also be established at the end of this chapter.

Observability of Dynamical Systems

Consider the following general nonlinear system:

ẋ(t) = f (x, u,t) y(t) = h(x)
(2.1)

with system state noted as

x [x 1 , • • • , x n ] T ∈ M, where M is a smooth manifold of R n ; u [u 1 , • • • , u m ] T ∈ U ⊂ R m represents input while y [y 1 , • • • , y p ] T ∈ Y ⊂ R p is output. The two mappings f [ f 1 , • • • , f n ] T (with f : M × U → M) and h [h 1 , • • • , h p ] T (with h : M → Y)
are the vectorfields defined on M. Denote x(t, x 0 , u) as the state solution of system (2.1) at instant t due to the initial state x 0 = x(t 0 ) and the input u over [t 0 ,t], and the corresponding output is denoted as y(t, x 0 , u).

Observability problem and definitions

In control theory, observability [START_REF] Canuto | Spacecraft dynamics and control[END_REF] concerns the problem of how to estimate the state vector x(t) from incomplete and possibly inaccurate observations (measurements), collected over a finite time t ft 0 < ∞. The observations on which to rely are the input and output temporal profiles u(t) and y(t) with t ∈ [t 0 ,t f ]. The following problem is generally considered: Observability problem: Given t 0 , find a final time t f > t 0 , such that x 0 can be uniquely determined from input u(t) and output y(t) with t ∈ [t 0 ,t f ].

As stated in [START_REF] Nijmeijer | Nonlinear dynamical control systems[END_REF] and [START_REF] Hermann | Nonlinear controllability and observability[END_REF], the observability of systems (2.1) is defined from the indistinguishability concept, which is: Definition 2.1 (Indistinguishable). For system (2.1), two initial states x 1 0 ∈ M and x 2 0 ∈ M are said to be indistinguishable if for any admissible system input u(t) (for all t > 0), the corresponding outputs y(t, x 1 0 , u) and y(t, x 2 0 , u) are equal. According to Definition 2.1, the observability definition is then given as: Definition 2.2 (Observable). For different initial states of system (2.1), if there is no indistinguishable pair {x 1 0 , x 2 0 }, then system (2.1) is said to be observable at x 0 . Moreover, if the system is observable at every x ∈ M, then we say that system (2.1) is observable.

Note that the observability defined above is a global concept, which means globally observable. Since our aim afterward is to focus on the rank condition study, the observability in this case will inherently be a local condition for nonlinear systems, which leads to the following definitions : Definition 2.3 (V -Indistinguishable). Given two initial states x 1 0 ∈ M and x 2 0 ∈ M, let V be a subset of M containing x 1 0 as well as x 2 0 , we say x 1 0 is V -indistinguishable from x 2 0 if for every admissible input u(t) (all t > 0) whose trajectories x(t, x 1 0 , u) and x(t, x 2 0 , u) both remain in V , the corresponding outputs y(t, x 1 0 , u) and y(t, x 2 0 , u) are the same. Definition 2.4 (Locally Observable). The system (2.1) is called locally observable at x 0 if there exists a neighborhood W around x 0 such that for every neighborhood V ⊂ W of x 0 , y(t, x 1 0 , u) = y(t, x 2 0 , u) implies x 1 0 = x 2 0 . If the system (2.1) is locally observable at each x 0 , then we call it locally observable.

Observability of Dynamical Systems

Remark 2.1 (Detectability). System (2.1) is said to be detectable if all the unobservable states are stable.

Notice that another concept, stability (stable), is mentioned in Remark 2.1. It will be analyzed later according to the specific study case, which means it will not be detailed here. Please see [START_REF] Nijmeijer | Nonlinear dynamical control systems[END_REF], [START_REF] Hermann | Nonlinear controllability and observability[END_REF], [START_REF] Fossard | Nonlinear Systems[END_REF] and the references therein for more details if needed. The same strategy was also used for the concept of controllability (controllable), which is a dual concept of observability. Roughly speaking, the concept of controllability denotes the ability to move a system around in its entire configuration space using only certain admissible manipulations. The exact definition may varies slightly within the framework or the type of models applied.

So far, we have presented the general and fundamental concepts of observability. As we known, observability indicates whether or not we can uniquely determine the system state x(t) from the input u and measurements y = h(x), which can be directly measured. As a matter of fact, the derivatives of output y = h(x) also contain the information of system state x(t). This fact leads us to the Lie derivatives-based "rank condition" study for observability in next subsection.

Observability rank condition

Considering system (2.1), let us assume that input u = u c is constant. Firstly we define Definition 2.5 (Observation Space). The observation space O for system (2.1) is defined as the linear vectorial subspace (over R) of functions on M, including output function h ∈ R p , and all its repeated Lie derivatives

{L 1 f u h, L 2 f u h, • • • , L k f u h} with respect to vectorfields f u = f (x, u) with u = u c being fixed.
According to the definition of Lie derivative, it is easy to see that h = L 0 f u h, then we can write the observation space O(x, u c ) as follows:

O(x, u c ) :=       L 0 f u h L 1 f u h . . . L k f u h       =       y ẏ . . . y (k)       (2.2)
where y (k) represents the k-th time derivative of y. Note dO(x, u c )| x as the differential space of O(x, u c ) and dO(x, u c )| x=x 0 as the differential space evaluated at x 0 . The observability rank condition is thus denoted as:

Chapter 2 On Some Nonlinear Observers (NLOs) and Fault Diagnosis Definition 2.6 (Observability Rank Condition). Suppose u = u c , system (2.1) is said to be locally weakly observable at x 0 if the dimension of dO(x 0 , u c ) equals to n, namely

dim{dO(x, u c )| x=x 0 } = rank       dL 0 f u h dL 1 f u h . . . dL k f u h       = rank       dy d ẏ . . . dy (k)       = n (2.3)
In addition, the system is said to satisfy the observability rank condition if the above rank condition (2.3) holds for all x 0 .

The rank condition states that dO(x 0 , u c ) contains n linearly independent vectors. It may be worth mentioning that k = n -1 when the dynamical system is linear thanks to Cayley-Hamilton theorem, while there does not exist a fixed value k for nonlinear dynamical system in general. In fact, if we try to see the rank condition from the linear algebraic point of view, it is easy to understand that it means the kernel of dO(x 0 , u c ) is reduced to the zero subspace. That is why both the output h and its Lie derivatives

L i f u h (i = 1, 2, • • • , n-1)
in the direction of f u will change as the state x change, this is exactly what we called "observable".

Remark 2.2. The following remarks are worth mentioning concerning the observability rank condition in Definition 2.6 :

1. It is only a "local" condition for nonlinear systems.

2. Unlike the linear case, it is only a necessary but "non" sufficient condition to build an observer for a nonlinear system. It may need to take into account the input properties in some cases.

Assumption 2.1 (Universal Inputs Assumption). It can be seen that the control input u = u c , taken as a constant during the entire illustrations above and there is no doubt that input u will affect the observability of nonlinear system (2.1). However, since the nonlinear systems considered in this thesis are assumed in the case where control input u are "universal inputs", the influence of input u on observability will no longer be detailed here. Please refer to [START_REF] Hermann | Nonlinear controllability and observability[END_REF] for a synthesis of this problem.

Up to nowadays, there is still no complete theory that allows to design an observer for general observable systems. Assumption 2.1 brings us to an interesting class of nonlinear systems, which are observable in the following sens [START_REF] Hammouri | Nonlinear observers for locally uniformly observable systems[END_REF]]: Definition 2.7 (Uniformly Observable). If the observability of system (2.1) does NOT depend on the input control u(t), i.e. all the inputs are universal, then system (2.1) is said to be uniformly observable. If for any t > 0, input u is universal on [0,t], then system (2.1) is uniformly locally observable.

So far, the aforementioned part is the general rank condition of observability for general nonlinear system (2. 1). In what follows, we will spend a little bit more space in this section to show the counterpart of observability rank condition in the cases of linear time-invariant (LTI) systems and linear time-varying (LTV) systems, which have been known as two typical linear systems.

Consider the following LTV systems with x ∈ R n , u ∈ R m and y ∈ R p :

LTV systems:

ẋ(t) = A(t)x(t) + B(t)u(t) y(t) = C(t)x(t) (2.
O ob (t 0 ,t f ) := t f t 0 Φ T (θ ,t 0 )C T (θ )C(θ )Φ(θ ,t 0 ) dθ (2.5)
is nonsingular. Here, Φ(θ ,t 0 ) ∈ R n×n is so-called state transition matrix from t 0 to θ with θ ∈ [t 0 ,t f ], which satisfies the following differential equation:

   dΦ(θ ,t 0 ) dθ = A(θ )Φ(θ ,t 0 ) Φ(t 0 ,t 0 ) = I n I n is identity matrix of size n (2.6)
Remark 2.3. Various methods of calculating state transition matrix Φ(•, •) can be found in the literatures, see [START_REF] Kailath | Linear systems[END_REF] and [START_REF] Chen | Linear system theory and design[END_REF] for example. It may be worth mentioning here the properties of state transition matrix Φ(•, •), which includes

• Φ -1 (t 2 ,t 1 ) = Φ(t 1 ,t 2 ) for any t 1 ,t 2 ∈ [t 0 ,t f ]; • Φ(t 1 ,t 3 )Φ(t 3 ,t 2 ) = Φ(t 1 ,t 2 ) for any t 1 ,t 2 ,t 3 ∈ [t 0 ,t f ]; • Φ(t,t) = I n holds for all t ∈ [t 0 ,t f ]; • Φ(•, •) is never singular.
For LTV systems (2.4), Definition 2.8 tells us that it is impossible to distinguish the initial state x 1 0 from x 2 0 if x 1 0x 2 0 lies in the kernel of O ob (t 0 ,t f ). Also, it is easy to see that the observability gramian matrix O ob (t 0 ,t f ) has the following properties:

Chapter 2 On Some Nonlinear Observers (NLOs) and Fault Diagnosis

• O ob (t 0 ,t f ) is symmetric;

• For all t f > t 0 , O ob (t 0 ,t f ) is positive semidefinite; Suppose the matrices A(•) and C(•) in (2.4) have real-analytic entries, then the observability rank condition in 2.6 is equivalent as follows Definition 2.9 (Observability Rank Condition for LTV Systems [START_REF] Sontag | Mathematical control theory[END_REF]]). System (2.4) is said to be observable on [t 0 ,t f ] if there exists an arbitrary t ∈ [t 0 ,t f ], so that

dim{dO} = rank       C 0 (t) C 1 (t) . . . C k (t)       = n for some k (2.7) with    C 0 (t) := C(0) i = 0 C i+1 (t) := C i (t)A(t) + d dt C i (t) i ≥ 1
Now consider that system (2.4) has the constant parameters A, B and C, namely it becomes the LTI systems LTI systems:

ẋ(t) = Ax(t) + Bu(t) y(t) = Cx(t) (2.8)
Based upon the observability rank condition above for LTV systems, the equivalent studies for LTI systems (2.8) are easy to obtain as: Definition 2.10 (Observability Gramian for LTI). System (2.8) is said to be observable if and only if the following n × n matrix

O ob (t 0 ,t) := t t 0 e A T (θ -t 0 ) C T Ce A(θ -t 0 ) dθ (2.9)
is nonsingular. It is worth noting that the term e A(θ -t 0 ) in (2.9) is the state transition matrix Φ(θ ,t 0 ) from t 0 to θ in the case of LTI systems.

Some Optimization-based Nonlinear Observers

Definition 2.11 (Observability Rank Condition for LTI Systems). System (2.8) is said to be observable if

dim{dO} = rank       C CA . . . CA n-1       = n (2.10)
Note that there is no state x involved in the observability rank condition (2.7) and (2.10) for LTV and LTI systems. It means that unlike the nonlinear case, the observability rank condition for linear system is global, which is well-known as the "Kalman rank condition" [START_REF] Kalman | Mathematical description of linear dynamical systems[END_REF]].

Some Optimization-based Nonlinear Observers

Among all the observer design methods for nonlinear systems, the optimization-based moving horizon technique is the one on which we are going to focus in this section. More specifically, this kind of method turns the state estimation problem into an optimization problem, which generally aims at minimizing the difference between real measurement and the predicted one obtained by using some suitable algorithms on a predetermined moving time horizon. Newton's method and least square method (Levenberg-Marquardt algorithm) are the two main optimization approaches for the NLOs introduced in this section. For the sake of brevity, we have included only some of the many techniques and the references listed in this part are therefore by no means exhaustive.

Moving horizon observers (MHOs)

The name of moving horizon observers (MHOs) was first used by H. Michalska and D. Q. Mayne in the early 1990s [Mayne andMichalska 1992, Michalska and[START_REF] Michalska | Moving horizon observers[END_REF], which deals with the continuous-time nonlinear systems (2.1) recalled here as:

ẋ(t) = f (x, u) (2.11a) y(t) = h(x) (2.11b)
where the uniform observability assumption in Definition 2.7 is required and the functions f and h are assumed locally Lipschitz continuous. Then an on-line approximation of a cost function J is defined under the form of Euclidean norm, which described the difference between the real system output and estimated one over the interval [t 1 ,t 2 ] as:

J(w;t 1 ,t 2 ) t 2 t 1
∥h(X(s; w,t 1 ))y(s)∥ 2 ds (2.12) here y(•) is the real system output with the true (but unknown) initial condition x 0 at initial time instant t 0 . Let X(•; w,t 1 ) represent the solution of (2.11a) passing through state w at time instant t 1 , so it is obvious that x

(t) = X(t, x 0 ,t 0 ), i.e. y(•) = h(x(•)) = h(X(•; x 0 ,t 0 )).
The minimization strategy for the cost function (2.12) is well defined using a gradient-based descent approach on the interval [t -T,t] where there always exists a better estimation w(t) for state x(t) of each interval in the case of no measurement noise and no model discrepancy [START_REF] Michalska | Moving horizon observers and observer-based control[END_REF].

It should be mentioned that the above optimization problem has to be solved at each sampling time instant, this may present some serious drawbacks in the case that one may not be able to control the required time to satisfy a given estimation accuracy. However, this kind of observer design idea establishes a cornerstone for the numerous contributions on continuous-time nonlinear observer synthesis, such as approaches based Newton's method [START_REF] Moraal | Nonlinear Observer Design: Theory and Applications to Automotive Control[END_REF]Grizzle 1995, Zimmer 1993a;1994], which we will get back to detail in next subsection. Moreover, the improved and developed MHOs were studied and revisited by Alamir, which the developed ideas are closely connected to those when the real time implementation of model predictive control is addressed, see [START_REF] Alamir | Optimization based non-linear observers revisited[END_REF] and [START_REF] Alamir | Nonlinear moving horizon observers: Theory and real-time implementation[END_REF] for more information.

At the end of this part, we would like to share our opinion on the name "moving horizon observer (MHO)": Michalska and Mayne indeed used MHO for their proposed observer, however this name should not be only dedicated to this single method. We believe that all the observers synthesis approaches based on a moving horizon technique can be called MHO. MHO speaks for a class of observers, for instance, finite memory observer is also a kind of MHO, so are the observers introduced in next section.

Nonlinear observers based on Newton's method

Newton's method or in some cases a quasi-Newton method was first introduced for the observer design to discrete-time systems by [START_REF] Grizzle | Newton, observers and nonlinear discrete-time control[END_REF] and then improved in [START_REF] Moraal | Nonlinear Observer Design: Theory and Applications to Automotive Control[END_REF] and [START_REF] Moraal | Observer design for nonlinear systems with discretetime measurements[END_REF]. Meanwhile, Zimmer has also proposed the nonlinear observer design methods for continuous-time systems. In article Zimmer [1993a], the gradient descent method is applied to deal with the problem of minimizing a cost function (a large consumption of computation time). In [START_REF] Zimmer | Newton step observer for nonlinear feedback systems[END_REF] and [START_REF] Zimmer | State observation by on-line minimization[END_REF], Newton's method is taken to minimize the cost function by using 2.3 Some Optimization-based Nonlinear Observers curvature information (i.e. the second derivative) to take a more direct path than gradient descent. Let's take Newton's method in [START_REF] Zimmer | State observation by on-line minimization[END_REF] as an example to explain it in detail.

The considered nonlinear system is described by:

ẋ = f (x, u) y = g(x)
(2.13)

where the vectors x, u and the scalar y represent the system state, control input and output (measurement), respectively.

Assumption 2.2. f and g are twice differentiable on non-empty set S ⊂ R n .

Assumption 2.3. System (2.13) is locally uniformly observable. As introduced in Definition 2.7, "uniformly" means that the control input u is completely accessible.

Consider a strongly observable (2.13) system, an interval I 0 := [0, T ] and the output of the system y(•) on this interval. The goal is to estimate an initial value x 0 for x 0, * (real value), such that y(t; 0, x 0 ) = y(t; 0, x 0, * ) for all t ∈ I 0 .

The cost function N is defined by:

N (x 0 , x 0, * ) := 1 2 T 0 y(t; 0, x 0 ) -y(t; 0, x 0, * ) 2 dt (2.14)
here, x 0 is an estimate of x 0, * . Since the system (2.13) is observable, so x 0 and x 0, * are distinguishable, namely, N (x 0 , x 0, * ) = 0 ⇔ x 0 = x 0, * (2.15)

Take the partial derivative of (2.14) with respect to x 0 , we have

D 1 N (x 0 , x 0, * ) = T 0 y(t; 0, x 0 ) -y(t; 0, x 0, * ) ∂ y(t; 0, x 0 ) ∂ x 0 dt = T 0 y(t; 0, x 0 ) -y(t; 0, x 0, * ) ∂ g x(t; 0, x 0 ) ∂ x ∂ x(t; 0, x 0 ) ∂ x 0 T dt (2.16)
If N (x 0 , x 0, * ) is convex near x 0, * , the following equivalence is then valid:

N (x 0 , x 0, * ) = 0 ⇔ D 1 N (x 0 , x 0, * ) = 0 (2.17)
The common way to check the convexity of N (x 0 , x 0, * ) near x 0, * is to examine whether the Hessian matrix

D 2 1 N (x 0 , x 0, * ) := ∂ 2 N (x 0 , x 0, * ) (∂ x 0 ) 2 (2.18)
is positive-definite at x 0 = x 0, * . Fortunately, Lemma 2.1 below gives a necessary and sufficient condition such that D 2 1 N (x 0 , x 0, * ) is a Hilbert matrix and thus positive-definite (see [START_REF] Zimmer | State observation by on-line minimization[END_REF] for details).

Lemma 2.1. Given x 1 ∈ S, the Hessian matrix D 2 1 N (x 1 , x 1 ) is said to be positive define if and only if the following linearized system is observable in I

0 := [0, T ]      ż = ∂ f (x(t; 0, x 1 )) ∂ x z v = ∂ g(x(t; 0, x 1 )) ∂ x z (2.19)
Then, Newton's method is applied to approximate x 0, * . Recall that Newton's method generates a sequence of x 0 using recursion as

x 0 new = x 0 old -D 2 1 N (x 0 old , x 0, * ) -1 D 1 N (x 0 old , x 0, * ) (2.20)
Note that before applying Newton's method, one can always check whether D 2 1 N (x 0 , x 0, * ) is invertible and by Lemma 2 (omitted here) in [START_REF] Zimmer | State observation by on-line minimization[END_REF].

The conditions for terminating the iteration are

N (x 0 new , x 0, * ) < κN (x 0 old , x 0, * ) (2.21)
Now all the state estimations x(•) over the interval I 0 := [0, T ] can be calculated with the initial value x 0 new obtained by (2.20) and (2.21) as follows:

x(t) := X(t; 0, x 0 new ) t ∈ I 0 (2.22) here X(•; 0, x 0 ) represents the solution of system (2.13) by the four-order Runge-Kutta (RK4) method with the initial value x 0 . Notation: Given T > 0, for all k ∈ N 0 , define

I k := [t k ,t k + T ] t k := kT
Without loss of generality, Zimmer's method can be carried out over an arbitrary interval

I k , k ∈ N 0 . Let η(x k , x k, * ) = x k -D 2 1 N (x k , x k, * ) -1 D 1 N (x k , x k, * ) (2.23)
where x k represents the final estimate of x k, * at time t k after terminating of iteration. Also

x k, * := X(kT ; 0, x 0, * )) k ∈ N 0

Repeat the above procedure for each interval I k , k ∈ N 0 , a sequence of x k is generated recursively by:

x k := X(kT ; (k -1)T, η(x k-1 , x k-1, * )) = X(T ; 0, η(x k-1 , x k-1, * )) k ∈ N (2.24)
These sequences yield

   lim k→∞ ∥x k -x(t k ; 0, x 0, * )∥ = 0 lim k→∞ ∥η(x k , x k, * ) -x(t k ; 0, x 0, * )∥ = 0
(2.25) if x 0 and x 0, * are close enough. Finally, the sequence {η(x k , x k, * )} k∈N will be completed towards a piecewise differentiable function by:

R + → R n t → x N (t) := X(t; kT, η(x k , x k, * )) t ∈ [kT, (k + 1)T ) (2.26)
The true state x(•) is approximated by (2.26) in the following norm sense: lim t→∞ ∥x(t)x(t; 0, x 0, * )∥ = 0

(2.27) if x 0 and x 0, * are close enough. Figure 2.1 illustrates the idea described above This method shows a great performance for nonlinear system (2.13) when there is no control inputs. However, with the presence of control input u, the convergence has only been proven for special control input functions [START_REF] Zimmer | Newton step observer for nonlinear feedback systems[END_REF]]. Meanwhile, this approach has already been successfully applied to systems with multiple outputs as well in [START_REF] Zimmer | Newton-verfahren zur zustandsapproximation nichtlinearer systerne[END_REF], where the cost function in (2.14) need to be defined in a quadratic form like N (x 0 , x 0, * ) := 1 2 T 0 y(t; 0, x 0 )y(t; 0, x 0, * ) T y(t; 0, x 0 )y(t; 0, x 0, * ) dt Note that the Hessian matrix (2.18) of the cost function (2.14) is required in order to perform the Newton step of this method, this may be hard to realize in practical or be Chapter 2 On Some Nonlinear Observers (NLOs) and Fault Diagnosis Time Figure 2.1: The illustration of Zimmer's method [START_REF] Zimmer | State observation by on-line minimization[END_REF] impossible to perform when the nonlinear functions f and g are not twice differentiable. This method is not in general used even though it shows great theoretical significance.

So far, we've learned that the cost function in gradient descent method is minimized by updating the parameter in the steepest-descent direction which can be computationally expensive if the curvature in different directions is very distinct for the given function. Meanwhile, the (quasi-) Newton's methods remedy this problem but show more complexity due to the calculation of Hessian matrix and performs much worse when the parameters are far from their optimal values. Is there a method that can acts more like a gradient descent method when the parameters are far from their optimal value, and acts more like the (quasi-) Newton's methods when parameters are close to their optimal value? This brings us to the Levenberg-Marquardt algorithm in next subsection.

Nonlinear observers based on Levenberg-Marquardt algorithm

Levenberg-Marquardt (L-M) algorithm was first developed by K. Levenberg [START_REF] Levenberg | A method for the solution of certain non-linear problems in least squares[END_REF]] in the early 1940's and then was rediscovered in 1960s by D. Marquardt [START_REF] Marquardt | An algorithm for least-squares estimation of nonlinear parameters[END_REF]], which the L-M algorithm adaptively updates the parameter between the gradient descent method and the Gauss-Newton method. Although many variations of L-M algorithm have been published during these years [START_REF] Kanzow | Levenberg-marquardt methods with strong local convergence properties for solving nonlinear equations with convex constraints[END_REF][START_REF] Pujol | The solution of nonlinear inverse problems and the levenberg-marquardt method[END_REF][START_REF] Wilamowski | Improved computation for levenberg-marquardt training[END_REF][START_REF] Transtrum | Improvements to the levenberg-marquardt algorithm for nonlinear least-squares minimization[END_REF][START_REF] Bellavia | A levenberg-marquardt method for large nonlinear least-squares problems with dynamic accuracy in functions and gradients[END_REF]], the observer-related one in [START_REF] Abdelhedi | Levenberg-marquardt and moving horizon estimation for the synthesis of nonlinear observers[END_REF] is what we are going to discussed in this section.

Some Optimization-based Nonlinear Observers

The nonlinear system (2.1) is considered in [START_REF] Abdelhedi | Levenberg-marquardt and moving horizon estimation for the synthesis of nonlinear observers[END_REF], rewritten here

ẋ(t) = f (x, u) y(t) = h(x)
(2.28) let us consider L bounded observations collected at regular time interval T e such that the length of horizon lh = (L -1)T e . Then the optimization problem can be stated as the minimization of the following cost function:

J(x) = 1 2 L ∑ k=1 (y t k -ŷt k ) 2 (2.29)
where y is the measured output and ŷ is the estimated one. t k is the beginning of the horizon so that t ∈ [t k ,t k + lh]. We know that (2.29) has an optimum at x if the the following condition is verified:

dJ dx = 0 (2.30)
The function J(x k ) in (2.29) may be locally approximated through a second-order Taylor series expansion as

J(x k+1 ) = J(x k + dx k ) ≈ J(x k ) + ∇J • dx k + 1 2 dx T k • Hess(i, j) • dx k (2.31) that is dJ(x k ) ≈ ∇J • dx k + 1 2 dx T k • Hess(i, j) • dx k (2.32) with ∇J = ∂ J ∂ x x=x k = -2 L ∑ k=1 (y t k -ŷt k ) ∂ ŷt k ∂ x k (2.33a) Hess(i, j) = ∂ 2 J ∂ x i ∂ x j = 2 L ∑ k=1 ∂ ŷt k (x i ) ∂ x j 2 -2 L ∑ k=1 (y t k -ŷt k ) ∂ ŷt k ∂ x i ∂ x j (2.33b)
Apply the condition in (2.30) to (2.32), we get

∇J + Hess(i, j) • dx k = 0 (2.34)
then we straightly have the state variation dx k as

dx k = x k+1 -x k = -Hess(i, j) -1 ∇J (2.35)
Chapter 2 On Some Nonlinear Observers (NLOs) and Fault Diagnosis

It is easy to see that the second term of Hessian matrix in (2.33b) can make the Hessian matrix negative. Therefore, the main idea of L-M method is to replace the second term of Hessian matrix by a diagonal matrix, namely

x k+1 = x k -[Hess * (i, j) + λ k I] -1 ∇J (2.36) with Hess * (i, j) = 2 L ∑ k=1 ∂ ŷt k (x i ) ∂ x j 2
(2.37)

represents the Hessian matrix in (2.33b) without the second term. λ k is called damping parameter or relaxation coefficient, which actually adjusts the eigenvalues of Hessian matrix. When λ k is a small value, L-M works like Gauss-Newton method while large value of λ k results in a gradient descent like method. The choice of λ k in each iteration is simply by multiplying or dividing a certain multiples of the last iteration in [START_REF] Abdelhedi | Levenberg-marquardt and moving horizon estimation for the synthesis of nonlinear observers[END_REF].

For more information about the best choice for the damping parameter, see [START_REF] Transtrum | Improvements to the levenberg-marquardt algorithm for nonlinear least-squares minimization[END_REF].

Nonlinear Observers with Continuous-Discrete Model

As we know, the dynamics of most engineering systems are naturally in continuous time (CT), such as trajectories of vehicles, flow of electric current etc. It is therefore more convenient and accurate to modeling the physical processes in continuous time by nonlinear ordinary differential equations (ODEs). However, observations or measurements are usually taken by sampling at discrete time (DT) instants since digital sensors are commonly used for engineering systems in practice. For this reason, a significant amount of observer designs has been investigated in the literature based on continuous-discrete (CD) modeling over the past few decades, where the dynamic for systems is expressed based on CT model and the measurement is described through DT modeling.

Early years, Jazwinski [START_REF] Jazwinski | Stochastic processes and filtering theory[END_REF]] presented a CD Kalman filter to deal with the filtering problem for CT stochastic nonlinear systems with discrete measurements. Then, Deza et al [START_REF] Deza | High gain estimation for nonlinear systems[END_REF]] developed the CD high-gain observer with a corrected gain obtained through the integration of CD Riccati equation. Inspired by these earlier works, a large number of papers have been published in recent decades, focusing on CD model based nonlinear observer design subject in both theoretical and practical applied aspects. For instance, a CD observer is designed in [START_REF] Ali | Continuous-discrete timeobserver design for state and disturbance estimation of electro-hydraulic actuator systems[END_REF] for the Electro-Hydraulic Actuators (EHA) system subject to mechanical and hydraulic disturbances. Ling and Kravaris [START_REF] Ling | Multirate sampled-data observer design based on a continuous-time design[END_REF] developed a robust nonlinear CD observer to deal with the multi-rate sampling measurements by building an inter-sample predictor, which has been proved robust with respect to perturbations and inherits all the good properties of the CT implementation as long as the maximum sampling period does not exceed a certain threshold. An efficient linear matrix inequalities (LMIs)-based tool was built in [START_REF] Dinh | Continuous-discrete time observer design for lipschitz systems with sampled measurements[END_REF] to solve the problem of observer synthesis for a class of globally Lipschitz systems in small dimensions. In addition, it is worth mentioning here that there are several research scholars in France whose works have been realized on observer synthesis based on CD model over years. Their relative researches widely include from interval observer design [START_REF] Mazenc | Construction of interval observers for continuous-time systems with discrete measurements[END_REF], nonlinear observer design [START_REF] Farza | Continuous-discrete-time observers for a class of uniformly observable systems[END_REF][START_REF] Mazenc | Design of continuous-discrete observers for time-varying nonlinear systems[END_REF][START_REF] Farza | Adaptive observer design for a class of nonlinear systems. application to speed sensorless induction motor[END_REF] to system stabilization [Mazenc and[START_REF] Mazenc | Predictor-based sampled-data stabilization via continuous-discrete observers[END_REF][START_REF] Mazenc | Predictor-based sampled-data exponential stabilization through continuous-discrete observers[END_REF], observer synthesis for systems with delayed inputs or measurements [START_REF] M'saad | Identification of continuous-time linear systems with time-delay[END_REF][START_REF] Tréangle | Observer design for a class of disturbed nonlinear systems with time-varying delayed outputs using mixed time-continuous and sampled measurements[END_REF][START_REF] Mazenc | Continuous discrete sequential observers for timevarying systems under sampling and input delays[END_REF].

Based upon the aforementioned researches of CD nonlinear observer synthesis and taking into consideration our intention of diagnostic application, there is no doubt that our interests in this thesis will be focused on the CD model as well, more precisely, the mathematical model of the systems considered in our work will be represented by CD model.

Fault Diagnosis

There are countless literatures and surveys focusing on fault diagnosis [START_REF] Isermann | Process fault detection based on modeling and estimation methods-a survey[END_REF][START_REF] Frank | Fault diagnosis in dynamic systems via state estimation -a survey[END_REF][START_REF] Frank | Fault diagnosis in dynamic systems using analytical and knowledgebased redundancy[END_REF][START_REF] Frank | Survey of robust residual generation and evaluation methods in observer-based fault detection systems[END_REF][START_REF] Isermann | Model-based fault-detection and diagnosis -status and applications[END_REF][START_REF] Blanke | Diagnosis and Fault-Tolerant Control[END_REF][START_REF] Ding | Model-based fault diagnosis techniques[END_REF][START_REF] Wang | Model-based Health Monitoring of Hybrid Systems[END_REF][START_REF] Termeche | Augmented analytical redundancy relations to improve the fault isolation[END_REF]. In this section, we intend to introduce some basic but necessary definitions and notations of fault diagnosis, as well as the general diagnostic steps. However, one may immediately realize by examining the literatures that the notations or terminology of this domain is not coherent. The definitions and notations about fault diagnosis given in this thesis are suggested by the IFAC (International Federation of Automatic Control) SAFEPROCESS 1 Technical Committee [Van Schrick 1997, Isermann and[START_REF] Isermann | Trends in the application of model-based fault detection and diagnosis of technical processes[END_REF]. Without being exhaustive, the generalized fault detection and isolation (FDI) method based on analytical redundancy will also be briefly presented, together with some observer-based diagnosis approaches at the end of this section.

Basic definitions / notations and general diagnostic procedure

According to [START_REF] Isermann | Trends in the application of model-based fault detection and diagnosis of technical processes[END_REF], the engineering terminology fault is defined as "An unauthorized deviation of at least one characteristic property or parameter of the system from the acceptable / usual / standard condition".

As shown in Figure 2.2, one of the most common fault classification is given by the aim of fault tolerant control [START_REF] Blanke | Diagnosis and Fault-Tolerant Control[END_REF]] as

• Actuator faults: faults that affect (interrupt or modify) the system control inputs.

• Sensor faults: such faults introduce substantial errors to sensor readings.

• Plant faults: faults that change the dynamical I/O properties of systems.

Real System Sensors Actuators

Physical Process u(t)

Actuator faults

Sensor faults

Plant faults

Parameter variations; Noise y(t) Note that a fault modifies or changes the performance (for example: operation mode) of the component in an unacceptable way. It is worth knowing that through certain fault-tolerant control, the faulty system may be able to stay operational even with the presence of fault. In the meantime, another notation failure, which is easily confused with fault, is defined in [START_REF] Isermann | Trends in the application of model-based fault detection and diagnosis of technical processes[END_REF] as: A permanent interruption of a system's ability to perform a required function under specified operating conditions. Unlike a fault, a failure interprets the incapability in functional level, which means a system or component is not able to achieve its function and then must be shut off. Apparently, a failure is an irrecoverable event. In order to understand the two notations better, it may be useful to recall that one important role of fault tolerant control is to prevent a fault from causing a failure at the system level.

Let us continue to see what the general diagnostic steps are in order to perform fault diagnosis. As Isermann stated in [START_REF] Isermann | Trends in the application of model-based fault detection and diagnosis of technical processes[END_REF], fault diagnosis consists of determining the type, amplitude, location and occur time of the fault. The diagnostic procedure follows with fault detection, isolation and identification:

• Fault detection: determine whether there occurs a fault or not; if yes, determine the time instant of fault occurrence. This step is very important since early fault detection can possibly prevent the system from a catastrophic failure.

• Fault isolation: determine in which component the fault is. This step aims to locating the fault, which means to find out the possible fault candidate(s) that can explain the undesired behavior of system. Sometimes there is only one unique fault candidate, unfortunately this case is not always possible. Then the objective of this step need to get a possible fault set.

• Fault identification and estimation: identify the fault type and maybe estimate the fault magnitude. This step is also used to see how severe the fault is, then decide whether to perform fault accommodation or to replace the faulty component directly.

Based upon the above general diagnosis procedure, various approaches and researches have been proposed then developed during the last two decades. All these fault diagnosis methods can be broadly categorized into two classes: model-based methods and data driven methods [START_REF] Wang | Model-based Health Monitoring of Hybrid Systems[END_REF]. With no doubt, our contribution lies in the first class, more specifically, observer-based FDI approach using analytical redundancy, which is going to be detailed in next subsection.

Generalized fault diagnosis methods using analytical redundancy

As interpreted in [START_REF] Venkatasubramanian | A review of process fault detection and diagnosis: Part i: Quantitative model-based methods[END_REF], the model-based diagnostic methods can still be classified into two categories: qualitative and quantitative. We currently show no interest to the former category in this thesis. On the contrary, the quantitative one is what we will focus on. Moreover, as most of the quantitative model-based approaches have used, the general input-output state-space model will be investigated in this thesis.

In the domain of automatic control, different FDI methods may use different kinds of dynamical models or might have different assumptions regarding the available measurement information. However, these approaches all follow one common principle: consistency check [START_REF] Blanke | Diagnosis and Fault-Tolerant Control[END_REF]]. The inconsistency between the actual system behavior and the expected nominal behavior might show a fault potential. Therefore, some form of redundancy is required in order to check for consistency. Actually, hardware redundancy and analytical redundancy are two kinds of redundancies that have been usually used in practical. The former one asks for multiple hardware equipments (mostly sensors), which has been applied in mission and safety-critical systems such as digital fly-by-wire flight systems and nuclear reactors [START_REF] Kratz | Detection of sensor failures using instruments and analytic redundancies[END_REF][START_REF] Goupil | AIRBUS state of the art and practices on FDI and FTC in flight control system[END_REF][START_REF] Hao | Sensor fault detection and isolation over wireless sensor network based on hardware redundancy[END_REF]. The analytical redundancy on the other hand utilize a mathematical model of the targeted system in cooperate with certain estimation techniques [START_REF] Hwang | A survey of fault detection, isolation, and reconfiguration methods[END_REF]]. The generalized architecture of FDI methods based on analytical redundancy is shown in Figure 2.3, which is inherited from [START_REF] Isermann | Process fault detection based on modeling and estimation methods-a survey[END_REF] and [START_REF] Frank | Fault diagnosis in dynamic systems using analytical and knowledgebased redundancy[END_REF]. Note that there are two major techniques within the framework of model-based FDI methods: state estimation (parity space approach, observer or filter) and parameter estimation. Eventually, these methods all end up by generating an "indicator", either a residual or the parity. This indicator is exactly the result of the inconsistency check that we have mentioned before.

The path where our contributions will go through is noted in blue in Figure 2.3, the related fault diagnosis procedures is equivalently remarked in green on the right side of Figure 2.3. Observer-based FDI methods will be briefly but accordingly summarized in the next subsection. As for the rest of approaches such as parity space method, parameter estimation and filter, please refer to [START_REF] Chow | Analytical redundancy and the design of robust failure detection systems[END_REF], [START_REF] Isermann | Fault diagnosis of machines via parameter estimation and knowledge processing[END_REF], [START_REF] Kinnaert | Fault diagnosis based on analytical models for linear and nonlinear systems -a tutorial[END_REF], [START_REF] Almasri | Parameter estimation based-fdi method enhancement with mixed particle filter[END_REF] and the references therein.

Real System

Model of Observed System

State Estimation ---------------------------------

Observer-based fault diagnosis methods

As model-based FDI methods received considerable attention, observer-based approach is the most extensively used one among model-based fault diagnosis for decades [START_REF] Chen | Design of unknown input observers and robust fault detection filters[END_REF][START_REF] Hammouri | Observer-based approach to fault detection and isolation for nonlinear systems[END_REF][START_REF] Shields | Observer-based residual generation for fault diagnosis for non-affine non-linear polynomial systems[END_REF][START_REF] Zhang | Observer-Based Fault Estimation and Accomodation for Dynamic Systems[END_REF]]. The consistency check indicator in this case is what we usually called residual, which represents the difference between estimate measurements and the real ones collected through the output of real system. Observer-based FDI method has been widely used in many fields such as PEM fuel cell [START_REF] Bougatef | Fault detection of a pemfc system based on delayed lpv observer[END_REF]], heat-exchanger/reactor system [START_REF] Han | Nonlinear observer based fault diagnosis for an innovative intensified heat-exchanger/reactor[END_REF], continuous stirred tank reactor (CSTR) [START_REF] Li | Diagnostic observer design for t-s fuzzy systems: Application to real-time-weighted fault-detection approach[END_REF], induction motor [START_REF] Toumi | Observer-based fault diagnosis and field oriented fault tolerant control of induction motor with stator inter-turn fault[END_REF][START_REF] El Merraoui | Real time observer-based stator fault diagnosis for im[END_REF] etc. Among all these observer-based applications, there includes unknown-input observer (UIO)-based fault diagnosis, adaptive observer (AO)-based fault diagnosis, sliding-mode observer (SMO)-based fault diagnosis etc [Zhang et al. 2016a].

UIO-based fault diagnosis methods play a significant role since the unknown input (UI) can represent modeling uncertainties, perturbations, faults, etc [START_REF] Marx | Unknown input observer for lpv systems[END_REF], which a physic system may commonly include. Over past decades, UIOs design for linear systems has been developed and applied to fault diagnosis by a lot of researchers or scientific group [START_REF] Hou | Disturbance decoupled observer design: a unified viewpoint[END_REF][START_REF] Johnson | Dual-mode robust fault estimation for switched linear systems with state jumps[END_REF][START_REF] Zhang | Robust unknown input observer-based fault estimation of leader-follower linear multi-agent systems[END_REF][START_REF] Tan | Extended results on robust state estimation and fault detection[END_REF]. As for nonlinear UIO-based fault diagnosis, Yang et al. presented an approach for actuator FDI by a designed nonlinear UIO with linearized error dynamics [START_REF] Yang | Monitoring and diagnostics of a class of nonlinear systems using a nonlinear unknown input observer[END_REF]. A new development based on decomposition techniques are introduced in [START_REF] Koshkouei | Nonlinear unknown input observer design for nonlinear systems: A new method[END_REF], which is constrained by certain conditions. L. Meyer et al. propose in Meyer et al. [2018] an interval UIO by using the derivative of the output vector under some existence assumptions, which also intends to separate the UI from interval error bounds. Meanwhile, Zhang et al. [START_REF] Zhang | Unknown input observer design for one-sided lipschitz nonlinear systems[END_REF]] have managed to designed a full-order and reduced-order UIO for oneside Lipschitz system by the Lyapunov-based linear matrix inequality (LMI) method, which the asymptotic convergence is guaranteed. In [START_REF] Koenig | Observer design for unknown input nonlinear descriptor systems via convex optimization[END_REF], an algebraic method is used to design an observers for nonlinear descriptor systems in presence of UI. As summarized in [START_REF] Alenezi | Adaptive unknown input and state observers[END_REF], the existing UIO design methods can be divided into two categories: (1) perform state estimation despite the existence of UI (structural decoupling techniques); (2) estimate both state and UI simultaneously via an augmented observer with some assumptions on the dynamics of UI. In this thesis, our contribution will take a place in the latter category in order to perform actuator fault estimation.

As examined in [Heffes 1966, Toda and[START_REF] Toda | Performance bounds for continuous-time filters in the presence of modeling errors[END_REF], observer-based (all model-based methods) diagnosis method will be affected to divergence due to the accumulation of modeling uncertainties. Furthermore, state estimation based on infinite memory (i.e. all the process history) may result in the insensitivity to recent measurements which might have the clues of a fault in incipient stage [START_REF] Graton | Finite memory observers for linear timevarying systems: Theory and diagnosis applications[END_REF]]. Thus, the corresponding researches like fading filter [START_REF] Sorenson | Recursive fading memory filtering[END_REF] and finite memory observer are naturally explored. Finite memory observer (FMO) was first proposed by Medvedev [START_REF] Medvedev | A continuous finite-memory deadbeat observer[END_REF] for linear system in the deterministic framework, which indicates that this observer is extremely efficient for state estimation. Afterwards, the robustness and sensitivity of this approach were addressed by Nuninger [Nuninger et al. 1998] and Graton [START_REF] Graton | Finite memory observers for linear timevarying systems: Theory and diagnosis applications[END_REF]. Researchers in Kratz's team have then continued to synthesize this observer in fault diagnosis of linear system [START_REF] Thuillier | Finite memory observer -based sensor fault detection and isolation for system when measurements are correlated with process noise[END_REF]] and hybrid system [START_REF] Kajdan | Fault detection of a nonlinear switching system using finite memory observers[END_REF]]. All these previous researches of finite memory observer for linear systems reveal that this specific moving horizon technique provides a great potential in state estimation as well as in fault diagnosis. Well, this is how this moving horizon technique has successfully attracted our attention and then became the foundation of our contribution in this thesis.

Conclusion

In this chapter, after briefly recalling the observation problems of nonlinear systems followed by different observability definitions and observability rank condition, we then passed to present some typical optimization-based observer synthesis for continuous-time nonlinear systems using moving horizon techniques, such as Newton's method based and least square method based. However, it should be noted that this bibliography review is not exhaustive but accordingly introduced. Meanwhile, the researches on nonlinear observer design with continuous-discrete model are being reviewed in order to see which type of model is commonly used for the application-oriented observer synthesis. At the end part of this chapter, the basic but necessary definitions and procedures of fault diagnosis are established together with a brief review of the observer-based fault diagnosis methods.

Through the studies of different types (algorithm-wise or application-wise) of nonlinear observers, we realize that among these existing methods, "half" of them have a theoretical significance but are really hard to apply in practical while the other "half" are ale to be well applicable but show no theoretical significance. Indeed, the nonlinear observer design still has a long way to go and luckily we are going to begin our first step on this road in next chapter. It may be a small step, at least we have already begun.

Chapter 3 Nonlinear Continuous-Discrete Finite Memory Observer (CD-FMO) Design

Introduction

Résumé en français :

Nous venons de voir au chapitre précédent que l'observateur joue un rôle important dans la réalisation d'un diagnostic performant. L'évolution technologique et numérique permet des communications et des interactions de plus en plus nombreuses entre les différents composants constituant les processus industriels, ou systèmes industriels, qui deviennent ainsi de plus en plus complexes. Il est aujourd'hui très important de pouvoir assurer un fonctionnement efficace à ces processus, c'est-à-dire assurer le diagnostic de défaut. Ce besoin de diagnostic sûr est devenu un problème important pour les processus industriels ainsi que un sujet de recherche à la mode. Inspiré d'approches existantes, nous allons concevoir dans ce chapitre un observateur non linéaire pour une classe de systèmes non linéaires modélisés par un modèle continu-discret (CD) en présence de bruit de processus et de bruit de mesures. L'observateur développé sera construit sur la méthodologie réalisée pour le système linéaire ayant abouti à la conception de l'observateur à mémoire finie (FMO) [Nuninger et al. 1998]. Nous démontrerons les propriétés de l'erreur d'estimation dans le cas déterministe alors que les mêmes études pour le cas stochastique seront réalisées par simulations de Monte Carlo (MC). La stratégie du choix de la longueur de la fenêtre d'observateur sera également donnée en fonction de la variance de l'erreur d'estimation. Concernant l'application, les résidus classiques sont générés pour implémenter la détection des défauts par l'algorithme de la somme cumulée (CUSUM), qui permet de détecter efficacement de petites dérives sur la moyenne. Finalement, un banc d'observateur est utilisé pour réaliser la localisation des défauts capteurs et/ou défauts actionneurs pour un bras robotique. Les résultats des simulations montreront que la méthode proposée fournit une détection de défaut assez efficace. La robustesse de cet observateur non linéaire vis-à-vis des bruit de mesure est également étudiée dans ce chapitre.

Introduction

We can see from the previous chapter that state observer plays an important role in fault diagnosis over the past decades. As the real engineering systems turn into more and more complex, not only the corresponding demand of observer design is growing but also the fault diagnosis for complex nonlinear systems have gained increasing consideration. Inspired by these existing approaches, we aim in this chapter to develop a continuous-discrete finite memory observer (CD-FMO) for a class of nonlinear dynamical systems modeled by ordinary differential equations (ODEs) with discrete measurements. The nonlinear systems under consideration are at least locally Lipschitz, which guarantees the existence and uniqueness of solution at each time instant. The proposed nonlinear CD-FMO uses a finite number of Chapter 3 Nonlinear Continuous-Discrete Finite Memory Observer (CD-FMO) Design collected measurements to estimate the system state in the presence of measurement noise. Besides, a one-step prediction algorithm incorporated with an iterative-update scheme is performed to solve the integral problem caused by system nonlinearity, and an analysis of the numerical integration approximation error is given. The properties of estimation performance have been further proved in deterministic case and been analyzed by Monte Carlo (MC) simulation in stochastic cases. It's worth noting that the presented method has a finite-time convergence while most nonlinear observers are usually asymptotically convergent. Another advantage is that the proposed CD-FMO has no initial value problem. For the application purpose, residuals are generated to implement fault detection cooperated with Cumulative Sum (CUSUM) algorithm, while a bank of CD-FMOs is adopted to realize fault isolation for different sensor and actuator faults of the considered nonlinear robotic arm. The robustness and effectiveness of the proposed approach are illustrated via the simulation results.

Nonlinear CD-FMO Design

At first, a description of the nonlinear system is made, then the design of the proposed nonlinear CD-FMO together with an iterative update algorithm for numerically approximating integration due to the nonlinearity of system. In addition, we also give a detailed proof of the finite-time convergence. The criteria of how to choose the window length is also stated in the end of this section.

Problem statement

We consider a class of continuous-discrete nonlinear systems described by the following state-space equations:

ẋ(t) = Ax(t) + Bu(t) + Φ(x) + Gω(t) (3.1a) y(k) = Cx(k) + v(k) (3.1b)
where x ∈ R n , y ∈ R p and u ∈ R q are continuous state vector, discrete measurement vector and continuous input vector, respectively. T s is the sampling period of measurement (i.e.

∃k ∈ N | t = k × T s ). A ∈ R n×n , B ∈ R n×q , G ∈ R n×n and C ∈ R p×n are known matrices. The nonlinearity Φ(x) is a nonlinear function with respect to state x. Φ(x) is at least locally Lipschitz, i.e. ∥Φ(a) -Φ(b)∥ ≤ κ∥a -b∥ (3.2)
where Lipschitz constant κ > 0. Vectors v and ω represent Gaussian measurement noise and Gaussian process noise, respectively, and v and ω are independent with the following properties: Remark 3.1. The authors in [START_REF] Kou | Finite-time observer for nonlinear dynamic systems[END_REF]] have proven that the observability of a nonlinear dynamical system is a necessary condition that there exists a finite-time observer for the system.

E[ω(t)] = 0 (3.3a) E[ω(t 1 )ω T (t 2 )] = Q • δ (t 1 -t 2 ) (3.3b) E[v(k)] = 0 (3.3c) E[v(k 1 )v T (k 2 )] = R • δ k 1 ,k 2 (3.
We are able to conclude from this remark that if we can build a finite-time observer for a nonlinear system, then this nonlinear system is observable.

Formulation of CD-FMO

Suppose that at each frozen time instant t, the discrete measurements are collected in the interval [tτ i ,t], where τ i = i × T s , and i = 0, 1, ..., L -1. Here L is called window length.

Use the square matrix exponential e -At as a factor and integrating (3.1a), we can give the relation between the states in two different time instant t and tτ i as:

x(t) = e Aτ i x(t -τ i ) + t t-τ i e A(t-θ ) Bu(θ ) dθ + t t-τ i e A(t-θ ) Φ(x(θ )) dθ + t t-τ i e A(t-θ ) Gω(θ ) dθ (3.4)
Chapter 3 Nonlinear Continuous-Discrete Finite Memory Observer (CD-FMO) Design Then, pre-multiply (3.4) by the matrix Ce -Aτ i and taking into account the measurement equation (3.1b) at time instant tτ i , we obtain:

Ce -Aτ i x(t) = y(t -τ i ) -v(t -τ i ) + α t-τ i ,t + β t-τ i ,t + γ t-τ i ,t (3.5) with α t-τ i ,t = t t-τ i Ce A(t-τ i -θ ) Bu(θ ) dθ (3.6a) β t-τ i ,t = t t-τ i Ce A(t-τ i -θ ) Φ(x(θ )) dθ (3.6b) γ t-τ i ,t = t t-τ i Ce A(t-τ i -θ ) Gω(θ ) dθ (3.6c)
Applying (3.5) with (3.6) for each measurement in the time window [tτ L-1 ,t], then a finite number of augmented measurements can be expressed in terms of system state x(t) as following linear equation:

Y L -V L = W L x(t) (3.7)
where

Y L =       y(t -τ 0 ) + α t-τ 0 ,t + β t-τ 0 ,t y(t -τ 1 ) + α t-τ 1 ,t + β t-τ 1 ,t . . . y(t -τ L-1 ) + α t-τ L-1 ,t + β t-τ L-1 ,t       ; W L =       Ce -Aτ 0 Ce -Aτ 1 . . . Ce -Aτ L-1       ; V L =       v(t -τ 0 ) -γ t-τ 0 ,t v(t -τ 1 ) -γ t-τ 1 ,t . . . v(t -τ L-1 ) -γ t-τ L-1 ,t       .
It is straightforward that the noise component V L has zero mean, i.e. E (V L ) = 0. Then the variance matrix P (see Appendix A for detailed presentation) is block symmetric as

P = E (V L -E(V L )) (V L -E(V L )) T = E V L V T L =       S 0 S 0 • • • S 0 S 0 S 1 • • • S 1 . . . . . . . . . . . . S 0 S 1 • • • S L-1       +       R 0 • • • 0 0 R . . . . . . . . . . . . . . . 0 0 • • • 0 R       (3.8)
where the block elements S k (k min[i, j] = 0, 1, ..., L -1) represent the following integral calculation [START_REF] Medvedev | Parity space method: a continuous time approach[END_REF]]:

S k = 0 -τ k Ce As GQG T e A T s C T ds
Now, the state estimation x(t) at time instant t can be obtained by minimizing the following cost function for (3.7):

J(x) = 1 2 ∥Y L -W L x(t)∥ 2 P -1
(3.9)

with the solution in the sense of least-squares as:

x(t) = arg min J(x) = W T L P -1 W L -1 W T L P -1 ŶL = Ω -1 L W T L P -1 ŶL Ω L W T L P -1 W L (3.10) where ŶL =       y(t -τ 0 ) + α t-τ 0 ,t + βt-τ 0 ,t y(t -τ 1 ) + α t-τ 1 ,t + βt-τ 1 ,t . . . y(t -τ L-1 ) + α t-τ L-1 ,t + βt-τ L-1 ,t       (3.11a) βt-τ i ,t = t t-τ i Ce A(t-τ i -θ ) Φ( x(θ )) dθ (3.11b)
It can be seen that the existence condition of x(t) in (3.10) is given by the existence of matrix Ω -1 L . This condition is then given by the rank of matrix W L , i.e., rank(W L ) = n = dim(x), which is guaranteed by the following assumption:

Assumption 3.1. The pair (A,C) is observable.

According to (3.10)-(3.11), we obtain the analytical form of state estimation x(t) for considered nonlinear systems (3.1). The calculation of two integral terms α t-τ i ,t and βt-τ i ,t in (3.11a) are then detailed afterward.

Analytical calculation of the integral α t-τ i ,t

It is obvious to see from (3.6a) that all the elements contained inside the integral are known and it is easy to have an analytical solution by some useful softwares with symbolic computation such as Maple, Mathematica, etc. If the mathematical expression of input u(t) at each instant is unknown, we can still get the solution by putting the element u(t) as a factor of integral under the assumption that u(t) is sampled as zero-order hold and thus remains to be constant between two consecutive sampling instants, which is usually true since most controllers of actual systems are digital computers in practice.

3.2.2.2 Iterative algorithm for solving the integral βt-τ i ,t

In order to compute βt-τ i ,t , we might note that it is impossible to have an exact analytical solution. Since we can see from (3.11b) that there is the term "Φ( x(θ ))" in the integral. In order to analytically calculate βt-τ i ,t , we must know the exact trajectory of " x(θ )" between instant tτ i and t, which unfortunately is what we seek to know (via the estimation x(t) in (3.10)). Hence, a one-step prediction together with iterative-update algorithm is designed to obtain the approximate solution of βt-τ i ,t by Newton-Cotes formulas [START_REF] Atkinson | An introduction to numerical analysis[END_REF]].

In each time window [tτ L-1 ,t], we define the measurement set

Z L = {y(t -τ i ); i = 0, 1, • • • , L -1} and estimation set X L = { x(t -τ j ); j = 1, • • • , L -1}.
It should be noted here that there is no case j = 0 since all the elements in XL are obtained by previous window and x(tτ 0 ) = x(t) is exactly what we aim to estimate by current window. Therefore, a one-step prediction of state x at instant t, noted as x * (t), has been performed by using the tangent slope ẋ(t -∆t) with a small time interval ∆t = T s as:

x * (t) = x(t -∆t) + ẋ(t -∆t)∆t = x(t -T s ) + ẋ(t -T s )T s = x(t -τ 1 ) + [A x(t -τ 1 ) + Bu(t -τ 1 ) + Φ( x(t -τ 1 ))] T s (3.12)
x * (t) is then iteratively updated by (3.10) -(3.11), which makes the final estimation x(t) after all iterations. The iterations are stopped when there is no significant change between two consecutive iterations or the maximum iteration numbers N max has been reached, namely

x(t) -x * (t) ≤ ε or m ≥ N max (with acceptable error)
Furthermore, the first-order Newton-Cotes formulas, which yields trapezoidal rule, is employed in this paper to numerically approximate the integral term βt-τ i ,t in (3.11a). For the purpose of reducing the massive computing burden in each iteration, we notice from (3.11b) that βt-τ i ,t can be divided as follows by the Segment Addition Postulate [START_REF] Nomizu | Affine differential geometry[END_REF] for integral calculus:

βt-τ i ,t = βt-τ i ,t-τ 1 + βt-τ 1 ,t (3.13) 3.2 Nonlinear CD-FMO Design with βt-τ i ,t-τ 1 = t-τ 1 t-τ i Ce A(t-τ i -θ ) Φ( x(θ )) dθ (3.14a) βt-τ 1 ,t = t t-τ 1 Ce A(t-τ 1 -θ ) Φ( x(θ )) dθ (3.14b) let g( x(θ )) = Ce A(t-τ i -θ ) Φ( x(θ )) (3.15)
we know that the previous estimation set

{ x(t -τ j )}( j = 1, • • • , L -1
) is unchanged during each iteration of updating x * (t), which leads to g( x(tτ j )) by (3.15) unchanged. As a consequence, βt-τ i ,t-τ 1 by (3.14a) also remains the same at each iteration. Therefore, as it is shown in Figure 3.1, we only need to recalculate the term βt-τ 1 ,t in (3.13) at each iteration. In this way, the unnecessary calculation burden caused by iteration can be dramatically reduced when using Newton-Cotes formulas to calculate the numerical integration.

 

1 , i t t       1 , t t             î A t g x Ce x         g     3 ĝ x t       2 ˆL g x t        2 ĝ x t       * ĝ x t t 1 t   2 t   3 t   2 L t    1 L t    0    1 , ˆi t t      1 , ˆt t        1 ĝ x t       1 ˆL g x t    Time window: L Time / s Figure 3.1: Calculation framework of βt-τ i ,t in each interval [t -τ i ,t]
For the sake of overall understanding, the summarized algorithm of the proposed nonlinear observer CD-FMO is shown in Algorithm 1.

Estimation property of CD-FMO

Theorem 3.1. If nonlinear system (3.1) satisfies Assumption 3.1, in the case of noise-free and fault-free, the property of estimation by presented CD-FMO are unbiased

x(t) = x(t) t ∈ [L × T s , +∞)
Algorithm 1 The implementation of nonlinear CD-FMO algorithm 1: Initialization: at each time instant t, given the iteration threshold ε, the maximum number of iteration N max , the measurement set

Z L = {y(t -τ i )} with i = 0, 1, • • • , L -1 and the previous estimation set X L = { x(t -τ j )} with j = 1, • • • , L -1. 2: Calculate g( x(t -τ j ))
for each member in X L by (3.15). 3: Compute integral βt-τ j ,t-τ 1 in (3.14a) by Newton Cotes. 4: One-step prediction: using x(tτ 1 ) to perform a one-step prediction of x(t) via (3.12), noted as x * (t). Proof. In the case noise-free and fault-free, according to (3.1)-(3.7), the proposed CD-FMO (3.10) can be rewritten for the deterministic case as following:

x(t) = W T L W L -1 W T L ŶL = Λ -1 L L-1 ∑ i=0 e -A T τ i C T y(t -τ i ) + α t-τ i ,t + βt-τ i ,t (3.16)
with

Λ L = W T L W L = L-1 ∑ i=0 e -A T τ i C T Ce -Aτ i (3.17)
then, as stated in (3.5) and regardless of noise term v(tτ i ), y(tτ i ) can be given as:

y(t -τ i ) = Ce -Aτ i x(t) -α t-τ i ,t -β t-τ i ,t (3.18)
3.2 Nonlinear CD-FMO Design by replacing the term y(tτ i ) in (3.16) by (3.18) and taking into account (3.17), we have

x(t) = Λ -1 L L-1 ∑ i=0 e -A T τ i C T Ce -Aτ i x(t) -β t-τ i ,t + βt-τ i ,t = Λ -1 L L-1 ∑ i=0 e -A T τ i C T Ce -Aτ i x(t) + Λ -1 L L-1 ∑ i=0 e -A T τ i C T βt-τ i ,t -β t-τ i ,t = x(t) + Λ -1 L L-1 ∑ i=0 e -A T τ i C T βt-τ i ,t -β t-τ i ,t (3.19)
In order to prove Theorem 1, we know that the following equivalence can be obtained directly:

x(t) = x(t) ⇔ ∥ x(t) -x(t)∥ = 0 t ∈ [L × T s , +∞) (3.20) from (3.19
), the norm of x(t)x(t) can be therefore expressed as follows:

∥ x(t) -x(t)∥ = Λ -1 L L-1 ∑ i=0 e -A T τ i C T βt-τ i ,t -β t-τ i ,t ≤ L-1 ∑ i=0 Λ -1 L e -A T τ i C T βt-τ i ,t -β t-τ i ,t (3.21)
then, based upon (3.6b) and (3.11b), by using Lipschitz condition (3.2), the matrix norm properties and the triangle inequality for integrals [START_REF] Rudin | Real and complex analysis[END_REF]], (3.21) can be further derived as:

∥ x(t) -x(t)∥ ≤ L-1 ∑ i=0 Λ -1 L e -A T τ i C T t t-τ i Ce A(t-τ i -θ ) [Φ( x(θ )) -Φ(x(θ ))] dθ ≤ L-1 ∑ i=0 t t-τ i Λ -1 L e -A T τ i C T Ce A(t-τ i -θ ) [Φ( x(θ )) -Φ(x(θ ))] dθ ≤ L t t-τ L-1 Λ -1 L e -A T τ L-1 C T Ce A(t-τ L-1 -θ ) [Φ( x(θ )) -Φ(x(θ ))] dθ ≤ L t t-τ L-1 Λ -1 L e -A T τ L-1 C T Ce A(t-τ L-1 -θ ) ∥Φ( x(θ )) -Φ(x(θ ))∥ dθ ≤ L t t-τ L-1 Λ -1 L e -A T τ L-1 C T Ce A(t-τ L-1 -θ ) κ ∥ x(θ ) -x(θ )∥ dθ ≤ L t L×T s Λ -1 L e -A T τ L-1 C T Ce A(t-τ L-1 -θ ) κ ∥ x(θ ) -x(θ )∥ dθ ≤ 0 + L t L×T s Λ -1 L e -A T τ L-1 C T Ce A(t-τ L-1 -θ ) κ ∥ x(θ ) -x(θ )∥ dθ (3.22)
Chapter 3 Nonlinear Continuous-Discrete Finite Memory Observer (CD-FMO) Design then, the Gronwall inequality [START_REF] Bellman | The stability of solutions of linear differential equations[END_REF]] yields

∥ x(t) -x(t)∥ ≤ 0 • e t L×T s L Λ -1 L e -A T τ L-1 C T Ce A(t-τ L-1 -θ ) κ dθ ≤ 0 (3.23) hence, ∥ x(t) -x(t)∥ ≤ 0 ⇒ ∥ x(t) -x(t)∥ = 0 ⇒ x(t) = x(t)
The proof is completed.

Remark 3.2. Theorem 3.1 states that x(t) = x(t) is always true when t ≥ L × T s , that is

(1) The proposed CD-FMO is a dead-beat observer in the case of noise-free and fault-free, the finite-time convergence is L × T s (one window-size).

(2) There is no estimation when t < L × T s . In other words, there is no initial value problem (IVP) for the presented nonlinear observer, which gives us another advantage for application in physics or other sciences.

Remark 3.3. It is clear that Theorem 3.1 shows the unbiased estimation property of the proposed CD-FMO in deterministic case. In stochastic case, namely in the presence of the process noise and measurement noise, the unbiased property

E ( x(t) -x(t)) = 0 or E ( x(t)) = x(t) (3.24)
can also be demonstrated in the similar manner, see Appendix B for more details.

Analytical choice of the window length L

As it is shown in (3.10) and (3.11), at each time instant t, the state estimation x(t) is related to the window length L. Thus, it is necessary to interpret how to select an appropriate window length L. Here we are going to explain this by defining the "minimal length L min " and "maximal length L max ", as it has been shown in [START_REF] Graton | Finite memory observers for linear timevarying systems: Theory and diagnosis applications[END_REF]].

Minimal length L min

The minimal window length L min is chosen to assure the existence of the proposed CD-FMO by (3.10). As we have already discussed before, this condition is then given by the rank of matrix W L , i.e., rank(W L ) = n = dim(x), which is already guaranteed by Assumption 3.1. However, L min is just used to valid the Assumption 3.1, it is definitely not the optimal window length, as showed in the latter subsection.

Illustrative Example: a Single-link Robot

Maximal length L max

Here it should be noticed that, theoretically speaking, there is no maximum window length L max for CD-FMO. The greater the length L, the better the estimation x(t), which is reasonable since the amount of measurement information augments as the window length increases. However, after a certain size, the contribution of additional information by increasing window length is not significant enough to decrease estimation error. Therefore in this chapter, we take "the maximum eigenvalue of covariance matrix Σ x" of estimation error x = xx as an indicator to select maximum window length L max . Given a selected threshold of estimation error tolerance ϒ Tol , L max is defined as:

L max = arg min L {max(eig(Σ x)) ≤ ϒ Tol } (3.25)
which is the smallest window length when the largest eigenvalue of Σ x is smaller than error tolerance threshold ϒ Tol . This part will be further analyzed in next section with an illustrative example.

Illustrative Example: a Single-link Robot

In this section, we consider a nonlinear single-link robotic arm, which has an elastic joint rotating in a vertical plane [START_REF] Zhang | Design and analysis of a fault isolation scheme for a class of uncertain nonlinear systems[END_REF]]. The nonlinear state-space model is described here as:

ẋ(t) = Ax(t) + Bu(t) + Φ(x) + Gω(t) y(k) = Cx(k) + v(k) with x = [x 1 x 2 x 3 x 4 ] T .
Here, components x 1 and x 3 are the displacement of link and rotor respectively while components x 2 and x 4 represent the velocity. The measurement noise v ∼ N (0, R) and the process noise ω(t) = 0. The initial conditions for the robotic arm system is x(0) = (1 1 1 1) T . The input control u(t) = 2 sin(2t), which is the torque provided by the motor. All the other related matrices are given as:

A =       0 1 0 0 -k J l -f l J l k J l 0 0 0 0 1 k J m 0 -k J m -f m J m       ; B =       0 0 0 1 J m       ; Φ(x) =       0 -mgl J l sin x 1 0 0       ;
Chapter 3 Nonlinear Continuous-Discrete Finite Memory Observer (CD-FMO) Design

C =    1 0 0 0 0 0 1 0 0 0 0 1    ; R =    4 × 10 -4 0 0 0 4 × 10 -4 0 0 0 9 × 10 -4    .
the simulation scenario is performed according to the parameters shown in Table 3.1.

Table 3.1: Physical parameters (in SI units)

elastic constant k = 2 link mass m = 4
viscous friction coefficient for motor f m = 1 motor inertia J m = 1 viscous friction coefficient for link f l = 0.5 link inertia J l = 2 mass center l = 0.5 link inertia g c = 9.8

Selection of the window length L

As shown in (3.25), we take "max(eig(Σ x))" as an indicator to select L max . It can be seen from Figure 3.2 that the maximum eigenvalue of Σ x is asymptotically convergent as window length L increases, which indicates that the estimation performance provided by the presented CD-FMO well improves while the window length augments. After window length L 1 = 13, the decrease of the curve is much less significant, which is normal since there is few additional information can be provided by increasing the window length. This is also why the proposed observer is called "finite memory". Nevertheless, starting from L 2 = 20, the curve shows a slight trend of going up, which is a normal phenomenon because the approximation error of Newton-Cotes formulas (used in (3.11a) for integral term βt-τ i ,t ) will also get bigger as L increases. In order to get a better diagnosis performance, we choose L = 15 for all the analysis and diagnosis later in this chapter, which is well between L 1 and L 2 .

Numerical integration approximation error analysis

In order to perform state estimate via (3.10), we choose trapezoidal rule to approximate the integral term βt-τ i ,t in (3.11a), so it is necessary to give the approximation error bound. We recall the following lemma: Lemma 3.1. Given a definite integral I = b a f (x) dx, the approximation error of trapezoidal rule is [START_REF] Atkinson | An introduction to numerical analysis[END_REF]]

O(I) = - (b -a) 3 12 f ′′ (η), η ∈ [a, b]
We can have the following expression of x(t) by rewriting (3.10),

x(t) = W T L P -1 W L -1 W T L P -1 ŶL = Λ -1 L L-1 ∑ i=0 e -A T τ i C T R -1 y(t -τ i ) + α t-τ i ,t + βt-τ i ,t
together with (3.11b) and (3.15), we extract the integral term related to βt-τ i ,t and noted as:

B t-τ i ,t = Λ -1 L L-1 ∑ i=0 e -A T τ i C T R -1 t t-τ i g( x(θ )) dθ
we can see from B t-τ i ,t that the calculation of approximation error by using trapezoidal rule can be divided into two steps:

1. Calculate the upper approximation error bound |O(Int)| with Int = t t-T s g( x(θ )) dθ .

2. Calculate cumulative error bound Γ nc as i varies in the summation ∑ L-1 i=0 .

For Step 1, the bound of approximation error O(Int) can be given by Lemme 3.1 as:

|O(Int)| ≤ T 3 s 12 max θ ∈[t-T s ,t] g ′′ ( x(θ ))
from the expression (3.15), we can get the first and second derivatives of g as following:

g ′ ( x(θ )) = dg( x(θ )) dθ = -Ce A(t-τ i -θ ) AΦ( x(θ )) +Ce A(t-τ i -θ ) dΦ( x(θ )) d x d x(θ ) dθ g ′′ ( x(θ )) = d 2 g( x(θ )) dθ 2 = dg ′ ( x(θ )) dθ = Ce A(t-τ i -θ ) A 2 Φ( x(θ )) -2Ce A(t-τ i -θ ) A dΦ( x(θ )) d x d x(θ ) dθ +Ce A(t-τ i -θ ) d dθ dΦ( x(θ )) d x d x(θ ) dθ
By calculating the norm of g ′′ together with the parameters defined at the beginning of this section, we have

|O(Int)| ≤ 2.5228 × 10 -6 ζ t-T s ,t
For Step 2, we can directly get the cumulative error bound Γ nc as i varies in the summation ∑ L-1 i=0 as follows:

Γ nc ≤ 0 + ζ t-T s ,t + 2ζ t-T s ,t + • • • + (L -1)ζ t-T s ,t = L(L -1) 2 ζ t-T s ,t = 2.6489 × 10 -4 (L = 15)
meanwhile, the maximum element of standard deviation (SD) σ of measurement noise, noted as σ max , is given by:

σ σ T = R ⇒ σ = R 1/2 =    2 × 10 -2 0 0 0 2 × 10 -2 0 0 0 3 × 10 -2    ⇒ σ max = 3 × 10 -2
It is obvious that Γ nc ≪ σ max , which means that the approximation error for numerical integration is drowned in measurement noise. As a result, we can conclude that our estimation is correct with window length L = 15.

State estimation performance

It can be clearly seen from Figure 3.3 that four-dimensional system state x is reconstructed correctly under the presence of measurement noise, the proposed CD-FMO provides great performance of state estimation. Besides, Figure 3.3 also depicts how the accuracy of state estimation getting much better as window length L getting longer, which is another consistent result with respect to Figure 3.2.

Unbiased estimation property analysis in stochastic case

The unbiased estimation property of presented CD-FMO in stochastic case is evaluated by root-mean-square error (RMSE) criteria together with monte carlo (MC) simulation, where 3.3 Illustrative Example: a Single-link Robot RMSE is defined as:

RMSE = 1 N mc N mc ∑ i=1 ( x(i) -x) 2
here, N mc represents MC simulation times. The state estimation by running multiple MC simulation is therefore defined in the average sense:

E ( x(t)) xmean = 1 N mc N mc ∑ i=1 x(i)
Let N mc take the values 100 and 500 respectively. By taking the component x 1 as an example, it can be seen from Figure 3.4 that during the MC simulations, the estimation upper and lower bounds of x is quiet small, which means that the state estimation by proposed CD-FMO varies within a small range around real state x in the presence of measurement noise. Moreover, the estimation xmean obtained with N mc = 500 is closer to true value than the one by N mc = 100, which is logical since MC simulation performed a series of repeated random sampling of Gaussian measurement noise, the larger the sampling size, the closer the mean value of noise is to zero.

The unbiased estimation property has also been examined by the RMSE with different N mc in Figure 3.5. We can see that the RMSEs are close to zero, meanwhile the RMSE of To summarize what has been mentioned above, we have established by monte carlo simulation that state estimation given by the presented nonlinear observer CD-FMO in the stochastic case is also unbiased, i.e. E ( x(t)x(t)) = 0. This property provides a good precondition for the fault diagnosis after-step.

Illustrative Example: a Single-link Robot

Robustness analysis with respect to measurement noise

We are going to analyze the robustness of CD-FMO against measurement noise through three scenarios shown in Table 3.2. Measurement noise varies from σ to 0.5σ and 1.5σ (±50 %) respectively while the parameter setting of observer doesn't change, which means that the proposed observer (3.10) has an inconsistency between y(•) in (3.11a) and noise parameter P.

= 0.5σ SD = σ R = σ 2 P = diag(R , R , • • • , R) Scenario 2 SD = σ Scenario 3 SD = 1.5σ
By taking x 1 as an example, it can be seen from Figure 3.6(a) that the state estimations x1 can still well follow the trajectory of true state x 1 even if the measurement noise has ±50 % variations, which shows the robustness of CD-FMO with respect to measurement noise. In addition, we can see from the RMSEs in Figure 3.6(b) that state estimation of Scenario 1 is better than Scenario 2. It is logical because of the following reason: we have chosen L = 15 for the considered robotic arm system. In fact, CD-FMO with L = 15 in Scenario 2 has already performed a little bit role of "filter" for this nonlinear system. As shown in Figure 3.2, when L = 15, max(eig(Σ x)) = 0.0157, while the minimum noise level in this case (minimum non-zero value of σ ) is 0.02, i.e. σ min = 0.02. The fact of max(eig(Σ x)) ≤ σ min means that the largest dispersion of estimation is still smaller than the minimum noise level, which is the performance of a filter. Accordingly, when we use the same window length for an even more lower noise level, i.e. Scenario 1, the presented CD-FMO will still perform as a filter, maybe filter even more. That is why we get a better estimation even when CD-FMO "over-estimate" the real noise level.

Application to Fault Diagnosis

In this section, we are going to apply the proposed CD-FMO to perform the fault diagnosis of the considered nonlinear single-link robotic arm system. In order to deal with all faults in the same simulation launch, we suppose that each fault only occurs during certain period [T fs , T fe ], therefore we use the following function to characterize the fault duration:

Π(t, T fs , T fe ) = H(t -T fs ) -H(t -T fe )
where H(•) is Heaviside step function. In this paper, we injected two kinds of typical faults:

1. Sensor bias: a sudden bias is one of the abrupt sensor faults, which is modeled as:

y f = y + Π(t, T fs , T fe )I∆
• A bias on y 1 (F1):

I = I y 1 = [1 0 0] T ; ∆ = ∆ y 1 =

Fault detection

In this section, residuals are chosen as fault indicators and it is defined as:

r(t) = y(t) -C x(t)
with t = k × T s , which checks the consistency of real measurements of system and measurements estimated by the proposed CD-FMO. We use both residual r and the Cumulative Sum (CUSUM) of r for the reason that CUSUM algorithm is well-known as the efficiency of detecting small change in the mean of a sequence. As introduced in [START_REF] Montgomery | Introduction to statistical quality control[END_REF]] in the case of control chart case, the upper CUSUM G + (k) and lower CUSUM G -(k) of residuals sequences r(k) (with mean µ r and SD σ r ) are defined as:

   G + (k) = max {0, G + (k -1) + r(k) -µ r -K} G -(k) = min {0, G -(k -1) + r(k) -µ r + K} with the starting value G + (0) = G -(0) = 0.
The detection criterion is:

G + (k) > H or G -(k) < -H
In order to quickly detect the small shift in mean, the parameters of CUSUM algorithm is set as: K = 1 2 σ r and H = 3σ r . In the presence of measurement noise, CUSUM algorithm can improve the performance of diagnosis. For example in Figure 3.7(a), the change of residual r 1 is not very obvious during fault F3 occurs, but it can be clearly seen from the CUSUM of r 1 in Figure 3.7(b). CUSUM can also help to detect the incipient fault such as F4 more quickly, as shown in Figure 3.7(a) and Figure 3.7(b). Fault signature of residual r and fault detection instant T d with respect to different faults are therefore given in Table 3.3. These results reveal that the proposed CD-FMO has a good and effective performance in both sensor and actuator fault detection for the single-link robotic arm.

Fault isolation

It can be obviously seen from Table 3.3 that fault F2 is isolable as it has a unique fault signature [0, 1, 1]. On the other hand, the remaining three faults F1, F3 and F4 can not be isolated because of the identical fault signature [1,0,1]. Hence in this subsection we aim at solving this problem by using Generalized observer scheme (GOS) [START_REF] Frank | Fault diagnosis in dynamic systems via state estimation -a survey[END_REF]] with another additional observer (CD-FMO 2), while CD-FMO 1 is the same as the previous part.

The structure of a bank of CD-FMO is illustrated in Figure 3.8. CD-FMO 2 is constructed by 3.4, we can obviously see that the three indistinguishable faults F1, F3 and F4, which have identical fault signature [1, 0, 1] by CD-FMO 1, become isolable with [1, 1, 0], [0, 1, 1] and [0, 1, 0] by CD-FMO 2. It means that by applying the GOS structure, the presented CD-FMO can also accomplish the objective of fault isolation effectively.

Fault Signature CD-FMO 2 r 1 r 2 r 3 r ′ 1 r ′ 2 r ′ 3 A bias on y 1 (F1) 1 0 1 1 1 0 A bias on y 2 (F2) 0 1 1 0 1 0 A bias on y 3 (F3) 1 0 1 0 1 1 Actuator fault (F4) 1 0 1 0 1 0

Conclusion

In this chapter, a nonlinear observer has been proposed to perform state estimation and fault diagnosis for a class of continuous-discrete nonlinear dynamical systems. The performance of state estimation is excellent and can be significantly improved by choosing a larger window length. Also the presented approach has a finite-time convergence, which is a great advantage from the perspective of FDI. Simulations have illustrated that the proposed method provides a quite effective fault detection for sensor and actuator faults, which can also show the robustness of this nonlinear observer against the measurement noise. Meanwhile, by using the bank of observers, we are able to deal with the isolation of faults with identical fault signature. It is very worth noting that the proposed observer structure can also be apply to the following cases:

• Estimation instant is not synchronized with measurement instant, i.e. we are able to obtain the state estimation x(t) with t ∈ (kT s , (k + 1)T s );

• The sampling period of measurement is not a constant, i.e. T s ̸ = constant.

The work in this chapter has been published in [START_REF] Zhang | A continuous-discrete finite memory observer design for a class of nonlinear systems: Application to fault diagnosis[END_REF]. During the development and synthesis of CD-FMO in this chapter, the following questions have been also raised in our mind as the perspectives:

1. Can we further perform fault estimation for the actuator fault? Since the actuator serves as the controller of system, where a faulty actuator may make the plant uncontrollable, it is obviously not enough to just detect that there exists an actuator fault.

2. What if a sensor fault and an actuator fault occur at the same time? How can we provide more evidences to help locating or isolating the multiple simultaneous faults.

3. Instead of having a time-invariant system, what if the considered nonlinear system (3.1) is time-varying? It is worth noting that the linearization of a time-invariant nonlinear system often (when the nominal solution is not a constant) gives rise to time-varying systems, and this is actually one of the chief ways time-varying systems are encountered in system analysis.

As we have always believed, there are no problems, only solutions. Let's find the solutions to the above three problems in the rest chapters of this thesis.

Introduction

Résumé en français :

Afin de répondre aux deux premières questions posées à la fin du chapitre précédent, nous allons construire dans ce chapitre un observateur à mémoire finie (CD-UI-FMO) en temps continu et à mesures discrètes pour les systèmes non linéaires dynamiques avec une entrée inconnue. Les systèmes non linéaires considérés sont toujours au moins localement Lipschitz et représentés par des équations différentielles ordinaires (ODEs) avec des bruits de processus et des bruits de mesure. Le CD-UI-FMO proposé est conçu par un modèle augmenté dans le but d'estimer simultanément les états du système et les entrées inconnues. La preuve de l'estimation non biaisée dans le cas déterministe est la même que celle dans le chapitre précédent, et elle sera également montrée via des simulations de Monte Carlo (MC) dans le cas stochastique. De plus, les observateurs non linéaires présentés sont appliqués avec succès à la détection et l'estimation de défaut actionneur pour un bras robotique à l'aide de l'algorithme à moyenne mobile avec pondération exponentielle (EWMA). Les résultats de la simulation mettent en évidence l'efficacité de l'approche proposée dans ce chapitre.

Introduction

In order to answer the first two questions raised at the end of the precedent chapter, we intend to design a continuous-discrete unknown input finite memory observer (CD-UI-FMO) for dynamical nonlinear systems with unknown input in this chapter. The nonlinear systems under consideration are still at least locally Lipschitz and represented by ordinary differential equations (ODEs) with process noise while measurements are sampled at discrete-time instants with measurement noises. By an augmented model, the proposed CD-UI-FMO is designed with the aim to simultaneously estimate system states and unknown inputs. The unbiased estimation property appears to remain the same as the previous chapter and it will be shown via Monte Carlo (MC) simulations as well in stochastic case. Moreover, the presented nonlinear observer are successfully applied to an actuator fault detection and estimation for a single-link joint robotic arm incorporating with the exponentially weighted moving average (EWMA) algorithm. Illustrative simulation results highlight the effectiveness of the proposed approach in this chapter.

CD-UI-FMO Design

For the sake of clarity, it should be noting that some notations may be changed in this chapter even if they represent the same variables compared to those in the previous chapter.

Problem statement

Let us consider the same class of nonlinear systems (3.1) as in precedent chapter with process noise and measurement noise, while another term is added to represent the unknown input. It is therefore modeled by the following state space equation: Consider the original system

Ẋ(t) = A 0 X(t) + B 0 u(t) + Φ(X) + Fξ (t) + G 0 ω X (t) y(k) = C 0 X(k) + v(k) (4.1)
where X(t) ∈ R n X , u(t) ∈ R n u and y(k) ∈ R n y are respectively continuous-time state vector, continuous-time input vector and discrete-time measurement vector, ∃k ∈

N | t = k × T s , T s is the sampling period of measurements. A 0 ∈ R n X ×n X , B 0 ∈ R n X ×n u , G 0 ∈ R n X
×n ω X and C 0 ∈ R n y ×n X are known matrices. The term ξ (t) represents the unknown input, F ∈ R n X ×n ξ is referring as the unknown input distribution matrix. The nonlinearity Φ(X) is a nonlinear function with respect to state X. Finally, the mutually independent vectors ω X ∈ R n ω X and v ∈ R n y respectively represent process noise and measurement noise, which assume to be zero mean Gaussian noises, namely:

E[ω X (t)] = 0 E[ω X (t 1 )ω T X (t 2 )] = Q X • δ (t 1 -t 2 ) E[v(k)] = 0 E[v(k 1 )v T (k 2 )] = R • δ k 1 ,k 2
here, δ (•) is Dirac delta function and δ i, j denotes Kronecker delta function. The proposed CD-UI-FMO for nonlinear systems (4.1) will be detailed under the following assumptions:

ξ (t) = D ξ ω ξ (t)
where D ξ ∈ R n ξ ×n ω ξ is the noise distribution matrix and ω ξ (t) ∈ R n ω ξ , which is also independent of v, is zero mean Gaussian process noise for ξ with properties as follows:

E[ω ξ (t)] = 0 E[ω ξ (t 1 )ω T ξ (t 2 )] = Q ξ • δ (t 1 -t 2 )
4.2 CD-UI-FMO Design

Mathematical model for CD-UI-FMO

Based on (4.1) and the dynamic of unknown input ξ (t) in Assumption 4.2, an augmented model can be built as follows:

ẋ(t) = Ax(t) + Bu(t) + φ (x) + Gω(t) (4.2a) y(k) = Cx(k) + v(k) (4.2b) with x(t) = X(t) ξ (t) ; A = A 0 F 0 0 ; B = B 0 0 ; φ (x) = Φ(X) 0
ω(t) = ω X (t) ω ξ (t) ; G = G 0 0 0 D ξ ; C = C 0 0 ; Q = Q X q T q Q ξ .
where the Lipschtiz condition in Assumption 4.1 is well-preserved for φ (x) according to the definition of the norm. The augmented process noise ω(t) is also Gaussian with the covariance Q, i.e.

E[ω(t)] = 0 E[ω(t 1 )ω T (t 2 )] = Q • δ (t 1 -t 2 )
Note that the covariance term q in matrix Q equals to zero when the UI process noise ω ξ is uncorrelated with the state process noise ω X , i.e. when correlation coefficient ρ ω ξ ω X = 0.

The proposed CD-UI-FMO will be designed based upon the mathematical model (4.2) in next subsection. Before we start, the remark and assumption of observability are recalled again as follows:

Remark 4.1 [START_REF] Kou | Finite-time observer for nonlinear dynamic systems[END_REF]). The observability of a nonlinear system is a necessary condition that there exists a finite-time observer.

Formulation of CD-UI-FMO

At each frozen time instant t, suppose that the discrete measurements are collected at L time instants tτ i with i = 0, 1, ..., L -1. It is evident that [tτ L-1 ,t] determines a time window where window length is L.

Firstly, both side of (4.2a) are pre-multiplied by an matrix exponential integrating factor e -At . Then, by directly integrating and rearranging, the relation of states between two different time instant t and tτ i can be given as:

x(t) = e Aτ i x(t -τ i ) + t t-τ i e A(t-θ ) Bu(θ ) dθ + t t-τ i e A(t-θ ) φ (x(θ )) dθ + t t-τ i e A(t-θ ) Gω(θ ) dθ (4.3)
then, we obtain the following equation via left multiplying the matrix Ce -Aτ i to (4.3) and taking into account the measurement equation ( 4.2b) at time instant tτ i :

Ce -Aτ i x(t) = y(t -τ i ) -v(t -τ i ) + α t-τ i ,t + β t-τ i ,t + γ t-τ i ,t (4.4) with α t-τ i ,t = t t-τ i Ce A(t-τ i -θ ) Bu(θ ) dθ (4.5a) β t-τ i ,t = t t-τ i Ce A(t-τ i -θ ) φ (x(θ )) dθ (4.5b) γ t-τ i ,t = t t-τ i Ce A(t-τ i -θ ) Gω(θ ) dθ (4.5c)
By applying equation (4.4) for every instant within the time window [tτ L-1 ,t], a finite number of concatenated measurements can be denoted with respected to the system state x(t) as the following linear equation:

Y L = W L x(t) +V L (4.6)
where

Y L =       y(t -τ 0 ) + α t-τ 0 ,t + β t-τ 0 ,t y(t -τ 1 ) + α t-τ 1 ,t + β t-τ 1 ,t . . . y(t -τ L-1 ) + α t-τ L-1 ,t + β t-τ L-1 ,t       ; V L =       v(t -τ 0 ) -γ t-τ 0 ,t v(t -τ 1 ) -γ t-τ 1 ,t . . . v(t -τ L-1 -γ t-τ L-1 ,t       ; W L =       Ce -Aτ 0 Ce -Aτ 1 . . . Ce -Aτ L-1      
.

It is evident that the noise term V L has zero mean, i.e. E(V L ) = 0. According to the properties of ω(t) and v(k), together with Fubini's theorem and the shifting property of the Dirac delta function, the covariance matrix P, which is block symmetric, may be computed 4.3 Application to a Single-link Flexible Joint Robotic Arm as the following form (see Appendix A for more details):

P = E (V L -E(V L )) (V L -E(V L )) T =       S 0 S 0 • • • S 0 S 0 S 1 • • • S 1 . . . . . . . . . . . . S 0 S 1 • • • S L-1       +       R 0 • • • 0 0 R . . . . . . . . . . . . . . . 0 0 • • • 0 R       (4.7)
where the block elements S k (k min[i, j] = 0, 1, ..., L -1) represent the following integral [START_REF] Medvedev | Parity space method: a continuous time approach[END_REF]]:

S k = 0 -τ k Ce As GQG T e A T s C T ds
Using the method of least-squares, the state estimation x(t) at time instant t is obtained from the solution of (4.6) as follows:

x(t) = W T L P -1 W L -1 W T L P -1 ŶL (4.8) with ŶL =       y(t -τ 0 ) + α t-τ 0 ,t + βt-τ 0 ,t y(t -τ 1 ) + α t-τ 1 ,t + βt-τ 1 ,t . . . y(t -τ L-1 ) + α t-τ L-1 ,t + βt-τ L-1 ,t       (4.9a) βt-τ i ,t = t t-τ i Ce A(t-τ i -θ ) φ ( x(θ )) dθ (4.9b)

Unbiased estimation property of CD-UI-FMO

Note that the formulation of CD-UI-FMO from (4.3) to (4.9) is very similar to CD-FMO in the previous chapter. Therefore, for the sake of brevity, please refer to section 3.2.3 and Appendix B for the theoretical proofs of unbiased estimation property in deterministic case and in stochastic case.

Application to a Single-link Flexible Joint Robotic Arm

The dynamics (4.1) cover a broad class of systems with unknown input, and the proposed CD-UI-FMO is validated in this section through a single-link flexible joint robotic arm.

Nonlinear augmented continuous-discrete state-space model

The state-space model is described by (4.1) where components X 1 and X 3 are the displacement of link and rotor respectively while components X 2 and X 4 represent the velocity. The control input u(t) = 2 sin(t), which represents the torque provided by the motor. The unknown input, ξ (t) = -0.5 sin(t), represents two different kinds of actuator fault in this example, which is injected during 0.15s ∼ 0.35s. The standard derivation (SD) for all the process noise are set as σ p X = σ p ξ = 0.01, i.e. Q X = 0.01 2 • I 4 and Q ξ = 0.01 2 . The noise distribution matrices G 0 and D ξ are set to the identity matrix with proper dimensions. The matrices and parameters are set as follows with the physical parameters shown in Table 4.1:

A 0 =       0 1 0 0 -k J l -f l J l k J l 0 0 0 0 1 k J m 0 -k J m -f m J m       ; Φ(X) =       0 -mlg c J l sin X 1 0 0       ; B 0 = F =       0 0 0 1 J m       ; Q X = σ 2 p X •I 4 =      0.01 2 0 0 0 0 0.01 2 0 0 0 0 0.01 2 0 0 0 0 0.01 2      ; Q ξ = σ 2 p ξ = 0.01 2 ; G 0 =    1 0 0 0 1 0 0 0 1    ; D ξ = 1; C 0 =    1 0 0 0 0 0 1 0 0 0 0 1    ; R = σ m σ T m =    0.0002 2 0 0 0 0.0002 2 0 0 0 0.0003 2    .
Table 4.1: Physical parameters (in SI units) elastic constant k = 2 link mass m = 4 viscous friction coefficient for motor f m = 1 motor inertia J m = 1 viscous friction coefficient for link f l = 0.5 link inertia J l = 2 mass center l = 0.5 link inertia g c = 9.8

Then, an augmented model can be constructed for CD-UI-FMO according to (4.2) as Augmented Model:

ẋ(t) = Ax(t) + Bu(t) + φ (x) + Gω(t) y(k) = Cx(k) + v(k)
; A = A 0 F 0 0 ; B = B 0 0 ; φ (x) = Φ(X) 0 ; ω(t) = ω X (t) ω ξ (t) ; G = G 0 0 0 D ξ ; Q σ p σ T p = Q X q T q Q ξ ; C = C 0 0 ; R σ m σ T m .
In next subsection, we will evaluate by simulations the performances of CD-UI-FMO on the unbiased state estimation property, the actuator fault (unknown input) detection and estimation based upon this augmented model in stochastic case, which includes the presence of measurement noise and process noise.

Optimal window length selection

As shown in Figure 4.1, with the augmented model, the variance of the estimation error x = xx by the proposed CD-UI-FMO decreases asymptotically as the window length increasing. Meanwhile, it can also be seen in Figure 4.1 that the speed of the convergence became slower compared to the estimation error with the nominal model described as follows Nominal Model: Since the nominal model does not need to deal with the unknown input during the state estimation while the CD-UI-FMO by the augmented model perform the state estimation and UI estimation in the same time, it seems logical that the convergence speed of the estimation error slows down. L = 10 will be chosen in the rest of this chapter if it is not indicated.

Ẋ(t) = A 0 X(t) + B 0 u(t) + Φ(X) + G 0 ω X (t) y(k) = C 0 X(k) + v(k)

Residuals comparison between different models

Give the notation of residual as r(t) = y(t) -C x(t), the residuals obtained by the nominal model and the augmented model are shown in 35s causes a significant bias on r 2 , a slight mean shift on r 3 and no influence at all on r 1 when using the nominal model. According to the given UI distribution matrix F, we can see that the unknown input ξ has only been distributed to the fourth component ẋ4 . Then

Application to a Single-link Flexible Joint Robotic Arm

through the matrices A 0 and C 0 , we know that r 1 is not sensible to the UI injection in this example. The different influence level on r 2 and r 3 is caused by the numerical sensitivity coefficient, which will be theoretically studied in the future work. On the other hand, it is shown that the residuals are well centered around zero when using the proposed CD-UI-FMO, which appears to be normal since the UI is simultaneously estimated by the augmented state.

Unbiased estimation performance in stochastic case

The unbiased estimation performance will be illustrated in two aspects: using one single Monte Carlo (MC) run to show the convergence of estimation accuracy through different window length L; using multiple Monte Carlo simulations show the unbiased performance of both system state and unknown input estimations.

State and UI estimations by different window length

Let us firstly see the state estimations of x 2 in Figure 4.5 and UI estimations in Figure 4.6, it is quite obvious that the estimation performance has been significantly improved as the window length L increases, which means that the estimation accuracies converge with the augment of window length L, as depicted in Figure 4.1. This is only logical since the longer the window length L is, the more amount of information it will contain in each horizon, and therefore the better estimations we will obtain. Indeed, "information amount" can be said as the key essence of the moving horizon techniques introduced in previous chapter.

On the other hand, it seems that the same conclusions can not been straightforwardly drawn to the state estimations of components x 1 , x 3 and x 4 shown in Figure 4.7 -Figure 4.9. As a matter of fact, it is necessary to notice that these three components (x 1 , x 3 and x 4 ) are exactly the states we measured (see matrices C and C 0 ). This means that the informations of 3 Application to a Single-link Flexible Joint Robotic Arm these three states can be obtained at all sampling instants, and it is absolutely normal to have a good estimation performance on these three components even if the window length is set as L = 1. Moreover, since L = 1 is "long" enough to have good estimations on x 1 , x 3 and x 4 , when we employ L ≥ 1, the filter role with respect to the noises can be seen as well by examining Figure 4.7 -Figure 4.9 (see subsection 3.3.3.2 of Chapter 3).

Unbiased state and UI estimations by MC simulations

As studied in previous chapter, we are going to perform multiple Monte Carlo simulations in this subsection to illustrate the unbiased estimation property of the proposed CD-UI-FMO in stochastic case, namely

x E ( x(t)) = x(t) t ∈ [L × T s , +∞) with E ( x(t)) = 1 N mc N mc ∑ i=1 x(i)
here, N mc represents Monte Carlo simulation times. N mc = 500 runs of Monte Carlo simulations have been performed to the nonlinear system with unknown input and the state estimation results are shown in Figure 4.10 -Figure 4.13. It can be seen by examining the figures that the original system states, with the presence of both noises (process and measurement) and unknown input, are well estimated around the true value with no bias after performing 500 runs of MC simulations. Among the 500 different state estimations obtained by 500 runs of MC simulation, let us take the one with maximum dispersion above true state value as the upper bound of the estimations while the one with the maximum dispersion below the true state value is taken as the lower bound. In stochastic case, these two bounds happen to define an state estimation envelope where all the possible estimations are supposed to be within this envelope. Now let us examine Figure 4.10 -Figure 4.13 again, it can be seen that the estimation upper and lower bounds of the proposed CD-UI-FMO are within a very small range, which means that even with the noises and UI, all the possible state estimations by CD-UI-FMO vary within a small range around real state x. This might be another verification of sufficient estimation precision of the proposed observer.

Finally, the unbiased estimation performance of the unknown input (actuator fault) ξ = -0.5 sin(t) can be seen from Figure 4.14. The expectation of ξ with 500 runs of MC simulation is well located around the true value.

Robustness with respect to measurement and process noises

In this section, the robustness of CD-UI-FMO with respect to measurement noise and process noise is analyzed through different scenarios in Table 4.2, which the standard derivation of measurement noise (SD m ) and process noise (SD p ) vary ±50 % while the noise configurations of CD-UI-FMO doesn't change. The robustness analysis intends to see whether the proposed CD-UI-FMO has the ability to resist noise variations without adapting its noise configuration.

SD m = σ m ; R = σ m σ T m SD p = σ p ; Q = σ p σ T p Scenario 1 (S1) SD m = 0.5σ m ; SD p = 0.5σ p Scenario 2 (S2) SD m = 1.5σ m ; SD p = 0.5σ p Scenario 3 (S3) SD m = 0.5σ m ; SD p = 1.5σ p Scenario 4 (S4) SD m = 1.5σ m ; SD p = 1.5σ p
By performing N mc = 500 runs of MC simulations with L = 10, the state estimation performance as well as the RMSEs are respectively shown in Figure 4.15 and Figure 4.16. The robustness of CD-UI-FMO can be well validated by examining these figures. Furthermore, it can be seen from the RMSEs in Figure 4.16 that estimation performance of Scenario S1 is always better than Scenario S4 since "overestimate" the real noises level is always better than "underestimate" the noises level for CD-UI-FMO. In the meantime, the RMSEs amplitudes also decrease one by one from "top" to "bottom" due to the filter performance explicated in subsection 3.3.3.2 of Chapter 3. It can be seen from Figure 4.17 that the unknown input estimation by CD-UI-FMO is robust as well facing different noise scenarios. The above analyses is performed when the state process noise ω X and unknown input process noise ω ξ are uncorrelated, i.e. ρ ω ξ ω X = 0. What will happen when ρ ω ξ ω X ̸ = 0 ? Let us take ρ ω ξ ω X = 0.1, in which case the process noise covariance matrix Q is no longer diagonal. Take the unmeasured component x 2 as a representative of state estimation, the performance of both state and UI estimations are shown in Figure 4.18 and Figure 4.19 with L = 3 and N mc = 500. It seems that the performance of CD-UI-FMO remains good even with the correlated process noises, which can be consider as another efficient performance. Now, let us only see Figure 4.6 and Figure 4.14 for the UI estimations obtained by the proposed CD-UI-FMO. On the one hand, the UI estimation in Figure 4.6 based on just one single run of MC simulation turns out to be a little bit noisy. Even though it can be improved by augmenting the window length L, the compute time and calculation burdens will augmented as well. On the other hand, the UI estimation performance in Figure 4.14 is indeed great due to the multiples runs of Monte Carlo simulations, however one of the major disadvantages of MC simulation is also the computer time. In addition, the UI in this thesis represents an actuator fault (without loss of generality). As previously introduced in Chapter 2 (see subsection 2.5.1), the first step of fault diagnosis procedure is fault detection, which is required to be as quick as possible. Apparently in this case, we do not want to perform fault detection neither after "multiple MC simulations" nor "with a super large window length L". This brings us to the exponentially weighted moving average (EWMA) algorithm introduced in next part, which trades off between fault estimation and fault detection effectiveness.

Actuator fault diagnosis by the EWMA algorithm

The exponentially weighted moving average (EWMA) [START_REF] Cisar | Ewma algorithm in network practice[END_REF] algorithm is employed in this subsection to perform fault detection and estimation thanks to its wellknown efficiency and filtering ability when facing small shifts. Compared to the previously used Cumulative Sum (CUSUM) algorithm, which is the sum of the entire process history, a weighted sum of the recent history would be more meaningful for the purpose of fault estimation (filtering). Furthermore, it might be worth noting that CUSUM algorithm uses the cumulative sum of the shift signal to detect the fault while EWMA uses directly the fault signal itself to perform the fault detection.

The following notations of EWMA are firstly defined: the upper control limit (UCL), center line (CL) and lower control limit (LCL) are defined as follows:

             UCL = µ target + n • σ target λ 2 -λ [1 -(1 -λ ) 2i ] CL = µ target LCL = µ target -n • σ target λ 2 -λ [1 -(1 -λ ) 2i ] (4.10)
where i represent the sampling numbers (observations), the mean and SD values of the target data are respectively noted as µ target and σ target . n is used to represent the control limits (with respect to σ target ) of the target data while the weighted constant is represented by λ .

Actuator fault detection and estimation

Without the loss of generality, the target data is shown in Figure 4.20, which is the estimated actuator fault ξ obtained by CD-UI-FMO. Under the usual three-sigma rule, the detection by CUSUM is also shown in Figure 4.21. However, as mentioned before, CUSUM performs the fault detection by the cumulative sum of mean deviations and does not directly process the target data (actuator fault), we are going to use EWMA algorithm in next step. In order to use the EWMA algorithm to deal with the estimated actuator fault ξ with i = 500 data, the target mean and SD values are set as µ target = 0 and σ target = 0.0517, which obtains by CD-UI-FMO under the same parameter configuration as in Figure 4.20 but fault-free (ξ = 0). The control limits n = 3 (three-sigma rule), the weighted constant

λ = 0.2.
It is shown in Figure 4.22 that the detection of actuator fault starting instant is 0.157s with a delay of 0.007s since the real fault occurring instant is 0.15s. It is worth mentioned here that in this example five consecutive alarm points is used as decision rule for detecting pattern, "less than five consecutive points" is said to be a false alarm. It should be noted that the detecting pattern of false alarm can be adapted accordingly for actual physical systems [START_REF] Montgomery | Introduction to statistical quality control[END_REF]].

Minimum detection amplitude analyses

A ramp-shape actuator fault is injected to find out the minimum actuator fault amplitude for EWMA to detect with n = 3 and λ = 0.2 while µ target = 0 and σ target = 0.0517. The detection alarm by EWMA is shown in Figure 4.24, in which the detection instant is t = 0.168s. Meanwhile, as shown in Figure 4.25, the corresponding actuator fault estimated via EWMA at t = 0.168s is ξmin = 0.0504, which is said to be the minimal detection amplitude. As a

ξ λ =0.2 bound = µ target ± n • σ target λ 2 -λ [1 -(1 -λ ) 2i ] = 0 ± 3 • 0.0517 0.2 2 -0.2 [1 -(1 -λ ) 2×500 ] = ±3 • 0.0517 1 9 = 0.0517 it is straightforward that ξmin ≈ ξ λ =0.2 bound = 1 • σ target
Let us inject a constant actuator fault ξ min = 0.055 during 0.15s ∼ 0.35s to test the obtained minimal amplitude. The fault detection results are shown in Figure 4.26 and Figure 71 4.3 Application to a Single-link Flexible Joint Robotic Arm 4.27. We can see that the detection instant has a delay of 0.013s and also the detection alarm is not "steady", it all seems normal since it is the minimal fault amplitude. Note that the parameter λ determines the rate at which "older" data enter into the calculation of the EWMA statistic. By the choice of weighting factor λ , the EWMA algorithm can be designed sensitive to a even small fault. For instant, if we take λ = 0.1 for actuator fault ξ min = 0.055, the detection alarm is shown in Figure 4.28, which becomes very "neat". However, the detection instant has a 1ms more delay which appears logical since a smaller value of λ gives more weight to older data. In fact, when λ = 0.1, the detection bound becomes as

ξ λ =0.1 bound = µ target ± n • σ target λ 2 -λ [1 -(1 -λ ) 2i ] = 0 ± 3 • 0.0517 0.1 2 -0.1 [1 -(1 -λ ) 2×500 ] = ±3 • 0.0517 1 19 = 0.0356 < ξ min = 0.055
that's why the detection alarm in Figure 4.28 is much more better than Figure 4.26 for actuator fault ξ min = 0.055. In order to further prove this, an actuator fault ξ * = 0.04 is used as

ξ λ =0.1 bound < ξ * < ξ λ =0.2 bound
It can be clearly seen from Figure 4.29 and Figure 4.30 that a detection alarm with 0.011s delay is provided when λ = 0.1 while there is no detection with λ = 0.2 as the alarms are always followed immediately by no alarms. Note that the EWMA control chart is very effective and can be designed by the configuration of the value of λ as well as control limit n, which there is no explicit formula of the best λ for a specific fault. It is possible to choose these parameters differently according to the practical requirements, such as noise level or fault-tolerant criteria.

Simultaneous actuator and sensor faults diagnosis

The proposed CD-UI-FMO in this chapter can also be used to provide more evidences for locating or isolating the simultaneous sensor and actuator faults. In order to show this performance, the following different multiple simultaneous faults (MSF) and single fault (SF) scenarios are used during 0.15s ∼ 0.35s: In the meantime, the following two fault patterns are defined according to the fault shape reflected on residuals r(t) = y(t)ŷ(t):

• Peak (p): the fault is reflected on residuals as the form of a peak;

• Bias (b): the fault is reflected on residuals as the form of a bias or shift.

CD-FMO (proposed in Chapter 3) with nominal model and CD-UI-FMO with augmented model are used to perform the fault diagnosis for the above three MSF scenarios, and the residuals r(t) are respectively as

r(t) = y(t) -ŷ(t) = y(t) -C 0 X(t) CD-FMO (with Nominal Model) y(t) -C x(t) CD-UI-FMO (with Augmented Model)
The resulting residuals obtained for different scenarios are respectively given from By examining the residuals of the upper left part of Table .4.3, one may notice that if we only use the fault signature of CD-FMO, the multiple simultaneous faults MSF1 and MFS2 can not at all be isolated from each other since they have the same signature [1,1,1]. Let us now take into account the fault pattern as well, with which seems that MSF1 can be distinguished from MSF2 since the fault pattern on r 3 is "p&b" for MSF1 and "b" for MSF2. However, we still need more evidences to answer the following questions: Is it a multiple simultaneous faults? a sensor fault? an actuator fault? This brings us to the proposed CD-UI-FMO in this chapter. CD-FMO

Fault Signature (Pattern) CD-UI-FMO r 1 r 2 r 3 r 1 r 2 r 3 MSF1 1 (p) 1 (p&b) 1 (p&b) 1 (p) 1 (p) 1 (p) MSF2 1 (p) 1 (p&b) 1 (b) 1 (p) 1 (p) 0 MSF3 0 1 (p&b) 1 (p&b) 0 0 1 (p) SF1 1 (p) 1 (p) 1 (p) 1 (p) 1 (p) 1 (p
) SF2 1 (p) 1 (p) 0 1 (p) 1 (p) 0 SF3 0 1 (p) 1 (p) 0 0 1 (p) SF4 0 1 (b) 1 (b) 0 0 0
The residuals obtained by using CD-UI-FMO is given in the upper right part of Table .4.3. It can be seen that all the pattern "b" disappear while there only remains fault pattern "p" for MSF1, MSF2 and MSF3. Since CD-UI-FMO is designed to estimate the unknown input of systems, we can't help but thinking that the pattern "b" might represent an actuator fault while "p" signifies a sensor fault. The reason why pattern "b" disappears on residual is that the actuator fault is estimated and compensated by CD-UI-FMO.

This diagnosis result can be confirmed by the results of using CD-UI-FMO to diagnose the four single faults SF1-SF4, which are in the lower right part of Table .4.3. The signature [0, 0, 0] for SF4 confirms that the compensated fault by CD-UI-FMO in MSF1-MSF3 is indeed actuator fault. Moreover, the fact that the signatures for SF1-SF3 are the same as MSF1-MSF3 proves there are also sensor faults in MSF1-MSF3. More specifically, according to the distinguished signatures [1,1,1], [1,1,0] and [0, 0, 1], it can be known that there is sensor fault ∆ y 1 in MSF1, ∆ y 2 for MSF2 and ∆ y 3 for MSF3. So far, the detection and isolation of the multiple simultaneous faults are realized with the aid of proposed CD-UI-FMO.

Conclusion

A nonlinear unknown input finite memory observer has been developed in this chapter for a class of nonlinear Lipschitz systems with the aim of providing simultaneous unknown input and state estimations in the presence of both process and measurement noises. The unbiased estimation property can be proved theoretically and has also been fully verified in detail by Monte Carlo simulations in stochastic case. Additionally, the proposed method has been applied to implement actuator fault detection and estimation for a single-link robotic arm with the aid of the EWMA algorithm. Finally the proposed CD-UI-FMO is successfully used to perform fault diagnosis for multiple simultaneous sensor and actuator faults. The state and unknown input estimations accuracy and the effectiveness of fault diagnosis are well illustrated via the simulation results.

Introduction

Résumé en français :

Dans ce chapitre, nous continuons à répondre à la troisième question posée à la fin du chapitre 3, qui concerne la synthèse d'observateur à mémoire finie pour une classe de systèmes non linéaires variant au cours du temps. À notre connaissance, l'observateur à mémoire finie n'a été conçu pour aucun système en temps-continu et à temps-variant, ni pour les systèmes linéaires en temps-continu, ni pour les systèmes non linéaires. Nous allons donc présenter la synthèse d'observateur à mémoire finie en commençant par des systèmes continus linéaires variant au cours du temps, une extension aux systèmes non linéaire sera ensuite réalisée sur la base du cas linéaire et de ce qui a été réalisé dans Chapitre 3. Un observateur à mémoire finie sera tout d'abord développé pour les systèmes linéaires à temps-variant en présence de bruit de processus et bruit de mesure. Nous démontrerons que l'observateur proposé est le meilleur estimateur non-biaisé parmi tous les estimateurs linéaires. De plus, une formulation séquentielle de la variance de l'erreur d'estimation en fonction de la longueur de fenêtre sera également montrée, ce qui conduit naturellement à une stratégie pour sélectionner une longueur optimale. Un exemple numérique sera fourni pour vérifier les démonstrations théoriques dans le cas linéaire. L'observateur proposé sera ensuite étendu aux systèmes non linéaires par une brève démonstration et un exemple numérique à la fin de ce chapitre.

Introduction

In this chapter, we continue to deal with the third question raised at the end of Chapter 3, which requires to design a finite memory observer for a class of continuous-discrete nonlinear time-varying (CD-NLTV) systems. As far as we know, the finite memory observer (FMO) has not (yet) been designed for any continuous time-varying system, neither for continuous linear time-varying (LTV) systems, nor for nonlinear time-varying (NLTV) systems. Therefore, we choose to illustrate our design method by beginning with continuous-discrete linear time-varying (CD-LTV) systems, then a further extension to NLTV systems will be carried out based upon the CD-LTV case and what has been realized in Chapter 3.

A robust finite memory observer will be firstly developed for the CD-LTV systems in the presence of both process and measurement noises. It is going to be theoretically proved that the state estimation is unbiased and has the smallest dispersion among all the linear estimators, namely best linear unbiased estimator (BLUE). In addition, a sequential form of the estimation error variance with respect to window length will be demonstrated as well, which naturally leads to be a theoretical strategy of selecting the window length. A numerical simulation example will be provided to verify the aforementioned theoretical proofs for CD-LTV systems. Finally, the proposed observer will be extended to the CD-NLTV systems through a brief demonstration and a numerical example at the end of this chapter.

5.2 FMO Design for CD-LTV Systems (CD-LTV-FMO)

Problem statement and preliminaries

Let us consider the following continuous-time LTV systems with discrete measurements:

ẋ(t) = A(t)x(t) + B(t)u(t) + ω(t)
(5.1a)

y(k) = C(t k )x(t k ) + v(k) (5.1b)
where x ∈ R n , y ∈ R p and u ∈ R q are continuous state vector, discrete measurement vector and continuous input vector, respectively. Without loss of generality, we assume that we have a constant sampling period of measurement T s , i.e. ∃k ∈ N | t k = k × T s . In the meantime, A(•) ∈ R n×n , B(•) ∈ R n×q and C(•) ∈ R p×n are known matrices and are functions of time. Vectors v and ω represent Gaussian measurement noise and Gaussian process noise, respectively. v and ω are independent with the following properties:

E[ω(t)] = 0 (5.2a) E[ω(t 1 )ω T (t 2 )] = Q • δ (t 1 -t 2 ) (5.2b) E[v(k)] = 0 (5.2c) E[v(k 1 )v T (k 2 )] = R • δ k 1 ,k 2 (5.2d)
where δ (•) is Dirac delta function and δ i, j is Kronecker delta function. The proposed observer will be detailed in next section under the following assumptions:

Assumption 5.1. The matrix A(•) is piecewise continuous or at least satisfies the integral condition (see [START_REF] Coddington | Theory of ordinary differential equations[END_REF]).

   dΦ(θ ,t 0 ) dθ = A(θ )Φ(θ ,t 0 ) Φ(t 0 ,t 0 ) = I n I n is identity matrix of size n
This means that for any unknown initial state x(0), there exists a finite t f > t 0 so that it is possible to determine x(0) by knowing inputs u and outputs y.

Remark 5.1. It may be worth recalling here some properties of state transition matrix Φ(•, •) as follows [START_REF] Chen | Linear system theory and design[END_REF][START_REF] Kailath | Linear systems[END_REF]] :

• Φ -1 (t 2 ,t 1 ) = Φ(t 1 ,t 2 ) for any t 1 ,t 2 ∈ [t 0 ,t f ]; • Φ(t 1 ,t 3 )Φ(t 3 ,t 2 ) = Φ(t 1 ,t 2 ) for any t 1 ,t 2 ,t 3 ∈ [t 0 ,t f ];
• Φ(t,t) = I n holds for all t ∈ [t 0 ,t f ];

• Φ(•, •) is never singular.

• Given x(t 0 ) at any time t 0 , the state at any other time t is given by the mapping:

x(t) = Φ(t,t 0 )x(t 0 ).

CD-LTV-FMO formulation

At each frozen time instant t, suppose that the discrete measurements are collected at L time instants tτ i with i = 0, 1, ..., L -1. It is evident that [tτ L-1 ,t] determines a time window. By using the state transition matrix Φ(t,tτ i ), the state mapping between x(t) and x(tτ i ) can be obtained as [START_REF] Kailath | Linear systems[END_REF]]:

x(t) = Φ(t,t -τ i )x(t -τ i ) + t t-τ i Φ(t, θ )B(θ )u(θ ) dθ + t t-τ i Φ(t, θ )ω(θ ) dθ (5.3)
Then, both sides of (5.3) are left multiplied by a coefficient matrix C(tτ i )Φ -1 (t,tτ i ), which gives (5.4) Take into consideration the measurement equation ( 5.1b) at time instant tτ i , then rearrange with the properties of Φ(•, •) in Remark 5.1, we have

C(t -τ i )Φ -1 (t,t -τ i )x(t) = C(t -τ i )x(t -τ i ) + t t-τ i C(t -τ i )Φ -1 (t,t -τ i )Φ(t, θ )B(θ )u(θ ) dθ + t t-τ i C(t -τ i )Φ -1 (t,t -τ i )Φ(t, θ )ω(θ ) dθ
C(t -τ i )Φ(t -τ i ,t)x(t) = y(t -τ i ) + α [t-τ i ,t] -β [t-τ i ,t]
(5.5)

with

α [t-τ i ,t] = t t-τ i C(t -τ i )Φ(t -τ i , θ )B(θ )u(θ ) dθ β [t-τ i ,t] = v(t -τ i ) - t t-τ i C(t -τ i )Φ(t -τ i , θ )ω(θ ) dθ
Now we write equation (5.5) for every instant tτ i with i = 0, 1, ..., L -1 in the time window [tτ L-1 ,t], a finite number of concatenated measurements can be denoted with respected to the system state at instant t, namely x(t), as the following linear equation:

Y L (t) = W L (t)x(t) +V L (t) (5.6)
where

Y L (t) =       y(t -τ 0 ) + α [t-τ 0 ,t] y(t -τ 1 ) + α [t-τ 1 ,t] . . . y(t -τ L-1 ) + α [t-τ L-1 ,t]       ; W L (t) =       C(t -τ 0 )Φ(t -τ 0 ,t) C(t -τ 1 )Φ(t -τ 1 ,t) . . . C(t -τ L-1 )Φ(t -τ L-1 ,t)       ; V L (t) =       β [t-τ 0 ,t] β [t-τ 1 ,t] . . . β [t-τ L-1 ,t]       .
It is evident that the noise term V L (t) has zero mean, that's E(V L (t)) = 0. In a manner similar to the calculation in Appendix A, the variance matrix P L (t) can be computed as follows [START_REF] Medvedev | Parity space method: a continuous time approach[END_REF]]:

P L (t) = E (V L (t) -E(V L (t))) (V L (t) -E(V L (t))) T = E V L (t)V T L (t) =       S 0 (t) S 0 (t) • • • S 0 (t) S 0 (t) S 1 (t) • • • S 1 (t) . . . . . . . . . . . . S 0 (t) S 1 (t) • • • S L-1 (t)       +       R 0 • • • 0 0 R . . . . . . . . . . . . . . . 0 0 • • • 0 R       (5.7)
where the block elements S k (t) (k min[i, j] = 0, 1, ..., L -1) represents the following integral:

S k (t) = t t-τ k C(t -τ k )Φ(t -τ k , θ )QΦ T (t -τ k , θ )C T (t -τ k ) dθ
Using the method of least-squares, the state estimation xL (t) at time instant t could be directly obtained from the least-squares solution of (5.6) as follows:

xL (t) = arg min 1 2 ||Y L (t) -W L (t)x(t)|| 2 P -1 L (t) = W T L (t)P -1 L (t)W L (t) -1 W T L (t)P -1 L (t)Y L (t)
(5.8) so far we obtain the analytical form of the finite memory observer for continuous-time LTV systems with discrete noisy measurements.

Estimation Error Properties Analyses: "BLUE"

In this section, we are going to theoretically prove by two steps that the proposed CD-LTV-FMO is the best linear unbiased estimator (BLUE) for the continuous-discrete linear time-varying systems.

Unbiased estimation proof

Theorem 5.1. Let ε L (t) = xL (t)x(t),t ≥ L × T s , represents the estimation error of the proposed observer (5.8), then in the case of fault-free, we have:

E[ε L (t)] = 0 Var(ε L (t)) = W T L (t)P -1 L (t)W L (t) -1
Proof. First of all, the following equation can be directly obtained according to (5. 5)

y(t -τ i ) = C(t -τ i )Φ(t -τ i ,t)x(t) -α [t-τ i ,t] + β [t-τ i ,t]
(5.9)

Let ρ i with i = 1, 2, • • • , L -1 represents the i-th column of matrix P -1 L (t), namely

P -1 L (t) [ρ 1 (t), ρ 2 (t), • • • , ρ L-1 (t)]
(5.10)

Consider rewriting x(t) in (5.8) as follows: .11) where

xL (t) = Ω -1 L (t) L-1 ∑ i=0 Φ T (t -τ i ,t)C T (t -τ i )ρ i (t) y(t -τ i ) + α [t-τ i ,t] (5
Ω L (t) W T L (t)P -1 L (t)W L (t) = L-1 ∑ i=0 Φ T (t -τ i ,t)C T (t -τ i )ρ i (t)C(t -τ i )Φ(t -τ i ,t) (5.12)
Substitute y(tτ i ) in ( 5.11) with (5.9), then rearrange it with (5.12) as

xL (t) = Ω -1 L (t) L-1 ∑ i=0 Φ T (t -τ i ,t)C T (t -τ i )ρ i (t) C(t -τ i )Φ(t -τ i ,t)x(t) + β [t-τ i ,t] = Ω -1 L (t)Ω L (t)x(t) + Ω -1 L (t) L-1 ∑ i=0 Φ T (t -τ i ,t)C T (t -τ i )ρ i (t)β [t-τ i ,t] = x(t) + Ω -1 L (t) L-1 ∑ i=0 Φ T (t -τ i ,t)C T (t -τ i )ρ i (t)β [t-τ i ,t] = x(t) + Ω -1 L (t)W T L (t)P -1 L (t)V L (t) which yields ε L (t) = xL (t) -x(t) = Ω -1 L (t)W T L (t)P -1 L (t)V L (t) (5.13)
Since only the noise term V L (t) in ε L (t) is stochastic and V L (t) is independent of the other terms, it is then straightforward that (5.14) we know that (Ω -1 L (t)) T = Ω -1 L (t) and (P -1 L (t)) T = P -1 L (t) are symmetric, then

E[ε L (t)] = Ω -1 L (t)W T L (t)P -1 L (t)E[V L (t)] = 0
Var(ε L (t)) = E[(ε L (t) -E[ε L (t)]) (ε L (t) -E[ε L (t)]) T ] = E[ε L (t)ε T L (t)] = E[Ω -1 L (t)W T L (t)P -1 L (t)V L (t)V T L (t)P -1 L (t)W L (t)Ω -1 L (t)] = Ω -1 L (t)W T L (t)P -1 L (t)E[V L (t)V T L (t)]P -1 L (t)W L (t)Ω -1 L (t) = Ω -1 L (t)W T L (t)P -1 L (t)P L (t)P -1 L (t)W L (t)Ω -1 L (t) = Ω -1 L (t)Ω L (t)Ω -1 L (t) = Ω -1 L (t) (5.15)
The proof is completed.

which yields

ε * (t) = A (t)V L (t) (5.17)
Now let us suppose the linear combination A (t) has the following form 5.18) where Γ(t) ∈ R n×pL is a non-zero matrix. Then,

A (t) = Ω -1 L (t)W T L (t)P -1 L (t) + Γ(t) (
A (t)W L (t) = Ω -1 L (t)W T L (t)P -1 L (t)W L (t) + Γ(t)W L (t) = Ω -1 L (t)Ω L (t) + Γ(t)W L (t) = I n + Γ(t)W L (t) (5.19)
according to equation ( 5.16), we obtain

Γ(t)W L (t) = 0 (5.20)
Calculate the variance of estimation error ε * (t) based on (5.17) and ( 5.20) as follows

Var(ε * (t)) = E ε * (t)ε * T (t) = A (t)E V L (t)V T L (t) A T (t) = Ω -1 L (t)W T L (t)P -1 L (t) + Γ(t) P L (t) Ω -1 L (t)W T L (t)P -1 L (t) + Γ(t) T = Ω -1 L (t)W T L (t)P -1 L (t)P L (t)P -1 L (t)W L (t)Ω -1 L (t) + Γ(t)Γ T (t) + Ω -1 L (t)W T L (t)P -1 L (t)P L (t)Γ T (t) + Γ(t)P L (t)P -1 L (t)W L (t)Ω -1 L (t) = Ω -1 L (t)Ω L (t)Ω -1 L (t) + Γ(t)Γ T (t) + 0 + 0 = Ω -1 L (t) + Γ(t)Γ T (t) + 0 + 0 = Var(ε L (t)) + Γ(t)Γ T (t) (5.21)
we know that Γ(t)Γ T (t) is a positive semi-definite matrix, then it is obvious that Var(ε * (t)) exceeds Var(ε L (t)) by a positive semi-definite matrix, namely

Var(ε * (t)) ≥ Var(ε L (t))
The proof is completed.

For continuous-time LTV systems (2.4), it is well known that Kalman filter (KF) is also one of the optimal estimators under certain conditions. This means that the variance of estimation error Σ(t) by KF described by the following continuous-time Riccati equation Chapter 5 CD-FMO Design for Nonlinear Time-Varying (NLTV) Systems Note that the related variances of ε L (t) and ε L+1 (t) can be expressed as

Var(ε L (t)) = Ω -1 L (t) = W T L (t)P -1 L (t)W L (t) -1 Var(ε L+1 (t)) = Ω -1 L+1 (t) = W T L+1 (t)P -1 L+1 W L+1 (t) -1
(5.25) In order to give the sequential formulation between Ω -1 L (t) and Ω -1 L+1 (t), decompose the matrices W L+1 (t) and P L+1 (t) based on (5.6) and (5.7) as follows

0   L t 1    L t 2 t   1 t   t
W L+1 (t) = W L (t) w(t -τ L ) P L+1 (t) = P L (t) η(t) η T (t) r T (t)r(t) (5.26) where w(t -τ L ) = C(t -τ L )Φ(t -τ L ,t) r T (t)r(t) = S L (t) + R η(t) = [S 0 (t), S 1 (t), • • • , S L-1 (t)] T
(5.27) Since P L+1 (t) and P L (t) are positive-definite and invertible, the Schur complement of P L (t) in P L+1 (t), namely

S(t) := P L+1 (t)\P L (t) = r T (t)r(t) -η T (t)P -1 L (t)η(t)
(5.28) is positive-definite (see Appendix C) and then invertible. Since r T (t)r(t) and η T (t)P -1 L (t)η(t) in (5.28) are all symmetric matrices, S(t) is thus symmetric as well. Therefore, the inverse of S(t) is symmetric positive-definite and it can then be decomposed via Cholesky decomposition [START_REF] Higham | Cholesky factorization[END_REF]] as S -1 (t) = U(t)U T (t) (5.29) where U(t) is a lower triangular matrix with positive diagonal entries. Use the block-wise inversion [START_REF] Zhang | The Schur complement and its applications[END_REF]], the inverse of P L+1 (t) can be obtained as 5.30) according to (5.26) and ( 5.30), we can then write Ω L+1 (t) as

P -1 L+1 (t) = P -1 L (t) 0 0 0 + -P -1 L (t)η(t) I U(t)U T (t) -η T (t)P -1 L (t) I (
Ω L+1 (t) = W T L+1 (t)P -1 L+1 W L+1 (t) = W T L (t) w T (t -τ L ) P -1 L (t) 0 0 0 + -P -1 L (t)η(t) I U(t)U T (t) -η T (t)P -1 L (t) I W L (t) w(t -τ L ) = W T L (t)P -1 L (t)W L (t) + D(t)D T (t) = Ω L (t) + D(t)D T (t) (5.31) with D(t) = -W T L (t)P -1 L (t)η(t) + w T (t -τ L ) U(t) (5.32)
then, the Sherman-Morrison-Woodbury identity [START_REF] Press | Numerical recipes[END_REF] yields 5.33) It can be clearly seen from ( 5.33) that state estimation error variance of the proposed CD-LTV-FMO, i.e. Ω -1 L (t), satisfies the matrix riccati equation with respect to window length L. Based on the properties of RE stated in De Souza et al. [1986], the variance will asymptotically converge to a minimal value as the window length L increases. That means when L increases to a certain size, the estimation error will no longer decrease, which seems reasonable since the amount of information brought by increasing the window length is not significant any more. Hence, L max can be decided either by the maximum tolerance on the dispersion of estimation or by the L chosen when the following condition meets:

Ω -1 L+1 (t) = Ω L (t) + D(t)D T (t) -1 = Ω -1 L (t) -Ω -1 L (t)D(t) I + D T (t)Ω -1 L (t)D(t) -1 D T (t)Ω -1 L (t) (
Ω -1 L+1 (t) ≈ Ω -1 L (t)
.

Numerical Example

Consider a continuous-time LTV system [START_REF] Kailath | Linear systems[END_REF]], described by a state-space model as

ẋ(t) = A(t)x(t) + B(t)u(t) + ω(t) y(k) = Cx(t k ) + v(k)
with the other parameters settings are as follows:

A(t) = -4/t -2/t 2 1 0 ; B(t) = 1 0 ; Q = σ p σ T p = 1 × 10 -5 0 0 1 × 10 -5 ; u(t) = sin(20t); C(t) = (0 1); R = σ 2 m = 2 × 10 -5 2 .
Before all the analyses, we firstly determine the state transition matrix as:

Φ(t, θ ) = -θ 2 t -2 + 2θ 3 t -3 -2θt -2 + 2θ 2 t -3 θ 2 t -1 -θ 3 t -2 2θt -1 -θ 2 t -2
which one can check it by calculating dΦ(t,θ ) dt and verifying that it is certainly equal to A(t)Φ(t, θ ).

Influence of window length L

It can be seen that the proposed CD-LTV-FMO (5.8) depends on the window length L. The influence of L is going to analyses through different aspects with the given example.

Variance of estimation error with respect to L

As theoretically proved in (5.15), the estimation variance of the proposed CD-LTV-FMO is Ω -1 L (t), it can been clearly seen from Figure 5.2(a) that at any time instant t, Ω -1 L (t) converges asymptotically while window length L augments. Now let us note Σ L (t) = (x(t)x(t))(x(t)x(t)) T as the practical estimation variance at each instant t calculated by the numerical simulation, the same convergence with respect to window length L can then be found in Figure 5.2(b). Comparing to 5.2(a), the "roughness" of Σ L (t) at beginning (when L = 2) is due to the numerical computation in the presence of noises. Take t = 0.3s as an example, a detailed comparison between Σ L (t) and Ω -1 L (t) with respect to window length L is shown in Figure 5.2(c). In the steady phase, Σ L (t) = Ω -1 L (t) is validated, which means Ω -1 L (t) is surely validated as the variance of the proposed CD-LTV-FMO. 5.4, the influence of window length L on the state estimation performance can be examined by comparing with true state. The longer the window length L is, the better the estimation is. Meanwhile, as usual, the window length L apparently has more impact on the state component without measurement since the measured state has enough amount of significant information and the unmeasured one mainly obtains the information by augmenting window length L. As a matter of fact, the proposed CD-LTV-FMO can be seen as a Kalman filter when the estimate variance no longer significantly changes as the window length L gets longer, namely

Ω -1 L+1 (t) ≈ Ω -1 L (t).
In this case, thanks to the property of KF, the proposed CD-LTV-FMO is again validated as "BLUE".

Unbiased estimation performance via MC simulations

In order to evaluate the state estimation performance in stochastic case, we employ Monte Carlo (MC) simulations to check the unbiased property of the proposed observer, which is defined as

x(t) E [ xL (t)] = 1 N mc N mc ∑ i=1 x(i) L
where N mc is the number of MC runs. Let us set window length L = 6 and N mc = 500. It can be seen by Figure 5.6 and Figure 5.7 that after performing 500 runs of MC simulations, the estimation expectation x2 (t) is well rebuilt at the true value of state. Meanwhile, the upper and lower bounds of the sate estimation by the presented observer are also established in Figure 5.7, which are very close to the true state and vary in a quite small range. The RMSE (root-mean-square error) is further used as another measure of the differences between estimations and true value, which is defined as RMSE =

1

N mc ∑ N mc i=1 ( x(i) L -x) 2 .
According to different window length and different numbers of MC runs, we establish six simulation scenarios, as shown in Table .5.1.

N mc = 100 (N1) N mc = 500 (N2) L = 2 (L1) "L1N1" "L1N2" L = 6 (L2) "L2N1" "L2N2" L = 10 (L3) "L3N1" "L3N2"
The RMSEs for different scenarios are depicted in Figure 5.8 and Figure 5.9. Through the examination of figure, the estimation performance may be analyzed in two aspects. On the one hand, seeing that there are three pairs of curves in the figure, the conclusion of "the longer the window length is, the better the estimations are" can be drawn immediately by comparing the RMSEs of the three pairs. This is by the way another evidence for the influence of window length.

FMO Design for CD-NLTV Systems (CD-NLTV-FMO)

On the other hand, examine the two different RMSEs of each pair, it can be obviously seen that how the increase of MC simulation times improves the state estimation accuracy, which appears to be reasonable since the more MC simulation times, the less uncertainty caused by process and measurement noises. Based on what we have designed above, the aim in this section is to extend the proposed CD-LTV-FMO to a class of continuous-discrete nonlinear time-varying (CD-NLTV) systems.

CD-NLTV-FMO formulation

The considered CD nonlinear time-varying (CD-NLTV) systems are described as

ẋ(t) = A(t)x(t) + B(t)u(t) + f (x,t) + ω(t) (5.34a) y(k) = Cx(t k ) + v(k) (5.34b)
where x ∈ R n , y ∈ R p and u ∈ R q are continuous state vector, discrete measurement vector and continuous input vector, respectively. The sampling period of measurement is T s , i.e. ∃k ∈ N | t k = k × T s . In the meantime, A(•) ∈ R n×n , B(•) ∈ R n×q and C(•) ∈ R p×n are known matrices and are functions of time. The Gaussian measurement noise v and Gaussian process noise ω are independent with the properties described in (5.2). The nonlinearity f (x,t) is at least locally Lipschitz with respect to x, i.e.

∥ f (a,t) -f (b,t)∥ ≤ κ∥a -b∥ (κ is Lipschtiz constant)
The continuous-discrete finite memory observer for NLTV systems, i.e. CD-NLTV-FMO, is then designed by the similar manner as the linear time-varying case based on Assumption 5.1, Assumption 5.2, Definition 5.1 and Remark 5.1.

Consider the time window [tτ L-1 ,t], use the state transition matrix Φ(t,tτ i ) to express the state mapping between x(t) and x(tτ i ) as:

x(t) = Φ(t,t -τ i )x(t -τ i ) + t t-τ i Φ(t, θ )B(θ )u(θ ) dθ + t t-τ i Φ(t, θ )ω(θ ) dθ + t t-τ i Φ(t, θ ) f (x(θ ), θ ) dθ (5.35)
both sides of (5.35) are left multiplied by a coefficient matrix C(tτ i )Φ -1 (t,tτ i ), then take into consideration the measurement equation ( 5.34b) at time instant tτ i and rearrange with the properties of Φ(•, •) in Remark 5.1, we have

C(t -τ i )Φ(t -τ i ,t)x(t) = y(t -τ i ) + α [t-τ i ,t] -β [t-τ i ,t] + γ [t-τ i ,t] (5.36) with α [t-τ i ,t] = t t-τ i C(t -τ i )Φ(t -τ i , θ )B(θ )u(θ ) dθ β [t-τ i ,t] = v(t -τ i ) - t t-τ i C(t -τ i )Φ(t -τ i , θ )ω(θ ) dθ γ [t-τ i ,t] = t t-τ i C(t -τ i )Φ(t -τ i , θ ) f (x(θ ), θ ) dθ
Write equation (5.36) for every instant in time window [tτ L-1 ,t], a concatenated measurements can be denoted as the following linear equation: 5.37) where

Y L (t) = W L (t)x(t) +V L (t) (
Y L (t) =       y(t -τ 0 ) + α [t-τ 0 ,t] + γ [t-τ 0 ,t] y(t -τ 1 ) + α [t-τ 1 ,t] + γ [t-τ 1 ,t] . . . y(t -τ L-1 ) + α [t-τ L-1 ,t] + γ [t-τ L-1 ,t]       ; W L (t) =      
C(tτ 0 )Φ(tτ 0 ,t) C(tτ 1 )Φ(tτ 1 ,t) . . .

C(t -τ L-1 )Φ(t -τ L-1 ,t)       ; V L (t) =       β [t-τ 0 ,t] β [t-τ 1 ,t]
. . .

β [t-τ L-1 ,t]       .
It can be seen that the noise term V L (t) is the same as the LTV case, so the variance matrix P L (t) remains the same as in (5.7).the least-squares solution of (5.6) Note that the calculation method for the integral term γ[t-τ i ,t] is the same as the one proposed in subsection 3.2.2.2 of Chapter 3. As a matter of fact, the CD-NLTV-FMO design idea can be referred as a combination of the CD-FMO designed in Chapter 3 and the FMO desgned for CD-LTV systems in section 5.2.

Numerical simulation

Consider adding a nonlinearity term f (x,t) to the numerical example in section 5.5 while the other matrices and parameters remain unchanged: f (x,t) = 0.1t sin(x 1 )

Influence of window length L

The same as the CD-FMO design for nonlinear systems in Chapter 3, the theoretical estimate variance of the nonlinear time-varying (NLTV) systems (5.34) cannot be currently given in this thesis. However, the practical estimate variance Σ L (t) = (x(t)x(t))(x(t)x(t)) T numerically calculated by simulations at each instant t is given in Figure 5.10, from which we can still see the convergence of variance influenced by the size of window L. Note that we obtain the same convergence results as for the LTV systems case in the previous section. In next subsection, we are going to compare the estimate variance with the nonlinear Kalman filter, more specifically first-order extended Kalman filter (EKF), by the solution of the Riccati equation.

Comparison with the estimate variance of EKF

According to [START_REF] Simon | Optimal state estimation[END_REF], the Riccati equations stated in (5.22) and (5.23) becomes as follows for the first-order EKF estimate variance P(t) of nonlinear time-varying systems (NLTV) (5.34a) where

F(t) = ∂ g ∂ x x(t) = A(t) + ∂ f ∂ x x(t)
The estimate variance Σ L (t) of the proposed CD-NLTV-FMO is compared to both the first-order CT-EKF and CD-EKF through the solutions of Riccati equation in Figure 5.12. The same performance can be seen as in Figure 5.5. However, the estimation accuracy for NLTV systems is not as great as LTV systems due to the approximation of the nonlinearity.

Conclusion

This chapter is dedicated to design a robust observer for continuous-discrete linear timevarying systems in noisy environment. The proposed approach has been proved unbiased and has the smallest estimation variance compared to other linear estimators. Furthermore, the presented observer also benefits from the finite time convergence and no initial value problem, which turns out to be an important consideration for practical applications. Like other moving horizon methods, the influence of window length has also been analysed both theoretically and practically, which a strategy of choosing the window length by dispersion tolerance are also provided. Monte Carlo simulations are used to examine and verify the performance of the proposed method, which appears to be robust and effective. It may be worth noting here that the presented observer framework also works well when sampling period is not a constant. At the last part, the proposed observer has been extended to the nonlinear time-varying systems through a brief demonstration and a numerical example is employed to test the performance of CD-FMO-NLTV.

Chapter 6

Conclusions and Future Research

This thesis set out to design a nonlinear observer for a class of dynamical systems, which are modeled by continuous-time nonlinear ordinary differential equations and are measured at discrete-time instant, and to use the proposed observer to perform fault diagnosis. A graphical representation of the structure of this thesis is given in Figure 6.1. In this final chapter, we will review the result conclusions of the work in this thesis, as well as discuss directions for future research.

Conclusions

The main conclusions of this thesis, which are drawn based on the obtained results, are summarized as follows in two aspects:

Nonlinear observer design

• Time-invariant: inspired by the existing optimization-based nonlinear observer design methods and the observer-based fault diagnosis approaches illustrated in Chapter 2, we have first devoted Chapter 3 to develop a continuous-discrete finite memory observer (CD-FMO) for a class of nonlinear dynamical time-invariant systems the presence of process and measurement noise, where a one-step prediction algorithm incorporated with an iterative-update scheme is performed to solve the integral problem caused by system nonlinearity, and an analysis of the unbiased estimation property has been proved in both deterministic and stochastic case.

• Unknown input: since there still remains a lack of study on nonlinear continuousdiscrete observer design with the presence of unknown input, process and measurement noises at the same time, which serves as the motivation of the presented work in Chapter 4, where a CD-UI-FMO is constructed based upon an augmented model aiming to simultaneously estimate system state and UI of a noisy process with noisy measurements. The unbiased property of CD-FMO remains for the CD-UI-FMO case. • Incomplete or missing measurements: missing measurements are a common occurrence in practical physical system and can cause a significant effect on system behavior. The proposed nonlinear finite memory observer is supposed to be robust with respect to the measurement missing thanks to the role of term "τ i " in the formulation structure.

It will be analyzed and validated in detail in the future.

• Extension to general nonlinear systems: since we have already built the observer for a class of nonlinear systems (3.1) and for time-varying nonlinear systems (5.34), it might be possible to extend the presented method to general nonlinear systems under the form x = f (x, u,t) by using the first-order or second-order Taylor expansion around x = x(t) at each instant t.

Fault diagnosis aspect

• Detection sensitivity analyses with respect to different fault and noises: in order to analyze the diagnostic performance according to different parameter configuration, it is then necessary to study the sensibility of the chosen fault indicator (for instance, the generated "residual" is used as fault indicator in this thesis) vis-a-vis different fault and noises. This type of study can help us theoretically analyze in advance the minimum fault detection amplitude with the known configuration parameters.

• Integrate with the qualitative model-based diagnosis methods: qualitative modelbased diagnosis methods sometimes cannot give a correct diagnosis of failures. For example, fault tree (FT) will give false diagnosis when the true failure component is not the one whose failure rate is the highest. Since the failure rate is based on historical data and the current scenario may be more complex. To ensure more correct diagnosis results, one has to use other real-time information, where the proposed observer might give some help with it. It will make a significant improvement if the proposed observer can share information with the qualitative model-based methods.

This above non-exhaustive list forms our future research directions.

E ( x(t) -x(t)) = 0 t ∈ [L × T s , +∞) (B.1)
First of all, let K i with i = 0, 1, • • • , L -1 represents the i + 1 th column of the inverse of variance matrix P -1 in (3.8) as follows

P -1 [K 0 , K 1 , • • • , K L-1 ] (B.2)
Recall the proposed CD-FMO here as in (3.9) and (3.10)

x(t) = arg min J(x) = W T L P -1 W L -1 W ∑ i=0 e -A T τ i C T K i Ce -Aτ i x(t) + v(tτ i )β t-τ i ,tγ t-τ i ,t + βt-τ i ,t

1 + 2 + 3 (B.9)

with 1 = Ω -1 L L-1
∑ i=0 e -A T τ i C T K i Ce -Aτ i x(t)

(B.6) = x(t) (B.10a) 2 = Ω -1 L L-1 ∑ i=0 e -A T τ i C T K i [v(t -τ i ) -γ t-τ i ,t ] → E 2 = 0 (B.10b) 3 = Ω -1 L L-1
∑ i=0 e -A T τ i C T K i βt-τ i ,tβ t-τ i ,t (B.10c)

According to (B.9) and (B.10), the following mathematical expectation is given

E ( x(t) -x(t)) = E 2 + 3 = E 3 = E Ω -1 L L-1 ∑ i=0 e -A T τ i C T K i βt-τ i ,t -β t-τ i ,t = Ω -1 L L-1 ∑ i=0 e -A T τ i C T K i E βt-τ i ,t -β t-τ i ,t (
E ( x(t) -x(t)) = Ω -1 L L-1 ∑ i=0 e -A T τ i C T K i E βt-τ i ,t -β t-τ i ,t = Ω -1 L L-1 ∑ i=0 e -A T τ i C T K i E t t-τ i Ce A(t-τ i -θ ) [Φ( x(θ )) -Φ(x(θ ))] dθ = Ω -1 L L-1 ∑ i=0 e -A T τ i C T K i t t-τ i Ce A(t-τ i -θ ) E [Φ( x(θ )) -Φ(x(θ ))] dθ = Ω -1 L L-1 ∑ i=0 e -A T τ i C T K i t t-τ i Ce A(t-τ i -θ ) • dΦ dx x • E ( x(θ ) -x(θ )) dθ (B.14)
Note that t ∈ [L × T s , ∞), let us take the norm for both side of (B.14), we have

∥E ( x(t) -x(t))∥ = Ω -1 L L-1 ∑ i=0 e -A T τ i C T K i t t-τ i Ce A(t-τ i -θ ) • dΦ dx x • E ( x(θ ) -x(θ )) dθ ≤ L-1 ∑ i=0 Ω -1 L e -A T τ i C T K i t t-τ i Ce A(t-τ i -θ ) • dΦ dx x • E ( x(θ ) -x(θ )) dθ ≤ L-1 ∑ i=0 t t-τ i Ω -1 L e -A T τ i C T K i Ce A(t-τ i -θ ) • dΦ dx x • E ( x(θ ) -x(θ )) dθ ≤ L t t-τ L-1 Ω -1 L e -A T τ L-1 C T K i Ce A(t-τ L-1 -θ ) • dΦ dx x • E ( x(θ ) -x(θ )) dθ ≤ L t t-τ L-1 Ω -1 L e -A T τ L-1 C T K i Ce A(t-τ L-1 -θ ) • dΦ dx x ∥E ( x(θ ) -x(θ ))∥ dθ ≤ L t L×T s Ω -1 L e -A T τ L-1 C T K i Ce A(t-τ L-1 -θ ) •

Figure 2 . 2 :

 22 Figure 2.2: Distinctions of different faults types

Figure 2

 2 Figure 2.3: Generalized architecture of FDI by analytical redundancy[START_REF] Isermann | Process fault detection based on modeling and estimation methods-a survey[END_REF]

3. 4

 4 Application to Fault Diagnosis . . . . . . . . . . . . . . . . . . . . . . 45 3.4.1 Fault detection . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.4.2 Fault isolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Figure 3

 3 Figure 3.2: Convergence of covariance Σ x with window length L increases

x 4 Figure 3 . 3 :

 433 Figure 3.3: State estimation x with different window length L

Figure 3 . 4 :x 4 Figure 3 . 5 :

 34435 Figure 3.4: Upper (lower) bound and mean of x1 between N mc = 100 and N mc = 500

  Figure 3.6: Robustness analysis in different measurement noise scenarios

  0.15; fault period [T fs , T fe ] = [0.5s, 1.0s]. • A bias on y 2 (F2): I = I y 2 = [0 1 0] T ; ∆ = ∆ y 2 = 0.15; fault period [T fs , T fe ] = [2.0s, 2.5s]. • A bias on y 3 (F3): I = I y 3 = [0 0 1] T ; ∆ = ∆ y 3 = -0.15; fault period [T fs , T fe ] = [3.5s, 4.0s].2. Actuator fault: the actuator fault is modeled as: u f = Π(t, T fs , T fe )(1ρ)u where ρ ∈ [0, 1] describes control loss level. ρ = 0 means there is no actuator fault whereas ρ = 1 signifies that the control is completely lost.• Actuator fault (F4): ρ = 0.6; [T fs , T fe ] =[5.0s, 5.5s].

Figure 3 . 7 :Figure 3

 373 Figure 3.7: Residual r and the CUSUM of r

Assumption 4 . 1 .

 41 The nonlinear function Φ(X) satisfies the Lipschitz condition with Lipschitz constant κ as follow: ∥Φ(a) -Φ(b)∥ ≤ κ∥a -b∥ Assumption 4.2. The dynamic of unknown input ξ (t) is given as:

Assumption 4 . 3 .

 43 The pair (A,C) is observable.

Figure 4 . 1 :

 41 Figure 4.1: Convergence of the estimation error as window length L increasing

  Figure 4.2 -Figure 4.4 in the presence of the unknown input. By examining the figures, we can see that the unknown input injected during

Figure 4 .

 4 Figure 4.2: Residual r 1 comparison between nominal model and augmented model

Figure 4 . 3 :

 43 Figure 4.3: Residual r 2 comparison between nominal model and augmented model

Figure 4 . 4 :

 44 Figure 4.4: Residual r 3 comparison between nominal model and augmented model

Figure 4 . 5 :Figure 4

 454 Figure 4.5: State estimations of x 2 through different window length L

Figure 4 . 7 :Figure 4 . 8 :Figure 4 . 9 :

 474849 Figure 4.7: State estimations of x 1 through different window length L

Figure 4 .Figure 4 .Figure 4 .

 444 Figure 4.12: Expectation x3 and upper and lower bound of x3 with N mc =

Figure 4 .

 4 Figure 4.16: RMSE of x via different noise scenarios

Figure 4 .

 4 Figure 4.17: Robustness analyses of UI ξ via different noise scenarios

Figure 4 .

 4 Figure 4.18: State estimation with correlated process noise

  Figure 4.20: Actuator fault estimate by CD-UI-FMO

Figure 4 .

 4 Figure 4.21: Actuator fault detection by CUSUM

Figure 4 .Figure 4 .

 44 Figure 4.22: EWMA detection alarm of actuator fault ξ = -0.5 sin(t)

Figure 4 .

 4 Figure 4.24: EWMA detection alarm of ramp-shape actuator fault

Figure 4 .

 4 Figure 4.25: Minimal amplitude estimate by EWMA

Figure 4 .Figure 4 .

 44 Figure 4.26: EWMA detection alarm of minimum amplitude (λ = 0.2)

Figure 4 .

 4 Figure 4.28: EWMA detection alarm of minimum amplitude (λ = 0.1)

Figure 4 .Figure 4 . 1 4. 4

 4414 Figure 4.29: Detection alarm of ξ * = 0.04 with λ = 0.2

1 .

 1 MSF1: Sensor fault ∆ y 1 = 0.01 (a bias on y 1 ) & Actuator fault ξ = -0.4; 2. MSF2: Sensor fault ∆ y 2 = 0.01 (a bias on y 2 ) & Actuator fault ξ = -0.4; 3. MSF3: Sensor fault ∆ y 3 = -0.01 (a bias on y 3 ) & Actuator fault ξ = -0.4; 4. SF1: Sensor fault ∆ y 1 = 0.01 (a bias on y 1 ); 5. SF2: Sensor fault ∆ y 2 = 0.01 (a bias on y 2 ) ; 6. SF3: Sensor fault ∆ y 3 = -0.01 (a bias on y 3 ); 7. SF4: Actuator fault ξ = -0.4.

Figure 4 .

 4 31 to Figure 4.37 and then summarized in Table.

  4.3.

Figure 4 .

 4 Figure 4.31: Residual r under the presence of MSF1

Figure 4 .Figure 4 .

 44 Figure 4.32: Residual r under the presence of MSF2

Figure 4 .

 4 Figure 4.34: Residual r under the presence of SF1

Figure 4 .Figure 4 .

 44 Figure 4.35: Residual r under the presence of SF2

Figure 4 .

 4 Figure 4.37: Residual r under the presence of SF4

Figure 5 . 1 :

 51 Figure 5.1: Illustration of state estimations x(t) with different window length L

Figure 5 . 2 :

 52 Figure 5.2: Convergence of variance with different window length L

Figure 5 . 4 :

 54 Figure 5.3: x1 over window length L

4 'Figure 5 . 5 :

 455 Figure 5.5: Estimate variance comparison with respect to KF by Riccati equation

Figure 5 . 7 :

 57 Figure 5.7: Expectation and bounds (upper / lower) of x 2 (LTV)

Figure 5 Figure 5

 55 Figure 5.8: RMSE of x1 with different L and different N mc (LTV)

Figure 5 .Figure 5 .

 55 Figure 5.10: Practical variance Σ L (t) of CD-NLTV-FMO with respect to L

5. 6 FMO

 6 Design for CD-NLTV Systems (CD-NLTV-FMO)

4 'Figure 5 .Figure 5 .Figure 5 .Figure 5 .

 45555 Figure 5.12: Estimate variance comparison with respect to EKF by Riccati equation

Figure 6 . 1 :

 61 Figure 6.1: Conclusions and structure of the chapters in the thesis

(

  T τ i C T K i y(tτ i ) + α t-τ i ,t + βt-τ i ,t T τ i C T K i Ce

  Proof of the unbiased estimation property in stochastic caseBefore we go any further, let us write the first-order Taylor series of the nonlinear term Φ( x) at the point x = x as followsΦ( x) ≈ Φ(x) + dΦ dx x • ( xx) (B.12) since x is not stochastic, it is then straightforward that E [Φ( x) -Φ(x)] = dΦ dx x • E ( xx) (B.13) write (B.11) with (B.4a), (B.8a) and (B.13) as

  Observer-based fault diagnosis methods . . . . . . . . . . . . . . 2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

	2.5.1 Basic definitions / notations and general diagnostic procedure . .
	2.5.2 Generalized fault diagnosis methods using analytical redundancy
	2.5.3

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Observability of Dynamical Systems . . . . . . . . . . . . . . . . . . . 2.2.1 Observability problem and definitions . . . . . . . . . . . . . . . 2.2.2 Observability rank condition . . . . . . . . . . . . . . . . . . . . 2.3 Some Optimization-based Nonlinear Observers . . . . . . . . . . . . . 2.3.1 Moving horizon observers (MHOs) . . . . . . . . . . . . . . . . 2.3.2 Nonlinear observers based on Newton's method . . . . . . . . . . 2.3.3 Nonlinear observers based on Levenberg-Marquardt algorithm . . 2.4 Nonlinear Observers with Continuous-Discrete Model . . . . . . . . . 2.5 Fault Diagnosis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

  "If we knew what it was we were doing, it would not be called research, would it?" -Albert Einstein

	Contents of chapter

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.2 Nonlinear CD-FMO Design . . . . . . . . . . . . . . . . . . . . . . . . 29 3.2.1 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.2.2 Formulation of CD-FMO . . . . . . . . . . . . . . . . . . . . . . 30 3.2.2.1 Analytical calculation of the integral α t-τ i ,t . . . . . . . 32 3.2.2.2 Iterative algorithm for solving the integral βt-τ i ,t . . . . 33 3.2.3 Estimation property of CD-FMO . . . . . . . . . . . . . . . . . . 34 3.2.4 Analytical choice of the window length L . . . . . . . . . . . . . 37 3.2.4.1 Minimal length L min . . . . . . . . . . . . . . . . . . . 37 3.2.4.2 Maximal length L max . . . . . . . . . . . . . . . . . . 38 3.3 Illustrative Example: a Single-link Robot . . . . . . . . . . . . . . . . 38 3.3.1 Selection of the window length L . . . . . . . . . . . . . . . . . 39 3.3.2 Numerical integration approximation error analysis . . . . . . . . 40 3.3.3 State estimation performance . . . . . . . . . . . . . . . . . . . . 41 3.3.3.1 Unbiased estimation property analysis in stochastic case 41 3.3.3.2 Robustness analysis with respect to measurement noise 44

  3d) here δ (•) is Dirac delta function and δ i, j is Kronecker delta function. It should be noted that the continuous-discrete systems like (3.1) naturally exist when continuous process are measured via digital sensors. Without loss of generality, we are going to present a nonlinear observer design where the estimation instant is synchronized 1 with the measurements instant since it is exactly what is needed under the background of diagnosis. The proposed CD-FMO will be detailed in next subsection. Before we start, we introduce the following remark first:

5 :

 5 Updating by iteration: 6: for each iteration m = 1 : N max do Calculate the entire integral βt-τ i ,t via (3.13). Moving the time window: reset t = t + T s , and go back to step 1 for next time instant.

	7:	Calculate g( x * (t)) with the x * (t) predicted in step 3.
	8:	Calculate βt-τ 1 ,t in (3.14b) by Newton Cotes formulas.
	9:
	10:	Update x * (t) to x(t) via (3.10):
	11:
	12:	Terminate the iteration, go forward to step 17.
	13:	else
	14:
	15:	end if
	16: end for
	17:

if x(t)x * (t) ≤ ε or m ≥ N max then Reset m = m + 1, x * (t) = x(t), return to step 7.

Table 3 .

 3 2: Different scenarios of SD for measurement noise Measurement noise settings CD-FMO parameter settings Scenario 1 SD

Table 3 .

 3 3: Fault signature for different faults

	Different Faults	Start Instant T fs (vertical dash line)	End Instant T fe (vertical solid line)	Signature Detect Instant T d r 1 r 2 r 3
	F1	0.5s	1.0s	1 0 1	0.5s
	F2	2.0s	2.5s	0 1 1	2.0s
	F3	3.5s	4.0s	1 0 1	3.5s
	F4	5.0s	5.5s	1 0 1	5.05s

Table 3 .

 3 4: Fault signature by CD-FMO 1 and CD-FMO 2

		Fault Signature
	Different Faults	CD-FMO 1

Table 4

 4

	.2: Different scenarios of process noise and measurement noise
	Noises scenarios of actual system	CD-UI-FMO parameter settings
	Scenario 0 (S0) SD

m = σ m ; SD p = σ p

Table 4 .

 4 3: Multiple Simultaneous Faults Diagnosis by CD-UI-FMO

	Fault Signature (Pattern)
	Scenarios

  Assumption 5.2. The pair (A(•),C(•)) of CD-LTV systems (5.1) is observable. if and only if there exists a finite t f > t 0 such that O ob (t 0 ,t f ) ∈ R n×n , defined by O ob (t 0 ,t f ) := is nonsingular. Here, Φ(θ ,t 0 ) ∈ R n×n is the state transition matrix from t 0 to θ with θ ∈ [t 0 ,t f ], which satisfies the following differential equation:

	5.2 FMO Design for CD-LTV Systems (CD-LTV-FMO)
	Meanwhile, Definition 2.8 and Remark 2.3 in Chapter 2 are recalled here:
	Definition 5.1 (Observability Gramian). The pair (A(•),C(•)) is said to be observable at
	time t 0 t f t 0	Φ T (θ ,t 0 )C T (θ )C(θ )Φ(θ ,t 0 ) dθ

Table 5 .

 5 1: MC simulation scenarios with respect to L and N mc

  , noted asẋ(t) = A(t)x(t) + B(t)u(t) + f (x,t) + ω(t) g(x, u,t, ω) CT-RE: Ṗ(t) = F(t)P(t) + P(t)F T (t) -P(t)C T (t)R -1 C(t)P(t) + Q(t) CD-RE: Ṗ(t) = F(t)P(t) + P(t)F T (t) + Q(t)

  τ 0 ) + α t-τ 0 ,t + βt-τ 0 ,t y(tτ 1 ) + α t-τ 1 ,t + βt-τ 1 ,t . . . y(tτ L-1 ) + α t-τ L-1 ,t + βt-τ L-1 ,t

					T L P -1	ŶL	(B.3)
	where		y(t 			Ce -Aτ 0	
	ŶL =	    		    	; W L =	     Ce -Aτ L-1 Ce -Aτ 1 . . .	    
	with
			βt-τ i ,t =	t t-τ i

  τ i ) = Ce -Aτ i x(t) + v(tτ i )α t-τ i ,tβ t-τ i ,tγ t-τ i ,t

			-Aτ i	(B.6)
	from (3.5) and (3.6) we have
	y(t (B.7)
	where
	β t-τ i ,t =	t-τ i t	Ce A(t-τ i -θ ) Φ(x(θ )) dθ	(B.8a)
	γ t-τ i ,t =	t-τ i t	Ce A(t-τ

i -θ ) Gω(θ ) dθ (B.8b) replace y(tτ i ) in (B.5) by (B.7), we obtain x(t) = Ω -1 L L-1

https://tc.ifac-control.org/6/4/terminology/terminology-in-the-area-of-fault-management

The proposed nonlinear CD-FMO also works well for the case that the estimation instant is not synchronized with measurement instant.

Acknowledgements

The work presented in this PhD thesis was supported by the China Scholarship Council (CSC), and has been carried out within Laboratory PRISME (Pluridisciplinaire de Recherche

. My gratitude extends to Mrs Gabriela Iuliana BARA and Mrs Samira EL YACOUBI for their kind acceptances to review my work. Thanks to Mr.

Chapter 5

It may be worth to compare the above unbiased proof for time-varying systems with the proof of Theorem 3.1 in Chapter 3 and the one in Appendix B for nonlinear time-invariant systems, which appears to base on the same "spirit" through different "manners".

Corollary 5.1. From Theorem 5.1, it can be easily conclude that in the case of noise-free (without process noise and measurement noise) and fault-free, the proposed observer has the following estimation property :

Remark 5.2. According to Theorem 5.1 and Corollary 5.1, it may be worth noting that the performance of the proposed observer doesn't depend on the initial condition. The fact of no initial value problem (IVP) gives the presented observer another advantage for real applications.

Note that the proposed observer (5.8) is the least square solution of linear equation (5.6), which is the linear combination of Y L (t). We have also proved that it is an unbiased linear estimator in both deterministic and stochastic case. Now we are going to give in the next subsection the variance property of the estimation error.

Smallest variance proof

Theorem 5.2. The proposed observer has the smallest variance (least dispersion) among all the other unbiased linear estimator. That is to say, if x * (t) is another unbiased linear estimation of linear equation (5.6)

Proof. Since x * (t) is another unbiased linear estimation of (5.6), then let x *

where A (t) ∈ R n×pL represents the linear combination. ε * (t) is then calculated as

since E[V L (t)] = 0 and x * (t) is another unbiased linear estimation, i.e. E(ε * (t)) = 0, then

(CT-RE), will converge to Cramér-Rao Lower Bound (CRLB).

CT-RE:

In the meantime, for continuous-discrete LTV systems like (5.1), we should not include the R term in (5.22) because we are integrating P between measurement times, during which we do not have any measurements. Another way of looking at it is that in between measurement times we have measurements with infinite covariance (R = ∞), so the R-related term on the right side of (5.22) goes to zero. This gives us the following CD-RE for the time-update variance equation of CD Kalman filter [START_REF] Simon | Optimal state estimation[END_REF]]:

The variance comparison between the proposed CD-LTV-FMO and KF will be later given in the simulation section by an numerical example of LTV system.

Optimal Window Length Selection Strategy

There follows the section in which we are going to provide a window length selection strategy by showing how to determine "the minimal length" L min and "the maximum length" L max , as been also used in [START_REF] Graton | Finite memory observers for linear timevarying systems: Theory and diagnosis applications[END_REF].

First of all, let L min is chosen to guarantee the possibility that the proposed observer (5.8) exists, which depends whether the matrix multiplication term W T L (t)P -1 L (t)W L (t) in (5.8) is invertible. This means that each column of matrix W L (t) should be linearly independent. Nonetheless, this condition will always been satisfied thanks to Assumption 5.2. Therefore, L min should be chosen to make sure that Assumption 5.2 is validated.

Since the variance of state estimation error Var(ε L (t)) depends on L, we are therefore going to show the idea of selecting L max by studying how does L has an influence on Var(ε L (t)), the estimation performance indicator of the proposed observer. Let us note

where xL (t) and xL+1 (t), as illustrated in Figure 5.1, respectively represent the estimations with different window length, Chapter 6 Conclusions and Future Research • Time-varying: As for time-varying systems, an observer has been designed for both linear and nonlinear cases in Chapter 5, respectively called CD-LTV-FMO and CD-NLTV-FMO. A detailed proofs are provided for linear case while the formulation has been further extended to nonlinear case and illustrated by a given example.

Fault diagnosis

• Detection,isolation and estimation: for the application purpose, residuals generated by the proposed CD-FMO are employed in Chapter 3 to implement fault detection cooperated with the CUSUM algorithm, while a bank of CD-FMOs is adopted to realize fault isolation for different sensor and actuator faults of a considered nonlinear robotic systems. In the meantime, the proposed CD-UI-FMO structure in Chapter 4 has been applied to actuator fault estimation when the actuator fault is modeled as an unknown input of systems.

• Multiple simultaneous faults detection and identification: thanks to the unknown input estimation ability of CD-UI-FMO, fault detection and identification have been performed in Chapter 4 as well for multiple simultaneous sensor and actuator faults as the actuator fault has been estimated and compensated by CD-UI-FMO.

The above mentioned conclusions have been verified and validated by numerical simulations in this thesis. The provided results showed the good performance on both state estimation and fault diagnosis, as well as efficiency and robustness with respect to diverse noises.

Future Research

Even though my PhD journey is going to end very soon, my research is still going on. There still remains much research to be done in the future as the succession and continuity of the work in this thesis. These future research directions are also going to be listed in two aspects as follows:

Observer design aspect

A note on notation

Three dots between delimiters (parenthesis, brackets, or braces) means that the quantity between the delimiters is the same as the quantity between the previous set of identical delimiters in the same equation. For example,

According to the definition of Variance and zero mean, i.e. E (V L ) = 0, we have

I

where the elements p i, j are indexed by 0 ≤ i ≤ L -1 and 0 ≤ j ≤ L -1. Using the noise properties of w and v in (3.3), we know that p i, j satisfy

Note that term 4 in the above equation needs to be deduced. In order to derived 4 , the following corollary of Fubini's theorem is introduced first

II

Since the Dirac delta function is even, then δ (θ 1θ 2 ) = δ (θ 2θ 1 ). Furthermore, the shifting property of Dirac delta function is recalled as follows:

Corollary 2 ( [START_REF] Hassani | Mathematical methods[END_REF]). For any real or complex valued continuous function f on R, the Dirac delta satisfies the shifting property

Then, let k min[i, j] = 0, 1, ..., L -1, Corollary 2 allows us to continue deriving (A.4) as follows

-τ k Ce As GQG T e A T s C T ds, based upon (A.1)-(A.5), we finally obtain

Note that block matrix P is symmetric, which is only logic since it is a variance matrix.

III

Appendix B

Proof of the unbiased estimation property of the CD-FMO in stochastic case

This appendix intends to give a detailed proof that the state estimation of the proposed CD-FMO (3.10) in Chapter 3 is unbiased in stochastic case, namely then, the Gronwall inequality [START_REF] Bellman | The stability of solutions of linear differential equations[END_REF]] yields

The proof is completed.

VIII

Appendix C

Proof of the "positive definiteness" for the Schur complement S(t) := P L+1 (t)\P L (t)

In order to prove S(t) = r T (t)r(t)η T (t)P -1 L (t)η(t) in (5.28) is positive-definite, let us firstly note P L+1 (t) in (5.26) as follows

then we need to prove that when M and A are positive-definite and invertible, the Schur complement of A in M, i.e. A\M = D -CA -1 B, are positive-definite.

Since M and A are positive-definite and invertible, then det(M) > 0 and det(A -1 ) > 0. Therefore, the following equation naturally holds

Take the determinant on both sides of (C.2) as

Since det(M) > 0 and det(A -1 ) > 0, we then obtain from (C.3) that

namely,

which proves that the Schur complement of P L (t), S(t), is positive-definite.

IX

Tingting ZHANG Abstract: The aim of this thesis is to design a nonlinear observer as a diagnostic tool for continuous-time nonlinear systems with discrete-time measurements. We begin with the study of some observability notions concerning the considered nonlinear systems, following by the presents of three typical optimization-based nonlinear observers and observer-based diagnostic methods. Inspired by the existing approaches, a finite memory observer is then designed for a class of nonlinear systems in the presence of both process and measurement noises in order to perform fault detection and isolation of sensor and actuator faults. In the second part, a nonlinear observer based on augmented model is then designed to simultaneously estimate both system states and unknown inputs. The robustness with respect to the diverse noises is studied, as well as the study of the minimum amplitude of fault for the detection. The EWMA algorithm was also introduced and analyzed for its performance in detection. Multiple simultaneous faults are also detected and identified in this part. At the end of this thesis, a finite memory observer is designed for the nonlinear time-varying systems on the basis of the fundamental synthesis for linear time-varying systems.