Elise Delavaud

Paul Cupillard

Gaetano Festa

Jean-Pierre Vilotte

3D SPECTRAL ELEMENT METHOD SIMULATIONS OF THE SEISMIC RESPONSE IN THE CARACAS BASIN

This paper presents the first results of 3D simulations of the seismic response in the Caracas basin using the spectral element method. The seismic response of the basin is obtained by the excitation of an incident plane wave which is efficiently implemented via its traction and velocity at the rock sediments interface. A generalized filtering perfectly matched layer allows for a better absorption of surface waves in elongated basins. Our simulations clearly show combined effects from the topography and the basin, with diffracted waves coming from the mountains, and low frequency energy trapped in the deepest part of the basin. Spectral ratio exhibits focalization and 3D amplification effects.

Introduction

3D simulation presents challenges for assessing the seismic response of heterogeneous media with topography and internal interfaces, such as sedimentary basins. In such models, where most of the energy is transported by diffracted waves, the 3D geometry and the local geological features can result in significant amplifications and trapping of specific frequencies. These 3D effects must be taken into account for strong motion prediction and seismic risk assessment and illustrate the need for detailed models, and flexible numerical techniques [START_REF] Sanchez-Sesma | The assessment of strong ground motion: What lies ahead[END_REF].

Many techniques have been developed to model elastic-wave propagation. The finitedifference methods have been widely used for smooth and steep topography [START_REF] Olsen | Site amplification in the Los Angeles basin from three-dimensional modeling of ground motion[END_REF]. Despite some recent improvements, they suffer from serious limitations, in particular when it comes to accurate computation of surface waves and the handling of complex geometries and strong heterogeneities. The spectral element method (SEM), introduced in seismology ten years ago [START_REF] Komatitsch | The spectral-element method: an efficient tool to simulate the seismic response of 2D and 3D geological structures[END_REF]) is a high-order variational technique that has already proven to be a powerful tool at regional [START_REF] Komatitsch | Simulations of ground motion in the Los Angeles basin based upon the spectral-element method[END_REF] and global scales [START_REF] Capdeville | Coupling the spectral element method with a modal solution for elastic wave propagation in global Earth models[END_REF][START_REF] Chaljub | Solving elastodynamics in a fluid-solid heterogeneous sphere: a parallel spectral element approximation on non-conforming grids[END_REF]. It provides solutions to the increasing demand for geometrical flexibility and for accuracy for both body and surface waves. The SEM is well adapted to tackle problems linked to the seismic response in heterogeneous media like basins. It exhibits a high accuracy, in particular for the resolution of surface waves, and the interaction between body waves and the topography of the main geological discontinuities.

Caracas is built on a typical alluvium basin and bordered by a strong topographic barrier, the Avila mountain range. It has been the subject of many studies since site effects have been revealed during the 1967 Mw=6.6 Caracas earthquake [START_REF] Bolton Seed | Relationships between soil conditions and building damage in the 1967 Caracas Earthquake[END_REF]. Geological observations, drill holes, seismic surveys and gravimetric measurements have been recently re-analyzed to provide, with new seismic refraction profiles done by FUNVISIS in 2001, a reliable model of the basin for the sediments' thickness and velocity.

This paper describes first new developments of the SEM used for the modeling of the seismic response in basins, a generalized filtering perfectly matched layer for a better absorption of surface waves and an efficient implementation of incident plane waves. The difficulties associated with unstructured mesh generation, the most problematic aspect of the modeling process, are discussed further.

Numerical technique

The SEM is a high order method based on the approximation of the variational formulation of elastodynamics. A domain is decomposed into hexahedral finite elements, mapped on a reference cube where Gauss Lobatto Legendre points (GLL) are defined. The wave field is represented in terms of high-degree Lagrange polynomials on the GLL interpolation points. The choice of a quadrature rule at the same GLL points results in a diagonal mass matrix, allowing for an explicit time integration Newmark scheme. The system solved at each time step can be written as follows

2 / 1 1 2 / 1 2 / 1 int 2 / 1 , 1 1
) ) ( (

+ + + + + - + ∆ + = + - ∆ + = n n n n T n n ext n n tV U U B U F F tM V V λ (1)
where U and V denote the vectors containing the displacement and the velocity at all the nodes, and λ the traction defined only at the interfaces where a Neumann condition is imposed, T B being an interface matrix. M is the diagonal mass matrix, ext F and int F external and internal forces, respectively. Superscript n stands for the time step.

Due to the geometrical flexibility the SEM shares with finite-element methods, it can handle complex geometries with an unstructured mesh, which respects the main discontinuities and topological features. Special attention should be paid to the mesh generation. Through the Courant number, the time step is linked to the space discretization and can become tremendously small if the mesh contains small or exceptionally deformed elements. This situation is frequently found when dealing with rough interfaces and topography. Deformed elements should be especially avoided as the effect of jacobian high variations on the numerical accuracy is not well known.

Generalized Filtering Perfectly Matched Layer

When modeling low frequencies in thin media such as basins, instabilities can arise from the interaction between propagating evanescent waves and the perfectly matched layer (PML) used as absorbing boundary condition at the bottom of the model. These effects come from the surface waves which have lost their characteristic exponential feature in the standard PML (Festa and Vilotte, 2005;Festa et al., 2005). In very long simulations, these instabilities affect the total volume, and can significantly pollute the records even at the free surface. To avoid this problem, the bottom of the model should be put at a sufficient depth, but it will increase the computational cost. A modification of the PML by adding a cut-off and an overdamping enables the evanescent wave characteristics for low frequencies in these layers to be preserved without producing any spurious effect. The computational domain size can be thus optimized and calculation costs reduced.

Plane wave implementation

The seismic response in the basin is assessed through an incident plane wave excitation. It is usually introduced as an initial condition, but this classical implementation exhibits inconveniences. The coupling between this initial condition and the absorbing conditions imposed at the borders of the model can produce diffraction phenomena. Moreover, the computational domain needs to be enlarged compared to the domain of interest when an inclined wave is considered. We introduce an incident plane wave with the SEM directly at the rock sediments interface and at the free surface of the rock part. A decomposition technique is applied: the elastodynamic equation is separately solved in both domains, but not on the same fields.

• In the rock, the wave is decomposed into an incident, analytically known, wave field and into a diffracted wave field which is numerically calculated. • In the sediments, total field is computed.

• Communication between the domains is insured at the rock sediments interface ( Γ ) by a continuity condition on the velocities and the tractions. This strategy leads to the following system of equations

Γ + = Γ - = - - ∆ + = + - ∆ + = + + + + + + + + - + + + - + on B B B on V V V B U F M t V V B U F M t V V n r i T n r d T n s T n r d n s n r i n s T n s s n s n s n r d T n r d r n r d n r d 2 / 1 , 2 / 1 , 2 / 1 , 1 , 1 , 1 , 2 / 1 , 2 / 1 , int 1 , 1 , 2 / 1 , 2 / 1 , 1 , 1 , ) ) ( ( ) ( ) ) ( ( ) ( λ λ λ λ λ (2)
where r and s superscripts stand for rock and sediments, i and d indexes for incident and diffracted fields, respectively. For the rock domain, F contains the internal forces and the contribution of the incident wave at the free surface.

The complete solution is retrieved by adding the analytical part. Propagation of the incoming wave field is thus avoided, suppressing spurious effects at the boundaries. The restriction of the calculation domain to the study area allows for a gain in computation time.

Seismic response in the Caracas basin

Caracas basin

The city of Caracas is located about 13km from the coast of northern Venezuela in an alluvial valley of the coastal mountains. The Caracas basin is narrow, long and almost flat, about 25km long in east-west direction and 3-4km wide, bordered to the north by a formidable topographic barrier, the Avila mountain range, and to the south by smaller hills. Alluvium materials like sand, gravel and stiff clay have been deposited by the Guaire River and alluvial fans coming from the Avila range.

The major seismic activity comes from a strike slip fault system associated with the movement along the Caribbean-South America plate boundary zone. During the 1967 magnitude Mw=6.6 Caracas earthquake, which occurred 25km northwest of the city, four buildings collapsed and 300 people were killed. Strong site effects were revealed and in particular, a close relationship between the building damage and the soil thickness was established. Following this event, various studies have been carried out to characterize the structure of the Caracas basin. A reliable subsurface model is now available thanks to recent geological and geophysical investigations. A total of 166 drill holes, seismic refraction results [START_REF] Inc | Seismic Investigations in the valley of Caracas and the Lithoral Central[END_REF], and gravimetric measurements were re-analyzed [START_REF] Kantak | Perfil geológico longitudinal al valle de Caracas, con énfasis en los sedimentos más recientes, para fines de microzonificación sísmica[END_REF]. In order to provide more data on the sedimentary thickness and seismic velocities, seismic refraction measurements were carried out in the region of study by FUNVISIS in July 2001 [START_REF] Sánchez | Mediciones sísmica profundas y gravimétricas en Caracas para la determinación del espesor de sedimentos y velocidades sísmica como aporte para el estudio de Microzonificación Sísmica[END_REF].

Figure 1. Map of sediment thickness in the Caracas valley based on geological observations, drill

holes and geophysical data [START_REF] Inc | Seismic Investigations in the valley of Caracas and the Lithoral Central[END_REF][START_REF] Kantak | Perfil geológico longitudinal al valle de Caracas, con énfasis en los sedimentos más recientes, para fines de microzonificación sísmica[END_REF][START_REF] Sánchez | Mediciones sísmica profundas y gravimétricas en Caracas para la determinación del espesor de sedimentos y velocidades sísmica como aporte para el estudio de Microzonificación Sísmica[END_REF].

In Figure 1, the two north-south profiles correspond to the areas where the damage in the 1967 earthquake was concentrated: San Bernardino (SB), profile NS-01 and Los Palos Grandes (LPG), profile NS-03. They coincide with the deepest parts of the basin, about 100m for SB and 320m for LPG. To assess the fundamental periods of vibration within the valley, microtremors have been measured during the 1990's [START_REF] Abeki | Microtremor observations in Caracas city, Venezuela[END_REF][START_REF] Duval | Caracas, Venezuela, Site Effect Determination with Microtremors[END_REF] using either the Nakamura method or H/V ratio. This data has been recently densified in a 250m grid. Predominant periods derived from H/V ratio correlate well with the 1D sediment thicknesses, the highest values corresponding to the two deepest regions, SB and LPG, with periods of about 1s and 2s, respectively. The data we used for our simulations has been supplied by FUNVISIS.

The city of Caracas continues to grow. New construction within the valley and a recent along the slopes increase the need for a precise microzonation and risk assessment. 2D simulations along the LPG profile [START_REF] Semblat | Seismic site effects in a deep alluvial basin : numerical analysis by the boundary element method[END_REF] show strong amplifications in the deepest part of the alluvial deposit at low frequencies and further away from it for higher frequencies. 3D simulations are attempting to evaluate the importance of 3D effects (amplification, focalization and interference phenomena) on the seismic response coming from the geological and geometrical features of the basin, the surrounding steep topography and their mutual interaction.

Basin model

For the simulations, we selected a volume of 23km long (east-west direction), 17km wide (north-south direction) with a maximum 4.5km height, including the Avila mountain range and the hills in the south. The 3D unstructured mesh generation remains a difficult part of the modeling process. Even though mesh generators like CUBIT offer new perspectives, assessing the quality of an unstructured hexahedral mesh both in terms of geometrical and numerical accuracy is still a serious challenge. As a first step we simplified the basin structure, shape and velocity. Shallow parts of less than 30m as well as some secondary basins haven't been incorporated, and basin edges have been cut at a depth of 70m (Figure 2). These simplifications are reasonable for the low frequency simulations we made where 70m corresponds to a 1D fundamental frequency of 3Hz. No anelastic attenuation is taken into account, and homogeneous velocities in the rock and sediments are considered (Table I). The unstructured mesh composed of 38394 elements was generated with CUBIT (Figure 2). It respects the steep topography and the velocity discontinuities with a conforming coarsening from the basin to the external boundaries of the model. Our 3D code is a MPI parallel version. Simulations were made with a polynomial degree of 6, with a time step of 0.6ms corresponding to a Courant number of 0.2. A 1s signal simulation was obtained in about 1h elapsed time on 16 Opteron 256 processors.

3D simulations

We studied the basin response through its excitation by a north-south polarized plane S wave with a 1Hz central frequency. Polarization and frequency correspond to the case of maximum amplification for a 3D Gaussian shape hill [START_REF] Komatitsch | The spectral-element method: an efficient tool to simulate the seismic response of 2D and 3D geological structures[END_REF]. The effects from the topography and the basin can be clearly observed. We focused in particular on the SB and LPG profiles (Figure 3).

The topographic effects are characterized by an amplification of motion on the mountain and diffracted waves being composed of a surface P wave traveling below the surface and a Rayleigh wave. The basin effects consist of trapping and amplification of the wave field in the basin, which produce more complex and longer signals. This basin effect is stronger in LPG where sediments are deeper. Most of the energy remains here due to amplified 2D and 3D focusing effects.

A 3D visualization of the velocity field modulus exhibits both 2D and 3D effects from the mountain. A P wave front parallel to the major axis travels firstly from the top of the topography towards the basin then Rayleigh waves follow, and focus in the LPG area.

San Bernardino NS Velocity comp. The Spectral ratio (SR), defined as the ratio of the Fourier amplitude spectrum of the observed movement at the surface with the incident one, is presented along the LPG profile and for the entire basin (Figure 4). Looking at the entire LPG profile, the highest amplifications are clearly localized in the basin. The influence of the topography is negligible compared to the effects of the basin, even if a small peak at the top of the mountain can be noticed. Frequency peaks follow well the thickness of the sediments: the associated frequency value increases as the sediments depth decreases. The deepest part shows an amplification at around 0.8Hz (the 1D fundamental frequency associated to our velocity model is 0.66Hz). Amplifications can be then observed at the borders of the basin, corresponding to higher frequencies. These observations correlate well with the H/V ratio results on microtremor found by [START_REF] Duval | Caracas, Venezuela, Site Effect Determination with Microtremors[END_REF]. Other amplification peaks can be found along the profile as a consequence of 2D/3D focusing effects. SR results for the total basin show that up to 2.5Hz, the highest amplifications are concentrated in the deepest part of the basin, the LPG area. A clear amplification is observed at 1Hz, then thicker parts are excited at higher frequencies. Above 1.5Hz, localized peaks reveal 2D/3D effects. At these frequencies, amplifications also occur in the SB area and interactions with the east part of the basin are expected.

Conclusions

We have presented an application of 3D SEM simulations of the seismic response in the Caracas valley. These have been carried out on an unstructured mesh taking into account the regional topography and the basin geometry. Given the limitations of the low frequency simplified model we considered, we observed 2D and 3D geometrical effects. Diffracted waves are generated by the mountain topography going into the basin, where high amplifications occur in the deepest area. Focusing effects are also observed in thicker sections at higher frequencies. This first step provides encouraging results for further simulations based on a more realistic model in terms of geometry, velocity structures and anelastic attenuation. A scenario study depending on the incidence, polarization and time function of the source at frequencies up to 5Hz should enable us to analyze and evaluate more precisely the 3D effects. An ongoing work aims to quantify the geometrical heterogeneity of the unstructured mesh and its influence in time and space on the numerical resolution.

Figure 2 .

 2 Figure 2. On the left, the 3D model used for our simulations. The basin is in red. On the right, the unstructured mesh realized with CUBIT (colors outside the basin correspond to an artificial cut for controlling the elements size).

Figure 3 .

 3 Figure 3. NS velocity component due to a NS polarized plane S wave along SB and LPG profiles.

Figure 4 .

 4 Figure 4. Spectral ratio for the NS velocity component due to a NS polarized plane S wave, along the LPG profile including the topography (left), along the LPG profile restricted to the basin section (middle) and for the entire basin for frequencies from 1 to 2.5Hz (right).

Table I

 I

		. velocities and densities in the basin model
	Sediments Rock	P velocity 2300 m/s 3800 m/s	S velocity 850 m/s 2200 m/s	density 2400 kg/m 3 2700 kg/m 3


Andrea Cristofaro
email: andrea.cristofaro@unicam.it

Sergio Galeani
email: sergio.galeani@uniroma2.it

Simona Onori
email: sonori@clemson.edu

Luca Zaccarian
email: zaccarian@laas.fr

A switched and scheduled design for model recovery anti-windup of linear plants

Keywords: model recovery anti-windup, scheduled stabilizer, switched stabilizer, bounded control

published or not. The documents may come

A switched and scheduled design for model recovery anti-windup of linear plants

Introduction

Actuator saturation is one of the most frequent hard nonlinearities encountered in control system implementation. When applying any type of control design to a real plant, the control engineer needs to account for the effects of the maximum and minimum control effort allowable for the actuator available on the experimental system. It was observed already from the 1940s that the presence of saturation often caused undesirable behavior and that suitable fixes were necessary to address the arising problem (this fact was actually one of the main motivations for the absolute stability results of the 1950s and the sector properties of static nonlinearities).

A popular approach to deal with saturation, which dates back to the 1960s (see, e.g., [START_REF] Fertik | Direct digital control algorithm with antiwindup feature[END_REF]), is the so-called anti-windup approach, wherein an "unconstrained controller" is assumed to perform desirably in the absence of saturation (equivalently, when signals are small enough on the saturated closed-loop) and "anti-windup" corrections are necessary for larger signals that would interest the flat region of the saturation, thereby causing performance loss and often instability. The ultimate goals of anti-windup designs are that: 1) the unconstrained behavior imposed by the "unconstrained controller" on the plant is fully reproduced whenever saturation is not activated (namely for small enough signals); 2) for larger signals the performance is close (as close as possible) to the unconstrained one, and stability is retained as much as possible given the hard limits imposed by the presence of saturation. Initially, anti-windup control was mainly an application driven discipline and most of the results available in the literature were not applicable to general classes of control systems, rather being specific naive solutions for some experimental problems. Toward the end of the 1980s, this lack of formality in the field has been pointed out (see, [START_REF] Doyle | Control of plants with input saturation nonlinearities[END_REF]) and the need of general solutions to the anti-windup augmentation problem led to in an increase of the research effort in the following years. Comprehensive surveys of the first anti-windup solutions are given in [START_REF] Hanus | Antiwindup and bumpless transfer: a survey[END_REF][START_REF] Åström | Integrator windup and how to avoid it[END_REF][START_REF] Kothare | A unified framework for the study of anti-windup designs[END_REF]. Starting from the mid 1990s, several high performance solutions to the antiwindup problem started to appear. Many of these rely on the socalled reference governor or command governor scheme which is most directly applicable to discrete-time control systems (see, e.g., [START_REF] Gilbert | Fast reference governors for systems with state and control constraints and disturbance inputs[END_REF][START_REF] Angeli | Command governors for constrained nonlinear systems[END_REF] and references therein). Other ones are mostly related to continuous-time cases and rely on the use of linear matrix inequalities (LMIs) together with absolute stability and generalized absolute stability concepts (see, e.g., [START_REF] Mulder | Multivariable antiwindup controller synthesis using linear matrix inequalities[END_REF][START_REF] Gomes Da Silva | Anti-windup design with guaranteed regions of stability for discrete-time linear systems[END_REF][START_REF] Hu | Anti-windup synthesis for linear control systems with input saturation: achieving regional, nonlinear performance[END_REF] and references therein). Some very detailed presentations of the above historical overview of anti-windup can be found in the surveys [START_REF] Galeani | A tutorial on modern anti-windup design[END_REF][START_REF] Tarbouriech | Anti-windup design: an overview of some recent advances and open problems[END_REF] and in the two books [START_REF] Zaccarian | Modern anti-windup synthesis: control augmentation for actuator saturation[END_REF][START_REF] Tarbouriech | Stability and stabilization of linear systems with saturating actuators[END_REF].

Among the many aspects that make the anti-windup design challenging is that it inherits well-known intrinsic limitations already known for bounded stabilization. In particular, it was already proved in [START_REF] Sontag | An algebraic approach to bounded controllability of linear systems[END_REF] that global bounded asymptotic stabilization of a linear plant can only be obtained if the plant has eigenvalues in the closed left half plane. Global exponential stabilization via bounded input can then only be obtained if the plant is already globally exponentially stable to begin with. Due to this reason, much of the research effort has gone in the direction of giving local guarantees with an intrinsic trade-off between size of the operating region and the aggressiveness of the control law (a notable alternative approach being the one where global nonlinear stabilizers were designed [START_REF] Megretski | L 2 BIBO output feedback stabilization with saturated control[END_REF][START_REF] Teel | Global stabilization and restricted tracking for multiple integrators with bounded controls[END_REF]). In this regard, interpolation of controllers has been recognized as a fruitful approach for achieving better regional performances, using e.g. parametrized Riccati equations or switching laws [START_REF] Gf Wredenhagen | Piecewise-linear lq control for systems with input constraints[END_REF][START_REF] Henrion | Piecewise-linear robust control of systems with input constraints[END_REF][START_REF] Gomes | Local stabilization of discrete-time linear systems with saturating controls: an lmi-based approach[END_REF]. From the early years of the 21 st century, the idea of scheduling different stabilizers was pursued in a number of works [START_REF] Kose | Scheduled controllers for linear systems with bounded actuators[END_REF][START_REF] Jabbari | Rate and magnitude-bounded actuators: scheduled output feedback design[END_REF][START_REF] Sajjadi-Kia | Multi-stage anti-windup compensation for open-loop stable plants[END_REF][START_REF] Sajjadi-Kia | Controllers for linear systems with bounded actuators: Slab scheduling and anti-windup[END_REF][START_REF] Valmorbida | Nonlinear static state feedback for saturated linear plants via a polynomial approach[END_REF], and this type of sophisticated generalization of the standard anti-windup paradigm (the one where a simple gain modifies the controller dynamics, driven by the excess of saturation) also led to some different type of anticipatory/delayed anti-windup actions (which can be seen as some kind of multi-stage nonlinear compensation) reported in [START_REF] Wu | Dynamic anti-windup design in anticipation of actuator saturation[END_REF][START_REF] Sajjadi-Kia | Multi-stage anti-windup compensation for open-loop stable plants[END_REF][START_REF] Wu | On immediate, delayed and anticipatory activation of anti-windup mechanism: static anti-windup case[END_REF][START_REF] Turner | An alternative approach to anti-windup in anticipation of actuator saturation[END_REF][START_REF] Turner | A non-square sector condition and its application in deferred-action anti-windup compensator design[END_REF] and references therein.

A relevant general approach to the solution of the antiwindup problem is that proposed in the pioneering paper [START_REF] Teel | The L 2 anti-windup problem: Its definition and solution[END_REF] (and the twin paper [START_REF] Teel | Uniting local and global controllers[END_REF]). In that paper, the authors proposed a framework for anti-windup design wherein the solution to the anti-windup problem (comprising the two properties qualitatively stated above) was obtained from the solution of a simpler external stabilization problem from an input-matched disturbance consisting in the energy that the unconstrained response would have spent outside of the saturation if the saturation limits were not there. That framework was therein denoted as "L 2 anti-windup solution" and was shown to solve a specific characterization of the qualitative anti-windup goals listed above, called "L 2 anti-windup problem". The parallel thread of work reported in [START_REF] Mulder | Multivariable antiwindup controller synthesis using linear matrix inequalities[END_REF][START_REF] Gomes Da Silva | Anti-windup design with guaranteed regions of stability for discrete-time linear systems[END_REF][START_REF] Hu | Anti-windup synthesis for linear control systems with input saturation: achieving regional, nonlinear performance[END_REF] and commented above, aimed indeed at minimizing the input-output L 2 gain of the closed loop with anti-windup augmentation, therefore more recently (and also in the book [START_REF] Zaccarian | Modern anti-windup synthesis: control augmentation for actuator saturation[END_REF]) it was decided to rename this approach "model recovery anti-windup" (MRAW) because through a model-based compensation MRAW is effective at recovering the unconstrained response.

In [START_REF] Teel | The L 2 anti-windup problem: Its definition and solution[END_REF], a preliminary solution was given to the MRAW problem, while indicating that many improvements upon that solution would have been possible via the many degrees of freedom still available in the proposed framework. Then, several extensions have been proposed in later years each of them having different performance and stability guarantees and being applicable to different classes of systems. Among these, [START_REF] Teel | Anti-windup for an F-16's daisy chain control allocator[END_REF][START_REF] Teel | Uniting local and global controllers for the Caltech ducted fan[END_REF][START_REF] Teel | An anti-windup strategy for active vibration isolation systems[END_REF][START_REF] Todeschini | Nonlinear pressure control for BBW systems via dead zone and anti-windup compensation[END_REF][START_REF] Zaccarian | Anti-windup for marginally stable plants and its application to open water channel control systems[END_REF][START_REF] Donnarumma | Antiwindup synthesis of heading and speed regulators for ship control with actuator saturation[END_REF] report some application studies illustrating the effectiveness of MRAW on practical engineering problems, while in [START_REF] Teel | Anti-windup for exponentially unstable linear systems[END_REF][START_REF] Galeani | Constructive nonlinear antiwindup design for exponentially unstable linear plants[END_REF], the approach was extended to give a nonlinear solution when dealing with exponentially unstable plants, that inevitably lead to non-global results due to the above mentioned intrinsic limitations. MRAW was also extended to rate-saturated plants and applied to relevant flight control problems in [START_REF] Teel | Anti-windup for an F-16's daisy chain control allocator[END_REF][START_REF] Barbu | Nonlinear anti-windup for manual flight control[END_REF][START_REF] Forni | Model recovery anti-windup for continuous-time rate and magnitude saturated linear plants[END_REF]. In [START_REF] Zaccarian | A common framework for anti-windup, bumpless transfer and reliable designs[END_REF], the same framework was applied to bumpless transfer and this idea was further extended and better characterized in [START_REF] Zaccarian | The L 2 (l 2 ) bumpless transfer problem: its definition and solution[END_REF]. In [START_REF] Grimm | The l 2 anti-windup problem for discrete-time linear systems: definition and solutions[END_REF] the ideas in [START_REF] Teel | The L 2 anti-windup problem: Its definition and solution[END_REF] were extended to the discrete-time case while in [START_REF] Zaccarian | L 2 anti-windup for linear deadtime systems[END_REF] they were extended to dead-time plants. A sampled-data implementation of the general approach in [START_REF] Teel | The L 2 anti-windup problem: Its definition and solution[END_REF] was proposed in [START_REF] Bemporad | Anti-windup synthesis via sampled-data piecewise affine optimal control[END_REF], based on a suitable MPC solution of a discrete-time problem associated to the continuous-time plant. Moreover, a nonlinear scheduled implementation based on hysteresis switching was proposed in [START_REF] Zaccarian | Nonlinear scheduled anti-windup design for linear systems[END_REF], although no guarantees were given there on the size of the switching regions.

In this paper, we show that it is possible to adopt multi-stage designs within the MRAW framework, thereby obtaining high performance unconstrained response recovery through the arising aggressive nonlinear anti-windup action. Mimicking the approach in [START_REF] Kose | Scheduled controllers for linear systems with bounded actuators[END_REF][START_REF] Jabbari | Rate and magnitude-bounded actuators: scheduled output feedback design[END_REF][START_REF] Sajjadi-Kia | Multi-stage anti-windup compensation for open-loop stable plants[END_REF][START_REF] Sajjadi-Kia | Controllers for linear systems with bounded actuators: Slab scheduling and anti-windup[END_REF], we base the switching/scheduling mechanism on the definition of a suitable set of nested ellipsoids, each of them associated to a choice of stabilizing gains, and then we propose: 1) A hybrid reformulation of the hysteresis switching solution first proposed in [START_REF] Zaccarian | Nonlinear scheduled anti-windup design for linear systems[END_REF] and then revised and improved in our preliminary conference paper [START_REF] Galeani | Regional, semiglobal, global nonlinear anti-windup via switching design[END_REF]; as compared to those preliminary results the hybrid formulation provides robust versions of the established asymptotic stability, that stems from the structural properties of the formulation, in addition to a clearer description of the hysteresis mechanism.

2) A scheduling mechanism corresponding to an improved version of the approach proposed in our preliminary work [START_REF] Galeani | Nonlinear scheduled control for linear systems subject to saturation with application to anti-windup control[END_REF][START_REF] Galeani | Nonlinear L 2 antiwindup for enlarged stability regions and regional performance[END_REF]. In particular, as compared to those results, here we choose the feedback gains based on a desirable trade-off between speed of convergence and size of the region of attraction, in addition to well characterizing the nonlinear algebraic loop, already introduced in [START_REF] Zaccarian | A common framework for anti-windup, bumpless transfer and reliable designs[END_REF], but assuming a slightly different expression in this new formulation.

Different from [START_REF] Galeani | Nonlinear scheduled control for linear systems subject to saturation with application to anti-windup control[END_REF][START_REF] Galeani | Nonlinear L 2 antiwindup for enlarged stability regions and regional performance[END_REF] where we mainly focus on antiwindup goals, we clarify here that the proposed construction is best characterized as a state-feedback bounded stabilizer, whose ideal application context is the anti-windup one, because in nonlinear model recovery anti-windup, the state of the dynamic augmentation (which must be driven to zero) is available for feedback. We regard this solution as the most advanced MRAW design strategy in terms of trade-off between size of the region of attraction and performance of the control design (which is characterized here in terms of the exponential rate of convergence of the response). As compared to our preliminary results in [START_REF] Galeani | Regional, semiglobal, global nonlinear anti-windup via switching design[END_REF][START_REF] Galeani | Nonlinear scheduled control for linear systems subject to saturation with application to anti-windup control[END_REF][START_REF] Galeani | Nonlinear L 2 antiwindup for enlarged stability regions and regional performance[END_REF], we also report here missing proofs, we revise and improve the generation of the ellipsoids that depends here on more intuitive parameters for which we provide a tuning rationale, and the design of the switching/scheduling law. Finally, we illustrate the results on novel examples.

The paper is organized as follows. In Section 2 we propose the construction of the nested ellipsoids establishing the switching/scheduling mechanism proposed subsequently. In Section 3 we show how those ellipsoids can be used for the design of a hysteresis switching stabilizer. Parallel results are given in Section 4 that presents a Lipschitz scheduled stabilizer. All these ingredients are combined in Section 5 that explains how these designs should be incorporated in the MRAW architecture. Finally, Section 6 discusses a few simulation examples with the switching and scheduling approaches. Conclusions and future works are reported in Section 7, while a few technical proofs are reported in the appendix.

Notation: Given a square matrix M, He M := M + M T 2. A family of stabilizers in nested stability regions 2.1. Global or regional bounded state feedback stabilization of a linear plant Consider the following linear plant:

ẋ = Ax + B sat ū(u), (1)
with state x ∈ R n and input u ∈ R n u and subject to the symmetric decentralized magnitude saturation nonlinearity sat ū with saturation limits ū = ū1

• • • ūn u T , defined as sat ū(u) = sat ū1 (u 1 ) • • • sat ūnu (u n u ) T
, where sat ūi (s) = max{-ū i , min{ū i , s}}, i = 1, . . . , n u , is the scalar symmetric saturation function with limit ūi .

In this paper we are interested in designing nonlinear switched and scheduled state-feedback control laws for plant (1) capable of inducing high-performance of the arising closed loop. It is well known (see, e.g., [START_REF] Sontag | An algebraic approach to bounded controllability of linear systems[END_REF]) that plant (1) is globally stabilizable from the bounded input if and only if A has no eigenvalue with positive real part (namely, the open-loop dynamics is not exponentially unstable). Due to this reason, we start off our design by focusing on a suitable stabilizer taking the form

u = K x + L sat m (u), (2)
and ensuring a reasonably large domain of attraction, depending on the open-loop dynamics of (1). Stabilizer (2) requires the solution of a nonlinear algebraic loop, which is rather simple with a small number of inputs is at hand but may become rather complicated in the presence of many saturated inputs (see [57, §2.3.7] for a discussion about how to solve these algebraic loops). The algebraic loop has been shown to generally lead to improved responses (see, e.g., [START_REF] Mulder | Multivariable antiwindup controller synthesis using linear matrix inequalities[END_REF]) but can be easily avoided (at the expense of a potentially deteriorated performance) by fixing X 2 = 0 in the optimization presented below in [START_REF] Barbu | Nonlinear anti-windup for manual flight control[END_REF]. In this special case, variants of the corresponding results have been presented in [START_REF] Gomes Da | Anti-windup design with guaranteed regions of stability: an LMI-based approach[END_REF] (see also [START_REF] Tarbouriech | Stability and stabilization of linear systems with saturating actuators[END_REF]). The following proposition provides a design guideline for the selection of the gains in (2).

Proposition 1. Consider the dynamical system (1), a scalar λ ≥ 0, and the following set of LMIs in the variables {Q, X 1 , X 2 , U, Y, ρ -2 }:

Q = Q T ≥ ρ 2 I, U > 0 diagonal (3a) He (A + λI)Q + BX 1 B(X 2 -U) X 1 + Y X 2 -U < 0, (3b)
m 2 k Y (k) Y (k) Q ≥ 0, k = 1, . . . , n u , (3c)
where Y (k) denotes the k-th row of matrix Y. Then given any solution to (3), the feedback stabilizer (2) with parameters (K, L) = (K 1 , L 1 ) with

K 1 = (I -X 2 U -1 ) -1 X 1 Q -1 , L 1 = -(I -X 2 U -1 ) -1 X 2 U -1 , (4)
guarantees 1) well-posedness of the algebraic loop in (2), namely existence of a unique (nonlinear) globally Lipschitz function ϕ such that u = ϕ(x); 2) exponential stability of the origin with exponential rate λ 1 = λ for (1), (2) with basin of attraction larger than the set (where we use Q 1 = Q):

E 1 := E(Q -1 1 ) := {x T Q -1 1 x ≤ 1}, (5)
which is forward invariant and contains the ball ρB = {|x| ≤ ρ};

3) global exponential stability of the origin with exponential rate λ, if Y = 0. Moreover, the following can be proven about the feasibility of (3): 3. In all cases, LMIs (3) are feasible for a sufficiently small value of ρ.

Remark 1. (Different optimization goals) While the focus of Proposition 1 is mostly on the size of the domain of attraction estimate (called stability region henceforth), it should be recognized that increasing the value of the certified exponential rate λ will in general reduce the size of the stability region so that a suitable trade-off should be performed. The goal of the switching and scheduled solutions in this paper is to partially overcome this trade-off by suitably increasing the performance level λ as the plant state gets closer to the origin.

In conditions (3) and throughout the rest of this paper, we focus on the optimality criterion associated to the exponential rate λ in (3b). However, different performance metrics can be also optimized using the approach proposed here. For example, in [START_REF] Galeani | Regional, semiglobal, global nonlinear anti-windup via switching design[END_REF] the goal was the minimization of the L 2 gain from an exogenous input w to a performance output z of an extended plant dynamics of the form:

ẋ = Ax + B sat ū(u) + B w w, z = Cx + D sat ū(u) + D w w.
This different performance goal can then be optimized by minimizing the L 2 gain estimate γ and replacing condition (3) by the conditions in [12, eqs. (7)]. This modification affects many of the derivations in the rest of this section, but the corresponding results are a straightforward extension of the techniques developed here.

•

Proof of Proposition 1. Proposition 1 is a slight modification of the results presented in [12, §II.B] and then recalled in [START_REF] Galeani | Nonlinear scheduled control for linear systems subject to saturation with application to anti-windup control[END_REF]Prop. 2]. The proof follows standard LMI derivations based on the generalized sector conditions proposed in [START_REF] Gomes Da | Anti-windup design with guaranteed regions of stability: an LMI-based approach[END_REF][START_REF] Hu | An analysis and design method for linear systems subject to actuator saturation and disturbance[END_REF] and corresponding to the fact that for any pair of inputs u, w ∈ R n u and any diagonal positive matrix W > 0, the following holds:

dz(w) = 0 ⇒ dz(u) T W(u -dz(u) + w) ≥ 0, (6)
where dz(u) := usat(u). In order to exploit condition (6), we may rewrite the plant input selection in (2) as follows:

u = (I -L) -1 K x -(I -L) -1 L dz(u) = X 1 Q -1 x + X 2 U -1 dz(u), (7)
where the second equality can be verified by inverting (4) and doing some straightforward simplifications. The following relation is then straightforward from ( 7):

ẋ = Ax + B sat ū(u) = (A + BX 1 Q -1 )x + B(X 2 U -1 -I) dz(u). ( 8
)
Consider now formulation ( 7), ( 8) of the closed loop and notice that using the Lyapunov function candidate V(x) = x T Q -1 x and selecting w = Hx, with H to be specified, and W = U -1 in ( 6), we may write:

dz(Hx) = 0 ⇒ V(x) ≤ V(x)+2 dz(u) T U -1 (u-dz(u)+Hx), (9)
whose rightmost term can be written as:

2 x dz(u) T Q -1 (A + BX 1 Q -1 ) Q -1 B(X 2 U -1 -I) U -1 (X 1 Q -1 + H) U -1 (X 2 U -1 -I)
x dz(u) [START_REF] Forni | Model recovery anti-windup for continuous-time rate and magnitude saturated linear plants[END_REF] where the first block line contains terms arising from V and the second block line contains terms from the generalized sector conditions. Pre-and post-multiplying the matrix appearing in [START_REF] Forni | Model recovery anti-windup for continuous-time rate and magnitude saturated linear plants[END_REF] by diag Q, U and selecting

H = Y Q -1 , we obtain that inequality (3b) implies dz(Y Q -1 x) = 0 ⇒ V(x) ≤ -2γV(x). (11)
Consider now inequality (3c), and note that, with the selections above, it implies:

|m -1 k H (k) x| 2 ≤ x T Q -1 x, ∀k = 1, . . . , n u ,
which, combined with (11), implies

V(x) ≤ 1 ⇒ dz(Hx) = 0 ⇒ V(x) ≤ -2γV(x), (12)
thus clearly proving items (2) and (3) of the statement from standard comparison and Lyapunov theorems. Moreover, let x belong to ρB, i.e. |x| 2 ≤ ρ 2 . Then, from Q -1 ≤ ρ -2 I, the following chain of inequalities holds

x T Q -1 x ≤ x T ρ -2 I x = ρ -2 |x| 2 ≤ 1,
thus showing the inclusion ρB ⊆ E 1 .

Regarding item (1), the proof is based on the following technical results Lemma 1. Given a positive-definite diagonal matrix W ∈ R n u ×n u and a square matrix S ∈ R n u ×n u with det(I -S ) 0, if 2W + WS (I -S ) -1 + (I -S T ) -1 S T W > 0, then I -S ∆ is nonsingular for all matrices ∆ included in the set

O n u := {∆ : ∆ = diag(δ 1 , ..., δ n u ), δ k ∈ [0, 1] ∀k}.
Lemma 2. Consider a locally Lipschitz map F : R n → R n and assume that the differential inclusion JF(z) ∈ M holds for a.e. z ∈ R n , where JF(z) denotes the Jacobian of F and M ⊂ R n×n is a compact and convex subset such that det(M) 0 ∀M ∈ M. Then there exists a unique globally Lipschitz function G :

R n → R n with F(G(z)) = z ∀z ∈ R n .
The proof of Lemma 1 is given in Appendix A, while the proof of Lemma 2 can be found in [START_REF] Zaccarian | A common framework for anti-windup, bumpless transfer and reliable designs[END_REF]Prop. 2]. Let us focus now on item [START_REF] Angeli | Command governors for constrained nonlinear systems[END_REF]. From the LMI (3b) one can infer that

X 2 + X T 2 -2U < 0 ( 13
)
as the bottom right term of the matrix must be negative-definite.

On the other hand, it follows from (4) that the matrix X 2 satisfies

X 2 = -L 1 (I -L 1 ) -1 U,
and hence inequality ( 13) can be rewritten in terms of L 1 as

-L 1 (I -L 1 ) -1 U -U(I -L T 1 ) -1 L T 1 -2U < 0
or equivalently as

-WL 1 (I -L 1 ) -1 -(I -L T 1 ) -1 L T 1 W -2W < 0 ( 14
)
where we have set W = U -1 . Consider the map F(u) := u -L 1 sat ū(u) and observe that, since the derivative of each component of the saturation function is either 0 or 1 a.e., the inclusion

JF(u) ∈ co({I -L 1 diag(δ 1 , ..., δ n u ), δ k ∈ {0, 1} ∀k}) = {I -L 1 ∆, ∆ ∈ O n u } =: M
holds, where co(E) denotes the closed convex hull of the set E and O n u is defined in Lemma 1. Due to [START_REF] Galeani | A tutorial on modern anti-windup design[END_REF], the conditions of Lemma 1 are met and thus the set M, which is compact and convex by construction, contains nonsingular matrices only. The conclusion follows then by applying Lemma 2, which establishes that there exists a unique globally Lipschitz map G(z) with F(G(z)) = z and

ϕ(x) -L 1 ϕ(x) = K 1 x,
where we have set ϕ

(x) = G(K 1 x).
Only the feasibility of the LMIs remains to be addressed. Since by assumption the plant is stabilizable, there exist a feedback gain K 0 and two positive definite matrices,

Q 0 = Q T 0 > 0, R 0 = R T 0 > 0, with He[(A + λI)Q 0 + BK 0 Q 0 ] = -R 0 . Selecting Q = υQ 0 , Y = K 0 Q = υK 0 Q 0 , X 1 = -Y + (X 2 - U) T B T and 2U > X T 2 + X 2 ,
it can be easily seen that inequality (3b) is satisfied for any υ > 0 because the left-hand side matrix becomes block diagonal. In the case of open-loop exponential stability (item (1)), the inequalities are trivially satisfied with the choice

Y = 0, X T 1 = -B(X 2 -U), ρ 2 = σ min (Q), 2U > X 2 + X T 2 , He[(A + λI)Q] < 0.
Regarding item (3), observe that condition (3c) is equivalent to

m 2 k -υY (k) 0 Q -1 0 Y (k) 0 ≥ 0 k = 1, ..., n u
where we have denoted by

Y (k) 0 the k th -row of Y 0 . Then, it is sufficient to pick υ ≤ υ := min k=1,...,n u m 2 k Y (k) 0 Q -1 0 Y (k) 0
which indeed implies the feasibility for any ρ 2 ≤ υσ min (Q 0 ). Item (2) corresponds to semi-global stabilization, and requires a few more manipulations. Define A λ = A + λI and note that by assumption it has eigenvalues in the closed left half plane. Following [25, Proposition 2], it can be proven that there exists γ 0 > 0 such that, for any ζ > 0 there exists

Q = Q T > ζI with He                A λ Q - γ 2 0 2 BB T - B 2 0 - I 2                < 0. (15)
Therefore, setting

Y = -(γ 2 0 /2)B T , X 1 = -Y, U = X 2 + I/2
, where X 2 is an arbitrary diagonal matrix with X 2 > -I/2, it is easily checked that (3b) is satisfied because it coincides with [START_REF] Galeani | Constructive nonlinear antiwindup design for exponentially unstable linear plants[END_REF]. Moreover, also inequalities (3a) and (3c) are satisfied by selecting [START_REF] Galeani | Constructive nonlinear antiwindup design for exponentially unstable linear plants[END_REF]. ♦

ζ ≥ max ρ 2 , max k∈{1,...,n u } |Y (k) | m k in

Generation of nested stabilizers

Proposition 1 is used in this section as a starting point for the generation of a family of N nested ellipsoids

E 1 ⊃ • • • ⊃ E N ,
each one of them associated with a pair of gains (K i , L i ) to be used in control law (2), and ensuring exponential stability of the origin from the corresponding ellipsoid. In particular, the region E 1 in (5) corresponds to the largest of these ellipsoids and an algorithm is proposed to generate a number of nested ellipsoids within E 1 . The reason for designing several gains in nested sets is that the closer the trajectory is to the origin, the higher gain can be used with stability guarantees, thereby improving the small and medium signal performance of the stabilizer. The solid and bold level sets in Figure 1 represent an example of two nested sets arising from the algorithm proposed in this section. This construction has common features with that of [START_REF] Zaccarian | Nonlinear scheduled anti-windup design for linear systems[END_REF], even though no generalized sector conditions where used in that work, thereby resulting in a conservative design.

Moreover, the construction is a suitable combination of our preliminary results in [START_REF] Galeani | Regional, semiglobal, global nonlinear anti-windup via switching design[END_REF] (where a switching law was used) and [START_REF] Galeani | Nonlinear L 2 antiwindup for enlarged stability regions and regional performance[END_REF][START_REF] Galeani | Nonlinear scheduled control for linear systems subject to saturation with application to anti-windup control[END_REF] (where scheduled laws were used and a similar algorithm to the one reported here is proposed). Here, once the family of nested sets with the corresponding gains is in place, we present switched and scheduled nonlinear stabilizers in Sections 3 and 4, respectively. Given the outer ellipsoid E 1 as per ( 5), the proposed nested regions construction is given in Algorithm 1, which is based on two key parameters:

E i-1 βE i-1 E i β 2 σ i-2 B β 2 σ i-1 B
1. a shrinking factor β ∈ (0, 1) providing outer and inner constraints for set E i graphically represented in Figure 1, based on set E i-1 and corresponding to (16c) for the outer constraint (dashed-dotted boundary in Figure 1) and to (16d) for the inner constraint (dashed boundary in Figure 1). Figure 1 reports in bold the shape of set E i that respects these constraints, given the set E i-1 reported in thin solid line; 2. a linearity radius η > 0 specifying a ball ηB = {|x| ≤ η},

where the nested control design should provide a linear feedback. For the algorithm to make sense it is necessary that ηB is contained in the interior of E 1 , which holds if and only if η 2 < σ min (Q 1 ).

The following result establishes desirable features of Algorithm 1.

Proposition 2. Consider any solution (λ 1 , Q 1 , K 1 , L 1 ) arising from the construction in Proposition 1 (see ( 4) and ( 5)), any shrinking factor β ∈ (0, 1) and any η > 0 satisfying η 2 ≤ σ min (Q 1 ).

Then Algorithm 1 always terminates in a finite number N of steps and guarantees:

E 1 ⊃ E 2 • • • ⊃ E N , (18)
λ 1 ≤ λ 2 • • • ≤ λ N . ( 19
)
Moreover, for each i ∈ {1, . . . , N}, the closed loop (1), (2) with K = K i and L = L i has a well posed algebraic loop, is exponentially stable to the origin, with basin of attraction containing the set

E i = {x T Q -1 i x ≤ 1}
, and is also such that the set E i is forward invariant.

Algorithm 1 Nested regions and stabilizers generation.

Input: Matrix Q 1 in (5), gains (K 1 , L 1 ) in ( 4) and λ 1 from Proposition 1. A shrinking factor β ∈ (0, 1) and a linear feedback radius η > 0 satisfying η 2 < σ min (Q 1 ). Initialization: Set i = 1 and σ 1 = σ min (Q 1 ).

Iteration: i = i + 1
Optimization: and solve the following generalized eigenvalue problem:

max λ,Q=Q T ,Y,X 1 ,X 2 ,U λ subject to: (16a) U > 0 diagonal (16b) Q < βQ i-1 (16c) Q > β 2 σ i-1 I (16d) He (A + λI)Q + BX 1 B(X 2 -U) X 1 + Y X 2 -U < 0, ( 16e
)
m 2 k Y (k) Y (k) Q ≥ 0, k = 1, . . . , n u , (16f)
Selection: Given the optimal solution to optimization [START_REF] Gilbert | Fast reference governors for systems with state and control constraints and disturbance inputs[END_REF],

set λ i = λ, Q i = Q, σ i = σ min (Q i ) and: K i = (I -X 2 U -1 ) -1 X 1 Q -1 , L i = -(I -X 2 U -1 ) -1 X 2 U -1 . ( 17
)
until η 2 > σ i . Output: Set N = i -1, and output N tuples (λ , Q , K , L ), = 1, . . . , N.
Finally, for a fixed value of η, integer N is a function of scalar β satisfying:

lim β→0 N = 1, lim β→1 N = +∞. ( 20
)
Proof. From the feasibility conditions of the LMIs established in Proposition 1, it follows that the optimization problem in the algorithm is well-posed, in the sense that it always admits a solution λ i ≥ λ i-1 . Moreover, the strict inclusion E i ⊂ E i-1 is guaranteed by (16c): 16d) ensure that the algorithm terminates after N steps. Indeed one has β 2 σ i-1 < σ i < βσ i-1 , and hence

E i = {x T Q -1 i x ≤ 1} ⊂ {x T β -1 Q -1 i-1 x ≤ 1} ⊂ {x T Q -1 i-1 x ≤ 1} = E i-1 . Conditions (16c)-(
β 2i-2 σ 1 < σ i < β i-1 σ 1 . (21)
Taking logarithms of both sides of η 2 ≤ σ N < β N-1 σ 1 , we get that the largest integer N with η 2 ≤ σ N (which comes from the termination condition) must satisfy the estimate

N < log η 2 -log σ 1 log β + 1,
which implies the left limit in (20) because η 2 < σ 1 by definition. Consider now again the termination condition and the left inequality in [START_REF] Hanus | Antiwindup and bumpless transfer: a survey[END_REF], which yields β 2((N+1)-1) < σ N+1 ≤ η 2 . Taking logarithms again, we get

N > log η 2 -log σ 1 2 log β ,
which implies the right limit in [START_REF] Grimm | The l 2 anti-windup problem for discrete-time linear systems: definition and solutions[END_REF]. ♦ Remark 2. In light of Proposition 2, from a practical point of view, Algorithm 1 should be applied by first fixing the local convergence rate λ 1 and solving the LMIs (3) in Proposition 1 with the goal of maximizing the size of Q 1 (e.g., maximizing its trace). Alternatively, if a certain size ᾱ of the outer region is required, one may fix Q 1 ≥ α 2 I and then maximize λ 1 by solving a generalized eigenvalue problem. Then the linearity radius η is typically selected as significantly smaller than √ σ min (Q 1 ) (so as to leave a large enough doughnut for the nesting of the ellipsoidal sets before the termination condition), and then the shrinking factor β is adjusted by running the algorithm for several values of it with the goal of obtaining a reasonable number N of ellipsoids. In particular, the result [START_REF] Grimm | The l 2 anti-windup problem for discrete-time linear systems: definition and solutions[END_REF] clarifies that increasing β one expects an increase of N, while decreasing β one expects a decrease of N. A larger N leads to improved performance, due to a more sophisticated control solution, but also increases the computational complexity in terms of the number of gains that need to be stored in the control system.

•

Hysteresis switching stabilization

In the previous two sections, a sequence of nested sets E 1 ⊃ • • • ⊃ E N has been determined by way of Algorithm 1, each of them associate with a pair of gains (K i , L i ) and with a guaranteed convergence rate λ i ≥ 0 satisfying the non-decreasing condition [START_REF] Gomes Da | Anti-windup design with guaranteed regions of stability: an LMI-based approach[END_REF].

Here, generalizing the approach of [START_REF] Zaccarian | Nonlinear scheduled anti-windup design for linear systems[END_REF] and [START_REF] Galeani | Regional, semiglobal, global nonlinear anti-windup via switching design[END_REF], in light of the recent formalism in [START_REF] Goebel | Hybrid Dynamical Systems: modeling, stability, and robustness[END_REF] for the representation of hybrid dynamical systems, we propose a hysteresis switching law that exploits the fact that by construction, for each q = 2, . . . , N, set E q (see the outer bold gray level set in Figure 1 for an example) is a subset of the β-restriction (see the dashed level set in Figure 1 for an example) of set E q-1 (see the inner bold gray level set in Figure 1 for an example). The intuitive idea behind the hysteresis switching law is to switch to less aggressive control gains (that is, from the gains associated to E q+1 to the gains associated to E q ) when the state x crosses the boundary of E q+1 , but to allow a switch to more aggressive gains (that is, from the gains associated to E q to the gains associated to E q+1 ) only when the state x crosses the boundary of βE q+1 ; the finite separation between the boundaries of E q+1 and βE q+1 provides some level of robustness to noise and prevents chattering problems.

To this aim, we introduce a logic state variable q ∈ {1, . . . , N} in our control algorithm, and implement a q-dependent version of the linear stabilizer (2), as follows:

u = K q x + L q sat m (u), (22a)
where state q evolves according to the following hybrid dynamics:

q = 0, x ∈ C q q + = q + 1, x ∈ D + q q + = q -1, x ∈ D - q , (22b)
6

E q+1 E q βE q+1 D + q D - q C q Figure 2:
The hysteresis switching mechanism behind control law [START_REF] Henrion | Piecewise-linear robust control of systems with input constraints[END_REF], [START_REF] Herrmann | Some new results on anti-windup-conditioning using the Weston-Postlethwaite approach[END_REF].

where, according to the intuitive representation in Figure 2, increase and decrease of q is enabled, respectively, whenever the state (x, q) belongs to the following sets:

D - q := {x ∈ R n : x T Q -1 q x ≥ 1}, q ∈ {2, . . . , N} D + q := {x ∈ R n : x T Q -1 q+1 x ≤ β}, q ∈ {1, . . . , N -1}, D - 1 := ∅, D + N := ∅, (23a
) and the set C q in (22b) is selected as the closure of the set where jumps (or updates) of q are not allowed:

C q = R n \ D + q ∪ D - q . (23b)
As customary with the representation in [START_REF] Goebel | Hybrid Dynamical Systems: modeling, stability, and robustness[END_REF] for hybrid systems, the so-called flow set of the hybrid dynamics corresponds to the subset of the state space Ξ = R n ×{1, . . . , N} where flow of the solutions is allowed, and the so-called jump set of the hybrid dynamics corresponds to the subset of the state space Ξ where jump of the solutions is allowed. Taking a similar approach to [17, eq. (1.10)], they correspond to:

C = q∈{1,...,N} C q × {q} D = q∈{1,...,N} (D - q ∪ D + q ) × {q}, (23c)
and the flow and jump maps are directly derived from (22b) due to the fact that the sets D - q , D + q , q = 1, . . . , N are all disjoint. The next result is the first main theorem of this paper establishing useful properties of the hysteresis switching stabilizer. An appealing feature of this theorem, as compared to previous approaches, such as those in [START_REF] Zaccarian | Nonlinear scheduled anti-windup design for linear systems[END_REF][START_REF] Galeani | Regional, semiglobal, global nonlinear anti-windup via switching design[END_REF] is that the well posedness of the hybrid formulation in [START_REF] Henrion | Piecewise-linear robust control of systems with input constraints[END_REF], [START_REF] Herrmann | Some new results on anti-windup-conditioning using the Weston-Postlethwaite approach[END_REF] allows obtaining intrinsic robustness of the stated stability results, by way of the general results in [START_REF] Goebel | Hybrid Dynamical Systems: modeling, stability, and robustness[END_REF]Thm 7.21].

Theorem 1. Consider plant (1), a set of initial parameters (Q 1 , K 1 , L 1 , λ 1 ) coming from Proposition 1 and a family of nested sets

(Q i , K i , L i , λ i ), i = 1, . . . , N coming from Algo- rithm 1.
Then the hybrid hysteresis switching state-feedback control law [START_REF] Henrion | Piecewise-linear robust control of systems with input constraints[END_REF], [START_REF] Herrmann | Some new results on anti-windup-conditioning using the Weston-Postlethwaite approach[END_REF] for plant [START_REF] Angeli | Command governors for constrained nonlinear systems[END_REF] is such that 1. complete solutions exist for all initial conditions, and are all eventually continuous;

2. if parameters (Q 1 , K 1 , L 1 , λ 1
) came from a solution to (3) of Proposition 1 with Y = 0, then the origin is globally exponentially stable; 3. the exponential convergence rate increases as the solution approaches the origin. In particular, there exists a scalar M ≥ 1, such that, for all solutions, the following holds:

x(0, 0) ∈ βE k ⇒ |x(t, j)| ≤ M exp(-λ k t)|x(0, 0)|, (24)
for all (t, j) ∈ dom x; 4. the origin is exponentially stable for the closed loop with region of attraction including the set E 1 .

Proof. First note that dynamics (1), ( 22), ( 23) satisfies the hybrid basic conditions in [17, Assumption 6.5] because C and D are closed, the flow map is continuous and the jump map can be continuously extended. Then existence and completeness of maximal solutions are guaranteed by [17, Proposition 6.10] and the well posedness of (1), (2) established in Proposition 1 (guaranteeing that item (b) of Proposition 6.10 in [START_REF] Goebel | Hybrid Dynamical Systems: modeling, stability, and robustness[END_REF] never occurs), together with the fact that item (c) of Proposition 6.10 never occurs because C ∪ D = R n × {1, ..., N}. The fact that all solutions are eventually continuous is proved below.

Consider now any solution ξ = (x, q) to ( 1),( 22), [START_REF] Herrmann | Some new results on anti-windup-conditioning using the Weston-Postlethwaite approach[END_REF]. Let ξ(0, 0) = (x 0 , q 0 ) and denote by R 1 the region of attraction of the closed-loop system with (K, L) = (K 1 , L 1 ) which, by construction, contains the ellipsoid E 1 (see Proposition 1). Two cases can be considered:

a) x 0 ∈ C q 0 b) x 0 C q 0
In case a), the initial state belongs to the set E q 0 which is forward invariant for the closed-loop dynamics with gains (K, L) = (K q 0 , L q 0 ) (again from Proposition 1). As a consequence, introducing the coefficients

c i := λ max (Q i ) λ min (Q i ) ≥ 1 ∀i = 1, ..., N,
the following exponential estimate is in force

|x(t, 0)| ≤ c q 0 exp(-λ q 0 t)|x(0, 0)| ( 25
)
for any t ≥ 0 such that (t, 0) ∈ dom ξ (because this means that the solution never jumps thereby remaining in C q 0 ). If the supremum of such t is ∞, then bound (25) proves the claim. Otherwise, the solution reaches the boundary ∂(βE q 0 +1 ) at some finite time (t 1 , 0) ∈ dom ξ such that (t 1 , 1) ∈ dom ξ and q(t 1 , 1) = q 0 + 1, x(t 1 , 1) ∈ ∂(βE q 0 +1 ) ⊂ E q 0 +1 because q jumps from q(t 1 , 0) = q 0 to q(t 1 , 1) = q 0 + 1. Since E q 0 +1 is forward invariant for continuous dynamics with gains (K, L) = (K q 0 +1 , L q 0 +1 ), one has |x(t, 1)| ≤ c q 0 +1 exp(-λ q 0 +1 (tt 1 ))|x(t 1 , 0)| for any t ≥ t 1 satisfying (t, 1) ∈ dom ξ, and thus, incorporating (25) and using [START_REF] Gomes Da | Anti-windup design with guaranteed regions of stability: an LMI-based approach[END_REF],

|x(t, j)| ≤ c q 0 +1 c q 0 exp(-λ q 0 t)|x(0, 0)| ( 26
)
for any t ≥ 0 such that x(t, j) ∈ dom ξ with j ∈ {0, 1}.

The above argument can be repeated ≤ N -1 times, thus proving items (2) and (3) of the theorem. Moreover, note that all selections of K q and L q exponentially stabilize the origin. Then all solutions evolving in E q , q < N, eventually reach the neighborhood of the origin βE q and jump, eventually reaching βE N-1 ⊂ int E N . From there the evolution is continuous, thus proving item (1). Summarizing, it has been shown that

|x(t, j)| ≤ M exp(-λ q 0 t)|x(0, 0)| ∀t ≥ 0, j = 0, 1, ..., , (27)
where the constant M is given by

M = Π N i=1 c i . (28)
Let us consider now case b). If x(0, 0) ∈ E q for some q ∈ {1, ..., N}, the jump map (22b) guarantees that after, ≤ |q 0 -q| ≤ N -1 jumps, the hybrid state ξ(0, ) is such that

x(0, ) ∈ C q, q(0, ) = q.

Therefore one turns back to the scenario of case a), with x(0, 0) replaced by x(0, ). This shows that ( 24) is fulfilled with M given by [START_REF] Kothare | A unified framework for the study of anti-windup designs[END_REF]. We point out that clearly it is possible to find some sharper estimates on the bound M.

The only case that has not been addressed yet is x(0, 0) E 1 , which is considered in item (4) for the case where (K 1 , L 1 ) are globally exponentially stabilizing. In this case, it is trivial to see that the solution performs a finite number of jumps until q becomes 1. From there, Proposition 1 gives an exponential bound until x ∈ ∂(βE 1 ), and then the analysis above applies. ♦ Remark 3. (Global, semi-global and regional results) Theorem 1 (as well as Theorem 2 given in the next section) guarantees different stability properties on the closed loop, depending on how the family of solutions to (3) were determined. In particular, according to the results of Proposition 1 (applied with λ = 0), for exponentially stable plants it will be possible to choose the 1-st solution in such a way that the corresponding gains are globally exponentially stabilizing (namely with Y = 0), so that global exponential stability will hold for the hysteresis switching feedback scheme. Alternatively, for marginally stable/unstable plants it is possible to achieve semiglobal exponential stability by choosing the largest invariant set (corresponding to the 1-st solution) arbitrarily large. Finally, for exponentially unstable plants, it will only be possible to achieve regional exponential stability, possibly maximizing the size of the largest invariant set E 1 : indeed, it is well known (see, e.g., [START_REF] Sontag | An algebraic approach to bounded controllability of linear systems[END_REF]) that exponentially unstable plants with bounded control inputs have null controllability regions that are bounded in the exponentially unstable directions, so that they cannot be globally nor semiglobally stabilized. •

Scheduled stabilization

As opposed to the switching solution proposed in the previous section, we now use a different strategy for exploiting the nested family of stabilizers determined in Section 2.2, which does not require any switching (thus avoiding discontinuities in the control input). This solution is based on interpolating the control law as suggested in [START_REF] Galeani | Nonlinear L 2 antiwindup for enlarged stability regions and regional performance[END_REF][START_REF] Galeani | Nonlinear scheduled control for linear systems subject to saturation with application to anti-windup control[END_REF], which is resemblant of the techniques in [START_REF] Kose | Scheduled controllers for linear systems with bounded actuators[END_REF][START_REF] Lin | Low-and-high gain design technique for linear systems subject to input saturation-a direct method[END_REF]. To this aim, the integer logic variable q of the previous section is here converted into a scheduling parameter belonging to the real interval [1, N]. In particular, the matrices X 1,i , X 2,i and Q i , i = 1, . . . , N obtained from Algorithm 1 can be used to define a Lipschitz nonlinear control law as follows. Let q := max{1, floor(q)} q := min{N, ceil(q)}, [START_REF] Lin | Low-and-high gain design technique for linear systems subject to input saturation-a direct method[END_REF] (so that for q ∈ [1, N], q denotes the smallest integer larger than q, whereas q denotes the largest integer smaller than q), and define interpolated variables through convex combination:

Q(q) = ( q -q)Q q + (q -q )Q q (30a)
U(q) = ( qq)U q + (qq )U q (30b)

X 1 (q) = ( qq)X 1 q + (qq )X 1 q (30c)

X 2 (q) = ( q -q)X 2 q + (q -q )X 2 q . ( 30d
)
Then we may use the real parameters q ∈ [1, N] (whose selection is specified below), to define an implicit, nonlinear statefeedback control law

u = K s (q(x))x + L s (q(x)) sat m (u) ( 31
)
where subscript "s" stands for "scheduled" and we define, similar to (4),

K s (q) = (I -X 2 (q)U(q) -1 )X 1 (q)Q(q) -1 L s (q) = -(I -X 2 (q)U(q) -1 )X 2 (q)U(q) -1 . ( 32
)
Finally, the function x → q(x) is selected as follows:

q(x) =              1 if x E 1 N if x ∈ E N q * (x) if x ∈ E 1 \ E N , (33a)
where x → q * (x) is the unique solution to the following implicit equation, whose existence and properties are guaranteed by the next lemma (whose proof is given in the appendix):

x T Q -1 (q * (x))x = 1, x ∈ E 1 \ E N . ( 33b
)
Lemma 3. Equation (33b) implicitly defines a unique, bounded and Lipschitz continuous function q * (x) in E 1 \ E N .

We are now ready to state our second main result, establishing desirable properties of the Lipschitz nonlinear scheduled stabilizer.

Theorem 2. Consider plant (1), a set of initial parameters (Q 1 , X 11 , X 21 , U 1 , λ 1 ) coming from Proposition 1 and a family of nested sets

(Q i , X 1i , X 2i , U i , λ i ), i = 2, . . . , N coming from Al- gorithm 1.
Then the Lipschitz nonlinear state-feedback control law ( 30)-( 33) is such that:

1. the origin is exponentially stable for the closed loop with region of attraction including the set

E 1 ; 2. if parameters (Q 1 , X 1 , X 2 , U 1 , λ 1 ) came from a solution to
(3) of Proposition 1 with Y = 0, then the origin is globally exponentially stable; 3. the exponential convergence rate increases as the solution approaches the origin. In particular, there exists a positive scalar M such that, for all solutions, the following holds:

x(0) ∈ E k ⇒ |x(t)| ≤ M exp(-λ k t)|x(0)|, (34)
for all t ≥ 0.

Proof. Consider the Lipschitz Lyapunov function candidate V(x, q(x)) := Ṽ(x, q(x)) + Nq(x),

where Ṽ(x, q(x)) = x T Q -1 (q(x))x and q(x) is assigned by [START_REF] Sajjadi-Kia | Controllers for linear systems with bounded actuators: Slab scheduling and anti-windup[END_REF] and is Lipschitz thanks to Lemma 3. From [START_REF] Sajjadi-Kia | Controllers for linear systems with bounded actuators: Slab scheduling and anti-windup[END_REF], the scheduling parameter q(x) is constant in E c 1 := R n \ E 1 and in E N , where it is equal to 1 and N respectively. Hence one has V(x, q(x)) = V(x, q(x))

for almost all x ∈ E c

1 ∪ E N . Moreover, following the derivations leading to [START_REF] Galeani | Regional, semiglobal, global nonlinear anti-windup via switching design[END_REF], there exists γ > 0 satisfying for almost all

x ∈ E N V(x, q(x)) = V(x, q(x)) ≤ -2γ Ṽ(x, q(x)). ( 35
)
On the other hand, from (33b) the identity Ṽ(x, q(x)) = Ṽ(x, q * (x)) = 1 [START_REF] Tarbouriech | Anti-windup design: an overview of some recent advances and open problems[END_REF] holds whenever x ∈ E 1 \E N , and therefore, if the state x belongs to such set, the derivative of V(x, q(x)) reduces to V(x, q(x)) = q(x) =q * (x), where we also used the definition in [START_REF] Sajjadi-Kia | Controllers for linear systems with bounded actuators: Slab scheduling and anti-windup[END_REF]. Differentiating both sides of (36), we obtain ∇ x Ṽ(x, q * (x)) = 0, which can be developed to get

q * (x) = -x T ∂Q -1 (q) ∂q q=q * (x) x -1 2x T Q -1 (q * (x)) ẋ.
As a consequence we obtain:

V(x, q(x)) = -q * (x) = 2x T Q -1 (q * (x))(Ax + B sat m (u)) x T M(q * (x))x , (37)
where for almost all q ∈ [1, N] (namely for all non-integer values of q in that interval) 2

M(q) := Q -1 (q)(Q q -Q q )Q -1 (q) > 0. ( 38
)
Due to convexity, LMIs (3) are still satisfied by the interpolated variables [START_REF] Megretski | L 2 BIBO output feedback stabilization with saturated control[END_REF], hence, selecting the control input as the scheduled feedback [START_REF] Mulder | Multivariable antiwindup controller synthesis using linear matrix inequalities[END_REF] and adapting the proof of item (2) in Proposition 1 to bound the numerator in [START_REF] Teel | On assigning the derivative of a disturbance attenuation control lyapunov function[END_REF], we obtain for some scalar γ > 0

V(x, q(x)) ≤ -2 γ M Ṽ(x, q(x)) (39)
for almost all x ∈ E 1 \ E N , where

M := sup x ∈ E 1 \ E N q ∈ [1, N] x T M(q)x.
The existence of a uniformly negative upper bound for V(x, q(x)) for almost all x ∈ E 1 implies exponential stability of the origin with region of attraction containing E 1 , by following the derivations in [37, page 99], thus proving item [START_REF] Angeli | Command governors for constrained nonlinear systems[END_REF]. Item (2) follows very similar steps. Indeed, from item (1) of Proposition 1, we have that bound (35) also holds for almost all x ∈ E c 1 , and therefore the negative upper bound for V(x, q(x)) holds almost globally. The argument in [START_REF] Teel | On assigning the derivative of a disturbance attenuation control lyapunov function[END_REF] then implies global exponential stability of the origin. Consider now item (3). To prove the increasing behavior of convergence rate one can make the preliminary observation that, given the sequence of solutions (Q i , X 1i , X 2i , U i , λ i ) of ( 16) for i = 1, ..., N coming from Algorithm 1, inequality (16e) with Q = Q i is still satisfied by any λ k with i ≥ k ≥ 1. By convexity, the same feature is transferred to the interpolated variables [START_REF] Megretski | L 2 BIBO output feedback stabilization with saturated control[END_REF], i.e.

He

(

A + λ k I)Q(q) + BX 1 (q) B(X 2 (q) -U(q)) X 1 (q) + Y X 2 (q) -U(q) < 0 (40)
for any q ≥ k ≥ 1. On the other hand, the function q * (x) defining the scheduled parameter has been proven to be nondecreasing along the solution (see conditions [START_REF] Teel | On assigning the derivative of a disturbance attenuation control lyapunov function[END_REF] and ( 39)). This property together with inequality (40) guarantees that, if the initial state x(0) belongs to the ellipsoid E k for some k ∈ {1, ..., N}, then the convergence rate of the solution x(t) is not smaller than λ k by condition [START_REF] Gomes Da | Anti-windup design with guaranteed regions of stability: an LMI-based approach[END_REF], thus also proving item (3). ♦

2 One clearly sees that M(q) = dQ -1 (q) dq : this identity follows from [START_REF] Megretski | L 2 BIBO output feedback stabilization with saturated control[END_REF] and keeping in mind that for a matrix function q → P(q), we have that P(q)P -1 (q) = 1, implies ∂P(q) ∂q = -P(q) ∂P -1 (q) ∂q P(q). See the proof of Lemma 3 for further details about identity [START_REF] Teel | On assigning the derivative of a disturbance attenuation control lyapunov function[END_REF] and positive definiteness of M(q). Remark 4. (Computation of the scheduling parameter q * ) Since the algebraic solution of equation (33b) (namely, the function x → q * (x)) does not appear to be easy to find, 3 a dynamic selection for q can be employed, which integrates the plant dynamics to asymptotically estimate the correct value of q * . While this can be done e.g. by bisection methods, in [START_REF] Galeani | Nonlinear L 2 antiwindup for enlarged stability regions and regional performance[END_REF][START_REF] Galeani | Nonlinear scheduled control for linear systems subject to saturation with application to anti-windup control[END_REF] a dynamical approach was suggested, which is based on suitably integrating the dynamics of q * (x) implicitly defined by condition (33b). In particular, condition (33b) can be first derived with respect to time, to obtain

d dt x T Q -1 (q)x = 2x T Q -1 (q) ẋ + x T dQ -1 (q) dq x q = 0, ( * )
from which it is possible to compute q and write a dynamic equation governing a new state q. In particular, taking M(q) as in [START_REF] Teel | Global stabilization and restricted tracking for multiple integrators with bounded controls[END_REF] and defining g(x, q) := (x T Q -1 (q)x -1), a robust implementation of dynamics q with feedback injection g(x, q), where scalar > 0 is arbitrary and may be tuned for optimized performance, is given by

q =                      max{0, -g(x, q)}, if q ≤ 1 2x T Q -1 (q)(Ax + B sat m (u)) x T M(q)x -g(x, q), if q ∈ (1, N) min{0, -g(x, q)}, if q ≥ N, (41)
with initial state q(0) ∈ [1, N]. It can be shown that, under the stated assumptions, the right hand side of ( 41) is bounded and well defined almost everywhere, and that the interval [1, N] is forward invariant for q (that is, the state q never leaves such interval). In fact, a more precise description of the q dynamics would require the use of differential inclusions because Q(•) in ( 30) is Lipschitz so that its generalized derivative should be used in place of matrix M. To keep the discussion simple, we abuse notation and treat the q dynamics as a differential equation. To better appreciate how the algorithm works in practice, few more comments are in order. First, the computations in (41) require the knowledge of the state x, which might appear limiting; however, in the antiwindup application such state x is exactly the state of the antiwindup compensator, and then it is explicitly available.

Next, the role of the terms appearing in the right hand side of (41) can be explained as follows. In all cases, the term -g(x, q) has the role of correcting a "wrong" initialization of the state q in (41), since the initial value of q(0) usually does not satisfy the condition x T (0)Q -1 (q(0))x(0) = 1 (the "correct" initialization would require the solution of (33b) to be already known); the term -g(x, q) is a stabilizing feedback which disappears as x T Q -1 (q)x gets closer and closer to 1. For the case q ∈ (1, N), 3 Several solutions have been proposed to determine α by on line search methods and other computationally intensive approaches; see e.g. [START_REF] Megretski | L 2 BIBO output feedback stabilization with saturated control[END_REF][START_REF] Kose | Scheduled controllers for linear systems with bounded actuators[END_REF] and references therein.

the other term appearing in the right hand side of ( 41) is exactly the dynamics that, according to equation ( * ) above, ensures forward invariance of the set {(x, q) : x T Q -1 (q)x = 1}, namely, it ensures that if x T Q -1 (q)x = 1 at some time t 0 , then the motion of q is suitably coordinated with the motion of x so to keep the relation x T Q -1 (q)x = 1 satisfied also for all t ≥ t 0 . Now, recalling that by [START_REF] Lin | Low-and-high gain design technique for linear systems subject to input saturation-a direct method[END_REF] the value of q is assumed to range in the interval [1, N], the max operator is used in the case q ≤ 1 to avoid that the value of q decreases below 1 once it has reached such a lower bound, and similarly the min operator is used in the case q ≥ N to avoid that the value of q increases above N (as mentioned, the discontinuity introduced by such terms can be formally dealt with by using differential inclusions). Note that, although the set [1, N] can be formally shown to be forward invariant under [START_REF] Teel | Uniting local and global controllers for the Caltech ducted fan[END_REF], allowing q < 1 or q > N in the right hand side of (41) ensures robustness to numerical errors. Finally, from the above discussion it is clear that the role of the parameter in g(x, q) is to make the feedback action exerted by g(x, q) in ( 41) more or less aggressive, thus influencing how fast the condition x T Q -1 (q)x = 1 is achieved. Hence, in practice the value of should be chosen large enough to ensure that q quickly becomes a good solution of x T Q -1 (q)x = 1; practical limits on how large the value of can be taken are clearly given by the possible numerical integration problems that might arise due to excessively large values. Note that, since only affects a fast initial transient, its effect on the performance of the antiwindup compensation are quite negligible.

•

Application to nonlinear model recovery anti-windup

In this section, similar to [START_REF] Galeani | Regional, semiglobal, global nonlinear anti-windup via switching design[END_REF][START_REF] Galeani | Nonlinear L 2 antiwindup for enlarged stability regions and regional performance[END_REF][START_REF] Galeani | Nonlinear scheduled control for linear systems subject to saturation with application to anti-windup control[END_REF], and following the paradigm in [START_REF] Teel | The L 2 anti-windup problem: Its definition and solution[END_REF] and [57, Part II], we apply the proposed constructions to model recovery anti-windup design, which is capable of transforming a stability and performance recovery problem into a stabilization problem (applied to the mismatch dynamics). To this end, we consider a linear plant given by:

P          ẋp = Ax p + Bu p + B w w y = C y x p + D y u p + D yw w z = C z x p + D z u p + D zw w (42)
where x p ∈ R n p is the plant state, u p ∈ R n u is the control input, w ∈ R n w is an exogenous input (possibly containing disturbances, references and measurement noise), y ∈ R n y is the measurement output and z ∈ R n z is the performance output.

As customary in the anti-windup setting, we assume that a (linear) controller 4 has been designed to induce desirable performance when interconnected to the plant without saturation:

C ẋc = A c x c + B cu u c + B cw w y c = C c x c + D cu u c + D cw w (43)
where x c ∈ R n c is the controller state and y c ∈ R n u is the controller output.

In the case without input saturation, we call unconstrained closed-loop system the direct feedback interconnection between the controller [START_REF] Teel | Uniting local and global controllers[END_REF] and the plant (42) via the equations

u = y c , u c = y. (44)
A necessary assumption for anti-windup design is that the unconstrained closed-loop system (42), ( 43), ( 44) is well posed and internally stable. The so-called saturated closed-loop system corresponds to the interconnection between [START_REF] Teel | The L 2 anti-windup problem: Its definition and solution[END_REF], [START_REF] Teel | Uniting local and global controllers[END_REF] and the level ū ∈ R n u >0 saturated interconnection

u = sat ū(y c ), u c = y, (45)
where each element of ū indicates the (symmetric) saturation level of the corresponding input channel.

The MRAW compensation scheme first proposed in [START_REF] Teel | The L 2 anti-windup problem: Its definition and solution[END_REF], later revisited in [START_REF] Zaccarian | A common framework for anti-windup, bumpless transfer and reliable designs[END_REF] (therein it was called "L 2 anti-windup") and finally well surveyed in [57, Part II], corresponds to inserting the following anti-windup filter, or mismatch dynamics, in the closed loop:

AW      ẋaw = Ax aw + B sat m (y c + v) -y c , y aw = C y x aw + D y (sat m (y c + v) -y c , (46)
with x aw (0) = 0 and where v is a signal to be designed. The saturation level m is defined as

m := (1 -ε)ū, (47)
where ε ∈ (0, 1) is a given arbitrarily small constant, so that sat m corresponds to a restricted saturation with respect to sat ū. The restriction of the saturation level by an (arbitrarily small) quantity ε is actually a necessary condition for stabilization and is therefore key to being able to prove Lemma 4 below. See [START_REF] Teel | The L 2 anti-windup problem: Its definition and solution[END_REF]Remark 2.2] for an example where this assumption is necessary. Note that this may appear somewhat close in spirit to the recent works on anticipatory anti-windup designs (see, e.g., [START_REF] Wu | On immediate, delayed and anticipatory activation of anti-windup mechanism: static anti-windup case[END_REF]), but its spirit is actually different because one typically selects ε as a small positive scalar. The interconnection of [START_REF] Turner | A non-square sector condition and its application in deferred-action anti-windup compensator design[END_REF] to the closed loop system (42), ( 43) via the anti-windup interconnection

u c = y -y aw , u = sat m (y c + v) (48)
yields the so called anti-windup closed-loop system, shown in Fig. 4. In [START_REF] Teel | The L 2 anti-windup problem: Its definition and solution[END_REF][START_REF] Zaccarian | A common framework for anti-windup, bumpless transfer and reliable designs[END_REF] the anti-windup design goals were formally stated as follows. Denote by u l , z l the linear controller output and the plant performance output of the unconstrained closed-loop system and by u a , z a the corresponding signals of the anti-windup closed-loop system. Then the signal v should be selected in such a way that the following two properties hold:

P AW + + C v 1 v 2 + + - + u c y c u y w w
1. if u l never exceeds the saturation bounds η, then z a coincides with z l (shortly, u l ≡ sat η (u l ) =⇒ z a ≡ z l ); 2. if dz m (u l ) (i.e. u lsat m (u l ), the portion of the control input in the linear closed loop exceeding the restricted saturation bounds5 m) is an L 2 signal then the difference between z a and z l is an L 2 signal (shortly, dz m (u l ) ∈ L 2 =⇒ (z az l ) ∈ L 2 ).

In the past decade, several papers have been published describing different selections for the signal v inducing various levels of stability and performance on the anti-windup closedloop system. In particular, the first technique proposed in [START_REF] Teel | The L 2 anti-windup problem: Its definition and solution[END_REF] corresponded to selecting v = k(x aw ) as a static nonlinear function of the anti-windup compensator state. In the special case where the plant was not unstable, this selection was linear and corresponded to v = K x aw with K selected based on suitable passivity properties to ensure stability (note that this selections had interesting connections with the scheme proposed in [START_REF] Weston | Linear conditioning schemes for systems containing saturating actuators[END_REF]). Later on, in [START_REF] Zaccarian | A common framework for anti-windup, bumpless transfer and reliable designs[END_REF], which dealt with exponentially stable plants, for performance improvement this selection was chosen as v = K x aw + L(sat m (y c + v)y c ), thus introducing a nonlinear algebraic loop in the scheme, and determining L so that this loop was well posed (this improved selection has interesting connections with the work in [START_REF] Herrmann | Some new results on anti-windup-conditioning using the Weston-Postlethwaite approach[END_REF]). Nonlinear selections of the signal v within the MRAW framework were also proposed later. In particular, in [START_REF] Bemporad | Anti-windup synthesis via sampled-data piecewise affine optimal control[END_REF], v was a sampled data signal chosen as a piecewise affine function of the state x aw by relying on explicit solutions to the discrete-time Receding Horizon Control (RHC) strategy. In [START_REF] Teel | Anti-windup for exponentially unstable linear systems[END_REF] and [START_REF] Galeani | Constructive nonlinear antiwindup design for exponentially unstable linear plants[END_REF], a nonlinear function also involving measurements of some plant states was used to obtain large operating regions in the presence of exponentially unstable modes in the plant dynamics. Finally, in [START_REF] Zaccarian | Nonlinear scheduled anti-windup design for linear systems[END_REF], a hysteresis switching approach was proposed to enforce improved transients when the plant is exponentially stable. This follow-up approaches are those of [START_REF] Galeani | Regional, semiglobal, global nonlinear anti-windup via switching design[END_REF][START_REF] Galeani | Nonlinear scheduled control for linear systems subject to saturation with application to anti-windup control[END_REF][START_REF] Galeani | Nonlinear L 2 antiwindup for enlarged stability regions and regional performance[END_REF], partly reorganized and presented in this paper.

The following lemma shows that the stability properties of the anti-windup closed-loop are equivalent to the stability properties of a simpler, input saturated system. Lemma 4. Given m as in (47), if the system

ẋaw = Ax aw + B sat m (v) (49)
with the (nonlinear and implicit) selection

v = f (x aw , sat m (v)), (50)
is well posed, has a globally Lipschitz right hand side and is globally (respectively, regionally) asymptotically stable, then the anti-windup closed-loop system ( 46), ( 42), ( 43), ( 48) is well-posed and globally (respectively, regionally) asymptotically stable.

Proof.

The proof is similar to the proofs in [START_REF] Teel | The L 2 anti-windup problem: Its definition and solution[END_REF] and [START_REF] Zaccarian | A common framework for anti-windup, bumpless transfer and reliable designs[END_REF], and is based on the fact that rewriting the anti-windup closed-loop system dynamics ( 46), ( 42), ( 43), [START_REF] Valmorbida | Nonlinear static state feedback for saturated linear plants via a polynomial approach[END_REF] in the coordinates (xx aw , x c , x aw ), a cascade structure is revealed where a first subsystem (with state (xx aw , x c )) reproducing the linear unconstrained closed-loop system drives the anti-windup compensator dynamics [START_REF] Turner | A non-square sector condition and its application in deferred-action anti-windup compensator design[END_REF], [START_REF] Gf Wredenhagen | Piecewise-linear lq control for systems with input constraints[END_REF], via the controller output y c . This second subsystem (which may be nonlinear due to the nonlinear nature of v 1 in ( 50)) can be written as in [START_REF] Weston | Linear conditioning schemes for systems containing saturating actuators[END_REF] with a disturbance signal whose norm can be bounded by a constant times dz η (y c ) (see [START_REF] Teel | The L 2 anti-windup problem: Its definition and solution[END_REF] for details). Global (respectively, regional) stability of the cascaded system then follows from the forward completeness of each subsystem which, in turns, derives from the global Lipschitz property of the right hand side. ♦ Corollary 1. Selecting the auxiliary input v as the solution of one of the algebraic loops

v = K q x aw + L q sat m (v), v = K s (q(x aw ))x aw + L s (q(x aw ))sat m (v)
with gains assigned either by the hysteresis switching scheme as in ( 22)- [START_REF] Herrmann | Some new results on anti-windup-conditioning using the Weston-Postlethwaite approach[END_REF] or by the scheduled scheme as in ( 31)-( 32)-( 33), the anti-windup closed-loop system (46), ( 42), ( 43), ( 48) is guaranteed to be well-posed and regionally asymptotically stable. Moreover, if the open-loop plant is exponentially stable, then the anti-windup closed-loop system is globally asymptotically stable.

Remark 5. The performance improvement guaranteed by the proposed design of v in Corollary 1 corresponds to establishing suitable increasingly fast convergence rates of x aw to zero, that can be established because the results of Theorems 1 and 2 carry over to the anti-windup dynamics. Following the derivations in [START_REF] Teel | The L 2 anti-windup problem: Its definition and solution[END_REF] and the later work [START_REF] Zaccarian | A common framework for anti-windup, bumpless transfer and reliable designs[END_REF], it is possible indeed to relate the anti-windup state x aw to the mismatch xx l between the actual plant response and the "unconstrained" response that would have been experienced in the absence of saturation (the information about this unconstrained response is actually stored in the anti-windup dynamics ( 46)). While it is beyond the scope of this paper to precisely characterize such a performance improvement, it is worth mentioning that proceeding in similar ways to [START_REF] Teel | The L 2 anti-windup problem: Its definition and solution[END_REF][START_REF] Zaccarian | A common framework for anti-windup, bumpless transfer and reliable designs[END_REF][START_REF] Zaccarian | Nonlinear scheduled anti-windup design for linear systems[END_REF] (see also [START_REF] Zaccarian | Modern anti-windup synthesis: control augmentation for actuator saturation[END_REF]Part 3]) it is possible to relate the estimate of the domain of attraction provided by Theorems 1 and 2 to the energy spent by the unconstrained input u l outside the restricted saturation bounds (i.e., the L 2 norm of u l -sat(u l )). This characterization is beyond the scope of this paper, which instead is mainly focused on the design of the stabilizer v and its qualitative properties. •

Examples

Mass-spring-damper system

We consider the same exponentially stable example used in [START_REF] Galeani | Regional, semiglobal, global nonlinear anti-windup via switching design[END_REF], i.e. a mass-spring system whose equations of motion are

ẋ = 1 0 -k/m -f /m x + 0 1/m sat 1 (u) y = z = [1 0]x
where x = (ξ, ξ) are the position and speed of the body of mass m = 1, and the elastic and damping coefficients are k = 1 and f = 0.005. The a priori given unconstrained controller in [START_REF] Zaccarian | A common framework for anti-windup, bumpless transfer and reliable designs[END_REF] is

y c = C f b (s)(C f f (s)r -u c ) with C f f (s) = 5 2s+5 , C f b = 500 (s+15) 2
s(s+80) . We follow the strategy suggested in Remark 2, we fix λ 1 = 1.2670 and then select the linear feedback radius η = 0.01. Setting the shrinking factor β = 0.6, a family of N = 11 nested ellipsoids has been generated using Algorithm 1, with associated exponential convergence rates λ i ranging from λ 1 = 1.2670 to λ N = 13.6663. The construction of ellipsoids has been performed in Matlab using the solver SeDuMi, and the total elapsed time of computation was 17 seconds. Switched and scheduled anti-windup compensators have been implemented as illustrated in Corollary 1. For the scheduled solution, the value = 60 has been used. As a comparison, an L 2 anti-windup compensator has also been considered using the gains K 1 , L 1 given by the algorithm. Figure 5 reports the arising responses. The switched and the scheduled constructions significantly outperform the linear anti-windup scheme. Moreover, from the middle plot it can be appreciated that the scheduled strategy reduces the unpleasant discontinuities at the plant input which characterize the switching technique. The lower plot compares the switching signal to the scheduling signal.

TAFA system

The example of an exponentially unstable system is also considered to further validate the proposed constructions. The linearized model of the short period longitudinal dynamics of TAFA (Tailless Advanced Fighter Aircraft) [START_REF] Barbu | Nonlinear anti-windup for manual flight control[END_REF] Setting β = 0.3, η = 0.01 and running Algorithm 1, a family of N = 5 nested ellipsoids is found. The corresponding exponential convergence rates are reported in Table 1. The construction of ellipsoids has been performed in Matlab using the solver Se-DuMi, and the total elapsed time of computation was 8 seconds. The results obtained with switched, scheduled and linear anti-windup compensators are reported in Figure 6. For the scheduled solution, the value = 40 has been used. It can be seen from the upper plot that both the switched and the scheduled scheme are characterized by improved performances with respect to the standard L 2 anti-windup, while the response of the saturated plant without compensation is divergent due to exponential instability. Middle and lower plots illustrate the behavior of control inputs and switching/scheduling signals, respectively.

Conclusions

We addressed model recovery anti-windup design with extreme performance induced by switching or scheduling antiwindup gains. The variations of the anti-windup gains was induced by a synthesis of a finite number of nested ellipsoidal sets, each of them associated to a pair of gains, and characterizing a suitable trade-off between exponential convergence rate and size of the ellipsoid. The switched solution is based on hybrid formalism involving a logic variable indicating what set of gain is active at each time, while the scheduled solution is based on a Lipschitz continuous scheduling among the finite set of available gains. Both approaches have been shown to lead to desirable closed-loop properties when combined with the antiwindup architecture, and the comparative effectiveness of the control solution has been shown on a few simulation studies.

Figure 1 :

 1 Figure 1: Two nested level sets arising from Algorithm 1.

Figure 3 :

 3 Figure 3: The nested sets and their interpolation using the parameter α.

Figure 4 :

 4 Figure 4: The anti-windup closed loop system.

8 .Figure 5 :

 85 Figure 5: Results for the mass-spring-damper system

Figure 6 :

 6 Figure 6: Results for the TAFA system

Table 1 :

 1 Exponential rates from Algorithm 1

	i	1	2	3	4	5
	λ i 0.3447 3.3173 4.6630 6.7759 10.1065

Since λ ≥ 0, this implies that A has no eigenvalues with positive real part.

We assume here that controller (43) be linear but our results extend straightforwardly to the case of nonlinear controllers.

As shown in[START_REF] Teel | The L 2 anti-windup problem: Its definition and solution[END_REF], it is necessary to consider such a restriction to correctly characterize the marginally stable/unstable case.

This work is supported by the Research Council of Norway through the Centres of Excellence funding scheme, project No. 223254 -AMOS.

Appendix A. Proof of Lemma 1

The proof is adapted from [START_REF] Zaccarian | A common framework for anti-windup, bumpless transfer and reliable designs[END_REF]Prop. 1]. Let us proceed by contradiction and suppose that ∆ ∈ O n u exists such that I -S ∆ is singular, e.g.

(I -S ∆)z = 0 (A.1)

for some z 0. Define z := (I -S )∆z; we claim that, since z 0 and (I -S ) is invertible, one has necessarily z 0. Moreover, the following identity holds

Observing that both W and ∆ c are diagonal, and that the entries of

However, as long as z 0, this contradicts the assumption 2W + WS (I -S ) -1 + (I -S T ) -1 S T W > 0.

Appendix B. Proof of Lemma 3

Set H(x, q) := x T Q -1 (q)x and observe that, by definition (30a), the mapping H(x, q) : R n ×R → R is smooth everywhere except for integer values of q. In particular, the derivative ∂H(x,q) ∂q exists almost everywhere and verifies

Then, by the nonsmooth implicit function theorem [5, Section 7.1], the equation H(x, q) = 1 defines a unique bounded and globally Lipschitz function q * (x) in E 1 \ E N , whose gradient exists almost everywhere and satisfies the identity

Finally one can observe that, for any integer q ∈ {1, ..., N} and x ∈ ∂E q , the identity q * (x) = q must hold thanks to uniqueness of the inverse function.