Jean-Pierre Richard

Diego

Matteo, Rosane, Jorge, Srinath, Xin, Nouha Antonio Safa

Andrey Sonia

Ma Ayoub

Vincent Karama

Philippe Vanheeghe

Christine Bernard Patrick

Jacques Gilles

Brigitte Hilaire

Keywords: 2, 1 Bilinear system state diagram

Science and Technology (HIAST) in Damascus, where I did my engineering studies, and the professors in the École Centrale de Nantes

Introduction xiii

Introduction

The use of digital technologies has been contributing significantly to the applicability of automatic control methods. Today, a digital computer is an essential part of almost any control loop. A classical sampled-data control system is shown in Fig. 1. It is constituted of a continuous-time plant (based on power exchanges and energy transformation), interacting in a feedback loop with a digital controller (based on a discrete-time control algorithm). The continuous-time signal corresponding to the output of the system is measured at sampling instants. The controller uses the sampled-data signal to calculate a corresponding control action. The interface between the continuous-time signals and the discrete-time signals is done by means of sample-and-hold devices.

In many present applications (such as cars, aircrafts, robots...) all these components are embedded and the control parts are deployed on several microcontrollers, which have to schedule their various tasks (measure, actuate, compute, communicate...) regarding to real-time specifications and expected performances. The complexity in the design of control algorithms is linked to the kind of modeling hypotheses one can accept as "sufficiently realistic". Among these, linearity of the process model and periodicity of the sampling have been supposed for a long time, mainly because sampled-data control However, these hypotheses are probably more related to the kind of control one can compute under theoretical guarantees, than to the physics of the processes or the nature of the controllers. When the interval between two successive sampling instances is timevarying, sampled-data systems become much more complex: basically, even guaranteeing stability in this case is not straightforward. It is also well recognized that, despite a rich and dense effort, nonlinear systems remain complex by nature and still constitute a research topic.

Nevertheless, from the engineer's point of view, the situation of a nonlinear process with an aperiodic controller has became the standard rather than the exception. Processes are expected to reach their maximum performance of speed, low power consumption, etc., and this makes non negligible nonlinear phenomena appear. In the same time, microprocessors have to schedule more tasks, including communication with each others. This increases their practical timing constraints and unavoidably generates imperfections on the sampling rates. Thus, already for linear systems and even more for nonlinear ones, it is of great interest to be able to determine an upper bound on the sampling intervals which guarantees the stability of a sampled-data controlled system. In the literature, this bound is referred to as the MASP, which acronym comes from Maximum Allowable Sampling Period.

However, note that in the case of aperiodic sampling, the period does not exist anymore.

Along this manuscript, we will keep this acronym but it will rather refer to a Maximum Allowable Sampling interval.

For the case of linear systems, several pioneering approaches exist for analyzing stability under aperiodic sampling. These approaches share the advantage of being constructive, thus quantitative estimations of the MASP may be provided. For the case of nonlinear systems, several generic methods exist. However, in practice it is not always clear how to apply them, and only few works provide a constructive tool for estimating the MASP.

Providing efficient methodology for computing the MASP in the nonlinear case is a challenging, open problem.

Objectives

The work in this thesis is dedicated to the following problem:

Find stability criteria for nonlinear sampled-data control systems, which provide a computable estimate of the MASP.

A particular attention will be first given to the case of bilinear system. These systems represent an intermediate between linear and nonlinear models. Their study is relevant in theory, since they may approximate various nonlinear systems. It is also relevant in practice, since they appear naturally in several application domains. We intend to study the stability of bilinear systems with aperiodic sampled-data control. This will allow for tackling the difficulties of nonlinear systems, while using the quasi-linear structure of the considered class of systems. Our goal is to provide constructive methods for this case.

Furthermore, we will show how the methodology can be extended for the much more general case of nonlinear systems affine in the control.

Structure of the thesis

The thesis is organized as follows.

Chapter 1

In the first chapter we intend to present an overview of sampled-data control techniques.

We introduce general sampled-data systems, and a very short history of using digital technology in control engineering. Then, we focus on the stability of sampled-data control systems with aperiodic sampling. Without being exhaustive, we present what we think to be the main methodologies for stability analysis in both the LTI and the nonlinear cases.

Chapter 2

The second chapter is dedicated to the local stability analysis of bilinear sampled-data systems, controlled via a linear state-feedback static controller, using a hybrid system methodology.

The proposed stability conditions are formulated as Linear Matrix Inequalities (LMIs).

Two constructive methods are considered. They are based on a hybrid system approach, which has been presented in Chapter 1. The first method is a specialization of a generic result used for the nonlinear case. The contribution here is to find a constructive way to apply this generic method for the particular case of bilinear systems. The second method is based on a direct search of a Lyapunov function using LMIs. The novelty here is to avoid some conservative upper bounds on the derivative of a Lyapunov function in the first method. The results of this chapter have been published in [START_REF] Omran | Stability of bilinear sampled-data systems with an emulation of static state feedback[END_REF].

Chapter 3

This chapter re-considers the problem of local stability of bilinear systems with aperiodic sampled-data linear state feedback control using a new approach. The method is based on the analysis of contractive invariant sets, and it is inspired by the dissipativity theory.

Local stability is investigated via an invariance property of some ellipsoidal sets. Statespace constraints are easily included in the analysis. The region of attraction is estimated by a certain level surface of a quadratic function, which can be interpreted as a discretetime Lyapunov function. An LMI optimization allows for choosing, among quadratic Lyapunov functions, the one which maximizes the MASP. The results of this chapter have been published in [START_REF] Omran | Local stability of bilinear systems with asynchronous sampling[END_REF][START_REF] Omran | Stability analysis of bilinear systems under aperiodic sampled-data control[END_REF].

Chapter 4

This chapter generalizes the results from Chapter 3 to the case of nonlinear sampleddata systems affine in the input. Assuming that a stabilizing continuous-time controller exists and has to be implemented digitally, we intend to provide sufficient asymptotic/exponential stability conditions for the obtained sampled-data system. The main idea of the chapter is to address the stability problem using the concept of exponential dissipativity. Furthermore, the result is particularized for the class of polynomial input-affine sampled-data systems, where stability may be tested numerically using Sum Of Squares (SOS) decomposition and semi-definite programming. The SOS techniques are used to derive storage and supply functions. The results of this chapter have been published in [START_REF] Omran | Stabilité des systèmes non linéaires sous échantillonnage apériodique[END_REF][START_REF] Omran | On the stability of input-affine nonlinear systems with sampled-data control[END_REF][START_REF] Omran | Stabilité des systèmes non linéaires sous échantillonnage apériodique[END_REF].

Personal Publications

Journals

• [START_REF] Omran | Stability analysis of bilinear systems under aperiodic sampled-data control[END_REF] H. Omran, L. Hetel, J.-P. Richard, and F. Lamnabhi-Lagarrigue. "Stability analysis of bilinear systems under aperiodic sampled-data control". Automatica.

Accepted.

• [START_REF] Omran | Stabilité des systèmes non linéaires sous échantillonnage apériodique[END_REF] H. Omran • [START_REF] Omran | On the stability of input-affine nonlinear systems with sampled-data control[END_REF] H. Omran, L. Hetel, J.-P. Richard, and F. Lamnabhi-Lagarrigue. "On the stability of input-affine nonlinear systems with sampled-data control". In European Control Conference (ECC), Zurich, Switzerland, pages 2585-2590, 2013.

• [START_REF] Omran | Stabilité des systèmes non linéaires sous échantillonnage apériodique[END_REF] H. Omran, L. Hetel, J. P. Richard, and F. Lamnabhi-Lagarrigue. "Stabilité des systèmes non linéaires sous échantillonnage apériodique". 5èmes Journées Doctorales / Journées Nationales MACS, Strasbourg, France, 2013.

Chapter 1

Sampled-data control systems

Introduction

In this chapter we intend to present an overview on sampled-data control. We introduce first general sampled-data systems, and a short history of using digital technology in control engineering. Then, we focus on the stability of sampled-data control systems with aperiodic sampling. We present the main methodologies for stability analysis in both the Linear Time-Invariant (LTI) and the nonlinear cases. Without being exhaustive, which would be neither possible nor useful, we try to give a structural survey of what we think to be the main results and issues in this domain.

Evolution of sampled-data control

Technological advances offer faster and wider range of innovation, yet exploiting them requires more research and engineering effort: automatic control did not escape it since digital technology appeared. The rapid development and growth of digital technologies have contributed significantly to the development of all engineering domains. Till the 1950's, control engineering was entirely depending on analog components, while today almost all control systems are digitally implemented. Making full use of the potentials of computers and networks in control needed a deep understanding of the emerging research domain. This issue has attracted the attention of researchers since the mid 20th century [START_REF] Jury | Synthesis and critical study of sampled-data control systems[END_REF]. In 1960, Rudolph Kalman stated the following [START_REF] Kalman | On the general theory of control systems[END_REF]:

In no small measure, the great technological progress in automatic control and communication systems during the past two decades has depended on advances and refinements in the mathematical study of such systems.

Conversely, the growth of technology brought forth many new problems (such as those related to using digital computers in control, etc.) to challenge the ingenuity and competence of research workers concerned with theoretical questions.

In the 1950's, computers were used for supervisory tasks, including scheduling, production planning and reporting. Analog control loops were needed anyway as early computers were unreliable, slow and expensive [START_REF] Åström | Computer-controlled systems: theory and design[END_REF]. Then, in the 1960's computers began to take the place of the analog devices in some large industrial systems. The first use of a digital computer for fully direct control of a process was initiated by Imperial Chemical Industries (ICI) who began to work in 1959 with the Ferranti Company on a Direct Digital Control (DDC) scheme for a soda ash plant at Fleetwood, Lancashire [START_REF] Bennett | Control and the digital computer: the early years[END_REF].

Late in the 1960's and in the 1970's, technological progress made it possible to produce smaller, cheaper and more reliable computers, with enhanced computing power [START_REF] Åström | Computer-controlled systems: theory and design[END_REF].

The development of minicomputers and microcomputers permitted to widen the domain of application of computers in control. It became possible to use them in smaller projects, and the number of computers used in control systems, has been increasing rapidly [START_REF] Auslander | Direct digital process control: Practice and algorithms for microprocessor application[END_REF].

Later, innovative efforts led to the crucial use of data networks in control systems. In 1986, Bosch introduced the Control Area Network (CAN) [START_REF] Baillieul | Control and communication challenges in networked real-time systems[END_REF], and nowadays several networks (Fieldbus, industrial Ethernet, etc) are used in control applications. The domain of application includes automotive industry, process control, teleoperation and others.

The advantages of using data networks in control are numerous: low-cost, avoidance of unnecessary wiring, ease of maintenance, flexibility of adding new modules to the control loop, etc. However, networks impose many imperfections that must be taken into account [START_REF] Baillieul | Control and communication challenges in networked real-time systems[END_REF][START_REF] Richard | Systèmes commandés en réseau. IC2 Systèmes Automatisés[END_REF][START_REF] Zhang | Network-induced constraints in networked control systems-a survey[END_REF]. This motivated a new domain of academic research called Networked Controlled Systems (NCSs) [START_REF] Hespanha | A survey of recent results in networked control systems[END_REF], where sampling belongs to the essential issues. controller is implemented using a digital computing unit, which can be a computer or a microcontroller, on-board or connected via a data network.

Sampled-data systems with aperiodic sampling

A schematic of a general sampled-data control system is shown in Fig. 1.1. It consists of continuous-time process with input u and output y, interconnected with a digital computing unit. The interface between the continuous-time process and the discrete-time controller is made using analog-to-digital (A/D) and digital-to-analog (D/A) converters.

The output y(t) which is a continuous-time signal, is converted to a discrete sequence {y(t k )} using an A/D converter. The computer then generates the control action which is the sequence {u(t k )}. This sequence is converted to a continuous-time input u(t) using a D/A converter. One way to do the D/A conversion is to keep the signal constant between two sampling instants, this mechanism is called zero-order hold (ZOH).

The digital controller must synchronize the sampling instants, receive the sampled measured value from the A/D converter, calculate the control action and send it to the D/A converter. This is commonly considered to be occurring in a periodic way, with constant sampling intervals. However, the intervals between two successive sampling instants may be varying due to practical constraints. In point-to-point digital control systems, jitter can be caused by clock inaccuracy, imperfect synchronization, computational delays, system architecture characteristics and real-time scheduling [START_REF] Wittenmark | Timing problems in real-time control systems[END_REF]. Aperiodic sampling intervals may also be encountered in NCSs, as constraints are induced by the network [START_REF] Richard | Systèmes commandés en réseau. IC2 Systèmes Automatisés[END_REF][START_REF] Zhang | Network-induced constraints in networked control systems-a survey[END_REF]. For examples, packet dropouts are almost inevitable in NCSs, especially in the case of wireless networks, and they cause variations of the sampling intervals. As a matter of fact, the sampling interval will be a multiple of the nominal one when packets are dropped out, as it can be seen in Fig. 1.2 (here, packets containing the samples 4, 7 and 8 are lost).

Solutions to such a problem can be obtained by means of choosing a hardware with more powerful capabilities. However, these solutions are usually expensive, and they may not always be available. One can also think of solutions from the computer science point of view, like improving the efficiency of calculations, optimizing the codes and enhancing the scheduling policies. Again, these solutions may be limited. From control theory point of view, the solution is to take the imperfections into account, and to design controllers that are less sensitive to the variations of the sampling intervals. Nevertheless, this requires studying systems with complex behaviors, and addressing challenging control problems. In fact, variations of the sampling intervals can have a major effect on stability and performance of sampled-data systems.

Besides, it must be mentioned that there exist several approaches that consider changing intentionally the sampling intervals, in order to sample as less as possible [START_REF] Anta | To sample or not to sample: Self-triggered control for nonlinear systems[END_REF][START_REF] Fiter | A state dependent sampling for linear state feedback[END_REF][START_REF] Heemels | An introduction to event-triggered and self-triggered control[END_REF][START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF].

In event-based control [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF], the sensor measures continuously the output of the system, but it sends the information to the controller only if specific conditions are satisfied. For example, if the difference between the currently measured value and the last transmitted one exceeds some threshold. In self-triggered control [START_REF] Anta | To sample or not to sample: Self-triggered control for nonlinear systems[END_REF], the next sampling instant is calculated as well as the control action, based on the current sampled value. The methodology in state-dependent sampling control [START_REF] Fiter | A state dependent sampling for linear state feedback[END_REF] is like the one in self-triggered control, the difference is that in the former method the next sampling interval is precalculated off-line, while in the latter one the calculations are real-time.

In this thesis we focus on the robustness aspects with respect to time-varying sampling intervals. This problem will be mathematically formalized in the following sections.

Analysis and controller design approaches for sampleddata systems

Because of the hybrid nature of sampled-data systems, there exist specific methods for controller design. Two approaches have attracted most of the attention in the literature: emulation and discrete-time. These approaches will be discussed later in Section 1.6.1

and Section 1.6.2, but we give here a rough overview of their main lines.

Emulation

The principle of the emulation approach is to design a continuous-time controller using one of the methods from continuous-time control theory. This is done while completely ignoring the sampling. Then, the controller is discretized using methods such as Euler, Runge-Kutta or Tustin. Finally, the discretized control law is implemented digitally using sample-and-hold devices such as ZOH. This is a popular and easily applied approach.

However, a fundamental question, which is important from both practical and academic points of view, needs to be addressed: how to choose the sampling period so that the system with the emulated controller, will have a satisfactory performance? Intuitively a "fast" sampling is needed, but an exact qualitative answer to this question is a very important issue.

Discrete-time

In this approach, an exact or an approximate discrete-time model of the plant is found first. Then, a discrete-time controller of the discrete-time model is designed and implemented using a ZOH. This method is straightforward for the case of LTI systems with a fixed sampling period, as an exact linear discrete-time model can be found. This case has been studied since the 1950's, leading to a mature discrete-time control theory for LTI systems. For other cases, it is usually harder to use this approach. For example, an exact discrete-time model of a nonlinear continuous-time plant is usually unavailable, and an approximation is often used in order to design the controller. However, in this case it is not guaranteed that the discrete-time controller, which is designed to stabilize the approximate discrete-time model of the plant, will also stabilize the sampled-data system.

Sampled-data approach

This approach is related to the emulation one. The main difference is the use of a discrete-time model of the plant. The approach takes into account the inter-sample behavior of the system, like in [START_REF] Chen | Optimal sampled-data control systems[END_REF] where lifting technique is used to study linear sampleddata systems. For more information about this approach see [START_REF] Chen | Optimal sampled-data control systems[END_REF][START_REF] Hetel | Discrete and intersample analysis of systems with aperiodic sampling[END_REF][START_REF] Monaco | Advanced tools for nonlinear sampled-data systems analysis and control[END_REF][START_REF] Zaccarian | On finite gain l p stability of nonlinear sampleddata systems[END_REF] and the references therein.

Stability analysis of LTI sampled-data systems

Consider the following LTI continuous-time plant:

ẋ(t) = A 0 x(t) + B 0 u(t), (1.1)
where x ∈ R n is the state vector, and u ∈ R m is the input vector. Assume that the following assumptions hold:

• The state vector x is available only on a set of sampling instants {t k } k∈N :

0 = t 0 < t 1 < • • • < t k < • • • ; t k ∈ R + , ∀k ∈ N; lim k→∞ t k = ∞. (1.2)
• The sampling intervals are time-varying, and they are bounded in the interval [h, h]:

0 < h ≤ t k+1 -t k ≤ h, ∀k ∈ N. (1.3)
• The control is a piecewise-constant:

u(t) = Kx(t k ), ∀t ∈ [t k , t k+1 ), (1.4)
for a matrix K with appropriate dimensions.

In the literature, the value h is often referred to as Maximum Allowable Sampling Period (MASP). Note that we are supposing that there is no transmission delays and no dataprocessing time. Under these assumptions, we obtain the closed-loop sampled-data system:

ẋ(t) = A 0 x(t) + B 0 Kx(t k ), ∀t ∈ [t k , t k+1 ), ∀k ∈ N. (1.5)
The solution x(•) of (1.5) at any instant t ∈ [t k , t k+1 ) is

x(t) = Φ(t -t k )x(t k ) + Γ(t -t k )u(t k ) = Φ(t -t k ) + Γ(t -t k )K x(t k ), where    Φ(t -t k ) := e A 0 (t-t k ) , Γ(t -t k ) := t-t k 0 e A 0 s B 0 ds. (1.6)
Then,

x(t) = Λ(t -t k )x(t k ), (1.7) with Λ(s) = Φ(s) + Γ(s)K. Denoting θ k := t k+1 -t k , x(k) := x(t k ) and u(k) := u(t k ) =
Kx(k), we get the discrete-time model

x(k + 1) = Φ(θ k )x(k) + Γ(θ k )u(k), (1.8)
which in closed-loop becomes

x(k + 1) = Λ(θ k )x(k). (1.9)
The controller is found via the emulation approach. First, the gain K is determined by classical continuous-time methods for the system (1.1). Then, it is discretized using a ZOH (1.4). Alternatively, this gain can be obtained by discrete-time methods for the model (1.8). This is called discrete-time approach.

Periodic sampling

Consider the sampled-data system (1.5) with periodic sampling, i.e. where the sampling instants satisfy

t k+1 -t k = T, ∀k ∈ N. (1.10)
In this case, an exact discrete-time model can be obtained from (1.9):

x(k + 1) = Λ(T )x(k).

(1.11)

The system (1.11) is a LTI discrete-time system. Well known necessary and sufficient conditions for its asymptotic stability are called in the following theorem. The case of periodic sampling is well understood, since control theory is well developed for discrete-time LTI systems. See also [START_REF] Chen | Optimal sampled-data control systems[END_REF][START_REF] Francis | Stability theory for linear time-invariant plants with periodic digital controllers[END_REF][START_REF] Franklin | Digital Control of Dynamic Systems[END_REF][START_REF] Åström | Computer-controlled systems: theory and design[END_REF] where other advanced topics can be found, such as: optimal control, robust controller design, identification, etc.

Aperiodic sampling

Consider the sampled-data system (1.5) with aperiodic sampling, i.e. where the sampling instants satisfy (1.3). Since 1989, much attention has been given to the stability analysis of such systems [START_REF] Ritchey | A stability criterion for asynchronous multirate linear systems[END_REF][START_REF] Voulgaris | Control of asynchronous sampled data systems[END_REF][START_REF] Wittenmark | Timing problems in real-time control systems[END_REF]. Control systems with aperiodic sampling are more complicated to study than the periodic case, as the variations of the sampling intervals can degrade the stability and the performance of sampled-data control systems. The following motivating example from [START_REF] Zhang | Stability analysis of networked control systems[END_REF] shows how variations of the sampling intervals can cause instability.

Example 1.1. Consider the LTI sampled-data system (1.5), where

A 0 = 1 3 2 1 , B 0 = 1 0.6 , K = -1 -6 .
(1.12)

In the case of constant sampling intervals, the sampled-data system is stable for both sampling intervals T 1 = 0.18 and T 2 = 0.54. This can be seen from Theorem 1.1, as the eigenvalues of the matrices Λ(T 1 ) and Λ(T 2 ) defined in (1.11) satisfy

|λ i (Λ(T 1 ))| = 0.7761, i = 1, 2; |λ i (Λ(T 2 ))| = 0.7083, i = 1, 2.
T 1 → T 2 → T 1 • • • .
An illustration of the systems evolution, for both of these constant sampling intervals, is

given in Fig 1 .3. One may think that alternating the sampling interval between T 1 and T 2 will not affect the stability. However, the sampled-data system with periodically time-

varying sampling intervals T 1 → T 2 → T 1 • • • is unstable, as it can be seen in Fig 1.4.
This example shows the importance of taking into consideration the variations of the sampling intervals, when analyzing the stability of sampled-data systems.

In what follows, we present different approaches in the literature which provide sufficient conditions for the stability of LTI sampled-data systems under aperiodic sampling.

Input-delay approach

This approach was first introduced in [START_REF] Mikheev | Asymptotic analysis of digital control systems[END_REF] and further developed in [START_REF] Fridman | Robust sampled-data stabilization of linear systems: an input delay approach[END_REF], and then in several other works like [START_REF] Fridman | A refined input delay approach to sampled-data control[END_REF][START_REF] Seuret | Exponential stabilization of delay neutral systems under sampled-data control[END_REF][START_REF] Seuret | Control of a remote system over network including delays and packet dropout[END_REF]. In this approach, the sampled-data system is modeled as a continuous-time system, with delayed control input. The basic idea in this approach is to write the sampled-data control (1.4) as a delayed control

u(t) = Kx(t k ) = Kx(t -τ (t)), τ (t) = t -t k , ∀t ∈ [t k , t k+1 ), (1.13)
where the delay is piecewise-linear, and satisfies τ (t) = 1 for t = t k , and τ (t k ) = 0. This delay indicates time that has passed since the last sampling instant, see Fig. 1.5. This permits to use tools for stability of systems with time-varying delays. Time-delay are described by means of functional differential equations.

Definition 1.2 (Retarded Functional Differential Equations). The general form of a retarded functional differential equation for a maximum delay h > 0 is ẋ = f (t, x t ), (1.14) where

x t 0 = φ(t 0 + θ), ∀θ ∈ [-h, 0], t t τ (t) = t - t k
x(t) ∈ R n and f : R × C [-h, 0], R n → R n . The notation C [a, b], R n denotes the set of continuous functions mapping the interval [a, b] into R n , and x t (θ) = x(t + θ), ∀θ ∈ [-h, 0].
Lyapunov methods are an efficient tool for stability analysis. In the case of delay-free systems, stability is guaranteed via the construction of a classical Lyapunov function, which is a positive definite function V (t, x(t)), whose time derivative is negative definite along the system trajectories. For a time-delay system, the evolution of the state at instant t is determined by x t , instead of x(t). Thus, it is natural to study the stability using a Lyapunov functional V (t, x t ).

Theorem 1.3 (Lyapunov-Krasovskii Stability Theorem [START_REF] Gu | Stability of Time-Delay Systems[END_REF]). Consider the continuous, non-decreasing functions α, β, γ : R + → R + , where α(s), β(s) are strictly positive and A functional which satisfies the hypothesis of Theorem 1.3 is called Lyapunov-Krasovskii Functional (LKF). An example of such LKF is given by:

satisfy α(0) = β(0) = 0. Suppose that the function f in (1.14) maps R × bounded set in C into a bounded set in R n . If there exists a differentiable functional V : R×C [a, b], R n → R such that α(|φ(0)|) ≤ V (t, φ) ≤ β( φ c ),
V (x t ) = x T (t)P x(t) + 0 -h t t+θ ẋT (s)U ẋ(s) ds dθ, (1.15)
where P > 0 and U > 0. This LKF has been used for several time-delay systems, and in particular for sampled-data systems [START_REF] Fridman | Robust sampled-data stabilization of linear systems: an input delay approach[END_REF]. We recall also the following discontinuous time-dependent Lyapunov functional from [START_REF] Fridman | A refined input delay approach to sampled-data control[END_REF]: Currently, an important effort of research is dedicated to finding and exploiting better ones.

V (t, x(t), ẋt ) = x T (t)P x(t) + (h -τ (t)) t t-τ (t)

Impulsive modeling approach

In this approach, the sampled-data system is modeled as an impulsive system. The stability is studied in a hybrid systems framework, using Lyapunov functions with discontinuities at the impulse times [START_REF] Briat | A looped-functional approach for robust stability analysis of linear impulsive systems[END_REF][START_REF] Naghshtabrizi | Exponential stability of impulsive systems with application to uncertain sampled-data systems[END_REF].

Definition 1.4 (Impulsive Systems [START_REF] Naghshtabrizi | Exponential stability of impulsive systems with application to uncertain sampled-data systems[END_REF]). Consider the system

ẋ(t) = f k (x(t), t), t = t k , ∀k ∈ N, (1.17)
x(t k ) = g k (x(t - k ), t k ), t = t k , ∀k ∈ N,
where f k , g k : R n × R → R are locally Lipschitz functions such that f (0, t) = 0, g(0, t) = 0, ∀t ≥ t 0 , with an impulse sequence t k which is strictly increasing in [t 0 , ∞) for some initial time t 0 .

The stability of the impulsive system (1.17) can be ensured by using Lyapunov methods, involving Lyapunov functions that are discontiuous at impulse instants. Recall the notation in (1.13) τ (t) = t -t k , ∀t ∈ [t k , t k+1 ). We state the following stability result from [START_REF] Naghshtabrizi | Exponential stability of impulsive systems with application to uncertain sampled-data systems[END_REF].

Theorem 1.5 ([82]). Assume that there exist positive scalars c 1 , c 2 , c 3 , b and a Lyapunov function V : R n × R → R, such that:

c 1 |x| b ≤ V (x, τ ) ≤ c 2 |x| b , ∀x ∈ R n , ∀τ ∈ [0, h].
Suppose that for any impulse sequence {t k } k∈N such that:

{t k | ǫ ≤ t k+1 -t k ≤ h, k ∈ N}, (1.18)
with some 0 ≤ ǫ ≤ h, the corresponding solution x(•) to (1.17) satisfies:

dV (x(t), τ (t)) dt ≤ -c 3 V (x(t), τ (t)), ∀t = t k , ∀k ∈ N, and
V (x(t k ), 0) ≤ lim t↑t k V (x(t), τ (t)), ∀k ∈ N.
Then, the equilibrium point x = 0 of system (1.17) is Globally Uniformly Exponentially Stable (GUES) over the class of sampling impulse instants (1.18), i.e. there exist c, λ > 0 such that for any sequence {t k } that belongs to the set (1.18):

|x(t)| ≤ c|x(t 0 )|e -λ(t-t 0 ) , ∀t ≥ t 0 .
In order to apply this method to the stability problem of sampled-data systems, system (1.5) is written as an impulsive system (1.17) with the state ξ(t) = [x T (t), z T (t)] T , where

z(t) = x(t k ), ∀t ∈ [t k , t k+1
). The dynamics of the system can be written as

ξ(t) = F ξ(t), t = t k , ∀k ∈ N, ξ(t k ) = x(t - k ) x(t - k ) , t = t k , ∀k ∈ N,
with the notation x(t -) = lim θ↑t x(θ), and

F := A 0 B 0 K 0 0 .
The stability analysis can be led in this hybrid framework, using time-varying discontinuous Lyapunov functions. For example, in [START_REF] Naghshtabrizi | Exponential stability of impulsive systems with application to uncertain sampled-data systems[END_REF] the following function is considered:

V (ξ(t)) = x T (t)P x(t) + ξ T (t) 0 -τ (t) (s + h)(F e F s ) T R(F e F s )ds ξ(t) +(h -τ (t))(x(t) -z(t)) T X(x(t) -z(t)),
where R := R 0 0 0 and P , R X are symmetric positive definite matrices. This discontinuous Lyapunov function is inspired by the Lyapunov-Krasovskii functional from the input-delay approach, like the one in [START_REF] Fridman | Robust sampled-data stabilization of linear systems: an input delay approach[END_REF]. Vice versa, this approach has also inspired the use of discontinuous Lyapunov functionals in the time-delay approach (see for example the functional (1.16)). Hybrid and input-delay approaches share the same advantages and drawbacks. Both of them are constructive, and LMI conditions are used to construct the Lyapunov functionals/functions. On the other hand, conservatism is added by the upper boundings introduced when studying the derivatives of Lyapunov functionals/functions.

Robust control theory approach

In this approach, sampling effect is seen as a perturbation, and tools from robust control theory are used to ensure stability. The main idea is to write the sampled-data system

(1.5) on each interval [t k , t k+1 ) as:

ẋ(t) = A 0 + B 0 K :=A x(t) + B 0 K :=B (x(t k ) -x(t) :=w(t)
).

(1.19) Then, the system can be represented equivalently by the feedback interconnection of the operator ∆ sh : y → w defined by:

w(t) = (∆ sh y)(t) = - t t k y(τ )dτ, ∀t ∈ [t k , t k+1 ), (1.20)
with the system

G G :=    ẋ(t) = Ax(t) + Bw(t), y(t) = Cx(t) + Dw(t), (1.21)
where C = A = A 0 + B 0 K and D = B = B 0 K, which yields y(t) = ẋ(t). Note that the nominal system (1.21) is LTI. It represents the dynamics of the continuous-time (delayfree) system with an additive input perturbation w(•). The operator ∆ sh captures both the effects of sampling and its variations. This can be seen in Fig. 1.6. The stability can then be studied by analyzing the equivalent model (1.21), (1.20). Small gain theory [START_REF] Gu | Stability of Time-Delay Systems[END_REF]

] ✛ ✲ ẋ(t) = A 0 x(t) + B 0 u(t) S H Kx(t k ) x(t) u(t) ✛ ✲ ẋ(t) = Ax(t) + Bw(t) y(t) = ẋ(t)
∆ sh y(t) w(t) constitutes an interesting tool in this framework: the stability of the interconnection G, ∆ sh is guaranteed if the following condition holds:

⇐⇒

∆ sh G < 1,
where G is the L 2 -induced norm of operator G : w → y, and it is equal to Ĝ

(s) ∞ the H ∞ norm of Ĝ(s) = s(sI -A) -1 B. ∆ sh is the L 2 -induced norm of operator
∆ sh : y → w. In order to check the small gain condition, ∆ sh must be estimated.

An estimate of the norm has been computed in [START_REF] Kao | Simple stability criteria for systems with time-varying delays[END_REF], with the purpose of studying the stability of single-input single-output time-delay systems, with a time-varying delay. As a matter of fact, a more general uncertain delay operator has been considered: Using this property, and the fact that the operator satisfies

∆ d : y(t) → w = (∆ d y)(t) = t t-
M ∆ d = ∆ d M for M ∈ R n×n ,
Mirkin [START_REF] Mirkin | Some remarks on the use of time-varying delay to model sample-andhold circuits[END_REF] provided the following small gain condition

∃ M ∈ R n×n , M > 0 such that M Ĝ(s)M -1 ∞ < 1 h . (1.23)
Interestingly, it is also shown that (1.23) is related to the condition in [START_REF] Fridman | Robust sampled-data stabilization of linear systems: an input delay approach[END_REF] which is obtained using the input-delay approach and the Lyapunov-Krasovskii functional (1.15).

The same LMI can be used to check both conditions. Mirkin then showed that the bound on the operator gain can be enhanced by exploiting the properties of ∆ sh .

Lemma 1.7 ( [START_REF] Mirkin | Some remarks on the use of time-varying delay to model sample-andhold circuits[END_REF]). The operator ∆ sh defined in (1.20) is bounded on L 2 and its L 2induced norm is

δ 0 = 2 π h, (1.24)
and thus

∆ sh z, ∆ sh z ≤ δ 2 0 z, z , for all z ∈ L 2 .
This bound on the norm is actually exact, and it is attained when t k+1 -t k = h. This leads to the following sufficient stability condition, improving (1.23):

∃ M ∈ R n×n , M > 0 such that M Ĝ(s)M -1 ∞ < π 2h . (1.25)
Note that π 2 ≈ 1.57, and thus the conservatism of (1.25) is reduced by about 57% with respect to (1.23). Fujioka [START_REF] Fujioka | Stability analysis of systems with aperiodic sample-and-hold devices[END_REF] showed that the operator ∆ sh also satisfies the following passivity-like property. Lemma 1.8 ([39]). The operator ∆ sh defined in (1.20) satisfies

∆ sh z, z ≤ 0, (1.26)
for all z ∈ L 2 .

The two above properties of ∆ sh are grouped into the following integral property for

0 ≤ Y = Y T ∈ R n×n , 0 < X = X T ∈ R n×n : ∞ 0 y(τ ) w(τ ) T δ 2 0 X -Y -Y -X y(τ ) w(τ ) dτ ≥ 0. (1.27)
Using the integral property (1.27), Fujioka [START_REF] Fujioka | Stability analysis of systems with aperiodic sample-and-hold devices[END_REF] has proposed a stability condition based on Integral Quadratic Constraints (IQCs) [START_REF] Megretski | System analysis via integral quadratic constraints[END_REF].

Theorem 1.9 ( [START_REF] Fujioka | Stability analysis of systems with aperiodic sample-and-hold devices[END_REF]). Suppose that

A = A 0 + B 0 K (1.19) is Hurwitz. The system (1.5) is GUAS if there exist ǫ > 0, 0 < X = X T ∈ R n×n , 0 ≤ Y = Y T ∈ R n×n satisfying Ĝ(jω) I T δ 2 0 X -Y -Y -X Ĝ(jω) I ≤ -ǫI, (1.28)
for all ω ∈ R.

Note that only input-output stability (L 2 -stability) is ensured under condition (1.28) in Theorem 1.9, as well as under (1.25) or (1.23). However, it has been shown in [START_REF] Fujioka | Stability analysis of systems with aperiodic sample-and-hold devices[END_REF] that when A is Hurwitz, the input-output stability implies the asymptotic stability of the origin x = 0.

Checking (1.28) in Theorem 1.9 requires verifying the condition at infinite number of points. The following equivalent LMI condition has been proposed using Kalman-Yakubovich-Popov Lemma [START_REF] Rantzer | On the Kalman-Yakubovich-Popov lemma[END_REF].

Theorem 1.10 ([39]). The system (1.5) is UGAS if there exist 0

< P = P T ∈ R n×n , 0 < X = X T ∈ R n×n , 0 ≤ Y = Y T ∈ R n×n satisfying A T P + P A P B B T P 0 + C D 0 I T δ 2 0 X -Y -Y -X C D 0 I < 0. (1.29)
The approach is interesting, because condition (1.29) is simple and has few variables. It enhances the applicability, especially from an engineering point of view. Nevertheless, it is only applicable to LTI systems, and it is not clear how to extended it to systems with time-varying parametric uncertainties. In Chapter 3, we will propose an extensible alternative via dissipativity theory

Convex-embedding approach

With the convex-embedding approach [START_REF] Cloosterman | Controller synthesis for networked control systems[END_REF][START_REF] Fujioka | Stability analysis for a class of networked/embedded control systems: A discrete-time approach[END_REF][START_REF] Hetel | Stabilization of arbitrary switched linear systems with unknown time-varying delays[END_REF], the stability is studied in the discretetime domain. Denote θ k = t k+1 -t k , and consider the discrete-time system (1.9)

x k+1 = Λ(θ k )x k , (1.30)
where, as in Section 1.5:

Λ(η) = e A 0 (η) + η 0 e A 0 s BdsK. (1.31)
The system (1.30) is Linear Parameter-Varying (LPV), where Λ(θ k ) is an exponential uncertainty with a time-varying parameter θ k ∈ [h, h]. Stability of (1.30) can be guaranteed by showing the existence of discrete-time Lyapunov functions. For example, the system is exponentially stable if one can find a quadratic Lyapunov function

V (x) = x T P x such that P > 0, Λ T (η)P Λ(η) -P < 0, ∀η ∈ [h, h]. (1.32)
Note however that verifying the previous condition requires verifying an infinite set of inequalities. The main idea here is to find a finite set of sufficient conditions for (1.32) by embedding the set W := {Λ(η), η ∈ [h, h]} in a larger set W, defined as the following convex hull with finite number of vertices Λ i , i = 1, • • • , N :

W W Λ 1 Λ 2 Λ n
W := {Λ(η), η ∈ [h, h]} ⊆ W := conv{Λ 1 , Λ 2 , • • • , Λ N }, (1.33)
= N i=1 α i Λ i | α i ≥ 0, i = 1, • • • , N, N i=1 α i = 1 .
This idea is illustrated in Fig. 1.7. The main difficulty in constructing the polytope W is the exponential nature of the uncertainty (1.31). Several approaches exist for the computation of the vertices Λ i . See for example [START_REF] Hetel | Stabilization of arbitrary switched linear systems with unknown time-varying delays[END_REF] for a Taylor series approximation, [START_REF] Fujioka | Stability analysis for a class of networked/embedded control systems: A discrete-time approach[END_REF][START_REF] Sala | Computer control under time-varying sampling period: An {LMI} gridding approach[END_REF] for a method using gridding and [START_REF] Cloosterman | Controller synthesis for networked control systems[END_REF] for another one based on the real Jordan form.

Using the vertices in (1.33), the infinite dimensional inequality problem in (1.32) is guaranteed to be satisfied if the following LMIs are satisfied

P > 0, Λ T i P Λ i -P < 0, i = 1, • • • , N, (1.34)
by applying twice the Schur complement lemma. Note that in the previous approaches, stability was guaranteed by means of a continuous-time Lyapunov function or functional (with or without discontinuities), while here a discrete-time Lyapunov is considered.

Thus, the stability is guaranteed for the sequence x(t k ), without consideration of the intersample behavior. However, in [START_REF] Fujioka | Stability analysis of systems with aperiodic sample-and-hold devices[END_REF], the following proposition has shown that for linear sampled-data system, the stability for the convergence of the state in continuoustime and in discrete-time, is equivalent.

Proposition 1.11 ([39]). Consider the sampled-data system (1.5) with (1.3). For a given x(t 0 ), the following conditions are equivalent:

1. lim t→∞ x(t) = 0 2. lim k→∞ x(t k ) = 0
In [START_REF] Hetel | Discrete and intersample analysis of systems with aperiodic sampling[END_REF] the convex embedding approach is extended to the continuous-time case, using a quasi-quadratic Lyapunov function. This method considers intersample behavior. It provides an accurate analysis, the precision of which can be tuned according to the accuracy of the polytopic approximation (1.33). However, this may increase the number of vertices, and therefore the computational complexity of the analysis. Furthermore, it seems difficult to adapt the method to systems with time-varying parametric uncertainties.

Stability analysis of nonlinear sampled-data systems

A system which includes a nonlinear plant and a sampled-data control law is called a nonlinear sampled-data control system. The linearization of the nonlinear model in the neighborhood of an operating point yields a linear approximation which permits to use tools from linear control theory. Nevertheless, the approximation is only valid sufficiently near the operating point, and the nonlinearity must be taken into account in order to analyze the stability. Either for the periodic or the aperiodic sampling cases, the nonlinear sampled-data control is less understood than the linear one. In the following, an overview of methods and tools for studying nonlinear sampled-data systems is presented. The main research lines are classified according to the way the controller is obtained. There are two main approaches: the emulation approach, and the discrete-time approach. The steps of these two approaches are given in Table 1.1 from [START_REF] Nešić | Sampled-data control of nonlinear systems: An overview of recent results[END_REF].

Emulation approach for nonlinear systems

In the emulation approach, it is assumed that some controller is designed in continuoustime. Then, this controller is discretized using one of the numerical methods, such as Euler, Runge-Kutta or Tustin [START_REF] Stuart | Dynamical Systems and Numerical Analysis[END_REF]. Finally, it is implemented using a sample-and-hold device. Thus, the controller design is separated from the sampling issue, and several [START_REF] Khalil | Nonlinear Systems[END_REF] can be used. On the other hand, in order to make the sampled-data system inherit the properties of the continuous-time system, fast sampling is required, and choosing the upper bound of the sampling interval is a critical question.

Consider the following plant:

ẋp (t) = f p (x p (t), u(t)), y(t) = g p (x p (t)), (1.35)
where x p is the plant state, u is the control input, y is the measured output. Suppose that stability in some sense (UGAS, ISS, etc) is guaranteed by the continuous-time controller:

ẋc (t) = f p (x c (t), y(t)), u(t) = g c (x c (t), y(t)), (1.36)
where x c is the controller state. The implementation of this controller using a ZOH yields:

t ∈ [t k , t k+1 ) : ẋp (t) = f p (x p (t), u(t k )), y(t) = g p (x p (t)), (1.37) t ∈ [t k , t k+1 ) : ẋc (t) = f c (x c (t), y(t k )), u(t) = g c (x c (t), y(t k )). (1.38)
Note that in this case, the controller is supposed to be calculated in continuous-time, as can be seen from (1.38). The values of y and u are transmitted on sampling instants t k .

When the controller is computed numerically, (1.38) is to be replaced by:

t k = kT, k ∈ N : x c (k + 1) = F c T (x c (k), y(k)), u(k) = g c (x c (k), y(k)), (1.39)
where T is a constant sampling interval, y(k) := y(t k ) and u(k) := u(t k ). Note that in this second model, periodic sampling is supposed. Moreover, the closed-loop system is determined by a differential equation which represents the continuous-time plant, and a difference equation which represents the discrete-time controller. The term F T c is obtained by calculating a discrete-time model of (1.38).

Qualitative properties of sampled-data systems under emulation

The choice of sampling intervals is a critical issue in the emulation approach. Intuitively, by choosing a sufficiently fast frequency of sampling, the stability will be preserved under sampled-data implementation. This conjecture has been confirmed in [START_REF] Herrmann | Discretization of sliding mode based control schemes[END_REF], for the case of input-affine systems: Theorem 1.12 ( [START_REF] Herrmann | Discretization of sliding mode based control schemes[END_REF]). Consider the system

ẋp (t) = f (t, x p (t)) + g(x p (t))u(t),
(1.40)

with x p ∈ R n , f : R × R n → R n and g : R n → R n , |g| ≤ c, c > 0 . Suppose that a control u C (x p (t)
) exists and stabilizes the system exponentially, and that f (•), g(•) and u C (•) are smooth with respect to t and x p . Furthermore, the continuous-time controlled system shall have the Lipschitz properties:

|f (t, x 1 ) -f (t, x 2 )| ≤ K f |x 1 -x 2 |, |u C (x 1 ) -u C (x 2 )| ≤ K u |x 1 -x 2 |.
Finally it is assumed that the open-loop system (1.40) has no finite escape time for any bounded input u. For discretized control let the state be available at well defined time

instants t k = t 0 +kT , k = 0, 1, • • • : x p (t k ) = x p (t)| t=t k . Consider that the control applied to (1.40) is discretized: u(t) = u C (x p (t k )), ∀t ∈ (t k , t k+1 ], u(t 0 ) = 0.
Then, there exists a sufficiently small T such that for any T < T the discretized control for the system (1.40) is stable.

Even thought Theorem 1.12 does not give an estimation of the stabilizing sampling frequency, it proves the interesting fact that the discretization of stabilizing continuoustime nonlinear control law with Lipschitz property preserves the stability of the initial nonlinear system, for constant and sufficiently small sampling intervals. This result has been generalized in [START_REF] Burlion | On the stability of a class of nonlinear hybrid systems[END_REF] to the case of time-varying sampling intervals, with dynamical control laws which are discretized using Euler approximation.

In the case of ISS, a similar result is presented in [START_REF] Teel | A note on input-to-state stability of sampleddata nonlinear systems[END_REF]. It shows that when the periodic sampling is sufficiently fast, ISS property of a nonlinear system is semi-globally practically preserved. The semi-global practical stability means that for any region of initial conditions, there exists a sufficiently small sampling period that asymptotically stabilizes the origin of the system. The result is based on exploiting a Razumikhin-type theorem for ISS.

Stability analysis based on linearization

For a special class of nonlinear sampled-data systems, it is shown in [START_REF] Hou | Some qualitative properties of sampled-data control systems[END_REF] that stability conditions can be obtained by analyzing a linearized model.

Theorem 1.13 ([52]

). Consider the sampled-data nonlinear system with a constant

sampling interval t k+1 -t k = T : t ∈ [t k , t k+1 ) : ẋp (t) = f (x p (t)) + B 0 x c (t k ), k ∈ N : x c (k + 1) = Cx c (k) + Dx p (k), (1.41)
where B 0 , C and D are real matrices with appropriate dimensions, and

x c (k) := x c (t k ),
x p (k) := x p (t k ). The function f : R n → R n is assumed to be continuously differentiable with f (0) = 0. Then, the equilibrium (x T p , x T c ) = (0, 0) of (1.41) is uniformly asymptotically stable, if the equilibrium of the linear sampled-data system

t ∈ [t k , t k+1 ) : ẋp (t) = A 0 x p (t) + B 0 x c (t k ), k ∈ N : x c (k + 1) = Cx c (k) + Dx p (k), (1.42)
is exponentially stable, where A 0 ∈ R n×n denotes the Jacobian of f at x p = 0

A 0 = ∂f ∂x p xp=0 .
The nature of the result is in the spirit of the Lyapunov's First Method [START_REF] Khalil | Nonlinear Systems[END_REF], as it permits to guarantee the stability of the equilibrium of the nonlinear system, by studying the stability of its linearization at the origin. However, in the same way, it does not provide any estimate of the domain of attraction. Note that the origin of the linear sampled-data system (1.42) is exponentially stable if and only if the matrix

Φ(T ) Γ(T ) D C
is Schur, where Φ(T ) and Γ(T ) are given in (1.6). This can be found directly from Theorem 1.1. This result has been generalized in [START_REF] Hu | Stability analysis of digital feedback control systems with time-varying sampling periods[END_REF] to the case of time-varying sampling intervals, with a more general class of nonlinear systems. In [START_REF] Lawrence | Proceedings of the 36th ieee conference on decision and control[END_REF], stability conditions are given, based on an appropriate linearization of the plant and of the controller. However, here again these methods do not provide any estimate of the domain of attraction.

Dissipation preservation under emulation

In [START_REF] Astolfi | Trends in nonlinear control[END_REF][START_REF] Beikzadeh | Dissipativity of nonlinear multirate sampleddata systems under emulation design[END_REF][START_REF] Laila | Open-and closed-loop dissipation inequalities under sampling and controller emulation[END_REF][START_REF] Nešić | Sampled-data control of nonlinear systems: An overview of recent results[END_REF], some results concerning the emulation approach were generalized and unified in a methodological framework, by considering the preservation of dissipation inequality under sampling. It is shown that if a continuous-time controller provides some dissipation properties, then the resulting sampled-data system satisfies similar properties in a semi-global practical sense.

Consider the general nonlinear plant:

ẋp = f p (x p , u, w), y = g p (x p ), (1.43)
where x p is the plant state, u is the control input, y is the measured output and w is the disturbance. Suppose that stability is guaranteed by the continuous-time dynamic output feedback:

ẋc = f c (x c , y), u = g c (x c ), (1.44)
where x c is the controller state. Consider the emulation of this controller, with a sequence of periodic sampling instants t k = kT , k ∈ N. In [START_REF] Astolfi | Trends in nonlinear control[END_REF], the discrete-time model of the closed-loop system with a ZOH is denoted as:

x p (k + 1) = F p T (x p (k), x c (k), w[k]), (1.45)
x c (k + 1) = F c T (x p (k), x c (k), w[k]),
where

x p (k) := x p (t k ), x c (k) := x c (t k ) and w[k] := {w(t) : t ∈ [t k , t k+1 ]}.
Theorem 1.14 ([5]). Suppose that there exists a differentiable storage function V (x p , x c ) and a continuous supply rate S(x p , x c , w) such that the following holds for all (x p , x c ) and w along (1.43) (1.44):

V = ∂V ∂x , f ≤ S(x p , x c , w), (1.46)
where x := (x p , x c ) and f := (f p , f c ). Then for any strictly positive numbers D > ν > 0 there exists T * > 0 such that for all T ∈ (0,

T * ), all (x p , x c , w) with |(x p (k), x c (k))| ≤ D, ess sup θ∈[t k ,t k+1 ] |w(θ)| ≤ D we have that (1.45) satisfies: ∆V T ≤ 1 T t k+1 t k S(x p (k), x c (k), w(t))dt + ν,
where

∆V := V x p (k + 1), x c (k + 1) -V x p (k), x c (k) .
The advantage of this method is that dissipation inequalities permit to study several properties of the sampled-data system with an emulated controller. These properties include stability, ISS, L p -stability, passivity, etc. See [START_REF] Nešić | Sampled-data control of nonlinear systems: An overview of recent results[END_REF] for an application of dissipation inequalities to the study of ISS and passivity. This preservation of dissipation is satisfied for sufficiently small sampling intervals upper bounded by T * . However, the result does not provide any quantitative estimate of T * .

Quantitative estimation of the MASP

The previous results are qualitative and prove some nice properties of sampled-data systems, for sufficiently small sampling intervals. However, they do not provide any method for estimating the maximum allowable sampling intervals, for which the stability properties are preserved. In the following, we review some works which provide such an estimation.

Hybrid system approach

In [START_REF] Nešić | Input-output stability properties of networked control systems[END_REF], L p -stability properties have been studied for NCS with scheduling protocols.

The results are based on the hybrid modeling approach and the small gain theorem, and they can be applied to the sampled-data case to calculate the MASP. In [START_REF] Carnevale | Further results on stability of networked control systems: a Lyapunov approach[END_REF][START_REF] Carnevale | A Lyapunov proof of an improved maximum allowable transfer interval for networked control systems[END_REF], the bound on the MASP has been improved, using a Lyapunov-based method, which result has been particularized to the sampled-data case in [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF]. Consider the plant:

ẋp = f p (x p , u), y = g p (x p ),
where x p is the plant state, u is the control input, y is the measured output. Suppose that asymptotic stability is guaranteed by the continuous-time output feedback:

ẋc = f c (x c , y), u = g c (x c ),
where x c is the controller state. The sampled-data implementation of the controller can be written in the following form:

ẋp = f p (x p , û), t ∈ [t k , t k+1 ), y = g p (x p ), ẋc = f c (x c , ŷ), t ∈ [t k , t k+1 ), u = g c (x c ), ẏ = 0, t ∈ [t k , t k+1 ), u = 0, t ∈ [t k , t k+1 ), ŷ(t + k ) = y(t k ), û(t + k ) = u(t k ), (1.47)
where x p and x c are respectively the states of the plant and of the controller, y is the plant output and u is the controller output; ŷ and û are the most recently transmitted plant and controller output values. In between sampling instants, the values of ŷ and û are held constant. Define the augmented state vector x(t) and the network-induced error e(t):

e(t) = e y (t) e u (t) := ŷ(t) -y(t) û(t) -u(t) ∈ R ne , x(t) := x p (t) x c (t) ∈ R nx . (1.48)
Note that the error vector is subject to resets at each sampling instant. The sampleddata system (1.47) can be written as a system with jumps:

ẋ = f (x, e) t ∈ [t k , t k+1 ), (1.49) ė = g(x, e) t ∈ [t k , t k+1 ), e(t + k ) = 0, with 0 < ǫ ≤ t k+1 -t k ≤ h, for all k ∈ N, x ∈ R nx , e ∈ R ne .
The functions f and g are obtained by direct calculations from the sampled-data system (1.47) (see [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF] and [START_REF] Nešić | Input-output stability properties of networked control systems[END_REF]):

f (x, e) := f p (x p , g c (x c ) + e u ) f c (x c , g p (x p ) + e y ) ; g(x, e) := - ∂gp ∂xp f p (x p , g c (x c ) + e u ) -∂gc ∂xc f c (x c , g p (x p ) + e y )
.

It should be noted that ẋ = f (x, 0) is the closed loop system without the sampleddata implementation. Considering a clock τ which evolves with respect to the sampling instants, system (1.49) can be written as the following hybrid system:

ẋ = f (x, e) ė = g(x, e) τ = 1        τ ∈ [0, h), x + = x e + = 0 τ + = 0        τ ∈ [ǫ, h], (1.50)
with x ∈ R nx , e ∈ R ne , τ ∈ R + , h ≥ ǫ > 0.
The following theorem provides a quantitative method to estimate the MASP, using the model (1.50).

Theorem 1.15 ([88]

). Suppose there exist ∆x , ∆e > 0, a locally Lipschitz function W : R ne → R + , a locally Lipschitz, positive definite, radially unbounded function V :

R nx → R + , real numbers L > 0, γ > 0, functions α W , α W ∈ K ∞ and a continuous, positive definite function ̺ such that, ∀e ∈ R ne : α W (|e|) ≤ W (e) ≤ α W (|e|),
and for almost all |x| ≤ ∆x and |e| ≤ ∆e :

∂W (e) ∂e , g(x, e) ≤ LW (e) + H(x, e),

∇V (x), f (x, e) < -̺(|x|) -̺(W (e)) -H 2 (x, e) + γ 2 W 2 (e).
Finally, consider that the MASP h satisfies 0 < ǫ ≤ h < T (γ, L), given by the following function:

T (γ, L) :=        1 Lr arctan(r) γ > L, 1 L γ = L, 1 Lr arctanh(r) γ < L, with r = γ 2 L 2 -1 .
Then, for all sampling intervals less than h the set {(x, e, τ ) : x = 0, e = 0} is UAS, i.e. there exist ∆ > 0 and β ∈ KL such that for each initial condition τ

(t 0 ) ∈ R + , | x(t 0 ), e(t 0 ) | ≤ ∆: x(t) e(t) ≤ β x(t 0 ) e(t 0 ) , t , ∀t ≥ t 0 .
To our best knowledge, Theorem 1.15 is among the first results providing an explicit formulation of the MASP. It is applicable for both constant and variable sampling intervals.

The proof is based on studying a hybrid Lyapunov function, and it addresses asymptotic/exponential stability. Moreover, it has the advantage of considering a general class of nonlinear systems. Nevertheless, it is not clear how to construct the functions V (x),

W (e) and H(x, e) which satisfy the hypotheses.

Time-delay approach

Recently, a new approach has been proposed by Mazenc et al. [START_REF] Mazenc | Robustness of nonlinear systems with respect to delay and sampling of the controls[END_REF], for the case of control affine non-autonomous systems. It is based on extending the idea in [START_REF] Fridman | Robust sampled-data stabilization of linear systems: an input delay approach[END_REF] for the case of LTI systems. This result considers the robustness of nonlinear systems, with respect to both sampling and delay. We state as follows an adaptation where only sampling is considered. Consider the nonlinear system:

ẋp (t) = f (t, x p (t)) + g(t, x p (t))u(t), (1.51)
with the state x p ∈ R n and the input u ∈ R m , and with functions f , g that are locally

Lipschitz with respect to x p and piecewise continuous in t. Assume that the C 1 controller u(t) = u c (t, x p ) is designed in order to make the system (1.51) UGAS. Moreover, assume that there exist a C 1 positive definite and radially unbounded function V , and a continuous positive definite function W such that:

W b (t, x p ) := - ∂V ∂t (t, x p ) + ∂V ∂x p (f (t, x p ) + g(t, x p )u c (t, x p )) (1.52) satisfies W b (t, x p ) ≥ W (x p ), for all t ≥ t 0 and x p ∈ R n . Also, consider u c (t, 0) = 0 for all t ∈ R. Hence, V is a strict Lyapunov function for ẋp = f (t, x p ) + g(t, x p )u c (t, x p ),
and one can fix class K ∞ functions α 1 and α 2 such that α

1 (|x p |) ≤ V (t, x p ) ≤ α 2 (|x p |),
for all t ≥ t 0 and x p ∈ R n . Define the function h(•) by: [START_REF] Mazenc | Robustness of nonlinear systems with respect to delay and sampling of the controls[END_REF]). Suppose that there exist constants c 1 , c 2 , c 3 and c 4 such that:

h(t, x p ) = ∂u c ∂t (t, x p ) + ∂u c ∂x p f (t, x p ) + g(t, x p )u c (t, x p ) . (1.53) Theorem 1.16 ([
| ∂u c ∂x p (t, x p )g(t, x p )| 2 ≤ c 1 , (1.54)
| ∂V ∂x p (t, x p )g(t, x p )| 2 ≤ c 2 , (1.55) |h(t, x p )| 2 ≤ c 3 W (x p ), (1.56)
| ∂V ∂x p (t, x p )g(t, x p )u c (t, x p )| 2 ≤ c 4 (V (t, x p ) + 1), (1.57)
hold for all t ≥ t 0 and x p ∈ R n . Consider the system (1.51) in closed-loop with:

u(t) = u c (t k , x p (t k )), t ∈ [t k , t k+1 ),
where the sequence {t k } satisfies t 0 = 0, 0 < h ≤ t k+1 -t k ≤ h, ∀k ∈ N. Then, the closed-loop system is UGAS if:

h ≤ 1 √ 4c 1 + 8c 2 c 3 . (1.58)
Note that the estimate of the MASP (1.58) is given directly in terms of the system dynamics, the control and the Lyapunov function. The stability is proven by means of a Lyapunov functional. However, it is not clear how conservative the result is.

Further notes and references on emulation approach

It must be mentioned that other works can be found in the literature. In [START_REF] Zheng | Fast sampling and stability of nonlinear sampled-data systems: Part 2[END_REF] an analytical relationship between sampling rates and the domains of attraction of the system is derived, for a special class of nonlinear sampled-data systems. In [START_REF] Karafyllis | Global stability results for systems under sampleddata control[END_REF], the input-delay approach is explored on the basis of vector Lyapunov functions. In [START_REF] Karafyllis | Nonlinear stabilization under sampled and delayed measurements, and with inputs subject to delay and zero-order hold[END_REF],

stabilization of nonlinear systems is considered, with inputs that are subject to both delays and sampling. It is shown that sampled-data feedback laws with a predictorbased delay compensation can guarantee global asymptotic stability for the closed-loop system. Results on global stabilization under sampled-data control can be found in [START_REF] Karafyllis | Global stabilization of feedforward systems under perturbations in sampling schedule[END_REF] for the case of feedforward system, based on a discontinuous feedback.

Discrete-time approach for nonlinear systems

The main motivation for considering direct discrete-time design, is to avoid the disadvantages of the emulation approach, among which is the necessity of a relatively fast sampling. Moreover, some properties such as dead-beat control [START_REF] Åström | Computer-controlled systems: theory and design[END_REF], can not be achieved in continuous-time. This approach has shown a promising potential to find better performing controllers, since sampling is taken into account for the design.

Discrete-time modeling

Consider the nonlinear continuous-time plant:

ẋp = f p (x p , u), x p (t 0 ) = x 0 , (1.59)
with a set of sampling instants 0 = t

0 < t 1 < • • • < t k < • • • which satisfy: t k+1 -t k = T, (1.60)
and with a sampled-data control u(t) = u(k) := u(t k ), ∀t ∈ [t k , t k+1 ). The relation between the states x p (k) := x p (t k ) is given by the exact discrete-time model of (1.59):

x p (k + 1) = x p (k) + t k t k+1 f (x p (s), u(k))ds (1.61) = F p T (x p (k), u(k)). (1.62)
In [START_REF] Monaco | Advanced tools for nonlinear sampled-data systems analysis and control[END_REF], the equivalent discrete-time model of the nonlinear continuous-time system is provided, using the formalism of asymptotic expansion. It is shown that solutions to non-autonomous differential equations can be described by their asymptotic expansion in powers of the sampling period. Considering the autonomous vector field f p , the differential equation

ẋp (t) = f p (x p (t)), (1.63)
is transformed, under sampling, into the difference equation

x p (k + 1) = F p T (x p (k)), (1.64)
with

F p T (x p ) = x p + i≥1 T i i! L i f (x p ), (1.65)
where the map F p T (x) : (0, T ]×R n → R n is deduced from the flow associated with (1.63). It is parameterized by the sampling period T , and the Lie derivatives of f p :

L i fp (x p ) = ∂L i-1 fp (x p ) ∂x p f (x p ) = f i p (x p ), L 0 fp (x p ) = f p (x p ).
(1.66)

A similar result using Lie derivatives may also be established, for non-autonomous systems (see [START_REF] Monaco | Advanced tools for nonlinear sampled-data systems analysis and control[END_REF] for details). Several works consider that it is possible to obtain an exact discrete-time model of the plant (1.62) (see for example [START_REF] Kreisselmeier | Numerical nonlinear regulator design[END_REF]). Nevertheless, this assumption is rarely applicable: as a matter of fact, calculating the discrete-time model of a nonlinear continuous-time plant, is a very hard problem. It requires an explicit analytic solution of a nonlinear differential equation. Alternatively, it is possible to consider a family of approximate models F p T which converge to the exact model when an approximation parameter (such as the sampling period) approaches to zero:

x p (k + 1) = F p T (x p (k), u(k)). (1.67)
Numerical approximation method permits to find such approximate discrete-time models. The Euler method is the easiest one, and it is the most popular in the literature.

Using this method, the discrete-time model (1.62) can be approximated by:

F p T (x p (k), u(k)) := x p (k) + T f p (x p (k), u(k)) (1.68)
Using series expansion methods is another way to find the approximate model. Consider again the system (1.63). Although the series expansion (1.65) is calculated over infinite terms to get the exact solution, it is pointed out in [START_REF] Monaco | From chronological calculus to exponential representations of continuous and discrete-time dynamics: A Liealgebraic approach[END_REF] that a truncation of the series may provide an efficient approximation. It yields an approximate model of order z in T , and with an error in O(T z+1 ). [START_REF] Monaco | From chronological calculus to exponential representations of continuous and discrete-time dynamics: A Liealgebraic approach[END_REF] also gives more details about series expansion, and the relation between continuous-time dynamics under holding devices, as well as discrete-time mappings.

Discrete-time controller design

Once the family of approximates (1.67) is calculated, classical discrete-time design methods are used to calculate a controller that stabilizes F p T :

x c (k + 1) = F c T (x c (k), x p (k)), (1.69) u(k) = G c T (x c (k), x p (k)).
At this stage, a critical question is whether the controller (1.69), which is designed in order to stabilize the approximate model (1.67), will also stabilize the exact one (1.62).

This must be guaranteed without knowing the exact model. Several examples in [START_REF] Nešić | A framework for stabilization of nonlinear sampled-data systems based on their approximate discrete-time models[END_REF] show that if the controller or the approximation is not chosen properly, then stability may not be preserved. The following case is taken from those examples.

Example 1.2 ([85]

). Consider the sampled-data control of a triple integrator

ẋ1 = x 2 , (1.70) ẋ2 = x 3 , ẋ3 = u.
Note that the exact LTI discrete-time model of this system can be computed. Nevertheless, an approximate model is considered in order to illustrate the main problem encountered in step (1.69). The Euler approximate discrete-time model is:

x 1 (k + 1) = x 1 (k) + T x 2 (k), (1.71)
x 2 (k + 1) = x 2 (k) + T x 3 (k), x 3 (k + 1) = x 3 (k) + T u(k).
A minimum-time dead-beat controller for the Euler discrete-time model is given by

u(k) = - x 1 (k) T 3 - 3x 2 (k) T 2 - 3x 3 (k) T . (1.72)
On the one hand, the closed-loop system (1.71) (1.72) has all poles equal zero for all T > 0, and hence the controller stabilizes asymptotically the Euler-based closed-loop system for all T > 0. On the other hand, the closed-loop system consisting of the exact discrete-time model of the triple integrator and the controller (1.72) is unstable for all T > 0.

Various conditions guaranteeing that (1.69) will stabilize (1.62) are presented in [START_REF] Astolfi | Trends in nonlinear control[END_REF][START_REF] Nešić | A framework for stabilization of nonlinear sampled-data systems based on their approximate discrete-time models[END_REF][START_REF] Nešić | Sufficient conditions for stabilization of sampled-data nonlinear systems via discrete-time approximations[END_REF]. We present here the conditions in [START_REF] Astolfi | Trends in nonlinear control[END_REF]. As stated by the authors, these conditions are strong, but they are relatively easy to state. Consider the following properties:

Definition 1.17 (Equi-Lipschitz Lyapunov function [START_REF] Astolfi | Trends in nonlinear control[END_REF]). Suppose that there exist a Lyapunov function V T , functions α 1 , α 2 , α 3 ∈ K ∞ , and T * > 0 such that for all T ∈ (0, T * ) and all x := (x c , x p ) we have:

α 1 (|x|) ≤ V T (x) ≤ α 2 (|x|), (1.73) ∆V a T ≤ -α 3 (|x|), (1.74)
where

∆V a := V T F p T (x p (k), u(k)), F c T (x c (k), x p (k)) -V T (x(k))
, and u(k) is defined in (1.69). Moreover, suppose that there exist L > 0 and T * > 0 such that for all T ∈ (0, T * ), x 1 , x 2 and all z we have:

|V T (x 1 , z) -V T (x 2 , z)| ≤ L|x 1 -x 2 |.
(1.75)

If V T satisfying (1.73), (1.74), (1.75) exists, it is called an equi-Lipschitz Lyapunov function for the system (1.67) (1.69).

Definition 1.18 (One-step consistency [START_REF] Astolfi | Trends in nonlinear control[END_REF]). Suppose that there exist T * and ρ, α 4 ∈ K such that for all T ∈ (0, T * ) and all x p , u we have:

| F p T (x p , u) -F p T (x p , u)| ≤ T ρ(T )α 2 (|(x p , u)|). (1.76)
Then F p T and F p T are said to be one-step consistent.

Definition 1.19 (Boundedness of G c T [START_REF] Astolfi | Trends in nonlinear control[END_REF]). Suppose there exist T * > 0 and α 5 ∈ K such that for all T ∈ (0, T * ) and all x := (x c , x p ) we have:

|G c T (x)| ≤ α 5 (|x|). (1.77)
Then G c T is said to be bounded uniformly in small T .

Theorem 1.20 ([5]

). Suppose that the following conditions hold:

1. There exists an equi-Lipschitz Lyapunov function for the closed-loop system (1.67),

(1.69).

2. F p T and F p T are one-step consistent.

G c

T is bounded uniformly in small T .

Then, there exists β ∈ KL such that for any positive numbers D, ν, T * > 0 such that for any T ∈ (0, T ) and any |x| ≤ D solutions of the exact closed-loop (1.62), (1.69) satisfy:

|x(k)| ≤ β(|x(0)|, kT ) + ν, k ≥ 0. (1.78)
Note that even if F p T is not known explicitly, consistency can still be checked. As a matter of fact, the conditions in Theorem 1.20 can be checked when the continuous-time system (1.59), the approximate model (1.67) and the controller (1.69) are available. The theorem provides a framework for controller design, but does not tell how to construct a stabilizing controller. In some particular cases, it is possible to design a controller that satisfy the conditions: for example, a backstepping control has been investigated in [START_REF] Nešić | Stabilization of sampled-data nonlinear systems via backstepping on their Euler approximate model[END_REF][START_REF] Postoyan | Robust backstepping for the Euler approximate model of sampled-data strict-feedback systems[END_REF].

For the more general case, one manner for approaching the controller synthesis problem is to redesign a continuous-time controller for sampled-data implementation [START_REF] Nešić | Lyapunov-based continuous-time nonlinear controller redesign for sampled-data implementation[END_REF][START_REF] Nešić | Stabilization of sampled-data nonlinear systems via backstepping on their Euler approximate model[END_REF][START_REF] Postoyan | Robust backstepping for the Euler approximate model of sampled-data strict-feedback systems[END_REF].

Assume that a continuous-time controller u c (x) has been designed for the closed-loop continuous-time system, together with a Lyapunov function V (•). Instead of a direct emulation, the following sampling period dependent controller can be implemented:

u sd (x) = u c (x) + N i=1 T i u i (x), (1.79)
where T is the sampling period, and u i (x) are extra terms that are determined through the redesign process. In [START_REF] Nešić | Stabilization of sampled-data nonlinear systems via backstepping on their Euler approximate model[END_REF][START_REF] Postoyan | Robust backstepping for the Euler approximate model of sampled-data strict-feedback systems[END_REF], u i (x) are determined using backstepping techniques.

See also [START_REF] Nešić | Lyapunov-based continuous-time nonlinear controller redesign for sampled-data implementation[END_REF], where Fliess series expansions of the first difference for V (•) along solutions of the system controlled by u sd (x), are used to determine u i (x). For application, see [START_REF] Laila | 3 sampled-data control of nonlinear systems[END_REF] where redesign methods have been considered for a jet engine, and an inverted pendulum examples.

Further notes on discrete-time approach

To end with this section, note that discrete-time approach was considered only for periodic sampling, even if it might be possible to extend it in same way to aperiodic cases.

Besides, it must be noted that structural properties of a given continuous-time plant may not be inherited to its discrete-time (exact or approximate) model [START_REF] Kazantzis | System-theoretic properties of sampled-data representations of nonlinear systems obtained via taylor methods[END_REF]. The affinity of the system in control and the minimum phase properties are among the properties that may be lost in the sampled-data model. However, the approach is complex, and does not have the attractive easiness of the emulation counterpart approach. At last, note that when using discrete-time methods, no inter-sample behavior is taken into account.

Consequently, the behavior of the sampled-data system, between sampling instants is not necessarily guaranteed.

Conclusion

In this chapter, an overview of results on stability and stabilization of sampled-data systems is presented. Attention has been given to robust stability analysis, with timevarying sampling intervals. It appears that robustness with respect to the time-variations of sampling intervals, is a very challenging problem for both linear and nonlinear systems.

For the case of linear systems, it is shown that several pioneering approaches exist in the literature. These approaches share the advantage of being constructive using LMIs, thus they are numerically tractable. In particular, the MASP that guarantees the stability of a given controller can be efficiently estimated. However, it is not clear how these methods can be extended to the nonlinear case.

For the case of nonlinear systems, the main results are classified into two categories:

emulation approach, and discrete-time approach. This classification takes into account the way the controller is synthesized. Concerning the main challenges of the discretetime approach, we underline the difficulty of constructing an accurate discrete-time model for a nonlinear plant. Another important challenge, is guaranteeing the stability of the closed-loop, with the limitation of using only an approximation of the discretetime model. In the emulation approach, the main difficulty is to provide a quantitative estimation of the MASP. Only few works provide a constructive method for estimating the MASP, which shows that the problem is more challenging in the nonlinear case, than the linear one.

In the following chapters, we intend to provide a contribution to the stability analysis of nonlinear systems under time-varying sampling intervals. The main objective is to provide tractable stability criteria, which allow for estimating the MASP.

We address first the case of bilinear systems, which represents a simple class of nonlinear systems, and can be considered as an intermediate between linear and nonlinear systems.

Two approaches are being considered for bilinear systems: the first one relies on the hybrid dynamical systems framework, while the second one is based on robust control theory.

After that, we will consider a more general class of nonlinear systems, with aperiodic sampled-data control.

Chapter 2

Stability of bilinear sampled-data systems -hybrid systems approach

Introduction

This chapter is dedicated to the local stability analyzis of bilinear sampled-data systems, controlled via a linear state feedback static controller, using a hybrid systems methodology. When a continuous-time controller is emulated, intuitively the stability will be preserved if the sampling intervals are sufficiently small. Nevertheless, this issue has been rarely addressed in a formal quantitative study for bilinear systems. Our purpose is to find a constructive way to calculate the MASP.

Two constructive methods are considered. They are both based on the hybrid systems framework, presented in Section 1.6.1.4. The first method is a specialization of the result used for the general nonlinear case [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF]. The contribution here is to find a constructive way to apply this generic method, for the particular case of bilinear systems. The second method is based on a direct search of a Lyapunov function using LMIs. The novelty here is to avoid some conservative upper bounds on the derivative of a Lyapunov function in the first method.

The chapter is organized as follows. First, bilinear systems are introduced in Section 2.2.

In Section 2.4, we formulate the problem under study. Section 2.5 is dedicated to system modeling. In Section 2.6, we introduce the main results, where sufficient conditions for the local stability of sampled-data bilinear systems are provided. Finally, the results are illustrated by means of a numerical example in Section 2.7.

Bilinear systems

Bilinear systems are considered as the "simplest" class of nonlinear systems. They are linear separately with respect to the state and the control, but not to both of them jointly. Since the beginning of 1970's, they have attracted the attention of many researchers [START_REF] Bruni | Bilinear systems: An appealing class of "nearly linear" systems in theory and applications[END_REF][START_REF] Elliott | Bilinear Control Systems Matrices in Action[END_REF][START_REF] Mohler | Natural bilinear control processes[END_REF][START_REF] Pardalos | Optimization and Control of Bilinear Systems[END_REF]. The associated state-space model is:

ẋ(t) = A 0 x(t) + m i=1 [u(t)] i N i x(t) + B 0 u(t), ∀t ≥ t 0 , (2.1)
where the state vector is x(t) ∈ R n , and the control input is u(t) ∈ R m . The term A 0 x is called the drift, B 0 u is the additive control and m i=1 [u] i N i x is the multiplicative control [START_REF] Elliott | Bilinear Control Systems Matrices in Action[END_REF].

Bilinear systems have applications in various domains since many processes can be modeled by this way. Examples of these processes are found in engineering application such as power electronics [START_REF] Hu | A nonlinear-system approach to analysis and design of power-electronic converters with saturation and bilinear terms[END_REF][START_REF] Sira-Ramírez | Control Design Techniques in Power Electronics Devices[END_REF], a.c. transmission systems [START_REF] Mohler | Bilinear control and application to flexible a.c. transmission systems[END_REF], controlled hydraulic systems [START_REF] Guo | A control scheme for bilinear systems and application to a secondary controlled hydraulic rotary drive[END_REF] and chemical processes [START_REF] Espana | Reduced order bilinear models for distillation columns[END_REF]. Bilinear systems can also be encountered in domains such as ecology, socio-economics, biology and immunology [START_REF] Mohler | Natural bilinear control processes[END_REF][START_REF] Mohler | Bilinear control processes: with applications to engineering, ecology, and medicine[END_REF], only to cite a few.

From the point of view of nonlinear systems theory, the study of bilinear systems is very interesting since such models offer a more accurate approximation to nonlinear systems than the classical linear ones. This can be seen in the added bilinear terms, in state and control, which may come from a Taylor series truncation: [START_REF] Mohler | An overview of bilinear system theory and applications[END_REF][START_REF] Pardalos | Optimization and Control of Bilinear Systems[END_REF] and the references therein give more insight to the approximation of more highly nonlinear systems by bilinear models. As a matter of fact, bilinear systems have also an interesting variable structure characteristic. For example, it has been shown in [START_REF] Mohler | Natural bilinear control processes[END_REF] that bilinear models have more powerful controllability properties than the linear ones. For information about structural properties, system characterization and solutions, see [START_REF] Mohler | Nonlinear Systems[END_REF].

Stabilization of bilinear systems

Even for such "simplest" class of nonlinear systems, the feedback stabilization of bilinear systems is a challenging problem, and several controller structures can be found in the literature [START_REF] Amato | Stabilization of bilinear systems via linear state-feedback control[END_REF][START_REF] Gutman | Stabilizing controllers for bilinear systems[END_REF][START_REF] Longchamp | Stable feedback control of bilinear systems[END_REF][START_REF] Mohler | Nonlinear Systems[END_REF][START_REF] Ryan | On asymptotically stabilizing feedback control of bilinear systems[END_REF][START_REF] Shen | Global feedback stabilization of multi-input bilinear systems[END_REF]. We mention as follows some of the notable approaches. Linear state feedback u = Kx has been proposed in several works [START_REF] Amato | Stabilization of bilinear systems via linear state-feedback control[END_REF][START_REF] Mohler | Nonlinear Systems[END_REF].

Quadratic controller has been considered in [START_REF] Gutman | Stabilizing controllers for bilinear systems[END_REF][START_REF] Mohler | Nonlinear Systems[END_REF][START_REF] Ryan | On asymptotically stabilizing feedback control of bilinear systems[END_REF], and improvements have been provided in the literature (see [START_REF] Chen | Normalised quadratic controls for a class of bilinear systems[END_REF][START_REF] Shen | Global feedback stabilization of multi-input bilinear systems[END_REF] for normalized quadratic control methods). In [70, 77] a discontinuous bang-bang controller has been proposed. In the special case of dyadic bilinear systems ẋ = A 0 x + m i=1 b i (c T i x + 1)u, several authors have considered stabilization using the so-called division controllers [START_REF] Chen | Exponentially stabilizing division controllers for dyadic bilinear systems[END_REF][START_REF] Gutman | Stabilizing controllers for bilinear systems[END_REF]. In [START_REF] Jerbi | Global feedback stabilization of new class of bilinear systems[END_REF], necessary and sufficient conditions for the global asymptotic stabilization by using a homogeneous feedback is provided for a class of bilinear systems (with scalar multiplicative control and no additive control). Sliding mode control has also been applied, see for example [START_REF] Tao | Design of sliding mode controllers for bilinear systems with time varying uncertainties[END_REF]. In [START_REF] Kang | Guaranteed cost control for bilinear systems by static output feedback[END_REF], a polynomial static output feedback controller has been proposed, with a guaranteed upper bound of a performance index. Global asymptotic stabilization using a hybrid controller has been proposed in [START_REF] Andrieu | Global asymptotic stabilization for a class of bilinear systems by hybrid output feedback[END_REF]. Finally, stabilisation of bilinear discrete-time systems using polyhedral Lyapunov functions, has been considered in [START_REF] Athanasopoulos | Unconstrained and constrained stabilisation of bilinear discrete-time systems using polyhedral Lyapunov functions[END_REF].

B 0 ✒✑ ✓✏ C 0 A 0 u + + + ẋ x y i [u] i N i x N (•)

Linear state-feedback control

The linear state feedback is an interesting solution because of its simplicity [START_REF] Amato | Stabilization of bilinear systems via linear state-feedback control[END_REF]. It is easily implemented, and several results address the problem of finding such controllers.

Unfortunately, in nontrivial cases it has been shown that it is usually impossible to stabilize globally the bilinear systems with linear feedback control [START_REF] Mohler | Nonlinear Systems[END_REF] (page. 39). As a matter of fact, in the scalar case (n = 1), it is impossible. For planar single-input systems (n = 2, m = 1), necessary and sufficient conditions are given in [START_REF] Koditschek | Stabilizability of second-order bilinear systems[END_REF]. To our best knowledge, the problem is not fully analyzed yet for n > 2.

Recently in [START_REF] Amato | Stabilization of bilinear systems via linear state-feedback control[END_REF][START_REF] Amato | Stabilization of bilinear systems via linear state feedback control[END_REF][START_REF] Tarbouriech | Control design for bilinear systems with a guaranteed region of stability: An LMI-based approach[END_REF], numerically tractable conditions have allowed for the design of a linear state feedback controller that ensures local asymptotic stabilization.

Theorem 2.1 ([1]

). Given the system (2.1) and the polytope containing the origin:

P c = conv{x 1 , x 2 , . . . , x p } (2.2) = {x ∈ R n : a T j x ≤ 1, j = 1, 2, . . . , r}. (2.3)
Then, a controller:

u(t) = Kx(t), K ∈ R m×n ,
which guarantees the asymptotic stability of the resulting closed-loop system, can be found if there exist scalars γ and c, a symmetric matrix P ∈ R n×n , and a matrix W ∈ R m×n such that

0 < γ < 1, c > 0, P > 0, 1 γa T j P c cP a j γ P c ≥ 0, j = 1, 2, . . . , r, 1 x T i x i cP ≥ 0, i = 1, 2, . . . , p, γ(A 0 P + P A T 0 ) + γ(B 0 W + W T B T 0 ) +        x T i N 1 x T i N 2 . . . x T i N m        W +W T N T 1 x i N T 2 x i • • • N T m x i < 0, i = 1, 2, . . . , p.
The controller is given by K = W P -1 and P c belongs to the domain of attraction of the equilibrium.

The LMI conditions depend on the vertices of the convex polytope P c (2.2), and the dual representation (2.3) where the polytope is presented by r hyperplanes. The proposed conditions are sufficient only for the local stabilization. Note that the above LMI conditions require the pair (A 0 , B 0 ) to be asymptotically stabilizable. However, this condition is not necessary for the stabilization of bilinear systems. This can be seen in following example.

Example 2.1.

ẋ = A 0 x + B 0 u + uN x, u = Kx, with A 0 = -1 0 0 0 ; B 0 = 0 0 ; N = 0 -1 1 0 ; K = 0 1 is equivalent to: ẋ1 = -x 1 -x 2 2 , ẋ2 = x 1 x 2 .
Even though the pair (A 0 , B 0 ) is not stabilizable, the system is still shown to be asymptotically stable using center manifold method 1 .

In spite of this academic example, this state feedback design strategy has shown its interest in practical applications [START_REF] Amato | Stabilization of bilinear systems via linear state-feedback control[END_REF][START_REF] Olalla | Optimal state-feedback control of bilinear dc-dc converters with guaranteed regions of stability[END_REF]. The question now is how to guarantee the stability of the closed loop with a discrete controller implementation.

Problem formulation

Consider the bilinear system (2.1). We suppose that the following assumptions hold:

A1 The control is a piecewise-constant control law

u(t) = Kx(t k ), ∀t ∈ [t k , t k+1 ),
with a set of sampling instants {t k } k∈N satisfying:

0 < ǫ ≤ t k+1 -t k ≤ h, ∀k ∈ N, (2.4)
where h is a given MASP.

A2 The pair A 0 , B 0 is stabilizable, and the linear feedback gain K ∈ R m×n is calculated so that the system (2.1) with the continuous state feedback u(t) = Kx(t) has a locally asymptotically stable equilibrium point at x = 0. The actual domain of attraction (a connected neighborhood of x = 0, see [START_REF] Gruyitch | Stability domains[END_REF]) is denoted D 0 .

A3 The state variables are subject to constraints defined by a polytopic set P ⊂ D 0 :

P = conv{x 1 , x 2 , . . . , x p } (2.5) = {x ∈ R n : a T j x ≤ 1, j = 1, 2, . . . , r} (2.6)
corresponding to an admissible set in the state-space 2 .

1 Jean-Pierre Richard, Lecture Notes: Systèmes Dynamiques, http://researchers.lille.inria.fr/ ~jrichard/pdfs/SystDynJPR2009_part3.pdf 2 The equivalence between the representations in (2.5) and (2.6) is given in [START_REF] Christophersen | Optimal Control of Constrained Piecewise Affine Systems[END_REF] (Theorem 1.29).

Under these assumptions, we obtain the closed-loop sampled-data system:

ẋ(t) = A 0 + m i=1 [Kx(t k )] i N i x(t) + B 0 Kx(t k ), ∀t ∈ [t k , t k+1 ), ∀k ∈ N. (2.7)
System (2.7) may also be written as follows

ẋ(t) = Ã[x(t), e(t)]x(t) + Be(t), ∀t ∈ [t k , t k+1 ) (2.8) with e(t) = x(t k ) -x(t), Ã[x, e] := A 0 + B 0 K + m i=1 [K(x + e)] i N i , (2.9)
and

B = B 0 K. (2.10)
The goal of the chapter is twofold. First, we would like to ensure that the obtained sampled-data system satisfies the state-space constraints (2.5) or (2.6) for any x 0 ∈ P.

Secondly, we would like to provide conditions that guarantee the asymptotic convergence of the system solutions to the origin.

Problem: Find a criterion for the local asymptotic stability of the equilibrium point x = 0 of the bilinear sampled-data system (2.7), together with an estimate E ⊂ P of the domain of attraction, such that for any initial condition x(t 0 ) ∈ E the system solutions satisfy x(t) ∈ P, ∀t > t 0 , and x(t) → 0.

Hybrid system framework

Several works about sampled-data systems [START_REF] Carnevale | A Lyapunov proof of an improved maximum allowable transfer interval for networked control systems[END_REF][START_REF] Forni | Event-triggered transmission for linear control over communication channels[END_REF][START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF] adopt the hybrid systems framework [START_REF] Goebel | Hybrid dynamical systems[END_REF]. A hybrid system H is a tuple (A, B, F, G), where A ⊆ R n and B ⊆ R n are, respectively, the flow set and the jump set, while F : R n → R n and G : R n → R n are, respectively, the flow map and jump map. The hybrid system is usually represented by:

H : ξ = F (ξ) ξ ∈ A ξ + = G(ξ) ξ ∈ B
The dynamics given by F , determines the continuous-time evolution (flow) of the state through A, while G determines the discrete-time evolution (jumps) in B. See [START_REF] Goebel | Hybrid dynamical systems[END_REF] for more details about hybrid dynamical systems.

In Section 1.6.1.4, it has been shown how the sampled-data system (1.47) can be represented by a hybrid model (1.50). In a similar way, we fit the sampled-data system (2.7) into a hybrid model. The system (2.7) is formulated similarly to (1.47) as follows:

ẋ = A 0 x(t) + m i=1 u i (t)N i x(t) + B 0 u(t), t ∈ [t k , t k+1 ), y = x, u = K ŷ, ẏ = 0, t ∈ [t k , t k+1 ), ŷ(t + k ) = y(t k ).
(2.11)

The hybrid model for this case is determined by

ẋ = f (x, e) = Ã[x, e]x + Be ė = g(x, e) = -Ã[x, e]x -Be τ = 1        τ ∈ [0, h) x + = x e + = 0 τ + = 0        τ ∈ [ǫ, h] (2.12)
with Ã[x, e] and B given in (2.9) and (2.10), and ǫ given in (2.4). Note that in contrast to the general case model, there is no û in (2.11). This is due to the fact that the considered controller is a static one. In this case, we may consider only one ZOH mechanism in the input side of the controller.

For the hybrid system (2.12), we are only interested in stability with respect to the variables x and e. We consider the following definition of stability with respect to the set {(x, e, τ ) : x = 0, e = 0}, adapted from [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF].

Definition 2.2. Consider the hybrid system (2.12). The set {(x, e, τ ) : x = 0, e = 0} is uniformly asymptotically stable (UAS) if there exist ∆ > 0 and β ∈ KL such that for each initial condition τ (t 0 ) ∈ R + , | x(t 0 ), e(t 0 ) | ≤ ∆:

x(t) e(t) ≤ β x(t 0 ) e(t 0 ) , t , ∀t ≥ t 0 .
(2.13)

Local stability and MASP estimation

In this section, we provide sufficient stability conditions for the considered case of sampled-data bilinear systems (2.7), or equivalently (2.12). The conditions are used to estimate an upper bound on the MASP. Two methods are to be considered. First, we introduce a method that is based on the application of results for general nonlinear sampled-data systems in [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF] (Method 1). Next, to avoid the use of conservative bounds in the previous method, we look directly for an underlying Lyapunov function by formalizing the conditions as LMIs (Method 2). In both of these methods, we will be dealing with local asymptotic stability. Consider the the polytope P defined in (2.5). If

x(t k ) is in the polytope P, then

A[x(t k )] := Ã[x(t), e(t)] ∈ conv{A 1 , A 2 , . . . , A p }, with A q = A[x q ] ∀q ∈ {1, 2, . . . , p}. (2.14)
Note that the set of barycentric coordinates that determine x(t k ) with respect to the vertex of the polytope P, determine also A[x(t k )] with respect to the vertices in (2.14).

This is due to the linearity of A[x(t k )] in x(t k ), and it can be seen as follows. If x(t k ) ∈ P, then there exist positive scalars

{λ q (t k )} p q=1 , p q=1 λ q = 1 (2.15) such that x(t k ) = p q=1 λ q x q hence p q=1 λ q A q = p q=1 λ q A 0 + B 0 K + m i=1 [Kx q ] i N i = A 0 + B 0 K + m i=1 K( p q=1 λ q x q ) i N i = A[x(t k )].

Method 1: adaptation of a result on general nonlinear sampleddata systems

The following theorem proposes stability conditions using an adaptation of the results in [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF] for the case of bilinear systems.

Theorem 2.3. Consider the bilinear sampled-data system (2.12), the polytope P in (2.5), the notations (2.14) and a function

T (γ, L) :=        1 Lr arctan(r) γ > L 1 L γ = L 1 Lr arctanh(r) γ < L (2.16) with r = γ 2 L 2 -1 (2.17)
where L is given by

L = 1 2 max{-λ min (B T + B), 0} (2.18)
and γ is the solution to the following optimization problem:

γ = min γ ′ (2.19)
satisfying the constraints ∃P ∈ R n×n a symmetric positive definite matrix , ∃γ ′ > 0 and ∃α > 0, such that

M lj = A T l P + P A l + 1 2 (A T l A j + A T j A l ) + αI P B * (α -γ ′2 )I < 0, ∀l, j ∈ {1, 2, ..., p}, (2.20)
where A l and A j are the vertices given in (2.14). Assume that the MASP is strictly bounded by T (γ, L), i.e. h < T (γ, L). Then, for the bilinear sampled-data system (2.12), the set {(x, e, τ ) : x = 0, e = 0} is locally uniformly asymptotically stable.

Proof. This proof is mainly based on an adaptation of Theorem 1 in [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF] to the bilinear case.

Let φ : [0, T ] → R be the solution to

φ = -2Lφ -γ(φ 2 + 1) φ(0) = λ -1 (2.21)
where λ ∈ (0, 1). We recall the following result.

Claim 2.6.1. [START_REF] Carnevale | A Lyapunov proof of an improved maximum allowable transfer interval for networked control systems[END_REF] φ(τ ) ∈ [λ, λ -1 ] for all τ ∈ [0, T ]. Moreover, we have that φ( T ) = λ for T given by

T (λ, γ, L) :=          1 Lr arctan r(1-λ) 2 λ 1+λ ( γ L -1)+1+λ γ > L 1 L 1-λ 1+λ γ = L 1 Lr arctanh r(1-λ) 2 λ 1+λ ( γ L -1)+1+λ γ < L (2.22)
with r is given in (2.17).

Consider the following notations

ξ := [x T , e T , τ ] T , (2.23)
F (ξ) := [f (x, e) T , g(x, e) T , 1] T .

(2.24)

Note that T (λ, γ, L) in (2.22) and T (γ, L) in (2.16) satisfy T (γ, L) = T (0, γ, L), and for a fixed L and γ we have that T (•, γ, L) is strictly decreasing. Hence, since the conditions of the theorem require h to be strictly smaller than T (γ, L), there exists λ ∈ (0, 1) such that h = T (λ, γ, L). For the considered value of λ, define the function

U (ξ) = V (x) + γφ(τ )W 2 (e) (2.25)
with a quadratic function V (x) = x T P x, and W (e) = |e|. The function U (ξ) will be used as a Lyapunov function. Note that

λ min (P )|x| 2 + λγ|e| 2 ≤ U (ξ) ≤ λ max (P )|x| 2 + λ -1 γ|e| 2 . (2.26)
The Lyapunov function is non-increasing at sampling instants as it can be seen from the following

U (ξ + ) = V (x + ) + γφ(τ + )W 2 (e + ) = V (x) ≤ V (x) + γφ(τ )W 2 (e) = U (ξ). ( 2

.27)

In order to treat the quantity ∇U (ξ), F (ξ) we need two inequalities that correspond to both ∂W (e) ∂e , g(x, e) and ∇V (x), f (x, e) . We get the first inequality as follows:

∂W (e) ∂e , g(x, e) = e T (t) W (e) [-A[x(t k )]x(t) -Be(t)] = - 1 2W (e) e T (t)(B T + B)e(t) - 1 W (e) e T A[x(t k )]x(t) ≤ 1 2 max{-λ min (B T + B), 0}W (
A T [x(t k )]P + P A[x(t k )] + 1 2 (A T [x(t k )]A j + A T j A[x(t k )]) + αI P B * (α -γ 2 )I < 0, ∀j ∈ {1, 2, ..., p}.
Similarly, by multiplying the resulting inequalities by λ j (t k ), and taking the sum we get

A T [x(t k )]P + P A[x(t k )] + A T [x(t k )]A[x(t k )] + αI P B B T P (α -γ 2 )I < 0. (2.31)
Define the continuous, positive definite function ̺(s) = αs 2 . From (2.31) and (2.30) the following inequality will be satisfied locally inside the addressed polytopic region

∇V (x), f (x, e) < -̺(|x|) -̺(W (e)) -H 2 (x, e) + γ 2 W 2 (e). (2.32)
From (2.28) and (2.32) we have

∇U (ξ), F (ξ) < -̺(|x|) -̺(W (e)) -H 2 (x, e) + γ 2 W 2 (e)
+2γφ(τ )W (e)(LW (e) + H(x, e))

-γW 2 (e)(2Lφ(τ ) + γ(φ 2 (τ ) + 1))

< -̺(|x|) -̺(W (e)) -H 2 (x, e) +2γφ(τ )W (e)H(x, e) -γ 2 W 2 (e)φ 2 (τ ) yielding ∇U (ξ), F (ξ) < -̺(|x|) -̺(W (e)).
(2.33)

The local stability is straightforward, since ̺(•) is positive definite.

Remark 2.4. In this method, the MASP is calculated by the expression (2.16), based on L and γ. L is calculated analytically, whereas γ is found by solving LMI conditions.

The optimization problem is a minimization of γ ′ because for any constant L, T (•, L) is a strictly decreasing function.

Remark 2.5. Note that since γ does not depend on L, and from the continuity of T (γ, •):

T (γ, 0) = lim L→0 T (γ, L) = lim L→0 arctan( γ 2 L 2 -1 ) γ 2 -L 2 = π 2γ .
Remark 2.6. The stability conditions presented in this theorem are based on the generic inequalities (2.32), (2.28) for nonlinear system presented in [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF]. Our contribution is to provide a constructive manner to apply this result to the case of bilinear systems.

We provide explicit forms of H(x, e), W (e), V (x), and we find L, γ that gives the upper bound on MASP. We provide as well, an LMI formulation that allows us to obtain sufficient stability condition. Note that in order to obtain LMI based stability conditions the approach has been adapted to the bilinear case: the function H(•, •) used here has been modified to depend both on the error e(t) and the state x(t), while in [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF] it is only a function of x.

Method 2: direct Lyapunov function approach

In the previous method, the stability conditions are obtained using upper estimations of the derivative of a Lyapunov function in (2.28) and (2.32). Such upper estimations may be found conservative. In order to avoid them, we provide as follows a second method which evaluates directly the derivative of the Lyapunov function.

Theorem 2.7. Consider the bilinear sampled-data system (2.12). Suppose that MASP is bounded by a value T , i.e. h ≤ T . Assume that there exist symmetric positive definite matrices P, Q, X, Y ∈ R n×n , such that the following LMIs are satisfied

A T l P + P A l + X P B -A T l Q * -B T Q -QB -1 T Q + Y < 0,
∀l ∈ {1, 2, ..., p}.

(2.34)

A T l P + P A l + X P B -A T l Q exp(-1) * [-B T Q -QB -1 T Q] exp(-1) + Y < 0, ∀l ∈ {1, 2, ..., p}. (2.35)
where A l are the vertices in given in (2.14). Then the set {(x, e, τ ) : x = 0, e = 0} of the bilinear sampled-data system (2.12) is locally uniformly asymptotically stable.

Proof. We consider the function

U ′ (ξ) = V ′ (x) + W ′ (τ, e) (2.36)
with V ′ (x) = x T P x, and W ′ (τ, e) = exp( -τ T )e T Qe. We recall the notations ξ and F (ξ) defined as in (2.23), (2.24). This Lyapunov function will be used to prove the stability of the hybrid system (2.12). It is inspired by the Lyapunov functions from [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF] and [START_REF] Carnevale | A Lyapunov proof of an improved maximum allowable transfer interval for networked control systems[END_REF].

From the fact that P > 0, Q > 0, we have that U ′ (ξ) satisfies

U ′ (ξ) ≥ λ min (P )|x| 2 + λ min (Q) exp(-1)|e| 2 , (2.37)
U ′ (ξ) ≤ λ max (P )|x| 2 + λ max (Q)|e| 2 . (2.38)
At sampling instants, U ′ (ξ) is non increasing

U ′ (ξ + ) = V ′ (x + ) + W ′ (τ + , e + ) = x T P x ≤ x T P x + W ′ (τ, e) = U ′ (ξ). (2.39)
In order to study the derivative of U ′ (ξ), we note that

∇U ′ (ξ), F (ξ) = ẋT P x + x T P ẋ - 1 T e T [Q exp( -τ T )]e + ėT [Q exp( -τ T )]e + e T [Q exp( -τ T )] ė.
by replacing ẋ and ė from (2.12) we have that

∇U ′ (ξ), F (ξ) =x T A T [x(t k )]P x + x T P A[x(t k )]x + e T B T P x + x T P Be -x T A T [x(t k )][Q exp( -τ T )]e -e T [Q exp( -τ T )]A[x(t k )]x -e T (B T Q + QB + Q T ) exp( -τ T )e
and we can write the following matrix form

∇U ′ (ξ), F (ξ) + x T Xx + e T Y e = (2.40) x e T A[x(t k )] T P + P A[x(t k )] + X P B -A[x(t k )] T Q exp(-τ T ) * [-B T Q -QB -Q T ] exp(-τ T ) + Y x e . (2
M µ 1 = A T [x(t k )]P + P A[x(t k )] + X P B -A T [x(t k )]Q * [-B T Q -QB -1 T Q] + Y < 0, (2.42)
M µ 2 = A T [x(t k )]P + P A[x(t k )] + X P B -A T [x(t k )]Q exp(-1) * [-B T Q -QB -1 T Q] exp(-1) + Y < 0. (2.
)M µ 1 + 1 -θ(τ ) M µ 2 x e < 0.
This yields

∇U ′ (ξ), F (ξ) < -x T Xx -e T Y e, ∀τ ∈ [0, T ]. (2.44)
From (2.38) we have that for any σ > 0,

-σU ′ ≥ -σλ max (P )|x| 2 -σλ max (Q)|e| 2 . (2.45)
Moreover, from the fact that X > 0, Y > 0 we have

-x T Xx -e T Y e ≤ -λ min (X)|x| 2 -λ min (Y )|e| 2 . ( 2

Numerical example

In this section we present a numerical comparison of the two proposed methods. Consider the example of bilinear systems in [START_REF] Amato | Stabilization of bilinear systems via linear state feedback control[END_REF] and [START_REF] Tarbouriech | Control design for bilinear systems with a guaranteed region of stability: An LMI-based approach[END_REF], where a continuous-time state feedback controllers has been computed in order to locally stabilize the bilinear system. The system is described by the matrices

A 0 =     -0.5 1.5 4
4.3 6.0 5.0 3.2 6.8 7.2

    ; B 0 =     -0.7 -1.3 0 -4.3 0.8 -1.5     N 1 =     -1 0 0 0 0 0 0 0 0     ; N 2 =     0 1 0 0 0 0 0 0 0     .
In [START_REF] Tarbouriech | Control design for bilinear systems with a guaranteed region of stability: An LMI-based approach[END_REF], the linear state feedback The results illustrate the reduction of conservatism in Method 2 with respect to Method 1.

K = 0.
Simulations show that the system is unstable for a larger sampling intervals. However, it is not clear how to improve the method in order to obtain a larger estimate of the MASP.

Conclusion

In this chapter, we have provided sufficient conditions for the local stability of bilinear sampled-data systems, controlled via a linear state feedback controller. We presented results for estimating the MASP that guarantees the local stability of the system. Two methods which are based on a hybrid system approach were considered. The first method is an adaptation of results on the general nonlinear case, while the second one is based on a direct search of a Lyapunov function for the hybrid model. The stability conditions, in both methods, were given in the form of LMIs, which are easily computationally tractable. The results were illustrated by a numerical example.

Chapter 3

Stability of bilinear sampled-data systems -dissipativity approach

Introduction

This chapter considers the problem of local stability of bilinear systems with aperiodic sampled-data linear state feedback control. This problem has been considered in Chapter 2, and we intend to address it using a new approach in this chapter. The method is based on the analysis of contractive invariant sets, and it is inspired by the dissipativity theory.

The notion of dissipativity was introduced by [START_REF] Willems | Dissipative dynamical systems. part I: general theory[END_REF]. Since its introduction, the dissipativity theory has been attracting an increasing attention. It can be used to study stability, passivity, robustness and other analysis and design problems. It was motivated by passivity properties of electrical circuits, and it can be seen as a generalized notion of abstract energy for dynamical systems. See the Appendix A for more details.

In this chapter, local stability of bilinear sampled-data systems will be investigated via an invariance property of some ellipsoidal sets [START_REF] Khalil | Nonlinear Systems[END_REF], [START_REF] Blanchini | Set invariance in control[END_REF]. The proposed method is inspired by the results of [START_REF] Fujioka | Stability analysis of systems with aperiodic sample-and-hold devices[END_REF] for the linear case, and by the dissipativity theory [START_REF] Willems | Dissipative dynamical systems. part I: general theory[END_REF][START_REF] Willems | Dissipative dynamical systems. part II: Linear systems with quadratic supply rates[END_REF].

State-space constraints are easily included in the analysis. It will be proven that the invariance property leads to local asymptotic stability, and the region of attraction will be estimated by a certain level surface of a quadratic function, which can be interpreted This chapter is organized as follows. Technical lemmas are presented in Section 3.2.

Sufficient conditions for the invariance and the local stability are given in Section 3.3.

Finally, the results are illustrated by means of two examples in Section 3.4.

Technical preliminaries

Consider again the problem formulation from Section 2.4. The bilinear system

ẋ(t) = A 0 x(t) + m i=1 [u(t)] i N i x(t) + B 0 u(t), ∀t > t 0 , x(t 0 ) = x 0 ,
with a sampled data state feedback u(t) = Kx(t k ) from equation (2.7), can be written as follows:

ẋ(t) = A 0 + B 0 K + m i=1 [Kx(t k )] i N i :=A(x(t k )) x(t) + B 0 K :=B (x(t k ) -x(t) :=w(t)
).

(3.1)

Defining C(x(t k )) = A(x(t k )) = A 0 + B 0 K + m i=1 [Kx(t k )] i N i , D = B = B 0 K, (3.2)
this shows that the closed-loop bilinear sampled-data system, can be represented by the feedback connection of the system

G :=    ẋ(t) = A(x(t k ))x(t) + Bw(t), y(t) = C(x(t k ))x(t) + Dw(t), (3.3)
with the operator ∆ sh : y → w,

w(t) = (∆ sh y)(t) = - t t k y(τ )dτ, ∀t ∈ [t k , t k+1 ).
(3.4)

The properties of the operator ∆ sh

We recall that the operator ∆ sh in (3.4) has been studied in the LTI systems context, and has two important properties. The first one concerns the gain, and the second is of the passivity-type. In [START_REF] Mirkin | Some remarks on the use of time-varying delay to model sample-andhold circuits[END_REF], it has been shown that the gain of the operator is is bounded on L 2 and its L 2 -induced norm satisfies ∆ sh ≤ δ 0 with δ 0 = 2 π h (see Lemma 1.7).

Moreover, it has been shown that for any v ∈ L 2 [0, h), the L 2 [0, h)-induced norm is also bounded by δ 0 :

t t k (∆ sh v) T (τ )(∆ sh v)(τ )dτ ≤ δ 2 0 t t k v T (τ )v(τ )dτ, ∀t ∈ [t k , t k+1 ). (3.5)
The passivity-type property is given in [START_REF] Fujioka | Stability analysis of systems with aperiodic sample-and-hold devices[END_REF], where it is shown that for any 0

≤ Y = Y T ∈ R n×n and v ∈ L 2 : Y ∆ sh v, v = ∞ 0 v T (τ )Y (∆ sh v)(τ )dτ ≤ 0.
This relation is based on the fact that for any v ∈ L 2 [0, h)

t t k v T (τ )Y (∆ sh v)(τ )dτ ≤ 0, ∀t ∈ [t k , t k+1 ).
In the LTI context, the two properties lead to LMI conditions for stability, which are based on Integral Quadratic Constraints (IQC) [START_REF] Megretski | System analysis via integral quadratic constraints[END_REF], and on the Kalman-Yakubovich-Popov lemma [START_REF] Rantzer | On the Kalman-Yakubovich-Popov lemma[END_REF]. The application of these techniques is restricted to the LTI case, and their extension to bilinear sampled-data systems is not direct. This is why we propose to use the operator's properties to define contractive invariant sets.

Two technical lemmas

The following technical lemmas are based on the work in [START_REF] Fujioka | Stability analysis of systems with aperiodic sample-and-hold devices[END_REF].

Lemma 3.1. Let ∆ sh be the operator defined in (3.4). Then, for any v ∈ L 2 [0, h) and 0 < X T = X ∈ R n×n , the following inequality holds:

I 1 (t) = t t k (∆ sh v) T (τ )X(∆ sh v)(τ ) -δ 2 0 v T (τ )Xv(τ ) dτ ≤ 0, ∀t ∈ [t k , t k+1 ). (3.6)
Proof. First of all, we note that since X T = X > 0, then there exists U ∈ R n×n such that X = U T U . For any t ∈ [t k , t k+1 ) one has

I 1 (t) = t t k U (∆ sh v)(τ ) T (U (∆ sh v)(τ )) -δ 2 0 U v(τ ) T U v(τ ) dτ.
From (3.4) we can see that U (∆ sh v) = ∆ sh (U v), then

I 1 (t) = t t k (∆ sh (U v))(τ ) T (∆ sh (U v))(τ ) -δ 2 0 U v(τ ) T U v(τ ) dτ.
Considering the vector z = U v ∈ L 2 [0, h), we have

I 1 (t) = t t k (∆ sh z) T (τ )(∆ sh z)(τ ) dτ -δ 2 0 t t k z T (τ )z(τ ) dτ
which can be seen to be negative directly from (3.5).

Lemma 3.2. Let ∆ sh be the operator defined in (3.4). Then, for any v ∈ L 2 [0, h) and 0 ≤ Y T = Y ∈ R n×n , the following inequality holds:

I 2 (t) = t t k (∆ sh v) T (τ )Y v(τ ) + v T (τ )Y (∆ sh v)(τ ) d τ ≤ 0, ∀t ∈ [t k , t k+1 ). (3.7) Proof. For any t ∈ (t k , t k+1 ) we have d dt (∆ sh v)(t) = -v(t), hence I 2 (t) = 2 t t k v T (τ )Y (∆ sh v)(τ ) dτ = - t t k d dτ (∆ sh v) T (τ )Y (∆ sh v)(τ ) dτ = -(∆ sh v) T (τ )Y (∆ sh v)(τ ) t t k = -(∆ sh v) T (t)Y (∆ sh v)(t) ≤ 0.

Stability results

In this section we give first a useful generic lemma concerning the positive invariance of nonlinear sampled-data systems, controlled by a linear state feedback. Then we provide the LMI conditions for the stability of bilinear sampled-data systems.

Invariance property

In the following, we derive sufficient conditions for the positive invariance (see [START_REF] Khalil | Nonlinear Systems[END_REF]) of some sub-level sets for a class of nonlinear sampled-data systems:

   ẋ(t) = f k x(t) + g k x(t) Kx(t k ), ∀t ∈ [t k , t k+1 ), ∀k ∈ N, t > t 0 , x(t 0 ) = x 0 , (3.8)
where K ∈ R m×n is the linear feedback gain and, for any k ∈ N, f k : R n → R n and g k : R n → R n×m are Lipschitz continuous functions1 . We also suppose that the state of system (3.8) does not exhibit impulsive behaviors at the sampling instants, thus the solution is everywhere continuous. The set of sampling instants {t k } k∈N satisfies (2.4). [START_REF] Khalil | Nonlinear Systems[END_REF]). Let x(t) be the solution of (3.8), the set E ⊂ R n is said to be positively invariant w.r.t. the system (3.8) if:

Definition 3.3 (Positively Invariant Set

∀t 0 ∈ R, x(t 0 ) ∈ E ⇒ x(t) ∈ E, ∀t ≥ t 0 .
Lemma 3.4. Consider the system (3.8), a differentiable positive definite function V :

R n → R + , and the function S(•, •) defined by the quadratic form

S( ẋ(t), x(t k ) -x(t)) = ẋ(t) x(t k ) -x(t) T -δ 2 0 X Y Y X ẋ(t)
x(t k ) -x(t) (3.9)

with δ 0 = 2 π h, 0 < X T = X ∈ R n×n , and 0 ≤ Y T = Y ∈ R n×n . Assume that: for ẋ(t k ) = 0, d dt V (x(t)) < S( ẋ(t), x(t k ) -x(t)), ∀t ∈ [t k , t k+1 ). (3.10)
For a positive scalar β, consider the sub-level set defined by:

L β := {x ∈ R n : V (x) ≤ β}. (3.11)
Then all the sub-level sets L V (x(t k )) are positively invariant and a) if ẋ(t k ) = 0, the sets L V (x(t k )) are in contraction :

V (x(t k+1 )) < V (x(t k )), ∀k ∈ N, s.t. ẋ(t k ) = 0. (3.12) b) if ẋ(t k ) = 0 then the sets L V (x(t k )) and L V (x(t k+1 )) are equal.
Proof. a) Note that the system in (3.8) can be written as

ẋ(t) = f k x(t) + g k x(t) Kx(t) F k (x(t)) + g k x(t) K G k (x(t)) (x(t k ) -x(t) w(t)
)

and thus it can be written as an interconnection of the system

N :=    ẋ(t) = F k x(t) + G k x(t) w(t) y(t) = ẋ(t)
with the operator ∆ sh : y → w given in (3.4)

implies that x(t) = x(t k ) and V (x(t)) = V (x(t k )), ∀t ∈ [t k , t k+1 ].
Note that for both points a) and b) we get the positive invariance of L V (x(t k )) , which completes the proof.

LMI stability conditions for bilinear sampled-data systems

In the next theorem, sufficient conditions are provided under the form of LMIs, for (2.7) to be locally asymptotically stable at x = 0, inside a given polytopic region P defined by (2.5) and (2.6). The result is based on the application of the Lemma 3.4.

Theorem 3.5. Consider the system (2.7), the equivalent representation (3.3) and (3.4).

Suppose there exist symmetric positive definite matrices X, Y, P ∈ R n×n , matrices P 2 , P 3 ∈ R n×n , and a scalar γ > 0 such that the following optimization problem is feasible γ * = min γ, under the constraints:

(3.16)

E j = γ a T j a j P ≥ 0, ∀j ∈ {1, 2, ..., r}, (3.17)
and

M q =     A T q P 2 + P T 2 A q P -P T 2 + A T q P 3 P T 2 B P -P 2 + P T 3 A q -P 3 -P T 3 + δ 2 0 X P T 3 B -Y B T P 2 B T P 3 -Y -X     < 0, (3.18)
∀q ∈ {1, 2, ..., p}

where the vertices {A q } q∈{1,2,••• ,p} are defined by

A q := A(x q ) = A 0 + B 0 K + m i=1 Kx q i N i (3.19)
with {x q } q∈{1,2,...,p} given in (2.5). Then the equilibrium x = 0 of (2.7) is locally asymptotically stable, and an estimate of its domain of attraction is given by the ellipsoid

E c * (P ) = {x ∈ R n : x T P x ≤ c * } ⊂ P, with c * = 1/γ * . (3.20)
Proof. The proof consists of two steps. First we show that the existence of a solution for (3.18) makes the quadratic function V (x) = x T P x satisfy the conditions of Lemma 3.4, and thus leads to the positive invariance of the sub-level sets L V (x(t k )) . In the second step, we show that this positive invariance leads to the local asymptotic stability in E c * (P ). In the proof, we consider the more general representation G ′ instead of G in (3.3):

G ′ :=          ẋ(t) = A(η k )x(t) + Bw(t), y(t) = C(η k )x(t) + Dw(t), η k ∈ P, ∀k ∈ N, (3.21)
C(η k ) = A(η k ) = A 0 + B 0 K + m i=1 [Kη k ] i N i , D = B = B 0 K Obviously, system G (3.3) corresponds to G ′ (3.21) in the particular case η k = x(t k ).
The interconnection of G ′ with the operator ∆ sh : y → w in (3.4) may also be expressed as:

ẋ(t) = A 0 + m i=1 [Kη k ] i N i x(t) + B 0 Kx(t k ), ∀t ∈ [t k , t k+1 ), ∀k ∈ N. (3.22)
Step 1 : Assume that η k is in the polytope P. Let {λ q (η k )} q∈{1,2,••• ,p} represent the barycentric coordinates of η k in P, i.e. η k = p q=1 λ q x q . The set of barycentric coordinates that determines η k with respect to the vertices of P, also determine A(η k ) with respect to the vertices in (3.19). This is due to the linearity of A(η k ) in η k . Multiplying each of the inequalities (3.18) by the appropriate λ q , and taking the sum over of the resulting inequalities yields:

    A T (η k )P 2 + P T 2 A(η k ) P -P T 2 + A T (η k )P 3 P T 2 B P -P 2 + P T 3 A(η k ) -P 3 -P T 3 + δ 2 0 X P T 3 B -Y B T P 2 B T P 3 -Y -X     < 0. (3.23)
Recall the notations with y(t) = ẋ(t), w(t) = x(t k ) -x(t) defined in (3.3) and (3.4), and the quadratic supply rate function S(•, •) defined in (3.9). Thus, for all [x T (t) y T (t) w T (t)] = 0, the LMI (3.23) implies:

2 x T P T 2 + y T P T 3 -y + A(η k )x + Bw + 2y T (t)P x(t) -S(y(t), w(t)) < 0, (3.24)
where we get the first term using the descriptor method [START_REF] Fridman | Robust sampled-data stabilization of linear systems: an input delay approach[END_REF]:

2 x T P T 2 + y T P T 3 -y + A(η k )x + Bw = 0.
From (3.24), we see that for η k ∈ P the inequality in (3.23) is equivalent to the condition:

V (x(t)) < S(y(t), w(t)), whenever [x T (t) y T (t) w T (t)] = 0.

(3.25)

Note that ẋ(t k ) = 0 implies that ẋ(t) = 0 for all t ∈ [t k , t k+1 ), therefore [x(t) T y(t) T w(t) T ] = 0, and Lemma 3.4 leads to the positive invariance of the sub-level sets L V (x(t k )) , and also

V (x(t k+1 )) < V (x(t k )) whenever ẋ(t k ) = 0, η k ∈ P. (3.26)
This shows the positive invariance of the sets L V (x(t k )) .

Step 2 : Now we show that the positive invariance property obtained in Step 1, leads to local asymptotic stability. From (3.22), the state evolution over the interval t ∈ [t k , t k+1 ) is:

x(t) = Λ η k , σ x(t k ), (3.27) with Λ (η k , σ) = e Ã0 (η k )σ + σ 0 e Ã0 (η k )(σ-s) B 0 Kds, Ã0 (η k ) = A 0 + m i=1 [Kη k ] i N i and σ = t -t k . From (3.26)
x T (t k+1 )P x(t k+1 ) < x T (t k )P x(t k ), whenever ẋ(t k ) = 0, η k ∈ P.

(3.28)

For any non-zero vector y ∈ R n , multiplying the LMI in (3.23) by [y T y T A T (η k ) 0] from the right, and by its transpose from the left yields:

y T A T (η k )P + P A(η k ) + δ 2 0 A T (η k )XA(η k ) y < 0, y = 0.
This shows that A(η k ) is Hurwitz for η k ∈ P, thus from (3.28) we have:

x T (t k ) Λ T η k , h k P Λ η k , h k -P x(t k ) < 0, for all x(t k ) = 0, η k ∈ P and 0 < h k ≤ h. Therefore Λ T η k , h k P Λ η k , h k -P is negative definite.
Given that P is positive definite, then there exists a sufficiently small ̺ > 0 which is independent of k, such that:

Λ T η k , h k P Λ η k , h k -P ≤ -̺P. Setting 0 < α = 1 -̺, 0 < α < 1, as a result we obtain V (x(t k )) ≤ α k V (x(t 0 )), which leads to lim k→∞ V (x(t k )) = 0 ⇒ lim k→∞ x(t k ) = 0.
From (3.27), since η k belongs to the compact set P and σ is bounded, then by the continuity of Λ, the image of Λ : x(t) = 0 and x = 0 is locally asymptotically stable for (3.22). The local asymptotic stability of (2.7) follows from the particular case η k = x(t k ). However, we still need to find a positive invariant set inside P. The desired ellipsoid (3.20) is found, as according to [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF] (page. 70). Note that E c * (P ) is the largest sub-level set of x T P x contained in the polytope P.

P × [ǫ, h) → R n×n is compact,
Remark 3.6. The last theorem provides sufficient, thus possibly conservative conditions for the local stability of bilinear sampled-data systems with state constraints. These conditions exploit dissipativity properties, and depend on the chosen supply rate function.

Besides, the obtained MASP depends on the choice of the analytical polytope.

Remark 3.7. For given P, K and h, the provided conditions represent LMIs, thus they are numerically tractable. Note that the set of LMI conditions in (3.18) require the pair (A 0 , B 0 ) to be stabilizable. Thus, the open-loop system can be unstable. Numerical examples of the proposed approach will be given in the following section.

    , N 1 =     -1 0 0 0 0 0 0 0 0     , N 2 =     0 1 0 0 0 0 0 0 0     .
In [START_REF] Tarbouriech | Control design for bilinear systems with a guaranteed region of stability: An LMI-based approach[END_REF], the state feedback Our objective here is to find a MASP for which the local stability of the bilinear system with aperiodic sampled-data control is guaranteed, while satisfying the set constraints defined by P. Using the method of Theorem 2.3, we find that the LMI conditions in Considering the initial state x 0 = [-0.8 + 0.2 + 0.25] T , the time evolution of the state is shown in Fig. 3.2. The random sequence of sampling periods satisfies the hypothesis in (2.4) with h = 0.051. The stability is ensured as the initial state is located inside E c * (P ).

K = 0.
Numerical solutions starting from the same initial conditions, show that for a uniform sampling interval t k+1 -t k = 0.09 the solution of the system becomes unbounded (see Fig. 3.3). This gives an idea about the conservatism induced by the proposed analysis method.

Considering the same box P, other methods are used to find the MASP that ensures the stability, and a comparison is given in Table 3. the preliminary results in (c) is due to the use of the descriptor method in formalizing the LMI conditions, which avoids some conservative cross products. Note that in this example, dissipativity-based techniques give better estimation than hybrid ones.

Example 2: DC-DC Power Converter

Consider the buck-boost converter in Fig. 3.4, where a pulse width modulator is used to adjust the duty cycle of the switching device. Consider the average-value model of the converter [START_REF] Hu | A nonlinear-system approach to analysis and design of power-electronic converters with saturation and bilinear terms[END_REF][START_REF] Sira-Ramírez | Control Design Techniques in Power Electronics Devices[END_REF]:

ẋ = DA 1 + (1 -D)A 2 x + DB 1 + (1 -D)B 2 v.
In the system state x = [ īL vc ] T , īL is the average inductor current, and vc the average capacitor voltage. The system matrices are

A 1 = -R ON +R L L 0 0 -1 RC ; A 2 = -R L L 1 L -1 C -1 RC ;
B 1 = 1 L 0 0 0 ; B 2 = 0 -1 L 0 0 ; v = V DC v D .
R ON is the on-resistance of the switching device, v D is the diode voltage, and

V DC is the source voltage. D ∈ [D 1 , D 2 ] = [0, 1]
is system input representing the duty cycle.

The system is subjected to saturation due to the hard limits on the duty cycle. Several works have dealt with stability and stabilization of DC-DC converter. Examples like in [START_REF] Hu | A nonlinear-system approach to analysis and design of power-electronic converters with saturation and bilinear terms[END_REF] and [START_REF] Olalla | Optimal state-feedback control of bilinear dc-dc converters with guaranteed regions of stability[END_REF] consider a nonlinear systems approach to design continuous time statefeedback controller that achieves stabilization and tracking, and guarantees robustness with respect to bilinearities and saturation. However, less attention has been paid to study robustness with respect to sampled-data implementation. For a certain working point x0 , D 0 we have

0 = D 0 A 1 + (1 -D 0 )A 2 x0 + D 0 B 1 + (1 -D 0 )B 2 v.
Considering x = x -x0 , and the input signal u = D -D 0 , we can see that

ẋ = A 0 x + B 0 u + N ux, (3.29)
where

A 0 = (D 0 A 1 + (1 -D 0 )A 2 ), B 0 = ((A 1 -A 2 )x 0 + (B 1 -B 2 )v), and N = (A 1 -A 2 ).
Consider the following values

V DC = 6 V , R = 50 Ω, L = 20 mH, C = 220 µF , R ON = 0.08Ω, R L = 0.
34 Ω, and v D = 0.67 V . From the constraints over the duty cycle we see that u must be bounded by

-D 0 + D 1 ≤ u ≤ D 2 -D 0 . We consider D 0 = (D 1 + D 2 )/2
, which corresponds to the equilibrium point x0 = [+0.21 -5.17] and

|u| ≤ u max = (D 2 -D1)/2.
We are interested in the state-space region where a linear control u = K x is not saturated, i. the stabilization of the continuous-time system, we find the following controller

K = [-1.7329 + 0.0738].
Finally, in order to study the robustness with respect to aperiodic sampling, we apply Theorem 3.5. We find that the system is stable under sampled-data implementation of the feedback controller K with variable sampling periods bounded by h = 1.5 ms. The guaranteed domain of attraction E c * (P ) is given in (3.20), for c * = 37.81 × 10 3 and P = 10 3 554.9 -49.62 -49.62 14.01 .

The domain of attraction is shown in Fig. 3.5, together with simulations of the evolutions of the state of the sampled-data system. Different initial conditions are considered, and random variable sampling periods, bounded by h = 1.5 ms are used in the simulations.

Simulations show that by slightly increasing the sampling interval, the system becomes unstable. For example, with the initial condition x 0 = [-0.15 -1.7] T ∈ E c * (P ), we obtain an unstable behavior when choosing a constant sampling t k+1 -t k = 2.1 ms.

However, for the same initial condition the system state converges to the origin if the bound h = 1.5 ms is respected (as shown in Fig. 3.5).

Conclusion

In this chapter we have provided sufficient conditions for the local stability of bilinear sampled-data systems, when controlled via a linear state feedback. Polytopic state-space constraints have been included in the analysis. The local stability is guaranteed inside an ellipsoid contained in the addressed convex hull. The conditions for the stability analysis, as well as the estimate of the domain of attraction, were given in the form of LMIs, which makes them computationally tractable. The results have been illustrated by numerical examples, and compared to the exiting literature. Note that Lemma 3.8 treats a more general case of nonlinear systems. However, it only shows invarince property. In the next chapter, we intend to show how such a result can be extended in order to cover the asymptotic stability of a general class of nonlinear systems.

Chapter 4

Stability of input-affine nonlinear systems with sampled-data control

Introduction

This chapter is dedicated to the stability analysis of nonlinear sampled-data systems, which are affine in the input. Assuming that a stabilizing continuous-time controller exists and is to be implemented digitally, we intend to provide sufficient conditions for the sampled-data system to be asymptotically/exponentially stable. The main idea of the chapter is to extend the results from Chapter 3 using an approach inspired by the dissipativity theory.

In Chapter 3, local asymptotic stability of bilinear sampled-data systems controlled by a linear state feedback has been considered by using the analysis of contractive invariant sets and the dissipativity theory. The obtained results are constructive, but their extension for generic nonlinear systems does not seem to be trivial. Here we keep the objectives of Chapter 3, and enlarge them to the case of input-affine nonlinear systems. Dissipativity will constitute the keystone for the MASP estimation, and the robustness analysis with respect to the sampling jitters. The method will be applied to local and global analysis. Additionally, the particular case of polynomial systems will be studied in relation with SOS techniques. The result will be applied to a benchmark example from the literature in order to show the usefulness of the proposed stability conditions.

✛ ✲ ẋ = f (x) + g(x)u S H K(x(t k ))
x(t) u(t) The chapter is organized as follows. The problem under study is introduced in Section 4.2. In Section 4.3 the system is represented by an equivalent model which is adopted to our dissipativity analysis. Sufficient conditions for the asymptotic/exponential stability of affine nonlinear sampled-data systems are given in Section 4.4. Finally, illustrative examples are presented in Section 4.5.

Problem formulation

Consider the affine nonlinear control system given by ẋ(t) = f x(t) + g x(t) u(t), ∀t > t 0 , x(t 0 ) = x 0 , (

where x(t) ∈ R n and u(t) ∈ R m are the state and the input, respectively. The functions f : R n → R n with f (0) = 0, and g : R n → R n×m are sufficiently smooth to make the system well defined, i.e. for any x(t 0 ) and any admissible u(•), the existence and uniqueness of a solution is ensured on [t 0 , ∞). We suppose that a continuous-time controller u(t) = K x(t) stabilizes asymptotically/exponentially the equilibrium x = 0 of the system, where K : R n → R m is a continuously differentiable function.

We consider the emulation of the controller u = K(x) with the following assumptions:

• the set of uncertain sampling instants

{0 = t 0 < t 1 , . . . < t k < . . .} satisfies 0 < t k+1 -t k ≤ h, ∀k ∈ N,
for a given MASP h, and

lim k→∞ t k = ∞;
• the control input is then calculated based on the sampled-data version of the state: Under these assumptions, we obtain a closed-loop sampled-data system (see also Fig. 4.1):

u(t) = K x(t k ) , ∀t ∈ [t k , t k+1 ). (4.2)
✛ ✲ ẋ = fn(x) + gn(x)w y = ∂K ∂x ẋ ∆ sh y(t) w(t)
ẋ(t) = f x(t) + g x(t) K x(t k ) , ∀t ∈ [t k , t k+1 ), k ∈ N. (4.3)
We consider the following notions of stability:

Definition 4.1 ([64]
). The equilibrium point x = 0 of the system (4.3) is locally uniformly asymptotically stable, if there exists a class KL function β(•, •), such that

|x(t)| ≤ β(|x(t 0 )|, t -t 0 ), ∀t ≥ t 0 , ∀x(t 0 ) ∈ D 0 . (4.4)
In this case D 0 is an estimate of the domain of attraction of x = 0. The equilibrium point x = 0 is globally uniformly asymptotically stable if (4.4) is satisfied for any initial state x(t 0 ) ∈ R n (i.e. , D 0 = R n ).

Definition 4.2 ([64]

). The equilibrium point x = 0 of the system (4.3) is locally uniformly exponentially stable in a neighborhood D 0 of the equilibrium, if (4.4) is satisfied with

β(s, t) = cse -λt , c > 0, λ > 0.
In this case D 0 is an estimate of the domain of attraction of x = 0. The equilibrium point x = 0 is globally uniformly exponentially stable if this condition is satisfied for any initial state x(t 0 ) ∈ R n , (i.e. , D 0 = R n ).

Problem: Find a criterion for the local/global asymptotic/exponential stability of the equilibrium point x = 0 of the sampled-data system (4.3).

Robustness analysis representation

The system (4.3) can be written as

ẋ(t) = f n (x(t)) + g n (x(t))w(t), ∀t ∈ [t k , t k+1 ), k ∈ N, (4.5)
where f n (x) = f (x)+g(x)K(x), g n (x) = g(x) and w(t) = K x(t k ) -K x(t) . Note that f n (x) represents the dynamics of the nominal, continuous-time, closed-loop system, i.e.

the dynamics without the sampled-data implementation. From (4.5) the sampled-data system (4.3) can be represented by the equivalent feedback connection of

G :=    ẋ = f n (x) + g n (x)w, y = ∂K ∂x ẋ, (4.6)
with the operator ∆ sh : y → w

w(t) = (∆ sh y)(t) = - t t k y(τ )dτ, ∀t ∈ [t k , t k+1 ). (4.7)
This representation is shown in Fig. 4.2. Recall that the properties of the operator ∆ sh have been shown in Section 3.2.1.

Main results

Stability analysis

In the following we provide the main results of this chapter.

Theorem 4.3. Consider the sampled-data system (4.3) and the equivalent representation (4.6), (4.7). Consider the quadratic form:

S y, w = y w T -δ 2 0 X Y Y X y w , (4.8
)

with δ 0 = 2 π h, 0 < X T = X ∈ R m×m , and 0 ≤ Y T = Y ∈ R m×m .
Consider a neighborhood D ⊂ R n of the equilibrium point x = 0, and suppose that there exist a differentiable positive definite function V : D → R + , such that there exist α > 0 and class K functions β 1 and β 2 , verifying

β 1 (|x|) ≤ V (x) ≤ β 2 (|x|), ∀x ∈ D, (4.9)
and for and any x(t) ∈ D:

V x(t) + αV x(t) ≤ S y(t), w(t) , (4.10)
V x(t) + αV x(t) ≤ S y(t), w(t) e -αh . (4.11)

Then, the equilibrium x = 0 of the system (4.3) is locally uniformly asymptotically stable.

Moreover, consider the sub-level set defined by V (•) and a scalar c > 0

L c := {x ∈ R n : V (x) ≤ c}. (4.12)
Then the set L c * defined by the maximal sub-level set of V contained in D

c * = max Lc⊂D c (4.13)
is an estimate of the domain of attraction. Finally, if all the conditions are satisfied for D = R n , with class K ∞ functions β 1 and β 2 , then the equilibrium x = 0 is globally uniformly asymptotically stable.

Proof. To show the stability of the sampled-data system, we define first the function

W (t) = V x(t) e α(t-t k ) - t t k S y(τ ), w(τ ) dτ,
for any t ∈ [t k , t k+1 ). The conditions (4.10) and (4.11) are sufficient to have

Ẇ (t) ≤ 0, ∀t ∈ [t k , t k+1 ), ∀x(t) ∈ D. (4.14)
The last equation yields

V x(t) e α(t-t k ) - t t k S y(τ ), w(τ ) dτ ≤ V (t k ). (4.15)
From Lemma 3.1 and Lemma 3.2, it is easy to see that

V x(t) ≤ e -α(t-t k ) V x(t k ) , ∀t ∈ [t k , t k+1 ), ∀x(t) ∈ D. (4.16)
Clearly, the set L c * is positively invariant [START_REF] Khalil | Nonlinear Systems[END_REF], and it is the largest sub-level set contained in D. Consider an initial condition x 0 ∈ L c * . From the continuity of the solution x(t), (4.16) leads to V x(t) ≤ e -α(t-t 0 ) V x(t 0 ) , ∀t ≥ t 0 , ∀x 0 ∈ L c * . (4.17)

From (4.9) and (4.17), we see that for any solution with

x(t 0 ) ∈ L c * |x(t)| ≤ β -1 1 V x(t 0 ) e -α(t-t 0 ) ≤ β -1 1 β 2 (|x(t 0 )|)e -α(t-t 0 ) := β(|x(t 0 )|, t -t 0 ), ∀t ≥ t 0 , ∀x(t 0 ) ∈ L c * .
The function β(•, •) can be easily seen to be a class KL function. This shows that x = 0 is locally uniformly asymptotically stable. Finally, it is trivial to see that if all the conditions are satisfied for D = R n , with class K ∞ functions β 1 and β 2 , then x = 0 is globally uniformly asymptotically stable. This completes the proof.

Corollary 4.4. Suppose that all the conditions of Theorem 4.3 are satisfied with

β 1 (|x|) ≥ k 1 |x| q , β 2 (|x|) ≤ k 2 |x| q , for some k 1 , k 2 , q > 0. (4.18)
Then, the equilibrium x = 0 is locally exponentially stable. Moreover, the sub-level set L c * defined in (4.13) and (4.12), is an estimate of the domain of attraction. If the conditions hold for D = R n , then x = 0 is globally exponentially stable.

Proof. Following the same steps as in the proof of Theorem 4.3, we get V x(t) ≤ e -α(t-t 0 ) V x(t 0 ) , ∀t ≥ t 0 , ∀x 0 ∈ L c * .

Thus, from (4.9) and (4.18)

|x(t)| ≤ V x(t 0 ) e -α(t-t0) k 1 1/q ≤ k 2 |x(t 0 )| q e -α(t-t0) k 1 1/q = k 2 k 1 1/q |x(t 0 )|e -(α/q)(t-t0) , ∀t ≥ t 0 , ∀x(t 0 ) ∈ L c * .
This shows that x = 0 locally exponentially stable. If the conditions hold for D = R n , the proof of global exponential stability is trivial.

Remark 4.5. Considering the storage function V x(t) , the inequalities (4.10) and (4.11) show that (4.6) is exponentially dissipative with respect to the supply rates S y, w and e -αh S y, w respectively, with S defined in (4.8). See Section A.4 for the definitions of exponential dissipativity.

Sum of squares stability conditions for the class of polynomial systems

When the linear approximation fails, the dynamics of many physical phenomena can be modeled by polynomial differential equations. They are frequently found in several domains like process control, biology, robotics, and electrical systems. For this class of systems, SOS decomposition and semi-definite programming [START_REF] Parrilo | Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization[END_REF], are shown to be a useful tool. It has been used in several analysis and synthesis control problems [START_REF] Papachristodoulou | A tutorial on sum of squares techniques for systems analysis[END_REF].

In this section we specialize the previous result for the class of affine polynomial sampleddata systems, using SOS decomposition and semi-definite programming techniques. We formulate a constructive method to find a storage function and a supply rate, which satisfy the asymptotic/exponential stability conditions proposed in the previous section.

Let us consider the stability problem defined in Section 4.2 for the particular case where the f (x), g(x) and K(x) are polynomial functions. The system (4.6) will be defined by: Checking the non negativity of a polynomial is known to be a difficult problem. Recent methods relaxed this problem using semi-definite programming and the SOS decomposition [START_REF] Parrilo | Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization[END_REF]. The relaxation is based on checking whether a polynomial is a SOS, which is sufficient to ensure the semi-definite positivity.

   ẋ = F (x, w) y = G(x, w) (4.19)
Definition 4.6.

[100] A multivariate polynomial p(x) ∈ R[x] is said to be a sum of squares (SOS), if there exist some polynomials p i (x) ∈ R[x], i ∈ {1, . . . , M }, such that

p(x) = M i=1 p 2 i (x).
The relaxation is only sufficient, but there are suggestions in the literature which indicate that it is not too conservative (see [START_REF] Papachristodoulou | A tutorial on sum of squares techniques for systems analysis[END_REF] and the references therein). However, it must be noted that the computational complexity of the algorithms testing whether a polynomial p(x) is an SOS increases rapidly with the degree of p(x).

SOS techniques are shown to be very useful in systems analysis [START_REF] Papachristodoulou | A tutorial on sum of squares techniques for systems analysis[END_REF]. In the following, we reformalize Theorem 4.3 and Corollary 4.4 using the SOS method. The local applicability of the dissipativty inequalities inside a region D is ensured using a technique similar to the S-procedure [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF]. Note that when looking for a Lyapunov or a storage function, we need to ensure its positive definiteness. Thus, guaranteeing that it is an SOS is not sufficient, as it only guarantees its non negativity. To overcome this problem, we use the following proposition: with γ a positive number, and ǫ ij ≥ 0 for all i and j. Then the condition

V (x) -ϕ(x) is SOS, (4.21)
guarantees the positive definiteness of V (x).

with δ 0 = 2 π h, 0 < X T = X ∈ R m×m , 0 ≤ Y T = Y ∈ R m×m
, and ϕ(x) a positive definite polynomial defined in (4.20). Then, the equilibrium x = 0 of the system (4.3) is locally uniformly asymptotically stable. Moreover, the sub-level set L c * defined in (4.13) and (4.12), is an estimate of the domain of attraction. Finally, if (4.23) and (4.24) are SOS while µ l (x) = 0, for all l ∈ {1, 2, . . . , s}, then the equilibrium is globally uniformly asymptotically stable.

Proof. First, note that from (4.22) and Proposition 4.7, the function V (x) is ensured to be definite positive and radially unbounded (V (x) → ∞ when x → ∞). Therefore, using Lemma 4.3 from [START_REF] Khalil | Nonlinear Systems[END_REF], there exist class K functions β 1 and β 2 , such that

β 1 (|x|) ≤ V (x) ≤ β 2 (|x|), ∀x ∈ R n .
Moreover, when x ∈ D, i.e. µ l (x) ≥ 0 for all l ∈ {1, 2, . . . , s}, then from the non negativity of the SOS polynomials σ l (ξ) and ς l (ξ), we can see that ρ 1 (ξ) ≥ 0 ( resp. Corollary 4.9. Suppose that all the conditions of Corollary 4.8 are satisfied, and that the storage function V (x) satisfies

k 1 |x| q ≤ V (x) ≤ k 2 |x| q , ∀x ∈ R n . (4.25)
Then, the equilibrium x = 0 is locally exponentially stable. Moreover, the sub-level set L c * defined in (4.13) and (4.12), is an estimate of the domain of attraction. If the conditions hold for D = R n , then x = 0 is globally exponentially stable.

Proof. The proof follows the same steps as the one of Corollary 4.8. It is a direct result of Corollary 4.4.

Illustrative Examples

In the following, we apply the proposed method on two nonlinear systems. First we revisit the example in [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF]. We find the MASP which guarantees the global uniform asymptotic stability of the sampled-data system. Next, we consider another example that illustrates the applicability of the results for local exponential stability.

Example 1

Consider the following system from [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF] ẋ = dx 2 -x 3 + u, with a bounded time-varying |d| ≤ 1, and a stabilizing control u = K(x) = -2x.

Emulating this controller results in a sampled-data system that can be represented by the operator ∆ sh in (4.7), and a system (4.6) described by

   ẋ = dx 2 -x 3 -2x + w, y = -2(dx 2 -x 3 -2x + w).
We apply the Corollary 4.8 in order to find a storage function of the form V (x) = ax 2 + bx 4 , such that (4.22), (4.23) and (4.24) are SOS. We choose ϕ(x) = 10 -3 x 2 , α = 0.1 and h = 0.72. We intend to test the global stability of the closed-loop sampleddata system at the origin. In this case, the polynomials (4.23) and (4.24) take the form Previous works considered this example in the literature for estimating the MASP. In [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF], a bound of h = 0.368 is found. In [START_REF] Karafyllis | Global stability results for systems under sampleddata control[END_REF], the proposed upper bound is h = 0.1428.

ρ 1 (ξ) = -(2ax + 4bx 3 )(dx 2 -x 3 -2x + w) -α(ax 2 + ax 4 ) + -4δ 2 0 X(dx 2 -x 3 -2x + w) 2 -4Y (dx 2 -x 3 -2x + w)w + Xw 2 , (4.26) ρ 2 (ξ) = -(2ax + 4bx 3 )(dx 2 -x 3 -2x + w) -α(ax 2 + ax 4 ) + -4δ 2 0 X(dx 2 -x 3 -2x + w) 2 -4Y (dx 2 -x 3 -2x + w)w + Xw 2 e -αh , (4.27
The conditions proposed in this paper are found feasible for h = 0.72. State trajectory evolutions are shown in Fig 4 .4. It can be seen that the state trajectory is asymptotically stable when the sampling periods are inferior to the bound h = 0.72. Also, note that for a uniform sampling period of t k+1 -t k = 1.05, asymptotic stability is no longer guaranteed.

Example 2

Consider the following system ẋ = x 2 + (x -1)u, with the controller u = K(x) = x + 2x 2 , which stabilizes the system at the equilibrium point x = 0. Note that, in the continuous-time case, this equilibrium is only locally stable. Our purpose is to find the maximum value of h that guarantees the local exponential stability of x = 0, when the controller is emulated. We consider the neighborhood

x ∈ [-0.4, +0.4]. The sampled-data system can be represented by the operator ∆ sh in (4.7), and a system (4.6) described by

   ẋ = -x + 2x 3 + (x -1)w, y = (1 + 4x)(-x + 2x 3 + (x -1)w).
We consider applying Corollary 4.9 with a quadratic storage function V (x) = ax 2 . Note that V (x) satisfies (4.25) with k 1 = k 2 = a and q = 2. We choose ϕ(x) = 10 -3 x 2 , α = 0.25 and h = 0.6. The considered domain D is described by {x ∈ R : µ 1 (x) ≥ 0} with µ 1 (x) = (x + 0.4)(0.4 -x). The polynomials (4.23) and (4.24) are in this case

ρ 1 (ξ) = -σ 1 (ξ)µ 1 (x) -(2ax)(-x + 2x 3 + (x -1)w) -α(ax 2 ) + -δ 2 0 X(1 + 4x) 2 (-x + 2x 3 + (x -1)w) 2 + 2Y (1 + 4x)(-x + 2x 3 + (x -1)w)w + Xw 2 , (4.28) ρ 2 (ξ) = -ς 1 (ξ)µ 1 (x) -(2ax)(-x + 2x 3 + (x -1)w) -α(ax 2 ) + -δ 2 0 X(1 + 4x) 2 (-x + 2x 3 + (x -1)w) 2 + 2Y (1 + 4x)(-x + 2x 3 + (x -1)w)w + Xw 2 e -αh , (4.29

Conclusion

In this chapter we have provided sufficient conditions for the stability of nonlinear sampled-data systems, which are affine in the control. The main idea of the chapter is to use the dissipativity theory to provide an estimate of the MASP. The provided results can be used to analyze asymptotic/exponential stability, and can be applied locally or globally. The results are numerically tractable for the case of polynomial systems, with the use of SOS decomposition and semi-definite programming. The method is applied to a benchmark example from the literature, and it has been shown that it can provide a good estimate of the MASP. The novelty of this contribution is that it provides a quantitative estimate of the MASP using robust control tools based on the dissipativity theory.

General conclusion

This thesis has provided contributions to the stability analysis of nonlinear systems under aperiodic sampling. A continuous-time controller is supposed to be designed without taking the sampling into consideration, and it is emulated in discrete-time. The main objective was to provide tractable stability criteria which allow for estimating the Maximum Allowable Sampling Period (MASP)1 .

A particular attention has been given to the case of bilinear systems, which are a special class of nonlinear systems. They represent a challenging intermediate between linear and nonlinear systems, which is relevant in practical applications. The study of such systems allows for tackling the difficulties of nonlinear systems while exploiting their quasi-linear structure. New theoretical methods have been proposed for this class of systems. Afterwards, the results have been extended to more general classes of nonlinear systems. We describe, in what follows, the contributions of the thesis with a little more detail.

In Chapter 1, we proposed an overview of the techniques involved in sampled-data control, ranging from Lyapunov-Krasovskii functionals, impulsive modeling, small gain and convex-embedding approaches for LTI systems, to different emulation and discretetime approaches for nonlinear systems.

In Chapter 2, we have provided sufficient conditions for the local stability of bilinear sampled-data systems, controlled via a linear state feedback controller. New results for estimating the MASP that guarantees the local stability of the system are given. Two methods were considered via the hybrid system modeling approach. The first method [START_REF] Omran | Stability of bilinear sampled-data systems with an emulation of static state feedback[END_REF] is a constructive adaptation of a generic result for nonlinear case [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF], while the second one is based on a direct search of a Lyapunov function for the hybrid model [START_REF] Omran | Stability of bilinear sampled-data systems with an emulation of static state feedback[END_REF]. The stability conditions of both methods are given in the form of Linear Matrix Inequalities (LMIs), which are easily tractable in terms of computation.

In Chapter 3, the local stability of bilinear sampled-data systems has been investigated

using a new approach inspired by dissipativity [START_REF] Omran | Local stability of bilinear systems with asynchronous sampling[END_REF][START_REF] Omran | Stability analysis of bilinear systems under aperiodic sampled-data control[END_REF]. Sufficient conditions have been provided based on the analysis of contractive invariant sets. Polytopic state-space constraints have been included in the analysis. The local stability is guaranteed inside an ellipsoidal estimate of the domain of attraction. The stability analysis criteria, as well as the conditions for estimating the domain of attraction, are given in the form of LMIs.

In Chapter 4, we have provided sufficient conditions for the stability of nonlinear sampleddata systems, affine in the control. The main idea of this contribution is to extend the dissipativity-based results developed for bilinear systems to a more general nonlinear case [START_REF] Omran | Stabilité des systèmes non linéaires sous échantillonnage apériodique[END_REF][START_REF] Omran | On the stability of input-affine nonlinear systems with sampled-data control[END_REF][START_REF] Omran | Stabilité des systèmes non linéaires sous échantillonnage apériodique[END_REF]. The method provides a quantitative estimate of the MASP and can be used to analyze asymptotic/exponential stability. It is shown that the results are numerically tractable for the case of polynomial systems. In this case, the tractability refers to the use of SOS decomposition and semi-definite programming.

We believe that the results of this thesis reveal several perspectives, and emerging research directions can now be considered as follows.

First, the provided results contribute to stability analysis of Networked Control Systems (NCSs), as for such systems robustness with respect to aperiodic sampling is an essential issue. However, networks impose other communication imperfections that must also be taken into account: time-varying delays, constraints on the number of nodes accessing the network, and quantization. Extending our methodology in order to include these additional network imperfections would be of great interest.

Second, although the results we provided are shown to have rather low levels of conservatism, it is still possible to improve the numerical solvability of the proposed conditions.

These conditions can be enhanced by giving more insight into the mathematical model of the sampling effects. This would lead to new characterizations of supply functions used in the dissipativity-based approach. Moreover, information about the lower bound of the sampling interval could be useful in the analysis. Analyzing stability while taking into consideration both the upper and the lower bound on the sampling intervals could enhance the results.

Third, the present work addresses stability analysis for sampled-data systems with an emulated controller. It means we considered that a controller has been designed in continuous-time without taking the sampling into account. In the future, we may try to build on our progress in order to design (possibly more complex) sampled-data controllers. This constitutes a challenging issue.

Finally, the thesis was focused on the robust stability with respect to aperiodic sampling.

From this point of view, the variations of the sampling intervals are seen as perturbations.

Formulation du problème

On considère le système bilinéaire:

ẋ(t) = A 0 x(t) + m i=1 [u(t)] i N i x(t) + B 0 u(t), ∀t ≥ t 0 . (1)
On suppose que les hypothèses suivantes sont satisfaites:

A1 La commande est constante par morceaux

u(t) = Kx(t k ), ∀t ∈ [t k , t k+1 ),
avec un ensemble des instants d'échantillonnage {t k } k∈N qui satisfait:

0 < ǫ ≤ t k+1 -t k ≤ h, ∀k ∈ N, (2)
où h est un plus grand pas d'échantillonnage permis.

Le système bilinéaire échantillonné peut être représenté par:

ẋ = A 0 x(t) + m i=1 u i (t)N i x(t) + B 0 u(t), t ∈ [t k , t k+1 ), y = x, u = K ŷ, ẏ = 0, t ∈ [t k , t k+1 ), ŷ(t + k ) = y(t k ). (10)
Le système [START_REF] Bennett | Control and the digital computer: the early years[END_REF] peut être représenté par le modèle hybride suivant:

ẋ = f (x, e) = Ã[x, e]x + Be ė = g(x, e) = -Ã[x, e]x -Be τ = 1        τ ∈ [0, h) x + = x e + = 0 τ + = 0        τ ∈ [ǫ, h] (11)
Méthode 1:

Le théorème suivant est une adaptation du résultat de [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF] pour le cas bilinéaire.

Théorème .0.1. On considère le système [START_REF] Blanchini | Set invariance in control[END_REF], le polytope P dans (3), la notation (9) et une fonction T donnée par

T (γ, L) :=        1 Lr arctan(r) γ > L 1 L γ = L 1 Lr arctanh(r) γ < L (12) avec r = γ 2 L 2 -1 (13)
où L est donné par

L = 1 2 max{-λ min (B T + B), 0} (14)
et γ est la solution du problème d'optimisation suivant

γ = min γ ′ (15)
sous les contraintes ∃P ∈ R n×n , une matrice symétrique définie positive , ∃γ ′ > 0 et ∃α > 0, telle que

M lj = A T l P + P A l + 1 2 (A T l A j + A T j A l ) + αI P B * (α -γ ′2 )I < 0, ∀l, j ∈ {1, 2, ..., p}, (16)
où A l and A j sont des sommets donnés par [START_REF] Beikzadeh | Dissipativity of nonlinear multirate sampleddata systems under emulation design[END_REF]. On suppose que h < T (γ, L). Alors, pour le système [START_REF] Blanchini | Set invariance in control[END_REF], l'ensemble {(x, e, τ ) : x = 0, e = 0} est localement uniformément asymptotiquement stable.

Méthode 2:

Dans cette méthode, on cherche directement une fonction de Lyapunov pour le modèle hybride. L'objectif est d'éviter le conservatisme présent dans la méthode précédente, dû aux bornes supérieures sur la dérivée de la fonction de Lyapunov.

Théorème .0.2. On considère le système [START_REF] Blanchini | Set invariance in control[END_REF]. On suppose que h ≤ T . On suppose qu'il existe des matrices symétriques définies positives P, Q, X, Y ∈ R n×n telles que les LMIs suivantes sont satisfaites:

A T l P + P A l + X P B -A T l Q * -B T Q -QB -1 T Q + Y < 0, ∀l ∈ {1, 2, ..., p}, (17)
A T l P + P A l + X P B -A T l Q exp(-1) * [-B T Q -QB -1 T Q] exp(-1) + Y < 0, ∀l ∈ {1, 2, ..., p}, (18)
où A l des sommets donnés par [START_REF] Beikzadeh | Dissipativity of nonlinear multirate sampleddata systems under emulation design[END_REF]. Alors, pour le système [START_REF] Blanchini | Set invariance in control[END_REF], l'ensemble {(x, e, τ ) :

x = 0, e = 0} est localement uniformément asymptotiquement stable.

Chapitre 3

Ce chapitre est dédié à l'analyse de la stabilité locale des systèmes bilinéaires

(t) = A 0 + B 0 K + m i=1 [Kx(t k )] i N i A(x(t k )) x(t) + B 0 K B (x(t k ) -x(t) w(t)
).

On définit

C(x(t k )) = A(x(t k )) = A 0 + B 0 K + m i=1 [Kx(t k )] i N i , D = B = B 0 K, (19)
ce qui montre que le système échantillonné peut être representé par le bouclage du système

G :=    ẋ(t) = A(x(t k ))x(t) + Bw(t), y(t) = C(x(t k ))x(t) + Dw(t), (20)
avec l'opérateur ∆ sh : y → w,

w(t) = (∆ sh y)(t) = - t t k y(τ )dτ, ∀t ∈ [t k , t k+1 ). ( 21
)
On remarque que l'effet des variations des pas d'échantillonnage est modélisé par l'opérateur ∆ sh . Cette approche est considérée dans [START_REF] Mirkin | Some remarks on the use of time-varying delay to model sample-andhold circuits[END_REF] et [START_REF] Fujioka | Stability analysis of systems with aperiodic sample-and-hold devices[END_REF] avec l'objectif d'étudier la stabilité des systèmes échantillonnés linéaires. Dans [START_REF] Mirkin | Some remarks on the use of time-varying delay to model sample-andhold circuits[END_REF], une limite supérieure sur le gain de l'opérateur ∆ sh est trouvée. Il est montré que ∆ sh ≤ δ 0 avec δ 0 = 2 π h max . Cette limite est atteinte ( ∆ sh = δ 0 ) pour t k+1 -t k = h max . Des conditions de stabilité basées sur le théorème du petit gain sont fournies sous la forme de LMI. Dans [START_REF] Fujioka | Stability analysis of systems with aperiodic sample-and-hold devices[END_REF], la propriété précédente est associée à une propriété de passivité pour trouver des conditions moins contraignantes. Le résultat est basé sur des techniques de commande robuste, utilisant une approche fréquentielle et le lemme de Kalman-Yakubovich-Popov. Théorème .0.3. On considère le système [START_REF] Blanchini | Set invariance in control[END_REF], la représentation équivalente [START_REF] Chellaboina | New sufficient conditions for stability analysis of time delay systems using dissipativity theory[END_REF] et [START_REF] Chellaboina | A dissipative dynamical systems approach to stability analysis of time delay systems[END_REF].

On suppose qu'il existe des matrices symétriques définies positives X, Y, P ∈ R n×n , et des matrices P 2 , P 3 ∈ R n×n telles que le problème d'optimisation suivant admet une solution γ * = min

E j ≥0, Mq<0
γ, ∀j ∈ {1, 2, ..., r}, ∀q ∈ {1, 2, ..., p}

avec

E j = γ a T j a j P ≥ 0 (23)
et

M q =    
A T q P 2 + P T 2 A q P -P T 2 + A T q P 3 P T 2 B P -P 2 + P T 3 A q -P 3 -P T 3 + δ 2 0 X P T 3 B -Y B T P 2

B T P 3 -Y -X     < 0 ( 24
)
avec δ 0 = 2 π h max , et les sommets {A q } q∈{1,2,••• ,p} sont donnés par

A q := A(x q ) = A 0 + B 0 K + m i=1 Kx q i N i (25)
avec {x q } q∈{1,2,...,p} donné dans (3). Alors, l'équilibre x = 0 du système [START_REF] Blanchini | Set invariance in control[END_REF]

Formulation du problème

On considère le système non linéaire ẋ(t) = f x(t) + g x(t) u(t), ∀t > t 0 , x(t 0 ) = x 0 ,

où x(t) ∈ R n et u(t) ∈ R m sont respectivement l'état et l'entrée. Les fonctions f : R n → R n avec f (0) = 0, et g : R n×m → R n sont suffisamment lisses pour qu'à chaque x(t 0 ) et u(•) admissible corresponde une seule solution sur [t 0 , ∞). On suppose qu'il existe une commande u = K(x) qui stabilise l'équilibre en temps continu, où K : R n → R m est une fonction continûment différentiable. On considère l'émulation de la commande u = K(x) en supposant que:

• les instants d'échantillonnage {0 = t 0 < t 1 , . . . < t k < . . .} satisfont 0 < t k+1 -t k ≤ h, ∀k ∈ N, pour une borne supérieure finie h,

lim k→∞ t k = ∞;
• le contrôle est un retour d'état constant par morceaux:

u(t) = K x(t k ) , ∀t ∈ [t k , t k+1 ). ( 28
)
On obtient alors le système en boucle fermée:

ẋ(t) = f x(t) + g x(t) K x(t k ) , ∀t ∈ [t k , t k+1 ), k ∈ N. (29)
Problème : Notre objectif est de trouver un critère de stabilité asymptotique et exponentielle locale/globale de l'équilibre x = 0 du système non linéaire échantillonné [START_REF] Ebenbauer | Dissipation inequalities in systems theory: An introduction and recent results[END_REF].

Une représentation équivalente

On note que le système (29) s'écrit aussi : ẋ(t) = f n (x(t)) + g n (x(t))w(t), ∀t ∈ [t k , t k+1 ), k ∈ N,

où f n (x) = f (x) + g(x)K(x), g n (x) = g(x) et w(t) = K x(t k ) -K x(t) . On note que f n (x) représente la dynamique de boucle fermée en temps continu. L'équation [START_REF] Ebenbauer | Dissipation inequalities in systems theory: An introduction and recent results[END_REF] montre que le système échantillonné peut être représenté par le bouclage du système:

G :=    ẋ = f n (x) + g n (x)w, y = ∂K ∂x ẋ, (31)
avec l'opérateur ∆ sh : y → w w(t) = (∆ sh y)(t) = -

t t k y(τ )dτ, ∀t ∈ [t k , t k+1 ). ( 32
)
Les propriétés de l'opérateur ∆ sh sont présentées dans la Section 3.2.1. Nous considérons ici l'exploitation de ces propriétés afin de développer un critère de stabilité pour le contrôle échantillonné des systèmes non linéaires. L'approche s'inspire de la notion de dissipativité exponentielle.

Analyse de stabilité

On considère les définitions de stabilité suivantes.

Il est globalement uniformément asymptotiquement stable si (33) est satisfaite pour D 0 = R n .

Théorème .0.4. Soient le système non linéaire échantillonné (29) et sa présentation équivalente [START_REF] Espana | Reduced order bilinear models for distillation columns[END_REF], [START_REF] Fiter | A state dependent sampling for linear state feedback[END_REF]. On considère la forme quadratique:

S y(t), w(t) = y(t)

w(t) T -δ 2 0 X Y Y X y(t) w(t) , (34)
avec δ 0 = 2 π h, 0 < X T = X ∈ R m×m et 0 ≤ Y T = Y ∈ R m×m . Considérons un voisinage D ⊂ R n de l'équilibre x = 0 et une fonction différentiable définie positive V : D → R + , pour laquelle il existe α > 0 et des fonctions β 1 et β 2 de classe K, telles que:

β 1 (|x|) ≤ V (x) ≤ β 2 (|x|), ∀x ∈ D, (35)
et pour tout x(t) ∈ D, V satisfait:

V x(t) + αV x(t) ≤ S y(t), w(t) ,

V x(t) + αV x(t) ≤ S y(t), w(t) e -αh .

Alors, l'équilibre x = 0 du système (29) est localement uniformément asymptotiquement stable. De plus, considérons les ensembles définis par V (•) et un scalaire c > 0:

L c := {x ∈ R n : V (x) ≤ c}. (38)
Alors, l'ensemble L c * défini par la surface de niveau maximal de V contenue dans D:

c * = max Lc⊂D c (39)
est une estimation du domaine d'attraction de x = 0. Enfin, si toutes les conditions sont satisfaites pour D = R n , avec des fonctions β 1 et β 2 de classe K ∞ , alors x = 0 est globalement uniformément asymptotiquement stable.

Conclusion

Cette thèse a contribué à l'analyse de stabilité des systèmes non linéaires sous échantillonnage apériodique. Passive dynamical systems have several appealing properties which are used in optimal control, design, large-scale networks and others. See [START_REF] Byrnes | Passivity, feedback equivalence, and the global stabilization of minimum phase nonlinear systems[END_REF][START_REF] Nuno | Passivity-based control for bilateral teleoperation: A tutorial[END_REF][START_REF] Ortega | Passivity-based control of nonlinear systems: a tutorial[END_REF][START_REF] Ortega | Interconnection and damping assignment passivity-based control of port-controlled hamiltonian systems[END_REF][START_REF] Sepulchre | Constructive Nonlinear Control[END_REF][START_REF] Van Der Schaft | L 2 -Gain and Passivity Techniques in Nonlinear Control[END_REF] for more information about passivity and its applications.

L 2 -Gain:

Gain properties describe how a system attenuates or amplifies a class of input signals.

They are given by the quotient between some measures of output and input signals.

In control systems theory, Lebesgue integrable functions are often considered, and the The following theorem illustrates how an estimate of the L 2 -gain of a system can be obtained using dissipation inequalities. shown that it is also a necessary one. L 2 -gains properties have several applications in control theory. For example, they can be used to show the stability of interconnected systems using small gain theorem [START_REF] Khalil | Nonlinear Systems[END_REF]. Furthermore, they play an important role in H 2 theory and H ∞ theory [START_REF] Doyle | State-space solutions to standard h 2 and h ∞ control problems[END_REF][START_REF] Scherer | Linear matrix inequalities in control[END_REF].

It must be noted that there exist several properties which can be studied in the framework of dissipativity. These properties include stability, ISS and minimum phase behavior. See [START_REF] Ebenbauer | Dissipation inequalities in systems theory: An introduction and recent results[END_REF] for more information and references about this issue.

A.3 Kalman-Yakubovich-Popov Lemma

The Kalman-Yakubovich-Popov (KYP) Lemma was motivated by the absolute stability

Lur'e problem, and it has a very wide range of applications in control and systems theory including dissipativity, stability, absolute stability, optimal control, adaptive control and others [START_REF] Brogliato | Dissipative Systems Analysis and Control[END_REF]. The lemma originates from a stability criterion of nonlinear feedback systems given by Popov. Then, Yakubovich and Kalman introduced the celebrated lemma, which

shows that the frequency condition of Popov is equivalent to the existence of a Lyapunov function of certain simple form. See [START_REF] Rantzer | On the Kalman-Yakubovich-Popov lemma[END_REF] and references therein for more details.

The KYP lemma provide the following interesting result for LTI systems. Suppose that Σ L is controllable, and let S be the supply rate (A.10). Then, the following statements are equivalent.

1. There exists P T = P ∈ R n×n such that Theorem A.6 shows the equivalence between a frequency domain condition (A.9) and an LMI condition (A.8). Note that the condition (A.9) needs to be tested at an infinite number of points. However, using Theorem A.6 it is possible to verify the equivalent condition (A.8), which can be easily tested. This has many applications in control theory, such as in the IQCs stability theorem [START_REF] Megretski | System analysis via integral quadratic constraints[END_REF].

The following theorem, which is known as the nonlinear KYP Lemma, provides necessary and sufficient conditions for the system Σ to be dissipative with respect to Definition A.2.

Theorem A. 7 ([50]). Suppose that the Σ (A.1) is reachable from the origin. More precisely given any x 1 and t 1 , there exists t 0 ≤ t 1 and an admissible control u(•) such that the state can be driven from x(t 0 ) = 0 to x(t 1 ) = x with Q = Q T , R = R T . Then, the nonlinear system Σ is dissipative in the sense of Definition A.2 with respect to the supply rate (A.10) if and only if there exist functions V : R n → R, L : R n → R q , W : R n → R q×m (for some integer q), with V (•) differentiable, such that:

V (x) ≥ 0, V (0)

A.4 Exponential dissipativity

With the objective of generalizing the Strict Positive Real Lemma and the Strict Bounded Real Lemma to nonlinear systems, the notion of exponential dissipativity has been introduced in [START_REF] Chellaboina | Exponentially dissipative nonlinear dynamical systems a nonlinear extension of strict positive realness[END_REF].

Definition A.8 (Exponential Dissipativity [START_REF] Chellaboina | Exponentially dissipative nonlinear dynamical systems a nonlinear extension of strict positive realness[END_REF]). The system Σ (A.1) is exponentially dissipative with respect to the supply rate S(y, w), if there exists a continuous exponential storage function V : R n → R + and a constant α ≥ 0 satisfying: Note that Definition A.8 and Definition A.1 coincide when α = 0. When the storage function V is smooth, then the integral inequality (A.11) can be written as:

e αt 2 V
V (x(t)) + αV (x(t)) ≤ S(y(t), w(t)), t ≥ t 0 .

This notion has several interesting applications. In [START_REF] Chellaboina | Exponentially dissipative nonlinear dynamical systems a nonlinear extension of strict positive realness[END_REF], exponential dissipativity has been used to provide a nonlinear analog to the classical real positivity and small gain theorems for linear feedback systems. Moreover, it has been used to provide sufficient conditions for asymptotic stability of a time delay system [START_REF] Chellaboina | A dissipative dynamical systems approach to stability analysis of time delay systems[END_REF][START_REF] Chellaboina | New sufficient conditions for stability analysis of time delay systems using dissipativity theory[END_REF]. We also use it in this work (see Theorem 4.3).

Figure 1 :

 1 Figure 1: Sampled-data control system.

  , L. Hetel, J. P. Richard, and F. Lamnabhi-Lagarrigue. "Stabilité des systèmes non linéaires sous échantillonnage apériodique". Journal Européen des Systèmes Automatisés. Accepted. Conferences • [92] H. Omran, L. Hetel, and J.-P. Richard. "Local stability of bilinear systems with asynchronous sampling". In The 4th IFAC Conference on Analysis and Design of Hybrid Systems(ADHS), pages 19-24, Eindhoven, Netherlands, 2012. • [95] H. Omran, L. Hetel, J.-P. Richard, and F. Lamnabhi-Lagarrigue. "Stability of bilinear sampled-data systems with an emulation of static state feedback". In IEEE 51st Annual Conference on Decision and Control (CDC), pages 7541-7546, Maui, USA, 2012.

Figure 1 . 1 :

 11 Figure 1.1: Diagram of a sampled-data control system.

Figure 1 . 2 :

 12 Figure 1.2: Aperiodic sampling as a result of packet dropouts.

Figure 1 . 4 :

 14 Figure 1.4: Instability of the sampled-data system in Example 1.1 with alternating sampling intervals T 1 → T 2 → T 1 • • • .

Figure 1 . 5 :

 15 Figure 1.5: The piecewise-linear delay induced by sampling.

  and V (t, φ) ≤ γ( φ c ), where | • | denotes a norm over R n , and φ c = max a≤ξ≤b |φ(ξ)| is the associated continuous norm of φ ∈ C([a, b], R n ), then the origin of the system (1.14) is stable. If γ(s) > 0 for s > 0, then it is Uniformly Asymptotically Stable (UAS). If, in addition, lim s→∞ α(s) = ∞, then it is Globally Uniformly Asymptotically Stable (GUAS).

Figure 1 . 6 :

 16 Figure 1.6: Equivalent representation of the sampled-data system, from a robust control theory point of view.

Figure 1 . 7 :

 17 Figure 1.7: Embedding of the uncertainty set W in W.

Figure 2 . 1 :

 21 Figure 2.1: Bilinear system state diagram.

43 )

 43 For any τ ∈ [0, T ], we have that exp(-τ /T ) ∈ [exp(-1), 1]. Finally from (2.42), (2.43) and (2.40) there exists then θ(τ ) ∈ [0, 1] such that ∇U ′ (ξ), F (ξ) + x T Xx + e T Y e = x e T θ(τ

  as a discrete-time Lyapunov function. An LMI optimization allows for choosing, among quadratic Lyapunov functions, the one which maximizes the MASP. The results are illustrated by means of numerical examples.

3 Figure 3 . 1 :

 331 Figure 3.1: The polytope (blue boxes) and the corresponding region of stability E c * (P ).

Figure 3 . 2 :

 32 Figure 3.2: State evolution for the bilinear sampled-data system in Example 1, with a variable sampling which is bounded by h = 0.051.

( 3 .

 3 18) are feasible for h = 0.051, with P The domain of attraction E c * (P ) given in (3.20) for c * = 10.84 × 10 3 (see Fig.3.1).

1 .Figure 3 . 3 :

 133 Figure 3.3: State evolution for the bilinear sampled-data system in Example 1, with a constant sampling intervals t k+1 -t k = 0.09.

Theorem 2 . 5 h 5 . 4 ×Figure 3 . 4 :

 255434 Figure 3.4: Buck-boost converter.

Figure 3 . 5 :

 35 Figure 3.5: The domain of attraction E c * (P ) for the system (3.29) when controlled with the static feedback controller, in the aperiodically sampled-data case with h max = 1.5 ms. The curves in black are simulations of the sampled-data system, for different initial states.

Figure 4 . 1 :

 41 Figure 4.1: The sampled-data feedback control of an affine nonlinear system.

Figure 4 . 2 :

 42 Figure 4.2: The equivalent representation of the sampled-data system (4.3).

whereF

  (x, w) := f n (x) + g n (x)w, and G(x, w) := ∂K ∂x F (x, w).When looking for a polynomial storage function V (x), verifying the dissipativiy inequalities in Theorem 4.3 is a problem of checking the non negativity of polynomials. This can be seen from (4.8) and (4.19), as for the polynomial case (4.10) and (4.11) are, respectively, equivalent to 0≤ -∂V ∂x F (x, w) -αV (x) + -δ 2 0 G T (x, w)XG(x, w) + 2G T (x, w)Y w + w T Xw ,and0 ≤ -∂V ∂x F (x, w) -αV (x) + -δ 2 0 G T (x, w)XG(x, w) + 2G T (x, w)Y w + w T Xw e -αh ,for any x ∈ D. In fact, the right terms in the last inequalities can be written as polynomials of the form p(ξ) ≥ 0, with p(ξ) ∈ R[ξ], and ξ = (x, w).

Proposition 4 . 7 .ǫ

 47 [START_REF] Papachristodoulou | A tutorial on sum of squares techniques for systems analysis[END_REF] Given a polynomial V (x) ∈ R[x] of degree 2d, let ij > γ, ∀i = 1, . . . , n (4.20)

Corollary 4 . 8 .

 48 Assume that the functions f (x), g(x) and K(x) in the sampled-data system (4.3) are polynomial functions. Consider the equivalent representation (4.[START_REF] Carnevale | A Lyapunov proof of an improved maximum allowable transfer interval for networked control systems[END_REF] and (4.7). Let D = {x ∈ R n : µ l (x) ≥ 0, l = 1, 2, . . . , s} be a neighborhood of the origin x = 0. Suppose that there exist a polynomial function V (x) ∈ R[x], and sums of squares σ l (ξ) and ς l (ξ), with l ∈ {1, . . . , s} and ξ = (x, w), such that the following polynomials are SOSV (x) = V (x) -ϕ(x), (4.22) ρ 1 (ξ) = -s l=1 σ l (ξ)µ l (x) -∂V ∂x F (x, w) -αV (x) + -δ 2 0 G T (x, w)XG(x, w) + 2G T (x, w)Y w + w T Xw ,(4.23)ρ 2 (ξ) = -s l=1 ς l (ξ)µ l (x) -∂V ∂x F (x, w) -αV (x)+ -δ 2 0 G T (x, w)XG(x, w) + 2G T (x, w)Y w + w T Xw e -αh . (4.24)

ρ 2

 2 (ξ) ≥ 0). The later implies that the dissipativity condition (4.10) ( resp. (4.11)) is satisfied. Thus all the local stability conditions of Theorem 4.3 are satisfied. The case where (4.23) and (4.24) are SOS for µ l (x) = 0 ∀l ∈ {1, 2, . . . , s} satisfies obviously the global stability conditions in Theorem 4.3.

Figure 4 . 3 :

 43 Figure 4.3: Tradeoff between α (the exponential decay rate of the storage function), and the estimation of the MASP h.

Figure 4 . 4 :

 44 Figure 4.4: State trajectory evolution for two sequences of sampling intervals.

Figure 5 : 1 2

 512 Figure 5: Commande du système avec retour d'état échantillonné.

4

 4 est localement asymptotiquement stable et le domaine d'attraction est estimé par E c * [P ] = {x ∈ R n : x T P x ≤ c * } ⊂ P, with c * = 1/γ * . (26) Chapitre Dans ce chapitre on généralise les résultats du Chapitre 3 pour le cas des systèmes non linéaires affines en l'entrée. Nous supposons qu'il existe une commande stabilisante en temps continu. Lors de l'implémentation numérique de cette commande, il s'agit de trouver des conditions préservant la stabilité asymptotique/exponentielle sous échantillonnage. Les conditions sont formulées à la fois pour la stabilité globale et la stabilité locale. L'idée principale est d'aborder le problème dans le cadre de la dissipativité exponentielle. Le résultat est ensuite repris dans le cas spécifique des systèmes non linéaires polynomiaux, où les conditions de stabilité sont vérifiées numériquement en utilisant la décomposition en somme des carrés (SOS) et la programmation semi-définie.

Définition .0. 1 .

 1 Le point d'équilibre x = 0 de (29) est localement uniformément asymptotiquement stable dans un voisinage D 0 de l'équilibre, s'il existe une fonction β(•, •) de classe KL, telle que |x(t)| ≤ β(|x(t 0 )|, t -t 0 ), ∀t ≥ t 0 , ∀x(t 0 ) ∈ L.

  voltage , and i(t) is the corresponding drawn current. The inequality (A.4) captures the fact that the energy stored in the circuit at instant t 2 , cannot exceed the sum of what was already stored in the circuit at time t 1 , and the accumulated power over the interval [t 1 , t 2 ]. Definition A.4 corresponds to Definition A.2 with E(•) as a storage function, and the product of voltage and current as a supply function. This has motivated the general definition of passive dynamical systems: Definition A.3 (Passivity). System Σ (A.1) with p = m is said to be passive if it satisfies the Definition A.1 with the supply function S(y, w) = w T y.

L 2 -

 2 gain is defined based on the L 2 -norm: Definition A.4. Consider the system Σ (A.1) with j(•) = 0, that is:   ẋ = f (x) + g(x)w, y = h(x), (A.5)and with x(0) = 0. The system (A.5) has an L 2 -gain less or equal to γ ifsup 0< w L 2 <∞ y L 2 w L 2 ≤ γ.

Theorem A. 5 (

 5 [START_REF] Van Der Schaft | L 2 -Gain and Passivity Techniques in Nonlinear Control[END_REF]). The system (A.5) has an L 2 -gain less or equal to γ if there exists a positive definite and proper storage function V , such that the system is dissipative with respect to the supply rateS(y, w) = γ 2 |w| 2 -|y| 2 .(A.6) Theorem A.5 provides a sufficient condition. Moreover, for linear control systems, it is

Theorem A. 6 (

 6 [START_REF] Scherer | Linear matrix inequalities in control[END_REF]). Consider the following system Σ L defined byΣ L :=    ẋ = Ax + Bw, y = Cx + Dw, (A.7)

2 .

 2 For all ω ∈ R with det(jωI -A) = 0, the transfer function Ĝ(s) := C(sI -A) -1 B +

  Qy + 2y T Sw + w T Rw, (A.[START_REF] Bennett | Control and the digital computer: the early years[END_REF]

  = 0, ∇V T (x)f (x) = h T (x)Qh(x) -L T (x)L(x), 1 2 g T (x)∇V (x) = Ŝ(x)h(x) -W T (x)L(x), R(x) = W T (x)W (x),whereŜ(x) = Qj(x) + S, R(x)= R + j T (x)S + S T j(x) + j T (x)Qj(x).

Table 1 . 1 :

 11 The steps of the discrete-time and the emulation approaches.

	Emulation	Discrete-time
	continuous-time plant model continuous-time plant model
	↓ continuous-time controller	↓ discretize plant model
	↓ discretize controller	↓ discrete-time controller
	↓ implement the controller	↓ implement the controller

  ∇U ′ (ξ), F (ξ) < -x T Xx -e T Y e ≤ -σλ max (P )|x| 2 -σλ max (Q)|e| 2 Remark 2.8. In this method the MASP is found by solving a set of LMIs for the maximum value possible of T . The existence of a solution to the LMI conditions, guarantees the existence of a Lyapunov function that will yield the asymptotic stability. Note that the proposed conditions directly study the derivative of the Lyapunov function. Numerical examples will show the conservatism reduction in comparison with the approach in Method 1. Note that both the approach of Method 1 and Method 2 are robust not only to the sampled-data implementation but also to variations of the sampling intervals.

	then from (2.44), (2.45) and (2.46),
			≤ -σU ′ .	(2.48)
	Asymptotic stability follows using standard Lyapunov arguments.
						.46)
	If σ satisfies	0 < σ ≤ min	λ min (Y ) λ max (Q)	,	λ min (X) λ max (P )	(2.47)

Remark 2.9. Note that the local asymptotic stability of the hybrid system (2.12) implies the local asymptotic stability of (2.7). As a matter of fact, the established asymptotic stability is local in both Method 1 and Method 2, since the inequalities (2.33) and

(2.48)

are satisfied only inside the studied polytope P. Moreover, one can find an invariant set E ∈ P, such that for x(t 0 ) ∈ E one has |(x(t 0 ), e(t 0 ))| = |(x(t 0 ), 0)| ≤ ∆ for some ∆ > 0, for which the inequality (2.13) is satisfied.

  . Since ẋ(t k ) = 0, then, for any t ∈ (t k , t k+1 ) and for any k ∈ N, integrating (3.10) from t k to t yields V (x(t)) -V (x(t k )) < (t) and I 2 (t) given in (3.6) and (3.7) respectively. For t → t k+1 , we obtain I 1 (t k+1 ) + I 2 (t k+1 ) ≤ 0. Using (3.13) and(3.14) we see that V (x(t)) < V (x(t k )), ∀t ∈ (t k , t k+1 ], ∀k ∈ N.

		t
		t k	S(y(τ ), w(τ ))dτ.	(3.13)
	Then, from (3.9), Lemma 3.1 and Lemma 3.2 we find directly
	t
	t k	S(y(τ ), w(τ )) dτ = I 1 (t) + I 2 (t) ≤ 0, ∀t ∈ [t k , t k+1 )	(3.14)
	with I 1

b) Assume that ẋ(t k ) = 0. Due to the Lipschitz continuity of the vector field, and Theorem 3.2 in

[START_REF] Khalil | Nonlinear Systems[END_REF]

, for any interval [t k , t k+1 ) we have

∃s ∈ [t k , t k+1 ) s.t. ẋ(s) = 0 ⇒ ẋ(t) = 0, ∀t ∈ [t k , t k+1 ).

(3.15)

Thus, since the state x(t) is continuous at the sampling instants, if ẋ(t k ) = 0,

(3.15)

Table 3 . 1 :

 31 Estimation of the MASP that guarantees the local asymptotic stability of the system in Example 1.

  )

		0.72
		0.7
		0.68
	MASP	0.66
		0.64
		0.62
		0.1 0.6	0.2	0.3	0.4	0.5	α	0.6	0.7	0.8	0.9	1

  )where a, X, Y are decision variables, and σ 1 (ξ), ς 1 (ξ) are decision SOS polynomials. Using the software SOSTOOLS we find that (4.28) and (4.29) are SOS with a = 0.12015, X = 0.25506, Y = 0.88456 10 -2 . The decision SOS polynomials are σ 1 (ξ) = 0.62335 w 2 -0.3616 xw 2 + 1.6714 x 2 w 2 -0.67622 x 3 w + 2.0314 x 4 w + 3.228 x 6 , ς 1 (ξ) = 0.52025 w 2 -0.31686 xw 2 + 1.4349 x 2 w 2 -0.54824 x 3 w + 1.60754 x 4 w + 2.8846 x 6 . Thus all the conditions of Corollary 4.9 are satisfied, and x = 0 is locally exponentially stable. The estimation of the domain of attraction L c * can be easily seen to be equals to the studied domain [-0.4, +0.4].

  échantillonnés, contrôlés par un retour d'état statique. Ce problème a été considéré dans le Chapitre 2, mais l'objectif de ce chapitre est de le traiter en utilisant une nouvelle approche. Le problème de l'analyse de stabilité est étudié via une propriété d'invariance des sous ensembles ellipsoïdaux. La méthode présentée ici est inspirée par la théorie de

	la dissipativité.
	La notion de dissipativité a été introduite par Willems [124]. Depuis son introduction,
	cette approche a attiré beaucoup d'attention, car elle peut être utilisée pour étudier la
	stabilité, la passivité, la robustesse et d'autres problèmes d'analyse et de synthèse. Ces
	travaux sont inspirés par les propriétés de passivité des circuits électriques et peuvent être
	considérés comme la généralisation d'une notion abstraite d'énergie pour les systèmes
	dynamiques.
	L'équation (5) peut être écrite
	ẋ

  En adoptant une démarche d'émulation, un contrôleur en temps continu est tout d'abord synthétisé sans prendre l'échantillonnage en considération. Ensuite il est implémenté en temps discret. L'objectif principal est de fournir un critère de stabilité qui permet d'estimer le plus grand pas d'échantillonnage admissible. Dans ce travail nous nous sommes essentiellement concentrés sur les systèmes bilinéaires. Ils représentent un cas particulier des systèmes non linéaires, mais aussi un cas intermédiaire entre les systèmes linéaires et non linéaires généraux. Plusieurs méthodes théoriques ont été proposées pour ce cas. Ensuite, les résultats ont été étendus au cas non linéaire général (sous l'hypothèse affine en la commande).Nous sommes convaincus que les perspectives qui émergent des travaux présentés dans cette thèse sont multiples. Tout d'abord, les résultats de cette thèse représentent une contribution à l'analyse de stabilité des systèmes de commande en réseau, car ils traitent le problème d'échantillonnage apériodique. Un axe de recherche intéressant serait de considérer d'autres imperfections consists of resistors, capacitors, and inductors (refereed to as RLC circuit), satisfies the following property: E(t 2 ) -E(t 1 ) ≤ (s)i(s)ds, t 1 ≤ t 2 , (A.4)where E(t) is the energy stored in the circuit at instant t, v(t) is the applied input

	t 2
	t 1

v

  (x(t 2 )) -e αt 1 V (x(t 1 )) ≤ ≤ t 2 and all signals (w, y, x) which satisfy (A.1).

	t 2
	t 1	e αs S(y(s), w(s))ds,	(A.11)
	for all t 1

One can also consider less conservative conditions, i.e. local Lipschitz continuity, by adding boundedness conditions on the solutions of (3.8), see Theorem 3.3 in[START_REF] Khalil | Nonlinear Systems[END_REF].

Note that the term "period" is usually employed, but should rather be called "interval" since it contains the asynchronous sampling case.

Acknowledgements

have contributed to this work. It was supported by Highly-complex and networked control systems (HYCON) 2, which is a four-year European project coordinated by CNRS Research Director Françoise Lamnabhi-Lagarrigue.

The zero matrix in appropriate dimensions.

I

The identity matrix in appropriate dimensions.

v, z

The inner product. For v, z ∈ R n , v, z = z T v, and for v, z ∈ L 2 , v, z = Nevertheless, there exist various approaches where the sampling intervals are supposed to be controllable. These approaches include event-based control, self-triggered control and state-dependent sampling control: the idea is to guarantee stability while sampling as less as possible. Extending our results for these controlled sampling methodologies is another interesting research direction.

A2 La paire (A 0 , B 0 ) est stabilisable et le retour d'état linéaire u(t) = Kx(t) est calculé afin de stabiliser asymptotiquement localement l'origine du système [START_REF] Amato | Stabilization of bilinear systems via linear state-feedback control[END_REF].

Le domaine d'attraction est D 0 .

A3 Les variables d'état sont soumises à des contraintes données par un polytope P ⊂ D 0 :

Sous ces hypothèses, le système en boucle fermée est:

Le système (5) peut être représenté par

avec

et

Si x(t k ) est dans le polytope P, alors

Problème:

Trouver un critère de stabilité asymptotique locale de l'équilibre x = 0 du système (5), ainsi qu'une estimation du domaine d'attraction E ⊂ P de telle sorte que pour tout

x(t 0 ) ∈ E les solutions satisfont x(t) ∈ P, ∀t > t 0 , et x(t) → 0.

Appendix A Dissipative dynamical systems A.1 Introduction

The purpose of this appendix is to provide a brief presentation of the notion of dissipativity of dynamical systems. This notion was initiated by Willems [START_REF] Willems | Dissipative dynamical systems. part I: general theory[END_REF][START_REF] Willems | Dissipative dynamical systems. part II: Linear systems with quadratic supply rates[END_REF]. It was motivated by the concept of passivity from electrical networks theory. Dissipativity extends, in an abstract sense, the notion of energy. It can be seen as a generalization of Lyapunov functions technique, for input-output systems. Since the 1970's, dissipativity has been providing several useful tools for studying dynamical systems, and several researchers have been considering it (see the references [START_REF] Brogliato | Dissipative Systems Analysis and Control[END_REF][START_REF] Ebenbauer | Dissipation inequalities in systems theory: An introduction and recent results[END_REF][START_REF] Hill | The stability of nonlinear dissipative systems[END_REF][START_REF] Hill | Dissipative dynamical systems: Basic input-output and state properties[END_REF][START_REF] Willems | Dissipative dynamical systems. part I: general theory[END_REF][START_REF] Willems | Dissipative dynamical systems[END_REF], just to name a few). Consider the continuous-time dynamical system Σ described by the equations

where the values of the state x, the input w and the output y lie in R n , R m and R p , respectively. The functions in (A.1) are supposed to be smooth enough to guarantee the existence of a solution for any initial condition x(t 0 ) = x 0 ∈ R n . Moreover, they satisfy f (0) = 0 and h(0) = 0. Suppose there exists a function S(y, w) : R p × R m → R such that S(0, 0) = 0 and for all input-output pairs w ∈ R m and y ∈ R p , it satisfies

The following definition introduces the notion of dissipativity.

Definition A.1 (Dissipativity [START_REF] Brogliato | Dissipative Systems Analysis and Control[END_REF]). System Σ (A.1) is said to be dissipative with respect to the supply rate S(y, w), if there exists a storage function V (x) : R n → R + such that the following dissipation inequality holds:

for all t 1 ≤ t 2 and all signals (w, y, x) which satisfy (A.1).

Definition A.1 can be interpreted as follows. The positive semi-definite, memoryless storage function V (x), generalizes the notion of energy. The memoryless function S(y, w)

represents the rate at which power flows into the system. Finally, the dissipation inequality (A.2) shows that over the time interval [t 1 , t 2 ], the change of stored energy It must be noted that there exists a variety of definitions for dissipativity in the literature.

For example, the following definition is provided by Hill and Moylan:

). System Σ is dissipative with respect to the supply rate S(y, w), if for all admissible w(•) and all t ≥ t 0 one has t t 0 S(y(s), w(s))ds ≥ 0, (A.3) with x(t 0 ) = 0, and along trajectories of Σ.

See [START_REF] Brogliato | Dissipative Systems Analysis and Control[END_REF] and the references therein for relationships between several types of definitions.

A.2 Dynamical control properties via dissipativity

Using dissipativity allows for considering various properties of control systems from a single point of view. These properties have a wide range of applications in control and systems theory. Here, we point out some of these properties, and provide some useful references.

Passivity:

The concept of passivity was first studied in control theory by Popov in the 1960s. It was motivated by electrical networks theory. In specific, a single input electrical circuit which

Contribution to the control of nonlinear systems under aperiodic sampling

Abstract: This PhD thesis is dedicated to the stability analyzis of nonlinear systems under sampled-data control, with arbitrarily time-varying sampling intervals. When a controller is designed in continuous-time, and then implemented digitally (emulation approach), it is of great interest to provide stability criteria, and to estimate the bound on the sampling intervals which guarantees the stability of the sampled-data system. Whereas several works deal with linear models, the issue has been rarely addressed in a formal quantitative study in the nonlinear case.

First, an overview on sampled-data control is presented. Challenges and main methodologies for stability analysis are presented for both the linear time-invariant and the nonlinear cases. Then, local stability of bilinear sampled-data systems controlled by a linear state feedback is considered by using two approaches: the first one is based on hybrid systems theory; the second one is based on the analyzis of contractive invariant sets and is inspired by the dissipativity theory. Both approaches provide sufficient stability conditions in the form of LMI. Finally, the dissipativity-based stability conditions are extended for the more general case of nonlinear systems which are affine in the input, including the case of polynomial systems which leads to conditions in the form of sum of squares (SOS).

Keywords: Sampled-data systems, bilinear systems, nonlinear systems, hybrid dynamical systems, aperiodic sampling, stability, dissipativity, linear matrix inequalities (LMIs), sum of squares (SOS).

Contribution à la commande de systèmes non linéaires sous échantillonnage apériodique

Résumé: Cette thèse est dédiée à l'analyse de stabilité des systèmes non linéaires sous échantillonnage variant avec le temps. Lors de l'implémentation numérique d'un contrôleur qui est calculé en temps-continu (approche par émulation), il est d'un grand intérêt de fournir des critères de stabilité et d'estimer la borne supérieure de l'intervalle d'échantillonnage qui garantit la stabilité du système en temps discret. Plusieurs travaux récents ont abordé ces questions dans le cas de modèles linéaires, mais la question a rarement été abordée dans une étude quantitative et formelle pour les systèmes non linéaires.

Tout d'abord, le mémoire présente un aperçu sur les systèmes échantillonnés. Les défis et les principales méthodes pour l'analyse de stabilité sont présentés pour le cas des systèmes linéaires invariants dans le temps et celui des systèmes non linéaires. Ensuite, l'analyse de la stabilité locale des systèmes bilinéaires échantillonnés contrôlés par un retour d'état linéaire est considérée. Deux approches sont utilisées, la première basée sur la théorie des systèmes hybrides, la seconde basée sur l'analyse des ensembles invariants contractants. Cette dernière approche est inspirée par la théorie de la dissipativité. L'ensemble de ces résultats conduisent à des conditions suffisantes de stabilité exprimées sous forme LMI. Enfin, les conditions de stabilité basées sur la dissipativité sont étendues au cas des systèmes non linéaires affines en l'entrée. Les résultats sont ensuite repris dans le cas spécifique des systèmes non linéaires polynomiaux où les conditions de stabilité sont vérifiées numériquement en utilisant la décomposition en somme des carrés (SOS).

Mots-clés : Systèmes échantillonnés, systèmes bilinéaires, systèmes non linéaires, systèmes dynamiques hybrides, échantillonnage apériodique, stabilité, dissipativité, inégalités matricielles linéaires (LMIs), somme des carrés (SOS).