N Amini

J Auriol

F Bonnans

D Beauvois

C Bonnet

I Boussaada

A Chaillet

Y Chitour

S Diop

G Duc

D Dumur

S Font

A Girard

E Godoy

A. Iovine L Greco

F Lamnabhi-Lagarrigue

H Lhachemi

A Loria

M Makarov

C Maniu

P Mason

G Mazanti

F Mazenc

H Mounier

S.-I Niculescu

D Normand-Cyrot

S Olaru

R Ortega

E Panteley

W Pasillas-Lepine

L Pfeiffer

D Prandi

P Rodriguez-Ayerbe

G Sandou

H Siguerdidjane

S Tebbani

S Tliba

A Bertinet

P Debever

S Douesnard

M Giron

L Antunes

L Marquet

C Charles

M Paleczny

P Lepeltier

W Michiels

P Pepe

M Ahmadi

A Diab

L B Groff

N Michel

D Penco

A Papachristodoulou

S Bertrand

J M Gomes Da Silva

P Kvieska

J Davins-Valldaura

Isabelle Germain

Didier, Dimitri Denis

Frédéric Sophie

Lucie

Thomas

Luca

Vincent Alexandre

Sergio Luca

Daniele Laura

F Ferrante

S Duncan

R Drummond

Karush-Kuhn-Tucker Kkt

Kalman-Yakubovic-Popov Kyp

Z F Zames

ERA Estimate of the Region of Attraction. 39 ISS Input-to-State Stability. 41

Keywords: PWA Piecewise Affine. 45, 49 PWL Piecewise Linear. 89 PWQ Piecewise Quadratic. 45, 49 ReLU Rectifier Linear Unit. 95 SDP Semidefinite Programming. 7, 41 SISO Single-Input Single-Output. 37

for all the support from and the time spent with the MAC group at LAAS

v "La vraie division humaine est celle-ci : les lumineux et les ténébreux. Diminuer le nombre des ténébreux, augmenter le nombre des lumineux, voilà le but. C'est pourquoi nous crions : enseignement ! science ! Apprendre à lire, c'est allumer du feu ; toute syllabe épelée étincelle. Du reste qui dit lumière ne dit pas nécessairement joie. On souffre dans la lumière ; l'excès brûle. La flamme est ennemie de l'aile. Brûler sans cesser de voler, c'est là le prodige du génie."

Victor Hugo, Les Misérables Tome IV "Considerate la vostra semenza: fatti non foste a viver come bruti, ma per seguir virtute e canoscenza" Dante Alighieri, Inferno -Canto 26 "Reconhece a queda e não desanima Levanta, sacode a poeira e dá a volta por cima" Paolo Vanzolini, Volta por cima

Preface

This document presents a selection of research results obtained over the last six years, period coinciding with the beginning of my associate professorship at CentraleSupélec and my affiliation to the Laboratoire de Signaux et Systèmes in September 2015. The choice of the contents of the manuscript reflects the current most active research activities with collaborators and, in my view, also the most promising. This choice has allowed me to write a coherent document regarding the use of convex optimization to study classes of nonlinear feedback control systems. Nonetheless, the contents of the manuscript reflect the evolution of previous research in related topics, acquired since my Ph.D. in 2010.

The manuscript is divided into two parts; each part contains two chapters. The two parts are independent and can be read separately. It ends with a brief description of ongoing work and research perspectives followed by conclusions.

The presented results focus on particular classes of nonlinear systems that can be expressed as the interconnection of linear systems and static nonlinearities. The main problems we study are the stability and input-output analysis of nonlinear systems. The solutions to these problems are stability conditions that can be checked with convex optimization, namely semidefinite program. These conditions are sufficient stability conditions cast as inequalities where the unknowns are the parameters of Lyapunov function candidates. The numerical solutions provide the values of the parameters defining these functions.

In several instances, we show that the proposed conditions reduce the conservatism and simplify the analysis when compared to existing methods. The conservatism reduction is achieved thanks to particular choices of Lyapunov Function structures and their parametrization. Importantly, such functions are chosen to exploit the information available from the classes of nonlinearities we study: in the first part, we assume the nonlinear terms are slope-restricted, while in the second part, we consider nonlinear loops with several nonlinearities of a single type, the ramp function. The potential of the proposed framework is enforced by a result showing that ill-posed algebraic loop involving ramp functions yield set-valued discontinuous mappings.

The results in the Chapters 1, 3, 4 have already been published [START_REF] Valmorbida | Regional analysis of slope-restricted Lurie systems[END_REF], [START_REF] Groff | Stability analysis of piecewise affine discrete-time systems[END_REF] [175], while the results in Chapter 2 are the Discrete-time counterpart for the ones in Chapter 1, and are currently under review. While the main ideas are the same as in the published papers, the presentation here is expanded with different examples and figures included. The classes of systems we consider and the studied problems are summarized in Table 1. Some of the research topics I have also studied in recent years are not included in this text. Namely, results on Polynomial Optimization methods for nonlinear systems, on numerical methods for the stability analysis of Infinite Dimensional Systems, and on the analysis of time-varying systems and applications. The reason for excluding these topics and the associated publications was to narrow the scope of the manuscript. The publications related to these topics and other lines of research can be found in the reference list of Appendix B while current and past projects of supervised Ph.D. students are listed in Appendix C. system, where G is the linear system, M is the multiplier and φ(.) is the nonlinearity. 2.3 Maximum sector δ obtained by Proposition 2.1 for G 4 (z) as a function of the horizon N ∈ [START_REF] Adegbege | A framework for multivariable algebraic loops in linear anti-windup implementations[END_REF][START_REF] Barabanov | Absolute characteristic exponent of a class of linear nonstatinoary systems of differential equations[END_REF]. We also plot the bounds achieved using the Circle criterion, the Tsypkin Lyapunov function and the function from [START_REF] Park | A less conservative stability criterion for discrete-time Lur'e systems with sector and slope restrictions[END_REF]. The upper limit set by the Nyquist gain is also plotted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Regional stability of the second numerical example. Positive invariant sets of the Lyapunov function V (x) from (2.5) are shown in blue, with blue dashed lines corresponding to sub-level sets. Light blue corresponds to N = 4 and dark blue to N = 1. Black dashed lines denote the limits {x : ν 0 (x) = y, ν 0 (x) = y}. Initial conditions from the red region did not converge to the origin. . . . . . . . . . . . . . . . . . . 2.5 Worst-case input output gain ω bounding z 2 2 ≤ ω w 2 2 as a function of the horizon length N in V (x). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Feedback representation of a Lurie system is here used to represent PWA systems.

Class of systems

In this part, φ will be a vector of ramp functions. The term f 5 is a constant vector that is added to the output of the linear block G(z) to generate the input y of the nonlinearity φ. These partitions differ also in the number of sets defining them. On the left, d 1 = -d 2 > 0, we obtain a partition with four sets. On the right, with d 1 = d 2 < 0, we obtain a partition with six sets. The dashed line corresponds to the set where y 3 = 0. . . . 3.8 The tree structure corresponding to (3.50) using the composition rules of (3.48) singleton. The set of values for V (x, y) is given in red in coordinate axis. In this case, the regularized dynamics give a set-valued mapping illustrated here by the set x + a , which in turn gives the set of values for V (x + , y + )) -depicted in blue in the coordinate axis. The theorem asks for all points in the green set to be below the red set. In this case the single valued function W is guaranteed to decrease. . . . . . The identity matrix of dimension n.

M

The transpose of a matrix M ∈ R n×m .

M ⊥ Any matrix having as columns a basis of ker M .

M i,j

For M ∈ R n×m (i, j) entry of a matrix M v i For v ∈ R n ith entry of a vector v

y[k]
The value of a discrete-time signal y at instant k

We may drop the arguments of some functions when it is clear from the context.

General Introduction

Incorporating knowledge about nonlinearities and understanding the phenomena they induce is of utmost importance to the analysis and design of feedback control systems. Fundamental results put forward in the last century still shape the general approaches to study nonlinear control systems. The absolute stability framework, studying feedback loops composed of linear dynamical systems and static nonlinearities -also called Lurie systems -is one of these approaches. Early contributions indicated how linear analysis tools could be generalized to nonlinear systems and were first applied to the interconnections of linear time-invariant systems and sector-bounded nonlinearities. These methods proposed stability conditions and provided insightful ways to assess closed-loop behavior for classes of nonlinearities. However, in practice, the nonlinearities are either known or belong to subsets of the sector descriptions used as surrogates. Examples from practical control systems are actuator nonlinearities such as saturations, deadzones, quantization, relays, hysteresis. For this reason, the sector models become only a rough representation of the actual nonlinear elements. The unavoidable consequences of replacing specific nonlinearities by sector models is that we may only obtain conservative estimates of stability bounds, regions of attraction, and induced gains. Concerning control design, the use of sector only information may lead to feedback laws that underperform with the nonlinear actuators, or some optimal performance is only met in a reduced operating set.

One successful approach to reduce the set of nonlinearities by narrowing down the sector, was proposed for the saturation nonlinearity [START_REF] Hu | Stability and performance for saturated systems via quadratic and nonquadratic Lyapunov functions[END_REF][START_REF] Tarbouriech | Stability analysis and stabilization of systems presenting nested saturation[END_REF]. The local sector conditions allow to obtain invariant sets as estimates of the region of attraction of the origin as level sets of quadratic Lyapunov functions. The Lyapunov stability theory methods are perhaps the most straightforward and versatile stability analysis and control design methods for nonlinear systems. They allow to assess the robustness of solutions, to evaluate regional stability, and to evaluate performance degradation according to the levels of disturbances.

The stability conditions based on the theory introduced by Lyapunov were initially solved analytically, and, more recently, thanks to the advances in convex optimization, they could also be solved numerically. The interest in obtaining a Lyapunov Function (LF) numerically was already pointed out in [START_REF] Kalman | Control system analysis and design via the "second method" of Lyapunov: II discrete-time systems[END_REF] actually the "second method" is more accurately described as a point of view, a philosophy of approach, rather than a systematic method. At present, much depends on the ingenuity of the user. In the future, we can hope that systematic procedures will be made possible by machine computation.

However, obtaining numerical solutions depends on a parametrization of the stability certificates -the Lyapunov functions -by a finite number of parameters. The classical and simplest parametrization is the quadratic function, of which the existence is necessary and sufficient for the stability of linear systems. Unfortunately, quadratic functions are of limited interest whenever studying the stability of systems containing specific nonlinearities from which we can obtain a more detailed description than with sector inequalities. Defining the class of functions giving necessary and sufficient stability conditions for specific nonlinearities might be difficult. The reason for this is that the converse theorems leading to suitable classes of Lypunov functions require the knowledge of the solutions of the systems, which are, in general, difficult to obtain for nonlinear systems. Therefore, these converse results are scarce, and most of the proposals for Lyapunov functions candidates yield only are sufficient stability conditions.

For some systems, it is nonetheless possible to establish properties of the desired classes of LF. For instance, the characterization of the LFs is possible in the case of sector inequalities with arbitrary variations or switching systems [START_REF] Molchanov | Criteria of asymptotic stability of differential and difference inclusions encountered in control theory[END_REF]. The parametrization for numerical computations may not be immediate though. In the specific case of uncertain systems, a convex optimization approach can be obtained using homogeneous polynomials as LF and Sum-of-Squares program-

GENERAL INTRODUCTION

ming [START_REF] Chesi | Homogeneous Lyapunov functions for systems with structured uncertainties[END_REF]. Other classes of homogeneous functions have also been proposed, piecewise affine (PWA) functions for example.

Regarding the numerical computation of Lyapunov functions, linear programming and semidefinite programming were used to obtain stability certificates and performance assessment for nonlinear systems. These methods are based on the computation of LF parameters by setting the Lyapunov inequalities as constraints of an optimization problem. In comparison to analytical approaches solving Lyapunov or (generalizations of) Riccati equation, the solution to inequalities make possible to assess input-output properties, to introduce decay rate bounds, and to treat of uncertainties using differential inclusions in a straightforward manner.

More sophisticated LF candidates and the formulations of conditions to check the associated inequalities may take advantage of the information on the nonlinearity. With more general class of LF (or storage function) and the detailed description of the nonlinearity we may expect to obtain better performance estimates of reachable sets, nonlinear gains, and estimates of the basin of attraction.

On the other hand, nonlinearities may also be introduced by the control laws to improve the system's performance or for constraint handling. The use of optimization-based approaches for constraint handling might seem disconnected from more classical static feedbacks strategies. They have, however, been shown to belong to the class of piecewise affine (PWA) of nonlinear functions [START_REF] Bemporad | The explicit linear quadratic regulator for constrained systems[END_REF]. A unified view of these optimization based approaches as Model Predictive Control with the more classical approaches to treat the input saturation, such as the anti-windup compensator is still missing. In this manuscript we present mathematical tools leading to a common framework for the analysis of input-constrained systems, namely a PWA modelling and the description of the model in terms of a single affine function, the ramp function. Finally, the study of switched dynamics and their closed-loop strategies [START_REF] Shorten | Stability criteria for switched and hybrid systems[END_REF] has not been considered within the Lurie systems framework. We indicate how we plan to tie these classes of systems using the PWA system analysis presented here.

Outline of the manuscript

The manuscript is organized into two parts. Each part starts with an introduction where the motivation the studied problems is presented and a brief literature review is provided. Each part has two chapters and the first part also presents Notes and References for a more thorough review of the litterature on the absolute stability problem.

Part I presents results for the analysis of the sector and slope-restricted nonlinearities. For continuous-and discrete-time systems, we have proposed LF structures that encompasses the existing ones in the literature. The results also include the regional stability analysis with strategies to estimate the region of attraction of the origin. Chapter 1 presents a numerical formulation to treat slope-restricted nonlinearities with generalized quadratic plus integral terms. The main contribution of this work was to highlight that more straightforward conditions on the parameters of the Lyapunov function can be obtained, thus relaxing conditions the function parameters given by matrices and coefficients. Surprisingly, the relaxed positivity constraints of the generalized quadratic matrix we proposed, trace back to [START_REF] Yakubovich | The method of matrix inequalities in the stability theory of nonlinear control systems: II. Absolute stability in a class of nonlinearities with a condition on the derivative[END_REF]. To our knowledge, they had not been used with optimization-based methods for stability analysis.

In a similar vein, but for discrete-time settings, in Chapter 2, we present recent results on the generalization of Tsypkin criterion for slope restricted and monotone nonlinearities. Differently from the continuous time-case where the integral term appears naturally, the integral terms appear to replace an infinite sum. Monotonicity of the nonlinearity is required to treat these integral terms of the LF. The conditions on the parameters of a Lyapunov function for its positivity simplify existing results in the literature.

Part II presents the stability analysis for classes of PWA discrete-time systems. These results are built upon an implicit representation of the vector field. This implicit representation is described in terms of ramp functions and an algebraic loop.

Chapter 3 considers the analysis of PWA systems. The main contrast with the literature is on the representation of the PWA systems. Our main argument in this chapter is that a suitable representation based on ramp functions can lead to more straightforward stability and robustness analysis tools.

Chapter 4 considers the analysis of systems with input quantization. The key result in this chapter is to show that the quantization is obtained from an ill-posed algebraic loop involving two ramps. Thanks to the "two-ramp" model for the step, we formulate stability conditions for set-valued maps, that are solved numerically using the same tools as the ones used in the analysis of continuous PWA systems.

The manuscript ends with the exposition of some ongoing works and perspectives and a conclusion summarizing the presented material.

Part I

Lurie Systems with sector and slope restricted nonlinearities

Introduction of Part I

The stability analysis of feedback loops consisting of linear time-invariant systems and sector bounded nonlinearities is known as the absolute stability problem. The stability of the origin of this feedback interconnection can be studied via the passivity properties of its elements; in particular, two celebrated results are given by the Circle and Popov criteria [START_REF] Khalil | Nonlinear systems[END_REF], where the only assumption on the nonlinearity is that it belongs to a sector. This problem has its roots in [START_REF] Lurie | On the theory of stability of control systems[END_REF] and its importance is evident since actuator devices in control loops are modeled, in general, as static nonlinearities.

Assuming that the nonlinearity is only sector-bounded might be overly conservative whenever the nonlinearities are known, or their slopes can be bounded. The study of the class of sloperestricted nonlinear systems using the framework of absolute stability theory was first proposed in two papers; a frequency domain condition given in [START_REF] Dewey | On the stability of feedback systems with one differentiable nonlinear element[END_REF] and a geometrical condition based upon the construction of a Lyapunov function (LF) in [START_REF] Yakubovich | The method of matrix inequalities in the stability theory of nonlinear control systems: II. Absolute stability in a class of nonlinearities with a condition on the derivative[END_REF]. When compared to conditions of the sectorbounded only case, it is noted that the conditions on the parameters of the LF to gurantee its positivity were already relaxed in [START_REF] Yakubovich | The method of matrix inequalities in the stability theory of nonlinear control systems: II. Absolute stability in a class of nonlinearities with a condition on the derivative[END_REF].

In addition to the Lyapunov functions associated with the Circle and Popov criteria, different LFs have been proposed for studying Lurie systems: composite LFs [START_REF] Hu | Absolute stability with a generalized sector condition[END_REF]; LFs with quadratic components on both the nonlinearities and the states and Lurie-Postnikov terms were studied in [START_REF] Yakubovich | The method of matrix inequalities in the stability theory of nonlinear control systems: II. Absolute stability in a class of nonlinearities with a condition on the derivative[END_REF][START_REF] Suykens | An absolute stability criterion for the Lur'e problem with sector and slope restricted nonlinearities[END_REF][START_REF] Park | Stability criteria of sector-and slope-restricted Lur'e systems[END_REF][START_REF] Turner | Lyapunov functions and L 2 gain bounds for systems with slope restricted nonlinearities[END_REF]. For the quadratic LFs associated with the Circle criterion, the positivity of the LF is enforced with a positive-definite Lyapunov matrix [START_REF] Khalil | Nonlinear systems[END_REF]. In the case of LFs with a Lurie-Postnikov type term, associated with the Popov criterion, the positivity of the LF requires the positivity of the Lyapunov matrix and imposes the positivity of the coefficients in the Lurie-Postnikov integral terms for sector-bounded nonlinearities.

For the case of nonlinearities that are sector-and slope-bounded in a set containing the origin, we can obtain local certificates for gains, reachable sets, and estimates of the basin of attraction. Thanks to a local characterization of the nonlinearities, we can obtain tighter estimates of gain properties and study nonlinearities with unbounded discontinuities. Examples of systems modeled with unbounded nonlinearities include the driven Stirling engine [START_REF] Hauser | Dynamics of a driven Stirling engine[END_REF] and electrical energy storage devices known as supercapacitors [START_REF] Drummond | Low-order mathematical modelling of electric double layer supercapacitors using spectral methods[END_REF], modeled using the logarithm nonlinearity. Estimates of regions of attraction for sector bound nonlinear systems obtained with the Popov criterion have been considered in [START_REF] Weissenberger | Application of results from the absolute stability problem to the computation of finite stability domains[END_REF][START_REF] Walker | Finite regions of attraction for the problem of Lur'e[END_REF][START_REF] Sastry | Finite regions of attraction for the problem of Lur'e[END_REF], and more recently in [START_REF] Hindi | Analysis of linear systems with saturation using convex optimization[END_REF] using Semidefinite Programming (SDP), the same type of numerical stability conditions we pursue in this part.

A fundamental difference in the study of continuous-and discrete-time Lurie systems is that, for the global stability of DT systems, the least conservative Lyapunov function without assumptions on the slope is the quadratic function. It is thus fair to say that there remain gaps in the understanding of the absolute stability problem, which focuses on the stability analysis of Lurie systems. In particular, methods proposed for the stability analysis of discrete-time systems remain underdeveloped compared to those for continuous-time systems; for example, the widely adopted formulation of the Popov criterion in discrete time requires extra conditions on the nonlinearity, including monotonicity [START_REF] Tsypkin | Frequency criteria for the absolute stability of nonlinear sampled-data systems[END_REF].

Several recent applications, including the stability analysis of neural network-based control policies [START_REF] Chu | Bounds of the induced norm and model reduction errors for systems with repeated scalar nonlinearities[END_REF] and the convergence analysis of first-order optimization algorithms [START_REF] Lessard | Analysis and Design of Optimization Algorithms via Integral Quadratic Constraints[END_REF], can be understood within the context of discrete-time Lurie systems. These applications' impact has motivated us to revisit the absolute stability problem in discrete time to improve its theoretical understanding. Such a better understanding should allow even more complex classes of systems to be analyzed with less conservative performance certificates to be obtained.

This part also explores the stability analysis of discrete-time Lurie systems with slope-restricted nonlinearities as a step in this direction. A new class of Lyapunov function is proposed with a simplified structure than the current state-of-the-art [START_REF] Park | A less conservative stability criterion for discrete-time Lur'e systems with sector and slope restrictions[END_REF][START_REF] Carrasco | Convex searches for discretetime Zames-Falb multipliers[END_REF] and conditions are developed for the regional stability analysis and bounding input-output gains of these systems.

Summary of contributions of Part I

The results presented in this part focus on the local analysis of Lurie-type systems with sloperestricted nonlinearities. We develop quadratic LFs in both the state and the nonlinear terms and contain Lurie-Postnikov integral terms. We present conditions for the positivity of the LF that do not impose the positivity of the Lurie-Postnikov terms coefficients nor require that the quadratic terms on the nonlinearities are positive definite. We also present connections between our results and recent results in the literature that use similar LF structures.

The presented conditions for stability analysis and gain assessment are cast as dissipation inequalities. These inequalities are obtained upon inequalities associated with the sector and the slope bounds. In cases where the sector inequalities hold only locally, we discuss how to guarantee the inclusion of level sets in the region where the sector inequalities hold. These inclusion conditions allow us to estimate the region of attraction using contractive and invariant sets defined by some level sets of the computed LF. This allows us to analyze the effect of additive exogenous inputs and outputs to derive conditions for the computation of reachable sets and local induced gains.

Structure of Part I

Chapter 1 studies Lurie systems in continuous time. These systems are assumed to have local bounds for both sector and slope. We characterize these sector and slope bounds in terms of inequalities which are used to verify Lyapunov inequalities. The regional analysis is then performed thanks to inclusions conditions to guarantee that the level sets of the computed LF are within the set where the bounds hold. Importantly, we treat feedbacks containing direct transmission terms. We thus generalized conditions for the well-posedness of the algebraic loops containing sector and slope bounds. We also highlight the constraints of the convex optimization formulation used to illustrate the results with numerical examples.

In Chapter 2, we present strategies for the local stability analysis of discrete-time Lurie systems. Here, we provide a different Lyapunov function candidate than the one used in the continuous-time case for systems with slope restriction. The proposed function is built by considering the propagation from a point taken as the initial condition of the dynamics and considers a quadratic form with the nonlinearities with several integral terms, thus generalizing the Tsypkin structure [START_REF] Tsypkin | On the stability in the large of nonlinear sampled-data systems[END_REF][START_REF] Szegö | On the absolute stability of sampled-data control systems[END_REF]. The presented results generalize the LF structures existing in the literature with a reduced number of parameters in the LF. For clarity of presentation, we consider only the single-input and the single-output case without direct transmission terms in the feedback. The semidefinite programming formulations allow us to obtain numerical examples to illustrate the proposed stability conditions.

This part ends with Notes and References providing a bibliography review in the study of Lurie systems. This review focuses on the global and local stability analysis of Lurie systems. It discusses results using leading to convex optimization-based conditions, thereby providing a perspective on how the contents of the two chapters in this part relate to the literature on the topic.

Chapter 1

Analysis of Continuous-time Slope-Restricted Lurie Systems

Problem statement

Consider the linear time-invariant (LTI) system

   ẋ = Ax + Bφ(y) + B w w y = Cx + Dφ(y) + D w w z = C z x + D z φ(y) + D zw w (1.1) with x ∈ R n , y ∈ R m , w ∈ R mw , A ∈ R n×n , B ∈ R n×m , B w ∈ R n×mw , C ∈ R p×n , D ∈ R p×m , D w ∈ R p×mw , C z ∈ R pz×n , D ∈ R pz×m , D w ∈ R pz×mw . The nonlinearity φ : Y → R m , Y ⊆ R m , is assumed to be time-invariant, memoryless, Lipschitz on Y • , decentralized φ(y) = [φ 1 (y 1 ) φ 2 (y 2 ) . . . φ m (y m )] , (1.2a)
sector bounded φ i (y i )

y i ∈ [δ i , δ i ] ∀y ∈ Y 0 ⊆ Y (1.2b)
which implies φ(0) = 0, δ i ∈ R, δ i ∈ R, and slope restricted

∂φ i (y i ) ∈ [γ i , γ i ] ∀y ∈ Y 0 ⊆ Y, (1.2c)
where γ i ≤ δ i and δ i ≤ γ i . We also introduce the matrices

∆ := diag(δ 1 , . . . , δ m ), ∆ := diag(δ 1 , . . . , δ m ), Γ := diag(γ 1 , . . . , γ m ), Γ := diag(γ 1 , . . . , γ m ),
to compactly express the sector and slope bounds. The Lipschitz assumption on φ implies that ∂φ i (y i ) = dφi dyi almost everywhere, relaxing the requirement for the nonlinearity to be continuously differentiable [START_REF] Turner | Lyapunov functions and L 2 gain bounds for systems with slope restricted nonlinearities[END_REF]Section 2].

Conditions for Well Posedness of the algebraic loop

The well posedness of the algebraic loop in (1.1) is guaranteed if there exists a unique solution to the implicit equation Claim 1], for functions φ that are differentiable almost everywhere, the well-posedness of the loop is obtained if JF (y), the Jacobian of F , where it is defined, belongs to a compact and convex set of invertible matrices for almost all values of y (see [START_REF] Zaccarian | A common framework for anti-windup, bumpless transfer and reliable designs[END_REF]Proposition 2]).

F (y) := y -Dφ(y) = ζ, that is, a mapping y(ζ) satisfying F (y(ζ)) = ζ. Following [187,
The Jacobian of F (y) is given by JF (y) = I -D∂φ(y) a.e.. Thanks to the slope restriction of φ(y) in (1.2c), for almost all y, JF (y) ∈ M := co({I -DΓ, Γ ∈ G}), where

G := Γ ∈ D : Γ = diag(γ 1 , γ 2 , . . . , γ m ), γ i ∈ γ i , γ i , ∀i
and co(A) denotes the closed convex hull of the set A. From the above description we have that the set M is convex and compact, the proposition below sets conditions for the matrices in the set M to be nonsingular, thus guaranteeing that the solution to the algebraic loop exists and is unique. The only difference to the reasoning presented in [187, Proposition 2] is given by conditions related to the non-singularity of the Jacobian of F (y).

Proposition 1.1 Given a matrix D ∈ R m×m , if there exists a matrix W ∈ D m ≥0 such that 2W -W (I - DΓ) -1 D(Γ -Γ) -(Γ -Γ)D T W ((I -DΓ) -1 ) T > 0 then I -DΓ is nonsingular for all matrices Γ belonging to the set G. Proof. If (I -DΓ) is singular then there exists z ∈ R m , z = 0 such that 0 = (I -DΓ)z = ((I -DΓ) -D(Γ -Γ))z = (I -DΓ)z -D(Γ -Γ)(Γ -Γ) -1 (Γ -Γ))z. Define z = (Γ -Γ) -1 (Γ -Γ))z to obtain (I -DΓ) z -(I -DΓ) -1 D(Γ -Γ)z = 0.
Multiply the above expression on the left by zT

W (I -DΓ) -1 , to obtain zT W z -zT W (I -DΓ) -1 D(Γ -Γ)z = 0. Since for γ i ≤ γ i ≤ γ i , 1 ≥ (γ i -γ i ) -1 (γ i -γ i ) ≥ 0 we have zT W z = z T (Γ -Γ) -1 (Γ -Γ)W z ≥ z T (Γ -Γ) -2 (Γ -Γ) 2 W z = zT W z. Thus, if (I -DΓ) is singular we must have zT W - 1 2 He(W (I -DΓ) -1 D(Γ -Γ)) z ≤ 0,
which contradicts the inequality of the claim. Hence if the inequality in the claim holds the matrix (I -DΓ) is non-singular for any Γ ∈ G.

From the above result, the following assumption on matrix D and bounds Γ and Γ, will guarantee the well-posed of the algebraic loop.

Assumption 1.1: Well-posedness

There exists a matrix W ∈ D m ≥0 such that 2W -He(W (I -DΓ) -1 D(Γ -Γ)) > 0.

(1.3)

Provided Assumption 1.1 holds, we can define the following set

X 0 := {x ∈ R n | y ∈ Y 0 , F (y) = Cx} , (1.4)
where Y 0 ⊆ Y ⊆ R m corresponds to the set where the sector and the slope restrictions hold, as defined in (1.2). We also define the following set In case the sector and slope bounds (1.2b) and (1.2c) hold globally, i.e. Y 0 = R m , global properties will be obtained by setting

X W 0 := {(x, w) ∈ R n × R mw | y ∈ Y 0 , x ∈ X 0 , F (y) = Cx + D w w} . ( 1
X 0 = R n and X W 0 ∈ R n × R mw .

Sector inequalities

In this section we present inequalities related to the sector and slope bounds of the nonlinearities in system (1.1).

Define

s 1 : R m×m × R m × R m → R, s 2 : R m×m × R m × R m → R, s 3 : R m×m × R m × R m × R m × R m → R as s 1 (T, φ, θ) := (φ -∆θ) T ∆θ -φ s 2 (T, φ, θ) := (φ -Γθ) T Γθ -φ s 3 (T, φ 1 , φ 2 , θ 1 , θ 2 ) := ((φ 1 -φ 2 ) -Γ (θ 1 -θ 2 )) T Γ (θ 1 -θ 2 ) -(φ 1 -φ 2 ) .
The following lemma is associated with the sector boundedness of the functions φ i .

Lemma 1.1 If T 1 ∈ D m ≥0 and φ : R m → R m satisfies (1.
2), then

s 1 (T 1 , φ(θ), θ) ≥ 0 (1.6)
for all θ ∈ Y 0 .

Proof.

For T 1 ∈ D m we have s 1 (T 1 , φ(θ), θ) = m i=1 T 1(i,i) (φ i (θ i ) -δ i θ i (x)) δ i θ i (x) -φ i (θ i ) . s 1 (T 1 , φ(θ), θ) = m i=1 T 1(i,i) (φ i (θ) -δ i θ i (x)) δ i θ i (x) -φ i (θ i ) . If φ satisfies (1.2a), (1.2b), we have φi(θi) θi -δ i δ i -φi(θi) θi ≥ 0, which, when multiplied by θ 2 i gives (φ i (θ i ) -δ i θ i ) δ i θ i -φ i (θ i ) ≥ 0, i = 1, . . . , m for θ ∈ Y 0 . Since T 1(i,i) ≥ 0, and (1.
2) hold then (1.6) holds for all θ ∈ Y 0 .

In the following two lemmas, we consider θ : [0, ∞) → Y 0 , θ(t) ∈ C 1 (t) to obtain inequalities related to the the slope restrictions (1.2c) of φ.

Lemma 1.2 If T 2 ∈ D m
≥0 and φ : R m → R m satisfies (1.2), then

s 2 (T 2 , φ(θ), θ) ≥ 0 (1.7)
almost everywhere for θ ∈ Y 0 .

Proof. From (1.2c) we have

(∂φ i (θ i ) -γ i )(γ i -∂φ i (θ i )) ≥ 0. ∂φ i -γ i (γ i -∂φ i ) ≥ 0.

Multiplying this expression by θ2

i , gives

0 ≤ θ2 i ∂φ i (θ i ) -γ i (γ i -∂φ i (θ i )) = ∂φ i (θ i ) θi -γ i θi γ i θi -∂φ i (θ i ) θi = φi (θ i ) -γ i θi γ i θi -φi (θ i ) . (1.8) For T 2 ∈ D m ≥0 we have s 2 (T 2 , φ(θ), θ) = m i=1 T 2(i,i) φi (θ i ) -γ i θi (x) γ i θi (x) -φi (θ i )
, and, from (1.8) then (1.7) holds.

Lemma 1.3 If T 3 ∈ D m
≥0 and φ : R m → R m satisfies (1.2c), then

s 3 (T 3 , φ(θ 1 ), φ(θ 2 ), θ 1 , θ 2 ) ≥ 0 (1.9) for all θ 1 , θ 2 ∈ Y 0 . 12CHAPTER 1. ANALYSIS OF CONTINUOUS-TIME SLOPE-RESTRICTED LURIE SYSTEMS Proof. For i = 1, . . . , m, define φai (θ i ) := (φ i (θ i ) -γ i θ i ), φbi (θ i ) := (γ i θ i -φ i (θ i )).
Provided (1.2c) holds, then φai , φbi satisfy

∂ θi φai (θ i ) ≥ 0, ∂ θi φbi (θ i ) ≥ 0, hence, for any θ 1 , θ 2 ∈ Y 0 , φai (θ 1i ) -φai (θ 2i ) θ 1i -θ 2i ≥ 0, φbi (θ 1i ) -φbi (θ 2i ) θ 1i -θ 2i ≥ 0. (1.10) For T 3 ∈ D m ≥0 we have s 3 (T 3 , φ(θ 1 ), φ(θ 2 ), θ 1 , θ 2 ) = m i=1 T 3(i,i) (φ i (θ 1i ) -φ i (θ 2i )) -γ i (θ 1i -θ 2i ) (γ i (θ 1i -θ 2i ) -(φ i (θ 1i ) -φ i (θ 2i ))) = m i=1 T 3(i,i) φai (θ 1i ) -φai (θ 2i ) φbi (θ 1i ) -φbi (θ 2i ) = m i=1 T 3(i,i) (θ 1i -θ 2i ) 2 φai(θ1i)-φai(θ2i) θ1i-θ2i φbi (θ1i)-φbi (θ2i) θ1i-θ2i
Since T 3(i,i) ≥ 0 and (1.10) hold, then (1.9) holds.

The above lemma shows that the slope restriction with non-negative bounds satisfies the incremental sector boundedness property [START_REF] Zhang | Dynamic feedback synchronization of Lur'e networks via incremental sector boundedness[END_REF]Definition 1].

Regional Stability Analysis, Reachable Sets and Nonlinear Gains

This section is concerned with functions of the form

V (x) = V 0 (x) + m i=1 λ i ỹi(x) 0 (φ i (s) -δ i s) ds, (1.11a)
where with P 11 ∈ R n×n , P 12 ∈ R n×m , P 22 ∈ R m×m and λ ∈ R m . These functions will be considered as Lyapunov candidate functions for system (1.1). We refer to the integral terms in (1.11a) as the Lurie-Postnikov terms. For the sake of compactness of notation we use φ to denote φ(ỹ(x)).

V 0 (x) = x φ(ỹ(x
One straightforward way to enforce the positivity of V (x) is to impose P > 0 and λ i ≥ 0. The lemma below, instead, gives conditions for V to be positive definite without imposing positive-definiteness of P , nor the non-negativity of the coefficients λ i . The lemma uses only the sector properties of the nonlinearity φ. In [START_REF] Park | A revisited Tsypkin criterion for discrete-time nonlinear Lur'e systems with monotonic sector-restrictions[END_REF][START_REF] Heath | Lyapunov functions for the multivariable Popov criterion with indefinite multipliers[END_REF][START_REF] Aizerman | Absolute stability of regulator systems[END_REF], the relaxation of the non-negativity of the coefficients λ i however in these references, V 0 was considered with P 12 = 0, P 22 = 0.

matrix Λ ∈ D m ≥0 such that Λ ≥ -Λ, (1.12a) V 0 (x) - 1 2 ỹT (x)(∆ -∆) Λỹ(x) > 0, ∀x ∈ X 0 \ {0}, (1.12b) then V (x) > 0, ∀x ∈ X 0 \ {0} ⊂ R n .
Proof. Use (1.12a) to obtain a positive-definite lower bound for (1.11a) as follows. If Assumption 1.1 holds, the mapping ỹ : X 0 → Y 0 is well defined. We can then prove that V (x) is positivedefinite in X 0 by obtaining a positive-definite lower bound as follows 1.3. REGIONAL STABILITY ANALYSIS, REACHABLE SETS AND NONLINEAR GAINS 13

V (x) = V 0 (x) + m i=1 λ i ỹi(x) 0 (φ i (s) -δ i s)ds ≥ V 0 (x) - m i=1 λi ỹi(x) 0 (φ i (s) -δ i s)ds = V 0 (x) -1 2 ỹT (x)(∆ -∆) Λỹ(x) - m i=1 λi ỹi(x) 0 (φ i (s) -δ i s)ds = V 0 (x) -1 2 ỹT (x)(∆ -∆ -∆) Λỹ(x) - m i=1 λi ỹi(x) 0 φ i (s)ds = V 0 (x) -1 2 ỹT (x)(∆ -∆) Λỹ(x) + m i=1 λi ỹi(x) 0 ((δ i s -φ i (s))ds = V 0 (x) - 1 2 ỹT (x)(∆ -∆) Λỹ(x) >0 from (1.12b) + m i=1 λi ỹi(x) 0 (δ i s -φ i (s))ds
≥0 from Λ≥0 and (1.2b).

(1.13)

The following theorem presents conditions for the stability of the origin of Lurie system (1.1) with slope-restricted nonlinearities:

Theorem 1.1
For nonlinearities φ satisfying (1.2) if there exists a matrix P ∈ R (n+m)×(n+m) , matrices Λ ∈ D m , Λ, T j ∈ D m ≥0 , j ∈ {0, . . . , 4}, and a scalar ρ > 0 such that (1.12a) holds,

V 0 (x) - 1 2 ỹT (x)(∆ -∆) Λỹ(x) -s 1 (T 0 , φ, ỹ(x)) > 0 (1.14a) ∀x ∈ R n , φ ∈ R m , - ∇xV ∇ φ V , ẋ φ -Ψ(z, w) -s 1 (T 1 , φ, ỹ(x)) -s 1 (T 2 , φ, y(x, w)) -s 2 (T 3 , φ, ẏ( φ, x, φ, w)) -s 3 (T 4 , φ, φ, ỹ(x), y(x, w)) > 0 (1.14b) ∀x ∈ R n , φ ∈ R m , φ ∈ R m , φ ∈ R m , w ∈ R mw and E(V, ρ) ⊆ X 0 (1.14c)
hold with a) Ψ ≡ 0 and w ≡ 0 (which gives φ = φ so that s 3 ≡ 0 and allows us to set

T 2 = 0); b) Ψ(z, w) = w T w; c) Ψ(z, w) = w T w -η -2 z T z;
then a) (stability) the origin of (1.1) is locally asymptotically stable and E(V, ρ) is an estimate of its region of attraction. In the case X 0 = R n , the origin is globally asymptotically stable.

b) (reachable set) x(0) = 0 and

w 2 ≤ ρ 1 2 , (x(t), w(t)) ∈ X W 0 , so that x(t) ∈ E • (V, ρ) for all t ≥ 0; c) (local finite L 2 -gain) x(0) = 0 and w 2 ≤ ρ 1 2 , (x(t), w(t)) ∈ X W 0 , imply z 2 < η w 2 ,
that is, the induced L 2 gain from w to z is bounded by η for every input satisfying

w 2 ≤ ρ 1 2 . Proof. If (1.14a) holds, V 0 (x) - 1 2 ỹT (x)(∆ -∆) Λỹ(x) > s 1 (T 0 , φ, ỹ(x))
from Lemma 1.1 and s 1 (T 0 , φ(ỹ), ỹ) ≥ 0 holds for all x ∈ X 0 , thus (1.12b) holds. Following Lemma 1.4 if (1.12a) also holds, then

V (x) ≥ 0, ∀x ∈ X 0 . 14CHAPTER 1. ANALYSIS OF CONTINUOUS-TIME SLOPE-RESTRICTED LURIE SYSTEMS
We use V (x, φ, φ, φ, w) to express the time-derivative of V (x) along the trajectories of (1.1)

V (x, φ, φ, φ, w) = ∇ x V ∇ φV , Ax + Bφ + B w w φ .
From (1.14b) we have

-V (x, φ, φ, φ, w) -Ψ(z, w) > s 1 (T 1 , φ, ỹ(x, φ)) + s 1 (T 2 , φ, y(x, φ, w)) + s 2 (T 3 , φ, ẏ(x, φ, φ, w)) + s 3 (T 4 , φ, φ, ỹ(x, φ), y(x, φ, w)).
If (1.2) holds, the relations in Lemmas 1.1-1.3 give

-V (x, φ, φ, φ, w) -Ψ(z, w) > 0, ∀x ∈ X 0 . (1.15)
Thus if a) Ψ(z, w) ≡ 0, we have that V is negative for all x ∈ X 0 . Since from (1.14c) the time-derivative of V is negative along the trajectories of system (1.1) provided the sector inequalities hold, that is, provided the trajectories belong to the set X 0 which, from (1.14c) contains the set E(V, ρ). Following [107, Theorem 4.1], with (1.14a) and (1.14b) that hold in the sublevel set, E(V, ρ) is an invariant and contractive set and hence provides an estimate of the region of attraction of (1.1).

b) Ψ(z, w) = -w T w, x 0 = 0, integrate (1.15) from 0 to t * to obtain

t * 0 w T (τ )w(τ )dτ > V (t * ) since V (0) = 0. Hence, provided w 2 2 = t * 0 w T (τ )w(τ )dτ ≤ ρ we have that x(t * ) ∈ E • (V (x), ρ). From (1.14c) the sector inequalities hold so (1.14a) and (1.14b) hold. c) Ψ(z, w) = -w T w + η -2 z T z and x 0 = 0, integrate from 0 to t * to obtain t * 0 w T (τ )w(τ )dτ > t * 0 η -2 z T (τ )z(τ )dτ + V (x(t * )). Since V (x(t * )) ≥ 0, then w 2 2 > η -2 z 2 2 for any t * ∈ [0, ∞). From w 2 ≤ ρ 1 2 and
t * 0 ηz T (τ )z(τ )dτ ≥ 0 the above inequality implies V (x(t * )) < ρ, thus from (1.14c) we have x(t * ) ∈ X 0 for any t * ∈ [0, ∞), hence (1.14a) and (1.14b) hold for w 2 ≤ ρ 1 2 .

Remark 1.1

The use of Lemma 2 in the proof of Theorem 1.1, requires ỹ to be differentiable. From (1.11c) we have dỹ dt = C dx dt +D∂φ(ỹ) dỹ dt , which can be written as (I -D∂φ(ỹ)) dỹ dt = C dx dt . Thus if (I -D∂φ(ỹ)), is non-singular for all ỹ ∈ Y 0 , dỹ dt exists and is given by dỹ dt = (I -D∂φ(ỹ)) -1 C dx dt . From Proposition 1.1 we have that Assumption 1.1 guarantees the invertibility of (I -D∂φ(ỹ)) thus, the existence of dỹ dt .

Note that the set inclusion (1.14c) is required to guarantee that the sector inequalities in Lemmas 1.1-1.3 hold so that (1.14b) implies (1.15). Moreover, from Assumption 1.1 and the fact that (x(t), w(t)) ∈ X W 0 we have y(t) ∈ Y 0 ∀t ≥ 0. The condition on the disturbance (x(t), w(t)) ∈ X W 0 can be dropped in two cases: 1) for D w = 0, we have ỹ ≡ y and (1.14c) implies that y(t) ∈ Y 0 , for all t ≥ 0; 2) for the case Y 0 = R m , the inequalities from Lemmas 1.1-1.3 hold globally so (1.14c) is trivially satisfied.

A convenient property of the quadratic inequalities (1.14a)-(1.14b) is the affine dependence on P , Λ, Λ, T i , i = {0, . . . , 4}. Whenever the inclusion (1.14c) is also formulated in terms of affine inequalities on these variables and the system matrices (A, B, B w , C, D, D w , C z , D z , D zw ) and the sector and slope bounds ∆, ∆, Γ, Γ are given, we can set the problem of computing these variables as a convex semi-definite program. Numerical examples illustrate the solution to these convex semi-definite programs in Section 3.4 and the corresponding linear matrix inequalities (LMIs) are detailed in the Appendix.

Inclusion conditions

To satisfy local properties of (1.1) with Theorem 1.1 we have to guarantee the inclusion (1.14c). For sets of the form

X 0 = x ∈ R n | (ỹ j (x) -ỹj )(ỹ j (x) -ỹj ) ≤ 0, j = 1 . . . m , (1.16)
a condition for the set inclusion is provided by the following lemma.

Lemma 1.5

If there exist scalars α j > 0 such that

-α j (ỹ j (x) -ỹj )(ỹ j (x) -ỹj ) ≥ (ρ -V (x))
(1.17) j = 1, . . . , m then (1.14c) holds.

Proof. If the above inequality holds, then for all x satisfying (ρ-V (x)) ≥ 0 the inequality -(ỹ j (x)ỹj )(ỹ j (x) -ỹj ) ≥ 0 holds and x ∈ X 0 ∀x ∈ E(V, ρ), hence the set inclusion.

For the function V (x) in (1.11), the inequalities (1.17) become

-α j ỹj ỹj -ρ + α j (ỹ j + ỹj )ỹ j (x) -α j ỹ2 j (x) + V 0 (x) + m i=1 λ i ỹi 0 φ i (s) -δ i s ds ≥ 0, (1.18)
j = 1, . . . , m. The reason for expressing nonlinearities in quadratic-like forms is to frame the inclusion condition of Theorem 1.1 as a set of affine matrix inequalities on the unknown coefficients λ i . Whenever only its bounds are given, as in (1.2b), consider λi satisfying λ i ≥ -λi to obtain the following lower bound for the Lurie-Postnikov terms in (1.18) (see (1.13))

m i=1 λ i ỹi 0 φ i (s) -δ i s ds ≥ - 1 2 ỹT (x)(∆ -∆) Λỹ(x) ≥ 0. (1.19)
Finally, provided the inequalities

-α j ỹj ỹj -ρ + α j (ỹ j + ỹj )ỹ j (x) -α j ỹ2 j (x) + V 0 (x) - 1 2 ỹT (x)(∆ -∆) Λỹ(x) ≥ 0, (1.20)
j = 1, . . . , m, hold, we have that (1.18) holds and hence guarantees set inclusion (1.14c). A lower bound on the Lurie-Postnikov terms that guarantee inclusion conditions for sector nonlinearities similar to (1.19), was proposed in [START_REF] Hindi | Analysis of linear systems with saturation using convex optimization[END_REF].

Discussion on the proposed Lyapunov Function

The function (1.11) was introduced in [START_REF] Yakubovich | The method of matrix inequalities in the stability theory of nonlinear control systems: II. Absolute stability in a class of nonlinearities with a condition on the derivative[END_REF] to study single-input single-output (SISO) systems with slope-restricted nonlinearities satisfying γ = -∞ or γ = ∞. The main result in [START_REF] Yakubovich | The method of matrix inequalities in the stability theory of nonlinear control systems: II. Absolute stability in a class of nonlinearities with a condition on the derivative[END_REF] yields a graphical criterion involving the frequency response of the linear part. The same Lyapunov structure was used in [START_REF] Josselson | Absolute stability of control systems with many sector and slope-restricted non-linearities[END_REF] where the extension of the frequency domain criteria of [START_REF] Yakubovich | The method of matrix inequalities in the stability theory of nonlinear control systems: II. Absolute stability in a class of nonlinearities with a condition on the derivative[END_REF] to the MIMO case was proposed. As pointed out above, neither the Lurie-Postnikov coefficient λ nor the corresponding P 22 block (scalar in the SISO case) are required to be positive definite. The use of function V 0 (x), with φ(ỹ) was proposed in [START_REF] Dai | Piecewise-quadratic Lyapunov functions for systems with deadzones or saturations[END_REF] in the context of the analysis of systems with input saturation, where

P11 P12 P T
12 P22 > 0 was used to enforce the positive definiteness of V 0 (x). Convex optimization based approaches using the quadratic-like term V 0 in (1.11) have also been proposed [START_REF] Suykens | An absolute stability criterion for the Lur'e problem with sector and slope restricted nonlinearities[END_REF][START_REF] Park | Stability criteria of sector-and slope-restricted Lur'e systems[END_REF][START_REF] Turner | Lyapunov functions and L 2 gain bounds for systems with slope restricted nonlinearities[END_REF], although none of these references addresses the positivity of the LF as proposed by Lemma 1.4. In [START_REF] Suykens | An absolute stability criterion for the Lur'e problem with sector and slope restricted nonlinearities[END_REF], the positivity of (1.11) it is obtained by imposing P > 0 and Λ > 0 and the slope restriction is addressed by considering a norm-bounded inequality. In [START_REF] Park | Stability criteria of sector-and slope-restricted Lur'e systems[END_REF] and [START_REF] Turner | Lyapunov functions and L 2 gain bounds for systems with slope restricted nonlinearities[END_REF], the slope restriction is studied with the inequality of Lemma 1.2 and the proposed Lyapunov functions contain additional Lurie-Postnikov type terms with non-negative coefficients and impose P ≥ 0 ( P > 0 in [START_REF] Turner | Lyapunov functions and L 2 gain bounds for systems with slope restricted nonlinearities[END_REF]). The remark below shows that the additional terms on these references can be recast in the form (1.11) where the block P 22 is not sign-defined. In [START_REF] Park | Stability criteria of sector-and slope-restricted Lur'e systems[END_REF] and [START_REF] Turner | Lyapunov functions and L 2 gain bounds for systems with slope restricted nonlinearities[END_REF], Lyapunov function structures containing the term V 0 (x) as in (1.11b) were studied for the stability and induced L 2 gain analysis for system (1.1). When compared to (1.11a) the structures in [START_REF] Park | Stability criteria of sector-and slope-restricted Lur'e systems[END_REF] and [START_REF] Turner | Lyapunov functions and L 2 gain bounds for systems with slope restricted nonlinearities[END_REF] use additional integral terms. It is shown in [START_REF] Turner | Lyapunov functions and L 2 gain bounds for systems with slope restricted nonlinearities[END_REF] that some of the additional Lurie-Postnikov terms in [START_REF] Park | Stability criteria of sector-and slope-restricted Lur'e systems[END_REF] were redundant. We now discuss how (1.11a) compares with the LF of [START_REF] Turner | Lyapunov functions and L 2 gain bounds for systems with slope restricted nonlinearities[END_REF], which can be written as

V (x) = x φ T P x φ + 4 j=1 m i=1 µ j,i ỹi(x) 0 ḡj,i (s)ds (1.21) where ḡ1,i (s) = φ i (s), ḡ2,i (s) = δ i s -φ i (s), ḡ3,i (s) = (γ i -∂φ i (s)) s, ḡ4,i (s) = ∂φ i (s) δ i s -φ i (s) ,
P > 0, and µ j,i ≥ 0, i = 1, . . . , m, j = 1, . . . , 4. For φ satisfying (1.2) with δ i = γ i = 0, i = 1, . . . , m we clearly have g j,i (x) ≥ 0, j = 1, . . . , 4, i = 1, . . . , m. By using the relations ỹi 0

φ i (s)∂φ i (s)ds = 1 2 φ 2 i (ỹ i ) ỹi 0 ∂φ i (s)sds = φ i (ỹ i )ỹ i + ỹi 0 φ i (s)ds, it is straightforward to obtain 4 j=1 m i=1 g j,i (x) = x φ(ỹ(x)) T M x φ(ỹ(x)) + m i=1 (µ 1,i -µ 2,i +µ 3,i -δ i µ 4,i ) ỹi(x) 0 φ i (s)ds with M = C T 0 D T I ∆M2+ΓM3 1 2 (∆M4-M3) 1 2 (∆M4-M3) -1 2 M4 [ C D 0 I ]
where M j = diag(µ j,1 , . . . , µ j,m ), j = 1, . . . , 4. Thus (1.11a) is obtained from (1.21) by setting P = P + M and λ i = (µ 1,i -µ 2,i + µ 3,i -δ i µ 4,i ). Note that the matrix P + M is not necessarily positive definite since its lower, right diagonal block, P22 -1 2 M 4 , may not be positive definite. Note also that the Lurie-Postnikov term coefficients μi := (µ 1,i -µ 2,i + µ 3,i -δ i µ 4,i ) can also be negative since µ j,i ≥ 0 does not imply μi ≥ 0.

For the specific case of saturation or deadzone nonlinearities, the integral terms can be incorporated to the quadratic-like term V 0 . This fact has been observed in [START_REF] Dai | Piecewise-quadratic Lyapunov functions for systems with deadzones or saturations[END_REF]. In [START_REF] Garulli | Global stability and finite L 2mgain of saturated uncertain systems via piecewise polynomial Lyapunov functions[END_REF], the slope restriction of the deadzone is accounted for (see [START_REF] Garulli | Global stability and finite L 2mgain of saturated uncertain systems via piecewise polynomial Lyapunov functions[END_REF]Fact 2]). In both [START_REF] Dai | Piecewise-quadratic Lyapunov functions for systems with deadzones or saturations[END_REF] and [START_REF] Garulli | Global stability and finite L 2mgain of saturated uncertain systems via piecewise polynomial Lyapunov functions[END_REF], the positive definiteness of V 0 (x) is obtained by imposing P > 0.

LMIs from Theorem 1.1

The quadratic inequalities in Theorem 1.1 and the inequality (1.20), which is a sufficient condition for (1.14c), are equivalent to linear matrix inequalities (1.22) below, where a generic matrix M Ψ is introduced to represent terms Ψ(w, z) as Ψ(w, z) = ξ T M Ψ ξ with ξ = x φ φ φ w . The inequality (1.12a) appears in (1.22a), inequalities (1.14a) and (1.14b) correspond respectively to (1.22b) and (1.22c), and (1.20) corresponds to (1.22d).

Whenever the nonlinearity is known and the Lurie term is expressed as a quadratic form, ad hoc inequalities may replace (1.22d). Λ ≥ 0, Λ ≥ -Λ, T i ≥ 0, i = 0, . . . , 4, T c,j ≥ 0, j = 0, . . . , m, (1.22a)

P 11 P 12 P T 12 P 22 - 1 2
C T D T (∆ -∆) Λ C D + He 1 2 (∆C) T (∆D T -I m ) T T 0 ∆C (∆D T -I m ) > 0, (1.22b) -He                   P 11 P 12 P T 12 P 22 0 0 0 0 0 0       + 1 2       -(∆C) T (I -∆D) T 0 0 0       ΛC ΛD       A 0 0 B B w 0 0 I m 0 0 + 1 2       (∆C) T (∆D -I m ) T 0 0 0       T 1 ∆C (∆D -I m ) 0 0 0 + 1 2       (∆C) T 0 0 (∆D -I m ) T (∆D w ) T       T 2 ∆C 0 0 (∆D -I m ) ∆D w + 1 2       A T 0 0 0 0 I m B T 0 B T w 0       (ΓC) T (ΓD -I m ) T T 3 ΓC (ΓD -I m ) A 0 0 B B w 0 0 I m 0 0 + 1 2       0 (I -ΓD) 0 -(I -ΓD) ΓD w       T 4 0 (I -ΓD) 0 -(I -ΓD) ΓD w + M Ψ       > 0, (1.22c)      -(ỹ j ỹj + ρ) ỹj +ỹ j 2 C j ỹj +ỹ j 2 D j ỹj +ỹ j 2 C T j -C T j C j + P 11 -1 2 C T (∆ -∆) ΛC -C T j D j + P 12 -1 2 C T (∆ -∆) ΛD ỹj +ỹ j 2 D T j -D T j C j + P T 12 -1 2 D T (∆ -∆) ΛC -D T j D j + P 22 -1 2 D T (∆ -∆) ΛD      + He   1 2   0 (∆C) T (∆D T -I m ) T   T c,j 0 ∆C (∆D T -I m )   ≥ 0 j = 1, . . . , m. (1.22d)

Numerical Examples

In this section we present numerical solutions for the inequalities presented in Theorem 1.1. The computation of the stability certificates, reachable sets and local induced L 2 -gains are based on the solution to the SDPs obtained from the inequalities of Theorem 1.1. The associated constraints to the SDP we solve are detailed in the end of this section. For nonlinearities that yield sector and slope bounds that hold only locally, we guarantee the set inclusion (1.14c) by solving the inequalities (1.18) for the case where the nonlinearity is known and has an explicit quadratic-like representation, or, if it is only known to satisfy sector bounds we use a lower bound to the integral term and solve (1.20) otherwise.

Optimal sector and slope bounds

This example computes the maximum sector and slope restriction for the SISO system described by

G 1c (s) = 0.2s 2 s 4 + 0.4s 3 + 6s 2 + 0.1s + 1 18CHAPTER 1. ANALYSIS OF CONTINUOUS-TIME SLOPE-RESTRICTED LURIE SYSTEMS
. The sector and slope conditions are defined by a parameter , as δ = 0, δ = , γ = -0.5 , γ = 1.5 . Via a bissection algorithm, we obtain bounds for the parameter such that the global stability of system (1.1) is guaranteed. Table 1.1 gives the results comparing the bounds of V (x) to the bounds obtained with V 0 (x), together with the special cases of V given by V Q := x T P 11 x and 1.1: Maximum bound on parameter , denoted , for global stability of system G 1c (s).

V LP := x T P 11 x + m i=1 λ i yi 0 φ(s)ds . Table
V Q V LP V 0 V 0.730 1.272 0.730 2.422

Local Stability

When the nonlinearity that satisfies the sector condition is known, in some cases it is possible to explicitly write the Lurie-Postnikov term in a quadratic-like form. As an example, consider the nonlinearities ln(1 + ỹi ) and ỹi 1+ỹi ỹi

0 ln(1 + s) -δ i s ds = ln(1 + ỹi )(1 + ỹi ) -ỹi -1 2 δ i ỹ2 i ỹi 0 s 1+s -δ i s ds = -ln(1 + ỹi ) + ỹi -1 2 δ i ỹ2 i , (1.23)
which can be expressed as quadratic-like forms in the vector [1 ỹi ln(1 + ỹi )] . These nonlinearities present sector and slope bounds that hold only in the interval ỹj , ỹj as detailed in the note that for both ln(1 + ỹj ) and ỹj 1+ỹj , (1.2) holds with Y = (-1, ∞) thus Y 0 = ỹj , ỹj is defined with -1 < ỹj < 0 and 0 < ỹj . These bounds are used in the system below.

Consider the system

       ẋ1 = -x 2 + ln(1 + y 1 ) + 2 y2 1+y2 ẋ2 = x 1 -0.65x 2 + ln(1 + y 1 ) + y2 1+y2 y 1 = 0.1(x 1 + x 2 ) -0.2 y2 1+y2 y 2 = 0.1(x 2 -x 1 ).
This system can be readily put in the form (1.1) with φ 1 (y 1 ) = ln(1 + y 1 ), φ 2 (y 2 ) = y2 1+y2 . In order to compute a region of attraction of its origin, we fix the interval of interest y 1 ∈ [-. [START_REF] Aizerman | On a problem concerning the stability "in the large" of dynamical systems[END_REF][START_REF] Desoer | Feedback Systems: Input-Output Properties[END_REF], y 2 ∈ [-. [START_REF] Aizerman | Absolute stability of regulator systems[END_REF][START_REF] Desoer | Feedback Systems: Input-Output Properties[END_REF] thus defining the slope and sector bounds for the nonlinearities according to Table 1.2. We obtain the inclusion inequality (1.18) by explicitly computing the Lurie-Postnikov terms as in (1.23) and fixing ρ = 1. We then obtain an ERA by solving the convex optimization problem that minimizes T race(P 11 ) subject to (1.22a)-(1.22c), (1.18). The level sets obtained are depicted in Figure 1.1. Inner level sets of the LF are also depicted and show that incorporating the Lurie-Postnikov terms and the nonlinearities in V 0 may yield an asymmetric ERA with respect to the origin. Note also that the innermost level set resembles an ellipsoid, showing that close to the equilibrium point, the term x T P 11 x dominates the non-quadratic terms of the LF.

Gain Curves

This example computes upper bounds for the local induced L 2 gain η of an idealised Stirling engine presented in [80, eq. 3] with damping factor c = 50 and the nonlinearity φ(y) = y/(1 + y)

ẋ1 = x 2 -cx 1 -cw ẋ2 = - x 1 1 + x 1 y = x 1 z = x 1 .
The induced gains depend upon both the local domain and the magnitude of the disturbance whose norm is upper bounded by w 2 ≤ ρ 1 2 . For this example, the upper bound on the domain is set as ỹ = 0.5 and η is computed for each {ỹ, ρ} = {1, 2, 5, 6, 8} × 10 {-2, -1} . Figure 1.2 shows minimal upper bounds for η searched over the values of ỹ for fixed ρ. The bounds were computed using V (x) subject to (1.22a)-(1.22c), (1.18) and a local Popov criterion obtained using V LP (x) and the substitution of a lower bound for the LF given by V Q into (1.17), a similar method used in [START_REF] Hindi | Analysis of linear systems with saturation using convex optimization[END_REF]. Tighter bounds were obtained using V for all values.

Discussion: Sector bounds for global stability

As pointed out in Remark 1.2, a single Lurie-Postnikov term may replace the four non-negative Lurie-Postnikov terms associated to each input in the Lyapunov function studied in [START_REF] Turner | Lyapunov functions and L 2 gain bounds for systems with slope restricted nonlinearities[END_REF]Theorem 5]. However, in this chapter, these terms and the matrix P 22 are not necessarily non-negative. On the other hand, Theorem 1.1 offers a simpler expression for the same Lyapunov function, therefore no improvement over the existing bounds should be expected. Indeed, we have performed the global stability and gain computations for the examples in [START_REF] Carrasco | LMI searches for anticausal and noncausal rational Zames-Falb multipliers[END_REF] and [START_REF] Turner | Lyapunov functions and L 2 gain bounds for systems with slope restricted nonlinearities[END_REF] to illustrate the fact that the global analysis using the presented results yield the same results as the ones obtained with a more complex Lyapunov function. Indeed, the conditions of Theorem 1.1 matched the stability bounds obtained with the results of [START_REF] Park | Stability criteria of sector-and slope-restricted Lur'e systems[END_REF] for the balanced realization of all transfer functions in 20CHAPTER 1. ANALYSIS OF CONTINUOUS-TIME SLOPE-RESTRICTED LURIE SYSTEMS [START_REF] Carrasco | LMI searches for anticausal and noncausal rational Zames-Falb multipliers[END_REF]Table 3]. Similarly, the solution to the inequalities of Theorem 1.1 give the same L 2 gain bounds as the ones in [START_REF] Turner | Lyapunov functions and L 2 gain bounds for systems with slope restricted nonlinearities[END_REF]Theorem 5] for the systems defined in of [START_REF] Turner | Lyapunov functions and L 2 gain bounds for systems with slope restricted nonlinearities[END_REF]Table 2].

Conclusions

In this chapter, the stability analysis of Lurie type systems with slope-restricted nonlinearities was carried out using LFs that have a quadratic-like term on the state and the nonlinearity and Lurie-Postnikov type terms. We have proposed relaxed conditions for the positivity of the LF (cf. Lemma 1.4) and have used sector inequalities to propose conditions for the global and local properties of solutions to Lurie systems. Importantly, the LF structure allows for negative coefficients in the Lurie-Postnikov term.

Numerical solutions to the dissipation inequalities of the main result (cf. Theorem 1.1) can be obtained with the solutions to SDPs. The proposed numerical formulation is a convex optimisation problem since the SDP constraints are affine both on the Lyapunov/storage function coefficients and the multipliers associated to sector inequalities. The local stability analysis with the computation of ERAs and local gain analysis are illustrated with numerical examples.

Chapter 2

Analysis of Discrete-time Slope-Restricted Lurie Systems

Introduction

Consider a single-input single-output discrete-time Lurie system described by the feedback interconnection of a strictly proper, linear system, with transfer function G(z), and a function φ : R → R, as illustrated in Figure 2 Here, G(z) is assumed to admit a minimal state-space realisation

x[k + 1] = Ax[k] + Bφ(y[k]), (2.1a
)

y[k] = Cx[k], (2.1b)
with A ∈ R n×n , B ∈ R n×1 and C ∈ R 1×n . The nonlinearity φ is assumed to be static, φ(0) = 0 and sector bounded with sector [δ, δ]

(δσ -φ(σ))(φ(σ) -δσ) ≥ 0, ∀σ ∈ R, (2.2a)
as in φ(σ)/σ ∈ [δ, δ]. If δ > 0 the sector is said to be strict.

We say the nonlinearity is slope restricted if

φ(σ 1 ) -φ(σ 2 ) σ 1 -σ 2 ∈ [γ, γ], ∀σ 1 , σ 2 ∈ R (2.2b)
for some γ, γ ∈ R and monotonic if γ ≥ 0. Note that monotonicity can always be obtained by loop transformations whenever the nonlinearity is slope-bounded.

Generalised quadratic and integral Lyapunov functions

A class of Lyapunov function for discrete-time Lurie systems is presented. We use the LF in Theorem 2.1, the main result of the chapter, to study the stability of systems with monotone nonlinearities. Importantly, the parameters of the considered function, namely a matrix in a generalized quadratic form and the scaling terms in integrals, need not be positive definite.

Proposed Lyapunov Function

Given a function φ : R → R, define the vector function ν : R n → R N +1 , χ → ν(χ), with ν j denoting the j th element of ν as

ν j (χ) = Cχ, j = 0, CA j χ + j i=1 CA j-i Bφ(ν i-1 ), 1 ≤ j ≤ N. (2.3)
From the above definition, we have that ν j (χ) is the j th -step propagation of y in (2.1) from x[0] = χ. Let us also introduce the vector function ξ

N : R n → R n+N +1 ξ N (χ) =      χ φ(ν 0 (χ)) . . . φ(ν N (χ))      . (2.4)
For a fixed N ∈ N ∪ {0}, we can thus define the function

V : R n → R, x → V (x) V (x) = V 0 (x) + N j=0 λ j νj (x) 0 φ(s) ds (2.5) with V 0 (x) = ξ N (x) P ξ N (x), λ ∈ R N +1
, and P ∈ S n+N +1 . In the rest of the chapter, we may omit the dependence on x of ξ N and ν j to simplify the notation and may also avoid denoting explicitly the dependence of V (x) on N . The above function is composed of a generalised quadratic term V 0 (x) and the integral terms. It will be used as a Lyapunov candidate function to show stability of (2.1). Namely, we will formulate inequalities imposing conditions on its parameters P and λ for the stability of the origin of (2.1).

Discussion on the Proposed Function

We now discuss the structure of the above function and its connection to the Popov-type stability criteria for both continuous-and discrete-time Lurie systems.

Firstly, it is noted how the function generalizes the Tsypkin Lyapunov function for discrete-time systems

V Tsyp (x) = x P 0 x + η Cx 0 φ(σ) dσ (2.6)
which is obtained setting N = 0. The above V Tsyp first appeared in [START_REF] Szegö | On the absolute stability of sampled-data control systems[END_REF] with P 0 ∈ S n and parameter η ∈ R (not sign-defined).

Continuous-time systems

It can also be observed that (2.6) also appears from the derivation of Popov-type stability criteria of continuous-time Lurie systems using passivity. Roughly speaking, this stability criterion considers the loop-transformed Lurie system of Figure 2.2. It follows from passivity (under some additional assumptions, see [START_REF] Khalil | Nonlinear systems[END_REF]Chapter 6] for details) that the stability of this system can be inferred as long as both the upper linear subsystem and the lower nonlinear subsystem can be shown to be passive. Since the upper subsystem is linear, it admits a quadratic storage function, but the passivity of the nonlinear subsystem requires exploiting the sector bounds. Indeed, with the Popov multiplier M (s) = (1 + ηs), in the linear branch, passivity for the nonlinear branch can be shown provided there exists a storage function S : R → R satisfying dS(t) dt ≤ φ(y(t)) y(t) + η dy(t) dt .

Since the nonlinearity φ lies within the sector [0, δ], we have φ(y)y ≥ 0.

y(0) φ(σ) dσ = η y(T ) 0 φ(σ) dσ -η y(0) 0 φ(σ) dσ. (2.7)
Thus, by identifying terms in the above equation, we can use the function S(y) = η y 0 φ(σ) dσ as a storage function for the nonlinear branch of the loop-transformed system, and, since y = Cx, S can be expressed as the mapping x → S(Cx). The sum of a quadratic function as the storage function certifying the passivity of the linear subsystem and the integral term as the storage function for the nonlinear subsystem gives V Tysp as a candidate Lyapunov function for the continuous-time Lurie system.

Discrete-time systems

The following details a similar passivity-based analysis on the use of the function but for discretetime Lurie systems. For these systems, with an equivalent feedback structure to Figure 2.2, it is usual to substitute the continuous-time multiplier multiplier

(1 + ηs) by a discrete-time Popov multiplier M (z) = (1 + η(1 -z -1
)) [START_REF] Kapila | A multivariable extension of the Tsypkin criterion using a Lyapunov-function approach[END_REF][START_REF] Park | A revisited Tsypkin criterion for discrete-time nonlinear Lur'e systems with monotonic sector-restrictions[END_REF]. By replacing the structure of this multiplier by M (z) = 1 + η N i=0 c i z i for some real coefficients c i , the passivity of the nonlinear subsystem can be shown if we can find a function S : R → R satisfying

S(y[k]) -S(y[k -1]) = η N i=0 c i y[k + i] φ(y[k]),
or, in its summation form,

S(y[K]) -S(y[0]) = η K k=1 N i=0 c i y[k + i] φ(y k ), = -η ∞ k=K+1 N i=0 c i y[k + i] φ(y k ) + η ∞ k=0+1 N i=0 c i y[k + i] φ(y k ).

By identifying terms, the infinite sum S(y[

]) = -η ∞ k= +1 N i=0 c i y[k + i] φ(y k+1
) can be associated to a storage function. However, note that the above is in constrast with (2.7) since to compute the values of S for a given time instant, that is S(y[ ]), the signal y from the solution of the system must be known. For the continuous-time case, the integral in (2.7) allows for the dependence on time of the output signal y to be dropped.

It follows that adding a quadratic storage function S lin (x) = x P 0 x, P 0 ∈ S n 0 for the linear subsystem to the above expression (below we also replace y in the above sum by ν since we have ν j = y[j] from (2.3)), then corresponding Lyapunov function structure should be

V Pop (x) := x T P 0 x -η ∞ k=1 N i=0 c i ν k+i (x) φ(ν k ) (2.8)
which contains a sum, not an integral as in (2.6). By rearranging terms in the double sum and regrouping the terms ν k+i into a single index j = k + i, we obtain scalars cj , such that

∞ k=1 N i=0 c i ν k+i (x) φ(ν k ) = N j=1 cj ν j φ(ν j ) + ∞ j=N +1 cj ν j φ(ν j ).
The first term above can be written as a quadratic form in ξ N (x) namely

N j=1 cj ν j φ(ν j ) = ξ N (x) P T ξ N (x), with P T ∈ S n+N +1
. Hence, adding this first term to the quadratic function x P 0 x, we obtain a term as in V 0 (x) of (2.5). We are left with the sum ∞ j=N +1 cj ν j φ(ν j ). In case this remaining term is bounded, we can then consider the integrals of (2.5) as approximations of this infinite sum. Thus (2.5) gives an approximation of the function in (2.8).

In the above discussion, the storage function showing the passivity of the nonlinear branch of the continuous-time system was obtained using only the sector information. Unfortunately, when the sums are replaced by the integral terms for the discrete-time problem, it is no longer possible to carry out the stability analysis considering only sector information since theaw integrals need to be bounded by quadratic terms using slope information as in [START_REF] Tsypkin | On the stability in the large of nonlinear sampled-data systems[END_REF][START_REF] Kapila | A multivariable extension of the Tsypkin criterion using a Lyapunov-function approach[END_REF][START_REF] Park | A revisited Tsypkin criterion for discrete-time nonlinear Lur'e systems with monotonic sector-restrictions[END_REF]. We thus have to also assume monotonicity of the nonlinearity hereafter. We will use Lemma 2.1 below to bound integrals.

Remark 2.1

We now show that, V (x) in (2.5) includes the recently developed Lyapunov function of [START_REF] Park | A less conservative stability criterion for discrete-time Lur'e systems with sector and slope restrictions[END_REF] as a special case when N = 2. To show this, consider the Lyapunov function V (x) from [START_REF] Park | A less conservative stability criterion for discrete-time Lur'e systems with sector and slope restrictions[END_REF] which can be expressed as

V (x) = V 1 (x) + V 2 (x) + V 3 (x) + V 3 (x) (2.9) with parameters P ∈ S 2n+2 >0 , {m 1 , m 2 , n 1 , n 2 , n 3 , n 4 } ∈ R ≥0 and ζ =     x x[k + 1] φ(ν 0 (x)) φ(ν 1 (x))     defining V 1 (x) = ζ P ζ V 2 (x) = 2m 1 ν1(x) ν0(x) φ(σ) -φ(ν 0 (x))dσ + 2m 2 ν1(x) ν0(x) γ(σ -ν 0 (x)) -(φ(σ) -φ(ν 0 (x))dσ, = 2m 1 ν1 0 φ(σ)dσ - ν0 0 φ(σ)dσ -φ(ν 0 )(ν 1 -ν 0 ) + 2m 2 γ 2 (ν 2 1 -2ν 0 ν 1 + ν 2 0 ) - ν1 0 φ(σ)dσ + ν0 0 φ(σ)dσ + φ(ν 0 )(ν 1 -ν 0 ) , V 3 (x) = 2n 1 ν0(x) 0 φ(σ)dσ + 2n 2 ν0(x) 0 δσ -φ(σ) dσ, = 2n 1 ν0 0 φ(σ)dσ + 2n 2 δν 0 (x) 2 2 - ν0 0 φ(σ) dσ , V 3 (x) = 2n 3 ν1(x) 0 φ(σ)dσ + 2n 4 ν1(x) 0 δσ -φ(σ) dσ, = 2n 3 ν1 0 φ(σ)dσ + 2n 4 δν 1 (x) 2 2 - ν1 0 φ(σ) dσ .
Collecting terms in the integrals of V 2 , V 3 , and V 3 above, we have that V (x) can be written as (2.5) with N = 1. Indeed, we obtain

λ 0 = 2(-m 1 +m 2 +n 1 -n 2 ), λ 1 = 2(m 1 -m 2 +n 3 -n 4 ). Finally, noting that ζ = M ξ 1 with M =     0 0 I n 0 0 0 0 A B 0 0 0 0 1 0 0 0 0 0 1     ,
and

ν 0 = 0 0 C 0 0 ξ 1 , ν 1 = 0 0 CA CB 0 ξ 1 , φ(ν 0 ) = 0 1 0 ξ 1 .
We retrieve matrix P in (2.5) as a function of coefficients P , m 1 , m 2 , n 2 and n 4 by identifying terms in the expression below

ξ 1 P ξ 1 = ξ 1 M P M ξ 1 -2m 1 φ(ν 0 )(ν 1 -ν 0 ) + 2m 2 γ 2 (ν 2 1 -2ν 0 ν 1 + ν 2 0 ) + φ(ν 0 )(ν 1 -ν 0 ) + n 2 δν 2 0 + n 4 δν 2 1 .
It is further noted that another stability test was stated in [50, Thm 5(a), Sec 7, Chap VI] which does not require monotonicity of φ, instead only that φ satisfy a strict sector condition with δ > 0. As far as the authors are aware, no equivalent formulation in terms of Lyapunov functions for this result is known.

Stability analysis of Discrete-Time Lurie Systems

Stability conditions using (2.5) as the candidate Lyapunov function are now stated for the discretetime Lurie system (2.1) with a monotonic nonlinearity. We first introduce some quadratic constraints related to the sector-bounded and slope-restricted, monotone, nonlinearities. For a nonlinearity with sector bounds [δ, δ], we define

s sec (σ i ) := δσ i -φ(σ i ) (φ(σ i ) -δσ i ) ≥ 0 (2.11) ∀σ i ∈ R.
The relation below exploits (2.2b) and monotonicity of φ,

s slo (σ i , σ j ) := (γ(σ i -σ j ) -(φ i -φ j )) ((φ i -φ j )) ≥ 0, (2.12
)

∀σ i , σ j ∈ R, with φ i = φ(σ i ) and φ j = φ(σ j ).
The above inequality is obtained, from (2.2b) and monotonicity since we have σi-σj φi-φj ≥ 1 γ , which gives

(σi-σj )(φi-φj ) 2 (φi-φj ) ≥ (φi-φj ) 2 γ
. The inequalities in the lemma below are obtained using the slope restrictions

Lemma 2.1: [131, Lemma 1] If φ is slope restricted (2.2b) with γ ≥ 0 then ∀σ i , σ j ∈ R L(σ j , σ i ) ≤ σi σj φ(σ)dσ ≤ U (σ j , σ i ) (2.13)
where

L(σ j , σ i ) = φ(σ j )(σ i -σ j ) + 1 2γ (φ(σ i ) -φ(σ j )) 2 , U (σ j , σ i ) = φ(σ i )(σ i -σ j ) - 1 2γ (φ(σ i ) -φ(σ j )) 2 .
The bounds (2.13) give the inequality below

λ p L(σ j , σ i ) -λ n U (σ j , σ i ) ≤ (λ p -λ n ) σi σj φ(s) ds ≤ λ p U (σ j , σ i ) -λ n L(σ j , σ i ) (2.15)
that will be used to upper and lower bound the integral terms in the Lyapunov inequalities.

Global Stability Analysis

With the inequalities (2.11), (2.12), and (2.15) in hand, global stability conditions with the Lyapunov function V (x) can be formulated.

Theorem 2.1: Global Stability Analysis Discrete-Time Systems Consider the Lurie system of (2.1) with the nonlinearity φ both sector bounded (2.2a) and slope restricted (2.2b) with γ ≥ 0. If there exist

P ∈ S n+N +1 , {λ p , λ n } ∈ R N +1 ≥0 , {τ sec , ψ sec , θ sec } ∈ R N +1 ≥0 , {τ slo , ψ slo , θ slo } ∈ U N +1 ≥0 , 2 ≥ 1 > 0, and 3 > 0 such that, ∀x ∈ R n \ {0} 1 x 2 ≤ V (x), (2.16a) V (x) ≤ 2 x 2 , (2.16b) ∆V (x) ≤ -3 x 2 , (2.16c) with V (x) =V 0 (ξ N ) + N j=0 λ p j L(0, ν j ) -λ n j U (0, ν j ) - N j=-N τ sec j s sec (ν j ) - N -1 j=-N N g=j+1 τ slo j,g s slo (ν j , ν g ), (2.17a)
V (x) =V 0 (ξ N ) + N j=0 λ p j U (0, ν j ) -λ n j L(0, ν j ) + N j=-N ψ sec j s sec (ν j ) + N -1 j=-N N g=j+1 ψ slo j,g s slo (ν j , ν g ), (2.17b) ∆V (x) =∆V 0 (ξ N ) + N j=0 λ p j U (ν j , ν j+1 ) -λ n j L(ν j , ν j+1 ) + N +1 j=-N θ sec j s sec (ν j ) + N j=-N N +1 g=j+1 θ slo j,g s slo (ν j , ν g ), (2.17c)
then (2.1) is globally exponentially stable and

x[k] ∈ E(V, V (x[0])) with V as in (2.5) with λ = λ p -λ n .
Proof. From the quadratic bounds (2.11), (2.12) and the lower bound to the integral term in (2.15), it follows that V (x) ≤ V (x)∀x, with V defined by λ = λ p -λ n in (2.5). Thus (2.16a) implies

1 x ≤ V (x). Similarly, we show that V (x) is an upper bound to V (x) thus (2.16b) implies V (x) ≤ 2 x , hence 1 x 2 ≤ V (x) ≤ 2 x 2 . (2.18a) We have ∆V (x[k]) = V 0 (ξ N (x[k + 1])) -V 0 (ξ N (x[k])) + N i=0 λ i νi+1 νi φ(s)ds.
Using the inequalities (2.11), (2.12) and both bounds to the integral terms in (2.15) we conclude that ∆V (x) ≤ ∆V (x), hence the satisfaction of ∆V (x)

< -2 x 2 implies ∆V (x) < -2 x 2 . (2.18b)
To conclude, if the conditions of the theorem are satisfied, we use (2.18) to obtain

x[k] 2 ≤ 2 1 1 -3 2 k x[0] 2 . Remark 2.2
Since the inequalities that need to be checked in Theorem 2.1 are quadratic expressions, they can be cast as linear matrix inequalities (LMIs). They are obtained by expressing the terms in (2.17a), (2.17b) as quadratic forms in the vector ξ N , and in (2.17c) as quadratic forms in the vector ξ N +1 . These LMIs are solved to obtain the numerical results reported in Section 2.4. Due to space limitations the matrices corresponding to the LMIs are not reported here.

Remark 2.3

A key feature of Proposition 2.1 is related to the set of parameters defining V in (2.5). Note that the positivity of the elements of λ in V can be relaxed in Proposition 2.1, since they are given by λ = λ p -λ n where λ p and λ n are non-negative vectors. The sign definiteness of the matrix P can also be relaxed. These relaxations parallel the results by the authors for continuous-time Lurie systems [START_REF] Valmorbida | Regional analysis of slope-restricted Lurie systems[END_REF] and those with rational vector fields [START_REF] Drummond | Generalized absolute stability using Lyapunov functions with relaxed positivity conditions[END_REF] where the positivity of the LF parameters have also been relaxed.

Regional Stability Analysis

In many cases, a regional stability analysis is often desired since global stability may not be achieved for nonlinear systems. This is the case when the domain of the nonlinearity is not R (e.g. with φ(σ) = ln(1 + σ)) or when the region region of attraction of the origin is some set R 0 ⊂ R n with 0 ∈ R 0 . The previous section demonstrated how the function (2.5) could be used for a global stability analysis of a Lurie system, corresponding to a region of attraction of the origin given by R n . This section follows a similar approach to [START_REF] Valmorbida | Regional analysis of slope-restricted Lurie systems[END_REF] where the regional stability for continuous-time systems was studied, and provides conditions to obtain estimates of the region of attraction of the origin using (2.5). Such estimates will be invariant sets given by level sets of the Lyapunov function. We guarantee the inclusion of these level sets within a subset of the state space where (local) sector and slopes bounds for the nonlinearities hold. We characterize these sets by considering scalars y ≤ 0, y ≥ 0 that define

X 0 = {x ∈ R n : (y -ν 0 (x))(ν 0 (x) -y) ≥ 0},
that is, ∀x ∈ X 0 , ν 0 (x) ∈ y, y . The values y, y give the interval of the domain of the nonlinearity φ in (2.1), where sector and slope bounds will be assumed to hold.

We thus assume that ∀σ i ∈ y, y , we have

s sec,loc (σ i ) := δ loc (y, y)σ i -φ(σ i ) φ(σ i ) -δ loc (y, y)σ i ≥ 0 (2.19)
with δ ≤ δ loc (y, y) ≤ δ loc (y, y) ≤ δ, with the global sector bounds as in (2.2a) satisfied with δ and δ. Similarly, we assume ∀σ i , σ j ∈ y, y

s slo,loc (σ i , σ j ) := γ loc (y, y)(σ i -σ j ) -(φ i -φ j ) ((φ i -φ j ) -γ loc (y, y)(σ i -σ j )) ≥ 0, (2.20)
with 0 ≤ γ loc (y, y) ≤ γ loc (y, y) ≤ γ, with the global sector bounds as in (2.2b) satisfied with γ = 0 and γ. The use of sector and slope bounds depending on y and y can help reduce the conservatism in the estimates of the region of attraction obtained, as the numerical examples below will illustrate.

To guarantee that the above sector and slope inequalities are verified for all trajectories starting in an invariant level set of (2.5), we should establish a condition for the inclusion of a level set of (2.5) in a given set X 0 ⊂ R n . The lemma below provides this inclusion condition.

Lemma 2.2: [131, Lemma 1]

Given a function W : R n → R, if there exists a scalar α > 0 such that

α(y -ν 0 (x))(ν 0 (x) -y) ≥ ρ -W (x), (2.21)
then the following set inclusion holds

E(W, ρ) ⊆ X 0 .
Proof. If (2.21) holds, then whenever W (x) ≤ ρ, we have that the inequality (y-ν 0 (x))(ν 0 (x)-y) ≥ 0 is satisfied. Hence, x ∈ E(W, ρ) ⊆ X 0 , giving the set inclusion.

The inclusion condition of Lemma 2.2 and the stability conditions of Theorem 2.1 are combined in the following regional stability analysis result.

Theorem 2.2: Regional Stability Analysis Discrete-Time Systems Consider the Lurie system of (2.1) with the nonlinearity φ both sector bounded (2.2a) and slope restricted (2.2b) with γ ≥ 0 and bounds on the nonlinearity y, y defining the set X 0 and sector and slope bounds as δ loc (y, y), δ loc (y, y), γ loc (y, y), γ loc (y, y)

in (2.19)-(2.20). If there exist P ∈ S n+N +1 , {λ p , λ n } ∈ R N +1 ≥0 , {τ sec , ψ sec , θ sec } ∈ R N +1 ≥0 , {τ slo , ψ slo , θ slo } ∈ U N +1 ≥0
and 2 ≥ 1 > 0, and 3 > 0 such that, ∀x ∈ R n \ {0}, the inequalities in (2.16) hold with

V (x) =V 0 (ξ N ) + N j=0 λ p j L(0, ν j ) -λ n j U (0, ν j )- N j=-N τ sec j s sec,loc (ν j )- N -1 j=-N N g=j+1 τ slo j,g s slo,loc (ν j , ν g ), (2.22a) V (x) =V 0 (ξ N ) + N j=0 λ p j U (0, ν j ) -λ n j L(0, ν j )+ N j=-N ψ sec j s sec,loc (ν j )+ N -1 j=-N N g=j+1 ψ slo j,g s slo,loc (ν j , ν g ), (2.22b) ∆V (x) =∆V 0 (ξ N ) + N j=0 λ p j U (ν j , ν j+1 ) -λ n j L(ν j , ν j+1 ) + N +1 j=-N θ sec j s sec,loc (ν j ) + N j=-N N +1 g=j+1 θ slo j,g s slo,loc (ν j , ν g ), (2.22c)
and a scalar α > 0 such that

α(y -ν 0 (x))(ν 0 (x) -y) ≥ ρ -V (x), ∀x ∈ R n \ {0} (2.23)
holds, then all solutions of (2.1) satisfying x[0] ∈ E(V, ρ) ⊆ X 0 , with V defined by P and λ = λ p -λ n , also satisfy E(V (x[k]), ρ) ⊆ X 0 for all k ∈ N. Moreover, the origin of (2.1) is (locally) exponentially stable.

Proof. Since we have

V (x) ≥ V (x), then V (x) ≤ ρ implies V (x) ≤ ρ hence E(V, ρ) ⊆ E(V , ρ). Following Lemma 2.2, if (2.23) holds, we conclude that E(V , ρ) ⊆ X 0 thus implying E(V, ρ) ⊆ X 0 . Note also that if (2.16) holds, we have V (x) > 0, giving V (x[0]) > 0 and ∆V (x) < 0 gives ∆V (x[k]) < 0 ∀k ∈ N. Thus, for V (x[0]) ≤ ρ we get 0 < V (x[k]) ≤ ρ ∀k ∈ N. Since V (x) ≤ V (x), we also have that V (x[k]) ≤ ρ ∀k ∈ N. Hence, the set E(V, ρ
) is invariant and is contained in the set where ∆V (x) is strictly negative with an upper quadratic bound as in (2.16c). Exponential stability of the origin within the set E(V, ρ) can then be concluded.

Remark 2.4

The following particular case of the function (2.5) has been proposed in [START_REF] Gonzaga | Stability analysis of discrete-time Lur'e systems[END_REF] V GJD (x) = x T P x + λν 0 φ(ν 0 ) for the regional analysis of Lurie systems. Interestingly, with the above structure the stability analysis can be carried out using only sector bounds. Also, the level sets of the above function can be disconnected therefore yielding disconnected level sets for estimates of the region of attraction. On the other hand, the above function does not appear to be more effective than a simple quadratic function when assessing global stability.

Input-Output Analysis

This section considers the open Lurie system

x[k + 1] = Ax[k] + Bφ(y[k]) + B w w[k],
(2.24a)

y[k] = Cx[k], (2.24b
)

z[k] = C z x[k], (2.24c)
with the input given by an external disturbance w ∈ W ⊆ 2 , and z a performance output signal to be assessed. To evaluate the impact of the input signals w in z, in this section, we propose a strategy to compute gains yielding worst-case bounds of the form z 2 ω w 2 .

To compute the input-output induced gains we use the storage function is given by V as in (2.5). It is important to observe that (2.5) does not depend on w, since ν i , i ∈ {1, N } in (2.3) and the vector ξ N in (2.4) defining the expression V (x) depend only on its argument x and not on w. On the other hand, to analyse the input-output gains of (2.24), the forward difference ∆V

(x[k]) = V (x[k + 1]) -V (x[k]) has to be computed using x[k + 1] as in (2.24). To obtain V (x[k + 1]) we use (2.
3) and (2.24a) to arrive at the expressions

ν j (x[k+1]) = C (Ax[k] + Bφ(Cx[k]) + B w w[k]) j = 0 CA j (Ax[k] + Bφ(Cx[k]) + B w w[k]) + j i=1 CA j-i Bφ(ν i-1 (x[k + 1])), 1 ≤ j ≤ N.
We thus observe that

ν j (x[k + 1]) depends on w[k] which is different from ν j+1 (x[k]). Let us define ν + j (x[k], w[k]) := ν j (x[k + 1]
) that will help avoid expressions with multiple indexing. Using the above expression, we obtain

ν + j (x, w) = C (Ax + Bφ(ν 0 ) + B w w) , j = 0, CA j (Ax + Bφ(ν 0 ) + B w w) + j i=1 CA j-i Bφ(ν + i-1 ), 1 ≤ j ≤ N, (2.25)
and

ξ + N (x[k], w[k]) := ξ N (x[k + 1]), that is ξ + N (x, w) =      x + φ(ν + 0 ) . . . φ(ν + N )      . ( 2
∈ S n+N +1 , {λ p , λ n } ∈ R N +1 ≥0 , {τ sec , ψ sec , θ sec , θ sec+ } ∈ R N +1 ≥0 , {τ slo , ψ slo , θ slo , θ slo+ } ∈ U N +1 ≥0 2 ≥ 1 > 0, and 3 > 0 such that, ∀x ∈ R n \ {0}, (2.16a)-(2.16b) hold with (2.17a)-(2.17b), and ∆V (x) ≤ -(ω -2 )z 2 + w 2 , (2.27)
with ∆V (x) given by

∆V (x) =ξ + N (x) P ξ + N (x) -ξ N (x) P ξ N (x) + N j=0 λ p j U (ν j , ν + j ) -λ n j L(ν j , ν + j ) + N j=1 θ sec j s sec (ν j ) + θ sec+ j s sec (ν + j ) + N -1 j=1 N g=j+1 θ slo j,g s slo (ν j , ν g ) + θ slo+ j,g s slo (ν + j , ν + g ) (2.28) then z 2 ≤ ω w 2 , ∀w ∈ 2 . (2.29) Moreover, for x[0] = 0, we have that V (x[k]) ≤ w 2 2 , ∀k ∈ N. Proof. Since ∆V ≤ ∆V (x), from (2.27) we have ∆V (x[k]) ≤ -(ω -2 )z[k] 2 +w[k] 2 ∀k ∈ N.
Summing this expression from to 0 to k, we obtain,

V (x[k]) -V (x[0]) ≤ -(ω -2 ) k i=0 z 2 + k i=0 w 2 ∀k. Letting k → ∞, we get (ω -2 ) z 2 2 ≤ V (x[0]) + w 2 2 and since the bias term V (x[0]) satis- fies V (x[0]) ≥ 0, we have z 2 ≤ ω w 2 .
That is, the input-output induced 2 gain is bounded by ω.

Moreover, we have that

V (x[k]) + k i=0 (ω -2 ) z[i] 2 ≤ V (x[0]) + k i=0 w[i] 2 , ∀k and since z[i] ≥ 0 ∀i, if x[0] = 0 (thus V (x[0]) = 0), we obtain V (x[k]) ≤ k i=0 w[i] 2 ≤ ∞ i=0 w[i] 2 = w 2 2 , ∀k.
We thus conclude that, if w 2 2 ≤ ρ, that is, if the 2 norm of the input is bounded by a scalar √ ρ,

we have that ∀k ∈ N, x[k] ∈ E(V, ρ).

LMI conditions

This section illustrates how the relations (2.16a) and (2.16c) in Theorem 2.1. can be written in the generic quadratic with an affine dependence on P , λ p , λ n , τ sec , ψ sec , θ sec , τ slo , ψ slo , θ slo . Thanks to these generic quadratic forms, conditions expressed as LMI can be obtained to ensure (2.16a) and (2.16c). This is formalized below with corollary to Theorem 2.1. Similar formulations are obtained for theorems 2.2 and 2.3.

Corollary 2.1

Consider the Lurie system of (2.1) with the nonlinearity φ both sector bounded (2.2a) and slope restricted (2.2b) with γ ≥ 0. If there exist

P ∈ S n+N +1 , {λ p , λ n } ∈ R N +1 ≥0 , {τ sec , ψ sec , θ sec } ∈ R N +1 ≥0 , {τ slo , ψ slo , θ slo } ∈ U N +1 ≥0 , 2 ≥ 1 > 0, and 3 > 0 such that, ∀x ∈ R n \ {0} P -1 I nN + Ω 0 LU (N, λ p , λ n ) -Ω sec (N, τ sec ) -Ω slo (N, τ slo ) ≥ 0, (2.30a) 2 I nN -P -Ω 0 UL (N, λ p , λ n ) -Ω sec (N, ψ sec ) -Ω slo (N, ψ slo ) ≥ 0, (2.30b
)

-( Ω P Ω-diag(P, 0)-3 I n(N +1) +Ω UL (N, λ p , λ n )+Ω sec (N +1, θ sec )+Ω slo (N +1, θ slo )) ≥ 0, (2.30c) where I nN = diag(I n , 0 N,N ), I n(N +1) = diag(I n , 0 N +1,N +1 ) Ω = A B 0 n,N 0 N,n 0 N,1 I N .
and matrices Ω 0 LU , Ω UL , Ω sec , and Ω slo as in in (2.31), then (2.1) is globally exponentially stable and

x[k] ∈ E(V, V (x[0])) with V as in (2.5) with λ = λ p -λ n .
We the expressions Theorem 2.1 as quadratic expressions in vectors ξ N and ξ N +1 . The term involving the sector inequality

N j=0 τ sec j s sec (ν j ) = ξ N Ω sec (N, τ sec )ξ N (2.31a) with Ω sec (N, τ sec ) = 1 2 N j=0 τ sec j He (Ω sec,j ) (2.31b)
where

Ω sec,j =   0 n+j,n+N +1 (δ + δ) CA j CA j-1 B . . . CA 0 B -1 0 1,N +1-j 0 n+(N -j),n+N +1   -δδ             (CA j ) (CA j-1 B) . . . (CA 0 B)      0 N -j,1        CA j CA j-1 B . . . CA 0 B 0 1,N -j .
The term involving the slope inequality

N -1 j=0 N g=j+1 τ slo j,g s slo (ν j , ν g ) = ξ N Ω slo (N, τ slo )ξ N (2.31c)
where

Ω slo (N, τ slo ) = 1 2 N -1 j=0 N g=j+1 τ slo j,g He (Ω slo,j,g ) (2.31d)
with

Ω slo,j,g =     0 n+j,n+N +1 γ CA j CA j-1 B . . . CA 0 B 0 1,N -j 0 n+(N -j),n+N +1   -   0 n+g,n+N +1 γ CA g CA g-1 B . . . CA 0 B 0 1,N -g 0 n+(N -g),n+N +1   -    0 n+j,n+N +1 0 1,n e (N +1) j+1 0 n+(N -j),n+N +1    -    0 n+g,n+N +1 0 1,n e (N +1) g+1 0 n+(N -g),n+N +1              0 n+j,n+N +1 0 1,n e (N +1) j+1 0 n+(N -j),n+N +1    -    0 n+g,n+N +1 0 1,n e (N +1) g+1 0 n+(N -g),n+N +1       .
The bounds obtained using the integral terms, for (2.16a)

N j=0 λ p j L(0, ν j ) -λ n j U (0, ν j ) = ξ N Ω 0 LU (N, λ p , λ n )ξ N (2.31e)
where

Ω 0 LU (N, λ p , λ n ) = 1 2 N j=0 He Ω 0 LU,j (2.31f)
with

Ω 0 LU,j =   0 n+j,n+N +1 -λ n j CA j CA j-1 B . . . CA 0 B 1 2γ (λ p j + λ n j ) 0 1,N -j 0 n+(N -j),n+N +1   ; for (2.16b) N j=0 λ p j U (0, ν j ) -λ n j L(0, ν j ) = ξ N Ω 0 UL (N, λ p , λ n )ξ N (2.31g)
where

Ω 0 UL (N, λ p , λ n ) = 1 2 N j=0
He Ω 0 UL,j

(2.31h)

with

Ω 0 UL,j =   0 n+j,n+N +1 λ p j CA j CA j-1 B . . . CA 0 B -1 2γ (λ p j + λ n j ) 0 1,N -j 0 n+(N -j),n+N +1   ; for (4.18) N j=0 λ p j U (ν j , ν j+1 ) -λ n j L(ν j , ν j+1 ) = ξ N Ω UL (N, λ p , λ n )ξ N (2.31i)
where

Ω UL (N, λ p , λ n ) = 1 2 N j=0 He (Ω UL,j ) (2.31j) with Ω UL,j =     0 n+j,n+N +1 -λ n j C(A j+1 -A j ) C(A j -A j-1 )B . . . C(A -In)B -λ n j CA 0 B -1 2γ (λ p j + λ n j ) 1 γ (λ p j + λ n j ) 0 1,N -j λ p j C(A j+1 -A j ) C(A j -A j-1 )B . . . C(A -In)B CA 0 B -1 2γ (λ p j + λ n j ) 0 1,N -j 0 n+(N -j),n+N +1     .

Numerical examples

The proposed Lyapunov function structure is now evaluated through three numerical examples i) assessing the maximal achievable sector for a global analysis using benchmark LTI systems from the literature ii) computing estimates of the region of attraction, and iii) computing bounds for the worst-case input-output gains. The LMIs corresponding to each stability conditions were solved using YALMIP [START_REF] Löfberg | YALMIP : A toolbox for modeling and optimization in MATLAB[END_REF] and MOSEK [START_REF] Mosek Aps | The MOSEK optimization toolbox for MATLAB manual[END_REF].

Maximum Achievable Sector

We first evaluate the conditions for global stability of system (2.1) using (2.5) on minimal realizations of the seven systems given in Table 2.1 with equal sector and slope bounds, as in δ = γ = 0, and γ = δ. We then look for the maximum value of δ for which stability could be verified. The tests were carried out by using a sequence of increasing values of the integer N .

Table 2.2 compares the maximum achievable δ obtained by solving the inequalities in Theorem 2.1 against other modern methods, including the Zames-Falb multipliers of [START_REF] Carrasco | Convex searches for discretetime Zames-Falb multipliers[END_REF] and [START_REF] Turner | Discrete-time systems with slope restricted nonlinearities: Zames-Falb multiplier analysis using external positivity[END_REF], and the Lyapunov functions of [START_REF] Park | A less conservative stability criterion for discrete-time Lur'e systems with sector and slope restrictions[END_REF] and [START_REF] Ahmad | A less conservative LMI condition for stability of discrete-time systems with slope-restricted nonlinearities[END_REF].

For G 2 (z), G 3 (z) and G 4 (z), the proposed Lyapunov function V (x) provides less conservative sector bounds δ than the Lyapunov function V (x) from [START_REF] Park | A less conservative stability criterion for discrete-time Lur'e systems with sector and slope restrictions[END_REF], as in (2.9). Furthermore, as discussed in Remark 2.1, the sector bounds obtained with V (x) and N = 1 match the ones obtained with V (x). Thus, showing that V could encompass and generalise V . For G 2 (z), G 3 (z) and G 4 (z), extending the horizon length N of V beyond N = 1 led to some conservatism reduction in the achievable sector δ, with the horizon length yielding the maximum achievable sector reported in Table 2.2. Figure 2.3 illustrates the effect of the horizon length N on the achievable sector by showing the maximum achievable sector bound δ * for G 4 (z) as a function of the horizon length N , with a clear increase at N = 3. It must also be said that the Zames-Falb multipliers of [START_REF] Carrasco | Convex searches for discretetime Zames-Falb multipliers[END_REF] and [START_REF] Turner | Discrete-time systems with slope restricted nonlinearities: Zames-Falb multiplier analysis using external positivity[END_REF] could still achieve superior sector bounds for G 4 (z), G 5 (z) and G 7 (z), however, the Lyapunov function approach for stability analysis considered in this chapter still offers advantages. In particular, Lyapunov functions provide a more natural framework to conduct a regional analysis.

Example

Plant

G 1 (z) [2] 0.1z z 2 -1.8z+0.81 G 2 (z) [2] z 3 -1.95z 2 +0.9z+0.05 z 4 -2.8z 3 +3.5z 2 -2.412z+0.7209 G 3 (z) [2] - z 3 -1.95z 2 +0.9z+0.05 z 4 -2.8z 3 +3.5z 2 -2.412z+0.7209 G 4 (z) [2]
z 4 -1.5z 3 +0.5z 2 -0.5z+0.5 4.4z 5 -8.957z 4 +9.893z 3 -5.671z 2 +2.207z-0.5 G 5 (z) [START_REF] Ahmad | A less conservative LMI condition for stability of discrete-time systems with slope-restricted nonlinearities[END_REF] -0.5z+0.1 z 3 -0.9z 2 +0.79z+0.089 G 6 (z) [START_REF] Heath | Second-order counterexamples to the discretetime Kalman conjecture[END_REF] 2z+0.092 z 2 -0.5z G 7 (z) [START_REF] Carrasco | Convex searches for discretetime Zames-Falb multipliers[END_REF] 1.341z 4 -1.221z 3 +0.6285z 2 -0.5618z+0.1993 z 5 -0.935z 4 +0.7697z 3 -1.118z 2 +0.6917z-0.1352 [START_REF] Carrasco | Convex searches for discretetime Zames-Falb multipliers[END_REF]. . We also plot the bounds achieved using the Circle criterion, the Tsypkin Lyapunov function and the function from [START_REF] Park | A less conservative stability criterion for discrete-time Lur'e systems with sector and slope restrictions[END_REF]. The upper limit set by the Nyquist gain is also plotted.

Max δ Plant G 1 (z) G 2 (z) G 3 (z) G 4 (z) G 5 (z) G 6 (z) G 7 (

Regional Analysis

The second numerical example uses V (x) from (2.5) to estimate the region of attraction of the Lurie system (2.1). Consider a balanced realization of the plant G 6 (z) from Table 2.1, the polynomial Note that φ above is monotonic in the interval y, y and it is sector bounded and slope restricted with

p(σ) = cσ(σ -r 1 )(σ + r 1 )(σ -r 2 )(σ + r 2 ),
δ loc (y, y) = min σ∈[y,y] φ(σ) σ , δ loc (y, y) = max σ∈[y,y] φ(σ) σ , γ loc (y, y) = 0, γ loc (y, y) = max σ∈[y,y] dφ(σ) dσ .
Since the nonlinearity φ is a polynomial, the terms φ(ν j (x)) in the vector ξ N and the integrals of (2.5) also become polynomials on the variable x. In the following, the parameters of the nonlinearity and the interval were set to r 1 = 1, r 2 = 2, c = 8 × 10 -3 and y = -y = 5.28.

We formulate a semidefinite program using the inequalities in Theorem 2.2. To optimise the estimates of the region of attraction, we used the trace of the quadratic matrix defining V as the cost function. Figure 2.4 shows the estimates of the obtained region of attractions -denoted by the blue curves in the figure-obtained for horizon lengths N = 1 (dark blue) and N = 4 (light blue), with the blue dashed lines being sublevel sets of the corresponding V (x). The red area displays the set of initial values generating trajectories that did not converge to the origin and the black dashed lines correspond to {x : ν 0 (x) = y, ν 0 (x) = y}. The figure shows that increasing the horizon length N in V (x) can generate non-convex estimates of the region of attraction with larger volumes than those obtained using ellipsoidal sets.

Bounding the Worst Case Input-Output Gain

Thus final numerical example highlights the potential of V (x) in (2.5) for bounding the worstcase input-output gain of the Lurie system (2.24). Consider a balanced realization (A, B, C) of G 4 (z) from Table 2.1 and assume a global analysis (so X 0 = R n ). Furthermore, assume that the nonlinearity is bounded by δ = γ = 2.55 and δ = γ = 0 and take B w = B (as in the input vector of the disturbance equals that of the nonlinearity) and, similarly, C z = C. 3 defining the worst-case bound z 2 ≤ ω w 2 for all w 2 ∈ 2 as a function of the horizon length N of V (x). As N increased, there was a significant drop in ω, going from 6.08×10 3 with N = 1 to 3.13 × 10 1 at N = 4 before reaching a plateau. This noticeable drop in ω suggests longer horizons N in V (x) may prove important for tight bounds of the input-outputs of Lurie systems.

Conclusions

The absolute stability problem for discrete-time Lurie systems with monotonic nonlinearities was considered. A class of Lyapunov function composed of a generalised quadratic term plus a sum of Lurie-Postnikov type integral terms was proposed. It was shown that sign-definiteness of both the quadratic matrix of the Lyapunov function and the scalars in front of the various integral terms could be relaxed. It was also shown that the proposed Lyapunov function generalised existing Lyapunov function structures and its derivation from applying passivity theory to the feedback Lurie system was discussed. Numerical examples demonstrated the value of the proposed candidate Lyapunov functions for i) increasing the maximum achievable sector bound for verifying global stability, ii) estimating the region of attraction of the Lurie system, and iii) bounding the worstcase input-output gain of the system.

Notes and References

This part ends with notes and references on the absolute stability problem. Since its formulation in [START_REF] Lurie | On the theory of stability of control systems[END_REF], the absolute stability problem is at the origin of a significant number of contributions in the Automatic Control literature, including robust control analysis and synthesis and the analysis of nonlinear actuators, stability guarantees for optimization-based strategies. The short literature review presented below highlights some of the results motivating the problems studied in the first two chapters of the manuscript.

The absolute stability problem and the Lurie Lyapunov function

Motivated by the stability analysis of a continuous-time linear SISO system in feedback with a relay (on-off switch), Lurie and Postnikov proposed in [START_REF] Lurie | On the theory of stability of control systems[END_REF] a Lyapunov function to study the stability of the origin of the feedback of a class of linear systems and a class of nonlinearities described by a sector inequality. The two researchers were primarily motivated to study this problem related to several practical problems that appear as the interconnection of a linear system and nonlinear static elements. The most straightforward instances of such elements are actuator nonlinearities such as saturation or relay.

This simple description of a family of nonlinear functions raised a question, the Aizerman conjecture [START_REF] Aizerman | On a problem concerning the stability "in the large" of dynamical systems[END_REF], of whether the stability of the feedback of the same linear system replacing the nonlinearity by linear gains within the sector was sufficient to guarantee the stability for every nonlinearity within the sector. This conjecture can be stated as:

Conjecture 2.1: Aizerman Conjecture

The equilibrium state x = 0 of the system ẋ = Ax + bφ(y), y = cx, is absolutely stable on an arbitrary open sector L ( φ(y) y ∈ L) when the origin in the associated linear system ẋ = (A + bkc)x, asymptotically stable for all k ∈ L.

The conjecture is not valid in general, the first counterexamples being reported in [START_REF] Krasovskii | Theorems on stability of motions determined by a system of two equations[END_REF][START_REF] Pliss | Certain problems in the theory of stability of motion in the whole[END_REF][START_REF] Dewey | A note on Aizerman's conjecture[END_REF][START_REF] Fitts | Two counterexamples to Aizerman's conjecture[END_REF]. These refutations of the conjecture presented systems satisfying the conditions but showing periodic trajectories. Recent results propose the systematic construction of counterexamples [START_REF] Leonov | Hidden attractors in dynamical systems. From hidden oscillations in Hilbert-Kolmogorov, Aizerman and Kalman problems to hidden chaotic attractor in Chua circuits[END_REF].

The approaches developed to study the absolute stability problem in the '50s and the '60s are reported in [START_REF] Lurie | Some Nonlinear Problems in the Theory of Automatic Control[END_REF][START_REF] Letov | Stability in Nonlinear Control Systems[END_REF][START_REF] Aizerman | Absolute stability of regulator systems[END_REF][START_REF] Narendra | Frequency Domain Criteria for Absolute Stability[END_REF]. It is fair to credit the most important result from the early years to Popov, Yakubovic, and Kalman [START_REF] Popov | On absolute stability of non-linear automatic control systems[END_REF][START_REF] Yakubovich | On absolute stability of non-linear automatic control systems[END_REF][START_REF] Kalman | Lyapunov functions for the problem of Lur'e in automatic control[END_REF]. Their contribution became known as the Kalman-Yakubovic-Popov (KYP) Lemma or Positive Real Lemma. Two of these influential contributions are also presented in [START_REF] Basar | Control Theory: Twenty-Five Seminal Papers[END_REF] where the seminal papers [START_REF] Popov | On absolute stability of non-linear automatic control systems[END_REF] and [START_REF] Yakubovich | On absolute stability of non-linear automatic control systems[END_REF] are shortly presented by J. C. Willems and P. V. Kokotovic, respectively.

The KYP Lemma characterizes the set of linear time-invariant (LTI) SISO systems that are strictly positive real and, as a consequence, stable in closed-loop with passive time-invariant nonlinearities. For this class of systems, the KYP gives a quadratic LF. The KYP lemma can also be applied to loop-transformed systems, allowing to conclude on the stability of the feeback with an LF containing the integral term proposed in [START_REF] Lurie | On the theory of stability of control systems[END_REF]. The relation to the (strict) positive realness of the transfer function provides a stability criteria in terms of the frequency response of the LTI system. This frequency response characterization for SISO systems was introduced by Popov [START_REF] Popov | On absolute stability of non-linear automatic control systems[END_REF], the Popov criteria, and the proof of the lemma includes the case of systems with closed left half plane poles.

The two classical criteria for verifying the stability of Lurie systems, the circle, and the Popov criteria, provide graphical interpretations for the SISO case and associate a Lyapunov function to stable systems (a quadratic function for the circle criterion and a quadratic plus integral term for the Popov criterion). By establishing stability criteria with graphical interpretation allowed to connect to standard linear system analysis. When these results were proposed, the analysis tools used experimental models based on the frequency response.

Slope restrictions

Following the refutation of the Aizerman conjecture, there was still interest in using linear systems to infer the stability of nonlinear Lurie systems. This lead to the Kalman conjecture [START_REF] Kalman | Physical and mathematical mechanisms of instability in nonlinear automatic control systems[END_REF] that asks whether the linear stability for all gains in the sector would be sufficient to guarantee the stability of the feedback for all nonlinearities with slope restrictions in the sector. It can be stated as:

Conjecture 2.2: Kalman Conjecture
The equilibrium state x = 0 of the system ẋ = Ax + bφ(y), y = cx, is stable for all differentiable nonlinearities satisfying dφ(y) dy ∈ L on an arbitrary open sector L when the origin in the associated linear system ẋ = (A + bkc)x is asymptotically stable for all k ∈ L.

However, the Kalman conjecture was also disproved by counterexamples in continuous-time [START_REF] Fitts | Two counterexamples to Aizerman's conjecture[END_REF][START_REF] Barabanov | On the Kalman problem[END_REF], and the discrete-time case [START_REF] Heath | Second-order counterexamples to the discretetime Kalman conjecture[END_REF]. Since the Kalman conjecture is false, it is essential to formulate methods to assess the stability of SISO systems with slope-restricted nonlinearities. In general, the slope restriction does not impose differentiability of the nonlinearity. Continuous but nondifferentiable nonlinearities are appealing in practice since slope bounded nonlinearities appear in saturating actuators and other nonlinear elements modeled with globally Lipschitz functions.

For continuous-time systems, the first and rather general conditions involving slope restriction appeared in [START_REF] Yakubovich | The method of matrix inequalities in the stability theory of nonlinear control systems: II. Absolute stability in a class of nonlinearities with a condition on the derivative[END_REF], where inequalities characterizing the slope restriction were introduced. More recently, the stability analysis with Lyapunov functions, which exploit slope information, has received renewed attention [START_REF] Park | A revisited Tsypkin criterion for discrete-time nonlinear Lur'e systems with monotonic sector-restrictions[END_REF][START_REF] Haddad | Absolute stability criteria for multiple slope-restricted monotonic nonlinearities[END_REF].

The main contributions in the study of slope-restricted nonlinearities appeared in the context of the input-output analysis [START_REF] Zames | Stability conditions for systems with monotone and slope-restricted nonlinearities[END_REF][START_REF] Desoer | Feedback Systems: Input-Output Properties[END_REF]. The input-output methods drew inspiration from the result by Popov, by generalizing its loop transformations by including general transfer functions in the loop, the multipliers. Upon showing the existence of such multipliers, stability certificates are obtained. The methods leading to these multipliers offer an alternative to the Lyapunov inequalities yielding Lyapunov functions as stability certificates.

In short, by transforming the feedback loop with multipliers, we can set conditions on the passivity of the linear part combined with the multiplier to conclude upon the stability of the loop. Monotone and slope-restricted nonlinearities introduce a specific class of multipliers. For the class of monotone nonlinearities, these multipliers are called the Zames and Falb (ZF) multipliers [START_REF] Zames | Stability conditions for systems with monotone and slope-restricted nonlinearities[END_REF]. The survey [START_REF] Carrasco | Zames-Falb multipliers for absolute stability: From O'Shea's contribution to convex searches[END_REF] gives an overview of the first contributions to characterize this class of multipliers introduced in the works of O'Shea [START_REF] O'shea | An improved frequency-time domain stability criterion for autonomous continuous systems[END_REF] and Zames & Falb [START_REF] Zames | Stability conditions for systems with monotone and slope-restricted nonlinearities[END_REF]. The discrete-time counterpart of the ZF multipliers was introduced in [START_REF] Willems | Some new rearrangement inequalities having application in stability analysis[END_REF]. By using a particular class of multipliers, [START_REF] Fujii | Verification of the Aizerman and/or Kalman conjecture[END_REF] listed conditions on the parameters of 4th order systems that satisfy both Aizerman and Kalman conjectures. Regarding the computational formulation, numerical searches for the ZF multipliers were proposed in [START_REF] Safonov | Computer-aided stability analysis renders Popov criterion obsolete[END_REF]. Also, for the discrete-time case, several results were recently obtained [START_REF] Turner | Discrete-time systems with slope restricted nonlinearities: Zames-Falb multiplier analysis using external positivity[END_REF][START_REF] Carrasco | Convex searches for discretetime Zames-Falb multipliers[END_REF]. More recently, ZF multipliers could be extended to verify stability within some region of the state-space [START_REF] Fetzer | Zames-Falb multipliers for invariance[END_REF].

Assuming slope restrictions also help enlarge the set of LF that can be used to study the stability for time-varying nonlinearities. For instance, instead of using only a quadratic function, it is possible to use the integral Lurie term whenever the slope is restricted. In this case, quadratic bounds can be obtained to the term ẏ 0 φ(s)ds, that appear in the Lyapunov function derivative.

The discrete-time case

In [START_REF] Szegö | Sur la stabilité absolue d'un système d'equations aux differences finies[END_REF], a quadratic Lyapunov function was used to study the stability of a discrete-time Lurie system. In [START_REF] Szegö | On the absolute stability of sampled-data control systems[END_REF] Szegö introduced a Lyapunov function mimicking the Lurie function with an integral term used for continuous-time systems by then. The derivation of these time-domain conditions for DT systems were thus less formal than the CT counterpart. As a result, the stability conditions with the same LF structure require different assumptions on the nonlinearity. Indeed, for DT systems, it was also required to impose a slope constraint on the nonlinearity. Interestingly, already in this paper, the stability conditions allow the coefficient of the integral term in the LF to be a real number; namely, it is not required to be non-negative as in the continuous-time analysis (where no slope restriction is imposed). The discrete-time counterpart to the Popov frequencydomain tests for absolute stability was proposed by Tsypkin [START_REF] Tsypkin | On the stability in the large of nonlinear sampled-data systems[END_REF][START_REF] Tsypkin | Fundamentals of the Theory of Non-Linear Pulse Control Systems[END_REF]. A summary of these early contributions is presented in [START_REF] Larsen | A brief look at the Tsypkin criterion: from analysis to design[END_REF]. In both approaches, the LF function with integral terms and the discrete-time Popov multiplier [START_REF] Tsypkin | Fundamentals of the Theory of Non-Linear Pulse Control Systems[END_REF] the nonlinearity must be monotonic and slope restricted.

Technically, such a restriction bounds an integral of the nonlinearity between two instants by a quadratic expression. Even though the extensions from the continuous-time case led to the first stability analysis results in discrete-time, the choice of the Lyapunov function was not evident, as Szegö observed in [START_REF] Tsypkin | Fundamentals of the Theory of Non-Linear Pulse Control Systems[END_REF]Discussion]:

Even if we now have some fairly good results on this problem, the status of stability theory for sampled-data systems has still not reached a satisfactory stage as in the continuous case. Further work is needed, and improvements can be achieved by using the new method of Popov and very likely by using some more sophisticated Lyapunov functions.

Another critical aspect of these methods is that approximation of the integrals is crucial to reducing the conservatism of the stability conditions. The impact of the integral approximation when assessing the inequality involving the variation of the LF was discussed in [START_REF] Soliman | Absolute stability of a class of nonlinear sampled-data systems[END_REF]:

Various area inequalities can be found for the integral of

y[k+1] y[k]
φ(s)ds, and it appears that, by combining these results, a better and more versatile stability condition may be obtained. Further work is being done to obtain a suitable discrete equivalent of the area integral, yielding a better stability boundary.

Indeed, better approximations of the integral term were proposed by [START_REF] Ahmad | LMI-based stability criteria for discrete-time Lur'e systems with monotonic, sector-and slope-restricted nonlinearities[END_REF][START_REF] Park | An improved stability criterion for discrete-time Lur'e systems with sector-and slope-restrictions[END_REF], using the bounds on the slope of the nonlinearities.

Analysis methods based on optimization also renewed the interest in the stability of discretetime Lurie systems. The paper [START_REF] Haddad | Explicit construction of quadratic Lyapunov functions for the small gain, positivity, circle, and Popov theorems and their application to robust stability. part II: Discrete-time theory[END_REF] reinterprets the results from the previous decades within a Ricatti equation framework. The authors propose to study multivariable nonlinearities with multivariable sector inequality and decentralized slope inequalities. On the other hand, the LF therein is a direct generalization of the one introduced in [START_REF] Szegö | On the absolute stability of sampled-data control systems[END_REF], namely the sum of integrals, thus requiring the same type of approximations of integral terms. This sum of integrals in a MIMO decentralized nonlinearity was already adopted in [START_REF] Sharma | On the absolute stability of multivariable discrete-time nonlinear systems[END_REF]. The results in [START_REF] Haddad | Explicit construction of quadratic Lyapunov functions for the small gain, positivity, circle, and Popov theorems and their application to robust stability. part II: Discrete-time theory[END_REF] paved the way to LMI formulations since matrix inequalities later replaced the Riccati equations formulation. LMI results for discrete-time systems with slope restrictions are detailed in [START_REF] Kapila | A multivariable extension of the Tsypkin criterion using a Lyapunov-function approach[END_REF] and [START_REF] Park | A revisited Tsypkin criterion for discrete-time nonlinear Lur'e systems with monotonic sector-restrictions[END_REF]. Both papers seem to have neglected that the constraints on the positivity of the integral term could be dropped as in [START_REF] Szegö | On the absolute stability of sampled-data control systems[END_REF] (even though in [START_REF] Park | A revisited Tsypkin criterion for discrete-time nonlinear Lur'e systems with monotonic sector-restrictions[END_REF], the several integrals that are used make a simplified quadratic plus integral expression have real coefficients in a single integral, see Remark 2.1 of this manuscript). In the wake of these formulations, the LMI approaches have mainly imposed these coefficients to be positive since they provide a rather simple way of parameterizing the positivity of the LF.

Local analysis of Lurie Systems

In some practical cases, verifying the global stability of an equilibrium point is not possible for Lurie Systems, for example, the situation where the trajectories converge to an attractor such as a limit cycle or a chaotic trajectory or the cases where the set of initial conditions of converging trajectories to the equilibrium point is a subset of the state space. The main challenge in such cases is to characterize the set of initial conditions converging to the equilibrium either asymptotically or in finite time. Such a task can be carried out by solving the Zubov equations [START_REF] Zubov | Methods of AM Lyapunov and their Application[END_REF]. However, the Zubov method can be involved since it consists of a set of partial differential equations in R n . Alternatively, we can obtain estimates of it in terms of inner or outer approximations of the region of attraction, namely the set of all initial conditions generating trajectories converging to the equilibrium.

For the Absolute stability problem, a first result for the local stability appeared in [START_REF] Komarova | On Estimating the Attraction Regions of the Equilibrium States of Dynamic Systems By the Direct Lyapunov Method[END_REF] providing estimates of the region of attraction (ERA) in terms of Lyapunov level sets. The approach consisted of using bounds for the sector for an imposed region in the state space. A similar approach was proposed in [START_REF] Walker | Finite regions of attraction for the problem of Lur'e[END_REF] where the regions where the sector inequalities hold are also limited. In these early results, the ERA is given by a level set of an LF, and it is obtained by checking some set containment conditions. These set containment essentially guarantee that the Lyapunov inequalities hold within the set where the sector conditions hold. These additional set containment conditions can be obtained for specific nonlinearities. The results in [START_REF] Weissenberger | Application of results from the absolute stability problem to the computation of finite stability domains[END_REF] show how to obtain the maximum level set in terms of bounds of the nonlinearity input for a quadratic LF.

Saturation and quantization within the absolute stability framework

Even though the relay was the first nonlinearity to be studied in the context of absolute stability, the saturation nonlinearity is perhaps the most significant nonlinear static element in control systems [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF][START_REF] Hu | Control Systems with Actuator Saturation: Analysis and Design[END_REF]. Saturation appears due to limits in actuators for safety or technological reasons. A nonlinearity closely related to the saturation is the deadzone. Indeed, saturation and deadzones are alike for Lurie systems since they can be transformed into one another via a linear loop transformation.

In practice, to prevent actuators from reaching their physical limits, the sofware implements the saturation, and in these cases, an exact model is obtained for analysis and design purposes. On the other hand, we may consider the saturation as a class of static mappings within the more general framework of absolute stability for simplicity or because the actuator gains, limits, and the actual saturation mappings are not precisely known.

Even though the sector characterization of the saturation introduces conservatism in the analysis, it does not impose the analysis to be carried out only in the linear region and thus allows to study trajectories with input signals that undergo saturation. This flexibility is beneficial when computing estimates of the region of attraction or when assessing the local properties such as induced gains. Allowing the input signals to saturate is in contrast with approaches imposing the signals to remain in the non-saturating regions [START_REF] Jarvis-Wloszek | Control applications of sum of squares programming[END_REF].

The first results to consider saturation as a sector nonlinearity trace back [START_REF] Pearson | On the asymptotic stability of a class of saturating sampleddata systems[END_REF] where discretetime systems were studied using the Lyapunov function from Szegö [START_REF] Szegö | On the absolute stability of sampled-data control systems[END_REF]. The results are illustrated with saturating systems to highlight the benefits of the integral terms over a simple quadratic LF for the absolute stability problem.

More recently, optimization methods were used to compute add-ons to the feedback loops to reduce the performance degradation induced by saturation. These methods give formal proofs for the global or local stability of saturating feedback and introduce elements in the feedback loop that become active in the event of saturation. These elements, known as anti-windup (AW) compensators, had been initially introduced in industrial applications containing SISO loops with integral action and saturation in actuators to prevent the state of the analog devices, often implemented using capacitors, to "wind-up". The first strategies were empirical, following heuristics and relying on the operator experience (for details on the development of AW techniques, see the surveys [START_REF] Tarbouriech | Anti-windup design: an overview of some recent advances and open problems[END_REF][START_REF] Galeani | A tutorial on modern anti-windup design[END_REF]. With the use of sector conditions and semidefinite programming, the AW design can be systematic and allows to improve the performance of saturating systems whenever the input signal saturates [START_REF] Zaccarian | Modern anti-windup synthesis: control augmentation for actuator saturation[END_REF]. In the development of AW compensators, a refinement of the sector conditions for saturating nonlinearities was introduced. The proposed generalized sector condition is reported in [START_REF] Hu | Stability and performance for saturated systems via quadratic and nonquadratic Lyapunov functions[END_REF][START_REF] Gomes Da | Anti-windup design with guaranteed regions of stability for discrete-time linear systems[END_REF]. Roughly speaking, the idea behind this generalization is to narrow the sector within which the saturation lies. Unlike previous local analysis results for Lurie systems, this approach does not impose the set in which the narrowed sector holds, making it attractive in an optimization context since the parameters defining the set where it holds are decision variables of the problem. These parameters are thus used to enlarge estimates of the regions of attraction.

However, in most existing methods for the design of feedback gains for saturating systems or for the design of AW compensators, the LF structure is a simple quadratic function. Analyzing these feedback loops to compute gain curves or ERAs can benefit from more complex Lyapunov functions, such as LFs with the integral of the nonlinearity. In this case, it is important to observe that for a piecewise affine function, such as saturation input, the Lurie integral term becomes a quadratic [START_REF] Dai | Piecewise-quadratic Lyapunov functions for systems with deadzones or saturations[END_REF] term. The resulting LF is thus a generalized quadratic form involving both the state and the nonlinear function. However, one aspect in [START_REF] Dai | Piecewise-quadratic Lyapunov functions for systems with deadzones or saturations[END_REF][START_REF] Garulli | Global stability and finite L 2mgain of saturated uncertain systems via piecewise polynomial Lyapunov functions[END_REF] (also in [START_REF] Suykens | An absolute stability criterion for the Lur'e problem with sector and slope restricted nonlinearities[END_REF] for general Lurie systems) was overlooked when casting conditions for the positive definiteness of the LF with conditions on the matrix describing the generalized quadratic form. In these papers, the extended matrix was required to be positive definite. However, this is only a sufficient condition for the positivity of the LF, since in general, the matrix in the generalized quadratic forms needs not to be positive definite.

Relaxing the positivity of the extra blocks in generalized quadratic forms helps improve the ERA estimates and the local gains computation as highlighted in [START_REF] Valmorbida | Regional analysis of slope-restricted Lurie systems[END_REF] and [START_REF] Valmorbida | Regional L 2m gain analysis for linear saturating systems[END_REF]. We can mention [START_REF] Hu | A unified Lyapunov approach to analysis of oscillations and stability for systems with piecewise linear elements[END_REF] that already proposed the relaxation of the positive definiteness of the matrix in the generalized quadratic form.

Since the static, time-invariant nonlinearities within a sector need not be continuous in the absolute stability framework, it is also possible to analyze systems with relays and quantization. The absolute stability applied to systems with input quantization is pursued in the works by [START_REF] Ferrante | Stabilization of continuous-time linear systems subject to input quantization[END_REF][START_REF] Ferrante | On sensor quantization in linear control systems: Krasovskii solutions meet semidefinite programming[END_REF][START_REF] Jayawardhana | The circle criterion and input-to-state stability[END_REF][START_REF] Coutinho | Input and output quantized feedback linear systems[END_REF].

The case of MIMO Nonlinearities

A sector nonlinearity is any mapping of a single variable between two linear maps, the sector containing the nonlinearity can be easily depicted graphically. The bounds by linear maps are also easily described by an inequality. For MIMO nonlinear loops with a static MIMO nonlinearity, the graphical interpretation is no longer possible. The most common approach is then to directly characterize the nonlinearity by inequality as φ (y)(y -Kφ(y)) ≥ 0 for some positive semidefinite matrix K.

However, the MIMO nonlinearity can be a simple combination of several SISO terms; this is called the decentralized case. In practice, some nonlinearities are not decentralized, and examples are actuators that saturate in the norm of multivariable inputs. This way, the magnitude bounds for each channel depends on the current values of the outputs of the other channels.

Due to the difficulties in describing a multivariable sector inequality, the absolute stability problem for MIMO systems has not been studied in detail as the SISO case. Most of the existing results rely on sufficient conditions to compute Lyapunov functions and consider the decentralized case. In [START_REF] Anderson | Stability of control systems with multiple nonlinearities[END_REF] a proposal to generalize sector inequalities for MIMO nonlinearities was presented. More recently, the equivalence of the frequency-domain conditions and the time-domain conditions for the Circle Criterion in the MIMO setting is discussed in [START_REF] Lipkovich | Equivalence of MIMO circle criterion to existence of quadratic Lyapunov function[END_REF]. For systems with multiple slope-restricted nonlinearities, a frequency domain criterion generalizing previous results for SISO systems and the associated multipliers has been proposed in [START_REF] Halanay | Absolute stability of feedback systems with several differentiable non-linearities[END_REF][START_REF] Rotea | New results for analysis of systems with repeated nonlinearities[END_REF][START_REF] Safonov | Zames-Falb multipliers for MIMO nonlinearities[END_REF].

Generalizations of absolute stability

If instead of sector bounds, the nonlinearity is bounded by other nonlinear maps, inequalities generalizing sector inequalities can be formulated [START_REF] Hu | Absolute stability with a generalized sector condition[END_REF][START_REF] Martins | Uniform estimates of attracting sets of extended Lurie systems using LMIs[END_REF][START_REF] Hancock | Generalised absolute stability and sum of squares[END_REF][START_REF] Drummond | Generalized absolute stability using Lyapunov functions with relaxed positivity conditions[END_REF]. These approaches offer alternatives to linear sector description to reduce the set in which a given nonlinearity lies.

Also, for the case of nonlinearities that belong to a sector only outside a neighborhood of the origin, a robustness problem can be formulated by assuming a global sector and bounded disturbances with the notion of input-to-state stability (ISS) [START_REF] Jayawardhana | The circle criterion and input-to-state stability[END_REF]. The ISS framework is also used to study reachability under bounded disturbances [START_REF] Arcak | Input-to-state stability and boundedness in Lurie systems[END_REF].

Semidefinite programming for Lurie systems

The first methods to obtain the parameters of Lurie functions were based on the analytical solution to algebraic equations. These methods generalized the approaches based on the solution of a Lyapunov equation A P + P A = -Q, to show the stability of a linear time-invariant system. In the above equation, the matrix Q is a fixed positive definite matrix. However, it is important to keep in mind that the LF parameters showing the stability of a system may not be unique. Indeed, different choices for Q give different matrices P . The set of solutions is convex (an LMI set [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF]). Such a set characterizes all parameters of the Lyapunov matrix, satisfying the Lyapunov inequalities.

The existence of these convex sets motivates the use of inequalities instead of equations to obtain stability certificates as LF. The main advantage of the inequality description of the set of LFs for LTI systems is that it can be solved with Semidefinite Programming (SDP), a class of convex optimization problems. Thanks to the inequalities description, it is also simpler to use the finite-dimensional parametrization of Lyapunov functions to study uncertain systems, incorporate constraints in the nonlinear analysis (often described by set containment conditions), or add optimization goals to estimate input-output gains. The applications of SDP in Automatic Control increased its popularity and contributed to its development. The potential of LMIs for stability and performance analysis was observed in [START_REF] Willems | Least squares stationary optimal control and the algebraic Riccati equation[END_REF], [START_REF] Yakubovich | The method of matrix inequalities in the stability theory of nonlinear control systems: II. Absolute stability in a class of nonlinearities with a condition on the derivative[END_REF].

Furthermore, fundamental results trace back to Yakubovic with the formulation of the Sprocedure for quadratic forms, which permeates the LMI literature. The reader can refer to [22, Section 1.2, and Chapter 5 Notes and References] for an account of the relation between the study of Lurie systems and the development of numerical solutions to Lyapunov inequalities. Checking the positivity of the LF with Lurie integral terms for systems with sector bounded nonlinearities has an immediate SDP formulation. On the other hand, the application to other problems requires more insight into inclusion conditions. This is the case of local stability analysis, where the use of SDP to compute estimates of the region of attraction for Lurie Systems appeared in [START_REF] Hindi | Analysis of linear systems with saturation using convex optimization[END_REF], and [START_REF] Pittet | Stability regions for linear systems with saturating controls via circle and Popov criteria[END_REF].

Also, the solution to Lyapunov inequalities for control law design has led to stabilizability conditions solved by semidefinite programs. The local sector conditions for quadratic stabilizability have been successfully used for the local stabilization of linear saturating systems. More complex LFs and ZF multipliers have been used for the analysis, resulting in better stability bounds or performance estimates. For this reason, it appears that the use of more complex LF for nonlinear control design will help obtain better closed-loop performance. Such a generalization to other LF may lead to Anti-Windup strategies that will improve performance at the expense of a more complicated control law with possibly the need for online algebraic loop solutions.

Moreover, understanding the fundamental results of Lurie system analysis is crucial to understanding and contributing to emerging applications. Indeed, applications of SDP formulations for Lurie systems appear in the study of multi-agent with nonlinear interconnections [START_REF] Col | Regional H ∞ synchronization of identical linear multiagent systems under input saturation[END_REF][START_REF] Zhang | Dynamic feedback synchronization of Lur'e networks via incremental sector boundedness[END_REF][START_REF] Proskurnikov | Tsypkin and Jury-Lee Criteria for Synchronization and Stability of Discrete-Time Multiagent Systems[END_REF] in the study of systems with forced oscillations [START_REF] Pogromsky | A non-quadratic criterion for stability of forced oscillations[END_REF] or the design of numerical optimization algorithms [START_REF] Lessard | Analysis and Design of Optimization Algorithms via Integral Quadratic Constraints[END_REF].

Introduction of Part II

The widespread availability and the decreasing costs of digital devices have promoted the implementation of control systems in discrete time. In these control systems, the actuation devices introduce static nonlinearities such as saturation, deadzones, or quantizations, leading to piecewise affine (PWA) systems [START_REF] Gomes Da | Polyhedral regions of local stability for linear discrete-time systems with saturating controls[END_REF][START_REF] Lathuilière | Periodic orbits in planar linear systems with input saturation[END_REF]. Moreover, engineered systems such as nonlinear circuits [START_REF] Julian | High-level canonical piecewise linear representation using a simplicial partition[END_REF][START_REF] Kahlert | The complete canonical piecewise-linear representation. I. The geometry of the domain space[END_REF] and mechanical elements [START_REF] Kawamura | A modeling concept of a mechanical system having a piecewise linear spring property for its diagnosis[END_REF][START_REF] Md | Chaos and multiple periods in an unsymmetrical spring and damping system with clearance[END_REF] are suitably modeled by PWA systems. The Receding Horizon Optimal Control (ROHC) [START_REF] Bemporad | The explicit linear quadratic regulator for constrained systems[END_REF], or Model Predictive Control (MPC), also leads to PWA functions. This optimization-based strategy that allows handling state and input constraints easily, can be formulated as a multi-parametric linear or quadratic programs. The solutions to these problems are PWA functions that can be computed offline, the so-called Explicit Model Predictive Control (EMPC) [START_REF] Bemporad | The explicit linear quadratic regulator for constrained systems[END_REF][START_REF] Johansen | Hardware synthesis of explicit model predictive controllers[END_REF].

Nonlinear elements in feedback control systems may induce limit cycles, chaotic behavior, possibly leading to poor performance and instability. This is not different for the PWA systems. The stability analysis of systems presenting these nonlinearities introduces several theoretical challenges. In particular, within the framework of Lyapunov stability analysis, different steps for their study raise important questions: the choice of system representation, the choice of the Lyapunov function, and the methods for the verification of the Lyapunov inequalities.

The stability analysis of PWA systems traces back to [START_REF] Sontag | Nonlinear regulation: the piecewise linear approach[END_REF], where the following explicit representation was introduced for a PWA function f :

R n → R n f (x) = A i x + b i , ∀x ∈ Γ i ⊂ R n , (II.1) i = 1, . . . , n s , A i ∈ R n×n , b i ∈ R n with the sets Γ i defining a partition of R n , i.e. ∪ ns i=1 Γ i = R n .
The analysis continuous-time systems using the explicit representation (II.1), has been mainly studied with piecewise quadratic Lyapunov functions (LF) [START_REF] Johansson | Computation of piecewise quadratic Lyapunov functions for hybrid systems[END_REF][START_REF] Iervolino | Cone-copositive piecewise quadratic Lyapunov functions for conewise linear systems[END_REF][START_REF] Iervolino | Lyapunov stability for piecewise affine systems via cone-copositivity[END_REF]. In [START_REF] Johansson | Computation of piecewise quadratic Lyapunov functions for hybrid systems[END_REF], the sets Γ i are described by the intersections of half-spaces. In [START_REF] Iervolino | Cone-copositive piecewise quadratic Lyapunov functions for conewise linear systems[END_REF] the particular case of conewise linear systems is addressed. In [START_REF] Iervolino | Lyapunov stability for piecewise affine systems via cone-copositivity[END_REF], a representation of the polyhedral regions of the state space partition by vertices and cone rays is considered. The Lyapunov stability inequalities are tested with sufficient conditions based on the cone rays and the vertices representing each set Γ i in the partition.

On the other hand, the stability analysis of discrete-time PWA systems has been studied, for instance, in [START_REF] Feng | Stability analysis of piecewise discrete-time linear systems[END_REF][START_REF] Ferrari-Trecate | Analysis of discrete-time piecewise affine and hybrid systems[END_REF][START_REF] Hovd | Relaxing PWQ Lyapunov stability criteria for PWA systems[END_REF][START_REF] Rubagotti | A Lyapunov method for stability analysis of piecewise-affine systems over non-invariant domains[END_REF]. Similar to [START_REF] Johansson | Computation of piecewise quadratic Lyapunov functions for hybrid systems[END_REF], polyhedral partitions described by the intersections of half-spaces are considered in these works. Discontinuous piecewise quadratic Lyapunov functions can be studied for discrete-time systems. The additional difficulty in the analysis with a piecewise quadratic (PWQ) function for each set Γ i is that it requires the enumeration of all possible transitions between sets in the partition and the evaluation of the associated decrease of the LF.

A particular class of discontinuous PWA systems is motivated by the implementation of control laws in digital devices and networks that often require the input and output signals to be suitably encoded/decoded into finite alphabets. The limited number of values leads to a fundamental limitation in digital control systems: the quantization of sensors and actuators. An attempt to tackle the negative impact of quantization in control systems can be traced back to work by Kalman featured in [START_REF] Kalman | Nonlinear aspects of sampled-data control systems[END_REF], where stochastic methods were used to reduce the influence of quantization in the useful bandwidth. This approach can be effective whenever the level of specification is rather modest and the quantization somehow restrained.

The work by Delchamps [START_REF] Delchamps | Extracting state information from a quantized output record[END_REF][START_REF] Delchamps | Stabilizing a linear system with quantized state feedback[END_REF] introduced a different viewpoint by proposing to model the quantization by a static nonlinear function, the quantizer, mapping a real variable into a variable belonging to a countable set, thus enabling the analysis and the design of quantized control systems via deterministic nonlinear control theoretical tools. Consider a plant with n u ∈ N inputs taking values into the set

Q := {0, δ 1 } × {0, δ 2 } × . . . {0, δ nu },
where δ i ∈ R, for all i ∈ {1, 2, . . . , n u }, are some given levels, controlled by an static state feedback law v : R n → R nu . In this case, we can describe the quantizer by the function

Q : R n u → Q defined as Q(v) = ∆S(v)
∆ := diag{δ 1 , δ 2 , . . . , δ nu }, and S : R nu → R nu , S(u) := (s(u 1 ), s(u 2 ), . . . , s(u nu )) where, for all θ ∈ R,

s(θ) := 1 if θ > 0 0 if θ ≤ 0. (II.2)
This setup is rather general and includes the case in which actuators are subject to finite-level quantization.

The first attempts hinging upon this model approach for SISO systems are detailed in [START_REF] Miller | Quantization and overflow effects in digital implementations of linear dynamic controllers[END_REF]. In particular, in [START_REF] Miller | Quantization and overflow effects in digital implementations of linear dynamic controllers[END_REF], the authors study the problem of having quantized measurements in a linear control system by first bounding the quantization error and then by pursuing a Lyapunov approach to establish ultimate boundedness. Later on, this approach has been extended in [START_REF] Brockett | Quantized feedback stabilization of linear systems[END_REF] to general linear control systems with quantized measurements and in [START_REF] Liberzon | Hybrid feedback stabilization of systems with quantized signals[END_REF] to nonlinear systems in the presence of quantized control inputs or quantized measurements. In [START_REF] Fu | The sector bound approach to quantized feedback control[END_REF], the authors analyze discrete-time linear quantized control systems by encapsulating quantization error into a bounded sector. The stability analysis and controller design for quantized LTI continuous-time systems based on a sector bounded representation are addressed in [START_REF] Tarbouriech | Control design for quantized linear systems with saturations[END_REF][START_REF] Ferrante | Stabilization of continuous-time linear systems subject to input quantization[END_REF][START_REF] Ferrante | On sensor quantization in linear control systems: Krasovskii solutions meet semidefinite programming[END_REF]. Other results on quantized control systems can be found in [START_REF] Ceragioli | Discontinuous stabilization of nonlinear systems: Quantized and switching controls[END_REF][START_REF] Coutinho | Input and output quantized feedback linear systems[END_REF][START_REF] Liberzon | Switching in Systems and Control[END_REF][START_REF] Sur | Observers for linear systems with quantized outputs[END_REF] and in the very recent works [START_REF] Di Ferdinando | On practical stability preservation under fast sampling and accurate quantization of feedbacks for nonlinear time-delay systems[END_REF][START_REF] Wang | Quantized feedback control based on spherical polar coordinate quantizer[END_REF], showing a vivid interest in quantization in the controls community.

Summary of Contributions of Part II

This part studies discrete-time systems by introducing an implicit representation for PWA systems based on ramp functions. We follow the rationale of the absolute stability problem by recasting the class of nonlinear PWA systems as the feedback interconnection of a linear block and ramp functions, as in the figure below.

G(z) φ f 5 + y + Figure 2
.6: Feedback representation of a Lurie system is here used to represent PWA systems. In this part, φ will be a vector of ramp functions. The term f 5 is a constant vector that is added to the output of the linear block G(z) to generate the input y of the nonlinearity φ.

Since the nonlinearity in the feedback is known, instead of using a sector inequality, we use the optimality Karush-Kuhn-Tucker (KKT) conditions to characterize the ramp function. Thanks to this characterization, given by two linear inequalities and one quadratic identity, we can formulate conditions to verify generalized quadratic inequalities. These conditions are based on semidefiniteness contraints of matrices associated to the generalized quadratic forms.

The verification of generalized quadratic inequalities appears in stability conditions of PWA system using PWQ Lyapunov functions. These PWQ functions are generalized quadratic forms on the states and nonlinearity and the ramp function. Conditions for robust stability analysis are also presented. In this context, the use of polytopic sets to represent uncertainties helps describe uncertainty in the partition.

We also exploit the characterization of the ramp function and the verification of generalized quadratic inequalities in a slightly different context: to verify the positiviy of discontinuous PWQ function. Again, using KKT optimality conditions, we show how to describe a regularized step function. The description of the step function as KKT conditions, enable us to study systems with discontinuities. In this context, we propose global stability conditions for discrete-time systems with quantization of the inputs. This framework also allows us to compute discontinuous Lyapunov functions.

Structure of Part II

In Chapter 3, we investigate stability conditions for discrete-time PWA systems. With this aim, we introduce an implicit representation of PWA functions. The proposed representation allows us to avoid some shortcomings of the explicit representation (II.1). In particular, we use generalized quadratic forms to parametrize continuous piecewise quadratic Lyapunov functions. The stability of PWA systems can thus be assessed by evaluating Lyapunov stability conditions in terms of linear matrix inequalities (LMI) and does not require the enumeration of transitions between the partition sets. Furthermore, we show that the proposed representation easily copes with uncertainties in the partition, which is rather difficult with the existing methods.

Chapter 4 of this part focus on the analysis of systems with quantization. We propose a representation of a regularized step mapping based on an ill-posed algebraic loop containing two ramp functions. The relation to ramp functions is obtained thanks to the KKT necessary conditions for optimality. We carry out the stability analysis using generalized quadratic functions. In particular, we propose a Lyapunov function including the considered set-valued nonlinearity [START_REF] Dai | Piecewise-quadratic Lyapunov functions for systems with deadzones or saturations[END_REF][START_REF] Gonzaga | Stability analysis of discrete-time Lur'e systems[END_REF][START_REF] Camlibel | Lyapunov stability of complementarity and extended systems[END_REF][START_REF] Valmorbida | Regional analysis of slope-restricted Lurie systems[END_REF] that is quadratic in the plant state and the nonlinearity. We then propose sufficient conditions in the form of LMIs to certify global exponential stability of the origin of linear systems with input quantization. Those conditions can be efficiently checked by using semidefinite programming and numerical examples to illustrate the proposed method's effectiveness.

Chapter 3

Piecewise Affine Systems

This chapter is organized as follows: Section 3.1 presents the proposed implicit representation for PWA functions that rely on ramp functions, illustrates it with examples. Section 3.2 characterizes ramp functions in terms of quadratic identities and inequalities and presents conditions for verifying positivity of piecewise quadratic forms. In Section 3.3 we apply the positivity verification to formulate conditions for stability of discrete-time PWA systems using Piecewise Quadratic (PWQ) Lyapunov functions. Finally we illustrate the obtained results with numerical examples in Section 3.4 and present concluding remarks and perspectives in Section 3.6. The relation of the proposed implicit representation to other representations of PWA functions in the literature is discussed in Section 3.5.

Implicit Representation of Continuous PWA Functions

In this chapter we study the stability of piecewise affine discrete-time dynamical systems by

x[k + 1] = f (x[k]) (3.1)
where x ∈ R n is the state and f (x) is a continuous piecewise affine vector function.

The methods we propose here are based on the implicit representation for piecewise affine vector functions f : R n → R n as follows

f (x) = F 1 x + F 2 φ(y(x)) (3.2a) y(x) = F 3 x + F 4 φ(y(x)) + f 5 (3.2b) where x ∈ R n , y ∈ R ny , F 1 ∈ R n f ×n , F 2 ∈ R n f ×ny , F 3 ∈ R ny×n , F 4 ∈ R ny×ny , f 5 ∈
R ny and the vector function φ : R ny → R ny is defined elementwise by the ramp function as Some structures of matrix F 4 can yield explicit solutions to (3.2b). For instance, with strictly upper or lower triangular matrices we obtain recursive expressions of y allowing for an explicit dependence on x. However, in general, it is an implicit equation and we will assume its wellposedness (see Section 3.1.1).

φ i (y) = r(y i ) := 0 if y i ≤ 0 y i if y i > 0 , i = 1, . . . , n y (3.2c
We should also observe that thanks to the well-posdeness of equation (3.2b) and the continuity of the ramp functions in φ, f is a continuous function. We illustrate below two examples of (3.2).

Example 3.1. Consider (3.2) with n = 2,n y = 2,n f = 1 and

F 1 = 0 1 , F 2 = 1 1 , F 3 = -1 -1 1 -1 , F 4 = 0 -2 Γ 1 Γ 4 Γ 3 Γ 2
f (x) =          x 2 , x ∈ Γ 1 = {x ∈ R 2 | -x 1 ≤ x 2 ; x 1 ≤ x 2 }, -x 1 , x ∈ Γ 2 = {x ∈ R 2 | x 1 ≤ 0; x 2 ≤ -x 1 }, x 1 , x ∈ Γ 3 = {x ∈ R 2 | x 2 ≤ x 1 ; x 2 ≥ -5x 1 }, x 1 , x ∈ Γ 4 = {x ∈ R 2 | 0 ≤ x 1 ; x 2 ≤ -5x 1 }.
Note that the sets in the partition do not satisfy Γ i ∩ Γ j = ∅ for i = j. However, f is uniquely defined since the functions are continuous and coincide on the boundary of the sets.

Example 3.2. Given a matrix K ∈ R n f ×n and vectors µ ∈ R n f , µ ∈ R n f satisfying µ µ, the saturation function f (x) = sat [µ,µ] (Kx), is defined elementwise as follows f i (x) =      µ i x ∈ Γ 1 = {x ∈ R n | (Kx) i ≤ µ i } K i x x ∈ Γ 2 = {x ∈ R n | µ i ≤ (Kx) i ≤ µ i } µ i x ∈ Γ 3 = {x ∈ R n | (Kx) i ≥ µ i }, i = 1, . . . , n f .
For instance, with n = 2, n f = 1, K = -1 1 , µ = -1 and µ = 2 we obtain the partition depicted in Figure 3.3, corresponding to the explicit representation as in (II.1) given by

f (x) =        -1, x ∈ Γ 1 = {x ∈ R 2 | Kx ≤ -1} Kx, x ∈ Γ 2 = {x ∈ R 2 | -1 ≤ Kx ≤ 2} 2, x ∈ Γ 3 = {x ∈ R 2 | 2 ≤ Kx}. x 1 x 2 Γ 2 Γ 3 Γ 1 Figure 3.3: Partition of R 2 for f (x) defined in (3.4).
This function can be described as in (3.2) using

F 1 = K, F 2 = -I n f I n f F 3 = K -K , F 4 = 0 ny×ny f 5 = -µ µ (3.4)
with n y = 2n f and the induced partition of R 2 is expressed as

Γ 1 = {x ∈ R 2 | φ 2 (y(x)) ≥ 0} Γ 2 = {x ∈ R 2 | φ 1 (y(x)) = φ 2 (y(x)) = 0} Γ 3 = {x ∈ R 2 | φ 1 (y(x)) ≥ 0}.
To formulate stability conditions for piecewise affine systems (3.2) we will characterize the ramp function by a two inequalities and a complementarity relation. These relations will be obtained by expressing the ramp function as the solution to an optimization problem and will be detailed in Section 3.2. Thanks to these relations we will study the piecewise affine system as the feedback interconnection of a linear system and a nonlinearity. The difference with general framework for interconnection analysis, is that here we treat only the ramp function and not a set of functions as it is the case in the absolute stability. Also, thanks to the implicit representation, it will be easier to define piecewise quadratic Lyapunov function candidates by with generalized quadratic forms containing ramp function as detailed in the next section.

Different models for PWA functions have been proposed in the literature in the context of nonlinear circuits and control systems. A comparison among several models and results showing their equivalence are presented in [START_REF] Heemels | Equivalence of hybrid dynamical models[END_REF]. In Section 3.5, we relate the proposed model to other models that do not explicitly define the partition as (II.1).

Conditions for well-posedness

In Example 3.2 above we have F 4 = 0 giving an explicit solution to the equation (3.2b) expressed as

y = F 3 x + f 5 thus giving f (x) = F 1 x + F 2 φ(F 3 x + f 5 ).
In general, with F 4 = 0, as in Example 3.1 above, (3.2b) is an implicit equation and its wellposedness, namely the existence and uniqueness of a solution y for all x ∈ R n must be ensured. To ensure the well posedness of (3.2b), below we provide a condition for the well-posedness of the equation

F (y) := y -F 4 φ(y) = ζ ∀ζ ∈ R ny , (3.5)
In Section 1.1.1 we formulated a condition for the well posedness of slope-restricted nonlinearities. Note that the slope bounds for the vector of ramp functions φ are Γ = I ny and Γ = 0 ny . This way we obtain a particular case of the conditions in Proposition 1.1. We refer to this particular case, introduced in [187, Proposition 1], in the proposition below

If there exist a matrix W ∈ D ny , W > 0 such that 2W -W F 4 + F 4 W > 0 then (I -F 4 ∆) is non-singular ∀∆ ∈ D{∆ ∈ D ny | ∆ (i,i) ∈ [0, 1]}.
In the rest of the chapter, we will assume that the condition for well-posedness of (3.5) given in Proposition 3.1 holds. For implementation purposes, for instance when the PWA function has to be computed to generate a control input, a well posed equation (3.5) can be solved from the solution of a Linear Complementarity Problem [START_REF] Cottle | The Linear Complementarity Problem[END_REF] (see Remark 3.1 below).

Ramp Functions description from KKT conditions

Several results to verify the positivity of generalized quadratic forms involving sector nonlinearities rely on sector inequalities that hold either globally or locally [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF][START_REF] Hu | Stability and performance for saturated systems via quadratic and nonquadratic Lyapunov functions[END_REF]. These standard sector inequalities cover a broad class of nonlinearities lying in the considered sector. In the following, we show how to obtain an exact characterization of the ramp function (3.2c), by using some identities and inequalities.

It is possible to use the Karush-Kuhn-Tucker (KKT) optimality conditions to implicitly characterize nonlinearities in terms of identities and inequalities. Such an idea was detailed in [START_REF] Primbs | Kuhn-Tucker-based stability conditions for systems with saturation[END_REF] for the saturation nonlinearity. We illustrate this approach for the ramp function, which can be expressed as the solution to the optimization problem parameterized in θ as follows (3.9)

Proof. Since (3.8a) holds for all θ ∈ R and the elements of φ are ramp functions, that is φ i (y) = r(y i ) we have

s 1 (T 1 , y) = ny i=1 T i,i r(y i )(r(y i ) -y i ) = 0. Lemma 3.2
For any matrix M ∈ P (1+2ny)×(1+2ny) the vector function φ in (3.2c) satisfies the inequality

s 2 (M, y) :=   1 φ(y) φ(y) -y   M   1 φ(y) φ(y) -y   ≥ 0. (3.10) ∀y ∈ R ny .
Proof. Since (3.8b)-(3.8c) hold and φ i (y) = r(y i ), φ i (y) -y i = r(y i ) -y i , ∀i = 1, . . . n y and all entries of M are nonnegative scalars, it follows that s 2 (M, y) is a nonnegative scalar.

Remark 3.1

The above relations (3.8) can be used to obtain a solution to the algebraic loop (3.2b). With (3.8) and (3.2c), we have

(φ i -y i )φ i = 0 φ i ≥ 0 (φ i -y i ) ≥ 0, i =
((I -F 4 )φ -ζ) i φ i = 0 (3.12a) φ i ≥ 0 (3.12b) ((I -F 4 )φ -ζ) i ≥ 0. ( 3

Conditions for the Non-negativity of Generalized Quadratic Forms

In this section we use the above lemmas to set conditions to verify the positivity of generalized quadratic forms of the type h(x) = χ(x) Hχ(x).

(3.13) with H ∈ R n+2m×n+2m , χ(x) =   1 x φ(y(x))  
and y(x) as in (3.2b).

Proposition 3.2

Given a generalized quadratic form h(x) as in (3.13), if there exist matrices T ∈ D ny , M ∈ P (1+2ny)×(1+2ny) such that

h(x) + s 1 (T, y(x)) -s 2 (M, y(x)) ≥ 0, ∀x ∈ R n (3.14) then h(x) ≥ 0, ∀x ∈ R n . (3.15)
Proof. From Lemma 3.1, we have that s 1 (T, y(

x)) = 0 ∀x ∈ R n . If (3.14) is satisfied it follows that h(x) ≥ s 2 (M, y(x)), ∀x ∈ R n .
With Lemma 3.2 we conclude that h(x) ≥ 0, ∀x ∈ R n .

Remark 3.2

Setting conditions to verify the non-negativity of a generalized quadratic form as (3.13) by solving the inequality (3.14) makes possible the solution to the Lyapunov inequalities related to the stability of PWA systems. These inequalities are studied in the next section.

Using lemmas 3.1 and 3.2, we restrict the elements of a generic nonlinearity φ in the generalized quadratic forms treated in Proposition 3.2 to be ramp functions and not a set of of functions.

The remark below points out to the fact that other functions may satisfy the relation in Lemma 3.1.

Remark 3.3

Consider a set Ω ⊂ R satisfying the complementarity property that if θ ∈ Ω then -θ / ∈ Ω and consider the function

ρ Ω : R → R ρ Ω (θ) = 1 Ω (θ)θ (3.16)
where

1 Ω is the indicator function of a set Ω ∈ R, that is, 1 Ω (θ) = 1 if θ ∈ Ω, and 1 Ω (θ) = 0 if θ ∈ Ω c , with Ω c = R \ Ω.
The ramp function can be expressed in the above form with Ω = [0, ∞), i.e. r(θ) = ρ [0,∞) (θ). We also have r(-θ) = ρ (-∞,0) (-θ).

It follows that the complementarity relation (3.8) also holds if r is replaced by any function in the class (3.16). Indeed, using (3.16) we have ρ

Ω (θ) (ρ Ω (θ) -θ) = 1 Ω (θ)θ (1 Ω (θ)θ -θ) = 1 Ω (θ)θ (1 Ω c (θ)θ) = 0.
However, following (3.16) and the complementarity property of the set Ω above, the only instance of Ω satisfying ρ Ω (θ) ≥ 0 ∀θ ∈ R is Ω = [0, ∞), that is, the ramp function is the only nonlinearity in this class that satisfies (3.8b) and (3.8c) ∀θ ∈ R .

Stability Analysis of PWA Systems with PWQ Lyapunov Functions

In this section we apply the results for the verification of non-negativity of generalized quadratic forms presented in the previous section to study the stability of the origin of a discrete-time systems defined by the implicit PWA function (3.2). Consider discrete-time systems of the form

x + = f (x), (3.17)
where x ∈ R n , f : R n → R n is a PWA function defined by matrices F i , i = 1, . . . , 4, and f 5 as in (3.2) and x + is the value of the state at next time instant. We assume that φ(y(0)) = 0 and thus the origin is an equilibrium point, since f (0) = 0 in this case. The stability analysis of the origin of system (3.17), is studied with a continuous piecewise quadratic Lyapunov function given by a generalized quadratic form on x and the function φ(y(x)). Hence, differently from previous approaches, the definition of an explicit quadratic form on x for each set of the partition is not required. More precisely, we consider Lyapunov candidate functions V : R n → R ≥0 , V (0) = 0 given by

V (x) = x φ(y(x)) P x φ(y(x)) . (3.18)
with P = P1 P2 P 2 P3 , P 1 ∈ S n , P 2 ∈ R n×ny and P 3 ∈ S ny , and y(x) from (3.2b). The expression of the candidate Lyapunov function evaluated at x + for system (3.17) is given by

V (x + ) = x + φ(y(x + )) P x + φ(y(x + )) (3.19) with y + satisfying y + = F 3 F 1 x + F 3 F 2 φ(y(x)) + F 4 φ(y + ) + f 5 . (3.20)
The theorem below presents conditions for the global exponential stability of the origin of (3.17) using (3.18) as a Lyapunov function candidate.

Theorem 3.1

If there exist matrices P ∈ S (n+ny)×(n+ny) , T ∈ D ny , M ∈ P (1+2ny)×(1+2ny) , T u ∈ D ny , M u ∈ P (1+2ny)×(1+2ny) and positive scalars 1 and 2 such that

V (x) -1 x x + s 1 (T, y) -s 2 (M, y) ≥ 0 (3.21a) -V (x) + 2 x x + s 1 (T u , y) -s 2 (M u , y) ≥ 0 (3.21b)
matrices T ∈ D 2ny , M ∈ P (1+4ny)×(1+4ny) and a scalar η ∈ (0, 1) such that

-V (x + ) -(1 -η)V (x) + s 1 ( T , ỹ) -s 2 ( M , ỹ) ≥ 0 ( 3
1 x 2 ≤ V (x) ≤ 2 x 2 (3.24a) V (x + ) ≤ (1 -η)V (x). (3.24b)
Thus, (3.24) allows to conclude that x(k) ≤ Ce δk x(0) with C = ( 2 1 )

1 2 , δ = ln( √ 1 -η), ∀x(0) ∈ R n . Moreover, (3.24a) implies that V (x) is radially unbounded. Remark 3.4
The generalized quadratic form involving the the state and a nonlinearity as in (3.18) has been studied in the context of stability analysis of continuous-time linear complementarity systems [START_REF] Camlibel | Lyapunov stability of complementarity and extended systems[END_REF]. Here, the generic formulation presented in [START_REF] Camlibel | Lyapunov stability of complementarity and extended systems[END_REF] is used considering ramp functions, that also satisfy complementarity conditions. In that paper the authors suggest that their stability conditions could benefit from a numerical formulation exploiting copositivity conditions. The co-positivity is here accounted for by considering the inequalities of Lemma 3.2 in Theorem 3.1.

Remark 3.5

In case f 5i ≤ 0, ∀i ∈ {1, . . . , n y }, we easily show that the generalize quadratic form V (x) has a quadratic upper bound. To see this we first compute an upper bound for φ 2 . With y := y -f 5 , (3.2b) gives

y = F 3 x + F 4 φ(y + f 5 ). (3.25)
Assuming f 5i ≤ 0, ∀i ∈ {1, . . . , n y } the monotonicity of r gives 0 ≤ r(y i + f 5 ) ≤ r(y i ). We thus have that φ(y

+ f 5 ) = ∆y with some ∆ ∈ D := {∆ ∈ D ny | ∆ i,i ∈ [0, 1]}.
From the well-posedness assumption, we have that (I -F 4 ∆) is invertible for all ∆ ∈ D, thus using (3.25) we obtain y = (I -

F 4 ∆) -1 F 3 x and φ(y) = φ(y + f 5 ) = ∆y = ∆(I -F 4 ∆) -1 F 3 x, yielding φ(y(x)) ≤ σ x ,
with σ = max ∆∈D ∆(I -F 4 ∆) -1 F 3 . Finally, using (3.18), it follows that

V (x) ≤ P 1 x 2 + 2 P 2 x φ + P 3 φ 2 ≤ P 1 + 2σ P 2 + σ 2 P 3 x 2 (3.26)
thus showing that V (x) has an upper-bound of the form 2 x 2 .

LMI conditions

The relations (3.21) and (3.22) can be written in the generic quadratic form given by (3.13)- (3.14), where the corresponding matrices H present an affine dependence on the elements of matrix P . Hence, conditions in LMI form can be obtained to ensure (3.21) and (3.22). This is formalized in the following Corollary to Theorem 3.1.

Corollary 3.1

If there exist matrices P 1 ∈ S n , P 2 ∈ R n×ny , P 3 ∈ S ny , T ∈ D ny , T uj ∈ D ny , T ∈ D 2ny , symmetric matrices M ∈ P (1+2ny)×(1+2ny) , M u ∈ P (1+2ny)×(1+2ny) and M ∈ P (1+4ny)×(1+4ny) , and positive scalars 0 < η < 1, 1 and 2 such that the following LMIs are verified

I P I -1 I x I x + 1 2 He I φ T I φ-y -I χ M I χ ≥ 0 1+n+ny (3.27a) -I P I + 2 I x I x + 1 2 He I φ T u I φ-y -I χ M u I χ ≥ 0 1+n+ny (3.27b)
-

I + P I + + (1 -η)I 0 P I 0 + 1 2 He I φ T I φ-ỹ -I χ M I χ ≥ 0 1+n+2ny (3.27c)
where

I
I φ-ỹ = -f 5 -F 3 I ny -F 4 0 ny -f 5 -F 3 F 1 -F 3 F 2 I ny -F 4 , I χ =   1 0 1,n+2ny I φ I φ-ỹ   ,
then the origin of (3.17) is globally exponentially stable.

Proof. Consider V (x) defined as in (3.18). To show that if = Iχ(x),

x = I x χ(x) φ(y(x)) = I φ χ(x) φ(y(x)) -y(x) = I φ-y χ(x),   1 φ(y(x)) φ(y(x)) -y(x)   = I χ χ(x) and x + φ(y + (x)) = I + χ(x), x φ(y(x)) = I 0 χ(x), φ(ỹ(x)) = I φ χ(x) φ(ỹ(x)) -ỹ(x) = I φ-ỹ χ(x)   1 φ(ỹ(x)) φ(ỹ(x)) -ỹ(x)   = I χ χ(x).
We have χ(x)

I P I -1 I x I x + 1 2 He I φ T I φ-y -I χ M I χ χ(x) = V (x) -1 x x + s 1 (T, y) -s 2 (M, y) χ(x) -I P I + 2 I x I x + 1 2 He I φ T u I φ-y -I χ M u I χ χ(x) = -V (x) + 2 x x + s 1 (T u , y) -s 2 (M u , y) χ(x) -I + P I + + (1 -η)I 0 P I 0 + 1 2 He I φ T I φ-ỹ -I χ M I χ χ(x) = -V (x + ) -(1 -η)V (x) + s 1 ( T , ỹ) -s 2 ( M , ỹ)
Thus the matrix inequalities in (3.27) are imply the inequalities expressed as the generalized quadratic forms in (3.21) and (3.22).

Remark 3.6

Thanks to the expression (3.2) to express piecewise functions, all possible transitions between sets in the partition of system (3.17) are implicitly taken into account and no enumeration is required to set up the stability conditions. As a result, only three LMIs are needed to assess the stability of the PWA system. This is in sharp contrast with the results in the literature that use descriptions with polyhedral partitions expressed by hyperplanes (e.g. [START_REF] Johansson | Computation of piecewise quadratic Lyapunov functions for hybrid systems[END_REF][START_REF] Feng | Stability analysis of piecewise discrete-time linear systems[END_REF][START_REF] Ferrari-Trecate | Analysis of discrete-time piecewise affine and hybrid systems[END_REF]) or by vertices and cone rays (e.g [START_REF] Iervolino | Cone-copositive piecewise quadratic Lyapunov functions for conewise linear systems[END_REF][START_REF] Iervolino | Lyapunov stability for piecewise affine systems via cone-copositivity[END_REF]). Note that in the aforementioned works, an LMI has to relate each possible state transition from a region Γ j to a region Γ i . These inequalities are required to enforce the strictly decrease of the LF. Moreover, for each region Γ j , an LMI constraint is needed to ensure the positivity of the piecewise quadratic Lyapunov function.

Uncertain Systems Case

One major obstacle for the stability analysis using explicit representations of PWA systems concerns the presence of uncertainties in the partition. Uncertainties in the partition may occur for instance whenever PWA control laws as for instance the ones obtained with MPC contain rounding errors. Also, the sets of the partition can be modified or even be removed whenever the parameters defining them are uncertain. In this case, methods using explicit representation and with the enumeration of transitions can no longer be applied.

An important aspect of (3.2) is that handling uncertainties in the partition induced by is simpler since it can be cast as uncertainties on the matrices F 3 , F 4 and f 5 . This section studies the case of polytopic matrix uncertainties [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF] inducing uncertainties in the partition.

We consider the time-invariant differential inclusion

f (x) ∈ F 1 x + F 2 φ(y(x)) (3.31a) y(x) = F 3 x + F 4 φ(y(x)) + f 5 (3.31b)
with the system matrices satisfying

F 1 F 2 0 n,1 F 3 F 4 f 5 ∈ F F =    M ∈ R ny×1+n+ny | M = N j=1 α j F 1 j F 2j 0 n,1 F 3j F 4j f 5j , N j=1 α j = 1, α j ∈ [0 1], ∀j = 1, . . . , N    .
To treat the case where uncertainties affect all parameters of the system let us define where the first row is obtained from the identity y = φ(y) -(φ(y) -y), the second row is obtained from y + = φ(y + ) -(φ(y + ) -y + ). Let us define the following matrices considering the matrices in the vertices of the polytopic set F The theorem below presents conditions for the global exponential stability of the origin of an uncertain system (3.17) using a parameter-dependent function as a Lyapunov function candidate.

ζ :=     1 x φ(y) φ(y) -y     ξ :=           1 x x + φ(y) φ(y + ) φ(y) -y φ(y + ) -y +           (3.
S j = f 5j F 3j 0 ny,n F 4j -I ny I ny (3.36) Sj =   0 n,1 F 1j -I n F 2j 0 n,

Theorem 3.2

If there exist matrices P j ∈ S (n+ny)×(n+ny) , T j ∈ D ny , M j ∈ P (1+2ny)×(1+2ny) , T uj ∈ D ny , M uj ∈ P (1+2ny)×(1+2ny) , j = 1, . . . , N , matrices R ∈ R 1+n+2ny , R u ∈ R 1+n+2ny and positive scalars 1 and 2 such that

V j (x) -1 x x + s 1 (T j , y) -s 2 (M j , y) + ζ RS j ζ ≥ 0 (3.42a) -V j (x) + 2 x x + s 1 (T uj , y) -s 2 (M uj , y) + ζ R u S j ζ ≥ 0 (3.42b) j = 1, .
. . , N , matrices Tj ∈ D 2ny , Mi ∈ P (1+4ny)×(1+4ny) , j = 1, . . . , N , a matrix R ∈ R 1+2n+4ny and a scalar η ∈ (0, 1) such that

-V j (x + ) -(1 -η)V j (x) + s 1 ( Tj , ỹ) -s 2 ( Mj , ỹ) + ξ R Sj ξ ≥ 0 (3.43)
i = 1, . . . , N , then the origin of (3.17

α j V j (x) -1 x x + s 1 (T j , y) -s 2i (M j , y) + ζ RS j ζ = V (x) -1 x x + s 1 (T, y) -s 2i (M, y) N j=1 α j -V j (x) + 2 x x + s 1 (T uj , y) -s 2 (M uj , y) + ζ R u S j ζ = -V (x) + 2 x x + s 1 (T u , y) -s 2 (M u , y) N j=1 α j -V j (x + ) -(1 -η)V j (x) + s 1 ( Tj , ỹ) -s 2 ( Mj , ỹ) + ξ R Sj ξ = -V (x + ) -(1 -η)V (x) + s 1 ( T , ỹ) -s 2 ( M , ỹ)
thus, from the inequalities (3.42)-(3.43) we have

V (x) -1 x x + s 1 (T, y) -s 2i (M, y) ≥ 0 -V (x) + 2 x x + s 1 (T u , y) -s 2 (M u , y) ≥ 0 -V (x + ) -(1 -η)V (x) + s 1 ( T , ỹ) -s 2 ( M , ỹ) ≥ 0
and the proof is completed following the steps in the proof of Theorem 3.1.

LMI conditions for uncertain systems

The relations (3.42) and (3.43) can be written in the generic quadratic forms respectively on vectors ζ and ξ with an affine dependence on matrices P j , T j , T uj , Tj , M j , M uj , Mj , R and R u . Hence, conditions in LMI form can be obtained to ensure (3.42) and (3.43). This is formalized in the following corollary to Theorem 3.2.

Corollary 3.2

If there exist matrices P 1 ∈ S n , P 2 ∈ R n×ny , P 3 ∈ S ny , T ∈ D ny , T uj ∈ D ny , T ∈ D 2ny , symmetric matrices M ∈ P (1+2ny)×(1+2ny) , M u ∈ P (1+2ny)×(1+2ny) and M ∈ P (1+4ny)×(1+4ny) , R ∈ R 1+n+2ny , R u ∈ R 1+n+2ny , R ∈ R 1+2n+4ny and positive scalars 0 < η < 1, 1 and 2 such that the following LMIs are verified

Ī P j Ī -1 Ī x Īx + 1 2 He Ī φ T j Īφ-y -Ī χ M j Īχ + 1 2
He (RS j ) ≥ 0 1+n+ny

(3.44a)

-Ī P j Ī + 2 Ī x Īx + 1 2 He Ī φ T uj Īφ-y -Ī χ M uj Īχ + 1 2 He (R u S j ) ≥ 0 1+n+ny (3.44b) -Ī + P j Ī+ + (1 -η) Ī 0 P j Ī0 + 1 2 He Ī φ Tj Ī φ-ỹ -Ī χ Mj Īχ + 1 2
He R Sj ≥ 0 1+2n+4ny

Īχ =   1 0 1,2n+2ny Ī φ Ī φ-ỹ   ,
then the origin of (3.17) is globally exponentially stable.

Proof. Consider V (x) defined as in (3.18). To show that if (3.27a), (3.27b) and (3.27c) hold then the conditions (3.21) and (3.22) in Theorem 3.1 also hold. Consider vectors ζ and ξ in (3.32) and note that x φ(y(x))

= Īζ,

x = Īx ζ φ(y(x)) = Īφ ζ φ(y(x)) -y(x) = Īφ-y ζ,   1 φ(y(x)) φ(y(x)) -y(x)   = Īχ ζ and x + φ(y + (x)) = Ī+ ξ, x φ(y(x)) = Ī0 ξ, φ(ỹ(x)) = Ī φξ φ(ỹ(x)) -ỹ(x) = Ī φ-ỹ ξ   1 φ(ỹ(x)) φ(ỹ(x)) -ỹ(x)   = Īχ ξ.
We have

ζ Ī P Ī -1 Ī x Īx + 1 2 He Ī φ T Īφ-y -Ī χ M Īχ + 1 2 He (RS j ) ζ = V (x) -1 x x + s 1 (T, y) -s 2 (M, y) + ζ RS j ζ ζ -Ī P Ī + 2 Ī x Īx + 1 2 He Ī φ T u Īφ-y -Ī χ M u Īχ + 1 2 He (R u S j ) ζ = -V (x) + 2 x x + s 1 (T u , y) -s 2 (M u , y) + ζ R u S j ζ ξ -Ī + P Ī+ + (1 -η) Ī 0 P Ī0 + 1 2 He Ī φ T Ī φ-ỹ -Ī χ M Īχ + 1 2 He R Sj ξ = -V (x + ) -(1 -η)V (x) + s 1 ( T , ỹ) -s 2 ( M , ỹ) + ξ R Sj ξ

Numerical Examples

In this section, we illustrate the results of Theorem 3.1 with four numerical examples. In the first example, we demonstrate the global stability of a piecewise linear system. In the second example, we analyze the global stability of a linear system subject to actuator saturation. A third example treats a benchmark example of MPC control laws. Finally, an example illustrates how partition uncertainties are handled by considering the uncertain matrices in equation (3.2b)

Piecewise Quadratic function for Global Stability

Consider a piecewise linear system given by (3.17) with

F 1 = 0.5 0.1 + κ -1 0.5 F 2 = κ 1 1 0 0
and F 3 , F 4 and f 5 as in (3.3) and where κ is a scalar. We try to obtain the largest value of κ for which we can prove the stability of the origin of (3.17) with the above data. Note that obtained system is homogeneous of degree 1, namely f (λx) = λf (x) for λ ≥ 0. Applying the conditions of Theorem 3.1 through the LMI formulations in Corollary 3.1, we obtain the largest value allowing to show globally stable for κ = 0.699. For comparison, the dual problem formulation presented in [58, Section II] demonstrates that there does not exist a quadratic Lyapunov function for this system, that is V (x) = x P 1 x, with P 1 ∈ S n , that certifies the stability for κ ≥ 0.357, and through simulation, we observe that the origin of the system is stable for -0.35 < κ < 0.7. We also test the method proposed in [START_REF] Feng | Stability analysis of piecewise discrete-time linear systems[END_REF], using a piecewise quadratic Lyapunov function (using all possible transitions between sets). With that method, the stability limit for parameter was bounded by κ ≥ 0.51 , which shows that our conditions lead to less conservative results.

The computed Lyapunov function (3.18) for this system is defined by matrix Note that the matrix P is not positive definite. Indeed, the positive definiteness of matrix P is not imposed by the conditions in Theorem 3.1. However, since (3.21) holds we have that the Lyapunov function is guaranteed to be positive definite. Some trajectories of the system are shown in Figure 3.4, along with the level sets of the decreasing Lyapunov function.

P =     2.
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

Asymmetric Saturation

Consider the following system, taken from [START_REF] Drummond | Generalized absolute stability using Lyapunov functions with relaxed positivity conditions[END_REF], discretized with a sampling period of 100ms, and subject to asymmetric actuator saturation

x + = Ax + Bsat [-1,15] (Kx)
with A = 0.9464 0.0957 -0.9568 0.9033 , B = 0.0049 0.0959 , K = 9.9000 0.4950 .

From (3.4) we have that the right hand side of the above system is written as (3.17) with f (x) defined by

F 1 = A + BK, F 2 = -B B
and F 3 , F 4 and f 5 as in (3.4).

It can be shown (see [58, Section II]) that there does not exist a common quadratic Lyapunov function for the linear systems defined by A and (A + BK). Since the quadratic global stability of a linear system subject to a saturating linear state feedback imposes the existence of a common Lyapunov function for the open-loop and the closed-loop without saturation, we conclude that there is no quadratic function to assess the global stability of the origin of system [START_REF] Burgat | Stability and control of saturated linear systems[END_REF]. However, considering a piecewise quadratic Lyapunov function as in (3.18) and applying Theorem 1, we can certify that the origin is globally exponentially stable with This matrix was obtained from the solution to the LMIs described in Corollary 3.1.

In Figure 3.5, a trajectory of the system and the level sets of the decreasing Lyapunov function are depicted.

Explicit MPC law

Consider the following closed-loop system

x + = Ax + Bu, A = 0.7326 -0.0861 0.1722 0.9909 , B = 0.0609 0.0064 , with u given by the explicit MPC law computed in [START_REF] Bemporad | The explicit linear quadratic regulator for constrained systems[END_REF] leading to the explicit PWA representation (II.1) with the partition defined in Table 3.1.

The MPC control law can be expressed by the closed-loop system (3.17), with f (x) in (3.2) with

F 1 = A + BK 1 , F 2 = B 1 -1 1 -1 φ(y) F 3 =     K 2 -K 1 K 1 -K 2 -K 1 K 1     , F 4 =     0 0 0 0 1 0 0 0 -1 1 0 0 1 -1 1 0     , f T 5 = -0.6423 -0.6423 -2 -2 K 1 = -5.9220 -6.8883 , K 2 = -6.

Uncertainty in the Partition

Consider an uncertain piecewise linear system given by (3.17) with

F 1 = 0.85 0.25 -0.8 0.8 , F 2 = -1 1 1 0 0 0 , F 3 =   -0.15 0.15 0.15 -0.15 d 1 d 2   , F 4 = 0 3 , f 5 =   -1 -1 0   ,
For this example, the enumeration of possible transitions between the sets in the partition is not possible since the number of regions may vary according on the values of the parameters d 1 and d 2 , as can be seen in Figure 3.7.

x 1 x 2 Γ 3 Γ 4 Γ 2 Γ 1 x 1 x 2 Γ3 Γ4 Γ2 Γ1 Γ5 Γ6
      .
This example shows the that the proposed approach allows for a simpler formulation of the uncertainty analysis. Thanks to the implicit parametrization of the vector field and the Lyapunov functions we can easily formulate the conditions for the transitions between uncertain sets.

Relation to other representations

We present two PWA models that result in equation (3.2b) with a structured matrix F 4 . As discussed above, a structured matrix F 4 may give an explicit solution to (3.2b).

MMPS functions

This section relates the representation (3.2) to max-min-plus-scaling (MMPS) models, which are equivalent to other models discussed in [START_REF] Heemels | Equivalence of hybrid dynamical models[END_REF]. From the arguments detailed in [START_REF] Heemels | Equivalence of hybrid dynamical models[END_REF] we can then conclude on the equivalence to the other models studied in that paper.

An MMPS function is a mapping f : R n → R which is recursively defined by a grammar [START_REF] Schutter | Model predictive control for max-min-plus-scaling systems[END_REF] g(x, g k , g

):=(x i |α| max(g k , g )| min(g k , g )|g k ) + (g |βg k ) (3.48)
where g k and g are themselves MMPS expressions. The symbol "|" in expression (3.48) denotes an "or" operator (see details in [START_REF] Schutter | Model predictive control for max-min-plus-scaling systems[END_REF]).

• To obtain (3.2) from an MMPS expression it suffices to consider the identities max(g k , g

) = g k + r(g -g k ), min(g k , g ) = -max(-g k , -g ),
and perform the composition of terms using the corresponding expressions g i .

• To obtain an MMPS model from (3.2), we can write the ramp function as the MMPS function r(y i (x)) = max(0, y i (x)) (3.49) which is an expression (3.48) with g k = 0 and g = y i . Hence, using (3.49), we can write f (x) in (3.2) as an MMPS expression as in the example below.

Consider Example 2 above (3.4) with n f = 1, n = 2, giving the scalar equations

     f (x) = K 11 x 1 + K 12 x 2 -r(y 1 ) + r(y 2 ) y 1 = K 11 x 1 + K 12 x 2 -µ y 2 = -K 11 x 1 -K 12 x 2 + µ.
With (3.49), we obtain

f (x) = K 11 x 1 + K 12 x 2 -max(0, K 11 x 1 + K 12 x 2 -µ) + max(0, -K 11 x 1 -K 12 x 2 + µ), (3.50)
which can be expressed as an MMPS with

g 1 = x 1 , g 2 = x 2 , g 3 = K 11 g 1 , g 4 = K 12 g 2 , g 5 = g 3 + g 4 , g 6
= -µ, g 7 = µ, g 8 = (-1)g 5 , g 9 = g 5 + g 6 , g 10 = g 7 + g 8 , g 11 = max(0, g 9 ), g 12 = max(0, g 10 ), g 13 = (-1)g 11 g 14 = g 12 + g 13 , f (x) = g 5 + g 14 .

Using the expression (3.48) it is possible to extract a tree structure using the above expressions, where a node has two children nodes in case both terms in the sum (3.48) are not zero, this is the case for nodes g 5 , g 9 , g 10 , g 14 , and f . The nodes with a single child can be obtained by setting either α = 0 or β = 0 in (3.48). The end nodes correspond to the input values of x 1 and x 2 or constants as in g 6 and g 7 .

PWA Canonical Representation

We now relate the proposed representation (3.2) to the canonical representation for PWA functions [START_REF] Kahlert | The complete canonical piecewise-linear representation. I. The geometry of the domain space[END_REF][START_REF] Julian | The complete canonical piecewise-linear representation: functional form for minimal degenerate intersections[END_REF]. In particular, we show that from the canonical representation it is always possible to obtain a representation as in (3.2) with a lower triangular block structure for matrix F 4 . We briefly recall the main definitions of the canonical representation, as presented in [START_REF] Julian | The complete canonical piecewise-linear representation: functional form for minimal degenerate intersections[END_REF]. The basic element for obtaining the canonical form are the N h hyperplanes generating the partition of the state space. Each of these hyperplanes can be described by a PWA function with | • | the absolute value of a scalar argument, from which a family of nested functions γ k can be obtained using (3.52) and defining

q i : R n → R, i = 1, . . . , N h , as in {x ∈ R n | q i (x) = α i x + β i }, (3.51)
α i ∈ R n , β i ∈ R. Furthermore, a generating function γ : R × R → R is defined as follows γ(v 1 , v 2 ) = || -v 1 | + v 2 | -||v 2 | -v 1 | + | -v 1 | + |v 2 | -| -v 1 + v 2 | (3.52)
γ 0 (v 1 ) = v 1 , γ 1 (v 1 ) = γ(v 1 , v 1 ), γ 2 (v 1 , v 2 ) = γ(v 1 , v 2 ), γ 3 (v 1 , v 2 , v 3 ) = γ(v 1 , γ 2 (v 2 , v 3 )), . . . γ k (v 1 , v 2 , . . . , v k ) = γ(v 1 , γ k-1 (v 2 , . . . , v k )). (3.53)
As stated in [98, Theorem 1], any PWA function f : R n → R can be expressed by a canonical form of level k, k ≥ 1, as

f (x) = a x + b + k j=1 N k (j) =1 c j, γ j (d (j), ,m 1
q(j), 1 x, . . . , d (j), ,m j q(j), j x) (3.54) a ∈ R n , b ∈ R, c j, ∈ R, m ∈ N where each pair d (j), ,m k ∈ R, q(j), ,j k ∈ R 1×n , k = 1, . . . , j
, is associated to the hyperplanes as in (3.51), and j corresponds to the order of a degenerate intersection from which the arguments of function γ j are computed. Roughly speaking, a degenerate intersection allows to create partitions that are more general than parallel hyperplanes thanks to the nested use of the absolute value in (3.52). The parameter N k (j) denotes the number of degenerate intersections of order j in the partition, and m is the index associated to one of the degenerated intersections of level j -1 that creates the intersection of level j with index l. More details about how to obtain the parameters c j, ∈ R, m ∈ N, d (j), ,m k ∈ R, q(j), ,j k ∈ R 1×n in (3.54), using an example in R 3 , can be found in [98, Section IV].

To relate the above description to the proposed representation in terms of ramp functions note that ∀θ ∈ R we have

|θ| = 2r(θ) -θ r(-θ) = r(θ) -θ
we can rewrite the terms in (3.52) as

γ(v 1 , v 2 ) =2r (2r(v 1 ) -v 1 + v 2 ) -(2r(v 1 ) -v 1 + v 2 ) -2r (2r(v 2 ) -v 1 -v 2 ) -(2r(v 2 ) -v 1 -v 2 ) + 2r (v 1 ) -v 1 + 2r (v 2 ) -v 2 -2r (-v 1 + v 2 ) -(-v 1 + v 2 ) = -2r (-v 1 + v 2 ) + 2r (2r(v 1 ) -v 1 + v 2 ) -2r (2r(v 2 ) -v 1 -v 2 ) + 2v 1 -2v 2
which, in turn, is expressed as (3.2) as follows

γ(v 1 , v 2 ) = 2 -2 v 1 v 2 + 0 0 -2 2 -2 φ(y) y =       1 0 0 1 -1 1 -1 1 -1 -1       v 1 v 2 +       0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0       φ(y)
that is, (3.2) using a structured algebraic equation (block triangular matrix F 4 ) for which an explicit expression can be easily obtained.

For each expression of γ in the recursive definition of γ k in (3.53), we can use the above matrices to obtain (3.54) using ramp functions. We can then compose the different terms in (3.53) to obtain vector functions f : R n → R n f , describing each level j, with vectors y (j) ∈ R n y(j) and corresponding vectors of ramp functions φ(y (j) ), leading to a representation (3.2)

Conclusions

In this chapter, we presented a framework for the stability analysis of discrete-time PWA systems. To this end, we introduced an implicit representation of PWA functions based on ramp functions. Since several other implicit models for PWA exist, we established the relation to some of these models.

The main advantage of the proposed representation concerns the analysis of PWA systems. Indeed, by exploiting some properties of ramp functions in the form of identities and inequalities, we verify Lyapunov inequalities related to piecewise quadratic Lyapunov functions candidates. This analysis is carried out by casting these inequalities in a generalized quadratic form depending on the ramp functions, leading to stability conditions given by LMIs. We illustrated numerical solutions of the proposed stability conditions in examples.

Importantly, in the proposed approach there is no need to define the quadratic function associated to each set of the partition since it is implicitly obtained with a generalized quadratic form. Moreover, there is no need to enumerate all possible transitions between sets in the partition. The use of continuous piecewise quadratic function is simpler than the methods in the literature since, thanks to the generalized quadratic forms involving ramp functions, no additional continuity conditions are required.

Regarding the verification of generalized quadratic forms, the use of properties associated to ramp functions applies only this class of function and therefore are less conservative than generic sector bounded conditions. For the case of uncertain PWA systems, we formulated stability tests allowing to consider uncertainties in the partition.

Chapter 4

Systems with Quantization nonlinearities

In this chapter, we focus on the stability analysis of discrete-time quantized control systems. The main goal of this chapter is to formulate sufficient conditions for global exponential stability analysis of systems with a finite number of quantization levels. The key result to obtain these conditions is to represent the regularized step function as an ill-posed algebraic loop containing ramp functions. Thanks to this representation and in contrast with the existing literature of quantized control systems, we do not rely on any sector bound approach and introduce a class of piecewise quadratic Lyapunov functions.

Linear Systems with input quantization

We consider a scenario in which a linear plant is controlled via an affine static state feedback law taking values into the set

Q := {0, δ 1 } × {0, δ 2 } × . . . {0, δ m },
where m ∈ N is the number of control inputs and δ i ∈ R, for all i ∈ {1, 2, . . . , m}, are some given levels. More specifically, we focus on the following class of nonlinear discrete-time systems

x + = Ax + B∆S(Kx + d) (4.1)
where

A ∈ R n×n , B ∈ R n×m , K ∈ R m×n , ∆ := diag{δ 1 , δ 2 , . . . , δ m }, d ∈ R m
, and S : R m → R m is defined as follows

S(u) :=      s(u 1 ) s(u 2 ) . . . s(u m )      (4.2)
where for all v ∈ R

s(v) := 1 if v > 0 0 if v ≤ 0 (4.3) v s(v) Figure 4.1: Step function s(v).
Due to the discontinuity of S at zero, (4.1) is a discontinuous dynamical system. In discrete-time dynamical systems, discontinuities do not lead to the technical problems found in their continuoustime counterparts (see, e.g., [START_REF] Cortés | Discontinuous dynamical systems[END_REF][START_REF] Ceragioli | Discontinuities and hysteresis in quantized average consensus[END_REF][START_REF] Ferrante | On sensor quantization in linear control systems: Krasovskii solutions meet semidefinite programming[END_REF][START_REF] Ferrante | Stabilization of continuous-time linear systems subject to input quantization[END_REF][START_REF] Filippov | Differential equations with discontinuous right-hand side[END_REF]). However, they generally lead to a lack of robustness, with stability properties being fragile in the presence of vanishing perturbations; see [START_REF] Goebel | Hybrid Dynamical Systems: Modeling, Stability, and Robustness[END_REF]Example 4.4,page 76].

To avoid this lack of robustness, we consider the following set-valued regularization of (4.1)

x + ∈ Ax + B∆S(Kx + d) (4.4a)
where the set-valued mapping S : R m ⇒ R m is defined as follows

S(u) :=      s(u 1 ) s(u 2 ) . . . s(u m )      (4.4b)
with, for all v ∈ R, Observe that, due to S(R m ) ⊂ S(R m ), solutions to (4.1) are solutions to (4.4a). Thus, stability properties concerning all solution to (4.4a) carry over (4.1).

s(v) :=      1 if v > 0 0 if v < 0 [0, 1] if v = 0. (4.4c) v s(v)

Remark 4.1

Clearly, S contains the so-called (discrete-time) Krasovskii regularization of the step function S, which writes as

S(u) :=      s(u 1 ) s(u 2 ) . . . s(u m )      , with, for all v ∈ R s(v) =      1 if v > 0 {0, 1} if v = 0 0 if v < 0.
see, e.g., [START_REF] Goebel | Hybrid Dynamical Systems: Modeling, Stability, and Robustness[END_REF]Definition 4.13]. Therefore, (4.4a) captures all possible solutions to (4.1) obtained by introducing vanishing state perturbations, i.e., Hermes solutions; see [START_REF] Goebel | Hybrid Dynamical Systems: Modeling, Stability, and Robustness[END_REF]Chapter 4]. Thus ensuring that the origin of (4.1) is robustly stable in the presence of vanishing perturbations, thereby making our results appealing in practice. The fact that we consider a larger regularization of (4.1) than its Krasovskii regularization is due to the approach we propose in Section 4.2 to represent S that inherently requires S(u) to be convex-valued for all u ∈ R m .

The following proposition concerns the solutions to (4.4a). For definitions of maximal and complete solutions and details about solutions to difference inclusions as (4.4a) see Section 4.3. Proposition 4.1. For any ξ ∈ R n , there exists a maximal solution σ to (4.4a) such that σ(0) = ξ. Moreover, σ is complete.

Proof. The proof follows simply from the fact that S is defined everywhere; see, e.g., [70, Proposition 2.10].

KKT characterization of the set-valued step mapping

In this subsection we illustrate the key result of this chapter. This result yields a tight characterization of the mapping s in (4.4c) in terms of linear and quadratic inequalities. To achieve this goal, we pursue a similar approach as in [START_REF] Primbs | Kuhn-Tucker-based stability conditions for systems with saturation[END_REF] and rely on optimization-based representation of the mapping s along with Karush-Kuhn-Tucker (KKT) optimality conditions.

Let us observe that, for all θ ∈ R, we can express (4.4c) as

s(θ) ∈ arg min w∈[0,1] -θw. (4.5)
Clearly, if θ < 0, one has s(θ) = 0, if θ > 0, one has s(θ) = 1, while when θ = 0, s(θ) ∈ [0, 1], which is consistent with (4.4c).

Building upon (4.5), we can obtain a characterization of the mapping s via the application of the Karush-Kuhn-Tucker (KKT) optimality conditions. This is formally stated in the result given next.

Theorem 4.1

Let s be defined as in (4.4c), θ ∈ R, and s ∈ R. Then, the following items are equivalent

(i) s ∈ s(u) (ii) there exist λ 1 , λ 2 ∈ R such that -θ -λ 1 + λ 2 = 0 (4.6a) λ 1 s = 0 (4.6b) λ 2 (1 -s) = 0 (4.6c) -λ 1 ≤ 0 (4.6d) -λ 2 ≤ 0 (4.6e) -s ≤ 0 (4.6f) -1 + s ≤ 0 (4.6g)
Proof. Proof of (i) =⇒ (ii). Using (4.5), it follows that

s(θ) ∈ arg min w∈[0,1] -θw (4.7)
The Lagrangian associated to (4.7) writes

L θ (w, λ) = -θw + λ 1 λ 2 -1 1 w + 0 -1 .
From the Karush-Khun-Tucker (KKT) necessary conditions for optimality, one has that for any optimal solution w to (4.7), there exists a unique λ := (λ 1 , λ 2 ) such that

d dw L θ (w , λ ) = 0 (4.8a) λ 1 w = 0, (4.8b) λ 2 (1 -w ) = 0 (4.8c) λ 1 ≥ 0, (4.8d)
λ 2 ≥ 0, (4.8e)
w ≥ 0, (

w ≤ 1 (4.8g) which reads as (4.6). Hence, recalling that s is an optimal solution to (4.7), i.e., w = s satisfies (4.8a), the implication is established.

Proof of (ii) =⇒ (i). This implication can be readily shown by observing that since (4.7) is convex (for al θ), the satisfaction of (4.6) (KKT conditions) implies (4.7). This establishes the result.

As a consequence of the fact that S(u) is a decentralized nonlinearity composed by set-valued mappings s, Theorem 4.1 shows that for all u ∈ R m and s ∈ S(u), there exist λ 1 , λ 2 ∈ R m such that satisfying the above relations for each channel.

The result given next shows that the multipliers λ 1 and λ 2 introduced in Theorem 4.1 can be expressed as ramp functions of the input u.

Lemma 4.1

The set valued step mapping (4.4c) is expressed in terms of ramp functions as

s(θ) = r(y 1 ) (4.9a)
y 1 y 2 = 1 0 θ + 1 -1 1 1 r(y 1 ) r(y 2 ) + 0 -1 . (4.9b)
Moreover we have r(θ) = r(y 2 ).

Proof. To obtain the above expression, we consider (4.6) and set s(θ) = s * the solution of the parameterized problem (4.5). Next, relate the complementarity inequalities (4.6b)-(4.6c) to the ramp function complementarity inequality (3.8a) of some inputs y 1 and y 2 to be determined. To this end, let us set the relations

λ 1 s = r(y 1 )(r(y 1 ) -y 1 ) (4.10a) λ 2 (1 -s) = r(y 2 )(r(y 2 ) -y 2 ). (4.10b)
and associate the following terms to λ 1 and λ 2 λ 1 = (r(y 1 ) -y 1 ) (4.11a)

λ 2 = r(y 2 ). (4.11b)
we have that the relations (4.10)-(4.11) hold if

r(y 1 ) = s (4.12a) (r(y 2 ) -y 2 ) = (1 -s). (4.12b)
where y 1 and y 2 are defined below. Moreover, the inequalities of (3.8b)-(3.8c) respectively hold for any y 1 and y 2 since the dual feasibility inequalities (4.6d)-(4.6e) hold. Similarly, the inequalities (3.8c)-(3.8b) respectively hold for any y 1 and y 2 since the primal feasibility inequalities (4.6f)-(4.6g) hold.

With (4.11), we have that (4.6a) gives -θ -(r(y 1 ) -y 1 ) + r(y 2 ) = 0, therefore

y 1 = θ + r(y 1 ) -r(y 2 ), (4.13a)
and by adding the two equations in in (4.12) we can eliminate s to obtain

y 2 = -1 + r(y 1 ) + r(y 2 ). (4.13b)
Therefore, with (4.12a), (4.13) we obtain (4.9). We now show that r(θ) = r(y 2 ). Note that (4.13a) gives r(y 2 ) = θ + (r(y 1 ) -y 1 ).

From the complementarity conditions (4.10a), we have 0 = (r(y 1 ) -y 1 )s.

i) Since for θ > 0, s(θ) = 1, we conclude that (r(y 1 ) -y 1 ) = 0, and using the above equation we obtain r(y 2 ) = θ for θ ≥ 0. (

ii) From (4.12a) we have that for θ < 0, r(y 1 ) = 0, thus (4.13b) gives y 2 = r(y 2 ) -1 of which the unique solution is y 2 = -1. Following the definition of the ramp function we have r(y 2 ) = 0. Thus r(y 2 ) = 0 for θ < 0. (

iii) Similarly, for θ = 0, we have r(y 1 ) ∈ [0, 1], hence y 2 = r(y 2 ) -(-r(y 1 ) + 1) = r(y 2 ) -β with β ∈ [0, 1] of which a solution can be any y 2 ∈ [-1, 0], which gives r(y 2 ) = 0. Thus, combined with (4.14), we have r(y 2 ) = r(θ).

Remark 4.2

The implicit equation (4.9b) is not well-posed since for θ = 0 any y 1 ∈ [0, 1] gives a solution.

The matrix (I 2 -F 4 ∆) with

F 4 = 1 -1 1 1
is not invertible for ∆ = [ 1 0 0 0 ] and for ∆ = [ 0 0 0 1 ]. Therefore the inequality provided in Proposition 3.1 as a condition for well-posedness cannot hold.

With the above characterization of the set-valued step s, we can write system (4.4a) in terms of ramp functions as a PWA system with the mapping

x + = F 1 x + F 2 φ(y) (4.15a)
y = F 3 x + F 4 φ(y) + f 5 (4.15b)
for some y ∈ y(x) = {y ∈ R 2m | y -F 4 φ(y) = F 3 x + f 5 }, with matrices

F 1 = A (4.15c) F 2 = B∆ I m ⊗ 1 0 (4.15d)
F 3 = K ⊗ 1 0 (4.15e)
F 4 = I m ⊗ 1 -1 1 1 (4.15f)
f 5 = d ⊗ 1 0 + 1 m,1 ⊗ 0 -1 . (4.15g)
From Lemma 4.1, we have that

φ(y(x)) =            s(K 1 x + d 1 ) r(K 1 x + d 1 ) s(K 2 x + d 2 ) r(K 2 x + d 2 ) . . . s(K m x + d m ) r(K m x + d m )            .

PWQ Lyapunov Functions for systems with input Quantization

Since the systems we are studying are set-valued dynamics of the form

x + ∈ G(x) (4.16)
where x ∈ R n is the system state and G : R n ⇒ R n is a set-valued map, we need to recall some definitions of solutions to this class of systems since, for a given initial conditions, they might not be unique. A solution to (4.16) is any function σ : dom σ → R n with dom σ = N ∩ {0, 1, . . . , J} for some J ∈ N ∪ {∞} such that for all j ∈ dom σ, with j + 1 ∈ dom σ, σ(j + 1) ∈ G(σ(j)). We say that a solution σ is maximal if it cannot be extended and it is complete if max dom σ = ∞.

In particular, in the setting of this chapter, namely for system (4.4a), maximal solutions to (4.16) are complete.

The following notion of global exponential stability is used in the chapter.

Definition 4.1

We say the the origin is globally exponentially stable for (4.16) if there exists λ, κ > 0 such that any maximal solution σ to (4.16) satisfies

σ(j) ≤ κe -λj σ(0) ∀j ∈ dom σ.
The result below provides sufficient conditions global exponential stability of a the origin of (4.16). Those conditions are formulated in terms of Lyapunov inequalities involving a set-valued function V .

Theorem 4.2

Suppose that there exists V : R n ⇒ R, and positive real numbers c 1 , c 2 , c 3 , and p such that

c 1 x p ≤ max V (x) ≤ c 2 x p ∀x ∈ R n , (4.17) max V (g) -max V (x) ≤ -c 3 x p ∀x ∈ R n , ∀g ∈ G(x). (4.18)
Then, the origin is globally exponentially stable for (4.16).

Proof. For all x ∈ R n , define W (x) := max V (x). The proof of the statement follows directly by observing that W is a standard single-valued Lyapunov function for (4.16).

Remark 4.3

The use of set-valued Lyapunov functions for stability analysis has also been pursued in [START_REF] Goebel | Set-valued Lyapunov functions for difference inclusions[END_REF][START_REF] Angeli | Stability of leaderless discrete-time multi-agent systems[END_REF]. However, the approach outlined here is tailored to the stability analysis of system (4.4a).

The conditions given in Theorem 4.2 are in general difficult to check. To formulate conditions that can be more easily verified we will propose a result that provides sufficient conditions to satisfy those in Theorem 4.2. The main reason for introducing the above theorem is that we would like to construct a set-valued function given by the generalized quadratic form involving the regularized step function, namely the set-valued mapping

V (x) = y∈y(x) x φ(y) P x φ(y) . (4.19)
with y(x) given by the set of solutions to the algebraic loop (4.15b). In Figure 4.4, we depict (4. [START_REF] Blanchini | Set invariance in control[END_REF] for all possible values of y in y for a function (4.9) for x ∈ R and m = 2 with K = 1 -2 and d = -1 -1 and matrix

P =       1 0 0 0 0 0 1 0 0 0 0 0 -0.8 0 0 0 0 0 -2 0 0 0 0 0 -0.5      
.

We are now in a position to state the main result of this chapter. To provide sufficient conditions for global exponential stability of (4.4a) in the form of matrix inequalities. To this end, we use the following Lyapunov function candidate with V as in (4.19). To formulate conditions for the decrease of W at each time step for all possible values of in the set-valued mapping defining the system, let us define the algebraic loop

W (x) = max V (x, y) (4.20) x V (x) 0 -1 1 -2 2 -3 3 -4 4
ȳ = F 3 F 3 F 1 x + F 4 0 ny F 3 F 2 F 4 φ(ȳ) + f 5 f 5 (4.21)
with matrices in (4.15), and with ȳ decomposed as

ȳ = ȳ0 ȳ+ 0 . (4.22)
Note that the above algebraic loop has a block-triangular structure for the matrix multiplying φ(ȳ). Thus the components ȳ0 do not depend on the components ȳ+ 0 . Since the above algebraic loop may have multiple solutions, given that the matrix F 4 is the same as in (4.15f), the set of its solutions is given by the set

ȳ(x) = ȳ ∈ R 4m | ȳ - F 4 0 ny F 3 F 2 F 4 φ(ȳ) = F 3 F 3 F 1 x + f 5 f 5 .
For a given ȳ ∈ ȳ(x), from (4.15a), we obtain

x + = F 1 x + F 2 φ(ȳ 0 ).

Theorem 4.3

If there exist matrices P ∈ S (n+2m)×(n+2m) , T ∈ D 2m , M ∈ P (1+2m)×(1+2m) , T u ∈ D 2m , M u ∈ P (1+2m)×(1+2m) and positive scalars 1 and 2 such that,

∀ x ∈ R n , ∀ y ∈ y(x), V (x, y) -1 x x + s 1 (T, y) -s 2 (M, y) ≥ 0 (4.23a) -V (x, y) + 2 x x + s 1 (T u , y) -s 2 (M u , y) ≥ 0 (4.23b) matrices T ∈ D 6m , M ∈ P (1+6m)×(1+6m) and a scalar η ∈ (0, 1) such that ∀ x ∈ R n , ∀ y ∈ y(x), ∀ ȳ ∈ ȳ(x) -V (F 1 x + F 2 φ(ȳ 0 ), ȳ+ 0 ) -V (x, y) + ηx x + s 1 ( T , ỹ) -s 2 ( M , ỹ) ≥ 0 (4.24)
with ỹ = ȳ y , y as in (4.15b) ȳ as in (4.21) then the origin of (4.15) is globally exponentially stable.

Proof. Take W (x) as in (4.20). Since the inequalities (4.23) hold for all y ∈ y they hold in particular for W (x)

1 x x ≤ W (x) ≤ 2 x x (4.25) Let us denote V (x + , ȳ(x + )) = V (F 1 x + F 2 φ(ȳ 0 ), ȳ+ 0 )
We have i) if y(x) contains only one point, namely y(x) = y(x), then W (x) = V (x, y(x)) and x + is uniquely defined. In this case, ȳ can either be a) a singleton, in which case

W (x + ) = V (x + , ȳ(x + )) b) a set, in which case, W (x + ) = max ȳ∈ȳ(x) V (x + , y + 0 )
in both cases, since the decrease in (4.24) should hold for all ȳ ∈ ȳ(x) it will hold, in particular, for the value of ȳ yielding W (x + ).

ii) if y(x) is a set we have W (x) = max y∈y(x)

V (x, y). In this case, ȳ is a set since ȳ0 also belongs to a set. For any value in this set, we have x + = F 1 x + F 2 φ(ȳ 0 ) for some value in the solution ȳ (note that here, since we consider any solution to (4.21), ȳ0 can take a value for ȳ0 different from the value of y used to define W (x)). From the value of ȳ0 , the value of ȳ+ 0 can either be a) a singleton, in which case

W (x + ) = V (x + , ȳ(x + )) b) a set, in which case, W (x + ) = max ȳ∈ȳ(x) V (x + , y + 0 ).
Since, from (4.24), the decrease is guaranteed for any value in the sets y and ȳ, if we choose any value within these sets the decrease is guaranteed, in particular when choosing the max over these sets, which gives, according to the above cases for W (x + ) and W (x),

W (x + ) -W (x) ≤ -ηx x (4.26)
∀x, ∀x + . Thus W (x) is a single-valued Lyapunov function.

x V (x, y(x))

V (xa , y (xa )) V (x + a , y(x + a ))
x + a x a The set of values for V (x, y) is given in red in coordinate axis. In this case, the regularized dynamics give a set-valued mapping illustrated here by the set x + a , which in turn gives the set of values for V (x + , y + )) -depicted in blue in the coordinate axis. The theorem asks for all points in the green set to be below the red set. In this case the single valued function W is guaranteed to decrease.

LMI conditions

The relations (4.23) and (4.24) can be written in the generic quadratic with an affine dependence on the elements of matrix P . Hence, conditions in LMI form can be obtained to ensure (4.23) and (4.24). This is formalized in the following corollary to Theorem 4.3.

Corollary 4.1

If there exist matrices P 1 ∈ S n , P 2 ∈ R n×ny , P 3 ∈ S ny , T ∈ D ny , T ∈ D 2ny , symmetric matrices M ∈ P (1+2ny)×(1+3ny) and M ∈ P (1+6ny)×(1+6ny) , and positive scalars 0 < η < 1, 1 and 2 such that the following LMIs are verified

I P I -1 I x I x + 1 2 He I φ T I φ-y -I χ M I χ ≥ 0 1+n+2ny (4.27a)
-I P I + 2 I x I x + 1 2 He I φ T u I φ-y -I χ M u I χ ≥ 0 1+n+2ny (4.27b)
-

I + P I + + (1 -η)I 0 P I 0 + 1 2 He I φ T I φ-ỹ -I χ M I χ ≥ 0 1+n+4ny (4.27c)
where

I = 0 n+ny,1 I n+ny , I x = 0 n,1 I n 0 n,ny , I φ = 0 ny,1+n I ny , I φ-y = -f 5 -F 3 (I ny -F 4 ) , I χ =   1 0 1,n+ny I φ I φ-y   , I + = 0 n,1
F 1 0 n,ny F 2 0 n,ny 0 ny,1 0 ny,n 0 ny 0 ny I ny , I 0 = 0 n+ny,1 I n+ny 0 n+ny,2ny ,

I φ = 0 3ny,1+n I 3ny , I φ-ỹ =   -f 5 -F 3 (I ny -F 4 ) 0 ny 0 ny -f 5 -F 3 0 ny (I ny -F 4 ) 0 ny -f 5 -F 3 F 1 0 ny -F 3 F 2 (I ny -F 4 )   , I χ =   1 0 1,n+3ny I φ I φ-ỹ   ,
then the origin of (4.1) is globally exponentially stable.

Proof. Consider V (x) defined as in (4.19). To show that if (4.27a), (4.27b) and (4.27c) hold then the conditions (4.23) and (4.24) in Theorem 4.3 also hold. Consider For this example, no common quadratic function exists to certify exponential stability of the matrices A and A + B K. This prevents from using a quadratic Lyapunov function to certify the global exponential stability of the origin. The proposed methodology instead enables to certify global exponential stability. We obtain

χ(x) =   1 x φ(y(x))   χ(x) =   1 x φ(ỹ(x))   with ỹ = y ȳ and note that x φ(y(x)) = Iχ(x), x = I x χ(x) φ(y(x)) = I φ χ(x) φ(y(x)) -y(x) = I φ-y χ(x),   1 φ(y(x)) φ(y(x)) -y(x)   = I χ χ(x) and x + φ(y + (x)) = I + χ(x), x φ(y(x)) = I 0 χ(x), φ(ỹ(x)) = I φ χ(x) φ(ỹ(x)) -ỹ(x) = I φ-ỹ χ(x)
P =         82.9572
3.8632 0.5475 0.0000 -0.5474 0.0000 3.8632 9.1279 0.0563 0.0000 -0.0563 0.0000 0.5475 0.0563 -0.0530 -0.8915 0.0530 0.0000 0.0000 0.0000 -0.8915 0.0000 0.0000 0.0000 -0.5474 -0.0563 0.0530 0.0000 -0.0530 -0.8915 0.0000 0.0000 0.0000 0.0000 -0.8915 0.0000

       
The Lyapunov function W (x) = max V (x) with V defined as in (4.4) is depicted in Figure 4.6. while Figure 4.7 depicts level sets of the corresponding function along with a trajectory of the system.

Binary Control

In this second example, we take B and K as in

Stabilization with a finite alphabet

In this example we select B = I 2 and consider a scenario in which the control input takes values in the set Q := {(0, 0), (0, -1), (-1, 0), (-1, -1)}, depending on the value of the state. In particular, we consider

ϕ(x) =                                      0 0 if x ∈ (-∞, 1] × (-∞, 1] -1 0 if x ∈ (1, ∞) × (-∞, 1] 0 -1 if x ∈ (-∞, 1] × [1, ∞) -1 -1 if x ∈ (1, ∞) × (1, ∞) (4.30)
To analyze (4.29) via the tools presented in this chapter, we rewrite ϕ as ϕ(x

) = S x 1 -1 x 2 -1 ,
where S is the step function defined in (4.4c). This enables us to rewrite (4.29) as (4.4a) with K = I 2 , , d = -1 2 , and ∆ = I 2 . We obtain

P =        
9.6376 -0.0289 -2.9186 -0.0000 0.9474 0.0000 -0.0289 1.0155 -0.5239 -0.0000 -0.3571 0.0000 -2.9186 -0.5239 1.6011 0.3795 -0.0009 0.0005 -0.0000 -0.0000 0.3795 -0.0000 -0.1719 0.0000 0.9474 -0.3571 -0.0009 -0.1719 0.4369 0.6290 0.0000 0.0000 0.0005 0.0000 0.6290 -0.0000

       
.

The Lyapunov function W (x) = max V (x) with V defined as in (4.4) is depicted in Figure 4.9. while Figure 4.10 depicts level sets of the corresponding function along with a trajectory of the system.

Conclusions

This chapter proposed a characterization of the set-valued step mapping based on quadratic/linear constraints stemming from KKT necessary conditions for optimality to study the stability of finitelevel quantized feedback control systems. Based on the proposed characterization, we use a generalized quadratic set-valued Lyapunov function to study the stability of linear systems in feedback with quantization functions. We give sufficient LMI conditions for the global exponential stability of the origin of the studied discontinuous nonlinear control systems. Three numerical illustrate the effectiveness of the methodology, which have highlighted the potential of our approach in systematically generating generalized quadratic Lyapunov functions.

An essential aspect of the proposed results is that it does not rely on sector representation of the nonlinearities. As discussed in Chapter 2, it is difficult to use nonquadratic Lyapunov functions to study systems with nonlinearities without slope restrictions. This chapter offers an alternative to the simple quadratic functions for global stability analysis of systems with discontinuous nonlinearities.

A preliminary version of the results in this chapter is presented in [START_REF] Valmorbida | On quantization in discrete-time control systems: Stability analysis of ternary controllers[END_REF]. The main difference here is that we carry out the analysis of a set-valued regularized version of the discontinuous dynamics in (4.1). This ensures that, despite the discontinuous nature of the right-hand side of (4.1), the stability properties certified via our methodology are robust to vanishing perturbations. Such an extension naturally leads to the use of set-valued Lyapunov functions, which require proper handling.

Perspectives and Concluding Remarks

systems, the dynamics on the boundary between regions must be studied separately [START_REF] Della Rossa | Max-min Lyapunov functions for switched systems and related differential inclusions[END_REF] as sliding modes can appear.

The main challenges to extend the results of Part II to continuous-time systems in an implicit form, with either continuous or discontinuous vector fields, are related to the use of more complex LF than simple quadratic functions. The main difficulties are related to the non-differentiability of the ramp function, as discussed below.

Consider the generalized quadratic forms as the LF candidates

V (x) = 1 2
x φ(ξ(x))

P 1 P 2 P 2 P 3 x φ(ξ(x)) (4.32a) ξ = C ξ x + D ξ φ(ξ) + e ξ , (4.32b)
where the parameters P 1 ∈ R n×n , P 2 ∈ R n×m , P 3 ∈ R m×m are to be computed, and the parameters C ξ ∈ R m×n , D ξ ∈ R m×m , e ξ ∈ R m×1 are given. The algebraic loop defining the value of ξ for each x, is assumed to be well-posed, thus resulting in a continuous function. The matrices in the above algebraic loop can be different from the matrices defining the partition of the continuous-time PWA system through variable y in (4.31). The difficulties of using the above function as an LF candidate are related to the fact that the ramp functions are not differentiable. Indeed, the expression for its derivative along the trajectories of (4.31) is V = x φ(ξ(x))

P 1 P 2 P 2 P 3 dx(t) dt dφ(ξ(x(t))) dt (4.33)
where the term dφ(ξ(x(t))) dt , is not differentiable ∀t for all possible trajectories since the ramp function defining the vector φ(ξ) is not differentiable at the origin.

Remark P.4

For the particular case of saturating systems, the above generalized quadratic forms were used in [START_REF] Dai | Piecewise-quadratic Lyapunov functions for systems with deadzones or saturations[END_REF][START_REF] Garulli | Global stability and finite L 2mgain of saturated uncertain systems via piecewise polynomial Lyapunov functions[END_REF][START_REF] Valmorbida | Regional L 2m gain analysis for linear saturating systems[END_REF]. In these papers, dφ(ξ(x)) dt is treated as an independent variable satisfying some identities in the same spirit as the sector inequalities. It is, however, important to show that the set where the function is not differentiable is not an invariant set, which roughly speaking, guarantees the decrease of the LF for almost all time. This way, it is still possible to use the above function with some conservatism. On the other hand, in light of the results of Part II, we can exploit the set-valued step mapping to represent the generalized derivative of the step as discussed below.

The following lemma, generalizes Lemma 4.1 and will be instrumental to represent regularized discontinuous functions and generalized derivatives.

Lemma P.2

The set-valued step mapping of magnitude ρ ≥ 0 on variable θ is expressed in terms of ramp functions as s(θ, ρ) = r(η 1 ) (4.34a)

η 1 η 2 = θ -ρ + 1 -1 1 1 r(η 1 ) r(η 2 ) . ( 4

.34b)

We do not provide the proof of the lemma as it follows closely the proof of Lemma 4.1.

As indicated in the above lemma, relation (4.37b) holds true only for non-negative values of the scalar ρ, which gives the step magnitude. For negative values of ρ, the following Lemma applies Lemma P.3

The set-valued step mapping of magnitude ρ ≤ 0 on variable θ is expressed in terms of ramp functions as s(θ, ρ) = -r(η 1 ) (4.35a) To allow for both positive and negative values ρ of the step we may then write separately two expressions for the positive and the negative magnitudes s(θ, ρ) := s(θ, r(ρ)) -s(θ, r(-ρ)) (4.36) which, using (4.34) to write one expression for s(θ, ρ) and another expression for s(θ, -ρ), becomes

η 1 η 2 = θ ρ + 1 -1 1 1 r(η 1 ) r(η 2 ) . ( 4
s(θ, ρ) = r(η 1 ) -r(η 3 ) (4.37a)     η 1 η 2 η 3 η 4     =     1 0 0 0 -1 0 1 0 0 0 0 -1       θ r(ρ) r(-ρ)   +     1 -1 0 0 1 1 0 0 0 0 1 -1 0 0 1 1         r(η 1 ) r(η 2 ) r(η 3 ) r(η 4 )     (4.37b) alternatively s(θ, ρ) = 1 0 -1 0 0 0         r(η 1 ) r(η 2 ) r(η 3 ) r(η 4 ) r(η 5 ) r(η 6 )         (4.38a)         η 1 η 2 η 3 η 4 η 5 η 6         =         1 0 0 0 1 0 0 0 0 1 0 -1         θ ρ +         1 -1 0 0 0 0 1 1 0 0 -1 0 0 0 1 -1 0 0 0 0 1 1 0 -1 0 0 0 0 0 0 0 0 0 0 0 0                 r(η 1 ) r(η 2 ) r(η 3 ) r(η 4 ) r(η 5 ) r(η 6 )         . (4.38b)
Note that the above general step function is also given by the product of a unit step function and its magnitude s(θ, ρ) = s(θ, 1)ρ.

(4.39)

The lemma below states that the term dφ(ξ(x)) dt in the expression of V in (4.33) is an implicit algebraic expression in terms of ramp functions. with N 0 = 1 0 -1 0 0 0 ,

N 11 =         1 0 1 0 0 0         N 12 =         0 0 0 0 1 -1         N 2 =         1 -1 0 0 0 0 1 1 0 0 -1 0 0 0 1 -1 0 0 0 0 1 1 0 -1 0 0 0 0 0 0 0 0 0 0 0 0        
.

The proof of the Lemma is given at the end of this chapter.

With the above results, we have V = x φ(ξ(x))

P 1 P 2 P 2 P 3 A B 0 n,n ξ 0 n,nη 0 n ξ ,n 0 n ξ ,ny 0 n ξ ,n ξ F     x φ(y(x)) φ(ξ(x)) φ(η(x))    
with y, ξ and η, given by the solution to the algebraic loops (4.31b), (4.32b), and (4.40b). These algebraic relations can be combined in a single algebraic equation as

  y ξ η   =   C C ξ C   x +   D 0 ny,n ξ 0 ny,nη 0 n ξ ,ny D ξ 0 n ξ ,nη Dy Dξ D     φ(y) φ(ξ) φ(η)   +   e e ξ ē   . (4.42)
Hence -V is a PWQ function, and its global non-negativity can be checked using the conditions in Proposition 3.2. The PWQ structure for the LF candidate may be used in different problems. Some of these are discussed in the sections below.

P.3 Linear switching and polytopic uncertain systems

Linear Differential/Difference Inclusions (LDI) [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF]Chapter 4] for CT and DT systems model a broad class of uncertain time-invariant and time-varying linear systems, including LPV systems with arbitrary variations, switching systems, and nonlinear systems with sector inequalities. It has been shown that the stability of the switching system is equivalent to the stability of the differential inclusion resulting from the polytopic convex hull of systems [START_REF] Molchanov | Criteria of asymptotic stability of differential and difference inclusions encountered in control theory[END_REF]. Also, for exponentially stable LDI, the existence of a convex Lyapunov function of quadratic growth has been shown to be necessary and sufficient [START_REF] Molchanov | Criteria of asymptotic stability of differential and difference inclusions encountered in control theory[END_REF] for stability. Notably, the universal classes that should be considered are norms [START_REF] Barabanov | Absolute characteristic exponent of a class of linear nonstatinoary systems of differential equations[END_REF]. Note that the convexity of the LF for these classes is in contrast with the class of switched systems, where non-convex functions might be needed [START_REF] Blanchini | Stabilizability of switched linear systems does not imply the existence of convex Lyapunov functions[END_REF] as discussed in the section below.

In [START_REF] Molchanov | Criteria of asymptotic stability of differential and difference inclusions encountered in control theory[END_REF], two classes of homogeneous LF functions are suggested to study differential inclusions: PWQ functions and polynomial forms. Even if the computation of these universal Lyapunov functions can be challenging, whenever the number of parameters is fixed, different computational methods [START_REF] Goebel | Conjugate convex Lyapunov functions for dual linear differential inclusions[END_REF][START_REF] Zelentsovsky | Nonquadratic Lyapunov functions for robust stability analysis of linear uncertain systems[END_REF][START_REF] Chesi | Homogeneous Lyapunov functions for systems with structured uncertainties[END_REF][START_REF] Xie | Piecewise Lyapunov functions for robust stability of linear time-varying systems[END_REF] have been put forward. For the max of quadratics proposed in [START_REF] Xie | Piecewise Lyapunov functions for robust stability of linear time-varying systems[END_REF], the stability conditions are cast as bilinear matrix inequalities. Based on the fact that the stability of dual LDI system is a necessary and sufficient condition for the stability of the primal LDI system [START_REF] Barabanov | Stability of inclusions of linear type[END_REF], in [START_REF] Goebel | Conjugate convex Lyapunov functions for dual linear differential inclusions[END_REF], the max of quadratics is used to show the stability of the dual LDI system whenever the same function does not show the stability of the primal LDI. An LF function for the primal LDI is then constructed from the convex conjugate function to the LF of the dual LDI. SDP solutions have been obtained for polynomial functions whenever the degree of the form is fixed [35]. The computation of LF is also pursued in [START_REF] Blanchini | Nonquadratic Lyapunov functions for robust control[END_REF] where polyhedral functions are searched with the solution to LP.

Moreover, for the class of uncertain polytopic systems, several methods consider quadratic growth LF with polynomial dependence on uncertain parameters describing the polytopic set instead of considering a parameter independent but not quadratic on the state [START_REF] Dasgupta | Lyapunov functions for uncertain systems with applications to the stability of time varying systems[END_REF][START_REF] Peaucelle | A new robust d-stability condition for real convex polytopic uncertainty[END_REF][START_REF] Daafouz | Parameter dependent Lyapunov functions for discrete time systems with time varying parametric uncertainties[END_REF][START_REF] Chesi | Polynomially parameter-dependent Lyapunov functions for robust stability of polytopic systems: an LMI approach[END_REF][START_REF] Oliveira | Parameter-dependent LMIs in robust analysis: Characterization of homogeneous polynomially parameter-dependent solutions via LMI relaxations[END_REF].

To study the stability of switching and uncertain systems in both CT and DT, we will parameterize homogeneous PWQ LF candidates as

V (x) = 1 2 φ(ξ(x)) I n ξ I n ξ P I n ξ I n ξ φ(ξ(x)) (4.43a) ξ = C ξ -C ξ x, (4.43b)
with a fixed matrix C ξ ∈ R n×n ξ , rank(C ξ ) = n and P ∈ S n ξ , such that V (x) = V (-x). The above form representation with ramp functions prevents additional constraints for the continuity of the LF at the boundaries as in [START_REF] Johansson | Computation of piecewise quadratic Lyapunov functions for hybrid systems[END_REF].

Another important motivation for the analysis with the PWQ function approach is its extension to design nonlinear feedback for switching and uncertain linear systems. The advantage of using a nonlinear, possibly discontinuous, feedback control law is to obtain better performance than linear feedback laws [20]. The performance can be measured as the increase of the achievable convergence rate or reduction of induced gains for systems with exogenous inputs. For continuous-time systems

as ẋ = Ax + Bu, (A, B) ∈ (A, B) ⊂ R n×n+m , (4.44)
we will focus on computing nonlinear control laws [START_REF] Goebel | Conjugate convex Lyapunov functions for dual linear differential inclusions[END_REF] as

u = K dV dx . (4.45)
with K ∈ R m×n . For the proposed class of PWQ systems, the resulting control laws will be homogeneous piecewise linear (PWL) functions, leading to a switched closed-loop system since the gain will depend on the partition induced by φ(ξ(x)). To see this, observe that

dV dx = φ(ξ(x)) I n ξ I n ξ P I n ξ I n ξ dφ(ξ(x)) dξ C ξ -C ξ . (4.46)
The discontinuity on dV dx appears due to the term dφ(ξ) dξ that is discontinuous in the set {x ∈ R n | ξ i (x) = 0, i = 1, . . . , n ξ }, ∀i ∈ n ξ . Note that dφ(ξi) dξi ∈ {0, 1} and dφ(ξi) dξj = 0 for i = j, thus the matrix dφ(ξ(x)) dξ is a diagonal matrix with diagonal elements in {0, 1}, thus giving

dφ(ξ) dξ = diag                     s(C ξ1 x, 1) . . . s(C ξn ξ x, 1) s(-C ξ1 x, 1) . . . s(-C ξn ξ x, 1)                     .
We then obtain

dV dx = φ(C ξ x) + φ(-C ξ x) P   diag       s(C ξ1 x, 1) . . . s(C ξn ξ x, 1)       -diag       s(-C ξ1 x, 1) . . . s(-C ξn ξ x, 1)          C ξ . (4.47)
since the terms φ(C ξ x) are PWL on x and the terms s(-C ξn ξ x, 1) are piecewise constant on x, the function dV dx is PWL with a conic partition defined by ξ = C ξ x. We will also study the characterization of the classes of systems that can be stabilized with the proposed switched laws. For the particular case of quadratic stabilizability of switching systems, these conditions have been proposed in [START_REF] Wicks | Construction of piecewise Lyapunov functions for stabilizing switched systems[END_REF].

P.4 Discontinuous PWA systems

Discontinuous vector fields appear in both CT and DT feedback loops due to different reasons:

• the plant dynamics changes according to some partition in the state space. This is the case for instance where some discontinuous forces, such as dry friction, appear in the system;

• technological constraints introduce only a finite number of levels in actuators or sensors as discussed in Chapter 4 for DT systems with input quantization;

• the control law is discontinuous. This type of discontinuity appears for performance improvement of the closed-loop either when finite-time convergence is sought or when sliding-mode control is adopted. Optimization-based control strategies can also introduce discontinuities, for instance, in the use of LP MPC [START_REF] Bemporad | Model predictive control based on linear programming -the explicit solution[END_REF].

The quantization control, perhaps the simplest case introducing discontinuities, has been studied as a sector nonlinearity [START_REF] Ferrante | Stabilization of continuous-time linear systems subject to input quantization[END_REF][START_REF] Fu | The sector bound approach to quantized feedback control[END_REF][START_REF] Coutinho | Input and output quantized feedback linear systems[END_REF]. These simple systems were then analyzed with quadratic LF in the CT and DT cases thanks to the sector inequalities. On the other hand, more complex discontinuous dynamics, such as the ones appearing in sliding-mode control laws [START_REF] Utkin | Sliding Modes in Control and Optimization[END_REF][START_REF] Shtessel | Sliding Mode Control and Observation[END_REF], are most often modeled as an explicit switched system. The models used for these discontinuous systems determine the methods for their analysis. In this context, we would like to formulate a unified framework to study discontinuous systems as discussed below.

Despite the progress in the study of switched systems [START_REF] Shorten | Stability criteria for switched and hybrid systems[END_REF], we point out the fact that the use of explicit representations has several flaws or difficulties. Perhaps the most significant one is the choice and parameterization of PWQ Lyapunov functions. Indeed, when formulating stability conditions, several constraints need to be explicitly imposed, such as the continuity of the function along the boundaries of the partition defining the LF (4.32a). Other difficulties are related to the test for the decrease of the LF only on the partition where it is active and the possible existence of sliding modes. Due to these reasons, the existing results lead to numerically tractable conditions only in simple instances. Moreover, the study of local stability, computation of induced gains in the presence of disturbances, and control design by extending approaches based on explicit representation are not appealing.

Using an algebraic loop to represent the set-valued step mapping allows us to describe other discontinuous mappings, such as the ones appearing in the switched systems. The advantage of the set-valued mappings is that they also provide a regularized version of the discontinuity since the set-valued maps appear only in the points of discontinuity [START_REF] Della Rossa | Max-min Lyapunov functions for switched systems and related differential inclusions[END_REF].

LF functions for switched systems need not be convex, as pointed out by [START_REF] Blanchini | Stabilizability of switched linear systems does not imply the existence of convex Lyapunov functions[END_REF]. Therefore, the general (non-differentiable) PWQ function defined in a partition should be parametrized. Note that the partition of the PWQ can be different from the partition of the system. Moreover, a single framework will simplify the numerical construction of Lyapunov conditions, reducing the number of constraints to be checked.

In the rest of this section, we give a simple example of using algebraic loops to represent discontinuous functions defined on conic partitions. Thanks to this representation, one can then use the (continuous) function in (4.32a) as the LF candidate for the switched system described by the proposed discontinuous functions. Following the discussion in the previous section, the derivatives of the PWQ LF have an implicit algebraic loop description. The approach to study this class of system would then consist of verifying just two inequalities: the Lyapunov inequality for positive-definiteness of LF and the negative-definiteness of its derivative. The method to check these inequalities is the same as in Proposition 3.2. With the obtained PWA implicit model and the more straightforward analysis conditions obtained, we will have a framework to study other problems such as the local stability (see Section P.5 below).

To motivate this problem, consider the following explicit planar CT switched system with two modes

ẋ = A 1 x if x 1 x 2 < 0 A 2 x if x 1 x 2 ≥ 0
that can be rewritten as

ẋ = A n x + 0 2,1 if x 1 x 2 < 0 A p x if x 1 x 2 ≥ 0 (4.48)
with A n = A 1 and A p = A 2 -A 1 . To write the linear expression A p x restricted to the set {x ∈ R 2 | x 1 x 2 ≥ 0} we can use an implicit equation to represent both x 1 and x 2 restricted to this set as in the lemma below. Lemma P.5

The mapping

x r (x) = H 1 φ(u) (4.49a) u = H 2 x + H 3 φ(u) (4.49b)
with H 1 = I 2 ⊗ 1 0 0 -1 0 0 ;

H 2 =     N 1 -N 1 N 2 -N 2     ; N 1 =   0 1 0 0 1 0   , N 2 =   1 0 0 0 0 1   ; H 3 = (I 4 ⊗ M ) ; M =   1 -1 0 1 1 -1 0 0 0   . satisfies x r (x) =    0 2,1 if x 1 x 2 < 0 co{0 2,1 , x} if x 1 x 2 = 0 x if x 1 x 2 > 0 (4.50)
The proof of the lemma is presented in the last section of this chapter.

With the above lemma, we have

A p x r (x) =    0 2,1 if x 1 x 2 < 0 co{0 2,1 , A p x} if x 1 x 2 = 0 A p x if x 1 x 2 > 0.
Hence, with (4.49), we have that (4.48) becomes the regularized switched system

ẋ = A n x + A p H 1 φ(u) (4.51a) u = H 2 x + H 3 φ(u), (4.51b)
which is a particular case of (4.31). Note that the above result is not restricted to a partition defined by the orthants. In the more general case where the switched planar system is defined in another conic partition defined by R ∈ S 2 as in ẋ

= A n x + 0 2,1 if x1 x2 < 0 A p x if x1 x2 ≥ 0 x = Rx (4.52)
the switched system is then given by ẋ

= A n x + A p R -1 H 1 φ(u) (4.53a) u = H 2 Rx + H 3 φ(u). ( 4

.53b)

Example To illustrate the above switched system, consider the following data taken from [START_REF] Blanchini | Stabilizability of switched linear systems does not imply the existence of convex Lyapunov functions[END_REF] A

n = 1 0 0 -1 , A p = γ -1 -1 1 γ + 1 , R = τ 1 1 (4.54) with τ = 1 γ+ √ γ 2 +1
and the numerical values of γ = 1.1, = 0.01. The partition is illustrated in

P.5 Conditions for the regional positivity of a PWQ function

To formulate conditions for the local stability of nonlinear PWA systems, we need a result that allows verifying the positivity of a PWQ function only in a set. This section presents a generalization of Proposition 3.2. To this aim, we first propose a straightforward generalization of Lemma 3.2 Given any function y : R n → R ny , and a set R ⊂ R n , for any matrix function M : R n → R (1+2ny)×(1+2ny) , satisfying

M (x) ∈ P (1+2ny)×(1+2ny) ∀x ∈ R, then s 2 (M (x), y(x)) :=   1 φ(y(x)) φ(y(x)) -y(x)   M (x)   1 φ(y(x)) φ(y(x)) -y(x)   ≥ 0 ∀x ∈ R. (4.55)
Given the structure of the generalized quadratic form in (4.55), we have that for particular structures of the matrix M (x) the resulting expression is still a generalized quadratic form. To observe this fact, consider the following structure for M (x)

M (x) = M 1,1 (x) M 1φ (x) M 1φ (x) M φ (4.56)
with M φ ∈ P 2ny×2ny and terms for the first row and column of M (x), M 1,1 : R n → R, M 1φ : R n → R 1×2ny given by

M 1,1 (x) =   1 φ(y(x)) φ(y(x)) -y(x)   M1,1   1 φ(y(x)) φ(y(x)) -y(x)   (4.57a) M 1φ (x) =   1 φ(y(x)) φ(y(x)) -y(x)   M1φ (4.57b)
with M1,1 ∈ R (1+2ny)×(1+2ny) and M1φ ∈ R (1+2ny)×(2ny) . With (4.56) we obtain for (4.55)

s 2 (M (x), y(x)) =   1 φ(y(x)) φ(y(x)) -y(x)   M   1 φ(y(x)) φ(y(x)) -y(x)   where M = M1,1 + He 0 1+2nn y ,1 M1φ + 0 0 1,2ny 0 2ny,1 M φ .
which is also a PWQ function. We can thus state the following corollary to the above lemma. Lemma P.7

Given any function y : R n → R ny , and matrices M1,1 ∈ R (1+2ny)×(1+2ny) , M 1φ : R n → R 1×2ny , as in (4.57), and a set R ⊂ R n , satisfying

M 1,1 (x) ∈ P, M 1φ (x) ∈ P 1×2ny ∀x ∈ R, (4.58)
then s 2 (M (x), y(x)) ≥ 0 ∀x ∈ R.

In the above lemma, (4.58) requires checking whether a piecewise quadratic function is positive within the set R. Let us consider a particular case of such a set R, described by the intersection PERSPECTIVES of a set of piecewise quadratic functions as

R =      x ∈ R n | r i (x) =   1 φ(y(x)) φ(y(x)) -y(x)   R i   1 φ(y(x)) φ(y(x)) -y(x)   ≥ 0, i = 1, . . . , n r      . (4.59)
The lemma below gives sufficient condition for set inclusion involving sets defined by PWQ functions.

Lemma P.8

Given a matrix S ∈ R (1+2ny)×(1+2ny) defining a PWQ function as

s(x) =   1 φ(y(x)) φ(y(x)) -y(x)   S   1 φ(y(x)) φ(y(x)) -y(x)   (4.60)
and matrices R i ∈ R (1+2ny)×(1+2ny) , i = 1, . . . , n r defining the set R as in (4.59), if there exist scalars β i ≥ 0, i = 1, . . . , n r such that

s(x) -β i r i (x) ≥ 0 ∀x ∈ R n , (4.61) then s(x) ≥ 0 ∀x ∈ R. Namely, {x ∈ R n | s(x) ≥ 0} ⊆ R.
Since the expressions in (4.61) are PWQ functions required to be globally non-negative, Proposition 3.2 can be used to to check the inequalities (4.61).

We can now present the following proposition to test the non-negativity of a PWQ function, paralleling the global results presented in Proposition 3.2.

Proposition P.1

Given a generalized quadratic form h (x) and a set R defined by matrices R i ∈ R (1+2ny)×(1+2ny) , i = 1, . . . , n r , if there exist matrices T ∈ D ny , M1,1 ∈ R (1+2ny)×(1+2ny) and M1φ ∈ R (1+2ny)×(1×2ny) , M φ ∈ P 2ny×2ny defining M (x) as in (4.56) such that

h (x) + s 1 (T, y(x)) -s 2 (M (x), y(x)) ≥ 0, ∀x ∈ R n (4.62)
and scalars β i ≥ 0, i = 1, . . . , n r , γ i,j ≥ 0, i = 1, . . . , n r , j = 1, . . . , 2n y such that

M 1,1 (x) -β i r i (x) ≥ 0 ∀x ∈ R n , (4.63a) M 1φj (x) -γ i,j r i (x) ≥ 0 ∀x ∈ R n , ∀j = 1, . . . , 2n y , (4.63b) then h(x) ≥ 0 ∀x ∈ R. (4.64)
The above proposition uses Lemma P.8 to guarantee (4.58), thus, according to Lemma P.7, that s 2 (M (x), y(x)) is not negative in the set R. Note that the entries of M 1φ as in (4.57b) are PWL functions, thus a particular case of PWQ function of the form (4.60), yielding (4.63b). The numerical verification of (4.62) and the conditions obtained with (4.63) and Proposition 3.2 can then be carried out with semidefinite programming following the lines of Corollary 3.1.

Proposition P.1 can be applied to any PWQ function. For the analysis of dynamical systems, its interest lies mainly in the local stability analysis. Whenever a region of interest R is fixed, similar conditions to Theorem 3.1 can be formulated, where the expressions for the positivity of the LF and the negativity of its derivative on the set R enter as the function h (x) in Proposition P.1. A second step in the regional analysis consists of computing an ERA using the inclusion conditions of Lemma P.8. These ERA can be determined by a level set of the LF included in the set R or by invariant sets that are not level sets of the LF as in [START_REF] Valmorbida | Region of attraction estimation using invariant sets and rational Lyapunov functions[END_REF].

P.6 Further research directions

In this section, we briefly mention other research directions stemming from the results presented in this manuscript.

• Robustness analysis to bounded disturbances. In Sections 3.3.2, 3.3.3, we introduced the analysis for uncertain PWA system given by polytopic differential inclusions. These results enabled studying systems with time-varying partitions. On the other hand, the robustness of the stability properties should also hold for exogenous disturbances. For DT systems, these exogenous disturbances may appear as in

x[k + 1] = F 1 x[k] + F 2 φ(y(x)) + w 1 [k] (4.65a) y(x[k]) = F 3 x[k] + F 4 φ(y(x[k])) + f 5 + w 2 [k] ( 4
(x[k]) = F 3 x[k] + F 4 φ(y(x[k])) + f 5 + w 1 [k]
where the value of w 1 indicates the error in the solution of the algebraic loop. The above equation indicates that the Input-to-state [START_REF] Sontag | New characterizations of input-to-state stability[END_REF] stability analysis of (4.65), suitable to bounded disturbances, is fundamental to evaluate the numerical errors related to the solution of the implicit equation.

• Model Predictive Control: Analysis and Implementation. A PWA control strategy issued from an optimization-based control is MPC [START_REF] Bemporad | The explicit linear quadratic regulator for constrained systems[END_REF]. For MPC strategies with QP or LP [START_REF] Bemporad | Model predictive control based on linear programming -the explicit solution[END_REF], our goal will be to directly obtain the control law in the form of an algebraic loop involving ramp functions without passing through its explicit representation [START_REF] Bemporad | The explicit linear quadratic regulator for constrained systems[END_REF][START_REF] Tøndel | Evaluation of piecewise affine control via binary search tree[END_REF]. Interestingly, the MPC can be readily written as a Linear Complementarity Problem (LCP) [START_REF] Camacho | Constrained Generalized Predictive Control[END_REF], we can then use the LCP form as the starting point to obtain an implicit equation using ramp functions.

The aim of such a representation will be twofold: for the analysis, it should allow studying directly closed-loop systems with MPC within the framework presented in this manuscript for PWA systems. We would also like to exploit the structured -given by the ramp functions -implicit equation to propose a strategy to solve the implicit equation issued from MPC to generate the control input. The root-finding strategies to solve the implicit equation exploiting the ramp functions will be compared with QP-based strategies.

• Neural Networks with ReLU activation functions

The results in Part II study PWA discrete-time systems thanks to the verification of the non-negativity of PWQ functions. An application of these techniques relates to the recent use of NN in different applications, including dynamical systems. An activation function used in NNs is the Rectifier Linear Unit ReLU, a niche term for ramp functions. Indeed, a DT linear dynamical systems in feedback with Rectifier Linear Unit Neural Network ReLU NN are PWA DT systems as the ones studied in Chapter 3. The study of dynamical systems with NN in the loop is not new [START_REF] Suykens | An absolute stability criterion for the Lur'e problem with sector and slope restricted nonlinearities[END_REF][START_REF] Barabanov | Stability analysis of discrete-time recurrent neural networks[END_REF] and have recently been revisited in [START_REF] Kim | Standard representation and unified stability analysis for dynamic artificial neural network models[END_REF], where the need for an LF to capture the slope properties was highlighted. Actually, already in [START_REF] Barabanov | Stability analysis of discrete-time recurrent neural networks[END_REF], it was observed that

In contrast, current stability methods can not distinguish between any two types of nonlinearities as long as these nonlinearities belong to the same sector, monotone and time-invariant. One avenue for future research is to develop methods enhancing discriminatory capabilities of the stability criteria for Recurrent NN, i.e., to permit them to distinguish between nonlinearities of different kinds, thereby mitigating their conservatism.

therefore, a tighter description of the activation function, like the one obtained with ramp functions, will help analyze the stability, reachability, and safety under disturbances.

Feedforward NNs can be expressed as explicit algebraic loops like the one detailed in (3.40). This fact has been considered to study dynamical NN controller in [START_REF] Kim | Standard representation and unified stability analysis for dynamic artificial neural network models[END_REF], and in [START_REF] Yin | Stability analysis using quadratic constraints for systems with neural network controllers[END_REF], where local sector bounds, similar to the ones used in Part I were applied to compute ERA.

On the other hand, for ReLU NN, the SDP formulations to solve Lyapunov inequalities can still be conservative since the positivity of the expressions can be formulated as a copositivity of a matrix [START_REF] Ebihara | l 2 induced norm analysis of discrete-time LTI systems for nonnegative input signals and its application to stability analysis of recurrent neural networks[END_REF]. Polynomial optimization should be considered to reduce this conservatism [START_REF] Parrilo | Structured Semidefinite Programs and Semialgebraic Geometry Methods in Robustness and Optimization[END_REF]Chapter 5]. Alternatives to LF analysis, with IQCs and ZF multipliers, should also be considered in this context [START_REF] Pauli | Linear systems with neural network nonlinearities: Improved stability analysis via acausal zames-falb multipliers[END_REF][START_REF] Yin | Stability analysis using quadratic constraints for systems with neural network controllers[END_REF].

Besides the stability analysis of dynamical systems with NN, another critical problem related to the robustness of NN classifiers is the estimation of Lipschitz constants of ReLU neural networks [START_REF] Chen | Semialgebraic optimization for lipschitz constants of ReLU networks[END_REF][START_REF] Pauli | Training robust neural networks using lipschitz bounds[END_REF]. Such a problem can be studied using the PWQ functions detailed in this manuscript. Moreover, fully interconnected NN with feedback paths lead to nonlinear implicit equations. Both structures -feedforward and feedback NN -can be used to approximate nonlinear functions, and the respective advantages are the simplicity to compute outputs and the compactness of representation. Recent results [START_REF] Ghaoui | Implicit deep learning[END_REF] investigate the implicit optimization, relying only on the implicit models. An understanding of ways to convert between the structured feedforward NN and fully implicit equations is still lacking.

• Linear programming to verify the non-negativity of PWL functions. Some results in the literature have used LP as the optimization approach for the study of linear and PWA system [START_REF] Blanchini | Set invariance in control[END_REF][START_REF] Milani | Piecewise-affine Lyapunov functions for discrete-time linear systems with saturating controls[END_REF][START_REF] Milani | Piecewise-affine Lyapunov functions for continuous-time linear systems with saturating controls[END_REF]. The use of LP contrasts with the results in this manuscript, where we have explored an SDP formulation to solve PWQ inequalities. For PWL inequalities, however, we would like to propose an LP-based method for inequality verification.

The main challenge in formulating LP tests for PWL inequalities based on implicit expressions is to add the complementarity condition. Namely the expression φ i (y)(φ i (y)-y) = 0 obtained to describe the ramp function, which is a quadratic expression on φ i (y) and y.

With an LP formulation at hand, we will extend the analysis of PWA systems with PWL LF candidates (yielding polyhedral LF), paralleling the SDP-based results presented in this manuscript for PWQ inequalities stemming from stability analysis. Also, the solution to dissipation inequalities using storage functions with linear growth seems to be the suitable tool for the assessment of induced L 1 and 1 gains.

• Develop a toolbox for PWQ programming. Algebraic operations on PWQ functions such as a sum of PWQ functions and products between PWL yielding PWQ may appear in the expressions we presented for stability analysis in this manuscript. So far, these expressions are obtained on a case-by-case basis.

To avoid the manual steps for the PWQ inequalities, we would like to develop a toolbox to manipulate these PWQ expressions. For instance, it should help automate merging two algebraic loops whenever the PWQ function contains ramp functions of different algebraic loops and help obtain the derivative of PWQ expressions.

Such a toolbox should also assist in the process of setting up sums of expressions containing variables to be set up as decision variables (Lyapunov function coefficient, for instance).

The main goal of such a numerical tool will be to let the user manipulate only scalarized expressions and define each algebraic loop in the PWQ and PWL functions. We believe this is a fundamental step to make the proposed analysis tools available to a larger public.

P.7 Proofs of the claims in the chapter

This section presents the proofs for some of the claims in the chapter.

dφ(ξ(x(t))) dt =         s ξ 1 , dξ1(x(t)) dt s ξ 2 , dξ2(x(t)) dt . . . s ξ m , dξm(x(t)) dt         . ( 4

.73)

With the above expression, we have that the time derivative of the vector ramp function is expressed as step functions whose level depends on the time derivative of each of its arguments. Thanks to the description of the step function as ramp functions, we now show that this derivative is also written as an algebraic loop involving ramp functions.

With (4.36) we have

s ξ i , dξ i (x(t)) dt = r(η i1 ) -r(η i3 ) (4.74a)     η i1 η i2 η i3 η i4     =     1 0 0 0 -1 0 1 0 0 0 0 -1       ξ i r( dξi(x(t)) dt ) r(-dξi(x(t)) dt )   +     1 -1 0 0 1 1 0 0 0 0 1 -1 0 0 1 1         r(η i1 ) r(η i2 ) r(η i3 ) r(η i4 )     (4.74b)
and, using (4.69), we obtain

s ξ i , dξ i (x(t)) dt = 1 0 -1 0 r(η i ) (4.75a) η i = Ci x + Dξi φ(ξ) + Dyi φ(y) + Di r(η) + ẽi +     1 -1 0 0 1 1 0 0 0 0 1 -1 0 0 1 1     r(η i ) (4.75b) with η i := η i1 η i2 η i3 η i4 , η := η 1 η 2 η 3 . . . η m and Ci =     1 0 1 0     C ξi +     0 -1 0 1     C ξi A Dξi =     1 0 1 0     D ξi Dyi =     0 -1 0 1     C ξi B Di =     0 -1 0 1     D ξi I m ⊗ 1 0 -1 0 , ẽi =     1 0 1 0     e ξi .
Finally, we can write dφ(ξ(x(t))) dt as the solution of the algebraic loop

dφ(ξ(x(t))) dt = F φ(η) (4.76a) η = Cx + Dξ φ(ξ) + Dy φ(y) + Dφ(η) + ē (4.76b) with F ∈ R m×4m , C ∈ R 4m×n , Dξ ∈ R 4m×m , Dξ ∈ R 4m×my , D ∈ R 4m×4m , and ē ∈ R 4m×1 as F = I m ⊗ 1 0 -1 0 C =     I m ⊗     1 0 1 0         C ξ +     I m ⊗     0 -1 0 1         C ξ A Dξ =     I m ⊗     1 0 1 0         D ξ Dy =     I m ⊗     0 -1 0 1         C ξ B D =     I m ⊗     0 -1 0 1              D ξ1 0 1,m . . . 0 1,m 0 1,m D ξ2 . . . 0 1,m . . . . . . . . . . . . 0 1,m 0 1,m . . . D ξm      I m ⊗ 1 0 -1 0 + I m ⊗     1 -1 0 0 1 1 0 0 0 0 1 -1 0 0 1 1     ē =     I m ⊗     1 0 1 0         e ξ .
P.7.2 Proof of Lemma P.5

Proof. We proceed by noticing that x 1 restricted to the set of where x 2 ≥ 0 can be expressed using the ramp function as x 1 = s(x 2 , 1)x 1 and, for x 1 ≥ 0, we have

x 1 = s(x 2 , 1)r(x 1 )
with (4.39), we obtain

x 1 = s(x 2 , r(x 1 )) that holds for {x ∈ R 2 | x 1 ≥ 0, x 2 ≥ 0}.
Using the expression for the step with arbitrary magnitude in (4.34), we obtain

x 1 = r(η 1 ) η 1 η 2 = x 2 -r(x 1 ) + 1 -1 1 1 r(η 1 ) r(η 2 )
that is

x 1 = r(η 1 ) (4.78a)   η 1 η 2 η 3   =   0 1 0 0 1 0   x 1 x 2 +   1 -1 0 1 1 -1 0 0 0     r(η 1 ) r(η 2 ) r(η 3 )   , (4.78b)
which holds for {x ∈ R

2 | x 1 ≥ 0, x 2 ≥ 0}.
Repeating the above steps, we have

x 2 = s(x 1 , r(x 2 )) (4.79)
that holds for {x ∈ R 2 | x 1 ≥ 0, x 2 ≥ 0}, which can be written as

x 2 = r(η 3 ) (4.80a)   η 4 η 5 η 6   =   1 0 0 0 0 1   x 1 x 2 +   1 -1 0 1 1 -1 0 0 0     r(η 4 ) r(η 5 ) r(η 6 )   , (4.80b)
Similarly, for {x ∈ R 2 | x 1 ≤ 0, x 2 ≤ 0}, we obtain

x 1 = -s(-x 2 , r(-x 1 )) x 2 = -s(-x 1 , r(-x 2 )).
which also give expressions similar to (4.78), (4.80). By merging the above expressions and the sets {x ∈ R

2 | x 1 ≥ 0, x 2 ≥ 0} and {x ∈ R 2 | x 1 ≤ 0, x 2 ≤ 0}, we obtain the following identity x 1 x 2 = 1 0 0 1 ⊗ 1 0 0 -1 0 0 φ(u) u =     N 1 -N 1 N 2 -N 2     x + (I 4 ⊗ M ) φ(u)
with

N 1 =   0 1 0 0 1 0   , N 2 =   1 0 0 0 0 1   , M =   1 -1 0 1 1 -1 0 0 0   that holds for {x ∈ R 2 | x 1 x 2 ≥ 0}.

Conclusions

This manuscript presented methods based on convex optimization for the stability analysis of Lurie nonlinear systems. A common feature of the presented results is a refined description of the nonlinear elements in the system. In Part I, the nonlinearities are sector-and slope-restricted, with the sector and slope bounds valid either globally or in a region containing the origin. In Part II, the nonlinearities are ramp or step functions that are precisely described thanks to relations obtained from KKT optimality conditions. In both parts, the machinery to verify that generalized quadratic forms are not negative share a common framework: to describe the nonlinearities by a set of inequalities and identities and to use these relations to bound the expression to be certified non-negative. Convex optimization problems are then obtained of which the constraints are the (linear) matrix inequalities extracted from generalized quadratic forms.

The results in Part I should allow for researchers and practitioners to carry out a more detailed analysis whenever the studied nonlinear functions are slope restricted. The central aspect that allows the proposed stability analysis conditions to improve over other Lyapunov-based stability conditions is the choice of LF functions structures. These choices make it possible to use inequalities related to slope information.

We can only claim the sufficiency of the proposed stability conditions. However, we these sufficient conditions allow for a convex optimization based calculation of stability certificates. Moreover, the numerical formulation requires as data only the bounds of the nonlinearities and the state-space representation of the linear part.

The proposed results pave the way to study the relevant problems of feedback law design with LF that are more complex than the quadratic ones. We highlight, however, that the sector and slope inequalities description were obtained for decentralized nonlinearities, thus corresponding to a SISO description of the nonlinear elements. In some applications, though, nonlinearities may present multiple channels that are not independent of one another. The study of these actuators with multiple inputs and outputs has not achieved the same maturity as the SISO or the decentralized case and is a topic for future work.

The results in Part II focus on the stability analysis of PWA systems described by an implicit representation. Chapter 3 introduced an implicit model to describe PWA systems as the feedback interconnection of an LTI system and static nonlinearities, more precisely, ramp functions. Such a representation offers a different view on the stability problem. The proposed numerical formulation exploits the implicit representation and avoids conditions requiring a preliminary reachability analysis of each set in the partition of a PWA system.

In Chapter 4, we show that the step function is obtained with an ill-posed algebraic loop involving two ramp functions. The key aspect of this result is that it enables the analysis of continuous and discontinuous generalized quadratic functions within the same framework.

We acknowledge that second-order systems presented in the numerical examples are used for illustration purposes. The potential of the proposed framework to assess the non-negativity of PWQ functions deserves a more thorough investigation. A study on the scalability of the developed methods should be carried out to understand the limiting dimensions of the number of states and the elements of the vector of ramp functions in the generalized quadratic form. Moreover, even though we obtained a sufficient condition exploiting the positivity of the ramp functions, an indepth analysis is required to understand whether other techniques for co-positivity verification can be less conservative. Also, in this manuscript, we have only proposed an SDP formulation of the problem; an open question is whether we can use similar inequalities describing the ramp function to study the non-negativity of PWL functions with an LP formulation.

Further work should also provide a better understanding of the particular use of the PWQ non-negativity tests in the analysis of discrete-time PWA systems in Part II. Unfortunately, we can not claim that the LFs we use in the results of Part II, inheriting the same partition of the system, are the most suitable class. The results of Chapter 2 show that adding future steps can improve the stability bounds for slope-restricted systems. If the same reasoning is applied to the systems in Part II, we should add future values of the nonlinearities into the LF, hence create a partition with additional sets. Also, the proposed Lyapunov methods and LF structures should be compared with IQC approaches.

As another perspective, we can also exploit the results of Chapter 2 in the analysis of systems with other SISO discontinuous nonlinearities. Clearly, functions composed of continuous Lipschitz functions and bounded discontinuities can be described as the sum of continuous function and step functions. Hence, provided these continuous functions belong to some class of sector-or sloperestricted nonlinearities, we can combine the results of the Part I with the results of Chapter 4 using the step function characterization proposed therein.

Perhaps the main question regarding Chapter 3 that is not answered in this manuscript is: how to obtain the proposed implicit representation for PWA systems? From a different perspective, we should also ask: how artificial is the explicit PWA representation? The answer to both questions should relate to the practical applications leading to PWA models. Our understanding is that the most complex PWA models in terms of the number of sets in the partition appear in optimizationbased schemes. In this context, instead of carrying out an additional step to obtain an explicit representation [START_REF] Bemporad | The explicit linear quadratic regulator for constrained systems[END_REF], we can exploit the KKT conditions associated with the problems leading to the PWA laws to directly obtain implicit representations.

Unmanned Aerial Vehicles (UAVs) quadrotors are versatile platforms capable of agile motion and stable hovering. The use of drones in civil application and industry has considerably increased in the last years, and is foreseen to continue growing. The design of autonomous UAVs should take into account safety and technological constraints, such as distance to obstacles, actuator limitations or real-time computational constraints for embedded implementation.

In this thesis, we focus on quadrotor control for applications in a cluttered environment, where we want to account for the presence of external disturbances. External disturbances and modelling mismatches can affect the execution of a mission and its impact on the closedloop trajectories must be assessed. A systematic way to assess the influence of disturbances is to compute invariant sets. The goal is to compute control laws that generate collision-free trajectories by bounding them within safe flight regions, characterized set-wise by invariant sets, where all constraints satisfaction is guaranteed. In particular, we study the design of control laws leading to invariant sets that are as small as possible.

Dr. Leonardo Broering Groff (Jun 2018-Sep 2020)

We study the problems of stability analysis of piecewise-affine (PWA) discrete-time systems, and trigger-function design for discrete-time event-triggered control systems. We propose a representation for piecewise-affine systems in terms of ramp functions, and we rely on Lyapunov theory for the stability analysis.

The proposed implicit piecewise-affine representation prevents the shortcomings of the existing stability analysis approaches of PWA systems. Namely, the need to enumerate regions and allowed transitions of the explicit representations. In this context, we can emphasize two benefits of the proposed approach: first, it makes possible the analysis of uncertainty in the partition and, thus, the transitions. Secondly, it enables the analysis of event-triggered control systems for the class of PWA systems since, for ETC, the transitions cannot be determined as a function of the state variables. The proposed representation, on the other hand, implicitly encodes the partition and the transitions.

The stability analysis is performed with Lyapunov theory techniques. We then present conditions for exponential stability. Thanks to the implicit representation, the use of piecewise quadratic Lyapunov functions candidates becomes simple. These conditions can be solved numerically using a linear matrix inequality formulation. The numerical analysis exploits quadratic expressions that describe ramp functions to verify the positiveness of extended quadratic forms.

For ETC, a piecewise quadratic trigger function defines the event generator. We find suitable parameters for the trigger function with an optimization procedure. As a result, this function uses the information on the partition to reduce the number of events, achieving better results than the standard quadratic trigger functions found in the literature The design of automobiles is moving towards the automation of driving and the development of driver assistance functions. Its main objectives are the safety of the driver, passengers, and road users and the comfort of the driver and passengers. The driving conditions of socalled autonomous vehicles include extremely varied real situations, to which it is necessary to respond in a precise and robust manner with limited intervention by the driver. In this context, the various control systems and laws must, in particular, respond to situations during:

• slow maneuvers, called weak dynamics, such as comfort maneuvers;

• fast maneuvers, called high dynamics, such as avoidance maneuvers.

The first versions of autonomous vehicles proposed by the various manufacturers of motor vehicles are developed with several assistance driving systems as, for instance, the system "Traffic Jam Pilot," which has for objective to control the vehicle in situations of traffic jam at low speed and highway at high speed.

Currently, autonomous vehicles must operate within a reasonably extended set defined by the radius of curvature of the trajectory and the longitudinal speed. Nevertheless, the existing architectures encounter significant difficulties in meeting the specifications for high dynamic maneuvers. In addition, all the different environmental conditions, such as grip, road profile, and the state of the vehicle, namely the mass and tires condition, among other factors, require that the control systems be robust in the face of the variation of a large number of parameters. The design of the control laws is finally confronted with the last source of uncertainties linked to the sensors of the environmental perception system (which provides the data for the generation of the reference trajectory on the one hand, and the measurements of the position of the vehicle with respect to its trajectory on the other hand). Indeed, these measurements are inherently noisy and affected by other sources of uncertainty, such as quantifications and biases.

In this context, the objectives of this thesis concern the methodological development of control laws for autonomous vehicles. ADAS systems are classified according to different levels of autonomy of a vehicle. The classification established by the Society of Automotive Engineers (SAE) is made up of six different levels, ranging from driving that requires full operator intervention to fully automated driving. The work of the thesis will focus on levels 3 and 4, where the systems must manage high dynamic situations without the assistance of the driver ("eyes-off" systems). In these levels of autonomy, it is not possible to limit the operating range only to nominal situations. The level of robustness must be significantly increased to cover the largest possible operating assembly, even in cases of low probability of occurrence use.

The work is organized in three phases:

1. Study of the transition from an autonomous vehicle level 2 to levels 3 and 4. The main goal is to assess the limitations of level 2 from both the point of view of available information and that of trajectory planning. Then we aim to refine the specifications and guide the choice of new control methods for autonomous vehicles. Recalling that in level 2, the driver must always be attentive and take control quickly in difficult situations or the event of an "eye-on" system problem.

2. Develop a new control law for high dynamic maneuvers (obstacle avoidance) explicitly designed for level 3 and 4 autonomous vehicles. This phase will initially consist of the vehicle's dynamic modeling (bicycle model, four-wheel model) and the methodological development of new structures/control methods. The development of control laws will have to consider aspects of trajectory planning, robustness to parametric variations related to the vehicle (e. g., mass, tire stiffness), and road grip, and the complexity of the solution, and by considering the criteria for its implementation (embeddability of control laws on current vehicle computers, for example). The automatic tuning/tune-up aspects should also be addressed to facilitate the transfer to the automotive industry of the proposed methods.

3. Validate the proposed control laws in a vehicle.

2.

Mr. Ali Diab (Sep 2019-present) Robust nonlinear control and filtering for steer by wire systems

The introduction of computing units in a network connecting sensor and actuators allows to develop the so called by-wire technologies, therefore allowing to remove mechanical links between the driving interfaces and the wheels of the vehicle. In the case of the steering system, the sensors are incremental encoders and the actuators are electric motors. These electrical drives are placed at the rack allowing the wheels to move and in the steering wheel allowing a reaction torque that provides the driver a feeling of the forces acting on the wheels. The electronic control unit computes the control signals sent to the two electric motors. The communication network makes it possible to connect the elements. They introduce however delays in the feedback loops. This set of components replaces the role of the steering column.

The two main interests of the steering wheel removal are a reduced risk in the case of an accident and an increased number of possible configurations for the interior of the vehicle. It also makes it easier the task of integrating driver and autopilot demands.

The goal of the thesis is to analyze the constraints imposed by the stability of the feedback system generating the steering wheel torque for steer-by-wire systems considering non-linear control laws. The studied problem is similar to the questions addressed in robotics, in the context of bilateral teleoperation, in which a human operator controls the position of a slave robot by acting on a master robot, returning a feeling of effort associated with the environment forces on the slave robot. In this scenario the position tracking problem can be

List of Figures 1 . 1

 11 Estimate of the region of attraction with the Lyapunov function (1.11) (dark blue). Trajectories asymptotically converging to the origin are shown in green, while diverging trajectories are depicted in red . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Induced L 2 gain bounds for an idealised Stirling engine. . . . . . . . . . . . . . . . 2.1 Feedback representation of a Lurie system. . . . . . . . . . . . . . . . . . . . . . . 2.2 Stability of the Lurie system can be shown via passivity of the above feedback

  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Ramp function r(y i ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Partition of R 2 for f (x) defined by (3.2), (3.3). . . . . . . . . . . . . . . . . . . . . 3.3 Partition of R 2 for f (x) defined in (3.4). . . . . . . . . . . . . . . . . . . . . . . . . 3.4 System trajectories and Lyapunov function level sets for Example I. . . . . . . . . 3.5 System trajectory and Lyapunov function level sets for Example II. . . . . . . . . . 3.6 One system trajectory and level sets of the Lyapunov function for Example III. . . 3.7 Two different partitions of R 2 are obtained for different values of d 1 and d 2 .

  .

4. 1

 1 Step function s(v). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Regularization of the step function s(v). . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Krasovskii regularization of the step function s(v). . . . . . . . . . . . . . . . . . . 4.4 Prototype of set-valued mapping V (x, y). . . . . . . . . . . . . . . . . . . . . . . . 4.5 Illustration of the conditions in Theorem 4.3 for the case in which y(x a ) is not a

. 4 . 6

 46 Function W (x) = max V (x) in Example 4.4.1. . . . . . . . . . . . . . . . . . . . . . xv xvi LIST OF FIGURES 4.7 Simulations in Example 4.4.1. Level sets of the function W (x) = max V (x) and the trajectory σ of (4.29) starting from (1, 0) (dashed-crossed line). The dashed line indicates the set where Kx + d = 0, namely, the set where the argument of the ramp function and the step function is equal to zero. . . . . . . . . . . . . . . . . . . . . 79 4.8 Level sets of the function W (x) in Example 4.4.2 and the trajectory of (4.29) starting from (1, 0) (dashed-crossed line). The dashed line indicates the set where Kx+d = 0, namely, the set where the argument of the ramp function and the step function is equal to zero. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.9 Lyapunov function W (x) for Example 4.4.3. . . . . . . . . . . . . . . . . . . . . . 81 4.10 Trajectories for the system along with the partitioning of the state space introduced in (4.30). Initial conditions are selected as (0.9, 3), (-1, 0.5). Note that the level sets of the computed Lyapunov function are disconnected. These disconnected sets appear, for instance in the set {x | x 1 ≥ 1, x 2 ≤ 1} close to the point (1, 1). . . . . 81 P.11 Regularization of the step function s(θ, ρ). . . . . . . . . . . . . . . . . . . . . . . . 87 P.12 Partition of R 2 of the vector field defined by R, where Γ n = {x | x1 x2 < 0, x = Rx}, Γ p = {x | x1 x2 ≥ 0, x = Rx}. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 P.13 Plots of the two components of f (x) = A n x + A p R -1 H 1 φ(u) in (4.53) with the data in (4.54) (plots were obtained with the PWA description, without the use of explicit representation). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 ≤ ρ . max The maximum of a set. ⊗ A ⊗ B indicates the Kronecker product of matrices A and B Ω The closure of the set Ω. σ(M ) The largest singular value of matrix M ∈ R n×m . ∂ The generalized derivative operator. rge f The image of the function f . xix xx LIST OF SYMBOLS diag(M, N ) For M ∈ R m×m N ∈ R n×n , the block diagonal matrix of dimensions m + n with diagonal blocks given by M and N . He For a matrix A ∈ R n×n , He(A) = A + A I n

. 5 )Problem 1 . 1 For system ( 1 . 1 ) 1 2

 511111 Under Assumption 1.1, this chapter provides a solution to the following problem. with φ satisfying (1.2): a) For w ≡ 0, certify the stability of the origin with an estimate of the region of attraction (ERA) contained in X 0 ; b) Compute reachable sets contained in X 0 for disturbances satisfying w ∈ {w ∈ L 2 | w 2 ≤ ρ }, and (x(t), w(t)) ∈ X W 0 ; c) Compute the (local) induced L 2 gains between w and z, with w ∈ {w ∈ L 2 | w 2 ≤ ρ 1 2 }, and (x(t), w(t)) ∈ X W 0 .

  the the solution of the implicit equation ỹ -Dφ(ỹ) = Cx.(1.11c)

Lemma 1. 4 Consider

 4 V in (1.11) defined by some P 11 ∈ R n×n , P 12 ∈ R n×m , P 22 ∈ R m×m and λ ∈ R m and where φ satisfies (1.2a)-(1.2b) and Assumption 1.1 holds. With Λ := diag(λ 1 , . . . , λ m ), if there exists a

16CHAPTER 1 .

 1 ANALYSIS OF CONTINUOUS-TIME SLOPE-RESTRICTED LURIE SYSTEMS Remark 1.2: Additional Lurie-Postnikov terms for slope-restricted nonlinearities

Figure 1 . 1 :

 11 Figure 1.1: Estimate of the region of attraction with the Lyapunov function (1.11) (dark blue). Trajectories asymptotically converging to the origin are shown in green, while diverging trajectories are depicted in red .

Figure 1 . 2 :

 12 Figure 1.2: Induced L 2 gain bounds for an idealised Stirling engine.

Figure 2 . 1 :

 21 Figure 2.1: Feedback representation of a Lurie system.

Figure 2 . 2 :

 22 Figure 2.2: Stability of the Lurie system can be shown via passivity of the above feedback system, where G is the linear system, M is the multiplier and φ(.) is the nonlinearity.

Figure 2 . 3 :

 23 Figure 2.3: Maximum sector δ obtained by Proposition 2.1 for G 4 (z) as a function of the horizon N ∈ [1, 10]. We also plot the bounds achieved using the Circle criterion, the Tsypkin Lyapunov function and the function from[START_REF] Park | A less conservative stability criterion for discrete-time Lur'e systems with sector and slope restrictions[END_REF]. The upper limit set by the Nyquist gain is also plotted.

  and γ p (y, y) = min σ∈[y,y] dp(σ) dσ defining the nonlinearity φ(σ) = p(σ) -γ p (y, y)σ.

Figure 2 . 4 :

 24 Figure 2.4: Regional stability of the second numerical example. Positive invariant sets of the Lyapunov function V (x) from (2.5) are shown in blue, with blue dashed lines corresponding to sub-level sets. Light blue corresponds to N = 4 and dark blue to N = 1. Black dashed lines denote the limits {x : ν 0 (x) = y, ν 0 (x) = y}. Initial conditions from the red region did not converge to the origin.

Figure 2 .

 2 Figure 2.5 shows the computed values of ω from Theorem 2.3 defining the worst-case bound z 2 ≤ ω w 2 for all w 2 ∈ 2 as a function of the horizon length N of V (x). As N increased, there was a significant drop in ω, going from 6.08×10 3 with N = 1 to 3.13 × 10 1 at N = 4 before reaching a plateau. This noticeable drop in ω suggests longer horizons N in V (x) may prove important for tight bounds of the input-outputs of Lurie systems.

Figure 2 . 5 :

 25 Figure 2.5: Worst-case input output gain ω bounding z 2 2 ≤ ω w 2 2 as a function of the horizon length N in V (x).

) as depicted in Figure 3 Figure 3 . 1 :

 331 Figure 3.1: Ramp function r(y i ).

Figure 3 . 2 :

 32 Figure 3.2: Partition of R 2 for f (x) defined by (3.2), (3.3).

Proposition 3 . 1 :

 31 [START_REF] Zaccarian | A common framework for anti-windup, bumpless transfer and reliable designs[END_REF] Proposition 1]

  conditions for optimality. These relations offer a characterization in terms of linear and quadratic identities and inequalities in three variables (θ, r, λ). To obtain a description in the variables (θ, r) one can use λ = (r -θ) above to obtain r(r -θ) function φ is composed of ramp functions, we can use (3.7) to prove the relations in the following lemmas Lemma 3.1 For any T ∈ D ny the function φ in (3.2c) satisfies s 1 (T, y) := φ (y)T (φ(y) -y) = 0, ∀y ∈ R ny .

  1, . . . , n y . Set ζ = F 3 x + f 5 in equation (3.2b), and use y i = (F 4 φ + ζ) i in the above expressions to obtain respectively

.12c) i = 1 ,

 1 . . . , n y . The problem of solving on φ the inequalities (3.12c), (3.12b), affine in φ, and equations (3.12a), quadratic in φ, is called a mixed Linear Complementarity Problem (LCP). For a given ζ, the solution φ to (3.12) provides a solution to the implicit equation y -F 4 φ(y) = ζ. Please refer to the Lemke algorithm presented in [1, Section 5.1] for a strategy to solve LCPs yielding solutions to algebraic loops. Also, as one should expect, the condition for the well posedness of LCPs in [1, Proposition 7.1] applied to (3.12) holds if the condition in Proposition 3.1 is satisfied.

4 , 1 F 2

 412 = 0 n+ny,1 I n+ny , I x = 0 n,1 I n 0 n,ny , I φ = 0 ny,1+n I ny , I φ-y = -f 5 -F 3 I ny -F 0 n,ny 0 ny,1 0 ny,n 0 ny I ny , I 0 = 0 n+ny,1 I n+ny 0 n+ny,ny , I φ = 0 2ny,1+n I 2ny ,

  (3.27a), (3.27b) and (3.27c) hold then the conditions (3.21) and (3.22) in Theorem 3.1 also hold. Consider χ(x) =

f 5 F 3 F 4 - 3 0

 5343 I ny I ny =:S φ ζ (3.33) obtained from the identity y = φ(y) -(φ(y) -y), and the identities below, obtained from (3.2). 0 = 0 n,1 F 1 -I n F 2 0 n,ny 0 n,ny 0 n,ny ny,n F 4 -I ny 0 ny I ny 0 ny f 5 0 ny,n F 3 0 ny F 4 -I ny 0 ny I ny

  note that, following the definition of the set F, for scalarsN j=1 α j = 1, α j ∈ [0 1], ∀j = 1, . . . , N , we have N j=1 α j S j = S φ (3

  ) is globally exponentially stable. Proof. The proof follows the same lines of the proof of Theorem 3.1, namely the satisfaction of (3.24) allows to show the exponential stability of the origin. Let us show that (3.42)-(3.43) imply the existence of a PWQ Lyapunov function for the systems defined by elements of the polytopic set F. Note that in (3.42)-(3.43) the system matrices appear only in the matrices S j and Sj . By defining Mj and using (3.39)-(3.40) we have that, along the trajectories of system (3.31) N j=1

( 3 .

 3 44c) j = 1, . . . , N , where Ī = 0 n+ny,1 I n+ny 0 n+ny,ny , Īx = 0 n,1 I n 0 n,2ny , Īφ = 0 ny,1+n I ny 0 ny,ny , Īφ-y = 0 ny,1+n 0 ny,ny I ny , n,1 0 n I n 0 n,4ny , ny 0 n,3ny 0 ny,1 0 ny,n 0 ny,n I ny 0 n,3ny , Ī φ = 0 2ny,1+2n I 2ny 0 2ny,2ny , Ī φ-ỹ = 0 2ny,1+2n 0 2ny,2ny I 2ny ,

2 Figure 3 . 4 :

 234 Figure 3.4: System trajectories and Lyapunov function level sets for Example I.

2 Figure 3 . 5 :

 235 Figure 3.5: System trajectory and Lyapunov function level sets for Example II.

By applying Theorem 1 ,

 1 we could obtain a quadratic Lyapunov function that certifies the global stability of the origin. Clearly, a quadratic function is obtained by setting P 2 = 0 and P 3 = 0 in (3.18). The computed values for V (x) = x P 1 x are given by P 1 = 0.9262 0.4674 0.4674 1.0815 . A trajectory and the level sets of the obtained Lyapunov function are shown in Figure 3.6.

Figure 3 . 6 :

 36 Figure 3.6: One system trajectory and level sets of the Lyapunov function for Example III.

Figure 3 . 7 :

 37 Figure 3.7: Two different partitions of R 2 are obtained for different values of d 1 and d 2 . These partitions differ also in the number of sets defining them. On the left, d 1 = -d 2 > 0, we obtain a partition with four sets. On the right, with d 1 = d 2 < 0, we obtain a partition with six sets. The dashed line corresponds to the set where y 3 = 0.

2 Figure 3 . 8 :

 238 Figure 3.8: The tree structure corresponding to (3.50) using the composition rules of (3.48).

Figure 4 . 2 :

 42 Figure 4.2: Regularization of the step function s(v).

Figure 4 . 3 :

 43 Figure 4.3: Krasovskii regularization of the step function s(v).

Figure 4 . 4 :

 44 Figure 4.4: Prototype of set-valued mapping V (x, y).

Figure 4 . 5 :

 45 Figure 4.5: Illustration of the conditions in Theorem 4.3 for the case in which y(x a ) is not a singleton. The set of values for V (x, y) is given in red in coordinate axis. In this case, the regularized dynamics give a set-valued mapping illustrated here by the set x + a , which in turn gives the set of values for V (x + , y + )) -depicted in blue in the coordinate axis. The theorem asks for all points in the green set to be below the red set. In this case the single valued function W is guaranteed to decrease.

Figure 4 . 6 :Figure 4 . 7 :

 4647 Figure 4.6: Function W (x) = max V (x) in Example 4.4.1.

Figure 4 . 8 :

 48 Figure 4.8: Level sets of the function W (x) in Example 4.4.2 and the trajectory of (4.29) starting from (1, 0) (dashed-crossed line). The dashed line indicates the set where Kx + d = 0, namely, the set where the argument of the ramp function and the step function is equal to zero.

Figure 4 .

 4 Figure 4.8 illustrates some level sets of the function W along with the solution to (4.29) starting from (1, 0). The picture clearly shows that the lack of symmetry of the nonlinearity s reflects on the function W .

Figure 4 . 9 :Figure 4 . 10 :

 49410 Figure 4.9: Lyapunov function W (x) for Example 4.4.3.

Figure P. 11 :

 11 Figure P.11: Regularization of the step function s(θ, ρ).

Lemma P. 4

 4 The time derivative of dφ(ξ(x)) dt can be expressed as a set-valued mapping defined by the implicit relationdφ(ξ(x(t))) dt = F φ(η) (4.40a) η = Cx + Dξ φ(ξ) + Dy φ(y) + ē + Dφ(η) (4.40b) with F ∈ R m×6m , C ∈ R 6m×n , Dξ ∈ R 6m×m , Dy ∈ R 6m×my , D ∈ R 6m×6m , and ē ∈ R 6m×1 as F = (I m ⊗ N 0 ) , (4.41a) C = (I m ⊗ N 11 ) C ξ + (I m ⊗ N 12 ) C ξ A (4.41b) Dξ = (I m ⊗ N 11 ) D ξ ,(4.41c) Dy = (I m ⊗ N 12 ) C ξ B, (4.41d) ē = (I m ⊗ N 11 ) e ξ , (4.41e) D = (I m ⊗ N 12 ) D ξ (I m ⊗ N 0 ) + I m ⊗ N 2 , (4.41f)

Figure P. 12

 12 Figure P.12 and the resulting functions are plotted in Figure P.13.

Figure P. 12 :

 12 Figure P.12: Partition of R 2 of the vector field defined by R, where Γ n = {x | x1 x2 < 0, x = Rx}, Γ p = {x | x1 x2 ≥ 0, x = Rx}.

Figure P. 13 :

 13 Figure P.13: Plots of the two components of f (x) = A n x + A p R -1 H 1 φ(u) in (4.53) with the data in (4.54) (plots were obtained with the PWA description, without the use of explicit representation).

C. 1 . 2

 12 Ongoing supervision -Thesis projects 1. Mr. Dario Penco (Mar 2019-present) Autonomous vehicle control: Robust and high performance control to enable high dynamics range maneuvering of autonomous vehicles

Table 1 :

 1 Studied problems in the manuscript: 1) Global stability analysis 2) Local stability analysis 3) L 2 Gains, 4) 2 Gains.

	Time	Vector field	Problems

table below Table 1 .

 below1

	2: Local sector and slope bounds for ln(1 + ỹj ) and ỹj 1+ỹj for X 0 as in (1.16) with ỹj > -1.
	φ(ỹ j )	δ	δ				γ	γ
	ln(1 + ỹj )	ln (1+ỹ j ) ỹj	ln (1+ỹ ỹj	j	)	1 1+ỹ j	1 1+ỹ	j
	ỹj	1	1				ỹj	ỹj
	1+ỹj	1+ỹ j	1+ỹ	j			(1+ỹ j ) 2	(1+ỹ	j	) 2

Table 2 .

 2 1: Various linear systems used as tests in the numerical examples. Set of examples taken from

Table 2 .

 2

2: Achievable maximum sectors for various tests. N * is the value of N in V (x) of (2.5) giving the maximum sector.

Table 3 .

 3 4159 -4.6953 .

	Inequalities defining the sets Γ i	Control law u i
	 -5.9220 -6.8883 		 2 
	   -1.5379 6.8296 5.9229 6.8883	   x ≤	   2 2   	-5.9220 -6.8883 x
		1.5379 -6.8296		2
	 -6.4159 -4.6953 			1.3577	
	 -0.0275 0.1220	 x ≤	 -0.0357	 -6.4159 -4.6953 x + 0.6423
		6.4159	4.6953				2.6423
		6.4159	4.6953				1.3577	
		0.0275 -0.1220  x ≤	 -0.0357	 -6.4159 -4.6953 x -0.6423
	-6.4159 -4.6953			2.6423
	 -3.4155 4.6452				2.6341	
		0.1044 0.1215	 x ≤	 -0.0353		2
		0.1259 0.0922			-0.0267
	0.0679 -0.0924 0.1259 0.0922	x ≤	-0.0524 -0.0519	2
	-0.0679 0.0924 -0.1259 -0.0922	x ≤	-0.0524 -0.0519	-2
		3.4155 -4.6452 			2.6341	
	 -0.1044 -0.1215  x ≤	 -0.0353		-2
	-0.1259 -0.0922		-0.0267

1: Explicit MPC law: inequalities defining the sets of the partition and the corresponding affine control law.

  .65b) where the signals w 1 or w 2 are assumed to be in a given set. The most common robustness analysis concentrate on signals in ∞ , 2 , namely bounded or energy-bounded signals. These signals may result from unmodeled dynamics or time-varying external signals. When the control law introduces the algebraic loop, we can reasonably suppose that real-time implementation constraints do not allow to solve the algebraic loop exactly. Instead, for a given x, an approximate solution of the equation y = F 3 x + F 4 φ(y) + f 5 would satisfy, at each time instant, a different equation y

  P.7.1 Proof of Lemma P.4Proof. Given that the ramp function is not differentiable for the values of x giving ξ i (x) = 0 we have to consider the generalized set-valued derivative for

								dφ(ξ1(x(t)))	
		dφ(ξ(x(t))) dt	=	    		dt dφ(ξ2(x(t))) dt . . . dφ(ξm(x(t)))	    	.	(4.66)
								dt
	We have
	dξ(x(t)) dt	= C ξ	d(x(t)) dt	+ D ξ	dφ(ξ(x(t))) dt	(4.67)
			= C ξ Ax + C ξ Bφ(y) + D ξ	dφ(ξ(x(t))) dt	(4.68)
	hence the ith entries of the vectors ξ and dξ(x(t)) dt	are given by
	ξ i = C ξi x + D ξi φ(ξ) + e ξi	(4.69a)
	dξ i (x(t)) dt	= C ξi Ax + C ξi Bφ(y) + D ξi	dφ(ξ(x(t))) dt	.	(4.69b)
	For each entry, the chain rule gives dφ(ξi(x(t))) dt	= dφ(ξi) dξi	dξi(x(t)) dt	. The generalized derivative of
	the ramp function is the step function of its argument, that is
				dφ(ξ i ) dξ i	= s(ξ i , 1)
	we then have, using (4.39),
		dφ(ξ i (x(t))) dt	=	dφ(ξ i ) dξ i	dξ i (x(t)) dt	(4.70)
					= s(ξ i , 1)	dξ i (x(t)) dt	(4.71)
					= s ξ i ,	dξ i (x(t)) dt	(4.72)
	thus, using (4.66),

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	-Responsabilité

University of Oxford, Oxford, United Kingdom. Janvier à Mars 2015 -Somerville College, University of Oxford, Tutorial teaching

CHAPTER 2. ANALYSIS OF DISCRETE-TIME SLOPE-RESTRICTED LURIE SYSTEMS

-1 0 , f 5 = 0 0 . (3.3)The corresponding partition of R 2 for (3.2) with the matrices in (3.3) is given byΓ 1 = {x ∈ R 2 |φ 1 (y(x)) = 0, φ 2 (y(x)) = 0}, Γ 2 = {x ∈ R 2 |φ 1 (y(x)) ≥ 0, φ 2 (y(x)) = 0}, Γ 3 = {x ∈ R 2 |φ 1 (y(x)) = 0, φ 2 (y(x)) ≥ 0}, Γ

= {x ∈ R 2 |φ 1 (y(x)) ≥ 0, φ 2 (y(x)) ≥ 0}.and is depicted in Figure3.2. An explicit representation for f (x) as in (II.1) is given by x 1x 2

Acknowledgements

for research and for providing opportunities to early career researchers. To the heads of department D. Dumur, G. Sandou for the opportunities, for the careful planning of the teachings, for the efforts on keeping the experimental activities and close ties with industry.

The research funding provided by the institutional funds from CentraleSupélec, Inria, CNRS, Università di Roma "Tor Vergata", and the funding agencies CNPQ, FAPESP, CAPES, EU through Alban Programme, EPSRC, Campus France, ANR, and ANRT.

The students of CentraleSupélec for making us improve

Part II

Analysis of Piecewise Affine

Discrete-Time Systems = -V (F 1 x + F 2 φ(ȳ 0 ), ȳ+ 0 ) -V (x, y) + ηx x + s 1 ( T , ỹ) -s 2 ( M , ỹ)

Thus the matrix inequalities in (4.27) imply the inequalities expressed as the generalized quadratic forms in (4.23) and (4.24).

Numerical Examples

In this section we showcase the effectiveness of the proposed methodology in some numerical examples. Specifically, we consider the following dynamical system In all examples we solve the conditions in (4.27) in YALMIP [START_REF] Löfberg | YALMIP : A toolbox for modeling and optimization in MATLAB[END_REF] and the solver SeDuMi [START_REF] Sturm | Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones[END_REF],

Ternary Control

In this first example, we pick B = 0.0049 0.0959 , K = 9.9 0.495 , and analyze the case of ternary control systems; see, e.g., [START_REF] De Persis | Robust self-triggered coordination with ternary controllers[END_REF][START_REF] Valmorbida | On quantization in discrete-time control systems: Stability analysis of ternary controllers[END_REF]. More specifically, we select ϕ(x) = τ ( Kx), where for all u ∈ R τ (u) :=

The regularized nonlinearity written in terms of the step function is then given by ϕ(x) = s( Kx -1) -s(-Kx -1) and system (4.29) can be rewritten as (4.4a) by taking

Perspectives

This chapter presents perspectives and ongoing work that build upon the results presented in Parts I-II. The ideas and claims introduced in this chapter will be presented without proof since some are still preliminary. The goal of this part is to highlight the potential of the results in the manuscript, mainly regarding the study of piecewise affine systems using the implicit representation presented in Part II.

P.1 Synthesis of feedback laws for slope-restricted nonlinearities

In the results presented in Part I for linear systems with slope-restricted nonlinearities, we proposed classes of Lyapunov/storage functions leading to stability/gain analysis based on convex optimization. In these results, we aimed to reduce the number of parameters in the LF. The natural question that can be raised regards the use of these functions to design control laws. Interestingly, in the conclusion of the survey [START_REF] Kokotovic | Constructive nonlinear control: a historical perspective[END_REF] it is observed:

Most of the surveyed tools and design procedures are analytical, while only a few relied on LMI computations. Symbolic and numerical procedures will strengthen analytical design methods.

However, systematic convex conditions for the control law design are challenging to formulate except for simple LF candidates. This is the case of quadratic functions, which can be used for the synthesis of state feedback laws. For this simple function, the conjugate quadratic function [START_REF] Goebel | Conjugate convex Lyapunov functions for dual linear differential inclusions[END_REF] allows for a change of coordinates followed by a change of variables yielding a convex optimization computation of the feedback gains.

In the context of input-saturating systems, we have recently proposed an iterative-based strategy to compute the feedback gains [START_REF] Queinnec | Design of saturating statefeedback with sign-indefinite quadratic forms[END_REF]. This strategy relies on the conditions of Finsler's lemma. However, to obtain a convex optimization formulation, we have to impose structure of a multiplier, which is a way to avoid the product of some terms which would otherwise give a non-convex set of constraints. We also proposed a similar convex-optimization based approach in [START_REF] Bertolin | An LMI-based iterative algorithm for state and output feedback stabilization of discrete-time Lur'e systems[END_REF] based on Lyapunov function stability conditions, and in [START_REF] Bertolin | An LMI-based iterative algorithm for state and output feedback stabilization of discrete-time Lur'e systems[END_REF], based on a Zames-Falb multiplier condition.

We would like next to exploit the proposed parameterization for the synthesis of state feedback gains with a direct convex formulation. These convex conditions will be investigated for both the continuous-time and the discrete-time cases.

P.2 PWQ Lyapunov functions for continuous-time systems

Consider the PWA continuous-time system, defined using the implicit PWA function with ramp functions as introduced in Part II for discrete-time systems.

In Chapter 4 we showed that the algebraic loop can be ill-posed, in which case a set-valued mapping is obtained and the resulting nonlinear function y : R n → R ny can be discontinuous. Since both continuous and discontinuous vector fields can be obtained with the above algebraic loop, it enables the study of both continuous or discontinuous [START_REF] Cortés | Discontinuous dynamical systems[END_REF] CT dynamics. The main objective of the perspectives presented in this section is to point out the potential of the implicit PWA representation to solve analysis problems that are difficult to be addressed with the usual explicit representation. Indeed, in the partition-based analysis using explicit PWA representation for CT

Appendices

Appendix A

Curriculum Vitae

Giorgio Valmorbida

Brazilian / Italian -Passport FH090334 (Brazil) Passport YA5266565 (Italy). Married, two children.

Contact information

Editorial Activities

• May 2020 to present Associate Editor, IMA Journal of Mathematical Control and Information.

• June 2020 to present Associate Editor, Journal of Control, Automation and Electrical Systems.

• June 2018 to present Associate Editor, Conference Editorial Board IEEE CSS.

• Main editor of the volume "Delays and Interconnections: Methodology, Algorithms and Applications" in the series Advances on Delays and Dynamics at Springer. Vol 10. 2019.

• Main editor of the volume "Incorporating constraints on the Analysis of Delay and Distributed Parameter Systems" In: Advances on Delays and Dynamics at Springer. In print.

Invited Talks • Organiser of the PhD students seminars of the "pôle Automatique" of the Laboratoire de Signaux et Systèmes.

PhD Supervision

Member of PhD panel

• January 2018 Member of panel of PhD defense of Fabien Niel "Modeling and control of a wing at low Reynolds number with high amplitude aeroelastic oscillations" ISAE -Institut Supérieur de l'Aéronautique et de l'Espace.

Member of PhD qualifying committee

• July 2021 Member of PhD qualifying committee of Mathieu Bajodek "Analysis and design of hetergeneous cyber-physical systems" École doctorale EDSYS.

• May 2021 Member of PhD qualifying committee of Daniel Denardi Huff "Stability Analysis and Stabilization of Linear Aperiodic Sampled-Data Sytems subject to Input Constraints" Double degree UFRGS/ Université de Grenoble Rhône-Alpes.

Projects

• Coordinator of the International Research Network SPa-DisCo "Systèmes à paramètres distribués et Contraintes". April 2017 to December 2021.

• Member of Project STIC AmSud, CoDysCo2 "Control of Dynamical Systems under Communication Constraints". January 2018 to December 2019.

• Member of Project ANR, HANDY "Hybrid And Networked Dynamical sYstems". September 2018 to September 2022.

• Coordinator of PHC Aurora Project " Control and Observation of Nonlinear Systems". January 2019 to December 2019.

List of Publications

Publications by the author of the manuscript as of December In the thesis, computational methods based on convex optimization, for the analysis of systems described by partial differential equations (PDEs), were proposed. Firstly, motivated by the integral inequalities encountered in the Lyapunov stability analysis of PDEs, a method based on sum-of-squares (SOS) programming was proposed to verify integral inequalities with polynomial integrands. This method parallels the schemes based on linear matrix inequalities (LMIs) for the analysis of linear systems and approaches based on SOS programming for the analysis of polynomial nonlinear systems.

Secondly, dissipation inequalities for input-state/output analysis of PDE systems were formulated. Similar to the case of systems described by ordinary differential equations (ODEs), it was demonstrated that the dissipation inequalities can be used to check inputstate/ output properties, such as passivity, reachability, induced norms, and input-to-state stability (ISS). Furthermore, it was shown that the proposed input-state/output analysis method based on dissipation inequalities allows one to infer properties of interconnected PDE-PDE or PDE-ODE systems. In this regard, interconnections at the boundaries and interconnections over the domain are considered. It is also shown that an appropriate choice of the storage functional structure casts the dissipation inequalities into integral inequalities, which can be checked via convex optimization.

Thirdly, a method was proposed for safety verification of PDE systems. That is, the problem of checking whether all the solutions of a PDE, starting from a given set of initial conditions, do not enter some undesired or unsafe set. The method hinges on an extension of barrier certificates to infinite-dimensional systems. To this end, a functional of the states of the PDE called the barrier functional is introduced. If this functional satisfies two inequalities along the solutions of the PDE, then the safety of the solutions is verified. If the barrier functional takes the form of an integral functional, the inequalities convert to integral inequalities that can be checked using convex optimization in the case of polynomial data. Furthermore, the proposed safety verification method was used for bounding output functionals of PDEs.

Finally, the tools developed in the thesis were applied to study the stability and input-output analysis problems of fluid flows. In particular, incompressible viscous flows with constant perturbations in one of the coordinates were studied. The stability and input-output analysis is based on Lyapunov and dissipativity theories, respectively, and subsumes exponential stability, energy amplification, worst case input amplification and ISS. It was shown that an appropriate choice of the Lyapunov/storage functional leads to integral inequalities with quadratic integrands. For polynomial base flows and polynomial data on flow geometry, the integral inequalities can be solved using convex optimization. This analysis includes both channel flows and pipe flows. For illustration, the proposed method was used for inputoutput analysis of several flows, including Taylor-Couette flow, plane Couette flow, plane Poiseuille flow and (pipe) Hagen-Poiseuille flow.

2. Dr. Nathan Michel (Nov 2016-Sep 2020) solved quite easily. On the other hand, for the force feedback, the problem is much more difficult since adding a force sensor is not always desirable (due to cost, reliability, and design constraints). Moreover, even with a force sensor, information transmission delays (between sensors, processors and actuators) can destabilize the loop encompassing position and effort control, especially when the forces transmitted between two robots are amplified (energy injection).

Our goal will be to design and analyze new control laws for steer-by-wire systems to increase (compared to current strategies) the delay margin of the system, similarly to what is done on interconnected systems, while allowing the injection of energy into it. These algorithms will rely on classical proportional/derivative control architectures, and will be augmented by a non-linear filtering of the assistance to reduce the destabilizing action associated with energy injection.

C.2 Joint publications with early career researchers, Ph.D. students and Post-Docs

The publication references correspond to the publications in the previous appendix.

In collaboration with supervised Ph.D. Students :

• Mohamadreza Ahmadi: J8, J14, J15, J16, C24, C27, C31, C33, C34;

• Nathan Michel: C13, C14, C15, C20;

• Leonardo B. Groff: J1, C11, C12;

• Dario Penco: C3, C4.

In collaboration with supervised MSc. Students :

• Thomas Lathuilière: J9, C16.

In collaboration with Post-Docs :

• Aditya Gahlawat: J5, C17;

• Ye Wang: J6, C18.

• Ross Drummond: J7, J10, C19, C21, C23

In collaboration with Ph.D. Students :

• Matthias G. Titton: C2, C10

• Leonardo Cabral: C1

• Ariádne L. J. Bertolin: J4, C7, C9;

• Mohammad Ali Abooshahab: C5;

• Marcio Lacerda: C25;