President Michel Cosnard

Laurent Nicod

This report is the sole result of the unit's evaluation by the expert committee, the composition of which is specified below. The assessments contained herein are the expression of an independent and collegial reviewing by the committee.

UNIT PRESENTATION

MANAGEMENT TEAM

The Inserm UMR 1152 is under the direction of Marina Pretolani.

HCERES NOMENCLATURE

GLOBAL ASSESSMENT OF THE UNIT

The Inserm UMR 1152 successfully investigates mechanisms as well as risk factors underlying acute and chronic lung diseases including severe asthma, chronic obstructive pulmonary disease (COPD)/emphysema, lung fibrotic diseases as well as different aspects and complications of lung transplantation. The unit is a founding member of the "Laboratory of Excellence (Labex) INFLAMEX" and of the "Départment Hospitalo-Universitaire" (DHU) FIRE (directed by a team leader of the unit).

The scientific international production is remarkable in the field of respiratory diseases, based on a real translational research enhanced by a useful geographic proximity between clinicians and experimental researchers. The interactions with the non-academic world are strong on the basis of significant support from industrial companies and 2 patents. Training is very good, with the benefit of financial help of Labex "INFLAMEX". Assessment of the unit's life and organization is outstanding with efficiency acknowledged by members of the unit and representatives of supervising bodies. The scientific strategy is remarkable, focused on studies of cellular and molecular mechanisms of respiratory diseases.

The evaluation reports of Hceres are available online : www.hceres.com

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INTRODUCTION

  HISTORY AND GEOGRAPHICAL LOCATION OF THE UNITThe Inserm UMR 1152,"Physiopathology and epidemiology of respiratory diseases" was created jointly by Inserm and Paris Diderot University on January 2005. The unit is located at the Faculty of Medicine, Xavier Bichat site, in the same campus than Bichat Hospital.

  Non-permanent professors and associate professors, including emeritusNon-permanent full time scientists, including emeritus, post-

HCERES scientific officer: Mr Jorge BOCZKOWSKI Representatives of supervising institutions and bodies: Mr

	Unit name:	Physiopathology and Epidemiology of Respiratory Diseases
	Unit acronym:	PHERE
	Requested label:	Inserm et Université Paris Diderot
	Application type:	Restructuration
	Current number:	UMR 1152
	Head of the unit (2017-2018):	Ms Marina PRETOLANI
	Project leader (2019-2023):	Ms Marina PRETOLANI
	Number of teams:	3
	COMMITTEE MEMBERS
	Chair:	Mr Laurent NICOD, CHUV University of Lausanne, Switzerland
	Experts:	Ms Marie-Laure FRANCO-MONTOYA, Inserm (supporting personnel)
		Mr François HUAUX, Université Catholique de Louvain, Belgium
		Mr Roger MARTHAN, Université de Bordeaux
		Mr David MONTANI, Université Paris-Sud (representative of CSS Inserm)
		Mr Jean-François MUIR, Université de Rouen (representative of CNU)

Christian BOITARD, Inserm Ms Laurence LOMME, Inserm Mr Matthieu RESCHE-RIGON, Paris Diderot

  SVE2 Biologie Cellulaire, Imagerie, Biologie Moléculaire, Biochimie, Génomique, Biologie Systémique, Développement, Biologie Structurale; SVE6 Santé Publique, Epidémiologie, Recherche Clinique; SVE5 Physiologie, Physiopathologie, Cardiologie, Pharmacologie, Endocrinologie, Cancer, Technologies Médicales; SVE3 Microbiologie, Immunité.

	SCIENTIFIC DOMAIN
	Respiratory pathophysiology.


Fabien Lauer
email: fabien.lauer@esstin.uhp-nancy.fr

Gérard Bloch
email: gerard.bloch@esstin.uhp-nancy.fr

Switched and PieceWise Nonlinear Hybrid System Identification

Hybrid system identification aims at both estimating the discrete state or mode for each data point, and the submodel governing the dynamics of the continuous state for each mode. The paper proposes a new method based on kernel regression and Support Vector Machines (SVM) to tackle this problem. The resulting algorithm is able to compute both the discrete state and the submodels in a single step, independently of the discrete state sequence that generated the data. In addition to previous works, nonlinear submodels are also considered, thus extending the class of systems on which the method can be applied from Piece-Wise Affine (PWA) and switched linear to PieceWise Smooth (PWS) and switched nonlinear systems with unknown nonlinearities. Piecewise systems with nonlinear boundaries between the modes are also considered with some preliminary results on this issue.

Introduction

Context. Hybrid systems are usually described by both a continuous state and a discrete state, where the vector field defining the evolution of the continuous state depends on the discrete state. In this framework, a system can be seen as switching between n different subsystems, which are usually modeled by AutoRegressive with eXogenuous inputs (ARX) models in the discrete-time case. Two types of identification problems may arise in this setting depending on whether the discrete state sequence that generated the data is known or not. If it is, then the problem can be simply recast as n common identification problems, each one using only the data for a given discrete state. However, in most cases this sequence is unknown and the problem becomes nontrivial.

Models of hybrid systems. The predicted output y t of a hybrid model in ARX form is given as a function of the continuous state x t = [u t-nc . . . u t-1 , y t-na . . . y t-1 ] T , containing the lagged inputs u t-j and outputs y t-j , and the discrete state λ t . Considering n submodels f j , the hybrid model is written as

y t = f λt (x t ) + e t , (1)
where e t is a noise term. Hybrid models can be classified with respect to the nature of the submodels f j and of the evolution of the discrete state λ t ∈ {1, . . . , n}. Table 1 defines the nomenclature that will be used in the paper. SARX and SNARX models assume that the system is arbitrarily switched. On the other hand, PWARX models consider a dependency between the discrete state and the continuous state. They can thus be defined by PieceWise Affine (PWA) maps of the type f (x) = f j (x), if x ∈ S j = {x : H j [x T 1] T ≤ 0}, j = 1, . . . , n, where the matrices H j represent a set of hyperplanes that define the polyhedral domains S j partitioning the continuous state space. Similarly, PWNARX models can be defined by PieceWise Smooth (PWS) maps, where f j are smooth nonlinear functions instead of affine functions. Extensions of the PWARX and PWNARX models to "nonlinearly piecewise" models, where the domains S j are no more constrained to be polyhedral, will be denoted NPWARX and NPW-NARX.

Related work. Five main approaches have been devised for hybrid system identification: the clustering-based approach [START_REF] Ferrari-Trecate | A clustering technique for the identification of piecewise affine systems[END_REF], the mixed integer programming based approach [START_REF] Roll | Identification of piecewise affine systems via mixed-integer programming[END_REF], the Bayesian approach [START_REF] Juloski | A Bayesian approach to identification of hybrid systems[END_REF], the bounded error approach [START_REF] Bemporad | A bounded-error approach to piecewise affine system identification[END_REF] and the algebraic approach [START_REF] Vidal | An algebraic geometric approach to the identification of a class of linear hybrid systems[END_REF][START_REF] Ma | Identification of deterministic switched ARX systems via identification of algebraic varieties[END_REF]. The first four focus on the problem of PieceWise Affine (PWA) system identification, where the discrete state depends on the continuous state. However, both the bounded error and Bayesian approaches can also be used to identify a broader class of systems, known as switched linear systems, where the discrete state evolves independently of the continuous state. The algebraic approach [START_REF] Vidal | An algebraic geometric approach to the identification of a class of linear hybrid systems[END_REF] focuses on this latter problem, but without taking the noise into account in its development. This leads to an algorithm very sensitive to noise, compared to the clustering-based or bounded error methods, as shown by [START_REF] Juloski | Comparison of four procedures for the identification of hybrid systems[END_REF]. Besides, the bounded error method [START_REF] Bemporad | A bounded-error approach to piecewise affine system identification[END_REF] provides a convenient way of dealing with noisy data by looking for a model with a predefined accuracy. However, the hyperparameters of the method, such as the model accuracy that determines the number of modes, may be difficult to tune to get a prescribed structure, e.g. if prior knowledge on the number of modes is available [START_REF] Juloski | Comparison of four procedures for the identification of hybrid systems[END_REF].

Tools and proposed method. The paper proposes a new method for hybrid system identification based on kernel regression and Support Vector Machines (SVMs).

Stemming from statistical learning theory, Support Vector Machines (SVMs) [START_REF] Vapnik | The nature of statistical learning theory[END_REF] quickly became a state-of-the-art tool for classification and are already com-monly used, either in their original form or through closely related methods [START_REF] Mangasarian | Generalized support vector machines[END_REF], in hybrid system identification to estimate the switching boundary between modes [START_REF] Ferrari-Trecate | A clustering technique for the identification of piecewise affine systems[END_REF][START_REF] Bemporad | A bounded-error approach to piecewise affine system identification[END_REF]. Based on the same theoretical concepts, Support Vector Regression (SVR) retains properties of SVMs, such as a good generalization ability from few samples, and offers an interesting alternative both for regression and system identification [START_REF] Drezet | Support vector machines for system identification[END_REF][START_REF] Mattera | Support vector machines for dynamic reconstruction of a chaotic system[END_REF][START_REF] Zhang | Nonlinear system identification based on an improved support vector regression estimator[END_REF]. SVR uses an ε-insensitive loss function which does not take into account errors that are less than ε [START_REF] Smola | A tutorial on support vector regression[END_REF]. This loss function ignoring errors below a predefined threshold is close in spirit to the bounded error approach. However, the origin is different. In learning theory, this effect is justified in order to minimize the generalization error of the model, whereas the bounded error approach was developed to allow the automatic determination of the number of linear submodels required to approximate a non-linear function with a given accuracy.

In the past decade, kernel methods have attracted much attention in a large variety of fields and applications: classification and pattern recognition, regression, density estimation, etc. Indeed, using kernel functions, many linear methods can be extended to the nonlinear case in an almost straightforward manner, while avoiding the curse of dimensionality by transposing the focus from the data dimension to the number of data.

The proposed method uses the SVR framework to estimate hybrid models with submodels in kernel expansion form. The resulting algorithm is able to compute both the discrete state and the submodels in a single step, independently of the discrete state sequence that generated the data. Nonlinear submodels with unknown types of nonlinearities can be easily treated, thus extending the class of systems on which the method can be applied from switched linear to switched nonlinear systems and from piecewise affine to piecewise smooth systems by considering models in SARX, SNARX, PWARX or PWNARX form. Nonlinearly PWA and PWS maps with nonlinear mode boundaries are also considered in the paper with some preliminary results using nonlinear SVM classifiers. The idea is that since the method estimates the discrete state without any assumption on the switching sequence, labeling data points generated from nonlinearly separable modes is possible.

Contribution. The paper proposes solutions for two problems that have not yet been extensively studied and solved in the literature: identification of hybrid systems switching between unknown nonlinear dynamics and identification of nonlinearly piecewise systems with nonlinear boundaries between the modes in the continuous state space.

Paper organization. The paper starts by some preliminaries on kernel functions and Support Vector Regression (Sect. 2.1) before using these to develop a hybrid system identification algorithm in Sect. 2.2. The problem of estimating nonlinear boundaries between modes is then discussed in Sect. 2.3 and Section 3 provides an interpretation of the method based on previous approaches from the literature. Finally, Section 4 gives some numerical examples of application.

Notations. All vectors are column vectors written in boldface and lowercase letters whereas matrices are boldface and uppercase. The vectors 0 and 1 are vectors of appropriate dimensions with all their components respectively equal to 0 and 1. For A ∈ IR d×m and B ∈ IR d×n containing d-dimensional sample vectors and the kernel function k : IR d × IR d → IR, the "kernel" K(A, B) maps IR d×m × IR d×n in IR m×n with K(A, B) i,j = k(A i , B j ), where A i and B j are the ith and jth columns of A and B. In particular, if x ∈ IR d is a column vector then K(x, B) is a row vector in IR 1×n . The matrix X ∈ IR N ×d contains all the training samples x i , i = 1, . . . , N , as rows. The vector y ∈ IR N gathers all the target values y i for these samples. The kernel matrix K(X T , X T ) will be written K for short.

Nonlinear Hybrid System Identification

This section presents a new method based on kernel regression and support vector machines (SVMs) for hybrid system identification. The basics of nonlinear function approximation by kernel methods are first recalled, before describing the proposed method itself. The section ends with a discussion on piecewise systems with nonlinear boundaries between modes.

Kernels and Support Vector Regression

A simple method to approximate a nonlinear function is to first map the data to a higher dimensional feature space and then perform linear regression in that space. This approach usually suffers from the so-called curse of dimensionality, which can however be avoided thanks to the "kernel trick" depicted below.

First, consider the nonlinear mapping Φ that maps the data x from the input space X ⊂ IR p to a vector Φ(x) in a feature space F . Assume now that the function f is given by an expansion based on the N training samples x i ∈ IR p in that feature space, i.e. f (x) = N i=1 α i Φ(x) T Φ(x i ) + b. Clearly, though being a nonlinear function in the input space, f is a linear function in F . Note that in order to compute f (x), it is not necessary to explicitly compute the images Φ(x i ) of the points but only the result of their inner product. This is the "kernel trick" which replaces the inner products between images of points by a kernel function k(x, x i ) = Φ(x) T Φ(x i ). In kernel regression, the training data (x i , y i ), i = 1, . . . , N , stacked in the matrix X and the vector y, are thus approximated by a kernel expansion

f (x) = N i=1 α i k(x, x i ) + b = K(x, X T )α + b , (2)
where α = [α 1 . . . α i . . . α N ] T and b are the parameters of the model and k(., .) is the kernel function. Typical kernel functions are the linear (k(x,

x i ) = x T x i ), Gaussian RBF (k(x, x i ) = exp(-x -x i 2
2 /2σ 2 ) and polynomial (k(x, x i ) = (x T x i + 1) d ) kernels. The kernel function defines the feature space F in which the data are implicitly mapped. The higher the dimension of F is, the higher the approximation capacity of the function f is, up to the universal approximation capacity obtained for an infinite feature space, as with Gaussian RBF kernels. It is also possible to build kernel functions from prior knowledge on the task at hand, see for instance [START_REF] Shawe-Taylor | Kernel Methods for Pattern Analysis[END_REF] for the properties of kernel functions and the construction of new kernels or [START_REF] Lauer | Incorporating prior knowledge in support vector machines for classification: a review[END_REF] for examples of application in pattern recognition. In the hybrid system identification framework, this can be useful for instance when the type of nonlinearity of a particular mode is known beforehand.

In kernel regression via linear programming (LP), the ℓ 1 -norm of the parameters α of the kernel expansion is minimized together with the ℓ 1 -norm of the errors

y i -f (x i ) by min (α,b) α 1 + C N i=1 |f (x i ) -y i | , (3)
where a hyperparameter C is introduced to tune the trade-off between the minimization of the model complexity (measured by α 1 ) and the error on the data (measured by

N i=1 |f (x i ) -y i |).
Minimizing the complexity of the model allows to control its generalization capacity. In practice, this amounts to penalize non-smooth functions and implements the general smoothness assumption that two samples close in input space tend to give the same output.

Instead of the ℓ 1 -norm of the errors, the ε-insensitive loss function, defined by [START_REF] Vapnik | The nature of statistical learning theory[END_REF] as

l(e) = |e| ε = 0 if |e| ≤ ε , |e| -ε otherwise , (4)
can also be used to yield Linear Programming Support Vector Regression (LP-SVR). This loss function builds a tube of insensitivity in which the errors are meaningless. Errors larger than the tube width1 ε are penalized linearly.

A possible formulation of the LP-SVR problem involves 4N + 1 design variables [START_REF] Smola | Linear programs for automatic accuracy control in regression[END_REF]. In the remaining of the paper, we will follow the approach of [START_REF] Mangasarian | Large scale kernel regression via linear programming[END_REF] that involves only 3N + 1 variables. Introducing two sets of optimization variables, in two positive slack vectors a and ξ, this problem can be implemented as a linear program solvable by standard optimization softwares such as the MATLAB linprog function. In this scheme, the LP-SVR problem may be written as min (α,b,ξ≥0,a≥0)

1 T a + C1 T ξ s.t. -ξ -ε1 ≤ Kα + b1 -y ≤ ε1 + ξ -a ≤ α ≤ a . (5)
The last set of constraints ensures that 1 T a, which is minimized, bounds α 1 .

In practice, sparsity is obtained as a certain number of parameters α i will tend to zero. The input vectors x i for which the corresponding α i are non-zero are called support vectors.

Hybrid System Identification with Kernels

The bounded error approach, developed by [START_REF] Bemporad | A bounded-error approach to piecewise affine system identification[END_REF] for the identification PWARX models, aims at finding a model with a predefined accuracy, i.e. that allows the error on all the training points (x i , y i ) to be bounded by

|y i -f λi (x i )| = |e i | ≤ δ, i = 1, . . . , N . (6)
The following presents a new method based on kernel regression to achieve this goal. As a direct benefit, nonlinear submodels f j are easily handled by the choice of the kernel functions, thus providing a method for the estimation of both piecewise and switched nonlinear ARX models. Following the SVR approach, submodels in kernel expansion form

f j (x) = N i=1 α ij k j (x, x i ) + b j = K j (x, X T )α j -b j , (7)
are trained by minimizing the ℓ 1 -norm of the parameters α j . As indicated by the subscript j, various kernel functions k j can be associated to the different models f j . This leads to vectors K j (x, X T ) and kernel matrices K j = K j (X T , X T ), as defined in the notations at end of the introduction. It is thus possible to take prior information into account such as the number of modes governed by linear dynamics or knowledge on the type of a particular nonlinearity. In this setting, the problem of training n models under the bounded error constraint may be written as

min αj ,bj ,aj ≥0 n j=1 1 T a j (8)
-δ1 ≤ y -K j α j -b j 1 ≤ δ1, ∀x i ∈ S j , j = 1, . . . , n, -a j ≤ α j ≤ a j , j = 1, . . . , n .

where y = [y 1 y 2 . . . y N ] T and the absolute error |e ij | = |f j (x i ) -y i | is constrained to be less than δ only for the model j corresponding to the discrete state λ i of the point x i . However, without further information on the classification of the data into modes, S j are unknown and the problem is intractable.

To circumvent this issue, consider the equivalent problem using the ε-insensitive loss function (4) for ε = δ implemented with slack variables ξ ij , i = 1, . . . , N , j = 1, . . . , n, stacked in the n vectors ξ j ∈ IR N as in [START_REF] Vidal | An algebraic geometric approach to the identification of a class of linear hybrid systems[END_REF]:

min αj ,bj ,aj ≥0,ξ j ≥0 n j=1 1 T a j (9)
-ξ j -δ1 ≤ y -K j α j -b j 1 ≤ δ1 + ξ j , j = 1, . . . , n, -a j ≤ α j ≤ a j , j = 1, . . . , n, n j=1 ξ ij = 0, i = 1, . . . , N .
The last equalities stand for the fact that all points must be estimated with accuracy δ by at least one submodel f j . In other words, for a given sample (x i , y i ), there is at least one j for which ξ ij = 0. As nonlinear equalities are not easy to deal with from an optimization point of view, these are approximated by min αj ,bj ,aj ≥0,ξ j ≥0 n j=1

1 T a j + C N i=1 n j=1 ξ ij (10)
-ξ j -δ1 ≤ y -K j α j -b j 1 ≤ δ1 + ξ j , j = 1, . . . , n, -a j ≤ α j ≤ a j , j = 1, . . . , n .
Solving this problem with a sufficiently large constant C leads to functions f j solutions of the former problem [START_REF] Vapnik | The nature of statistical learning theory[END_REF]. Moreover, the discrete state λ i , in which the system was for each data point x i is readily available from the variables ξ ij vanishing to zero as λi = j, for ξ ij = 0. The cases where the bounded error constraint is not satisfied, i.e. no ξ ij is zero, can be further discriminated by letting λi = arg min j (ξ ij ). On the other hand, for cases where more than one ξ ij is zero, the absolute error is considered and λi = arg min j |e ij |, with e ij = y i -f j (x i ).

In the case of PWARX or PWNARX models where the modes are linearly separable in the continuous state space, undetermined points can be reclassified after the training of separating hyperplanes (the boundaries between the sets S j ) based on the determined cases only. The linear classification issue is not discussed here due to size constraints and the reader is referred to [START_REF] Vapnik | The nature of statistical learning theory[END_REF] and [START_REF] Mangasarian | Generalized support vector machines[END_REF] for an introduction to state-of-the-art methods, whereas multi-class pattern recognition is considered for instance by [START_REF] Crammer | On the algorithmic implementation of multiclass kernelbased vector machines[END_REF]. In the next Section, extensions of the PWARX and PWNARX models to nonlinearly piecewise models, where the domains S j are no more constrained to be polyhedral, will be discussed.

An advantage of the proposed approach is the possibility to deal easily with a noise level that also switches with the model. In order to do so, multiple loss functions with different parameters δ j for each mode can be used and implemented in the following final problem min αj ,bj ,aj ≥0,ξ j ≥0 n j=1

1 T a j + C N i=1 n j=1 ξ ij (11)
-ξ j -δ j 1 ≤ y -K j α j -b j 1 ≤ δ j 1 + ξ j , j = 1, . . . , n, -a j ≤ α j ≤ a j , j = 1, . . . , n .

Another possible formulation with n × N less variables and constraints involves the minimization of the squares of the parameters α ij as min αj ,bj ,ξ j ≥0

n j=1 α T j α j + C N i=1 n j=1 ξ ij (12)
-ξ j -δ j 1 ≤ y -K j α j -b j 1 ≤ δ j 1 + ξ j , j = 1, . . . , n .

The solution of this problem is not sparse as the one of ( 11) but can usually be computed in less time.

Remark 1. In the case of a linear kernel k j (x, x i ) = x T x i , the parameters of the linear model f j (x) = w T j x + b j can be explicitly recovered by w j = X T α j . Remark 2. The hyperparameters of the method are the kernel types, the number n of modes, the bounds δ j , the regularization parameter C and the number of lagged inputs and outputs (dynamic order). They can be tuned on a subset of the data put aside for validation. When too few data are available, cross-validation techniques can be used. Moreover, the algorithms ( 11) and ( 12) can be extended to automatically tune the bounds δ j to the noise level by using a trick similar to the one introduced in ν-SVR [START_REF] Smola | Linear programs for automatic accuracy control in regression[END_REF][START_REF] Mangasarian | Large scale kernel regression via linear programming[END_REF]. This is studied in [START_REF] Lauer | A new hybrid system identification algorithm with automatic tuning[END_REF] for linear submodels and directly applicable to the problems above. Besides, the proposed method is well adapted when some basic prior knowledge on the system is available such as the number n of modes. However, due to the universal approximation capability of kernel models, the tuning of n is less crucial than when using linear or affine submodels. For piecewise maps, a good fit can be obtained with an underestimated n.

Remark 3. Problems ( 11) and ( 12) are linearly constrained nonlinear programs. They involve the minimization of a criterion composed of a linear [START_REF] Mattera | Support vector machines for dynamic reconstruction of a chaotic system[END_REF] or quadratic [START_REF] Zhang | Nonlinear system identification based on an improved support vector regression estimator[END_REF] term and a product of nonnegative variables subject to linear constraints. These problems are not convex and have multiple minima. This can be seen from their symmetric structure, leading to multiple solutions for simple permutations of models. All these solutions are acceptable and yield the same objective function value corresponding to a global optimum. However, care must be taken when using different kernels for different models, in which case permuting models is no more without effect and may lead to local minima.

A possible initialization of the optimization can be obtained by solving the feasibility problems corresponding to the constraints of ( 11) and [START_REF] Zhang | Nonlinear system identification based on an improved support vector regression estimator[END_REF], which are simple linear programs.

Nonlinear Boundaries Between Modes

A direct extension of the PWARX and PWNARX models, in which the discrete state is determined by a set of separating hyperplanes in the continuous state space, is obtained by introducing nonlinear boundaries or arbitrary regions (also pointed out in the conclusion of [START_REF] Bemporad | A bounded-error approach to piecewise affine system identification[END_REF]). In these "nonlinearly piecewise" models (denoted by NPWARX and NPWNARX, see Table 1 in the introduction), the discrete state is still a function of the continuous state, but the separating surfaces are no more restricted to hyperplanes. This can lead to a decrease of the number of submodels if the true system corresponds to this description. Indeed, in this case, the linear separability assumption may require to build multiple identical submodels for different regions of the continuous state space that are however governed by the same dynamics. Moreover, regrouping the data available in several regions of the continuous state space into one submodel may help to get better estimates for regions with few samples.

Nonlinear classification methods have to be used in this case and are readily available in number (KPCA, KFD, SVM...) [START_REF] Shawe-Taylor | Kernel Methods for Pattern Analysis[END_REF] thanks to the kernel trick used above. In particular, SVMs are similarly very easily extended to nonlinear classification by an appropriate choice of kernel function. Moreover, the final classifier is given as a sparse kernel expansion allowing for relatively fast estimation of the mode for a new sample. For the binary case (only 2 modes), the nonlinear separating surface S is given by

h(x) = N i=1 β i k c (x, x i ) + b c = 0 , (13)
where k c (., .) is a kernel function and the β i , b c are the trainable parameters of the classifier. Simply taking the sign of the function h yields the class of a pattern x, i.e. +1 if h(x) ≥ 0 and -1 otherwise. The method proposed in Sect. 2 can deal with nonlinearly piecewise maps (themselves either affine or nonlinear) without any modification and provide the labeling of the data, required to train a classifier, through λi . Indeed, any method (including the bounded-error, Bayesian or algebraic approaches) that estimates the discrete state without dependency on the continuous state, and thus without any assumption regarding the linear separability of the data in the continuous state space, can deal with nonlinearly piecewise maps. In practice, the procedure is as follows.

1. Train a hybrid model on the input-output data (x i , y i ), i = 1, . . . , N , by solving [START_REF] Mattera | Support vector machines for dynamic reconstruction of a chaotic system[END_REF] or ( 12), 2. Estimate the discrete states, e.g. by λi = arg min j |e ij |, i = 1, . . . , N , 3. Train a classifier on the labeled data (x i , λi ), i = 1, . . . , N .

Additionally, in a refinement step, the training points could be re-assigned to the different modes by the classifier and the submodels f j retrained one by one on the relevant data only.

An illustrative example of this procedure is given in Sect. 4.2, where the method is applied to estimate a Nonlinearly PieceWise Affine (NPWA) map.

Interpretation and Links with other Approaches

The proposed method can be interpreted as a bridge between the bounded error approach [START_REF] Bemporad | A bounded-error approach to piecewise affine system identification[END_REF], that can deal easily with noise, and the algebraic procedure [START_REF] Vidal | An algebraic geometric approach to the identification of a class of linear hybrid systems[END_REF], that can deal with arbitrarily switched systems, while providing nonlinear extensions to these. More precisely, it amounts to a bounded error relaxation of the hybrid decoupling constraint used in the algebraic procedure as follows.

The hybrid decoupling constraint of the algebraic procedure can be expressed as a function of the submodel errors, e ij = y i -f j (x i ), by

n j=1 e ij = 0, i = 1, . . . , N. (14)
These constraints account for the fact that there must be at least one of the submodels f j that can estimate the ith point with zero error. In the case of noisy data, these constraints cannot be satisfied for all the N points. Considering the bounded error approach, "decoupling" can be however enforced. The bounded error constraints (6) act similarly to [START_REF] Shawe-Taylor | Kernel Methods for Pattern Analysis[END_REF], though being less restrictive on the estimation error (with a threshold δ). Combining these two approaches results in constraints of the form

n j=1 [|e ij | ≥ δ] = 0, i = 1, . . . , N , (15)
where

Numerical Examples

The following presents three examples of application. The first one shows the simultaneous estimation of two functions, one linear and one nonlinear, from datasets overlapping in the input space, while the second one is concerned with the estimation of a nonlinearly piecewise affine (NPWA) map. The last example shows the identification of a SNARX model of a hybrid system arbitrarily switching between linear and nonlinear dynamics. For examples 1 and 3, the discrete state is arbitrarily switched and the type of nonlinearity is unknown. In all the examples, the problems are formulated as the optimization program [START_REF] Mattera | Support vector machines for dynamic reconstruction of a chaotic system[END_REF] and solved by the MATLAB function fmincon.

Switching Function with Unknown Nonlinearity

In this one-dimensional example, the data are generated by two models: a linear submodel y 1 (x) = ax + b + e = 2x -1 + e and a polynomial submodel y 2 (x) = 0.5x 2 + e, where e is a zero-mean Gaussian noise of standard deviation 0.5. The discrete state λ, determining which submodel is active, is independent of the variable x. Two data points, (x, y 1 (x)) and (x, y 2 (x)), are generated for 30 values of x in the interval ]-5, 1[. Beside these 60 data, the only prior knowledge is that one submodel is linear and the other is nonlinear. The aim of this example is to show that the proposed method can discriminate between the two submodels and correctly approximate each one without further knowledge on the type of nonlinearity. Figure 1

Nonlinearly Piecewise Affine Map Estimation

In this illustrative example, the problem is to estimate a Nonlinearly PieceWise Affine (NPWA) map defined as y = x + 0.5 + e, for x ∈ ] -∞, -1]

[1, ∞[, and y = -0.5x -1 + e, for x ∈ ] -1, 1[, where e is a zero-mean Gaussian noise of standard deviation 0.1. This problem could be solved by considering a PWA map with 3 modes linearly separable in the x variable, or, as proposed in this example, by considering only 2 modes with a nonlinear boundary between mode 1 and mode 2. Thus, two models with linear kernels are trained on N = 60 data points for δ 1 = δ 2 = 0.1 and C = 100. The resulting models, shown on Fig. 2, are y = 1.01x + 0.53 and y = -0.48x -1.02. All the points are associated to the correct model except for one point, (x i , y i ) = (-0.9, -0.31), close to the mode boundary. The training of a SVM classifier, with a polynomial kernel k c = (xx i + 1) 3 , on the data labeled by λi , i = 1, . . . , N, yields a nonlinear boundary S between the modes given by 2 support vectors x 1 = -3 and x 60 = 2.9. As the data x i are in IR, this nonlinear separating surface is a set of points defined as S = {x : h(x) = -0.24(-3x + 1) 3 -0.22(2.9x + 1) 3 + 10.3 = 0}, as shown on the right hand side of Fig. 2. This classifier yields no classification error with respect to the target labels λi .

Simulated Hybrid System Identification

Consider the hybrid system switching between mode 1: y t = -0.905y t-1 + 0.9u t-1 + e t , and mode 2: y t = -0.4y 2 t-1 + 0.5u t-1 + e t , where e t is a zeromean Gaussian noise of standard deviation σ b = 0.1. An output trajectory of N = 100 points of this system is generated with a random initial condition

XII -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 4 -3 -2 -1 0 1 2 0 2 4 6 8 10
Fig. 2. Nonlinear boundary between 2 modes. Left: Hybrid model (-) with its insensitivity tube (--) approximating the data points represented by '×' for λi = 1 and "•' for λi = 2. The estimated mode λi also appears on the x-axis as '×' for mode 1 and '•' for mode 2 to highlight the partition of the input space X . Right: Class labels λi (.) for each data point xi used to learn the nonlinear boundary S (•), defined as the zeros of h(x) (-).

y 0 , a random input sequence u t uniformly distributed in the interval [0, 1] and a mode switch from mode 1 to mode 2 at time t = 41. The signal-to-noise ratio of this trajectory is 12 dB corresponding to a variance of the noise free trajectory of 0.20 and a noise variance of 0.012. These data are then used to train a SNARX model with C = 1000, δ 1 = δ 2 = 0.1, a linear kernel k 1 and a RBF kernel k 2 with σ = 1. Thus, the only prior knowledge is that one submodel is linear and the other is nonlinear. The trajectory of the resulting model ŷt = f λt (ŷ t-1 , u t-1 ) is shown on Figure 3. The estimated parameters of the linear mode 1 are -0.929 and 0.960, to be compared to -0.905 and 0.9. The discrete state is estimated by λt = arg min j (ξ tj ). As shown at the bottom of Fig. 3, 22 classification errors occur on the whole trajectory. The effect of these errors is limited and their origin can be explained. Most of them occur on ambiguous points for which f 1 (y t-1 , u t-1 ) = f 2 (y t-1 , u t-1 ) ± (δ 1 + δ 2 ). Here, a switched system is identified, but note that in case of a piecewise system, these ambiguities could be removed by classifying the points with respect to a separating boundary in the continuous state space. The overall simulation error is RMSEsim = 1/N N t=1 (y t -ŷt ) 2 = 0.154, which is slightly more than the noise standard deviation σ b = 0.1. Only 8 support vectors with nonzero α ij are selected from the 100 training samples to build the kernel expansion f 2 .

Conclusion

In this paper, a new system identification method has been proposed to deal with nonlinear hybrid systems. In particular, this method is applicable to sys- tems switching between unknown nonlinear dynamics and nonlinearly piecewise systems with arbitrary nonlinear boundaries between the modes. It also bridges the gap between the bounded error approach and the algebraic procedure by making use of the ε-insensitive loss function proposed in the machine learning community for Support Vector Regression. Since no assumption on the discrete sequence that generates the data is required, arbitrarily switched systems can be treated as well as piecewise systems.

Future work will focus on the tuning of the hyperparameters and optimization issues as well as experiments with real life applications. Among other perspectives, the simultaneous estimation of the submodels and the boundaries between the modes for piecewise systems could be investigated.

  [•] = 1, if the bracketed expression is true, and 0 otherwise. Using these constraints, the absolute value of the error |e i | = min j |e ij | (assuming that λi = arg min j |e ij |) of the hybrid model is bounded by the threshold δ. Approximating [|e ij | ≥ δ] for all j by an ε-insensitive loss function and minimizing their product leads to the algorithm (11).

Fig. 1 .

 1 Fig. 1. One-dimensional example of the simultaneous estimation of two functions on noisy data. Points associated to the linear and RBF models are respectively represented by crosses (+) and diamonds (⋄).

Fig. 3 .

 3 Fig. 3. Simulated hybrid system identification. Top: Trajectory of the system (blue plain line) and of the model (red dash line) in simulation mode (only the initial condition and the input is given to the model). Bottom: Estimated discrete state λt.

Table 1 .

 1 Nomenclature of the hybrid models in ARX form

	ARX model	abbr.	models fj discrete state λt domains Sj
	PieceWise	PWARX	affine	function of x	polyhedral
	PieceWise Nonlinear	PWNARX nonlinear function of x	polyhedral
	Nonlinearly PieceWise	NPWARX affine	function of x	arbitrary
	Nonlinearly PieceWise Nonlinear NPWNARX nonlinear function of x	arbitrary
	Switched	SARX	affine	arbitrary
	Switched Nonlinear	SNARX	nonlinear arbitrary

Actually, ε does not stand for the tube width but for half of the tube section with respect to y.


This report is the sole result of the unit's evaluation by the expert committee, the composition of which is specified below. The assessments contained herein are the expression of an independent and collegial reviewing by the committee.

UNIT PRESENTATION

GLOBAL ASSESSMENT OF THE UNIT

The scientific output and the reputation of the unit in the field of respiratory diseases are outstanding, especially in the clinical setting, and the unit has the potential to reach the same level in the future in experimental research. The close and fruitful collaboration between basic scientists and clinicians is impressive and one of the major strengths of the unit. This is accompanied by very good Interactions with the nonacademic world, which could be further developed in regard to the scientific quality of the unit and its output. The high expertise in the unit, the impressive scientific output and the lively and supportive atmosphere make the unit very attractive not only for established scientists, but also for students who highly appreciate high quality teaching and mentoring by senior staff. The unit's projects are novel and original, and the expected results may potentially improve our understanding of ILD sarcoidosis. The addition of a new research axis focusing on airway engineering is very promising and fits into the scientific focus of the unit.

The evaluation reports of Hceres are available online : www.hceres.com

Evaluation of clusters of higher education and research institutions Evaluation of higher education and research institutions Evaluation of research Evaluation of doctoral schools Evaluation of programmes Evaluation abroad

INTRODUCTION

  HISTORY AND GEOGRAPHICAL LOCATION OF THE UNITThe unit EA 2363 belongs to the unit of formation and research "Santé, Médicine, Biologie humaine" at University Paris 13 and is situated in the University Campus of Bobigny in the Northern area of Paris. The unit was founded in 1992 and directed until 2013 by Mr Jean-Paul Richalet, focusing on cellular and functional responses to hypoxia. In 2014, Ms Carole Planes was appointed to the director of the unit. MANAGEMENT TEAM Ms Carole Planès is director of the research unit EA 2363 (Hypoxia & the lung).HCERES NOMENCLATURESVE5.SCIENTIFIC DOMAINThe scientific topic of the lab is monothematic, focusing on hypoxia and the lung. The unit consists of two research axes, pathophysiology of interstitial lung diseases (ILD) -hypoxia and other factors and modulators of physiological and pathological responses to hypoxia.Non-permanent professors and associate professors, including emeritusNon-permanent full time scientists, including emeritus, post-


President Michel Cosnard

Chairman Edward Feil

Pierre Mr

Gressens

Martine Hossaert

This report is the sole result of the unit's evaluation by the expert committee, the composition of which is specified below. The assessments contained herein are the expression of an independent and collegial reviewing by the committee.

UNIT PRESENTATION

MANAGEMENT TEAM

The director is Mr Erick Denamur, and the deputy director is Ms France Mentre.

HCERES NOMENCLATURE

SVE3 SCIENTIFIC DOMAIN

The focus of IAME is the epidemiology and evolution of bacterial and viral pathogens, with a strong focus on antimicrobial resistance.

UNIT WORKFORCE

GLOBAL ASSESSMENT OF THE UNIT

The greatest strength of the IAME unit is the outstanding quality of the scientific output. All five teams are headed by experts of high international repute, and there is no doubt this standard of excellence is reflected in all team members, including the young researchers. The unit therefore represents a highly valuable incubator in producing the next generation of world class scientists in the critically important field of infectious disease management, intersecting with basic evolutionary science.

The output of the unit has strong real-world impact, and there is clear evidence of close collaboration with industry and policy makers. The organisation of the teams and unit as a whole is also generally excellent, and there is a very positive atmosphere and team spirit. There is an effective management structure and regular meetings at all levels. There is clear evidence of training and support for PhD students. The organisation of this unit provides a unique opportunity for synergy between disparate disciplines. There are numerous examples of successful long-standing collaborations, and a synthetic approach is evident in the future strategies of the unit. Each team will clearly continue to be successful in developing their own fields of expertise, and continued cross-hybridisation of ideas and perspectives will help the unit to consolidate a coherent identity in the future.

There are few areas with potential room for improvement. Perhaps a greater emphasis should be made on PhD recruitment, particularly from fundamental sciences rather than clinical fields. Greater internationalisation and global visibility should also be a priority. There is a recognised need for more core or permanent staff in the unit particularly with respect to basic sciences, which would facilitate a more solid foundation for the overall growth of the unit in future years.

The evaluation reports of Hceres are available online : www.hceres.com

Evaluation of clusters of higher education and research institutions Evaluation of higher education and research institutions Evaluation of research Evaluation of doctoral schools Evaluation of programmes International evaluation and accreditation

INTRODUCTION

  HISTORY AND GEOGRAPHICAL LOCATION OF THE UNITThe IAME unit was formed in 2014 from a merger of two INSERM units. It is affiliated to two universities, Paris Diderot and Paris 13 and is located in the Bichat campus in the north of Paris.

HCERES scientific officer: Mr Théophile OHLMANN Representatives of supervising institutions and bodies:

	Unit name:	Infection, Antimicrobials, Modelling, Evolution
	Unit acronym:	IAME
	Requested label:
	Application type:	Renewal
	Current number:
	Head of the unit (2017-2018):	Mr Erick DENAMUR
	Project leader (2019-2023):	Mr Erick DENAMUR
	Number of teams or themes: 5
	COMMITTEE MEMBERS
	Chair:	Mr Edward FEIL, University of Bath, UK
	Experts:	Mr Ahmadou ALIOUM, Université de Bordeaux
		Ms Sandrine CHABAS, Université de Bordeaux-Inserm (supporting personnel)
		Mr Alain DOGLIO, CHU de Nice-Hôpital Pasteur (representative of INSERM
		CSS)
		Mr Bruno LACARELLE, CHU Marseille Timone
		Mr Pascal POIGNARD, CHU de Grenoble (representative of CNU)
		Mr Fabrice VAVRE, Université de Lyon (representative of CNRS)
		Ms Corinne ALBERTI, INSERM
		Mr François CREMIEUX, Hôpital Bichat
		Mr Jean-Luc DUMAS, Université Paris 13