Mohamad " Food?"

Sabrine Pedro Denise

Geoffrey " Geof- Freypedia"

Sabrina Laura

Mots-clés

systèmes dynamiques non linéaires -systèmes affine en l'état -capteurs non linéaires -mesures discrètes -robustesse au bruit -observateurs non linéaires -observateurs à grand gainobservateurs interconnectés -filtre de Kalman -stabilité de Lyapunov -stabilité entrée-àétat -petit-gain -inégalités matricielles linéaires

Résumé

L'estimation d'état d'un système non linéaire est essentielle pour la réussite des objectifs importants tels que : la surveillance, l'identification et le contrôle. Les observateurs sont des algorithmes qui estiment l'état actuel en utilisant, entre autres informations, les mesures effectuées par des capteurs. Le problème de synthèse d'observateur pour les systèmes non linéaires est un sujet de recherche majeur traité depuis plusieurs décennies dans le domaine de la théorie du contrôle. Récemment, les recherches ont également porté sur la synthèse des observateurs pour des modèles de plus en plus réels, qui peuvent prendre en compte des perturbations, des capteurs non linéaires et des sorties discrètes. Dans ce contexte, le but de cette thèse concerne la synthèse d'observateurs robustes pour certaines classes de systèmes non linéaires. Dans ce manuscrit, nous distinguons trois parties principales.

La première partie porte sur l'analyse des systèmes affines en état, affectés par le bruit, et l'estimation de l'état via le filtre de Kalman à grand gain. Les propriétés de convergence de cet observateur sont fortement influencées par deux variables : le paramètre de réglage du gain de l'observateur et l'entrée du système. Nous présentons un nouvel algorithme d'optimisation, basé sur une analyse de Lyapunov, qui adapte ces deux variables en fonction des perturbations affectant la dynamique et la sortie du système. La nouveauté de cette approche est qu'elle fournit une méthode systématique de réglage du gain et de sélection d'entrée simultanément ce qui améliore l'estimation de l'état en présence de telles perturbations et évite l'utilisation des méthodes basées de type essais-erreurs.

La deuxième partie concerne le problème de "redesign" d'observateurs pour des systèmes non linéaires sous une forme générale dont les sorties sont transformées par des fonctions non linéaires. En effet, l'observateur risque alors de ne pas estimer correctement l'état du système si elle ne prend pas en compte les non-linéarités des capteurs. Nous présentons une refonte d'observateur qui consiste en l'interconnexion de l'observateur originel avec un estimateur de sortie basé sur une inversion dynamique, et nous démontrons sa convergence asymptotique via des résultats du petit-gain. Nous illustrons notre méthode en utilisant deux classes de systèmes non linéaires courant dans la littérature : les systèmes affines en l'état avec injection de sortie, et les systèmes avec non-linéarité sous la forme canonique.

Enfin, la troisième partie étend notre approche présentée pour les systèmes continus aux systèmes dont les sorties sont non seulement transformées mais également discrétisées dans le temps. Cette propriété ajoutée introduit un défi important ; nous implémentons les techniques de "sample-and-hold" qui mènent à un gain de l'observateur basé sur des inégalités matricielles linéaires. La principale caractéristique des méthodes proposées est la possibilité d'adapter un grand nombre d'observateurs de la littérature à des scénarii plus réalistes. En effet, les capteurs classiques utilisés dans les applications d'ingénierie sont souvent non linéaires ou discrets, alors qu'une hypothèse récurrente dans la conception d'observateurs est la linéarité ou la continuité de la sortie. servers -Kalman filter -Lyapunov stability -input-to-state stability -small-gain -linear matrix inequalities General introduction

Context

Every physical system is composed of different parts that follow a common mechanism. Therefore, partial information can sometimes be used in order to estimate what we ignore. This is the fundamental idea underlying the concept of observer for a system. Here, we consider models of physical systems that are dynamic or time dependant and that are governed by differential equations.

The model of a physical system is a rigorous description of the corresponding phenomena usually based on fundamental laws. Reliable modeling should be accurate; a sufficiently good representation of reality often requires a nonlinear structure and the consideration of unknown factors such as disturbances. The objectives of system modeling are diverse and include: phenomenon understanding, prediction, optimization and, crucially, control.

Modern control theory relies on the so-called state-space representation of a system, where we can distinguish the possibly multi-dimensional variables: state or system evolution, input or control, and output or measurements. In this context, the implementation of control laws can require full or partial state information from the output or sensor. Internal information is also needed for other important tasks, such as: identification, decision-making or monitoring. The sensor capabilities, however, are often limited by either physical, technological, economical or even safety constrains.

Observers are algorithms that estimate the current state by using the system structure and the available measurements. In this thesis, such algorithms are given by adjacent dynamical systems and the convergence of the state estimation is with respect to the Euclidean norm.

Closely related notions to that of observer are the observability or detectability of a system, which distinguish those systems admitting an observer design.

The theory of observers has become an extremely rich field that lies at the interface between pure mathematical disciplines and practical applications. Although the design of observers for linear systems is considered a solved problem, its nonlinear counterpart has been under

intensive research in the control community for several decades. As a consequence, even though currently there is no systematic design method, observers have been developed for specific classes of nonlinear systems and under different assumptions. An extremely desired feature of observers is their robustness, this is due to their practical nature. Indeed, their estimations should be satisfactory even in the presence of system disturbances. This makes the problem of robust observer design for nonlinear systems pivotal in the field of automatic control.

Motivations

It is not trivial how to adapt the design of a given observer to disturbances affecting the system. This problem is evident for uniformly observable systems and their high-gain observer, whose design depends on a tuning parameter that can amplify measurement noise; several optimal or adaptive strategies have been developed in the literature in order to tune this observer correctly. On the other hand, the system input plays a central role in the observability of nonlinear systems, and the design of inputs that render a nonlinear system observable is usually intricate. In the case of input design for state-affine or bilinear systems, researchers have developed algorithms to construct sufficiently regular inputs with prescribed properties.

Surprisingly, no literature seems to aim for the simultaneous tuning and input selection in observers with the objective of improving state estimation in the presence of noise.

The focus in observer design is frequently on the nonlinearity of the system dynamics, however, modern sensors are constantly becoming more complex and output linearity might not match reality. This rises the question whether known observer designs can be adapted to an output nonlinearity. Similarly, we can also consider the case of discrete measurements, which cannot always be taken close enough in time; it is well-known that this can affect considerably the properties of an observer in the nonlinear case.

Contributions and thesis outline

The main contribution of this thesis is the design of three new observers for nonlinear systems, together with the proofs of their asymptotic convergence and of their robustness properties.

Chapter 2 provides a literature review on the subject of observability and observer design for nonlinear systems, especial attention is given to high-gain observers. The chapter also gathers central results from the input-to-state stability framework and introduces the notion of robust observer design. Some relevant and new preliminary results can be found at the end.

Chapter 3 proposes the first observer design, the systems under consideration are state-affine with dynamic and output disturbances. Although the observer structure is given by the known high-gain Kalman observer, our work develops an efficient optimization algorithm that improves the state estimation. That is, an offline algorithm for the simultaneous adaptation of the high-gain tuning and the system input to the disturbances affecting the system.

Chapter 4, which contains the main contribution of this thesis, introduces the second observer design as a general redesign method. More precisely, we first consider a general nonlinear system for which an observer design is available. We then redesign this observer in case the output is only measured after a nonlinear transformation. Our new observer is based on a dynamic inversion formula that avoids using the inverse of the transformation.

Chapter 5 provides the third observer design by generalizing the previous redesign method.

The difference being that now the output is not only measured through a nonlinear transformation but it is also discretized in time. Nevertheless, this redesign method differs from the previous one and it imposes alternative constraints on the observer gain.

Finally, Chapter 6 summarizes the conclusions from the present thesis and indicates future research directions.

Publications

The work in this thesis gave rise to the following publications:

1. F. González de Cossío, M. Nadri Chapter 2

Observer design for nonlinear systems

Introduction

In this thesis, we consider dynamical systems governed by ordinary differential equations (ODE) that are in the so-called state-space representation. That is, ⎧ ⎨ ⎩ ẋ(t) = f (x(t), u(t), t) y(t) = h(x(t), u(t), t), (2.1) where t ∈ R + represents time, x(t) ∈ R n the state, y(t) ∈ R p the output and u(t) ∈ R m an input assumed continuous unless otherwise stated. The structure is described by the functions f : R n × R m × R + → R n and h : R n × R m × R + → R p , and the system is assumed forward-complete: for each initial condition x(0) ∈ R n , system (2.1) has a unique solution defined on all R + .

The observer design problem consists in defining an adjacent system to estimate the unknown state x of system (2.1) by using its output y and its input u. Indeed, an observer has the form ⎧ ⎨ ⎩ ż(t) = f (z(t), y(t), u(t), t)

x(t) = ĥ(z(t), y(t), u(t), t),

where z(t) evolves in some Euclidean space and where x(t) ∈ R n should estimate the state, that is,

|x(t) -x(t)| → 0, as t → ∞.
The observer is called global if such an estimation holds for all initial conditions x(0), z(0).

We usually consider global observers and we denote the state estimation error by e(t) = x(t)x(t), ∀t ≥ 0.

A type of convergence that is commonly desired is the exponential, often given by an inequality of the form

|e(t)| ≤ c|e(0)| exp(-rt),
for some positive constants c and r. In order to simplify the notation, we sometimes omit the explicit time-dependence if no confusion is possible. Moreover, we identify n-tuples

(x 1 , . . . , x n ) ∈ R n with column matrices ⎡ ⎢ ⎢ ⎣ x 1 . . . x n ⎤ ⎥ ⎥ ⎦ ∈ R n×1 .
The internal information carried by the system state can be necessary for tasks such as: modeling, monitoring, diagnosis or, importantly, driving of the system through the classical control-loop. Therefore, observer design for nonlinear systems has been a research topic of utmost importance in the control community for several decades.

The main objective of this chapter is to introduce several popular observer designs from the literature with an emphasis on the so-called high-gain observers. We also discuss the closely related notions of observability of a nonlinear system and the corresponding representative system forms that facilitate the design of observers. Finally, we study the celebrated concept of input-to-state stability (ISS) and allied properties, which provide us with efficient tools for quantifying the robustness of observers with respect to system disturbances. Some selected results are gathered at the end, and will be used in our main result corresponding to Chapter 4.

Observer design for linear systems

The problem of observer design for linear systems was established and solved by: R.E. Kalman, 1960 [65] and D.G. Luenberger, 1964 [83]. However, the motivation behind the idea of using the output to estimate the state can be traced back to the so-called filtering in communication engineering and to the work of A. Kolmogorov, 1939 and N. Wiener, 1949 (more recent editions in [START_REF] Kolmogoroff | Interpolation und Extrapolation von stationären zufälligen Folgen[END_REF] and [START_REF] Wiener | Extrapolation, Interpolation, and Smoothing of Stationary Time Series[END_REF] respectively).

Let us suppose that system (2.1) is linear, that is, of the form

⎧ ⎨ ⎩ ẋ(t) = Ax(t) + Bu(t) y(t) = Cx(t), (2.2)
where A, B and C are constant real matrices of the appropriate dimensions. The celebrated Luenberger observer for system (2.2) is then given by

ẋ(t) = Ax(t) + Bu(t) -K(C x(t) -y(t)), (2.3)
where K is any matrix such that A -KC is Hurwitz, that is, all its eigenvalues have strictly negative real part. We say that system (2. On the other hand, we can also estimate the state in the case of bounded and continuous timevarying matrices A(t), B(t) and C(t) by using a high-gain version of the so-called Kalman filter. However, for a system of the form

⎧ ⎨ ⎩ ẋ(t) = A(t)x(t) + B(t)u(t) y(t) = C(t)x(t) (2.4)
it is known that a constant gain K as in (2.3) might not be enough to design an observer.

Instead, we set K(t) = S(t) -1 C(t) for S(t) given by the Riccati equation. The observer is described by

⎧ ⎨ ⎩ ẋ(t) = A(t)x(t) + B(t)u(t) -K(t)(C(t)x(t) -y(t)) Ṡ(t) = -θS(t) -A(t) S(t) -S(t)A(t) + C(t) C(t), (2.5)
where S(0) is a symmetric and positive definite matrix and where the tuning parameter is such that θ > 2 sup t≥0 |A(t)|. We require system (2.4) to satisfy the following property: there exist positive real numbers a > 0, T > 0 and t 0 ≥ T such that

t t-T Φ(s, t -T ) C(s) C(s)Φ(s, t -T )ds ≥ aI, ( 2.6)
for all t ≥ t 0 and where Φ is the transition matrix, namely, the solution to

⎧ ⎨ ⎩ ∂Φ ∂s (s, t) = A(s)Φ(s, t) Φ(t, t) = I,
for all s, t ∈ R + . Importantly, condition (2.6) is independent of the system input and the rate of convergence of the state estimation error can be tuned by θ.

Theorem 2.2 ([22]

). Consider system (2.4), suppose that A(t) and C(t) satisfy condition (2.6) and that θ is large enough. Then, the state estimation error from (2.5) converges exponentially to zero.

Remark 2.1. Another possibility is that the matrix A in system (2.4) depends on the system input u. We can proceed as before in order to design an observer for the so-called state-affine

system ⎧ ⎨ ⎩ ẋ(t) = A(u(t))x + B(u(t)) y(t) = Cx(t)
with similar convergence properties [START_REF] Hammouri | Observer synthesis for state-affine systems[END_REF]. However, we need to restrict the input set U by imposing the corresponding property (2.6) for the input dependent Φ u . System inputs satisfying this property are known as regularly persistent. Further details are given in Chapter 3.

Noise and Kalman filtering

The original Kalman filter [START_REF] Kalman | A New Approach to Linear Filtering and Prediction Problems[END_REF][START_REF] Kalman | New results in linear filtering and prediction theory[END_REF] differs from (2.5) in various ways: the equation of the gain is instead written for the inverse matrix, there is no high-gain parameter θ and it involves additional symmetric and positive definite matrices which have a stochastic interpretation when the system is affected by noise. In fact, it is known that the Kalman filter is the least squares estimator in a stochastic sense.

The purpose of this section is to present a deterministic interpretation of the Kalman filter as given in [START_REF] Willems | Deterministic least squares filtering[END_REF]. We consider a disturbed linear system given by

⎧ ⎨ ⎩ ẋ(t) = Ax(t) + Dd 1 (t) y(t) = Cx(t) + d 2 (t),
where A, C and D are constant matrices, d 1 ∈ L 2 loc,d represents dynamic uncertainties and d 2 ∈ L 2 loc,p measurement noise. The goal is to filter the noise from the observations in order to estimate the state x. Given an initial condition x(0) ∈ R n , the system state and output have the expressions:

x(t) = exp(At)x(0) + t 0 exp(A(t -s))Dd 1 (s)ds, (2.7
)

y(t) = C exp(At)x(0) + t 0 C exp(A(t -s))Dd 1 (s)ds + d 2 (t), (2.8)
for all t ∈ R + . The least square filtering problem can be set as follows. Suppose that we observe a particular system output ȳ on a time interval [0, T ] and that we collect all disturbances d 1 , d 2 and initial conditions x(0) such that (2.8) is satisfied on this interval (with ȳ). Then we aim to minimize over this collection the square sum:

|x(0)| 2 S + |d 1 | 2 2,T + |d 2 | 2 2,T , (2.9)
where

|d i | 2 2,T = T 0 |d i (s)| 2 ds < ∞, i = 1, 2
and |x(0)| 2 S = x(0) Sx(0) for a symmetric and positive definite matrix S. If a minimum is achieved at some triplet (d * 1 , d * 2 , x(0) * ), then we can use (2.7) to obtain the optimal estimate at time T as

x(T ) = exp(AT )x(0) * + T 0 exp(A(T -s))Dd * 1 (s)ds.
Therefore, the state estimate is chosen by using the most likely triplet in the sense that uncertainty is minimized. Computing x then requires to apply an optimization algorithm at each given time which can be computationally expensive. The author in [START_REF] Willems | Deterministic least squares filtering[END_REF] shows that this optimization can be solved in a much more efficient way and that the equations of the Kalman filter arise naturally in this context.

Theorem 2.3 ([110]

). Suppose that ȳ ∈ L 2 loc,p is observed and consider the system

⎧ ⎨ ⎩ ẋ = Ax + P C (ȳ -C x) Ṗ = AP + P A -P C CP + DD ,
with the initial conditions x(0) = 0 and P (0) = S -1 . Then, the least square filter with respect to (2.9) is given by x.

The proof of this theorem consists first in showing the equality

|x(0)| 2 S + |d 1 | 2 2,T + |d 2 | 2 2,T =|x(T ) -x(T )| 2 P (T ) -1 + |d 1 -D P -1 (x -x)| 2 2,T + |ȳ -C x| 2 2,T ,
where the triplet (x(0), d 1 , d 2 ) leads to the observed ȳ on [0, T ]. Thus,

|x(0)| 2 S + |d 1 | 2 2,T + |d 2 | 2 2,T ≥ |ȳ -C x| 2 2,T ,
where the right-hand side of this inequality is independent of the triplet. Finally, it is shown that there exists a unique triplet leading to ȳ and such that

|x(T ) -x(T )| 2 P (T ) -1 + |d 1 -D P -1 (x -x)| 2 2,T = 0,
hence, minimizing (2.9).

Observer design for nonlinear systems

The Luenberger and the Kalman observer presented in the previous section provide satisfactory solutions to the observation problem of time-invariant or time-varying linear systems.

However, the design of observers for nonlinear systems can be very challenging and is, in general, not systematic. A first approach to this problem can be made by considering the Extended Kalman Filter (EKF), which is based on linearizations along the system trajecto-ries. Indeed, consider a nonlinear system of the form

⎧ ⎨ ⎩ ẋ(t) = f (x(t), u(t)) y(t) = h(x(t)),
where f and h are of class C 1 . Then, we can proceed as before by copying the dynamics of the system and adding a correcting term

ẋ(t) = f (x(t), u(t)) -K(t)(h(x(t)) -y(t)),
where the dynamic gain K(t) is defined as for observer (2.5) but with:

A(t) = ∂f ∂x (x(t), u(t)), C(t) = ∂h ∂x (x(t)).
Unfortunately, this technique has a major drawback since the obtained observer is only local and a good initial guess is crucial for the successful estimation of the system state [START_REF] Reif | Nonlinear state observation using H_infinity-filtering Riccati design[END_REF]. Some other popular approaches to observer design for nonlinear systems include: the linearization of the error dynamics by nonlinear state transformations [START_REF] Krener | Linearization by output injection and nonlinear observers[END_REF] or the less conservative technique from [START_REF] Kazantzis | Nonlinear observer design using Lyapunov's auxiliary theorem[END_REF], exploiting the Lipschitz nonlinearities of the system by using Linear Matrix Inequalities (LMI's) [START_REF] Zemouche | Observers for a class of Lipschitz systems with extension to performance analysis[END_REF][START_REF] Abbaszadeh | Nonlinear observer design for one-sided Lipschitz systems[END_REF] or, instead, dominating this nonlinearities with the use of a high-gain parameter [START_REF] Gauthier | A simple observer for nonlinear systems. Applications to bioreactors[END_REF], using other types of nonlinearities such as monotonic [START_REF] Arcak | Nonlinear observers: a circle criterion design and robustness analysis[END_REF] or those satisfying incremental quadratic constraints [START_REF] Açıkmeşe | Observers for systems with nonlinearities satisfying incremental quadratic constraints[END_REF], and a large number of other different techniques and specific designs that constantly increases.

Observability of nonlinear systems

It is clear that for some systems the problem of observer design cannot be solved. For example, it can be shown that there exists a linear observer design for the linear system (2.2)

precisely when the pair (A, C) is detectable. The slightly stronger condition of observability is equivalent to the full-rank of the matrix given by

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ C CA . . . CA n-1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
, which allows us to implement the tunable observer (2.3). As we have seen, the notion of observability for a time-varying linear system like (2.4) cannot be directly extended and, instead, a condition such as (2.6) is needed. In general, the observability of a nonlinear system can depend on the system input. Consider a forward-complete nonlinear system of the form

⎧ ⎨ ⎩ ẋ = f (x, u) y = h(x, u), (2.10)
where x(t) ∈ R n , y(t) ∈ R p and u(t) ∈ R m , and where f and h are analytic functions satisfying f (0, 0) = 0 and h(0, 0) = 0. Let us denote by x(t; x 0 , u) the unique trajectory corresponding to the initial condition x 0 = x(0) and to the input u.

Definition 2.1. We say that an input u is universal for system (2.10) if for all initial conditions x 0 = x 0 there exists s ≥ 0 such that

h(x(s; x 0 , u), u(s)) = h(x(s; x 0 , u), u(s)). System (2.10) is uniformly observable if every input u ∈ L ∞ m is universal.
Remark 2.2. The universal property of an input can be lost over time. This gives rise to the notion of persistency. That is, the input u is persistent if there exist t 0 ≥ T > 0 such that for all t ≥ t 0 and for all initial conditions x t-T = x t-T we have

t t-T |h(x(s; x t-T , u), u(s)) -h(x(s; x t-T , u), u(s))| 2 ds > 0.
This notion is, of course, related to the regular persistence of an input as in Remark 2.1 and to allied properties from the next section such as local regularity.

A popular technique for observer design is to steer the system into a convenient form by a change of coordinates in order to facilitate the design of an observer, early work in this direction can be seen in [START_REF] Krener | Linearization by output injection and nonlinear observers[END_REF]. Suppose that we have a system in the specific form

⎧ ⎨ ⎩ ẋ = f (x) + g(x)u y = h(x), (2.11)
where x(t) ∈ R n , y(t) ∈ R and u(t) ∈ R, and for f, g : R n → R n and h : R n → R smooth functions. We define the transformation φ : R n → R n coordinate-wise for 1 ≤ i ≤ n as

φ i (x) = L i-1 f (h)(x), ∀x ∈ R n , ( 2.12)
where L i f (h) denotes the ith Lie derivative of h along f . The Lie derivative can be defined recursively by

L 0 f (h) = h, L f (h)(x) = ∂h ∂x (x)f (x), L i f (h) = L f (L i-1 f (h)).
⎧ ⎨ ⎩ ż = Az + ψ(z) + ω(z)u y = Cz,
where:

A = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 1 0 . . . . . . . . . 1 0 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , C = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 1 0 . . . 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , ψ(z) = ⎡ ⎢ ⎢ ⎣ 0 . . . ψ n (z) ⎤ ⎥ ⎥ ⎦ , ω(z) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ω 1 (z 1 ) ω 2 (z 1 , z 2 ) . . . ω n (z) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
and for some functions ψ n , ω 1 , . . . , ω n .

More generally, this result is valid for control-affine nonlinear systems with multi-dimensional inputs u(t) ∈ R m . The converse also holds: if the system can be steered into the specific form described in the previous theorem, then the system should be uniformly observable.

Theorem 2.4 is in fact a consequence of a more general result that requires the notion of canonical flag. Let us consider again the general nonlinear system described in (2.10). The study of the so-called observability normal forms is divided into two important dimension cases: p > m and m ≥ p. The former leads to the phase variable representation while the latter, studied next for p = 1, to the uniform observability canonical form.

In this case, the transformation φ defined in (2.12) depends on u and we can set

K i (x, u) = ker ∂φ i ∂x (x, u) ,
that is, the kernel of the Jacobian matrix. The uniform canonical flag condition [START_REF] Gauthier | Deterministic observation theory and applications[END_REF] applies to each i = 1, . . . , n and requires that for all x ∈ R n and all u ∈ R m :

dim(K i (x, u)) = n -i, K i (x, u) is independent of u, (2.13)
where dim stands for the dimension of the kernel subspace of R n . The following theorem provides global conditions for steering system (2.10) into a convenient triangular structure.

Let us define the function φ 0 : R n → R n as φ 0 (x) = φ(x, 0).

Theorem 2.5 ([48]

). Consider system (2.10) with p = 1 and suppose that it satisfies conditions (2.13) and that φ 0 is a global diffeomorphism. Then, the change of coordinates z = φ 0 (x)

takes system (2.10) into the form:

⎧ ⎨ ⎩ ż = f (z, u) y = h(z 1 , u),
for some functions f :

R n × R m → R n and h : R × R m → R such that: fi (z 1 , z 2 , . . . , z i+1 , u), fn (z, u) ∂ fi ∂z i+1 = 0, ∂ h ∂z 1 = 0, ∀z ∈ R n , ∀u ∈ R m .
An extensive summary of results concerning system representations that characterize observability can be consulted in [START_REF] Gauthier | Deterministic observation theory and applications[END_REF]. More recently, the authors in [START_REF] Besançon | An Immersion-Based Observer Design for Rank-Observable Nonlinear Systems[END_REF] also show that the class of systems for which there exist convenient forms (by immersion) can be extended to nonuniformly observable systems. Therefore, the problem of observer design for a large class of nonlinear systems reduces to specific system forms. The following section studies observer design for several representative systems.

High-gain observers

In nonlinear control, high-gain observers are a common state estimation tool due to their convenient structure. The pioneer work in [START_REF] Doyle | Robustness with Observers[END_REF] presents an observer adjustment of the Kalman filter, which is based on a fictitious noise that achieves a satisfactory linear control feedback.

As with modern high-gain techniques, this adjustment implies a trade-off between noise rejection and margin recovery.

An important research direction concerning high-gain observers was initiated in the work of [START_REF] Esfandiari | Output feedback stabilization of fully linearizable systems[END_REF]. It deals with the peaking phenomenon in the absence of global growth conditions and with the possible destabilization of the closed-loop system; the authors present an appropriate control function that saturates precisely during the peaking. A contemporary but different research direction consists in dominating the Lipschitz nonlinearities of the system with the gain, this results in two main observer constructions: the Luenberger type [START_REF] Gauthier | A simple observer for nonlinear systems. Applications to bioreactors[END_REF] or the Kalman type [START_REF] Deza | High gain estimation for nonlinear systems[END_REF].

Let us consider a nonlinear system in the representative form given in Theorem 2.5. As oppose to other systems considered in this section, this one allows a nonlinear output function. If we use the notation xi = (x 1 , . . . , x i ), ∀x ∈ R n then the system is described by

ẋ1 = f 1 ( x1 , x 2 , u) ẋ2 = f 2 ( x2 , x 3 , u) . . . ẋn-1 = f n-1 ( xn-1 , x n , u) ẋn = f n ( xn , u) y = h(x 1 , u), (2.14)
where x(t) ∈ R n , y(t) ∈ R and u(t) ∈ R m , and where f = (f 1 , . . . , f n ) and h are analytic.

Furthermore, we require each f i to be Lipschitz continuous with respect to xi , uniformly in x i+1 and u, and we assume the existence of positive constants α < β such that

α ≤ ∂f i ∂x i+1 ≤ β, α ≤ ∂h ∂x 1 ≤ β, ∀x ∈ R n , ∀u ∈ R m . (2.15)
The Luenberger-like observer from [START_REF] Gauthier | Deterministic observation theory and applications[END_REF] takes the form

ẋ = f (x, u) + Δ θ K(y -h(x 1 , u)), (2.16)
where θ > 0 and the constant gain matrix K = [k n-1 . . . k 0 ] are design parameters, and where

Δ θ = ⎡ ⎢ ⎢ ⎣ θ 0 . . . 0 θ n ⎤ ⎥ ⎥ ⎦ .
(2.17)

The design of K is intricate and it requires a preliminary result. Indeed, given any positive constants α < β and a collection of n continuous functions g i : R

+ → R + satisfying α ≤ g i (t) ≤ β, ∀t ≥ 0,
there exist real numbers k 0 , k 1 , . . . , k n-1 , a positive real number λ > 0 and a symmetric positive definite matrix S such that SA(t) + A(t) S ≤ -λI, ∀t ≥ 0 and where

A(t) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ -k n-1 g 1 (t) g 2 (t) 0 . . . 0 0 -k n-2 g 1 (t) 0 g 3 (t) . . . 0 0 . . . . . . . . . . . . . . . . . . -k 1 g 1 (t) 0 0 . . . 0 g n (t) -k 0 g 1 (t) 0 0 . . . 0 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ .
Crucially, the k i , λ and S depend only on the given α and β and not on the choice of the collection of functions g i . Since the proof of this result is inductive on n, it also provides an algorithm to find the aforementioned variables.

Finally, the design of K in (2.16) arises from the bounds in (2.15) and, in the proof of the observer convergence, the functions g i are defined as:

g 1 = ∂h ∂x 1 (δ 0 , u), g i+1 = ∂f i ∂x i+1 ( xi , δ i , u),
where δ i (t) ∈ R comes from the application of the mean value theorem on a convenient interval. As one might expect, a Lyapunov function for the re-scaled state estimation error i = θ n-i (x ixi ) is then given by V ( ) = S and there is the following result. We remark that how large θ has to be depends on several factors, such as the Lipschitz constants of f i and the state dimension n, which is a characteristic drawback of high-gain observers. Similarly, we can also design an observer for systems in the form given by Theorem 2.4. That is, consider the nonlinear system given by

⎧ ⎨ ⎩ ẋ = Ax + ζ(x, u) y = Cx, (2.18)
where:

A = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 1 0 . . . . . . . . . 1 0 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , C = 1 0 . . . 0 , ζ(x, u) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ζ 1 (x 1 , u) ζ 2 (x 1 , x 2 , u) . . . ζ n (x, u) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , ( 2
|ζ i ( xi , u) -ζ i ( xi , u)| ≤ c i | xi -xi |, ( 2.20)
where xi = (x 1 , . . . , x i ) and for all x, x ∈ R n and all u ∈ R m . The following observer was introduced in [START_REF] Gauthier | A simple observer for nonlinear systems. Applications to bioreactors[END_REF],

⎧ ⎨ ⎩ ẋ = Ax + ζ(x, u) + S -1 ∞ C (y -C x) 0 = -θS ∞ -A S ∞ -S ∞ A + C C, (2.21)
where θ > 0 is large enough and where the observability of the pair (A, C) ensures the existence of a (constant) symmetric and positive definite matrix S ∞ . This can be seen by noticing that S ∞ is the stationary solution of (2.5) for A and C constant matrices. In more detail, let us denote by S ∞,1 the solution of the second algebraic equation in (2.21) corresponding to θ = 1; which should not be confused with the actual tuning of the observer.

The authors in [START_REF] Gauthier | A simple observer for nonlinear systems. Applications to bioreactors[END_REF] then show that the state estimation error

e(t) = x(t) -x(t)
satisfies the inequality

|e(t)| ≤ k θ |e(0)| exp - θ 3 t , ∀t ≥ 0
as long as the parameter θ is large enough and for some k θ > 0. In fact, if we denote

c = max i c i , s = max i,j |(S ∞,1 ) i,j |
then the tuning lower bound takes the form

θ > 6nc √ s λ min (S ∞,1 )
,

where the right-hand side of the inequality is independent of θ. However, θ should dominate the product nc that depends on the system dimension and the Lipschitz constants of the functions ζ i .

Remark 2.3. The high-gain technique from [START_REF] Gauthier | A simple observer for nonlinear systems. Applications to bioreactors[END_REF] is generalized in [START_REF] Bornard | A high gain observer for a class of uniformly observable systems[END_REF]. The authors provide the more familiar observer form

ẋ = Ax + ζ(x, u) -Δ θ K(C x -y), (2.22)
where K is such that A -KC is Hurwitz, θ > 0 is large enough and Δ θ is as in (2.17).

Just as in the linear case, the natural extension of these results concerns time-varying continuous matrices. Moreover, let us suppose that the matrix in the system dynamics (2.18) depends continuously on the input so that we have

A(u) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0 a 1 (u) 0 . . . . . . . . . a n-1 (u) 0 0 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ , C(u) = c 1 (u) 0 . . . 0 ,
and the system

⎧ ⎨ ⎩ ẋ = A(u)x + ζ(x, u) y = C(u)x.
(2.23)

As in Remark 2.1, it is useful to restrict the set U of admissible inputs. However, this time we are interested in small time intervals. The property we look for is the local regularity of a bounded input u: there exist positive real numbers a > 0 and θ 0 > 0 such that for all θ ≥ θ 0 we have

t t-1 θ Φ u (s, t) C(u) C(u)Φ u (s, t)ds ≥ aθΔ -2 θ , ∀t ≥ 1 θ (2.24)
where Δ θ is as in (2.17) and, as before, Φ u is the transition matrix:

⎧ ⎨ ⎩ ∂Φu ∂s (s, t) = A(u(s))Φ u (s, t) Φ u (t, t) = I,
for all s, t ∈ R + . The authors in [START_REF] Besançon | An Immersion-Based Observer Design for Rank-Observable Nonlinear Systems[END_REF] develop an exponential observer for system (2.23) given by

⎧ ⎨ ⎩ ẋ = A(u)x + ζ(x, u) -Δ θ S -1 C(u) (C(u)x -y) Ṡ = -θ(γS + A(u) S + SA(u) -C(u) C(u)), (2.25)
for θ > 0 large enough and where γ > 0 guarantees the stability of the equation defining S(t). Indeed, it can be shown that the local regularity of u implies that there exists γ > 0 and α 1 , α 2 > 0 such that for all θ ≥ θ 0 we have The exponential rate of convergence of the state estimation error can then be arbitrarily set by choosing

α 1 I ≤ S(t) ≤ α 2 I,
θ > max 2α 2 γα 1 c, θ 0 ,
where c is the Lipschitz constant of ζ. Their work extends previous results presented in [START_REF] Bornard | Observability and observers[END_REF].

Remark 2.4. The authors in [START_REF] Dufour | Nonlinear observers synthesis based on strongly persistent inputs[END_REF] show that it is possible to design a constant gain observer (Luenberger-like) for system (2.23) when the system input is locally regular. Furthermore, they define the weaker notion of strongly persistent input for which, in turn, they exhibit an observer design with dynamic gain.

Observer design in the presence of noise

The nonlinearity ζ in the system dynamics of both (2.18) and (2.23) make us restrict our family of systems under consideration by imposing a specific form on the matrices A and C.

Furthermore, the convergence to zero of the state estimation error e requires us to tune the observer with θ larger than a value depending on the Lipschitz constant of ζ. Two important problems arise from this fact: (i) strong peaking in the transient of e, which can destabilize the control loop [START_REF] Khalil | High-gain observers in nonlinear feedback control[END_REF], (ii) the measurement noise amplification as studied in this section.

The authors in [START_REF] Vasiljevic | Error bounds in differentiation of noisy signals by high-gain observers[END_REF] study the known problem of real-time signal differentiation by using a high-gain observer in the presence of measurement noise. Their work exhibits and quantifies the trade-off between the estimation of the signal derivative and the amplification of measurement noise. Let us consider a chain of integrators of the form

⎧ ⎨ ⎩ ẋ = Ax + Bu (n) y = Cx + d, (2.26)
for A and C in the canonical form (2.19) and where B = 0 . . . 0 1 , the signal u : R + → R is of class C n and d : R + → R represents bounded measurement noise. Moreover, u and its derivatives are assumed to be bounded. Notice that the coordinates of x correspond to the derivatives of u, that is, a solution is given by (1) . . .

x = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ u u
u (n-1) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦
, where u (i) stands for the ith-derivative of u with respect to time. Therefore, the output measurements are precisely the signal u affected by noise and the task is to estimate the n -1 derivatives of u. The linear high-gain observer for system (2.26) used in [START_REF] Vasiljevic | Error bounds in differentiation of noisy signals by high-gain observers[END_REF] is given by

⎧ ⎨ ⎩ ẋi = xi+1 + α i -i (y -x1 ), 1 ≤ i ≤ n -1 ẋn = α n -n (y -x1 ), (2.27)
where the polynomial

s n + α 1 s n-1 + • • • + α n
is Hurwitz and where > 0 is a tuning parameter. The following theorem states that the estimation error can be bounded coordinate-wise.

Theorem 2.9 ([108]). Consider system (2.26), observer (2.27) and 1 ≤ i ≤ n. For an arbitrarily small quantity δ > 0 there exists T > 0 large enough such that:

|u (i-1) (t) -xi (t)| ≤ (p i+1 |u (n) | ∞ ) • n-i + (q i+1 |d| ∞ ) • -i + δ, ∀t ≥ T, (2.28)
where the constants p i , q i ≥ 0 depend only on α 1 , . . . α n .

The right-hand side of inequality (2.28) is a strictly convex function of > 0 that should be minimized. In the absence of measurement noise (d = 0), this is achieved by making → 0.

Otherwise, a unique minimum is reached at

* i = i n -i 1/n q i+1 |d| ∞ p i+1 |u (n) | ∞ 1/n
, which depends on the coordinate i under consideration. Nevertheless, their simulations show that * i does not vary much with i and that it is possible to choose * as the average of these values. Importantly, these simulations also show that minimizing the error bound is not far from minimizing the state estimation error itself. Therefore, the authors provide a systematic way of tuning their linear high-gain observer in the presence of measurement noise.

Instead of system (2.26) we can consider

⎧ ⎨ ⎩ ẋ = Ax + Bζ(x) y = Cx + d, (2.29)
where A, B and C remain in the canonical form as before, and where ζ : R n → R is Lipschitz continuous. In the case of no disturbances (d = 0), using the observer in (2.22) implies there

exist constants c 1 , c 2 > 0 such that |e(t)| ≤ c 1 θ n-1 |e(0)| exp(-c 2 θt), ∀t ≥ 0.
In particular, this means that we can set a large exponential decay rate at the price of a peaking of order n -1. Furthermore, in practice, it can be hard to implement observer (2.22) for large tuning θ and for large state dimension n. This is due to the fact that θ n appears in the observer gain. As a consequence, the authors in [START_REF] Astolfi | A High-Gain Nonlinear Observer with Limited Gain Power[END_REF] present a novel high-gain observer for system (2.29) with a limited gain up to power two and that preserves the same observer properties. Their innovative technique consists in increasing the observer state dimension to 2n -2 instead of n.

Although the observer developed in [START_REF] Astolfi | A High-Gain Nonlinear Observer with Limited Gain Power[END_REF] seems to have high-frequency noise rejection properties, it still suffers from the so-called peaking phenomenon. A useful methodology to face this challenge and the noise amplification itself consists in a dynamic adaptation of the observer gain. This gives rise to an observer design methodology known as adaptive observers.

The authors in [START_REF] Bullinger | An adaptive high-gain observer for nonlinear systems[END_REF] implement a dynamic gain tuning θ in the classic observer design for a system of the form (2.29). The noise is not taken into consideration and θ can only increase.

On the other hand, dynamic uncertainties and measurement noise is addressed in the work presented in [START_REF] Ahrens | High-gain observers in the presence of measurement noise: A switched-gain approach[END_REF]. The main idea is: (i) increase the tuning when the output error is large in order to achieve fast reconstruction, (ii) once it becomes small, decrease the tuning to balance the influence of both types of disturbances. Moreover, a switching delay timer is included to handle the peaking in the estimates. The authors in [START_REF] Boizot | An adaptive high-gain observer for nonlinear systems[END_REF] study gain adaptation for a system of the form (2.23) by a mix approach: the observer switches between Kalman filtering and high-gain techniques. There exists a vast literature concerning adaptive observers which also includes papers such as [START_REF] Andrieu | High gain observers with updated gain and homogeneous correction terms[END_REF][START_REF] Sanfelice | On the performance of high-gain observers with 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]] and many more.

Input-to-state stability and observers

In the previous section, we have seen the influence of different types of disturbances in the state estimation problem. However, we would like to have a general tool to determine if a given observer for a general nonlinear system has robustness properties with respect to measurement noise. The framework of Input-to-state stability (ISS) fits our needs since it quantifies the effect of external inputs: we can regard the error dynamics as a system whose input is precisely the measurement noise. This gives rise to the so-called ISS-observers or disturbance-to-error stable (DES) observers. In what follows, the operator sup is used to indicate either supremum or essential supremum, that is, up to a zero measure set.

Input-to-state stability

The concept of input-to-state stability was introduced by E. Sontag in his celebrated paper [START_REF] Sontag | Smooth stabilization implies coprime factorization[END_REF] as a test for the robustness of nonlinear systems to external perturbations, for a concise summary see [START_REF] Dashkovskiy | Input to state stability and allied system properties[END_REF]. As a motivational example, consider the linear system described by ẋ = Ax + Bu, (2.30) where x(t) ∈ R n is the state, u(t) ∈ R m the input and A and B constant matrices. The solution corresponding to the initial condition x(0) can be written as

x(t) = exp(At)x(0) + t 0 exp(A(t -s))Bu(s)ds, ∀t ≥ 0
and, if A is Hurwitz, we can dominate the system state with

|x(t)| ≤ c 1 exp(-c 2 t)|x(0)| + c 3 sup s∈[0,t] |u(s)|, ∀t ≥ 0. (2.31)
The positive constants c 1 , c 2 and c 3 are such that

|exp(At)| ≤ c 1 exp(-c 2 t), c 3 = c 1 c -1 2 |B|
and, thus, they are independent of x(0) and u. The first term in (2.31) quantifies the effect of the initial condition for short times, while the second term accounts for the input impact.

These ideas can be generalized for nonlinear systems by using comparison functions.

|x(t)| ≤ β(|x(0)|, t) + γ(|u| ∞ ) + c, ∀t ≥ 0.
(2.33)

The system is input-to-state stable (ISS) if the latter is satisfied with c = 0.

An alternative definition makes use of sup s∈[0,t] γ(|u(s)|) instead of γ(|u| ∞ ), this is justified by the causality of the system. In such a case, the space of inputs can be defined by local boundedness (L ∞ loc,m ).

Remark 2.5. We can recover a well-known stability property of system (2.32) if we fix u = 0.

Indeed, the equilibrium point x = 0 is stable if for each > 0 there exists δ > 0 such that

|x(0)| < δ =⇒ |x(t)| < ,
ẋ = f (x, k(x) + d),
where k stabilizes the system for d = 0. The author in [START_REF] Sontag | Smooth stabilization implies coprime factorization[END_REF] shows that, for control-affine systems, k can be modified in order to achieve ISS when regarding d as the system input.

The elegant formulation of ISS has now become a standard tool in the literature in order to study robustness of nonlinear systems with respect to inputs in a wide range of situations.

Showing directly the ISS of a system can be challenging. Therefore, the following definition and theorem provide a Lyapunov characterization of ISS.

Definition 2.4. A smooth function V : R n → R + is called an ISpS-Lyapunov function for system (2.32) if there exist functions α 1 , α 2 ∈ K ∞ and α 3 , χ ∈ K, and a constant c L ≥ 0
such that for all x ∈ R n and all u ∈ R m we have

α 1 (|x|) ≤ V (x) ≤ α 2 (|x|),
and |x| ≥ χ(|u|) + c L implies ∂V ∂x (x)f (x, u) ≤ -α 3 (|x|).

The function is called ISS-Lyapunov if the latter holds with c

L = 0. Theorem 2.10 ([103, 102]). System (2.

32) is ISpS if and only if it has an ISpS-Lyapunov function. The same equivalence holds for ISS.

As an example, the linear system (2.30) is ISS precisely when A is Hurwitz. We can construct the ISS-Lyapunov function as V (x) = x P x, where A P + P A < 0.

Small-gain theorem

It is often the case in control analysis that a plant is connected with feedback laws or adjacent systems. The stability of the resulting closed-loop can be establish by using the so-called small-gain results. The classical small-gain theorem [START_REF] Zames | On the input-output stability of time-varying nonlinear feedback systems[END_REF] arises in the context of input-output stability theory in the 1960s and guarantees the stability of the feedback loop if the loop gain is less than one. However, most of the early work concerning small-gain results considers normed-based linear gains. Further studies incorporate nonlinear function gains in the ISS framework [START_REF] Jiang | Small-gain theorem for ISS systems and applications[END_REF] giving rise to a vast literature in the topic.

Consider a system described by

⎧ ⎨ ⎩ ẋ = f (x, u) y = h(x, u), (2.34)
where x(t) ∈ R n is the state, u(t) ∈ R m an input and y(t) ∈ R p the output. The functions f and h are assumed smooth.

Definition 2.5. System (2.34) is input-to-output practically stable (IOpS) if there exist func-

tions β ∈ KL, γ ∈ K and a constant c d ≥ 0 such that for all x(0) ∈ R n and all u ∈ L ∞ m we have |y(t)| ≤ β(|x(0)|, t) + γ(|u| ∞ ) + c d , ∀t ∈ [0, T ),
where [0, T ) is the maximal interval of existence for x. If c d = 0, then the system is called input-to-output stable (IOS).

In the case that y = x, the previous definition corresponds to input-to-state practical stability, and to ISS if in addition c d = 0. The following definition is trivially satisfied in the case y = x. Definition 2.6. System (2.34) has the unboundedness observability (UO) property if there exist a function α ∈ K and a constant c D ≥ 0 such that for each initial condition

x(0) ∈ R n and each input u ∈ L ∞ m we have |x(t)| ≤ α |x(0)| + sup s∈[0,t] |(u(s), y(s))| + c D , ∀t ∈ [0, T ),
where [0, T ) is the maximal interval of existence for x.

Now let us consider an interconnected system given by

⎧ ⎨ ⎩ ẋ1 = f 1 (x 1 , y 2 , u 1 ), y 1 = h 1 (x 1 , y 2 , u 1 ) ẋ2 = f 2 (x 2 , y 1 , u 2 ), y 2 = h 2 (x 2 , y 1 , u 2 ) (2.35)
and suppose that the first subsystem is IOpS when considering y 2 and u 1 as the input. This results in an inequality of the form

|y 1 (t)| ≤ β 1 (|x 1 (0)|, t) + sup s∈[0,t] γ 11 (|y 2 (s)|) + γ 12 (|u 1 | ∞ ) + c d 1 .
Analogously, if the second subsystem is also IOpS we get

|y 2 (t)| ≤ β 2 (|x 2 (0)|, t) + sup s∈[0,t] γ 21 (|y 1 (s)|) + γ 22 (|u 2 | ∞ ) + c d 2 .
Theorem 2.11 ([64]). Suppose that each subsystem of (2.35) is IOpS as just described and that each subsystem has the UO property. If there exist functions ρ 1 , ρ 2 ∈ K ∞ and a constant r 0 ≥ 0 such that Theorem 2.11 can also be used to show the ISS of an interconnection of ISS subsystems. In such a case, the UO property is superfluous.

(I + ρ 2 ) • γ 21 • (I + ρ 1 ) • γ 11 (r) ≤ r, ∀r ≥ r 0 , ( 2
(I + ρ 1 ) • γ 11 • (I + ρ 2 ) • γ 21 (r) ≤ r, ∀r ≥ r 0 .

If both subsystems have the UO property with c

D 1 = c D 2 = 0, and if c d 1 = c d 2 = r 0 = 0, then the overall system (2.35) is IOS. Notice that
The small-gain theorem has not only a trajectory based formulation but also a Lyapunov one. Let us consider an interconnected system described by

⎧ ⎨ ⎩ ẋ1 = f 1 (x 1 , x 2 , u 1 ) ẋ2 = f 2 (x 1 , x 2 , u 2 ), (2.37)
where

x i (t) ∈ R n i , u i (t) ∈ R m i and f i : R n 1 × R n 2 × R m i → R n i are locally Lipschitz.
As before, we suppose that each subsystem is ISpS by using the Lyapunov functions V 1 and V 2 .

That is, V i for i = j satisfies for all

x 1 ∈ R n 1 , all x 2 ∈ R n 2 and all u i ∈ R m i : α i1 (|x i |) ≤ V i (x i ) ≤ α i2 (|x i |),
and V i (x i ) ≥ max{χ i (V j (x j )), γ i (|u i |) + c Li } implies ∂V i ∂x i (x i )f i (x 1 , x 2 , u i ) ≤ -α i3 (V i (x i )),
where the functions α i1 , α i2 , α i3 ∈ K ∞ , χ i , γ i ∈ K and the constant c Li ≥ 0. We can state the following result by using the gains χ 1 and χ 2 .

Theorem 2.12 ( [START_REF] Jiang | A Lyapunov formulation of the nonlinear small-gain theorem for interconnected ISS systems[END_REF]). Suppose that there exist Lyapunov functions V 1 and V 2 respectively for the first and second subsystem of (2.37) as described above. If there exists r 0 ≥ 0 such that

χ 1 • χ 2 (r) < r, ∀r > r 0 , ( 2
χ 2 • χ 1 (r) < r, ∀r > r 0 . If c L1 = c L2 = r 0 = 0, then the overall system (2.37) is ISS and a zero input (u 1 , u 2 ) = 0
implies global asymptotic stability.

Remark 2.8. The proof of Theorem 2.12 relies on the construction of an ISpS-Lyapunov function for the overall system, this function is defined on R n 1 +n 2 as

V (x 1 , x 2 ) = max{σ(V 1 (x 1 )), V 2 (x 2 )},
where σ ∈ K ∞ is continuously differentiable on the positive real line, with a positive derivative there and, if

χ 1 ∈ K ∞ , it satisfies χ 2 (r) < σ(r) < χ -1 1 (r), ∀r > 0.
Therefore, the function V is only locally Lipschitz on R n 1 +n 2 -{0} and differentiable almost everywhere. Nevertheless, this suffices to show the ISpS property.

A natural generalization of the nonlinear small-gain results concerns interconnections of more than two ISS subsystems as in [START_REF] Dashkovskiy | An ISS small gain theorem for general networks[END_REF]. In such a case, the interconnection gains are collected in a matrix and bounds on its image establish the small-gain condition. The corresponding Lyapunov based approach is also studied in [START_REF] Dashkovskiy | Small Gain Theorems for Large Scale Systems and Construction of ISS Lyapunov Functions[END_REF].

Observer robustness criterion

As we previously saw, the notion of ISS is a widely used tool for the study of the stability of nonlinear systems with inputs. On the other hand, the more general notion of IOS quantifies the effect of external perturbations on the system output. Another dual property is the so-called output-to-state stability (OSS) [START_REF] Sontag | Output-to-state stability and detectability of nonlinear systems[END_REF] which follows the same ideas and, in the linear case, corresponds to detectability of the system. The notion of input/output to state stability (IOSS), also studied in [START_REF] Sontag | Output-to-state stability and detectability of nonlinear systems[END_REF], combines the definitions of both ISS and OSS. The concept of IOSS, and its incremental version, are of particular interest to us given their connection with the so-called ISS-observers, observers that are robust with respect to measurement noise.

Here we consider a system described by

⎧ ⎨ ⎩ ẋ = f (x, u) y = h(x), (2.39)
where f : R n × R m → R n is continuous, locally Lipschitz on x uniformly on bounded u, f (0, 0) = 0 and h : R n → R p is continuously differentiable with h(0) = 0. We denote the trajectory and the output corresponding to the initial condition x 0 = x(0) and to the input u by x(t; x 0 , u) and y(t; x 0 , u) respectively. The following condition is stronger than IOSS.

, x0 ∈ R n it holds that |x(t; x 0 , u) -x(t; x0 , ū)| ≤ max{β(|x 0 -x0 |, t), sup s∈[0,t] γ 1 (|u(s) -ū(s)|), sup s∈[0,t] γ 2 (|y(s; x 0 , u) -y(s; x0 , ū)|)}, ∀t ∈ [0, T ),
where [0, T ) is the common domain of definition.

A full-order observer for system (2.39) subject to measurement noise d y and input noise d u is specified by the dynamics

ẋ = f (x, y + d y , u + d u ),
and its robustness can be quantified as follows: there exist β ∈ KL and γ 1 ,

γ 2 ∈ K such that for all u, d u ∈ L ∞ loc,m , all d y ∈ L ∞ loc,p and all x(0), x(0) ∈ R n we have |x(t) -x(t)| ≤ max{β(|x(0) -x(0)|, t), sup s∈[0,t] γ 1 (|d u (s)|), sup s∈[0,t] γ 2 (|d y (s)|)},
for all t ∈ R + in the maximal interval of existence for x (where x should exist). Such an ISS-observer makes the state estimation error converge to zero in the absence of noise and ensures a good estimation degradation in the case of disturbances. The following result sets a condition on the system for the existence of such an observer.

Proposition 2.1 ([104]). System (2.39) is i-IOSS if it has an ISS-observer as described above.

The authors in [START_REF] Shim | Nonlinear observer design via passivation of error dynamics[END_REF] first introduce a nonlinear observer design framework by using passi-vation of error dynamics. They then present a redesign of the observer injection term to achieve, under certain conditions, robustness with respect to measurement noise in an ISS sense as above. In the same context, the authors in [START_REF] Alessandri | Observer design for nonlinear systems by using Input-to-State Stability[END_REF] design an observer for a class of systems with Lipschitz nonlinearities and also consider its robustness with respect to both system uncertainties and measurement noise. An advantage of their observer is that it does not require state-space transformations. Indeed, consider a system of the form

⎧ ⎨ ⎩ ẋ = Ax + f (x) + D 1 d y = Cx + D 2 d, (2.40)
where x(t) ∈ R n and y(t) ∈ R p , and where d ∈ L ∞ loc,q is a time-varying disturbance. The matrices A, C, D 1 and D 2 are constant and such that (A, C) is observable, and

f : R n → R n is Lipschitz continuous: there exists a constant c f > 0 such that |f (x 1 ) -f (x 2 )| ≤ c f |x 1 -x 2 |, ∀x 1 , x 2 ∈ R n .
The work in [START_REF] Alessandri | Observer design for nonlinear systems by using Input-to-State Stability[END_REF] proposes a Luenberger-like observer given by

ẋ = Ax + f (x) + L(y -C x), (2.41)
where x is the state estimate and L is a gain matrix to be designed.

Theorem 2.13 ([5]). Consider system (2.40) and suppose that there exist α > 0, a symmetric and positive definite matrix P and a matrix L such that

(A -LC) P + P (A -LC) + αc 2 f I P P -α 2 I < 0, (2.42
)

then (2.41) is an ISS-observer.
Remark 2.9. The matrix inequality in (2.42) can be written as an LMI by using the change of variables L = P -1 K for a matrix K. However, the observer design is not always possible:

the existence of a matrix L such that A -LC is Hurwitz is a necessary condition for (2.42)

and a large Lipschitz constant c f can make the LMI unfeasible.

Some preliminary results

The purpose of this section is to develop the necessary tools for Chapter 4. It includes a useful reformulation of the small-gain theorem and some other results involving comparison functions. A comprehensive study of the function classes K and KL can be found in [START_REF] Kellett | A compendium of comparison function results[END_REF].

Remark 2.10. The function classes K and K ∞ are closed under composition. Also, functions in K ∞ are invertible and their inverses remain in the class. A frequently used triangle-type inequality for γ ∈ K is [START_REF] Freeman | Robust Nonlinear Control Design: State-Space and Lyapunov Techniques[END_REF]). For any locally Lipschitz function φ : R n 1 → R n 2 , there exist locally Lipschitz functions ϕ : R

γ(r 1 + r 2 ) ≤ γ(2r 1 ) + γ(2r 2 ), ∀r 1 , r 2 ∈ R + . Proposition 2.2 ([
n 1 → R + and α ∈ K ∞ such that |φ(z 1 ) -φ(z 2 )| ≤ ϕ(z 1 )α(|z 1 -z 2 |), ∀z 1 , z 2 ∈ R n 1 .
We now reformulate the definition of ISpS by considering families of function pairs, this will simplify our exposition of Chapter 4. As can be seen in the proofs that follow, the classical results hold in this setting.

Definition 2.8. Consider any family G formed by pairs of locally Lipschitz functions (z, w) :

I z,w = [0, T z,w ) → R n 1 × R n 2 .
We say that G is practically stable if there exist functions β ∈ KL and γ ∈ K and a constant c ≥ 0 such that for all (z, w) ∈ G we have

|z(t)| ≤ β(|z(0)|, t) + sup s∈[0,t] γ(|w(s)|) + c, ∀t ∈ I z,w . (2.43)
The functions β and γ and the constant c are respectively called decay rate, gain and constant of the practical stability.

In order to enlarge the class of admissible ISpS-Lyapunov functions, the next definition uses the fact that locally Lipschitz functions are differentiable almost everywhere (a.e.); this is known as Rademacher's theorem.

Definition 2.9. A function

V : R + × R n 1 → R + is a Lyapunov function for the family G if: (i) for each (z, w) ∈ G, V z = V (•, z(•)
) is everywhere continuous and locally Lipschitz outside

z -1 (0), (ii) there exist α 1 , α 2 ∈ K ∞ , α 3 , χ ∈ K and c L ≥ 0 such that for all (z, w) ∈ G: α 1 (|z(t)|) ≤ V z (t) ≤ α 2 (|z(t)|), (2.44
)

for all t ∈ I z,w and V z (t) ≤ -α 3 (|z(t)|), (2.45
)

if |z(t)| ≥ χ(|w(t)|) + c L and for a.e. t ∈ I z,w \ z -1 (0)
. The functions α i , the function χ and the constant c L are respectively called Lyapunov-bounds, gain and constant.

As usual, Lyapunov functions can be used for showing practical stability. The proof of the corresponding result goes along the lines of that in [START_REF] Sontag | On Characterizations of Input-to-State Stability with Respect to Compact Sets[END_REF] and it is partly based on a well known comparison principle we next state.

Lemma 2.1 ([80]

). For any α ∈ K there exists β ∈ KL with the following property. For every s ∈ (0, ∞] and for every locally Lipschitz function

ζ : [0, s) → R + the inequality ζ(t) ≤ -α(ζ(t)), for a.e. t ∈ [0, s) implies ζ(t) ≤ β(ζ(0), t), for every t ∈ [0, s).
Moreover, if we denote η(r) = -r 1 dq min{q, α(q)} then we can select

β(r, t) = ⎧ ⎨ ⎩ η -1 (η(r) + t), if r > 0 0, if r = 0.

Theorem 2.14 (Lyapunov function). Consider any family G formed by pairs of locally Lipschitz functions. Then G is practically stable if it has a Lyapunov function.

Proof. Fix for the moment (z, w) ∈ G and define the function

V z = V (•, z) and the subset S = {t ∈ [0, T z,w )|V z (t) ≤ Γ(t)},
where

Γ(t) = α 2 (χ( sup s∈[0,t] |w(s)|) + c L ). (2.46)
First, we prove by contradiction that for all t 0 ∈ R + we have that

t 0 ∈ S =⇒ t ∈ S, ∀t ∈ [t 0 , T z,w ). (2.47)
If this is not the case, then there is t 0 ∈ S such that

t * = inf{t ∈ [t 0 , T z,w )|V z (t) ≥ Γ(t) + } < ∞,
where > 0 is small enough. Notice that the continuity of V z and property (2.44) imply that

t * > t 0 and that α 2 (|z(t * )|) > Γ(t * ).
In particular, z(t * ) = 0. It follows from the definition of Γ in (2.46) and from property (2.45) that

V z (t) ≤ -α 3 (|z(t)|), for a.e. t ∈ B 1 (t * ) ⊆ [t 0 , T z,w ) \ z -1 (0)
and where 1 is a small enough radius. This means that V z does not increase on B 1 (t * ), which contradicts the choice of t * . We conclude that our assumption in (2.47) must be true.

Let us now define

s * = inf S ≤ ∞
and assume for now that s * is finite. Notice that [s * , T z,w ) ⊆ S, indeed, this is a consequence of s * ∈ S and of the claim shown in (2.47). Therefore, property (2.44) implies that

|z(t)| ≤ γ sup s∈[0,t] |w(s)| + c = sup s∈[0,t] γ(|w(s)|) + c, (2.48)
for all t ∈ [s * , T z,w ) and where

γ = α -1 1 (2α 2 (2χ)) ∈ K, c = α -1 1 (2α 2 (2c L )) ≥ 0.
On the other hand, it is clear that S ∩ [0, s * ) is empty and, as a consequence, also z -1 (0) ∩ [0, s * ). Then, properties (2.44) and (2.45) provide

V z (t) ≤ -α(V z (t)),
for a.e. t ∈ [0, s * ) and where α = α 3 (α -1 2 ) ∈ K. We can then apply Lemma 2.1 to get a function β ∈ KL such that V z (t) ≤ β(V z (0), t), for all t ∈ [0, s * ). Hence, property (2.44) implies that

|z(t)| ≤ β(|z(0)|, t), ( 2.49)
for all t ∈ [0, s * ) and where

β(r, t) = α -1 1 ( β(α 2 (r), t)),
for all r, t ∈ R + . It is straight-forward to check that β ∈ KL. We conclude the desired inequality in (2.43) by joining inequalities (2.48) and (2.49).

We proceed similarly if s * is infinite. The only difference being that inequality (2.49) is satisfied on the whole interval [0, T z,w ). Finally, notice that β, γ and c are independent from the choice of (z, w), and that c = 0 precisely when c L = 0.

The stability of the interconnection of ISpS systems can be achieved by imposing a small-gain condition. The proof of the corresponding result in our framework follows the ideas of [START_REF] Jiang | A Lyapunov formulation of the nonlinear small-gain theorem for interconnected ISS systems[END_REF] and, hence, it requires the next lemma.

Lemma 2.2 ([63]

). Consider functions σ 1 ∈ K and σ 2 ∈ K ∞ and suppose that σ 1 (r) < σ 2 (r), for all r > 0. Then, there exists a function σ ∈ K ∞ that is continuously differentiable on r > 0 and that satisfies:

0 < ∂σ ∂r (r), σ 1 (r) < σ(r) < σ 2 (r), ∀r > 0.
Remark 2.11. The fact that the derivative of σ is strictly positive on r > 0 is crucial. This is not necessarily true for an arbitrary function K ∞ ∩ C 1 as shown in [START_REF] Kellett | A compendium of comparison function results[END_REF]. Notice that the function σ at zero is no more than continuous and that it appears in the definition of the overall Lyapunov function V in the proof below. However, this does not cause any further problems given our specific definition of Lyapunov function.

In order to state the small-gain theorem in our setting, let us define the inverse set

G -1 = {(w, z)|(z, w) ∈ G}
and suppose that:

• there is a Lyapunov function for G with corresponding Lyapunov-bounds, gain and constant: α 11 , α 12 and α 13 , χ 1 and c L1 ,

• there is a Lyapunov function for G -1 with corresponding Lyapunov-bounds, gain and constant: α 21 , α 22 and α 23 , χ 2 and c L2 .

We can then define the mixed Lyapunov gains as:

χ m1 = α 12 2χ 1 (α -1 21 ) , χ m2 = α 22 2χ 2 (α -1 11 ) . ( 2
I z,w = [0, T z,w ) → R n 1 × R n 2 .
Suppose that there exist a Lyapunov function for G and a Lyapunov function for G -1 (defined above). Given the mixed Lyapunov gains in

(2.50), if χ 1 is of class K ∞ and if χ m1 (χ m2 (r)) < r, ∀r > 0 (2.51)
then G × {0} is practically stable. That is, there exist a function β ∈ KL and a constant c ≥ 0 such that for all (z, w) ∈ G we have

|(z(t), w(t))| ≤ β(|(z(0), w(0))|, t) + c, ∀t ∈ I z,w .
Proof. The small-gain condition in (2.51) implies the existence of σ as in Lemma 2.2, that is:

0 < ∂σ ∂r (r), χ m2 (r) < σ(r) < χ -1 m1 (r), ∀r > 0. (2.52)
By Theorem 2.14, it suffices to find a Lyapunov function for the family G × {0}. Consider the Lyapunov functions V 1 and V 2 , respectively for G and for G -1 , and set

V (t, (z, w)) = max{σ(V 1 (t, z)), V 2 (t, w)}, for all t ∈ R + and (z, w) ∈ R n 1 × R n 2 .
Let us fix for the moment any (z, w) ∈ G and, as before, the notation V z 1 , V w 2 and V z,w indicates evaluation on the corresponding trajectories.

It is clear that V z,w is everywhere continuous. Moreover, V z 1 and V w 2 are zero precisely where their corresponding trajectories are zero. Hence, V z,w is also locally Lipschitz on I z,w \ (z -1 (0) ∪ w -1 (0)). We consider the following partition of the interval I z,w :

D = {t ∈ I z,w |σ(V z 1 (t)) = V w 2 (t)}, D z = {t ∈ I z,w |σ(V z 1 (t)) > V w 2 (t)}, D w = {t ∈ I z,w |σ(V z 1 (t)) < V w 2 (t)}, (2.53)
in order to write

V z,w (t) = ⎧ ⎨ ⎩ σ(V z 1 (t)), if t ∈ D z V w 2 (t), if t ∈ D w ∪ D.
The open subsets D z and D w of I z,w are contained respectively in I z,w \z -1 (0) and I z,w \w -1 (0).

In particular, this implies that V z,w is also locally Lipschitz on D z ∪ D w and, consequently, on the whole I z,w \ (z, w) -1 (0) as needed.

The following verifies property (2.44) of V z,w by using:

α 1 = 1 2 min σ α 11 • 2 , α 21 • 2 ∈ K ∞ , α 2 = max{σ(α 12 ), α 22 } ∈ K ∞ .
Indeed, the corresponding properties of V z 1 and V w 2 imply:

α 1 (|(z(t), w(t))|) ≤ α 1 (2|z(t)|) + α 1 (2|w(t)|) ≤ max{σ(V z 1 (t)), V w 2 (t)} ≤ max{σ(α 12 (|z(t)|)), α 22 (|w(t)|)} ≤ α 2 (|(z(t), w(t))|),
for all t ∈ R + . We now continue by checking property (2.45) of V z,w . For this purpose, we select the function

α 3 = 1 2 min{α 1 , α2 } ∈ K,
where α1 (0) = 0 and

α1 = ∂σ ∂r (σ -1 (α 1 )) • α 13 (α -1 12 (σ -1 (α 1 ))), α2 = α 23 (α -1 22 (α 1 ))
and we also select the non-negative constant

c L = α -1 1 (σ(α 12 (2c L1 ))) + α -1 1 (α 12 (2c L2 )).
Fix a time t 0 ∈ I z,w \ (z, w) -1 (0) where V z,w , V z 1 and V w 2 are differentiable and suppose that

|(z(t 0 ), w(t 0 ))| ≥ c L . (2.54)
It then suffices to show the inequality

V z,w (t 0 ) ≤ -α 3 (|(z(t 0 ), w(t 0 ))|). (2.55)
Using (2.53), we have three possible cases:

1) Suppose that t 0 ∈ D z , in particular, z(t 0 ) = 0. There exists 1 > 0 such that

V z,w (t) = σ(V z 1 (t)), (2.56)
for all t ∈ I z,w ∩ B 1 (t 0 ). As a consequence,

V z,w (t 0 ) = ∂σ ∂r (V z 1 (t 0 )) • V z 1 (t 0 ). (2.57) Since V w 2 (t 0 ) < σ(V z 1 (t 0 )),
the inequality in (2.52) implies that

2χ 1 (α -1 21 (V w 2 (t 0 ))) ≤ α -1 12 (V z 1 (t 0 )).
From property (2.44) of V z 1 and V w 2 it follows that

χ 1 (|w(t 0 )|) ≤ χ 1 (α -1 21 (V w 2 (t 0 ))), α -1 12 (V z 1 (t 0 )) ≤ |z(t 0 )|
and, therefore,

|z(t 0 )| ≥ 2χ 1 (|w(t 0 )|). (2.58)
On the other hand, (2.54) and (2.56) together with the properties of α 1 imply

σ(V z 1 (t 0 )) ≥ σ(α 12 (2c L1 ))
and, hence,

|z(t 0 )| ≥ 2c L1 .
Summing this inequality with (2.58) leads to

|z(t 0 )| ≥ χ 1 (|w(t 0 )|) + c L1
and, using (2.56) and (2.57), property (2.45) of V z 1 implies:

V z,w (t 0 ) ≤ - ∂σ ∂r (σ -1 (V z,w (t 0 ))) • α 13 (α -1 12 (σ -1 (V z,w (t 0 )))) ≤ -α 1 (|(z(t 0 ), w(t 0 ))|) < -α 3 (|(z(t 0 ), w(t 0 ))|).
2) Suppose that t 0 ∈ D w , in particular, w(t 0 ) = 0. The procedure in this case follows the same ideas. We have that:

V z,w (t 0 ) = V w 2 (t 0 ), V z,w (t 0 ) = V w 2 (t 0 ). Inequality (2.52) implies 2χ 2 (α -1 11 (V z 1 (t 0 ))) ≤ α -1 22 (V w 2 (t 0 ))
and, together with assumption (2.54), we have

|w(t 0 )| ≥ χ 2 (|z(t 0 )|) + c L2 .
This in turn implies that

V z,w (t 0 ) ≤ -α 23 (α -1 22 (V z,w (t 0 ))) ≤ -α 2 (|(z(t 0 ), w(t 0 ))|) < -α 3 (|(z(t 0 ), w(t 0 ))|).
3) Now suppose that t 0 ∈ D so that

V z,w (t 0 ) = σ(V z 1 (t 0 )) = V w 2 (t 0 ).
(2.59)

In particular, both z(t 0 ) and w(t 0 ) differ from zero. Just as in the previous two cases, we have:

∂σ ∂r (V z 1 (t 0 )) • V z 1 (t 0 ) < -α 3 (|(z(t 0 ), w(t 0 ))|), V w 2 (t 0 ) < -α 3 (|(z(t 0 ), w(t 0 ))|).
As a consequence, there exists 2 > 0 such that

σ(V z 1 (t)) -σ(V z 1 (t 0 )) t -t 0 < -α 3 (|(z(t 0 ), w(t 0 ))|), V w 2 (t) -V w 2 (t 0 ) t -t 0 < -α 3 (|(z(t 0 ), w(t 0 ))|),
for all t ∈ I z,w ∩ B 2 (t 0 ), t = t 0 . Then (2.59) leads to

V z,w (t) -V z,w (t 0 ) t -t 0 < -α 3 (|(z(t 0 ), w(t 0 ))|),
for all such t, which implies (2.55). Finally, notice that the functions α 1 , α 2 and α 3 and the constant c L are all independent from the choice of (z, w), and that c = 0 precisely when

c L1 = c L2 = 0.

Conclusion

This chapter introduced the problem of observer design for dynamical systems. We presented the fundamental designs in the linear case known as the Luenberger observer and the Kalman filter, and we explored a deterministic interpretation of the latter. In the case of nonlinear systems, observability can depend on the input and this led to representative system structures for which it is possible to design observers more easily. In this context, we introduced the celebrated high-gain observers and we discussed the important issue of measurement noise amplification. The final part consisted on the theory of ISS and allied properties, on the corresponding small-gain theorems and on the quantification of observer robustness with respect to noise. We concluded by proving some useful results for Chapter 4.

of disturbance robustness of an observer for state-affine systems. This class of systems is not necessarily in canonical form. Two types of bounded time-varying disturbances are distinguished: measurement noise affecting the output of the system and model uncertainties in the dynamics of the system. We investigate the robustness to noise of the high-gain Kalman observer and quantify a bound for the limiting estimation error. This bound depends on both the high-gain parameter and the system input, which should be properly excited. We present a novel approach of simultaneous off-line design of the high-gain parameter and the system input by minimizing this bound. Therefore, our method provides a new degree of freedom in the problem of noise attenuation.

The chapter is organized as follows. In the preliminaries, we state the basic results concerning observer design for state-affine systems and introduce the important concept of regular persistence of an input. We then present selected input design strategies from the literature that aim to optimize the observability for this class of systems. This is followed by our main contribution of this chapter and by the corresponding illustration of our method. Some brief concluding remarks are gathered at the end.

Preliminaries

Observer design for state-affine systems

State-affine systems are systems of the form

⎧ ⎨ ⎩ ẋ = A(u)x + B(u) y = Cx, ( 3.1)
where x(t) ∈ R n is the state, y(t) ∈ R p the output, u(t) ∈ R m a bounded and measurable input, A : R m → R n×n and B : R m → R n are continuous functions and C is a constant matrix. These systems are not uniformly observable: fixing an input u in (3.1) leads to a time-varying linear system and, thus, the observability properties should arise in the Gramian form of (2.6). Systems described by (3.1) are quite general and, in particular, include the so-called bilinear systems, for which the authors in [START_REF] Bornard | Regularly persistent observer for bilinear systems[END_REF][START_REF] Couenne | Synthèses d'observateurs de systèmes affines en l'état-Etude du flot singulier des équations d'Euler[END_REF] present a high-gain version of the Kalman filter. Their results are then generalized to state-affine systems in [START_REF] Hammouri | Observer synthesis for state-affine systems[END_REF]. Since B does not play a fundamental role in the observability properties of system (3.1), we set

B(u) = 0, ∀u ∈ R m . (3.2)
The observer from [START_REF] Hammouri | Observer synthesis for state-affine systems[END_REF] takes the form

⎧ ⎨ ⎩ ẋ = A(u)x -S -1 C (C x -y) Ṡ = -θS -A(u) S -SA(u) + C C, ( 3.3)
where θ > 0 and S(0) is a symmetric and positive definite matrix. It is straight-forward to show that

S(t) = exp (-θt)Φ u (0, t) S(0)Φ u (0, t) + exp (-θt) t 0 exp (θs)Φ u (s, t) C CΦ u (s, t)ds, (3.4)
where Φ u is the transition matrix, namely, the solution on

R + × R + to ⎧ ⎨ ⎩ ∂Φu ∂s (s, t) = A(u(s))Φ u (s, t) Φ u (t, t) = I.
Therefore, we can see that the solution to the Riccati equation S(t) remains symmetric and positive definite for all t ∈ R + . The following input constraint [START_REF] Besançon | Nonlinear observers and applications[END_REF] keeps its minimum eigenvalue away from zero. In more detail, it can be shown that there exists a constant k θ,u > 0 such that the state estimation error e = xx is bounded by

|e(t)| 2 ≤ k θ,u |e(0)| 2 exp(-θt), ∀t ≥ t 0 .
As one might expect, the influence of the tuning parameter θ and of the input u appearing in (3.3) reflects on the decay rate and transient period of the state estimation error.

Optimal input design

A classical way to face the problem of dynamic and measurement disturbances in state estimation is to find a trade-off when tuning high-gain observers. However, the system input can also be implemented with the objective of improving the observer performance. The authors in [START_REF] Scola | Input optimization for Observability of State Affine Systems[END_REF] consider a discrete-time state-affine system with equations

⎧ ⎨ ⎩ x k+1 = A(u k )x k + B(u k ) y k = C(u k )x k , (3.6)
where k ≥ 0 is an integer, x k ∈ R n the state, y k ∈ R the output and u k ∈ R the input to be designed. The functions A(u k ), B(u k ) and C(u k ) are assumed to be bounded, and the transition matrix of the homogeneous part of the state equation is defined as

Φ u (k, l) = A(u k-1 )A(u k-2 ) . . . A(u l )
for k ≥ l. If each of these matrices is invertible, then we can set

Φ u (l, k) = Φ u (k, l) -1 . Definition 3.2. An input sequence (u k ) is regularly persistent for system (3.6) if A(u k ) are
invertible matrices and if there exist an integer k 0 ≥ 0 and constants α 1 , α 2 > 0 such that

α 1 I ≤ Γ u (k, k 0 ) ≤ α 2 I, ∀k ≥ k 0 ,
where

Γ u (k, k 0 ) = k i=k-k 0 Φ u (i, k) C(u i ) C(u i )Φ u (i, k).
For such an input, it is possible to design an exponential observer which is a counterpart of the high-gain Kalman-filter for the time-continuous case [START_REF] Ticlea | State and parameter estimation via discrete-time exponential forgetting factor observer[END_REF]. Other desired properties for the input are: to be bounded, to have bounded consecutive variations and to remain close to a reference value u ref .

The following algorithm from [START_REF] Scola | Input optimization for Observability of State Affine Systems[END_REF] guarantees a prescribed degree of observability α > 0 for system (3.6) under a cyclic input of period N > n:

min u N -1 k=0 (u ref -u k ) 2 , u min < u k < u max , 0 ≤ k ≤ N -1 |u k+1 -u k | < v max , 0 ≤ k ≤ N -2 Γ(u 0 , . . . , u N -1 ) > αI, Γ(u 1 , . . . , u N -1 , u 0 ) > αI, . . . Γ(u N -1 , u 0 , . . . , u N -2 ) > αI,
where the conditions on Γ ensure a lower bound at any shifted time. Their subsequent work extensions are not based on a Gramian characterization of observability but, instead, they focus on the observability of the whole system [START_REF] Scola | Optimizing Kalman optimal observer for state affine systems by input selection[END_REF]. The main idea being that input selection can be addressed by optimization when regarded as a control problem (observability-oriented optimal control formulation).

On the other hand, the authors in [START_REF] Dufour | An explicit optimal input design for first order systems identification[END_REF] design an explicit bang-bang controller for parameter identification based on a constrained Model Predictive Control (MPC) problem. Their goal is to design an input for an efficient estimation of the unknown time parameter T > 0 in the first order linear system

⎧ ⎨ ⎩ ẋ = -1 T x + G T u y = x,
where x(t) ∈ R is the state, which coincides with the output y(t), where u(t) ∈ R is the input to be designed and G > 0 the known static gain. The augmented model with state

x a (t) = (x(t), 1 T ) results in a state-affine system of the form ⎧ ⎨ ⎩ ẋa = A a (y, u)x a y = C a x a , A a (y, u) = 0 -y + Gu 0 0 , C a = 1 0
for which a Kalman-like observer [START_REF] Besançon | Nonlinear observers and applications[END_REF] can be designed under a persistent input to obtain the estimation xa (t) = (x(t), 1 T (t) ). Their technique consists in maximizing a norm of the sensitivities of x with respect to T , that is, of x s = ∂x ∂T described by

ẋs = 1 T 2 x - 1 T x s - G T 2 u,
where x s (0) = 0. The optimal input u * is then given on a sequence of sampling times (t k ) by the optimization problem:

max u inf ≤u(t k )≤usup t k +Np t k x s (t) 2 dt,
with u inf and u sup control bounds, N p the prediction horizon and where x s (t) is computed for t > t k by using the Laplace transform at t k and substituting T by its estimation T (t k ).

The authors then proceed and give an optimal control law u * that can be explicitly found offline and that results in an effective controller even for T varying on time periods.

Finally, the problem of constructing persistently exciting inputs for general nonlinear systems is addressed in [START_REF] Besançon | A Link between Output Time Derivatives and Persistent Excitation for Nonlinear Observers[END_REF], where the author proposes a relationship between the derivatives of output variables and the notion of persistent excitation. The general theory is then illustrated for the case of state-affine systems.

Robustness optimization based on Lyapunov techniques

The aim of this first part of our work is to optimally estimate the state vector of state-affine systems that are affected by noise. That is, we consider the system

⎧ ⎨ ⎩ ẋ(t) = A(u(t))x(t) + d 1 (t) y(t) = Cx(t) + d 2 (t), (3.7)
where

x(t) ∈ R n is the state, y(t) ∈ R the output, u(t) ∈ R a continuous input, d 1 (t) ∈ R n , d 2 (t) ∈ R
⎧ ⎨ ⎩ ẋ = A(u)x + S -1 C (y -C x) Ṡ = -θS -A(u) S -SA(u) + C C, ( 3.8)
where S(0) is symmetric and positive definite and θ > 0. Unfortunately, even for such choice of S(0), the eigenvalues of S(t) can still be arbitrarily close to zero for the so-called singular inputs; inputs where observability is lost. We add the next assumption to avoid this situation.

Assumption 3.1. There exist real positive constants c 1 (u, θ) and c 2 (u, θ) such that, if t is large enough, then we have

λ max (S(t) 2 ) ≤ c 1 (u, θ), λ min (S(t)) ≥ c 2 (u, θ).
(3.9)

The explicit dependencies of c i on u and θ are usually omitted to simplify the notation.

Although the state estimation error might not converge to zero when disturbances are considered, it can still be bounded. Our goal is then to find a quantifiable bound for the estimation

error e = x -x (3.10)
in such a case. By minimizing the estimation error bound over an admissible space, we will provide a systematic way to find an optimal excitation u and an optimal observer tuning θ in terms of noise robustness.

Bound on the estimation error

The following Gronwall-type lemma is instrumental to the proof of the bound on the state estimation error.

Lemma 3.1 ([14]

). Consider the real numbers t 0 < t 1 and suppose that f, g

: [t 0 , t 1 ] → R are continuous. If h : [t 0 , t 1 ] → R is of class C 1 and if ḣ(t) ≤ f (t)h(t) + g(t), ∀t ∈ [t 0 , t 1 ], then h(t) ≤ h(t 0 ) exp t t 0 f (s)ds + t t 0 exp t s f (r)dr g(s)ds, ∀t ∈ [t 0 , t 1 ].
For the sake of clarity, we denote the limiting disturbance magnitudes as

L 1 = lim sup t→∞ |d 1 (t)|, L 2 = lim sup t→∞ |d 2 (t)|.
lim sup t→∞ |e(t)| ≤ 2L 1 c 1 + 2L 2 c 1 λ max (C C) c 2 (θ √ c 1 + λ min (C C)) .
Proof. Consider the Lyapunov function V : R + × R n → R + defined by

V (t, e(t)) = e(t)

S(t)e(t),

and its derivative along the error trajectories given by

V =2e S ė + e Ṡe =2e S(A(u)x + d 1 -A(u)x -S -1 C (Cx + d 2 -C x)) + e (-θS -A(u) S -SA(u) + C C)e.
If we rearrange the equality shown above,

V =2e SA(u)e -2e C Ce -2e C d 2 + 2e Sd 1 -θe Se -e A(u) Se -e SA(u)e + e C Ce = -e C Ce -2e C d 2 + 2e Sd 1 -θe Se.
This implies then that

√V = 1 2 √ V (-e C Ce -2e C d 2 + 2e Sd 1 -θe Se).
On the other hand, by the Cauchy-Schwarz inequality we have

e Sd 1 ≤ |Se||d 1 |, -e C d 2 ≤ |Ce||d 2 |.
Using (3.9) and denoting

μ = c 1 c 2 > 0, μ 1 = λ max (C C) c 2 ≥ 0
we obtain:

|Se| 2 = e S 2 e ≤ μV,
|Ce| 2 = e C Ce ≤ μ 1 V.
Similarly,

λ min (C C) √ c 1 V ≤ e C Ce.
Putting all this together, we get that

√V ≤ √ V 2 (-θ -μ 2 ) + √ μ|d 1 | + √ μ 1 |d 2 |,
where

μ 2 = λ min (C C) √ c 1 ≥ 0.
An inequality with this form is quite useful since Lemma 3.1 implies that for t large enough,

V (t, e(t)) ≤ exp t 4 (-θ -μ 2 ) V t 2 , e t 2 + √ μ t t 2 exp (-θ -μ 2 ) (t -s) 2 |d 1 (s)|ds + √ μ 1 t t 2 exp (-θ -μ 2 ) (t -s) 2 |d 2 (s)|ds.
Notice that for any non-zero constant r,

t t 2 exp (r(t -s)) ds = - 1 r + 1 r exp r t 2
and since V , d 1 , d 2 are bounded, we can take the superior limit on both sides of the inequality to conclude that

lim sup t→∞ V (t, e(t)) ≤ L 1 2 √ μ θ + μ 2 + L 2 2 √ μ 1 θ + μ 2 .
Since also

|e| 2 ≤ 1 c 2 V,
the terms can be expanded and arranged to conclude the result. Assumption 3.2. The matrix functional A(u(t)) is bounded. This means that

Optimization of the bound

σ(u) = sup t≥0 |A(u(t))| < ∞. (3.12)
The explicit dependency of σ on u is usually omitted to simplify the notation.

The next lemma can be deduced from the work developed in [START_REF] Besançon | Observer Synthesis for a Class of Nonlinear Control Systems[END_REF]. The proof relies on the form of S(t) given in (3.4) and on basic properties of the transition matrix.

S(t) ≤ β 1 I,
where

β 1 = |C C| θ -2σ + |S(0)|.
Moreover, if u is regularly persistent for system (3.7) with respect to (T, α, t 0 ), then for all θ > 0 and all t ≥ t 0 we have

β 2 I ≤ S(t),
where

β 2 = α exp(-T (θ + 2σ)).
Remark 3.1. Lemma 3.2 implies that Assumption 3.1 is satisfied when u is regularly persistent. In fact, we can set the constants in (3.9) as:

c 1 = β 2 1 , c 2 = β 2 .
This selection of constants c 1 and c 2 , together with Theorem 3.2, provides immediately the following corollary. we have that

lim sup t→∞ |e(t)| ≤ L 1 2(θ -2σ)|S(0)| + 2|C C| αθ(θ -2σ) exp(-T (θ + 2σ)) + L 2 2 |C C| αθ exp(-T (θ + 2σ))
.

The bound in Corollary 3.1 has an advantage over the bound in Theorem 3.2. Namely, the influence of the tuning parameter θ and of the input u in the bound is now explicit. Our goal is to define a functional that assigns to each pair (u, θ) the bound given by Corollary 3.1. The optimal design of u and θ is then given by the minimization of such a functional.

Consider a set U of regularly persistent inputs for system (3.7) parametrized by a given bounded interval

P = [p min , p max ]
and suppose that

σ * = sup u∈U σ(u) < ∞,
where σ(u) is defined as in (3.12). This constraint on U is usually set in order to respect physical limitations of the system. In a similar way, the tuning parameter θ can have mag-nitude requirements and should verify the lower bound in Corollary 3.1 given by θ > 2σ * .

Consider then the bounded interval

Θ = [θ min , θ max ]
for some limit values θ max > θ min > 2σ * . In order to define a functional

J : Θ × P → R +
by using the bound in Corollary 3.1, it suffices to assign to each u ∈ U a selection of the regularly persistent parameters T and α. We do this in a natural way: for each u ∈ U select and fix T (u) such that the set

RP(u) = {α ∈ (0, ∞)|∃ t 0 > 0 satisfying (3.5)}
is non-empty. Then simply define

α(u) = sup RP(u). (3.13)
Of course, the actual implementation of this design strategy of u and θ is not trivial and needs to be adapted to the specific system under consideration. This strategy depends on the properties of the resulting functional J. For example, suppose the parameterization of U and the assignment T (u) are done in a continuous way. Then J results in a continuous functional with compact domain and a minimum is reached.

There exist several tools available in the software Matlab to perform the minimization. For example, if J is also strictly convex then the unique global minimum is easily obtained by the "fmincon" function, which is based on interior point algorithms [START_REF] Byrd | A trust region method based on interior point techniques for nonlinear programming[END_REF]. Next we summarize the off-line design strategy for finding the optimal input u * and the optimal tuning parameter θ * using such a functional J:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1.
Choose a regularly persistent input space U ,

5.

The optimal design is obtained by:

(θ * , p * ) = arg min θ∈Θ,p∈P J(θ, p),
where J is given by the bound in Corollary 3.1 and where u * ∈ U corresponds to p * .

(3.14)

The next section shows an academic example of this off-line design strategy. The dimension of the considered state-affine system is two and the input space U is set to be a family of parametrized cosine functions.

Illustration

Let us consider system (3.7) in the specific case of n = 2 and set

A(u(t)) = 0 u(t) 0 0 , C = 1 0 . (3.15)
Suppose that the physical limit of this system does not allow inputs with values outside the interval [-1, 1]. For any input u, we obtain in this case that

σ(u) = sup t≥0 |u(t)|
and we need to define U such that σ * = 1. On the other hand, the inputs have to be carefully selected since not every input of the system defined by (3.15) is regularly persistent [START_REF] Besançon | Nonlinear observers and applications[END_REF]. Let us then choose the parametrized input space

U = {u = cos (p u • t)|p u ∈ [p min , p max ]}.
It is clear that in practice a realistic input frequency is lower and upper bounded. We use a rough approximation of the optimal frequency: p min = 0.1 and p max = 14, and we set the limit tunings: θ min = 2.1 > 2σ * and θ max = 9.

We proceed to show that the inputs u ∈ U are regularly persistent with respect to their period

T (u) = 2π p u
and we give a formula for α(u). Since A(u(t))A(u(s)) = 0 for any t, s ≥ 0, the transition matrix is simply given by

Φ u (s, t) = exp s t A(u(τ ))dτ = 1 s t u(τ )dτ 0 1 ,
see for example [START_REF] Logemann | Ordinary Differential Equations[END_REF]. It follows that the observability Gramian in the left hand side of inequality (3.5) is the symmetric matrix G(u, t) described by

G(u, t) = ⎡ ⎣ T (u) t t-T (u) s t-T (u) cos(p u • τ )dτ ds ( ) t t-T (u) s t-T (u) cos(p u • τ )dτ 2 ds ⎤ ⎦ ,
where ( ) is defined due to symmetry.

Then u is regularly persistent if we can find α > 0 such that the matrix G(u, t) -αI has non-negative trace and non-negative determinant for all t ∈ [T, T + p u ]. After some simple computations, this is equivalent to stating

2π p u + π p 3 u -2α ≥ 0, α 2 - 2π p u + 3π p 3 u α + 2π 2 p 4 u ≥ 0. (3.16)
Finally, the largest admissible α(u) is the smallest root of the polynomial on the left hand side of the inequality in (3.16).

Let us now proceed with step five of the design strategy in (3.14). We suppose that the system is affected by a two-dimensional dynamic disturbance (3.17)

Our goal now is to implement the optimal input and tuning given by (3.17), and to compare the performance of this optimal selection against other combinations of inputs and tunings that may arise from classical trail-and-error methods. The measure of performance used here is given by the mean of the norm of the real estimation error over a fix time length. Several simulations of the system were run by setting an initial error derived from x(0) = (1, 1) and

x(0) = (0, 0). Then the mean of the norm of the estimation error was computed over 50 time units for different combinations of tunings and inputs. The results can be seen in The system states and different estimations can be seen in Figure 3.

Conclusion

In this chapter, we addressed the problem of robustness to measurement noise and model uncertainties of a high-gain observer for state-affine systems. For this, we first defined a cost functional based on an upper bound for the limiting estimation error that depends on the disturbances. We then formulated a feasible optimization algorithm for the off-line design of the input and the high-gain parameter, thus obtaining: (i) a new degree of freedom, given by the input, to improve the performance of the observer with respect to disturbances, (ii) a systematic way to directly tune the observer instead of using the classical trail-and-error method.

Figure 3.2: Influence of θ and p u on the state estimation. In dotted line the states, in solid their estimations. The optimal design corresponds to the third row from the top.

Our approach, however, does not specify in general how to verify if a given input is regularly persistent or how to compute the corresponding parameters arising from this property.

Therefore, further work needs to be done in this direction by combining our work with some of the results previously discussed in this chapter. Future studies can also focus on the development of an on-line design of the observer, this due to the speed of our optimization algorithm. Finally, this work opens a new perspective for optimal input design in parameter identification problems.

Chapter 4

Observer design for nonlinear systems with output transformation

Introduction

As discussed in Chapter 2, the usual observers for linear systems are the Luenberger observer and Kalman observer which can be extended to nonlinear systems in specific forms.

Therefore, one common strategy is to look for a coordinate transformation that steers the system into a convenient form [START_REF] Gauthier | Deterministic observation theory and applications[END_REF][START_REF] Besançon | Nonlinear observers and applications[END_REF]. Early important contributions in this direction include: a linear form with output injection [START_REF] Krener | Linearization by output injection and nonlinear observers[END_REF], a bilinear form with output injection and its Kalman-like observer [START_REF] Hammouri | Bilinearization up to output injection[END_REF][START_REF] Bornard | Regularly persistent observer for bilinear systems[END_REF], and a triangular form for uniformly observable system and its Luenberger-like or high-gain observer [START_REF] Gauthier | A simple observer for nonlinear systems. Applications to bioreactors[END_REF]. Even though these observer designs mainly concern nonlinear systems, their outputs tend to be linear functions of the state. Similarly, a common goal of sensor manufacturers is achieving linearity. This is often complicated and a large number of classical sensors exhibit nonlinear behavior [START_REF] Suranthiran | Signal Conditioning With Memory-Less Nonlinear Sensors[END_REF][START_REF] Silva | Sensors and actuators: control systems instrumentation[END_REF]. For example, this is the case in: fuel cell power systems [START_REF] Arcak | A Nonlinear Observer Design for Fuel Cell Hydrogen Estimation[END_REF], image restoration [START_REF] Tekalp | Image restoration with multiplicative noise: incorporating the sensor nonlinearity[END_REF], digital imaging [START_REF] Rush | Nonlinear sensors impact digital imaging[END_REF], combustion control in automobiles [START_REF] Nwagboso | Automotive Sensory Systems[END_REF] and engineering medicine [START_REF] Kothari | Capacitive sensors for measuring the pressure between the foot and shoe[END_REF].

A popular choice of an observer for general nonlinear systems with nonlinear output is the Extended Kalman Filter (EKF) in its deterministic version. Although simple and with good noise filtering properties, only local convergence is guaranteed. The observer proposed in [START_REF] Gauthier | Deterministic observation theory and applications[END_REF] is not limited in this sense, however, the system needs to be in observability canonical form with a smooth nonlinear output. A different approach is based on the celebrated Kazantzis-Kravaris observer [START_REF] Kazantzis | Nonlinear observer design using Lyapunov's auxiliary theorem[END_REF][START_REF] Andrieu | On the Existence of a Kazantzis-Kravaris/Luenberger Observer[END_REF], which adapts Luenberger's ideas to the nonlinear framework; the design problem reduces to solving a system of partial differential equations (PDEs).

On the other hand, observer design for systems with monotonic nonlinearities is studied in [START_REF] Arcak | Nonlinear observers: a circle criterion design and robustness analysis[END_REF], where the authors remove the Lipschitz condition and avoid high-gain by using the socalled circle criterion. In [START_REF] Fan | Observer design for systems with multivariable monotone nonlinearities[END_REF], they expand these results and also consider nonlinear outputs in the presence of model uncertainties. The work in [START_REF] Açıkmeşe | Observers for systems with nonlinearities satisfying incremental quadratic constraints[END_REF] instead deals with a more general type of nonlinearities, those satisfying incremental quadratic constraints. Finally, the authors in [START_REF] Lei | High-gain observers in the presence of sensor nonlinearities[END_REF] propose a simple observer design for systems in triangular form with a nonlinear output.

This output function is not necessarily differentiable but it must satisfy an incremental sector condition.

However, these results tend to require systems in specific forms and sufficient conditions for the existence of the corresponding coordinate transformations are usually strong [START_REF] Gauthier | Deterministic observation theory and applications[END_REF][START_REF] Besançon | Nonlinear observers and applications[END_REF]. Moreover, finding the right transformations can be difficult, especially in the multioutput case. Another limitation is that measurement noise is often ignored. As seen in Chapter 2, a natural framework for studying the robustness of an observer with respect to measurement noise is that of ISS. Indeed, we can consider the error dynamics as the state and the measurement noise as the input. In this context, ISS is also referred to as disturbanceto-error stability (DES) [START_REF] Shim | Nonlinear Observers Robust to Measurement Disturbances in an ISS Sense[END_REF].

In the main part of this chapter, we consider the problem of robust observer design for nonlinear systems in the presence of a nonlinear output transformation. Our work has been the subject of [START_REF] González De Cossío | Observer design for nonlinear systems with implicit ouput[END_REF][START_REF] González De Cossío | Observer design for nonlinear systems with output transformation[END_REF].

We first suppose that a DES observer has been designed for a given nonlinear system based on an output y. This design cannot be directly implemented if this output is not available and if we instead measure a nonlinear transformation ψ(y). Therefore, we propose a new interconnected observer that estimates both: y and the system state. We suppose that ψ is a local diffeomorphism, however, ψ can be difficult to invert or its inverse might not be available in closed form. In order to study the robustness of the new observer, we consider model uncertainties on y and measurement noise on ψ(y). We use small-gain arguments to show that the new observer is asymptotically convergent to a neighborhood of the origin that depends on the amplitude of the disturbances (or convergent to zero in their absence).

Our observer design is partially inspired by the Newton-Raphson method and [START_REF] Hammouri | An observer design for a class of implicit systems[END_REF], where the authors develop an explicit observer for systems defined on a manifold given by algebraic equations. Our work is also related to that of [START_REF] Grip | Observers for interconnected nonlinear and linear systems[END_REF], where they require a Lyapunov function for the error dynamics of an observer that cannot be directly used; the output is only measured through a second linear system. We apply our general results to two families of systems: state affine systems up to output injection and systems with additive triangular nonlinearity. These families differ considerably since they represent non-uniformly and uniformly observable systems respectively. We finish by providing numerical examples for both cases.

The organization of this chapter is as follows. In the preliminaries, we start by motivating the study of systems with nonlinear output by briefly discussing the challenges in the sensor industry. We continue with a literature review concerning important results in observer design for such kind of systems, in particular, we discuss the work of [START_REF] Grip | Observers for interconnected nonlinear and linear systems[END_REF]. We then present our main contribution, together with two case studies and the corresponding numerical examples.

Finally, we write a summary of our work and selected perspectives in the last section.

Preliminaries

Motivations

Sensors or transducers are fundamental for controlling industrial processes and the final outcome strongly depends on their quality [START_REF] Silva | Sensors and actuators: control systems instrumentation[END_REF]. Popular sensors in control systems include: tachometers, resolvers, piezoelectric devices, flow meters, thermistors, etc. These instruments convert the sensed physical signal into the form of the device output, just as a piezoelectric accelerometer senses acceleration and converts it into an electric charge. It is not surprising that, as technology advances, the need of more precise sensors is constantly increasing.

However, there are several undesirable properties of classical sensors such as noise, time delay, parameter drift and, importantly, nonlinearity. Fewer attention has been paid to the latter and usually rudimentary methods are applied in order to deal with nonlinearities, this includes look-up tables and calibration curves based on polynomial approximations [START_REF] Suranthiran | Signal Conditioning With Memory-Less Nonlinear Sensors[END_REF].

Similar techniques are useful in the context of Wiener models, which consist of linear dynamics followed by a static nonlinearity; see for example [START_REF] Abu-Rmileh | Wiener sliding-mode control for artificial pancreas: A new nonlinear approach to glucose regulation[END_REF] concerning glucose regulation and [START_REF] Gómez | Wiener model identification and predictive control of a pH neutralisation process[END_REF] that studies a pH neutralisation process.

Sensors are sometimes designed so that linearity between the system state and the output is achieved. This can be very challenging due to the nonlinear nature of most physical devices, for example in: image sensors [START_REF] Tekalp | Image restoration with multiplicative noise: incorporating the sensor nonlinearity[END_REF], fiber optic displacement sensors, biosensors [START_REF] Kothari | Capacitive sensors for measuring the pressure between the foot and shoe[END_REF] and oxygen sensors, which are used for closed-loop combustion control in vehicles. Although the wide-band version of the latter does achieve linearity, the price increases considerably.

An important drawback of the use of nonlinear sensors is distortion. The early work of [START_REF] Zames | Conservation of bandwidth in nonlinear operations[END_REF] concerns communication networks and suggests that nonlinear distortion can be tackled by using invertible nonlinear filtering. The authors in [START_REF] Jayasuriya | Nonlinear filtering of signals exhibiting large fluctuations in strength[END_REF] consider the problem of recovering a band-limited signal y(t) given by the measurements of a nonlinear sensor y ψ = ψ(y(t)). For example, y(t) can be a sum of cosine functions of different amplitudes and frequencies. It is clear that the signal y(t) cannot be recover unless we set specific conditions on ψ(y). The authors show that a sufficient condition for this is that ψ(0) = 0 and that ∂ψ ∂y (y) does not change sign.

Nonlinear output systems and observers

A turning process is a material removal process for creating rotational parts that consists of a turning machine and a cutting tool. Such a process can be modeled by a nonlinear system with an output resulting in a composition of a nonlinear function ψ and a linear function h, however, the nonlinearity is often compensated statistically. Instead, the authors in [START_REF] Rotella | Nonlinear Adaptive Control for Turning Processes[END_REF] develop an adaptive nonlinear control strategy in order to tackle the problem of turning force regulation by directly considering systems of the form

⎧ ⎨ ⎩ ẋ = f (x) + g(x)u y ψ = ψ(h(x)),
where the measured output y ψ is the main cutting force and where the value h(x) represents the feed in millimeters per revolution.

On the other hand, the authors in [START_REF] Arcak | A Nonlinear Observer Design for Fuel Cell Hydrogen Estimation[END_REF] develop an observer design to estimate the partial hydrogen pressure in the anode channel of a fuel cell. Their subsequent work in [START_REF] Fan | Observer design for systems with multivariable monotone nonlinearities[END_REF] generalizes these results by considering systems of the form

⎧ ⎨ ⎩ ẋ = Ax + Gγ 1 (Hx) + ρ(y 1 , y 2 , u) y 1 = C 1 x, y 2 = γ 2 (C 2 x), (4.1)
where x(t) ∈ R n is the state, y 1 (t) ∈ R r and y 2 (t) ∈ R p 2 the outputs, and u(t) ∈ R m an input. Here, the functions γ i and ρ are assumed known. We remark that system (4.1) can model a single-link flexible joint robot with the linear outputs: motor position and velocity.

However, the position sensor is known to be nonlinear and, thus, γ 2 plays an essential role in accurate modeling.

The functions γ i : R p i → R p i are assumed continuously differentiable, ρ : R r+p 2 × R m → R n locally Lipschitz and system (4.1) forward-complete. Furthermore, suppose that there exists a constant > 0 such that the following inequalities are satisfied:

∂γ 1 ∂v (v) + ∂γ 1 ∂v (v) ≥ 0, ∀v ∈ R p 1 ∂γ 2 ∂v (v) + ∂γ 2 ∂v (v) ≥ I, ∀v ∈ R p 2 . (4.2)
One possible approach to observer design for system (4.1) would be to use the inverse of γ 2 , which exists by (4.2), and to consider a linear output system as in [START_REF] Arcak | Nonlinear observers: a circle criterion design and robustness analysis[END_REF]. However, as the authors state, it can be impossible to find the analytical expression of γ -1 2 . Instead, they directly incorporate the nonlinearity in the observer as described by

ẋ =Ax + L 1 (C 1 x -y 1 ) + L 2 (γ 2 (C 2 x) -y 2 ) + Gγ 1 (H x + K 1 (C 1 x -y 1 )) + ρ(y 1 , y 2 , u), (4.3)
where L 1 , L 2 and K 1 are matrices designed in the next theorem.

(A + L 1 C 1 + 1 2 L 2 C 2 ) P + P (A + L 1 C 1 + 1 2 L 2 C 2 ) + ζI B B 0 ≤ 0, (4.4)
where

B = P G L 2 + λ 1 (H + K 1 C 1 ) λ 2 C 2 ,
then the estimation error from observer (4.3) converges exponentially to zero.

Remark 4.1. System (4.1) can be modified in different ways. For example, we can sum in the system dynamics a term Qθ with an unknown parameter θ, and use the nonlinear output y = γ 2 (Cx + θ). This gives rise to an adaptive extension of observer (4.3), which is especially useful in the context of fuel cell models [START_REF] Fan | Observer design for systems with multivariable monotone nonlinearities[END_REF]. Another modification consists on having a linear output, that is, a system of the form

⎧ ⎨ ⎩ ẋ = Ax + Gγ(Hx) + ρ(y, u) y = Cx,
where γ satisfies an inequality as the first one in (4.2). The corresponding observer (4.3) and condition (4.4) can be easily formulated in this case, in particular, the constant does not appear. Hence, in the original case, the output nonlinearity γ 2 is compensated in Theorem 4.1 by constraining the observer gains with a lower bound involving the derivative of γ 2 .

As discussed in Chapter 2, a popular approach for observer design for nonlinear systems consists in compensating the Lipschitz nonlinearities by using high-gain. Two popular research directions concern: (i) global results under global growth conditions [START_REF] Gauthier | A simple observer for nonlinear systems. Applications to bioreactors[END_REF], and (ii) the interactions between the peaking phenomena and the nonlinearities [START_REF] Esfandiari | Output feedback stabilization of fully linearizable systems[END_REF]. The proof of Theorem 4.1 follows a different approach which does not require global Lipschitz conditions and that avoids high-gain; this approach is instead based on the circle criterion. where x(t) ∈ R n , y(t) ∈ R and u(t) ∈ R, and the feedback interconnection

u(t) = -σ(y(t), t), ( 4.6)
for σ : R × R + → R a continuous nonlinear function that satisfies the sector condition

0 ≤ σ(y, t)y ≤ y 2 M, ∀y ∈ R, ∀t ∈ R + ,
for some constant M ∈ R. Furthermore, assume that the transfer function G of system (4.5)

is minimal and that 1 + MG is strictly positive real. In this case, the circle criterion is known as the positivity theorem and it guarantees the global exponential stability at the origin of interconnection (4.5)-(4.6), see for example [START_REF] Haddad | Nonlinear dynamical systems and control: a Lyapunov-based approach[END_REF].

The previously presented results from [START_REF] Fan | Observer design for systems with multivariable monotone nonlinearities[END_REF] and other related work are all unified in [START_REF] Açıkmeşe | Observers for systems with nonlinearities satisfying incremental quadratic constraints[END_REF][START_REF] Zhao | Exponential State Observers for Nonlinear Systems with Incremental Quadratic Constraints and Output Nonlinearities[END_REF],

where the authors characterize nonlinearities with incremental quadratic inequalities. More recently, systems with output nonlinearities and high-gain observers are studied in [START_REF] Lei | High-gain observers in the presence of sensor nonlinearities[END_REF]. The authors consider nonlinear systems of the form

⎧ ⎨ ⎩ ẋ = Ax + ζ(x, u) y ψ = ψ(Cx), (4.7)
where x 1 is typically measure through a nonlinear sensor y ψ = ψ(x 1 ). This time, the nonlinear transformation ψ is constrained by the incremental sector condition: there exists a constant

x(t) ∈ R n , y ψ (t) ∈ R, u(t) ∈ R m and
β > 0 such that (y 1 -y 2 )(ψ(y 1 ) -ψ(y 2 )) ≥ β(y 1 -y 2 ) 2 , ∀y 1 , y 2 ∈ R. (4.8)
Remark 4.3. The problem of observer design for system (4.7) can be solved with different approaches: (i) inverting the function ψ and using a standard high-gain observer, (ii) implementing the change of variables z 1 = ψ(x 1 ) and z i = żi-1 , for i = 1, and using the standard high-gain observer, (iii) applying the early results from [START_REF] Gauthier | Deterministic observation theory and applications[END_REF] which allow a nonlinear output, or (iv) using the just discussed results from [START_REF] Arcak | Nonlinear observers: a circle criterion design and robustness analysis[END_REF][START_REF] Fan | Observer design for systems with multivariable monotone nonlinearities[END_REF][START_REF] Açıkmeşe | Observers for systems with nonlinearities satisfying incremental quadratic constraints[END_REF]. However, these approaches require either a closed form for ψ -1 , smoothness assumptions on ψ and ζ, complicated LMI's to be solved, or other restrictive conditions.

The authors in [START_REF] Lei | High-gain observers in the presence of sensor nonlinearities[END_REF] propose a simple high-gain observer design, whose convergence is again inspired by the circle criterion, and that is given by

ẋ = Ax + ζ(x, u) -β -1 Δ θ K(ψ(C x) -y ψ ), (4.9)
where θ > 0 is large enough and Δ θ is the diagonal matrix with diagonal entries θ, θ 2 . . . , θ n . The matrix K is carefully selected as

K = ⎡ ⎢ ⎢ ⎣ α 1 . . . α n ⎤ ⎥ ⎥ ⎦ ,
where α i > 0 are chosen so that the following polynomial has only negative real roots, Finally, we discuss a result that does not involve an output nonlinearity but that concerns observer redesign in face of an output modification. The authors in [START_REF] Grip | Observers for interconnected nonlinear and linear systems[END_REF] develop a theoretical framework motivated by the problem of position, velocity and attitude estimation using integrated satellites and inertial measurements in navigation. They consider a cascade interconnection of a nonlinear system with a linear system:

s n + α 1 s n-1 + • • • + α n-1 s + α n .
⎧ ⎨ ⎩ ẋ = f (x, u) z = h(x, u), (4.10) ⎧ ⎨ ⎩ ẇ = Aw + B u u + B z z y = Cw + D u u + D z z, ( 4.11)
where The authors assume that an observer design is readily available for system (4.10) when considering z as the output, that is,

x(t) ∈ R n , z(t) ∈ R p , u(t) ∈ R m , w(t) ∈ R
ẋ = f (x, u) + g(x, z, u), (4.12)
where g is sufficiently smooth. This assumption can be formalized by considering the corresponding error dynamics in the following condition: there exist a continuously differentiable function V : R + × R n → R + and positive constants α 1 , α 2 , α 3 and α 4 such that for all where F(t, e) = f (x(t), u(t))f (x(t)e, u(t))g(x(t)e, z(t), u(t)). However, observer (4.12) cannot be directly implemented to estimate the state x since z is only available through system (4.11), that is, y is instead measured.

(t, e) ∈ R + × R n : α 1 |e| 2 ≤ V (t,
Let us also suppose that the functions g and

d(x, u, u) = ∂h ∂u (x, u) u + ∂h ∂x (x, u)f (x, u)
satisfy Lipschitz conditions, namely, there exist positive constants L 1 and L 2 such that:

|g(x, z(t), u(t)) -g(x, ẑ, u(t))| ≤ L 1 |z(t) -ẑ|, |d(x(t), u(t), u(t)) -d(x, u(t), u(t))| ≤ L 2 |x(t) -x|, (4.14)
for all t ∈ R + , all x ∈ R n and all ẑ ∈ R p . The observer design proposed in [START_REF] Grip | Observers for interconnected nonlinear and linear systems[END_REF] is given by the interconnection:

ẋ = f (x, u) + g(x, ẑ, u), ẑ = h(x, u) + ξ, ξ = - ∂h ∂x (x, u)g(x, ẑ, u) + K z (y -C ŵ -D u u -D z ẑ), ẇ = A ŵ + B u u + B z ẑ + K w (y -C ŵ -D u u -D z ẑ), (4.15)
where where:

K = [K w , K z ]
A = A B z 0 0 , B u = B u 0 , B d = 0 I , C = C D z .
Theorem

H(s) = (Is -A + KC) -1 B d .
It is then shown that, under additional hypotheses on A, B z , C and D z and for γ large enough, there is always such a choice of K. These additional hypotheses include the detectability of the pair (A, C).

Problem statement

The aim of this second part of our work concerns observer redesign to adapt a given observer to a nonlinear transformation of the system output. Let us consider a nonlinear system of the form

⎧ ⎨ ⎩ ẋ = f (x, u) y = h(x) + d, (4.16)
where x(t) ∈ R n is the state, y(t) ∈ R p the output, u(t) ∈ R m a continuous input and d(t) ∈ R p a disturbance. We suppose that f and h are of class C 2 , that the system is forward-complete and we denote the input set by u ∈ U.

We first assume that a "robust" observer has been designed for system (4.16) and that is given by

⎧ ⎨ ⎩ ẋ = f (x, g, y, u) ġ = G(g, u), (4.17)
where x(t) ∈ R n is the state estimation, g a dynamic gain and f is locally Lipschitz.

We then consider the case where y is not directly available for measurements. Instead, we measure a nonlinear transformation ψ(y) affected by noise. This situation arises frequently in engineering processes, where nonlinear sensor transformations are common. The new system takes the form For a general nonlinear system, there is no systematic way to adapt a given observer design to output transformations. We provide a novel method of observer redesign that faces this challenge.

⎧ ⎨ ⎩ ẋ = f (x, u) y ψ = ψ(y) + d ψ , ( 4
Remark 4.6. As an example, consider the uniformly observable systems. Many observers are easily designed for this class of systems, however, they are based on a linear output. If a nonlinear transformation of the output is instead measured, the convergence of these observers is no longer guaranteed. This example is studied in Section 4.6.2.

Remark 4.7. Other solutions to our problem include: (i) coordinate change to steer system (4.18) into a convenient form. This can be difficult and it is not systematic, especially for multi-output systems [START_REF] Gauthier | Deterministic observation theory and applications[END_REF][START_REF] Besançon | Nonlinear observers and applications[END_REF]; (ii) using the EKF for system (4.18) without guarantee of its global convergence; (iii) inverting ψ and using observer (4.17). Unfortunately, the inverse of ψ might not be available in closed form or it can be difficult to compute.

We cannot implement observer (4.17) since y is not directly known. However, we require this observer to be robust with respect to measurement noise. The corresponding error dynamics are given by ė = F(t, e, d), (4.19) where e = xx and for

F(t, e, d) =f (x(t), u(t)) -f (x(t) -e, g(t), h(x(t)) + d, u(t)), (4.20)
thus, what we need is the following assumption. on e = 0, and functions ᾱ1 , ᾱ2 ∈ K ∞ and ᾱ3 , χ ∈ K such that:

ᾱ1 (|e|) ≤ V (t, e) ≤ ᾱ2 (|e|),
for all t ∈ R + and all e ∈ R n , and such that:

∂ V ∂t (t, e) + ∂ V ∂e (t, e)F(t, e, d) ≤ -ᾱ3 (|e|), whenever |e| ≥ χ(|d|) and for all t ∈ R + , e ∈ R n -{0}, d ∈ R p and all x(0) ∈ R n .
Assumption 4.1 is equivalent to the ISS of system (4.19) as in [START_REF] Sontag | On Characterizations of Input-to-State Stability with Respect to Compact Sets[END_REF] or, for time-varying Lyapunov functions, [START_REF] Edwards | On input-to-state stability for time varying nonlinear systems[END_REF]. In this context, the ISS property guarantees the graceful degradation of observer (4.17) performance in the presence of measurement noise. Observers satisfying this property were first considered in [START_REF] Sontag | Output-to-state stability and detectability of nonlinear systems[END_REF] and they are known as disturbance-to-error stable (DES) observers [START_REF] Shim | Nonlinear Observers Robust to Measurement Disturbances in an ISS Sense[END_REF]. There are methods to determine if certain observers are DES [START_REF] Alessandri | Observer design for nonlinear systems by using Input-to-State Stability[END_REF] or to redesign them if they are not [START_REF] Shim | Nonlinear observer design via passivation of error dynamics[END_REF].

New observer design

Let us consider the system with the transformed output given in (4.18). Its proposed observer design is presented as the following interconnection:

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ẋ = f (x, g, ŷ, u) ġ = G(g, u)
ẏ = ĥ(x, ŷ, y ψ , u), (4.21) where f and G are as in (4.17),

ĥ(x, ŷ, y ψ , u) = ∂ψ ∂y (ŷ) -1 ∂ψ ∂y (h(x)) ∂h ∂x (x)f (x, u) + ∂ψ ∂y (ŷ) -1 ϕ(x, u)K(y ψ -ψ(ŷ)),
and ϕ : R n × R m → R + and K : R p → R p are locally Lipschitz functions defined next. We emphasize that the observer in (4.21) only requires knowledge of y ψ and not directly of y.

= f (x, u) y = h(x) + d y y ψ = ψ(y) + d ψ System (4.18) ẋ = f (x, g, y, u) ġ = G(g, u)
Observer (4.17)

y x ẏ = ĥ(x, ŷ, y ψ , u) ẋ = f (x, g, ŷ, u) ġ = G(g, u)
ŷ New observer (4.21)

x The new observer (4.21) is represented on the bottom part as an interconnected system. Observer (4.17) requires the unavailable output y, while the new observer uses the measurements y ψ instead.

y ψ u u u u
In order to define the function ϕ, let us first consider the function φ given by

φ(x, u, d 1 , d 2 , d 3 ) = ∂ψ ∂y (h(x) + d 1 ) ∂h ∂x (x)f (x, u) + d 2 + d 3 , ( 4
× R m → R + and α ∈ K ∞ such that, |φ(x, u, d 1 , d 2 , d 3 ) -φ(x, u, 0, 0, 0)| ≤ ϕ(x, u)α(|(x -x, d 1 , d 2 , d 3 )|) (4.23)
on all its domain and we can assume that ϕ ≥ 1.

Remark 4.9. The problem of finding α and ϕ as in (4.23) is simplified if ψ and each of the functions in expression (4.22) satisfy global Lipschitz conditions. We can then take α as the identity function and

ϕ(x, u) ≥ c ϕ (|f (x, u)| + 1),
for some c ϕ ≥ 1 and for all x ∈ R n and all u ∈ R m . There is a vast literature dealing with the computation of Lipschitz constants like those defining c ϕ . Another convenient case is when system (4.18) has bounded states, given the saturation techniques explained in Remark 4. [START_REF] Açıkmeşe | Observers for systems with nonlinearities satisfying incremental quadratic constraints[END_REF].

We now continue with the definition of K. Consider a function ρ ∈ K ∞ ∩ C 2 , a positive constant k and set K : R p → R p as

K(ξ) = ⎧ ⎨ ⎩ k ρ(|ξ|) |ξ| ξ, if ξ = 0 0, if ξ = 0. (4.24)
The choice of ρ guarantees that K is locally Lipschitz. Indeed, using that ρ(0) = 0 and L'Hôpital's rule, we can show that the function

⎧ ⎨ ⎩ ρ(r) r , if r > 0 ∂ρ ∂r (0), if r = 0
is continuously differentiable on R + . The function ρ provides a degree of freedom for the design of the new observer (4.21).

Main results and proofs

Given initial conditions x(0) ∈ R n and ŷ(0) ∈ R p , there is a corresponding maximal interval of existence [0, T ) for the unique solution (x, ŷ) of (4.21). We denote the estimation errors as

e(t) = x(t) -x(t), ξ(t) = y ψ (t) -ψ(ŷ(t)),
for all t ∈ [0, T ). Notice that the solution is unbounded if T is finite, see for example [START_REF] Cronin | Ordinary Differential Equations: Introduction and Qualitative Theory, Third Edition[END_REF].

We will see that this is not the case if ρ is properly chosen. The following results concern the definitions from Section 2.5.

G = {(ξ, e)|x(0), x(0) ∈ R n , ŷ(0) ∈ R p }, ( 4

.25)

where ξ = y ψψ(ŷ) and e = xx are defined on [0, T ). Moreover, if the disturbances d y and d ψ are both zero then the corresponding Lyapunov-constant can be chosen to be zero.

Proof. We propose the Lyapunov function simply as the norm. On the trajectory ξ, it takes the form

V ξ 1 (t) = |ξ(t)|,
for all t ∈ [0, T ). It is clear that property (2.44) is satisfied by defining both α 11 and α 12 as the identity function.

We next prove that property (2.45) is also satisfied on [0, T ) \ ξ -1 (0). From the definitions of φ in (4.22) and of the new observer in (4.21) we have that

V ξ 1 = ξ |ξ| ∂ψ ∂y (y) ẏ + ḋψ - ∂ψ ∂y (ŷ) ẏ = ξ |ξ| (φ(e + x, u, d y , ḋy , ḋψ ) -φ(x, u, 0, 0, 0)) - ξ |ξ| ϕ(x, u)K(ξ),
where ξ denotes the transpose of ξ. It then follows from the construction of ϕ and K, respectively in (4.23) and (4.24), and by Remark 2.10 that

V ξ 1 ≤ ϕ(x, u)α(|(e, d y , ḋy , ḋψ )|) -kϕ(x, u)ρ(|ξ|) ≤ ϕ(x, u)(α(2|e|) + α(c d ) -kρ(|ξ|)), (4.26)
where

c d = 2(|d y | ∞ + | ḋy | ∞ + | ḋψ | ∞ ). (4.27)
On the other hand, suppose that χ 1 ∈ K is given by

χ 1 (r) = ρ -1 (2α(2r)), (4.28)
for all r ∈ R + , and set the non-negative constant

c L1 = ρ -1 (2α(c d ))
.

If |ξ| ≥ χ 1 (|e|) + c L1 , then it follows that ρ(|ξ|) ≥ α(2|e|) + α(c d ).
Since k > 1 and ϕ ≥ 1, we get that the negative term ρ(|ξ|)kρ(|ξ|) dominates the last expression in (4.26). By using this and the inequalities in (4.26) we conclude that

V ξ 1 ≤ -(k -1)ρ(|ξ|)
and, as a consequence, we can define and, therefore, we can select α 23 = ᾱ3 to get

α 13 (r) = (k -1)ρ(
G -1 = {(e, ξ)|x(0), x(0) ∈ R n , ŷ(0) ∈ R p }, ( 4
V e 2 = ∂ V ∂t (•, e) + ∂ V ∂z (•, e) ė ≤ -α 23 (|e|), (4.30)
whenever |e| ≥ χ(|ŷh(x)|). We also define the class K function

χ 2 (r) = χ(2δ(2r)),
for all r ∈ R + , and the non-negative constant

c L2 = χ(2|d y | ∞ ) + χ(2δ(2|d ψ | ∞ )).
Assumption 4.2 then implies that

χ 2 (|ξ|) + c L2 ≥ χ(2δ(|ψ(y) -ψ(ŷ)|)) + χ(2|d y | ∞ ) ≥ χ(2|y -ŷ|) + χ(2|d y | ∞ ) ≥ χ(|y -ŷ| + |d y | ∞ ) ≥ χ(|ŷ -h(x)|)
which, together with (4.30), provides the needed property.

Finally, notice that c L2 is zero precisely when the disturbances d y and d ψ are zero.

We are now ready to state our main result. It establishes a condition on the gain ρ in order to guarantee the asymptotic convergence to a neighborhood of zero (or to zero itself) of the state estimation error given by the new observer in (4.21).

χ m1 = 2χ 1 (α -1 21 ), χ m2 = α 22 (2χ 2 ). (4.33)
It is then straight-forward to verify the equivalence of this condition with the inequality in (4.31). Indeed, by using the expression of χ 1 in (4.28) and the functions defined in the proof of Lemma 4.2 we have that The following section concerns two classical and widely used families of nonlinear systems and it helps to illustrate the design of the new observer.

2χ 1 (α -1 21 (α 22 (2χ 2 (r)))) <

Particular cases

The aim of this section is to study Theorem 4.4 when system (4.16) belongs to the family of: (i) state affine systems, (ii) systems with additive triangular nonlinearity. We add the following assumption on ψ throughout this section.

Assumption 4.3. The function ψ and its derivative are both Lipschitz continuous.

This assumption is not a critical design requirement but it simplifies the computations of ϕ and α in (4.23) as explained in Remark 4.9. Moreover, it can be easily met in the case of bounded states as shown in Remark 4.13 at the end of this section.

State-affine systems up to output injection

Consider A : R m → R n×n a continuous matrix functional, η : R p × R m → R n a nonlinear and continuous function and C ∈ R p×n a constant matrix. Our goal is to estimate the state of the system given by: The following observer [START_REF] Bornard | Regularly persistent observer for bilinear systems[END_REF][START_REF] Hammouri | Observer synthesis for state-affine systems[END_REF], which cannot be directly implemented, plays the role of observer (4.17) and it is described by:

⎧ ⎨ ⎩ ẋ = A(u)x + η(Cx, u) y ψ = ψ(y) + d ψ , ( 4
⎧ ⎨ ⎩ ẋ = A(u)x + η(y, u) + S -1 C (y -C x) Ṡ = -θS -A(u) S -SA(u) + C C, (4.35)
where θ > 0 is a tuning parameter and S(0) ∈ R n×n is a symmetric and positive definite matrix. It is known that S(t) maintains these properties for all t ∈ R + .

Notice that the equation defining S in (4.35) coincides with the one in [START_REF] Besançon | Observer Synthesis for a Class of Nonlinear Control Systems[END_REF]. As in Definition 3.1, we require the input u to be regularly persistent for the homogeneous part of system (4.34) given by

⎧ ⎨ ⎩ ẋ = A(u)x y = Cx,
and with respect to some a > 0, t 0 > 0 and T ≥ t 0 . Let us consider 0 < a ≤ λ min (S(0)) to simplify and denote

b 1 = sup |u|≤cu |A(u)|,
where |u| ∞ ≤ c u . If θ ≥ 3b 1 > 0, we can deduce the following bounds as in [START_REF] Besançon | Observer Synthesis for a Class of Nonlinear Control Systems[END_REF],

s 1 I n ≤ S(t) ≤ s 2 I n , (4.36)
for all t ∈ R + and where

s 1 = a exp(-t 0 (θ + 2b 1 )), s 2 = b -1 1 |C| 2 + |S(0)| are positive constants.
We next show that observer (4.35) satisfies Assumption 4.1. We define the function V :

R + × R n → R + as V (t, e) = e S(t)e,
for all t ∈ R + and e ∈ R n . According to (4.36), we can choose the class K ∞ functions:

ᾱ1 (r) = s 1 r 2 , ᾱ2 (r) = s 2 r 2 , ( 4.37)
for all r ∈ R + . Moreover, the following holds for all t ∈ R + , e ∈ R n and d ∈ R p . On one hand, the definition of S in (4.35) provides

∂ V ∂t (t, e) = -

θe S(t)ee A(u(t)) S(t)e e S(t)A(u(t))e + e C Ce

= -θe S(t)e -2e S(t)A(u(t))e + e C Ce, (

where we used that e A(u(t)) Se, as a real number, coincides with its transpose. On the other hand, F from (4.20) takes the form

F(t, e, d) = A(u(t))e -S(t) -1 C (Ce + d) + η(Cx(t), u(t)) -η(Cx(t) + d, u(t)).
It then follows that We now define the class K ∞ functions

∂ V ∂e (t, e)F(t, e, d) =2e S(t)F(t, e, d) =2e S(t)A(u(t))e -2e C (Ce + d) + 2e S(t)(η(Cx(t), u(t)) -η(Cx(t) + d, u(t))). ( 4
χ(r) = s -1 1 r, ᾱ3 (r) = s 1 (θ -b 2 )r 2 , ( 4.40)
where

θ > b 2 = 2(|C| + c η s 2 ).
Hence, the inequality |e| ≥ χ(|d|) implies that

∂ V ∂t (t, e) + ∂ V ∂e (t, e)F(t, e, d) ≤ s 1 (-θ + 2|C| + 2c η s 2 )|e| 2 = -ᾱ 3 (|e|)
and we are required to tune θ > max{3b 1 , b 2 }, which is possible since b 1 and b 2 are independent of θ.

We can now design the corresponding new observer (4.21). Notice that in this case, the function f is globally Lipschitz in x, uniformly in u such that |u| ≤ c U . Therefore, we are in the situation of Remark 4.9 and we can choose α as the identity function and

ϕ(x, u) ≥ c ϕ (|A(u)x + η(C x, u)| + 1), (4.41)
for some c ϕ ≥ 1 and for all x ∈ R n and |u| ≤ c u . By using the functions in (4.37) and (4.40)

and by using the small-gain condition in (4.31), we get

ρ(r) = 17(s -3 1 s 2 ) 1 2 δ(4r), (4.42)
for all r ∈ R + and where δ is given in Assumption 4.2. As a consequence, K in (4.24) is given by:

K(ξ) = ⎧ ⎨ ⎩ 17k(s -3 1 s 2 ) 1 2 δ(4|ξ|)|ξ| -1 • ξ, if ξ = 0 0, if ξ = 0, (4.43)
for k > 1. The observer in (4.21) takes the form:

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ẋ = A(u)x + η(ŷ, u) + S -1 C (ŷ -C x) ẏ = ∂ψ ∂y (ŷ) -1 ∂ψ ∂y (C x)C(A(u)x + η(C x, u)) + ∂ψ ∂y (ŷ) -1 ϕ(x, u)K(y ψ -ψ(ŷ)), (4.44)
where S is as in (4.35). We summarize our results in the following corollary. Finally, we compute the explicit decay rate β and the constant c in (4.45) as explained at the end of Section 4.4. The function ρ is already given in (4.42), this is Step 1). For Steps 2) and 3), we suppose that δ is linear with slope c δ > 0 so that:

δ(r) = c δ r, ρ(r) = c ρ r, c ρ = 68(s -3 1 s 2 ) 1 2 c δ ,
for all r ∈ R + . The mixed Lyapunov gains then take the form:

χ m1 (r) = c χ m1 r 1 2 , c χ m1 = 2s -1 2 1 c -1 ρ , χ m2 (r) = c χ m2 r 2 , c χ m2 = 64s -2 1 s 2 c 2 δ ,
for all r ∈ R + . As described in Section 2.5, we can select the function between the gains as

σ(r) = c σ r 2 , c σ = c -2 χ m1 + c χ m2 2 > 0,
for all r ∈ R + . Simple computations in Steps 4) and 5) show that the decay rate is given by

β(r, t) = r √ c 1 exp - c 2 c 1 t ,
for all r, t ∈ R + and with the positive constants:

c 1 = 8 max{s 2 , c σ } min{s 1 , c σ } , c 2 = 1 4 min{2(k -1)c ρ , (θ -b 2 )s 1 s -1 2 }.
Furthermore, the constant of the practical stability is

c = 16 c 1 s 1 min {s 1 , c σ } 1 2 (|d y | ∞ + 2c δ |d ψ | ∞ ) 1 2 + 64c -1 ρ c 1 c σ min {s 1 , c σ } 1 2 (|d y | ∞ + | ḋy | ∞ + | ḋψ | ∞ ).

Systems with additive triangular nonlinearity

Let us now consider the canonical matrices A ∈ R n and C ∈ R 1×n and a nonlinear and continuous triangular function ζ : R n × R m → R n in the form of (2.19). Our goal is to estimate the state of the system:

⎧ ⎨ ⎩ ẋ = Ax + ζ(x, u) y ψ = ψ(y) + d ψ ,
|ζ i ( xi , u) -ζ i ( xi , u)| ≤ c ζi | xi -xi |,
where xi is defined below (2.20) and for all x, x ∈ R n and all u ∈ R m .

As opposed to the state-affine case, the nonlinearity ζ in system (4.46) depends on the fullstate x which complicates its estimation. The observer in (2.21) plays the role of observer (4.17) and we recall its form

⎧ ⎨ ⎩ ẋ = Ax + ζ(x, u) + S -1 ∞ C (y -C x) 0 = -θS ∞ -A S ∞ -S ∞ A + C C, (4.47)
where θ > 0 is a tuning parameter. Similar to the previous case, the matrix S ∞ is symmetric and positive definite. Let us denote the eigenvalue extrema of S ∞ by

s 1 = λ min (S ∞ ), s 2 = λ max (S ∞ )
and define S ∞,1 as the solution of the second equation in (4.47) corresponding to the unitary tuning, that is,

S ∞,1 = θ -1 Δ θ S ∞ Δ θ , (4.48)
where Δ θ is as in (2.17). Finally, we name the following maxima:

c ζ = max i=1,...,n c ζi , s m = max i,j=1,...,n |(S ∞,1 ) i,j |. ( 4.49)
We next show that observer (4.47) satisfies Assumption 4.1 by using similar techniques to those of [START_REF] Gauthier | A simple observer for nonlinear systems. Applications to bioreactors[END_REF]. In this case, the input set U can be taken simply as the set of continuous functions u : R + → R m . We define the function V : R n → R + as the norm induced by S ∞ , that is,

V (e) = (e S ∞ e) 1 2 = |e| S∞ ,
for all e ∈ R n . Clearly, we can select the class K ∞ functions given by

ᾱ1 (r) = s 1 2 1 r, ᾱ2 (r) = s 1 2
2 r, for all r ∈ R + . Moreover, the following holds for all t ∈ R + , e ∈ R n -{0} and d ∈ R. The function F from (4.20) takes the form

F(t, e, d) = Ae -S -1 ∞ C (Ce + d) + ζ(x(t), u(t)) -ζ(x(t) -e, u(t)).
It then follows that

∂ V ∂e (e)F(t, e, d) = 1 2 |e| -1 S∞ (2e S ∞ F(t, e, d)) ≤ - θ 2 |e| S∞ -|e| -1 S∞ e C d + |e| -1 S∞ e S ∞ • (ζ(x(t), u(t)) -ζ(x(t) -e, u(t))) ≤ - θ 2 |e| S∞ + s -1 2 1 |C||d| + |ζ(x(t), u(t)) -ζ(x(t) -e, u(t))| S∞ . (4.50)
On the other hand, computing the induced norm and using (4.48) we have

|ζ(x(t), u(t)) -ζ(x(t) -e, u(t))| 2 S∞ = n i,j=1 θ θ i+j (S ∞,1 ) i,j ζ(i, t, e) ζ(j, t, e), ( 4.51)
where

ζ(i, t, e) = ζ i ( xi (t), u(t)) -ζ i ( xi (t) -ēi , u(t))
. Therefore, Assumption 4.6 implies:

|ζ(x(t), u(t)) -ζ(x(t) -e, u(t))| 2 S∞ ≤ n i,j=1 θs m c 2 ζ θ i+j | ēi || ēj | ≤ n 2 s m c 2 ζ λ min (S ∞,1 ) -1 |e| 2 S∞ ,
where we used the inequality |θ -i ēi | ≤ |Δ -1 θ e|. Putting (4.50) and (4.51) together,

∂ V ∂e (e)F(t, e, d) ≤ - θ 2 |e| S∞ + s -1 2 1 |C||d| + ns 1 2 m c ζ λ min (S ∞,1 ) -1 2 |e| S∞ ≤ -ᾱ 3 (|e|),
whenever |e| ≥ χ(|d|) and where:

χ(r) = s -1 1 r, ᾱ3 (r) = s 1 2 1 2 (θ -b)r,
for all r ∈ R + and for

θ > b = 2 ns 1 2 m c ζ λ min (S ∞,1 ) -1 2 + 1 .
Notice that such θ exists since S ∞,1 corresponds to the unitary tuning.

In order to construct the new observer (4.21) and given that we are in the case of Remark 4.9, we can choose α as the identity function and:

ϕ(x, u) ≥ c ϕ (|Ax + ζ(x, u)| + 1), ρ(r) = 17(s -3 1 s 2 ) 1 2 δ(4r)
for some c ϕ ≥ 1 and all x ∈ R n , u ∈ R m and r ∈ R + . The new observer (4.21) then takes the form:

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ẋ = Ax + ζ(x, u) + S -1 ∞ C (ŷ -C x) ẏ = ∂ψ ∂y (ŷ) -1 ∂ψ ∂y (C x)C(Ax + ζ(x, u)) + ∂ψ ∂y (ŷ) -1 ϕ(x, u)K(y ψ -ψ(ŷ)), (4.52)
where S is as in (4.47) and K as in (4.24) with k > 1.

m c ζ λ min (S ∞,1 ) -1 2 + 1 ,
where c ζ and s m are as in (4.49), there exist a function β ∈ KL and a constant c ≥ 0 such that for all x(0), x(0) ∈ R n and all ŷ(0) ∈ R the estimation errors e = x-x and ξ = y ψ -ψ(ŷ) are defined on R + and

|(ξ(t), e(t))| ≤ β(|(ξ(0), e(0))|, t) + c, ∀t ≥ 0.
Moreover, if the disturbances d y and d ψ are both zero then c is zero as well.

Proof. It follows from our previous development and from Theorem 4.4. Indeed, we showed that Assumption 4.1 is satisfied by using the specific forms of A and C and by using Assumption 4.

i : R i × R m → R as ζ s i (x 1 , . . . , x i , u) = ζ i (μ sat(μ -1 x 1 ), . . . , μ sat(μ -1 x i ), u),
where sat : R → R is given by sat(t) = min{1, |t|} sign(t). Replacing ζ by ζ s defines an equivalent system and the latter function satisfies Assumption 4.6. We can proceed similarly for ψ and Assumption 4.3 but using a non-constant and smooth saturation.

Numerical results

In this section, we study specific cases of systems (4.34) and (4.46) in order to provide numerical examples of Corollaries 4.1 and 4.2.

Example for state-affine systems

Let us suppose that system (4.34) is given by:

A(u) = 0 u 0 0 , η(y, u) = sin(y) u 2 , C = 1 0 ,
for all y, u ∈ R. The input and the nonlinear function are defined as:

u(t) = cos(t), ψ(y) = sin(y) + 2y.
The function ψ is a diffeomorphism whose inverse has no closed form, and it is straightforward to check that Assumptions 4.2, 4.3 and 4.4 are satisfied. Moreover, we can select δ as the identity function and it can be shown that u is regularly persistent, see for example [START_REF] González De Cossío | Optimal observer design for disturbed state affine systems[END_REF]. The lower bound of θ in Corollary 4.1 has a value of 6 when S(0) = I 2 . A simple computation shows that, according to (4.41) and (4.43), we can select

ϕ(x, u)K(ξ) = λ(|x 2 | + 1) • ξ, ( 4.53)
for all x ∈ R 2 and u, ξ ∈ R and where λ > 0 is to be tuned.

We initialize the system and the observers at: x(0) = (10, 0), x(0) = (4, 9) and ŷ(0) = C x(0) noise robustness. This is not surprising since noise amplification is a common problem for high-gain observers when their tuning parameter θ is large [START_REF] Ahrens | High-gain observers in the presence of measurement noise: A switched-gain approach[END_REF]. The noise, in contrast, is averaged out by the definition of ŷ.

These figures show that, for λ large enough, the new observer in (4.44) recovers the performance of observer (4.35). Furthermore, the design of the new observer (4.44) seems to be more robust against measurement noise.

Example for systems with triangular nonlinearity

Here we consider the example studied by the authors in [START_REF] Lei | High-gain observers in the presence of sensor nonlinearities[END_REF]. The system is in the form of (4.46) and it is described by:

A = 0 1 0 0 , C = 1 0 , ζ(x, u) = 0 -x 1 -2x 2 + ax 2 1 x 2 + u ,
for all x ∈ R 2 and u ∈ R and for some a > 0. They also define the input and the nonlinear output as:

u(t) = b sin(2t), ψ(y) = 1 3 y 3 - 1 2 y 2 + y,
for some b > 0. As opposed to [START_REF] Lei | High-gain observers in the presence of sensor nonlinearities[END_REF], we consider disturbances. Notice that ψ is a diffeomorphism whose derivative is bounded from below by 0.75 and, therefore, Assumption 4.2 is met. However, ψ does not satisfy Assumption 4.3 and ζ satisfies only locally Assumption 4.6.

As in [START_REF] Khalil | Nonlinear Control[END_REF] or [START_REF] Lei | High-gain observers in the presence of sensor nonlinearities[END_REF], for a = 0.25 and b = 0.2, every state starting at

X = {x ∈ R 2 |1.5x 2 1 + x 1 x 2 + 0.5x 2 2 ≤ √ 2}
remains in that set for all positive times. Therefore, in Remark 4.13 we can replace ψ and ζ by their saturated versions ψ s and ζ s . These new functions satisfy Assumptions 4.2, 4.3 and 4.6. Similar to the first case, we set

ϕ(x, u)K(ξ) = λ(|x 1 | + |x 2 | + |x 2 1 x2 | + 1) • ξ,
for all x ∈ R 2 and u, ξ ∈ R and where λ > 0. On the other hand, the observer from [START_REF] Lei | High-gain observers in the presence of sensor nonlinearities[END_REF] is given by

ż = Aẑ + 0 ζ 0 (ẑ, u) + (8/3) -1 (4/3) -2 (y ψ -ψ(C ẑ)), (4.54)
where

ζ 0 (ẑ, u) = -ẑ 1 -2ẑ 2 + a sat(ẑ 2 1 ẑ2 ) + u, y ψ = ψ(Cx + d y ) + d ψ and
where > 0 is to be tuned.

In order to make a fair comparison of observers (4.52) and (4.54): first we choose two far enough values of , then we try to improve or to match this performance by tuning θ and λ. We initialize the system and the observers at: x(0) = (1, -1), x(0) = ẑ(0) = (0, 0) and ŷ = C x(0) and we fix the value λ = 20. The results can be seen in Figures 4. [START_REF] Ahrens | High-gain observers in the presence of measurement noise: A switched-gain approach[END_REF] and observer (4.54) render quite similar estimations that converge to the system states. Furthermore, higher tuning of θ and -1 leads to faster state reconstruction at the price of higher peaking (right column). The third row shows that tuning θ does not have a very strong effect in the estimation of y given by ŷ. These figures suggest that both observers have similar performances if they are properly tuned. It is clear that observer (4.54) has a simpler design. Nevertheless, our methodology provides a new observer design for a much more general family of systems.

Conclusion

The method presented in this chapter concerns observer redesign for nonlinear systems in the presence of output transformations. The new observer consists in an interconnection of the original observer dynamics with an estimator of the unavailable output. By using the small-gain theorem, we showed that the new observer converges asymptotically to a neighborhood of zero that depends on the amplitude of the disturbances and their derivatives.

We then studied the new observer design for two major families of systems that differ in their observability properties. The simulations showed that the new observer recovers the performance of the initial observer and that its performance is comparable to that of other related observers. Unlike previous studies on observer design for systems with nonlinear outputs: (i) the generality of our approach considers systems without a specific form at the cost of a more intricate design, (ii) we quantified the influence of different types of system disturbances in our design and concluded its robustness.

Future work could focus on relaxing the conditions on the initial observer by considering weaker versions of the DES property, such as quasi-DES developed in [START_REF] Shim | Nonlinear Observers Robust to Measurement Disturbances in an ISS Sense[END_REF]. Another promising direction consists on using the results from [START_REF] Bernard | Expressing an Observer in Preferred Coordinates by Transforming an Injective Immersion into a Surjective Diffeomorphism[END_REF] concerning the augmentation of an immersion into a diffeomorphism. Indeed, this could be used to extend our observer redesign to the case of a system with bounded state and an output transformation ψ : R p 1 → R p 2 with p 1 < p 2 . However, our new observer would require a modification as in [START_REF] Astolfi | Output feedback stabilization for SISO nonlinear systems with an observer in the original coordinates[END_REF] in order to guarantee that ŷ remains in a convenient region. Another interesting work direction concerns observer design for nonlinear systems with implicit output. That is, we can suppose that the output y is measured and related to the state through an implicit equation of the form ψ(y, x) = 0; such a formulation arises, for example, in the field of dynamic vision [START_REF] Matveev | Observers for systems with implicit output[END_REF]. like design from [START_REF] Hammouri | Observer synthesis for state-affine systems[END_REF]. In this setting, a new interesting trade-off arises between the high-gain parameter and the measurement step size.

On the other hand, the authors in [START_REF] Raff | Observer with sample-and-hold updating for Lipschitz nonlinear systems with nonuniformly sampled measurements[END_REF] develop a Luenberger-like observer for Lipschitz nonlinear systems with a discrete output by implementing a sample-and-hold technique.

Their method, based on Lyapunov analyses, is inspired by sampled-data control techniques and LMI's. Instead of holding the last measurement, the authors in [START_REF] Karafyllis | From Continuous-Time Design to Sampled-Data Design of Observers[END_REF] suggest to use a given observer design for a general nonlinear system based on a continuous-time output and to interconnect it with an output predictor in case the measurements are instead discrete. This observer redesign preserves the robustness properties of the original observer for small enough sampling periods. In order to tackle sampling period constraints, the authors in [START_REF] Andrieu | Self-triggered continuous-discrete observer with updated sampling period[END_REF] develop a self-triggered observer design for systems with triangular nonlinearities and output measured at variable sampled times. That is, the sampling time is used as an additional tuning parameter of the observer. Finally, robust observer design for nonlinear networked control systems (NCS's) can be found in [START_REF] Maass | Observer design for non-linear networked control systems with persistently exciting protocols[END_REF]. The authors derive sufficient conditions on the network in order to preserve the stability of a given observer under communication constraints.

In this chapter, we extend our previous approach to deal simultaneously with both output nonlinearities and output non-uniform discretization in time. Our work has been the subject of a communication [START_REF] González De Cossío | Observer design for nonlinear systems with sampled and transformed measurements[END_REF].

One way to solve this problem consists in applying the results of the previous chapter together with the observer redesign from [START_REF] Karafyllis | From Continuous-Time Design to Sampled-Data Design of Observers[END_REF], which tackles the case of a discretized output. Nevertheless, this would result in an observer with three interconnections when it is expected that only two may suffice. Therefore, we instead propose to directly extend our observer design from Chapter 4 by implementing the output ψ(y(t k )) with sample-and-hold. The resulting interconnected observer consists of: (i) a subsystem with continuous dynamics based on the structure of the original observer, (ii) a subsystem with switched dynamics arising from the sample-and-hold techniques. Its convergence is shown by defining a Lyapunov-Krasovskii functional, based on [START_REF] Raff | Observer with sample-and-hold updating for Lipschitz nonlinear systems with nonuniformly sampled measurements[END_REF][START_REF] Naghshtabrizi | Exponential stability of impulsive systems with application to uncertain sampled-data systems[END_REF], and by using the small-gain theorem for switched systems [START_REF] Yang | A Lyapunov-based small-gain theorem for interconnected switched systems[END_REF].

This approach leads to two LMI's that depend, among other parameters, on the maximum time between two consecutive samplings.

The organization of this chapter is as follows. We first review the classical results involving nonlinear systems with a discrete output, this includes a general observer redesign method that addresses the transition from continuous to discrete measurements. We then proceed with the introduction of a Lyapunov functional that arises in the context of control and sampled-data systems, and we present in detail the known small-gain theorem for interconnected switched systems. The subsequent section includes our main contribution, which is illustrated by a case study and a numerical example. Finally, we gather some concluding remarks concerning our work in the last section.

Preliminaries

Observer design for systems with sampled output

Let us consider again the representative form for uniformly observable systems but this time with a sampled output, that is,

⎧ ⎨ ⎩ ẋ(t) = Ax(t) + ζ(x(t), u(t)) y(t k ) = Cx(t k ), (5.1)
where x(t) ∈ R n , y(t k ) ∈ R and u(t) ∈ R m , for A, C and ζ in the canonical form (2.19) and where (t k ) is a strictly increasing sequence in R + . We suppose that t 0 = 0 and t k+1 = t k + δ for a constant δ > 0 that can be chosen. Let us further assume that the lower triangular function ζ is Lipschitz continuous in x, uniformly in u, so that there is a constant

L > 0 such that |ζ(x, u) -ζ(x, u)| ≤ L|x -x|, ∀x, x ∈ R n , ∀u ∈ R m .
(5.2)

To simplify the notation, let us denote the left limit of a function f : [s, t) → R n at t as

f (t -) = lim r→t -f (r).
We have seen in Chapter 2 that system (5.1) admits a Luenberger-like observer in case of a continuous output y(t). The authors in [START_REF] Nadri | Observer Design for Uniformly Observable Systems With Sampled Measurements[END_REF] propose a constant gain observer for the discrete output case given by:

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ẋ(t) = Ax(t) + ζ(x(t), u(t)), t ∈ [t k , t k+1 ) x(t k+1 ) = x(t - k+1 ) -ρδS -1 ∞ C (C x(t - k+1 ) -y(t k+1
)) S ∞ = exp(-θδ) exp(-θδA )S ∞ exp(-θδA) + ρδC C, (5.3) where θ and ρ are positive tuning parameters and where S ∞ is symmetric and positive definite.

Crucially, this gain matrix is also constant.

Theorem 5.1 ([87]). Consider system (5.1) and suppose that ζ satisfies (5.2). There exist explicit functions α 1 : R + → R + and α 2 : R + → R + such that for every positive numbers r, δ and ρ, and for every θ satisfying max 1, 2L nα 2 (r)α 1 (r) -1 < θ ≤ δ -1 min{r, α 1 (r)}, the state estimation from observer (5.3) converges exponentially to zero. Remark 5.1. It can be shown that the matrix S ∞ arises as the limit of S(t k ) when k → ∞, that is, of the solution to the continuous-discrete Ricatti equation

⎧ ⎨ ⎩ Ṡ(t) = -θS(t) -A S(t) -S(t)A, t ∈ [t k , t k+1 ) S(t k+1 ) = S(t - k+1 ) + ρδC C,
where S(0) is symmetric and positive definite. Hence, the proof of Theorem 5.1 relies on bounds of the form

ρα 1 (r)I ≤ θ -1 Δ θ S(t)Δ θ ≤ ρα 2 (r)I.
A different study concerns continuous-discrete state-affine systems of the form

⎧ ⎨ ⎩ ẋ = A(u)x + B(u) y(t k ) = Cx(t k ), (5.4)
where x(t) ∈ R n , y(t k ) ∈ R p and u(t) ∈ R m , A : R m → R n×n and B : R m → R n are continuous functions, and C a constant matrix. As opposed to the previous case, the system output here is allowed to be multidimensional. Let us assume that the sequence (t k ) is as before and uniformly spaced by a constant δ > 0. The observer proposed in [START_REF] Nadri | Design of a continuous-discrete observer for state affine systems[END_REF] has the form

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ẋ(t) = A(u(t))x(t) + B(u(t)), t ∈ [t k , t k+1 ) x(t k+1 ) = x(t - k+1 ) -ρδ k S -1 (t k+1 )C (C x(t - k+1 ) -y(t k+1 )) Ṡ(t) = -θS(t) -A(u(t)) S(t) -S(t)A(u(t)), t ∈ [t k , t k+1 ) S(t k+1 ) = S(t - k+1 ) + δC C, (5.5)
where θ and ρ are positive tuning parameters and where S(0) is a symmetric and positive definite matrix. Notice that, as expected, the matrix gain is dynamic in this case. The previous results can be seen as extensions of their continuous output analogues presented in Chapters 2 and 3. An observer redesign method for adapting a general observer to a discrete output can be found in [START_REF] Karafyllis | From Continuous-Time Design to Sampled-Data Design of Observers[END_REF]. There, the authors first consider a forward-complete nonlinear system of the form

⎧ ⎨ ⎩ ẋ = f (x) y = h(x) + d, (5.6)
where x(t) ∈ R n , y(t) ∈ R and d(t) ∈ R, and for functions f and h, respectively of class C 1 and C 2 , such that f (0) = 0 and h(0) = 0. They assume that a robust observer is readily available for system (5.6) and that it is given by

⎧ ⎨ ⎩ ż = f (z, y) x = ĥ(z), (5.7)
where x(t) ∈ R n is the state estimation and z(t) ∈ R k , and where f and ĥ are functions of class C 1 satisfying f (0, 0) = 0 and ĥ(0) = 0.

The robustness of observer (5.7) with respect to measurement noise can be defined in an ISS style as in Chapter 2 but with some important differences. For this, let us introduce the function classes K + and N which respectively denote the positive and continuous functions ρ : R + → R + , and the non-decreasing continuous functions ρ : R + → R + such that ρ(0) = 0.

Definition 5.1. System (5.7) is called a robust observer for system (5.6) with respect to measurement errors if the following holds:

(i) there exist σ ∈ KL, γ, p ∈ N , μ ∈ K + and a ∈ K ∞ such that for all x(0) ∈ R n , all

z(0) ∈ R k and all d ∈ L ∞ loc,1 , |x(t) -x(t)| ≤ σ(|x(0)| + |z(0)|, t) + sup s∈[0,t] γ(|d(s)|), ∀t ≥ 0, |z(t)| ≤ μ(t) a(|x(0)| + |z(0)|) + sup s∈[0,t]
p(|d(s)|) , ∀t ≥ 0.

(ii) for all x(0) ∈ R n there exists z(0) ∈ R k such that,

d = 0 =⇒ x(t) = x(t), ∀t ≥ 0.
The goal is then to estimate the state of system (5.6) in the case of a discrete output, that is, of a system of the form

⎧ ⎨ ⎩ ẋ = f (x) y(t k ) = h(x(t k )) + d(t k ), (5.8)
where (t k ) is an increasing sequence in R + such that t 0 = 0 and t k+1t k ≤ r for a constant r > 0. The corresponding observer redesign proposed in [START_REF] Karafyllis | From Continuous-Time Design to Sampled-Data Design of Observers[END_REF] is given by

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ż(t) = f (z(t), w(t)), t ∈ [t k , t k+1 ), x(t) = ĥ(z(t)), t ∈ [t k , t k+1 ), ẇ(t) = L f h(x(t)), t ∈ [t k , t k+1 ), w(t k+1 ) = y(t k+1 ), (5.9)
where we recall that the Lie derivative of h along f is defined as L f h(x) = ∂h ∂x (x)f (x), for all x ∈ R n . Therefore, observer (5.9) uses the structure of observer (5.7) and, to deal with the output discretization, instead of y(t) it employs w(t) as an output predictor.

We say that observer (5.9) is a robust sampled-data observer for system (5.8) with respect to measurement errors if similar conditions to the ones from Definition 5.1 hold. Indeed, it suffices to consider a function τ in the sampling sequence:

t k+1 = t k + r exp(-τ (t k )), ∀k ≥ 0 (5.10)
and to do the following changes: part (i) also holds for all non-negative τ ∈ L ∞ loc,1 and |w(0)| is added to |x(0)| + |z(0)|, and part (ii) requires z(0) and w(0) to work for all such τ . Theorem 5. 3 ([68]). Suppose that system (5.6) is forward-complete and that it has a robust observer (5.7) with respect to measurement errors. Furthermore, suppose there exist a constant c ≥ 0 satisfying rc < 1, (5.11) and σ ∈ KL such that for all x(0

) ∈ R n , all z(0) ∈ R k and all d ∈ L ∞ loc,1 , |L f h(x(t)) -L f h(x(t))| ≤ σ(|x(0)| + |z(0)|, t) + c sup s∈[0,t] |d(s)|, ∀t ≥ 0,
where x is given by (5.7). Then, (5.9) is a robust sampled-data observer for system (5.8) with respect to measurement errors.

The previous theorem can be applied, for example, to the case of triangular Lipschitz systems

ẋ = Ax + ζ(x) -Δ θ K(C x -y),
where K is such that A -KC is Hurwitz, θ > 0 and Δ θ is as in (2.17). Thus, there exists a symmetric and positive definite matrix P and a constant μ > 0 such that

P (A -KC) + (A -KC) P + 2μI ≤ 0
and it can be shown that, for θ large enough, condition (5.11) constrains the sampling sequence with the inequality

2r(L + θ) |P ||K| μ λ max (P ) λ min (P ) < 1,
where r > 0 is as in (5.10). Notice that the latter illustrates the intuition behind Theorem 5.3, that is, it is possible to adapt an observer design for a continuous output to the case of relatively small sampling periods.

Stability of sampled-data systems

Sampled-data systems also arise as processes coupled with state-feedback controllers based on sample-and-hold blocks as in [START_REF] Naghshtabrizi | Exponential stability of impulsive systems with application to uncertain sampled-data systems[END_REF]. Consider a linear time-invariant system

ẋ = Ax + Bu,
where x(t) ∈ R n is the state and u(t) ∈ R m the input, and for A and B matrices of the appropriate dimensions. Let us suppose that (t k ) is a strictly increasing sampling sequence starting at t 0 = 0, that there is a constant δ > 0 such that t k+1t k ≤ δ, ∀k ≥ 0 (5.12) and let us denote the time since the last sampling as

ρ(t) = t -t k , t ∈ [t k , t k+1 ).
The piece-wise constant control u(t) = Kx(t k ), for t ∈ [t k , t k+1 ), is given by a gain matrix K and the closed-loop can be represented as an impulsive system with an enlarged state

ξ(t) = ξ 1 (t) ξ 2 (t) = x(t) x(t k ) , ∀t ∈ [t k , t k+1 ),
and whose dynamics are given by

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ξ(t) = F ξ(t), t ∈ [t k , t k+1 ), ξ(t k+1 ) = ⎡ ⎣ ξ 1 (t - k+1 ) ξ 1 (t - k+1 ) ⎤ ⎦ , ( 5.13)
where

F = A BK 0 0 .
Notice that, by continuity of x, we have in fact ξ 1 (t - k+1 ) = x(t k+1 ) and that at each sampling time t k+1 the control is restarted to ξ 2 (t k+1 ) = x(t k+1 ).

The authors in [START_REF] Naghshtabrizi | Exponential stability of impulsive systems with application to uncertain sampled-data systems[END_REF] propose a Lyapunov function V : R 2n × [0, δ] → R + for system (5.13) that consists of three parts:

V (ξ, ρ) = V 1 (ξ 1 ) + V 2 (ξ, ρ) + V 3 (ξ, ρ), where ξ = (ξ 1 , ξ 2 ) ∈ R n × R n and where V 1 (ξ 1 ) = ξ 1 P ξ 1 , V 2 (ξ, ρ) = ξ 0 -ρ (s + δ)(F exp(F s)) RF exp(F s)ds ξ V 3 (ξ, ρ) = (δ -ρ)(ξ 1 -ξ 2 ) Q(ξ 1 -ξ 2 ),
for symmetric and positive definite matrices P , R and Q to be designed, and for R = R 0 0 0 .

In particular, we have that

V 2 (ξ(t), ρ(t)) = t t-ρ(t) (δ -t + s) ẋ(s) R ẋ(s)ds, ∀t ≥ 0
which is closely related to Lyapunov functions appearing in delay differential equations and network control systems. For the following theorem, whose proof relies on V , we use the notation:

M 1 = P 0 F + F P 0 - I -I Q I -I -N I -I - I -I N + δ F R F, M 2 = I -I Q F + F Q I -I , F = A BK .
(5.14)

Theorem 5.4 ([88]). Consider system (5.13) and suppose there exist a matrix N and sym-metric and positive definite matrices P , R and Q such that

M 1 + δM 2 < 0, M 1 δN δN -δR < 0, (5.15)
where M 1 and M 2 are as in (5.14). Then, there exist positive constants c and λ such that for all ξ 1 (0) = ξ 2 (0) ∈ R n we have

|ξ(t)| ≤ c|ξ(0)| exp(-λt), ∀t ≥ 0.
Notice that the Lyapunov function along the trajectory V (ξ, ρ) is not standard in the sense that it is discontinuous at each sampling time t k . Nevertheless, it satisfies desirable properties whenever (5.15) is satisfied: there exist positive constants c 1 , c 2 and c 3 such that

c 1 |x(t)| 2 ≤ V (ξ(t), ρ(t)) ≤ c 2 |ξ(t)| 2 , ∀t ≥ 0, V (ξ(t), ρ(t)) ≤ -c 3 V (ξ(t), ρ(t)), ∀t ≥ 0.
Thus, V (ξ, ρ) is lower bounded by a function of the norm of x and not of ξ, which turns out to suffice. Crucially, it also holds

V (ξ(t k ), 0) ≤ lim t→t - k V (ξ(t), ρ(t)), ∀k ≥ 0.
In our work corresponding to this chapter, we will use a Lyapunov function that is partly inspired by V and the studies from [START_REF] Raff | Observer with sample-and-hold updating for Lipschitz nonlinear systems with nonuniformly sampled measurements[END_REF].

Remark 5.2. Letting δ tend to zero in (5.12) leads to a continuous control and the matrix conditions in (5.15) can be shown equivalent to (A + BK) P + P (A + BK) < 0 for a symmetric and positive definite matrix P . This is consistent with the stability of the closed-loop system ẋ = (A + BK)x.

Small-gain theorem for interconnected switched systems

A switched system consists of: (i) a family of dynamical systems indexed by an arbitrary set P and described by ẋ = f p (x, u), p ∈ P, where x(t) ∈ R n , u(t) ∈ R m and for f p locally Lipschitz such that f p (0, 0) = 0, (ii) a switching signal σ : R + → P which is a piece-wise constant and right-continuous function.

The switched system then has the form

ẋ(t) = f σ(t) (x(t), u(t)), t ≥ 0.
It is assumed that u ∈ L ∞ loc,m and that the set of switching time instants has no accumulation points.

Input-to-state stability has been studied for switched nonlinear systems since the early work of [START_REF] Liberzon | ISS and integral-ISS disturbance attenuation with bounded controls[END_REF] and continues to be a subject under intense research. The authors in [START_REF] Yang | A Lyapunov-based small-gain theorem for interconnected switched systems[END_REF] provide a Lyapunov based small-gain theorem for interconnected switched systems, and a trajectory approach can be found in [START_REF] Dashkovskiy | Trajectory-based small gain theorems for ISpS and ISS of large-scale networks of switched systems with arbitrary switchings[END_REF]. In what follows, we discuss the Lyapunov based result.

Consider systems given by

⎧ ⎨ ⎩ ẋ1 = f 1,σ 1 (x 1 , x 2 ) ẋ2 = f 2,σ 2 (x 2 , x 1 ), (5.16)
where x i ∈ R n i and for f i,σ i as above. Although it is possible to allow both switched systems to contain ISS and non-ISS subsystems, here we focus in the case that all subsystems are ISS.

Suppose that for each different i, j ∈ {1, 2} there exists a family of class C 1 functions V i,p i :

R n i → R + , p i ∈ P i , satisfying the following conditions:

1. ∃ α i,1 , α i,2 ∈ K ∞ such that for all x i ∈ R n i and all p i ∈ P i , α i,1 (|x i |) ≤ V i,p i (x i ) ≤ α i,2 (|x i |), 2. ∃ φ i ∈ K ∞ , c i > 0 such that for all x i ∈ R i , all x j ∈ R n j and all p i ∈ P i , |x i | ≥ φ i (|x j |) =⇒ ∂V i,p i ∂x i (x i )f i,p i (x i , x j ) ≤ -c i V i,p i (x i ),
3. ∃ μ i ≥ 1 such that for all x i ∈ R n i and all p i , q i ∈ P i ,

V i,p i (x i ) ≤ μ i V i,q i (x i ), 4. ∃ τ a,i > 0, N 0,i ∈ Z + such that for all t 2 ≥ t 1 ≥ 0, N i (t 2 , t 1 ) ≤ N 0,i + t 2 -t 1 τ a,i , ln(μ i ) τ a,i < c i ,
where

N i (t 2 , t 1 ) is the number of switchings of σ i in (t 1 , t 2 ].
Notice that conditions 1)-3) concern the dynamics of each switched system independently, while the small-gain condition is a joint requirement. On the other hand, 4) is a condition on the signals that limits their switching speed. We conclude this section with the statement of the theorem.

Theorem 5.5 (small-gain [START_REF] Yang | A Lyapunov-based small-gain theorem for interconnected switched systems[END_REF]). Consider system (5.16), suppose that 1) -4) hold and define the mixed gains

χ m i (r) = α i,2 (φ i (α -1 j,1 (r))) exp(N 0,i ln(μ i )), ∀r ≥ 0 for i, j ∈ {1, 2} such that i = j. If the small-gain condition χ m 1 (χ m 2 (r)) < r, ∀r > 0
is satisfied, then system (5.16) is globally asymptotically stable. That is, there exists β ∈ KL such that for all x 1 (0

) ∈ R n 1 and all x 2 (0) ∈ R n 2 , |(x 1 (t), x 2 (t))| ≤ β(|(x 1 (0), x 2 (0))|, t), ∀t ≥ 0.

Problem statement

The aim of this third and final part of our work is to extend the results from the previous chapter concerning observer redesign. In this case, we suppose that the system output is transformed by a nonlinear function and available only at discrete times. We consider a nonlinear system of the form

⎧ ⎨ ⎩ ẋ(t) = f (x(t)) y(t) = h(x(t)),
(5.17)

where x(t) ∈ R n is the state and y(t) ∈ R p the output. We assume that f and h are of class C 2 and that the system is forward-complete. We suppose that an observer for system (5.17)

has been designed and that it is described by ẋ(t) = f (x(t), y(t)), (5.18) where x is the state estimation. Similar to the problem studied in Chapter 4, y is considered not measured and, instead, the available output is a sampled and nonlinear transformation of y. This leads to a different and more complex nonlinear observer design problem.

Our goal is to design an observer for the system given by

⎧ ⎨ ⎩ ẋ(t) = f (x(t)) y ψ (t k ) = ψ(y(t k )), (5.19)
where (t k ) is an increasing sequence of sampling times such that t 0 = 0 and such that there exist positive constants δ and δ with

δ ≤ t k+1 -t k ≤ δ, ∀k ≥ 0. (5.20)
As in Chapter 4, our first assumption is that observer (5.18) is robust with respect to measurement noise in an ISS sense, that is, disturbance-to-error stable (DES). More precisely, if the output of (5.17) is affected by noise y = h(x)+d y , then the corresponding error dynamics e = xx are given by

ė(t) = F(t, e(t), d y (t)),
where

F : R + × R n × R p → R n is defined as F(t, e, d) = f (x(t)) -f (x(t) -e, h(x(t)) + d), ( 5.21)
and we have the following assumption.

Assumption 5.1. There exists a class C 1 function V : R n → R + such that the following conditions hold:

1. ∃ ᾱ1 , ᾱ2 ∈ K ∞ such that for all e ∈ R n , ᾱ1 (|e|) ≤ V (e) ≤ ᾱ2 (|e|), 2. ∃ χ ∈ K ∞ and a constant c > 0 such that for all t ∈ R + , e ∈ R n , d ∈ R p and x(0) ∈ R n , |e| ≥ χ(|d|) =⇒ ∂ V ∂e (e)F(t, e, d) ≤ -c V (e). Assumption
(x) = ∂ψ ∂y (h(x)) ∂h ∂x (x)f (x) is Lipschitz continuous. That is, there exists a constant c L > 0 such that |ϕ(x) -ϕ(x)| ≤ c L |x -x|, ∀x, x ∈ R n .
The system state is frequently bounded in physical estimation problems. Saturation techniques can then be used in order to satisfy Assumption 5.3, see for example [START_REF] Grip | Observers for interconnected nonlinear and linear systems[END_REF].

New observer design

Inspired by our work in Chapter 4, the proposed sample-and-hold observer for the continuousdiscrete system (5.19) is defined by the interconnection:

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ ẋ = f (x, ŷ) ẏ = ∂ψ ∂y (ŷ) -1 ∂ψ ∂y (h(x)) ∂h ∂x (x)f (x) + Kξ k ξ k (t) = y ψ (t k ) -ψ(ŷ(t k )), t ∈ [t k , t k+1 ), (5.22)
where f is as in Assumption 5.1 and K is a matrix to be designed. Notice that observer (5.22) consists of continuous dynamics interconnected with a switched system and that it only requires y ψ (t k ) = ψ(y(t k )) as an output measure. Our main result of this chapter is the following.

Theorem 5.6. Consider systems (5.19) and (5.22), and let Assumptions 5.1, 5.2 and 5.3 hold. Suppose that the following conditions are satisfied.

1. there exist a constant α > 0 and matrices: P = P > 0, K, N 1 , N 2 and N 3 in R p×p such that the LMI conditions (5.24) and (5.25) are satisfied with The previous theorem sets conditions ensuring, in particular, the asymptotic convergence to zero of e. Its proof, which consists of two lemmas, is shown in Section 5.5 and it is based on the small-gain theorem for switched systems [START_REF] Yang | A Lyapunov-based small-gain theorem for interconnected switched systems[END_REF]. Therefore, we search for ISS Lyapunov functions V 1 and V 2 by first considering e = xx as the state and ξ = y ψψ(ŷ) as the input and then by doing the opposite.

λ 1 = -α δ + 1, λ 2 = exp(-α δ), λ 3 = δ exp(-0.5α δ), 2. there exists a constant c > c 2 L /α such that ᾱ2 ( χ(c ψ c 1/2 λ min (P ) -1/2 ᾱ-1 1 (r))) < r, ∀r > 0. ( 5
Remark 5.3. The observer in (5.22) can be seen as an interconnection consisting of a regular and a switched subsystem, however, applying the general results from [START_REF] Yang | A Lyapunov-based small-gain theorem for interconnected switched systems[END_REF] requires careful analysis of their proofs. In particular: (i) the switching signal

σ(t) = k, ∀t ∈ [t k , t k+1 )
satisfies the so-called average dwell-time constraint, (ii) we use a single Lyapunov function V 2 but discontinuous at each time t k+1 ; the uniform ratio bound condition is replaced by (5.29),

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ (α -λ 1 )P + λ 2 (-N 1 -N 1 ) λ 1 P -(1 + δ) K + λ 2 (N 1 -N 2 ) (1+ δ)P + λ 2 (-N 3 ) 0 λ 1 P -(1 + δ) K + λ 2 (N 1 -N 2 ) -λ 1 P + δ( K + K ) + λ 2 (N 2 + N 2 ) -δP + λ 2 N 3 -δ K (1 + δ)P + λ 2 (-N 3 ) -δP + λ 2 N 3 -I δP 0 -δ K δP -δP ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ < 0, (5.24) ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ (α -λ 1 )P + λ 2 (-N 1 -N 1 ) λ 1 P -K + λ 2 (N 1 -N 2 ) P + λ 2 (-N 3 ) 0 λ 3 N 1 λ 1 P -K + λ 2 (N 1 -N 2 ) -λ 1 P + λ 2 (N 2 + N 2 ) λ 2 N 3 -δ K λ 3 N 2 P + λ 2 (-N 3 ) λ 2 N 3 -I δP λ 3 N 3 0 -δ K δP -δP 0 λ 3 N λ 3 N λ 3 N 0 -δP ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ < 0.
(5.25) and (iii) as in the proof of the original small-gain result [START_REF] Jiang | A Lyapunov formulation of the nonlinear small-gain theorem for interconnected ISS systems[END_REF], we can see that the condition:

there exists α 2,2 ∈ K ∞ such that V 2 (t) ≤ α 2,2 (|ξ(t)|), ∀t ≥ 0
is not crucial as long as the modified implications (5.26) and (5.28) hold, together with the corresponding small-gain condition.

Proofs of main results

In this section, we present the two Lyapunov function lemmas which are instrumental for the proof of Theorem 5.6.

There exists a Lyapunov function

V 1 from ξ = y ψ -ψ(ŷ) to e = x -x. That is, a class C 1 function V 1 : R n → R + such that the following conditions hold: 1. ∃ α 11 , α 12 ∈ K ∞ such that for all x(0) ∈ R n , all x(0) ∈ R n , all ŷ(0) ∈ R p and all t ∈ R + , α 11 (|e(t)|) ≤ V 1 (e(t)) ≤ α 12 (|e(t)|), 2. ∃ χ 1 ∈ K ∞ and a constant c 1 > 0 such that for all x(0) ∈ R n , all x(0) ∈ R n , all ŷ(0) ∈ R p and all t ∈ R + , |e(t)| ≥ χ 1 (|ξ(t)|) =⇒ V1 (e(t)) ≤ -c 1 V 1 (e(t)).
( and, therefore, we can select c 1 = c to get

V1 (e(t)) = ∂ V ∂z (e(t)) ė(t) ≤ -c V (e(t)) = -c 1 V 1 (e(t)), (5.27)
whenever |e(t)| ≥ χ(|ŷ(t)y(t)|). We also define the class K function

χ 1 (r) = χ(c ψ r), ∀r ≥ 0,
where c ψ is from Assumption 5.2 and hence

χ 1 (|ξ(t)|) = χ(c ψ |ψ(y(t)) -ψ(ŷ(t))|) ≥ χ(|y(t) -ŷ(t)|)
which, together with (5.27), provides the needed property.

The following lemma allows us to transform a nonlinear matrix inequality into an LMI at the cost of increasing the dimension.

Lemma 5.2. [Schur complement [START_REF] Abbasi | Robust state estimation for a class of uncertain nonlinear systems: Comparison of two approaches[END_REF]] Consider a symmetric real matrix Suppose there exist a constant α > 0 and matrices:

P = P > 0, K, N 1 , N 2 and N 3 in R p×p
such that the LMI conditions (5.24) and (5.25), with

λ 1 = -α δ + 1, λ 2 = exp(-α δ), λ 3 = δ exp(-0.5α δ),
are satisfied. If we set K = P -1 K, then there exists a Lyapunov-Krasovskii functional from e = xx to ξ = y ψψ(ŷ). That is, a function V 2 : R + → R + , depending on ξ and differentiable on t = t k , such that the following conditions hold:

1. ∃ α 21 ∈ K ∞ such that for all x(0) ∈ R n , all x(0) ∈ R n , all ŷ(0) ∈ R p and t ∈ R + , α 21 (|ξ(t)|) ≤ V 2 (t), 2. ∃ χ 2 ∈ K ∞ and a constant c 2 > 0 such that for all x(0) ∈ R n , all x(0) ∈ R n , all ŷ(0) ∈ R p and all t = t k , V 2 (t) ≥ χ 2 (|e(t)|) =⇒ V2 (t) ≤ -c 2 V 2 (t), (5.28)
3.

V 2 (t k+1 ) ≤ lim t→t - k+1 V 2 (t), ∀k ≥ 0. (5.29) Proof. We have that ξ(t) = φ(t) -Kξ k (t),
where ϕ : R + → R p is given by Assumption 5.3 and

φ(t) = ϕ(x(t)) -ϕ(x(t)), hence, |φ(t)| ≤ c L |e(t)|, ∀t ≥ 0. (5.30)
We use a slight modification of the Lyapunov-Krasovskii functional [START_REF] Naghshtabrizi | Exponential stability of impulsive systems with application to uncertain sampled-data systems[END_REF][START_REF] Raff | Observer with sample-and-hold updating for Lipschitz nonlinear systems with nonuniformly sampled measurements[END_REF]:

V 2 (t) =ξ(t) P ξ(t) + t t-τ (t) ( δ -t + s) exp(-α(t -s)) ξ(s) P ξ(s)ds + ( δ -τ (t))(ξ(t) -ξ k (t)) P (ξ(t) -ξ k (t)), (5.31)
where α > 0 and P ∈ R p×p symmetric and positive definite matrix are to be chosen and where τ (t) = tt k for t ∈ [t k , t k+1 ). Notice that the last two terms in (5.31) are non-negative and therefore condition 3) is satisfied. Since V 2 (t k ) = ξ(t k ) P ξ(t k ), we also have

λ min (P )|ξ(t k )| 2 ≤ V 2 (t k ) ≤ λ max (P )|ξ(t k )| 2 , ∀k ≥ 0.
Moreover, for the augmented state

ξ(t) = ξ(t) ξ k (t) φ(t)
we can show there exists a constant c 22 > 0 such that

λ min (P )|ξ(t)| 2 ≤ V 2 (t) ≤ c 22 max s∈[-τ (t),0] | ξ(t + s)| 2 , ∀t ≥ 0.
The augmented state ξ is used to express the time derivative of V 2 . Indeed, by the Leibniz integral rule we have that:

V2 (t) =2ξ(t) P ξ(t) + δ ξ(t) P ξ(t) -α t t-τ (t) ( δ -t + s) exp(-α(t -s)) ξ(s) P ξ(s)ds - t t-τ (t) exp(-α(t -s)) ξ(s) P ξ(s)ds + 2( δ -τ (t))(ξ(t) -ξ k (t)) P ξ(t) -(ξ(t) -ξ k (t)) P (ξ(t) -ξ k (t))
and, hence,

V2 (t) + αV 2 (t) ≤ ξ(t) Q 1 ξ(t) + δ ξ(t) Q 2 ξ(t) -exp(-α δ) t t-τ (t) ξ(s) P ξ(s)ds + (-α δ + 1) ξ(t) Q 3 ξ(t) + ( δ -τ (t)) ξ(t) Q 4 ξ(t) + α ξ(t) P ξ(t), (5.32)
where:

Q 1 = ⎡ ⎢ ⎢ ⎣ 0 -P K P -K P 0 0 P 0 0 ⎤ ⎥ ⎥ ⎦ , Q 2 = ⎡ ⎢ ⎢ ⎣ 0 -K I ⎤ ⎥ ⎥ ⎦ P 0 -K I , Q 3 = ⎡ ⎢ ⎢ ⎣ -P P 0 P -P 0 0 0 0 ⎤ ⎥ ⎥ ⎦ , Q 4 = ⎡ ⎢ ⎢ ⎣ 0 -P K P -K P PK + K P -P P -P 0 ⎤ ⎥ ⎥ ⎦ , P = ⎡ ⎢ ⎢ ⎣ P 0 0 0 0 0 0 0 0 ⎤ ⎥ ⎥ ⎦ .
By using

0 ≤ t t-τ (t) ξ(s) ξ(t) P -N -N NP -1 N ξ(s) ξ(t) ds, for any matrix N = [N 1 , N 2 , N 3 ] ∈ R 3p×p
, we can dominate the integral appearing in (5.32):

- t t-τ (t) ξ(s) P ξ(s)ds ≤ ξ(t) Q 6 ξ(t) + τ (t) ξ(t) NP -1 N ξ(t), (5.33)
where

Q 6 = ⎡ ⎢ ⎢ ⎣ -N 1 -N 1 N 1 -N 2 -N 3 N 1 -N 2 N 2 + N 2 N 3 -N 3 N 3 0 ⎤ ⎥ ⎥ ⎦ . Take any constant c > c 2 L α and suppose that V 2 (t) ≥ c|e(t)| 2 , ( 5.34)
that is, choose the gain χ 2 (r) = cr 2 , ∀r ≥ 0.

Let us denote

Q 5 = ⎡ ⎢ ⎢ ⎣ 0 0 0 0 0 0 0 0 -I ⎤ ⎥ ⎥ ⎦
so that by (5.30) and (5.34) we get

0 ≤ ξ(t) Q 5 ξ(t) + c 2 L c V 2 (t). (5.35)
Finally, we define the matrices

R 1 = Q 1 + δQ 2 + (-α δ + 1)Q 3 + Q 5 + exp(-α δ)Q 6 + α P, R 2 = exp(-α δ)NP -1 N , R 3 = Q 4 ,
in order to write from (5.32), (5.33) and (5.35) that

V2 (t) + α - c 2 L c V 2 (t) ≤ ξ(t) (R 1 + τ (t)R 2 + ( δ -τ (t))R 3 ) ξ(t).
As in [START_REF] Naghshtabrizi | Exponential stability of impulsive systems with application to uncertain sampled-data systems[END_REF], the time dependence in the matrix appearing in the right hand side above can be removed since the conditions:

R 1 + δR 3 < 0 and R 1 + δR 2 < 0, (5.36)
imply there exists a constant c R > 0 such that

V2 (t) + α - c 2 L c V 2 (t) ≤ -c R | ξ(t)| 2 < 0
and, therefore, we can choose

c 2 = α - c 2 L c > 0 to conclude V2 (t) ≤ -c 2 V 2 (t).
The conditions in (5.36) can be expressed as the LMI's in (5.24) and (5.25) by using Lemma 5.2 twice. First with:

S 11 = R 1 -δQ 2 + δQ 4 , S 12 = 0 -δP K δP , S 22 = -δP,
and then with:

S 11 = R 1 -δQ 2 , S 12 = ⎡ ⎢ ⎢ ⎣ 0 δλ 3 N 1 -δK P δλ 3 N 2 δP δλ 3 N 3 ⎤ ⎥ ⎥ ⎦ , S 22 = -δP 0 0 -δP ,
and by setting K = P K.

Proof of Theorem 5.6: Using Lemmas 5.1 and 5.3 define:

χ m 1 (r) = α 12 (χ 1 (α -1 21 (r))), ∀r ≥ 0, χ m 2 (r) = χ 2 (α -1 11 (r)), ∀r ≥ 0.
Then, the following implications hold for all t ∈ R + :

V 1 (e(t)) ≥ χ m 1 (V 2 (t)) =⇒ |e(t)| ≥ χ 1 (|ξ(t)|), V 2 (t) ≥ χ m 2 (V 1 (e(t))) =⇒ V 2 (t) ≥ χ 2 (|e(t)|).
As in the original theorem [START_REF] Jiang | A Lyapunov formulation of the nonlinear small-gain theorem for interconnected ISS systems[END_REF], the small-gain condition arises from the composition of the ISS gains when comparing directly V 1 with V 2 as above. That is, the gains are constrained by

χ m 1 (χ m 2 (r)) < r, ∀r > 0,
which is equivalent to the condition in (5.23). We can conclude the global asymptotic convergence of the estimation error of the new observer (5.22) by considering Remark 5.3 and by applying Theorem 5.5.

for all r ∈ R + , and with

θ > 2 ns 1 2 m c ζ λ min (S ∞,1 ) -1 2 + 1 ,
where s 1 , s 2 , s m , c ζ and S ∞,1 are as in Section 4.6.2. The rest of the hypotheses from Theorem 5.6 are illustrated in the numerical example of the following section.

Numerical example

We study the system dynamics as in [START_REF] Lei | High-gain observers in the presence of sensor nonlinearities[END_REF] but we consider a discrete output with a different nonlinearity. For this, suppose that system (5.38) is specified by:

A = 0 1 0 0 , C = 1 0 , ζ(x, u) = 0 -x 1 -2x 2 + ax 2 1 x 2 + u u(t) = b sin(2t),

Perspectives: the hybrid approach

The proof of Theorem 5.6 relies on the small-gain theorem for switched systems [START_REF] Yang | A Lyapunov-based small-gain theorem for interconnected switched systems[END_REF] presented in Section 5.2.3. Here, we discuss a possible framework for an alternative and more elegant approach to our previous results. The strategy in [START_REF] Yang | A Lyapunov-based small-gain theorem for interconnected switched systems[END_REF] consists in modeling the interconnection of switched systems in the socalled hybrid framework. Indeed, hybrid systems have both continuous and discrete features and can represent seemingly different systems including: impulsive, switched and sampleddata. The hybrid framework [START_REF] Goebel | Hybrid dynamical systems: modeling, stability, and robustness[END_REF] has now become standard and, partly due to its generality, increasingly popular in recent years. A hybrid system is a dynamical system denoted by The shared domain of definition should be a hybrid time domain E = dom(z) = dom(w) ⊆ R + × Z + , meaning: for each (T, J) ∈ E, E = E ∩ ([0, T ] × {0, 1, . . . , J}) is a compact hybrid domain, that is, E = ∪ J j=0 ([t j , t j+1 ], j) for some finite non-decreasing sequence of times 0 = t 0 , t 1 , . . . , t J +1 . Furthermore, each w(•, j) should be Lebesgue measurable and locally essentially bounded, and each z(•, j) locally absolutely continuous. Finally, z and w should satisfy: (i) (z(t, j), w(t, j)) ∈ C and ż(t, j) ∈ F (z(t, j), w(t, j)), for all j ∈ Z + and almost all t ∈ R + such that (t, j) ∈ dom(z), (ii) (z(t, j), w(t, j)) ∈ D and z(t, j + 1) ∈ G(z(t, j), w(t, j)), for all (t, j) ∈ dom(z) such that (t, j + 1) ∈ dom(z). Therefore, z(t, j) represents the system state after t time units and j jumps. Results involving existence and uniqueness of solutions for hybrid systems require careful analyses [START_REF] Goebel | Hybrid dynamical systems: modeling, stability, and robustness[END_REF], we assume local existence throughout the rest of this section.

Not only switched but also sampled-data systems can be modeled in the hybrid framework and, as detailed in the following remark, the latter could be a better fit for our setup.

Remark 5.4. The estimation error dynamics of (5.19) where e = xx, ξ = yŷ and ξ = ψ(y)ψ(ŷ), and where: The notion of Lyapunov function can be extended naturally to hybrid systems. In this case, the authors in [START_REF] Briat | Stability analysis and control of a class of LPV systems with piecewise constant parameters[END_REF] introduce the function V : R n → R + given by V (z) = x u P (τ, T ) x u ,

F 1 (t,
where P : [0, T 2 ] × [T 1 , T 2 ] → R (n-2)×(n-2) is a matrix functional such that each P (τ, T ) is symmetric and positive definite. Similarly, ISS can be defined in the hybrid context and its Lyapunov characterization can be found in [START_REF] Cai | Characterizations of input-to-state stability for hybrid systems[END_REF].

Definition 5.2. The hybrid system (5.40) is pre-input-to-state stable (pre-ISS) if there exist functions β ∈ KL and γ ∈ K ∞ such that all solutions satisfy |z(t, j)| ≤ max{β(|z(0, 0)|, t + j), γ(|w| (t,j) )}, ∀(t, j) ∈ dom(z).

(5.42)

The previous norm is defined as |w| (t,j) = max ess sup where N 1 (N 2 ) is the subset of dom(w) consisting of pairs (s, k) such that (s, k + 1) / ∈ (∈ ) dom(w) and s + k ≤ t + j, and where ess sup stands for the essential supremum. We say that system (5.40) is pre-globally asymptotically stable (pre-GAS) if (5.42) holds with w = 0.

Here, the prefix "pre" indicates that completeness of solutions to (5.40) is not included in the previous definition.

Remark 5.5. The idea is then to find a more adequate small-gain theorem to show the stability of the system in the previous remark. To our knowledge, possible alternatives are given by: (i) the very general small-gain results from [START_REF] Karafyllis | A Small-Gain Theorem for a Wide Class of Feedback Systems with Control Applications[END_REF], (ii) writing (5.41) as a fast-varying delay system with τ (t) = tt k , u(t) = Kx(tτ (t)) and using the corresponding small-gain theorem [START_REF] Dashkovskiy | Lyapunov-Razumikhin and Lyapunov-Krasovskii Theorems for Interconnected ISS Time-Delay Systems[END_REF], and (iii) modeling sampled-data systems in the hybrid framework and using the small-gain theorem [START_REF] Dashkovskiy | Input-to-state stability of interconnected hybrid systems[END_REF][START_REF] Liberzon | Lyapunov-Based Small-Gain Theorems for Hybrid Systems[END_REF]. We next discuss the results from the latter work.

An interconnected hybrid system without external inputs can be written as

⎧ ⎨ ⎩ ż1 ∈ F 1 (z 1 , z 2 ), ż2 ∈ F 2 (z 1 , z 2 ), (z 1 , z 2 ) ∈ C z + 1 ∈ G 1 (z 1 , z 2 ), z + 2 ∈ G 2 (z 1 , z 2 ), (z 1 , z 2 ) ∈ D, (5.43)
where C and D are closed subsets of R n 1 × R n 2 , and where F i and G i are set-valued maps from R n 1 × R n 2 to R n i . We make use of the Clarke derivative defined for a locally Lipschitz 1. ∃ α i,1 , α i,2 ∈ K ∞ such that for all z i ∈ R n i ,

α i,1 (|z i |) ≤ V i (z i ) ≤ α i,2 (|z i |),
2. ∃ χ i ∈ K ∞ and α i,3 ∈ K such that for all (z 1 , z 2 ) ∈ C and all w i ∈ F i (z 1 , z 2 ),

V i (z i ) ≥ χ i (V j (z j )) =⇒ V • i (z i ; w i ) ≤ -α i,3 (V i (z i )),
3. ∃ λ i ∈ K such that for all r > 0, all (z 1 , z 2 ) ∈ D and all w i ∈ G i (z 1 , z 2 ), λ i (r) < r, V i (w i ) ≤ max{λ i (V i (z i )), χ i (V j (z j ))}.

Theorem 5.7 ( [START_REF] Liberzon | Lyapunov-Based Small-Gain Theorems for Hybrid Systems[END_REF]). Consider the overall system (5.43) and suppose that 1)-3) hold. If the small-gain condition χ 1 (χ 2 (r)) < r, ∀r > 0 is satisfied, then the system is pre-GAS.

Applications of the small-gain theorem for hybrid systems arise in different contexts including:

the natural decomposition of a hybrid system, networked control systems, emulation with event-triggered sampling, and quantized-feedback control as shown in [START_REF] Liberzon | Lyapunov-Based Small-Gain Theorems for Hybrid Systems[END_REF].

To summarize: our main results corresponding to Chapter 5 consist in the construction of two

Lyapunov functions together with a small-gain argument, which we based on [START_REF] Yang | A Lyapunov-based small-gain theorem for interconnected switched systems[END_REF]. Although the switched system setting does not correspond directly to ours, the steps in the proof of their main result can still be followed to conclude the asymptotic convergence of our proposed observer. On the other hand, this section suggests that a possibly more suitable framework is that of hybrid systems. This is due to Theorem 5.7 together with the fact that hybrid systems can model adequately the estimation error dynamics implied by our observer. However, it is not clear yet how to construct the Lyapunov functions for the hybrid subsystems.

Conclusion

In this chapter, we proposed an observer redesign method that adapts a given observer to a discretized and nonlinearly transformed version of the system output. The main idea behind our method consists in interconnecting the given observer with a dynamic inversion of the nonlinear transformation based on sample-and-hold techniques. We formulated two LMI's and a small-gain type constraint that guarantee the asymptotic convergence to zero of the proposed observer. A simulation example illustrates the latter and also the effectiveness of our approach. Finally, we presented some brief ideas on an alternative approach to our results by using the hybrid framework. Future work can address the case of sampled and implicit outputs and the corresponding applications that arise in many fields such as machine vision.

Chapter 6

Conclusions and perspectives

Summary

The work presented in this thesis concerns observer design for nonlinear systems with robustness guarantees in the presence of disturbances. Three novel observer designs were introduced: a Kalman-like observer with optimal tuning and input selection concerning state-affine systems, an observer redesign for general nonlinear systems in the presence of nonlinear sensors, and an extension of this redesign in the case of sampled measurements.

Chapter 2 provided a general overview of observability and of observer design. We first discussed cornerstone results involving representative system forms, which then led to specific Luenberger-like and Kalman-like observer designs. We emphasized the advantages and disadvantages of high-gain observers, with a special focus on measurement noise sensitivity. It was then convenient to deepen in the input-to-state stability framework, first by studying the central results: Lyapunov and small-gain theorems, and then by introducing a related notion of observer robustness.

The problem of tuning high-gain observers has been widely studied in the literature, and the same holds for regularly persistent input design in the case of state-affine systems. Nevertheless, in Chapter 3, we developed a novel strategy for simultaneous tuning and input selection of a Kalman-like observer in the face of dynamic and output disturbances. This approach is based on an estimation error bound functional, given by Lyapunov analyses, and an algorithm to minimize it. The simulations were successful in the sense that the actual state estimation improved with the optimal design.

Observer design for state-affine systems and for many other classes of systems is usually based on linearity of the system output, however, nonlinear sensors arise frequently in practice. Chapter 4, which constitutes the main part of this thesis, presented a general and robust observer redesign approach to adapt a given observer for a nonlinear system to a nonlinear transformation of the system output. The main idea of the proposed approach is to consider an additional subsystem to estimate the output and to feed it back to the given observer, which is assumed measurement robust in the ISS sense. Thus, the new observer results in an interconnected system whose asymptotic convergence was shown by means of an adapted version of the Lyapunov based small-gain theorem. Our redesign was made explicit in detailed applications for two important classes of systems which differ considerably in their observability properties.

In many engineering applications, sensors provide measurements in discrete time and several observers have been designed in the literature under this constraint. In Chapter 5, we extended our observer redesign in order to handle sampled and transformed outputs.

This extension consists in a sample-and-hold technique together with the dynamic inversion equations developed in the previous chapter. As a consequence, the new observer is a continuous-discrete system whose gain arises as the solution of some LMI's, and whose asymptotic convergence was shown via the small-gain theorem for switched systems. We studied the new observer design for uniformly observable systems and illustrated the feasibility of the LMI's. Finally, we discussed the hybrid framework as an alternative approach for future work.

Perspectives

The work from Chapter 3 can be extended in different research directions. For example, the error minimization algorithm can be coupled with the design of regularly persistent inputs.

Or, if the main objective is to control the system, then a different functional could represent a trade-off between persistency of excitation and regulation constraints. Furthermore, it is interesting to consider our bounding strategy for state-affine systems up to triangular nonlinearity. With respect to the observer redesign presented in Chapter 4 or 5, new studies should target the main assumptions. The classes of systems which admit a robust observer in the ISS sense can be significantly enlarged when considering weaker stability notions. Our results could also be extended for an output transformation ψ : R p 1 → R p 2 with p 1 = p 2 , or for a time-varying ψ(y, t). In Chapter 5, further efforts can be done in order to improve the feasibility of the resulting LMI and to allow for larger sampling periods; new results might be possible within the hybrid system framework. As previously mentioned, these studies are a first step towards the case of sampled and implicit outputs arising in fields such as machine vision.

importantes, telles que : l'identification, la prise de décision ou la surveillance. Cependant, les capacités des capteurs sont souvent limitées par des contraintes physiques, technologiques, économiques ou même de sécurité.

Les observateurs sont des algorithmes qui estiment l'état actuel à l'aide de la structure du

Contents 5 . 6 . 1 2 λ

 5612 Uniformly observable systems . . . . . . . . . . . . . . . . . . . . . . 5.6.2 Numerical example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Perspectives: the hybrid approach . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Conclusions and perspectives 6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography A Résumé étendu en français A.1 Introduction générale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Chapitre 3 : design d'observateur optimal pour des systèmes affines en l'état perturbés . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3 Chapitre 4 : design d'observateur pour des systèmes non linéaires à transformation de sortie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4 Chapitre 5 : extension au cas de sortie discrète et transformée . . . . . . . . iii List of Figures 3.1 Example of dynamic and measurement disturbances affecting a state-affine system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 State estimation for a state-affine system using the Kalman-like observer with the optimal tuning and input . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Diagram of the observer redesign for a nonlinear system in the presence of a nonlinear output transformation . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 No disturbance case: estimation of the states of a state-affine system by the new observer based on ψ(y) and by the original observer based on y . . . . . 4.3 Disturbance case: estimation of the states of a state-affine system by the new observer based on ψ(y) and by the original observer based on y . . . . . . . 4.4 No disturbance case: estimation of the states of a system with triangular nonlinearity by the new observer and by the observer from [77], both based on ψ(y) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Disturbance case: estimation of the states of a system with triangular nonlinearity by the new observer and by the observer from [77], both based on ψ(y) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Estimation of the states of a system with triangular nonlinearity by the new observer based on ψ(y(t k )) and by the original observer based on y(t) . . . . A.1 Diagramme de la refonte de l'observateur pour un système non linéaire en présence d'une transformation de sortie non linéaire . . . . . . . . . . . . . . Nomenclature |f | ∞ sup t≥0 |f (t)|, where f : R + → R n is a function |M | spectral norm of the real matrix M or λ max (M M ) 1 max (M ) maximum eigenvalue of the real symmetric matrix M λ min (M ) minimum eigenvalue of the real symmetric matrix M Z + set of non-negative integers R + set of non-negative real numbers R n set of real numbers of n dimension R n×m set of real matrices of n × m dimension C k class of functions whose derivatives of order up to k are continuousf (t -) left limit of a function f : [s, t) → R n at t, or lim r→t -f (r) f (i) ith derivative of the function f : R + → R n f -1 (0) the set of t ∈ R + such that f (t) = 0 for a function f : R + →R n I Identity matrix of appropriate dimension L 2 loc,n space of measurable and locally square integrable functions f : R + → R n L ∞ n space of measurable and essentially bounded functions f : R + → R n L ∞ loc,n space of measurable and locally essentially bounded functions f : R + → R n M transpose of the matrix M M ≥ 0 the matrix M ∈ R n×n satisfies x Mx ≥ 0, for all x ∈ R n v Nomenclature M i,j element in the ith row and jth column of the matrix M a.e. almost everywhere (except for a Lebesgue zero measure subset) DES Disturbance-to-Error Stability EKF Extended Kalman Filter GAS Global Asymptotic Stability i-IOSS incremental Input/Output-to-State Stability IOpS Input-to-Output practical Stability IOS Input-to-Output Stability IOSS Input/Output-to-State Stability ISpS Input-to-State practical Stability ISS Input-to-State Stability LMI Linear Matrix Inequality MPC Model Predictive Control NCS Networked Control System ODE Ordinary Differential Equation OSS Output-to-State Stability PDE Partial Differential Equation pre-GAS pre-Global Asymptotic Stability pre-ISS pre-Input-to-State Stability UO Unboundedness Observability vi Chapter 1

Theorem 2 . 7 (

 27 [START_REF] Gauthier | A simple observer for nonlinear systems. Applications to bioreactors[END_REF]). Consider system(2.18) and suppose that the nonlinearity ζ satisfies (2.20) and that θ is large enough. Then, the state estimation error from (2.21) converges exponentially to zero.

  bounded and continuous disturbances, A : R → R n×n a continuous matrix functional and C ∈ R 1×n a constant matrix. We use the previously presented observer

(3. 11 ) 3 . 2 .

 1132 Theorem Consider systems (3.7) and (3.8) for fixed u and θ, and suppose that Assumption 3.1 is fulfilled. Based on the definitions given in (3.9), (3.10) and (3.11) the following inequality is satisfied,

Theorem 3 .

 3 2 provides a bound that can be minimized by playing simultaneously over positive tunings θ and admissible inputs u. To set this optimization problem correctly, we first need to specify a target function and its domain. We define the domain by considering inputs with the proper excitation and then redefine the bound in Theorem 3.2 as a function of u and θ by specifying an explicit choice of c 1 (u, θ) and c 2 (u, θ) in(3.9). This requires the concept of regular persistence of an input for the homogeneous part of system (3.7), see Definition 3.1.

Lemma 3 . 2 .

 32 Consider systems (3.7) and (3.8), and suppose Assumption 3.2 holds. For all θ > 2σ and t ≥ 0 we have

Corollary 3 . 1 .

 31 Consider systems (3.7) and(3.8), and suppose that Assumption 3.2 is satisfied. If u is regularly persistent for system (3.7) with respect to (T, α, t 0 ), then for any θ > 2σ

2 . 3 . 4 .

 234 Parameterize U continuously with P interval, Choose a tuning parameter space Θ, Set a continuous length assignment T (u) and compute α(u) as in (3.13),

20 and L 2 = 10 .

 20210 d 1 (t) represented, for simulation purposes, by pairs of uniform random numbers between 0 and √ 200. Similarly, the output measurements are corrupted by uniformly random noise between 0 and 10. Hence, L 1 = The plots of the disturbances are shown in Figure 3.1. We fix S(0) as the identity matrix and the functional J is fully determined. The solver outputs the optimal design: (θ * , p * u ) = (3.3024, 11.1651).

  2 for the first ten time units and for different inputs and tunings. Here we initialized the system at x(0) = (10, 0) and x(0) =[START_REF] Alessandri | Observer design for nonlinear systems by using Input-to-State Stability[END_REF][START_REF] González De Cossío | Observer design for nonlinear systems with implicit ouput[END_REF]. Small values of the tuning θ fail to approximate fast and correctly both states simultaneously (Figure3.2, first and second row from the top). But if θ is too large then the effect of the noise gets amplified (Figure3.2, bottom row, second column). The optimal combination θ * and u * is shown in the third row from the top of Figure3.2.

Figure 3 . 1 :

 31 Figure 3.1: The disturbances that affect the system. The first row corresponds to the two entrances of the dynamic disturbance d 1 . The second row corresponds to the output disturbance d 2 .

Theorem 4 . 1 (

 41 [START_REF] Fan | Observer design for systems with multivariable monotone nonlinearities[END_REF]). Consider system (4.1) and suppose that the nonlinearities γ 1 and γ 2 satisfy the inequalities in (4.2) for some constant > 0. If there exist a symmetric and positive definite matrix P , matrices L 1 , L 2 and K 1 , and constants ζ > 0, λ 1 > 0 and λ 2 > 0 such that

Remark 4 . 2 .

 42 The circle criterion, in its simplest version, considers any time-invariant linear system of the form ⎧

  where A, C and ζ have the canonical form (2.19). The function ψ : R → R is locally Lipschitz such that ψ(0) = 0 and the function ζ : R n × R m → R n is coordinate-wise globally Lipschitz in x, uniformly in u. If ζ i = 0 for all i = n, then system (4.7) can model different mechanical systems in which the position

  n l and y(t) ∈ R p l , where f and h and all signals are sufficiently smooth, and where (4.11) is composed by matrices of appropriate dimensions.

  is a gain matrix to be determined. Their strategy consists in estimating w e = [w , z ] by ŵe = [ ŵ , ẑ ] , this leads to consider the extended system ⎧ ⎨ ⎩ ẇe = Aw e + B u u + B d d(x, u, u) y = Cw e + D u u,

Assumption 4 . 1 .

 41 There exists a continuous function V (t, e) : R + × R n → R + , of class C 1

Figure 4 .

 4 Figure 4.1 compares observers (4.17) and (4.21) and illustrates the relation of these observers with system (4.18).

  ẋ

Figure 4 . 1 :

 41 Figure 4.1: System (4.18) and observer (4.17) are represented on the top part of the diagram.The new observer (4.21) is represented on the bottom part as an interconnected system. Observer (4.17) requires the unavailable output y, while the new observer uses the measurements y ψ instead.

Lemma 4 . 1 .

 41 Consider systems (4.18) and (4.21), with ϕ and K as in (4.23)-(4.24), and let Assumption 4.2 hold. If k > 1, then there exists a Lyapunov function for the family

. 29 )

 29 where e = xx and ξ = y ψψ(ŷ) are defined on [0, T ). Moreover, if the disturbances d y and d ψ are both zero then the corresponding Lyapunov-constant can be chosen to be zero. Proof. Consider the Lyapunov function V from Assumption 4.1 and the corresponding functions ᾱ1 , ᾱ2 , ᾱ3 and χ. Set V 2 = V , which on the trajectory takes the form V e 2 = V 2 (•, e), and define α 21 = ᾱ1 and α 22 = ᾱ2 . We next prove that property (2.45) holds on [0, T ) \ e -1 (0). From the definition of F in (4.20) we have that ė = F(•, e, ŷh(x))

Theorem 4 . 4 (

 44 Observer gain design). Consider systems (4.18) and (4.21), with ϕ and K as in (4.23)-(4.24), and let Assumptions 4.1 and 4.2 hold. For any k > 1 and any ρ ∈ K ∞ ∩ C 2 satisfying ρ(r) > 2α(2ᾱ -1 1 (ᾱ 2 (2 χ(2δ(4r))))), ∀r > 0 (4.31) there exist a class KL function β and a constant c ≥ 0 such that for all x(0), x(0) ∈ R n and all ŷ(0) ∈ R p the estimation errors e = xx and ξ = y ψψ(ŷ) are defined on R + and |(ξ(t), e(t))| ≤ β(|(ξ(0), e(0))|, t) + c, ∀t ≥ 0. (4.32) Moreover, if the disturbances d y and d ψ are both zero then c is zero as well. Proof. Using Lemmas 4.1 and 4.2, we deduce that there exist Lyapunov functions for the family G given in (4.25) and for the family G -1 given in (4.29). Since the first two Lyapunovbounds in the proof of Lemma 4.1 can be taken as the identity function, the small-gain condition in (2.51) is given by:

. 34 )Assumption 4 . 4 .

 3444 where y = Cx + d y and where ψ : R p → R p satisfies Assumption 4.2. We need the next condition on η. The function η is Lipschitz continuous with respect to its first entrance.That is, there exists a positive constant c η such that |η(y, u)η(ŷ, u)| ≤ c η |y -ŷ|, for all y, ŷ ∈ R p and all u ∈ R m .

Assumption 4 . 5 .Remark 4 . 12 .

 45412 The input set U consists of continuous, bounded and regularly persistent inputs u : R + → R m . Assumptions 4.4 and 4.5 concern the convergence properties of the initial observer (4.35) and, as shown below, substitute Assumption 4.1 in the case of state-affine systems up to output injection.

Corollary 4 . 1

 41 (state-affine case). Consider systems (4.34) and (4.44) and let Assumptions 4.2, 4.3, 4.4 and 4.5 hold. For any tuning parameter θ such that θ > max{3b 1 , 2|C| + 2c η (b -1 1 |C| 2 + |S(0)|)}, where b 1 = sup |u|≤cu |A(u)|, there exist a function β ∈ KL and a constant c ≥ 0 such that for all x(0), x(0) ∈ R n and all ŷ(0) ∈ R p , the estimation errors e = xx and ξ = y ψψ(ŷ) are defined on R + and |(ξ(t), e(t))| ≤ β(|(ξ(0), e(0))|, t) + c, ∀t ≥ 0. (4.45) Moreover, if the disturbances d y and d ψ are both zero then c is zero as well. Proof. It follows from our previous development and from Theorem 4.4. Indeed, we showed that Assumption 4.1 is satisfied by using Assumptions 4.4 and 4.5.

(4. 46 )Assumption 4 . 6 .

 4646 where y = Cx + d y and where ψ : R → R satisfies Assumption 4.2. As in Chapter 2, we add the following assumption on ζ. The function ζ is coordinate-wise Lipschitz continuous in the first entrance. That is, for each i = 1, . . . , n, there exists a positive constant c ζi such that

Corollary 4 . 2 (

 42 triangular case). Consider systems (4.46) and (4.52) and let Assumptions 4.2, 4.3 and 4.6 hold. For any tuning θ > 2 ns 1 2

Figure 4 . 2 :Discussions on Figure 4 . 2 :

 4242 Figure 4.2: Comparison of the new observer with the observer that uses y in the case: d y = 0, d ψ = 0. Estimation of the system states of (4.34) (blue solid) by observer (4.35) (red pointeddashed) and by the new observer (4.44) (black dashed). The columns correspond respectively to the tunings λ = 0.5 and λ = 2 in (4.53).

Figure 4 . 3 :

 43 Figure 4.3: Comparison of the new observer with the observer that uses y in the case: d y = 0, d ψ = 0. Estimation of the system states of (4.34) (blue solid) by observer (4.35) (red pointeddashed) and by the new observer (4.44) (black dashed). The columns correspond respectively to the tunings λ = 0.5 and λ = 2 in (4.53).

Figure 4 . 4 :

 44 Figure 4.4: Comparison of the new observer with the observer of J. Lei and H.K. Khalil [77] in the case: d y = 0, d ψ = 0. Estimation of the system states in (4.46) (blue solid) by observer (4.54) (green pointed-dashed) and by the new observer (4.52) (black dashed). The left column corresponds to the tunings θ = 2 and = 0.5 and the right column to θ = 9 and = 0.11.

  and 4.5. We simulate two cases: (i) d y and d ψ are both zero (Figure 4.4), (ii) d y and d ψ are uniformly distributed numbers respectively between ±0.3 and ±0.4 (Figure 4.5). Discussions on Figure 4.4: In the top two rows, we see that both the new observer (4.52)

Figure 4 . 5 :

 45 Figure 4.5: Comparison of the new observer with the observer of J. Lei and H.K. Khalil [77] in the case: d y = 0, d ψ = 0. Estimation of the system states in (4.46) (blue solid) by observer (4.54) (green pointed-dashed) and by the new observer (4.52) (black dashed). The left column corresponds to the tunings θ = 2 and = 0.5 and the right column to θ = 9 and = 0.11.

Theorem 5 . 2 (

 52 [START_REF] Nadri | Design of a continuous-discrete observer for state affine systems[END_REF]). Consider system(5.4) and suppose that u is a bounded and regularly persistent input as in Definition 3.1. If ρ ≥ 1 and if θ and δ -1 are large enough, then the state estimation error from observer (5.5) converges exponentially to zero.

  Ax + ζ(x) y = Cx, where A, C and ζ are in canonical form as in (2.19), and for ζ i Lipschitz continuous with constant L ≥ 0 and such that ζ i (0) = 0. Recall from Remark 2.3 the observer given by

. 23 )

 23 Then, choosing K = P -1 K, there exists β ∈ KL such that for all x(0), x(0) ∈ R n and all ŷ(0) ∈ R p the estimation errors e = xx and ξ = y ψψ(ŷ) are defined on R + and |(e(t), ξ(t))| ≤ β(|(e(0), ξ(0))|, t), ∀t ≥ 0.

Lemma 5 . 1 .

 51 Consider systems (5.19) and (5.22), and let Assumptions 5.1 and 5.2 hold.

. 26 )

 26 Proof. Consider the Lyapunov function V from Assumption 5.1 and the corresponding functions ᾱ1 , ᾱ2 , ᾱ3 and χ. Set V 1 = V and define α 21 = ᾱ1 and α 22 = ᾱ2 . From the definition of F in (5.21) we have that ė(t) = F(t, e(t), ŷ(t)y(t))

Fig. 5 . 1 .

 51 Fig. 5.1. The new observer corresponding to (5.22) correctly estimates the states from (5.38) only using ψ(y(t k )).

Figure 5 . 1 :

 51 Figure 5.1: First row: estimation of the system states of (5.37) (blue) by observer (5.39) (red, lower peaking) and by the new observer (5.22) (black, higher peaking). Second row: output and its estimation from (5.22), and the comparison of the unavailable output and the actual measurements ψ(y(t k )). The initial conditions are x(0) = (1, -1), x(0) = (0, 0), ŷ(0) = C x(0) and the sampling time distance δ = 0.1.

ż

  ∈ F (z, w), (z, w) ∈ C z + ∈ G(z, w), (z, w) ∈D,(5.40) where C and D are closed subsets of R n × R m , and where F and G are set-valued maps from R n × R m to R n . A solution pair to system (5.40) consists of a hybrid arc z : dom(z) → R n and a hybrid input w : dom(w) → R m , these notions stand for the following conditions.

T 1 ≤,

 1 t k+1t k ≤ T 2 , ∀k ≥ 0 and that the control law is specified by u = Kx, for a constant gain K. Suppose that n = m , then system (5.41) has the associated hybrid system (5.40) with n = 2n + 2 and: here τ is a timer that measures the time since the last jump and T represents the intersampling time; the flow and jump sets are defined asC = {z ∈ R n |τ ≤ T }, D = {z ∈ R n |τ ≥ T }.

(s,k)∈N 1 |w

 1 (s, k)|, sup (s,k)∈N 2 |w(s, k)| ,

  function f : R n → R and a vector v ∈ R n as f • (x; v) = lim sup h→0 + ,y→x f (y + hv)f (y) h ,which coincides with the directional derivative whenever f ∈ C 1 . Let us suppose that for different i, j ∈ {1, 2} there exist locally Lipschitz functions V i : R n i → R + satisfying the following conditions:

  système et des mesures disponibles. Dans cette thèse, de tels algorithmes sont donnés par des systèmes dynamiques adjacents et la convergence de l'estimation d'état est dans la norme Euclidienne. Des notions étroitement liées à l'observateur sont celles d'observabilité ou détectabilité d'un système, qui distinguent les systèmes admettant un modèle d'observateur. La théorie des observateurs est devenue un domaine extrêmement riche qui se situe à l'interface entre les disciplines des mathématiques pures et les applications pratiques. Bien que la conception d'observateurs pour les systèmes linéaires soit considérée comme un problème résolu, son homologue non linéaire fait l'objet de recherches intensives dans la communauté du contrôle depuis plusieurs décennies. En conséquence, bien qu'il n'existe actuellement aucune méthode de conception systématique, des observateurs ont été développés pour des classes spécifiques de systèmes non linéaires et sous différentes hypothèses. Une caractéristique extrêmement recherchée des observateurs est leur robustesse, en raison de leur caractère pratique. En effet, leurs estimations devraient être satisfaisantes même en présence de perturbations du système. Cela rend le problème de la conception d'observateur robuste pour les systèmes non linéaires crucial dans le domaine de la commande automatique. Motivations Il n'est pas anodin d'adapter la conception d'un observateur aux perturbations affectant le système. Ce problème est évident pour les systèmes uniformément observables et leur observateur à grand gain, dont la conception dépend d'un paramètre de réglage pouvant amplifier le bruit de mesure. Plusieurs stratégies d'ajustement optimal ou adaptatif ont été développées afin de régler cet observateur correctement. D'autre part, les entrées du système jouent un rôle central dans l'observabilité des systèmes non linéaires, et la conception des entrées permettant de rendre un système non linéaire observable est généralement complexe. Dans le cas de la conception des entrées pour les systèmes affine ou bilinéaire, les chercheurs ont mis au point des algorithmes pour construire des entrées suffisamment régulières avec les propriétés prescrites. De manière surprenante, aucune littérature ne semble avoir pour objectif le réglage simultané et la sélection des entrées chez les observateurs dans le but

  , P. Dufour, Observer design for nonlinear systems with output transformation. IEEE Transactions on Automatic Control. Accepted as full paper. 2. F. González de Cossío, M. Nadri, P. Dufour, Observer Design for nonlinear systems with sampled and transformed measurements. In 2019 IEEE Conference on Decision and Control (CDC), Nice, France. Accepted.

3. F. González de Cossío, M. Nadri, P. Dufour, Observer design for nonlinear systems with implicit output. In 2018 IEEE Conference on Decision and Control (CDC), Miami Beach, FL, pp. 2170-2175. 4. F. González de Cossío, M. Nadri, P. Dufour, Optimal observer design for disturbed state affine systems. In 2018 Annual American Control Conference (ACC), Milwaukee, WI, pp. 2733-2738.

  Definition 2.2. A function γ : R + → R + is of class K if γ(0) = 0 and if it is strictly increasing and continuous. It is of class K ∞ if it is also unbounded. A function β : R + ×R + → ∈ R n , the solution x is defined on R + and it holds that

	R + is of class KL if β(r, t) is of class K for each t ∈ R + and if β(r, t) decreases to zero as
	t → ∞ for each r ∈ R + . We use the notation γ ∈ K or γ ∈ K ∞ and β ∈ KL.
	For the following, we consider a nonlinear system of the form
	ẋ = f (x, u),	(2.32)
	where x(t) ∈ R n is the state, u(t) ∈ R m the input and f : R n × R m → R n a continuously
	differentiable function such that f (0, 0) = 0.
	Definition 2.3. System (2.32) is said to be input-to-state practically stable (ISpS) if there
	exist functions β ∈ KL and γ ∈ K and a constant c ≥ 0 such that for all inputs u ∈ L ∞ m and
	all initial conditions x(0)

Table 3

 3

	θ/p u	0.1	4	6	8	11.1	12	14
	0.01 269.3 157.7 169.7 177.9 186.5 187.9 190.2
	0.5	36.6 19.3 20.0 21.3 21.6 21.7 24.3
	1	20.8 11.0 11.1 11.7 11.6 11.4 12.9
	3.3 22.1	9.6	9.9	9.8	9.9	10.1 10.9
	5	24.1	9.8	10.2 10.2 10.4 10.7 11.6
	7	26.9 10.5 10.7 10.5 11.0 11.4 12.4
	9	29.1 11.3 11.4 11.1 11.7 12.1 13.3

.1. It is essential to notice that these values do not correspond to the functional J but to the actual estimation error. Although the optimization domain did not include θ < 2.1, some of these values were also simulated for comparison. Our method, summarized in

(3.14)

, provided a close to optimal observer performance with a mean estimation error of 9.9. Moreover, notice the almost convex behavior of the table column-wise and row-wise.

Table 3 .

 3 1: Mean norm of the real estimation error. The value 9.9, corresponding to our strategy design, is in the lower 10.2%.

  Assumption 4.2. The function ψ : R p → R p is of class C 2 and its Jacobian is invertible on all its domain. Moreover, there exists δ ∈ K ∞ ∩ C 2 such that Assumption 4.2 implies in particular the injectivity of ψ. It is satisfied, for example, if p = 1 and if | ∂ψ ∂y | is bounded from below by a positive constant. The authors in[START_REF] Lei | High-gain observers in the presence of sensor nonlinearities[END_REF] require this last condition to hold when the nonlinear output they study is continuously differentiable. Remark 4.8. Notice that Assumption 4.1 concerns the existence of a robust observer design for system (4.16) and, thus, depends on the properties of the functions f and h. However, Assumption 4.2 only constrains the output transformation ψ in system (4.18) and it is needed for redesigning the observer.

δ(|ψ(y)ψ(ŷ)|) ≥ |y -ŷ|, ∀y, ŷ ∈ R p .

  r), for all r ∈ R + . Finally, notice that if both disturbances d y and d ψ are the zero function then c d in (4.27) is zero and the same holds for the Lyapunov-constant c L1 . Lemma 4.2. Consider systems (4.18) and (4.21), with ϕ and K as in (4.23)-(4.24), and let Assumptions 4.1 and 4.2 hold. There exists a Lyapunov function for the family

  r, ∀r > 0 precisely when ρ satisfies (4.31). Hence, Theorem 2.15 concludes that G × {0} is practically stable. That is, there exist a function β ∈ KL and a constant c ≥ 0 such that (4.32) is satisfied on [0, T ). As a consequence, this interval is necessarily the whole R + . Finally, if both disturbances d y and d ψ are zero then in Lemmas 4.1 and 4.2 both Lyapunovconstants c L1 and c L2 are zero as well. The details of the proofs in Section 2.5 imply then Compute the corresponding decay rate β and constant c as shown in the proofs of Lemma 2.1 and Theorem 2.14 in Section 2.5.

	5.
	c = 0.
	satisfying the inequality in (4.31).
	2. Get the Lyapunov-bounds, gains and constants from Lemma 4.1 and Lemma 4.2.
	3. Consider the mixed gains in (4.33) and choose an in-between function σ as explained
	in Section 2.5.
	4. Compute the bounds, gain and constant of the Lyapunov function for G × {0} as in the
	proof of Theorem 2.15 in Section 2.5.

Remark 4.10. In practice, the design of the new observer

(4.21)

starts by proposing an observer as (4.17) and by finding a Lyapunov function V for its error dynamics, together with the functions ᾱ1 , ᾱ2 and χ from Assumption 4.1. We then require δ from Assumption 4.2 and the functions ϕ and α from (4.23) (see

Remark 4.9)

. Finally, the lower bound in

(4.31)

itself can be used to construct such a ρ and K is then given by (4.24). Remark 4.11. From the proofs of Lemmas 4.1 and 4.2, we have that the new observer recovers a type of DES property: there exist β ∈ KL and γ ∈ K such that for all x(0), x(0) ∈ R n and all ŷ(0) ∈ R p , and for all bounded, Lipschitz and differentiable d y and d ψ we have |(ξ(t), e(t))| ≤ β(|(ξ(0), e(0))|, t) + γ(|(d y , d ψ , ḋy , ḋψ )| ∞ ), ∀t ≥ 0.

Finally, notice that we can find the explicit decay rate β and constant c in (4.32) by following the next steps:

1. Find the Lyapunov-bounds and gain from Assumption 4.1. Use them to construct ρ

  and(5.22) can be written asF 1 (t, e(t), ξ(t)), t ≥ 0 ξ(t) = F 2 (t, e(t), ξ(t k )), t ∈ [t k , t k+1 )

	⎧ ⎨	ė(t) =
	⎩

  e, ξ) =f (x(t)) -f (x(t)e, y(t) -ξ), Motivated by the previous remark, a first step towards the reformulation of our problem is to reconsider the general sampled-data system from Section 5.2.2 described byẋ(t) = Ax(t) + Bu(t k ), t ∈ [t k , t k+1 ),(5.41)where x(t) ∈ R n and u(t) ∈ R m , A and B are constant matrices, and (t k ) is an increasing sequence of sampling times. Let us further assume that there exist positive real numbers T 1and T 2 such that

	F 2 (t, e, ξ k ) =	∂ψ ∂y	(h(x(t)))	∂h ∂x	(x(t))f (x(t))
		-	∂ψ ∂y	(h(x(t) -e))	∂h ∂x	(x(

t)e)f (x(t)e) -Kξ k ,

for a properly designed constant matrix K.

Acknowledgments

Chapter 3

Optimal observer design for disturbed state-affine systems

Introduction

We analyzed in the previous chapter one of the standard methods of nonlinear observer design known as high-gain. A large enough tuning parameter is used in the observer structure to moderate the rate of convergence of the state estimation error. However, this method has a major drawback in practice where disturbances tend to affect the system: measurement noise sensitivity [START_REF] Kwakernaak | Linear optimal control systems[END_REF]. Therefore, a common strategy is to formulate an adaptive gain. Simply put, the gain is high when the state needs to be reconstructed and then it drops down to prevent noise amplification, see for example [START_REF] Ahrens | High-gain observers in the presence of measurement noise: A switched-gain approach[END_REF]. We remark that the systems involved in most of these studies are uniformly observable and must be represented in their canonical observability form, which is not always easy to obtain.

On the other hand, a common strategy for parameter identification using an observer relies on designing an input that optimizes estimations [START_REF] Dufour | An explicit optimal input design for first order systems identification[END_REF][START_REF] Qian | Optimal input design for parameter estimation of nonlinear systems: case study of an unstable delta wing[END_REF]. In general, the selection of the input can have a considerable impact on the performance of an observer [START_REF] Besançon | Nonlinear observers and applications[END_REF] and designing inputs with different types of regularity is not a trivial task. Nevertheless, in practice, this is usually done in a heuristic way. Some notable exceptions can be seen in [START_REF] Scola | Input optimization for Observability of State Affine Systems[END_REF][START_REF] Scola | Optimizing Kalman optimal observer for state affine systems by input selection[END_REF] where the authors consider state-affine systems.

The main part of this chapter, which is based on our work [START_REF] González De Cossío | Optimal observer design for disturbed state affine systems[END_REF], considers the problem Chapter 5 Sampled-data observer design for nonlinear systems

Introduction

Recently, there has been an increasing interest in state estimation for nonlinear systems whose output is only available at discrete times. This and similar output constraints arise frequently in engineering applications within: networked systems [START_REF] Hespanha | A Survey of Recent Results in Networked Control Systems[END_REF], sampled-data systems and quantized systems [START_REF] Ferrante | Observer-based control for linear systems with quantized output[END_REF]. Indeed, a large number of modern control systems consist of subparts which are interconnected by limited digital communication networks. In this context, continuous and precise data is not an accurate representation of reality. Examples of network control systems include: mobile sensor networks, remote surgery, automated highway systems, etc. On the other hand, nonlinear quantization occurs when the output values are limited to a countable set. This is the case for sensors such as optical incremental encoders, which are known to destabilize the control loop.

The problem of observer design for systems with discrete output measurements can be traced back to the introduction of the continuous-discrete Kalman filter in a stochastic framework [START_REF] Jazwinski | Stochastic processes and filtering theory[END_REF], initially motivated by the problem of orbit determination of space vehicles. In the context of high-gain observers, the authors in [START_REF] Deza | High gain estimation for nonlinear systems[END_REF] adapt the well-known observer design for uniformly observable systems [START_REF] Gauthier | A simple observer for nonlinear systems. Applications to bioreactors[END_REF] to the case of a discrete output. It is later shown in [START_REF] Nadri | Observer Design for Uniformly Observable Systems With Sampled Measurements[END_REF] that such a hybrid design is also possible with a constant gain. The authors in [START_REF] Nadri | Design of a continuous-discrete observer for state affine systems[END_REF] propose an impulsive observer for state-affine systems with sampled output by modifying the Kalman-

Illustration

Uniformly observable systems

Consider the nonlinear system in triangular form given by ⎧ ⎨ ⎩ ẋ(t) = Ax(t) + ζ(x(t), u(t)) y(t) = Cx(t), (5.37) where the matrices A, C and the function ζ are as in (2.19). As in Chapter 4, our theory can be easily extended to the case when f (x, u) depends on a bounded and continuous input u(t) ∈ R m . We aim to estimate the state of system (5.37) by using a sampled and transformed version of the output, that is, we consider:

where the sequence (t k ) satisfies (5.20) and where ψ : R → R is a given function satisfying Assumption 5.2. We introduce the usual assumption on the nonlinearity of the dynamics. Assumption 5.4. For each i = 1, . . . , n, there exists a positive constant c ζi such that

We consider the observer for system (5.37) introduced in [START_REF] Gauthier | A simple observer for nonlinear systems. Applications to bioreactors[END_REF], (5.39) where θ > 0 and the matrix S ∞ is symmetric and positive definite. The same computations as in Chapter 4 then show that Assumption 5.1 is satisfied with the linear functions: D'un autre côté, une approche connue pour l'identification des paramètres à l'aide d'un observateur repose sur la conception d'une entrée, qui optimise les estimations (entrée riche) [START_REF] Dufour | An explicit optimal input design for first order systems identification[END_REF][START_REF] Qian | Optimal input design for parameter estimation of nonlinear systems: case study of an unstable delta wing[END_REF]. En général, la sélection de l'entrée peut avoir un impact considérable sur les performances d'un observateur [START_REF] Besançon | Nonlinear observers and applications[END_REF] et la conception d'entrées régulièrement persistantes n'est pas une tâche triviale. Néanmoins, en pratique, cela se fait généralement de manière heuristique.

Résumé étendu en français

Quelques exceptions notables peuvent être vues dans [START_REF] Scola | Input optimization for Observability of State Affine Systems[END_REF][START_REF] Scola | Optimizing Kalman optimal observer for state affine systems by input selection[END_REF] où les auteurs considèrent les systèmes affines en l'état.

Dans cette partie qui est basée sur notre travail [START_REF] González De Cossío | Optimal observer design for disturbed state affine systems[END_REF], on analyse le problème de la robustesse aux perturbations d'un observateur pour les systèmes affines en l'état. Cette classe de systèmes n'est pas nécessairement sous forme canonique. On distingue deux types de per-

Optimisation de la robustesse basée sur les techniques de Lyapunov

L'objectif de cette première partie de notre travail est d'estimer de manière optimale le vecteur d'état des systèmes affines en l'état affectés par le bruit. Autrement dit, nous considérons

une matrice constante. Nous utilisons l'observateur présenté dans le chapitre 3

où S(0) est une matrice symétrique définie positive et θ > 0. Malheureusement, même pour un tel choix de S(0), les valeurs propres de S(t) peuvent encore être arbitrairement proches de zéro pour les entrées dites singulières ; entrées où l'observabilité est perdue. Nous ajoutons l'hypothèse suivante pour éviter cette situation.

Hypothèse A.1. Il existe des constantes réelles positives c 1 (u, θ) et c 2 (u, θ) telles que, si t est suffisamment grand, alors nous avons

Les dépendances explicites de c i en u et θ sont généralement omises pour simplifier la notation.

En utilisant le lemme de Gronwall : lemme 3.1, nous pouvons montrer qu'une borne supérieure de l'erreur d'estimation peut être explicitée. Ce résultat est donné dans le théorème suivant.

Nous utilisons la notation : Hypothèse A.2. La matrice A(u(t)) est bornée. Cela signifie que,

La dépendance explicite de σ en u est généralement omise pour simplifier la notation.

Le lemme suivant peut être déduit du travail développé dans [START_REF] Besançon | Observer Synthesis for a Class of Nonlinear Control Systems[END_REF]. La preuve repose sur la forme de S(t) donnée dans (3.4) et sur les propriétés de base de la matrice de transition.

Lemme A.

De plus, si u est régulièrement persistant pour le système (A.1) par rapport à le triplet (T, α, t 0 ), alors pour tous les θ > 0 et tous les t ≥ t 0 nous avons

Remarque A.1. Lemme A.1 implique que l'hypothèse A.1 est satisfaite lorsque u est régulièrement persistant. En fait, nous pouvons définir les constantes dans (A.3) comme :

Cette sélection de constantes c 1 et c 2 , avec le théorème A.

. Il existe plusieurs outils disponibles dans le logiciel Matlab pour effectuer cette minimisation.

Considérons un ensemble

Par exemple, si J est également strictement convexe, le minimum global unique est facilement obtenu par la fonction "fmincon", qui est basée sur des algorithmes de points intérieurs [START_REF] Byrd | A trust region method based on interior point techniques for nonlinear programming[END_REF].

Nous résumons maintenant la stratégie de conception hors ligne de l'entrée optimale u * et du paramètre de réglage optimal θ * pour une telle J de la manière suivante : Les premières contributions importantes dans ce sens comprennent : une forme linéaire avec injection de sortie [START_REF] Krener | Linearization by output injection and nonlinear observers[END_REF], une forme bilinéaire avec injection de sortie et son observateur de type Kalman [START_REF] Hammouri | Bilinearization up to output injection[END_REF][START_REF] Bornard | Regularly persistent observer for bilinear systems[END_REF]

Formulation du problème

Considérons un système non linéaire de la forme

L'hypothèse A.3 est équivalente à l'ISS du système (A.11) comme dans [START_REF] Sontag | On Characterizations of Input-to-State Stability with Respect to Compact Sets[END_REF] ou, pour les fonctions Lyapunov variant dans le temps [START_REF] Edwards | On input-to-state stability for time varying nonlinear systems[END_REF]. Dans ce contexte, la propriété ISS garantit la dégradation contrôlée des performances de l'observateur (A.9) en présence de bruit de mesure. Les observateurs satisfaisant cette propriété ont d'abord été considérés dans [START_REF] Sontag | Output-to-state stability and detectability of nonlinear systems[END_REF] et ils sont connus comme observateurs stables de perturbation à l'erreur (DES) [START_REF] Shim | Nonlinear Observers Robust to Measurement Disturbances in an ISS Sense[END_REF]. Il existe des méthodes pour déterminer si certains observateurs sont DES [START_REF] Alessandri | Observer design for nonlinear systems by using Input-to-State Stability[END_REF] ou pour les repenser s'ils ne le sont pas [START_REF] Shim | Nonlinear observer design via passivation of error dynamics[END_REF].

Synthèse de l'observateur

Considérons le système avec la sortie transformée donnée dans (A.10). La conception proposée pour l'observateur est présentée comme l'interconnexion suivante :

Observateur (A.9)

ŷ Nouvel observateur (A.12)

x

Figure A.1 : Le système (A.10) et l'observateur (A.9) sont représentés dans la partie supérieure du diagramme. Le nouvel observateur (A.12) est représenté dans la partie inférieure comme un système inter-connecté. L'observateur (A.9) nécessite la sortie non disponible y, tandis que le nouvel observateur utilise à la place les mesures y ψ .

où f et G sont comme dans (A.9), ĥ(x, ŷ, Afin de définir la fonction ϕ, considérons d'abord la fonction φ donnée par Nous continuons maintenant avec la définition de K.

Résultat principal

Étant données les conditions initiales x(0 Une façon de résoudre ce problème consiste à appliquer les résultats de la partie précédente avec le "redesign" de l'observateur de [START_REF] Karafyllis | From Continuous-Time Design to Sampled-Data Design of Observers[END_REF], qui aborde le cas d'une sortie discrétisée. Néan-moins, cela se traduirait par un observateur avec trois interconnexions alors que l'on s'attend à ce que seulement deux puissent suffire. Par conséquent, nous proposons plutôt d'étendre directement notre conception d'observateur à partir du chapitre 4 en implémentant la sortie ψ(y(t k )) avec "sample-and-hold". L'observateur interconnecté résultant se compose de : (i) un sous-système avec une dynamique continue basée sur la structure de l'observateur d'origine, (ii) un sous-système avec une dynamique commutée découlant des techniques de "sampleand-hold". Sa convergence est montrée en définissant une fonctionnelle Lyapunov-Krasovskii, basée sur [START_REF] Raff | Observer with sample-and-hold updating for Lipschitz nonlinear systems with nonuniformly sampled measurements[END_REF][START_REF] Naghshtabrizi | Exponential stability of impulsive systems with application to uncertain sampled-data systems[END_REF], et en utilisant le théorème du petit-gain pour les systèmes commutés [START_REF] Yang | A Lyapunov-based small-gain theorem for interconnected switched systems[END_REF].

Cette approche conduit à deux LMIs qui dépendent, entre autres paramètres, du temps maximum entre deux échantillonnages consécutifs.

Notre objectif est de concevoir un observateur pour le système donné par Le théorème précédent pose des conditions assurant notamment la convergence asymptotique à zéro de e. Sa preuve, qui consiste en deux lemmes, est présentée dans la section 5.5 et est basée sur le théorème du petit-gain pour les systèmes commutés [START_REF] Yang | A Lyapunov-based small-gain theorem for interconnected switched systems[END_REF]. Par conséquent, nous recherchons les fonctions ISS Lyapunov V 1 et V 2 en considérant d'abord e = xx comme état et ξ = y ψψ(ŷ) comme entrée puis en faisant le contraire.

Remarque A.6. L'observateur dans (A. [START_REF] Besançon | An Immersion-Based Observer Design for Rank-Observable Nonlinear Systems[END_REF]) peut être vu comme une interconnexion composée d'un sous-système ordinaire et d'un sous-système commuté, cependant, l'application des résultats généraux de [START_REF] Yang | A Lyapunov-based small-gain theorem for interconnected switched systems[END_REF] nécessite une analyse minutieuse de leurs preuves.

Notre résultats ont été illustrés sur un cas important de système non linéaires (voir chapitre 5).