The joint research team, TRIPOP, between INRIA Grenoble Rhône–Alpes, Grenoble INP and CNRS, part of the Laboratoire Jean Kuntzmann (LJK UMR 5224) is a follow up of the BIPOP team (2003–2017). The team is mainly concerned by the modeling, the mathematical analysis, the simulation and the control of nonsmooth dynamical systems. Nonsmooth dynamics concerns the study of the time evolution of systems that are not smooth in the mathematical sense, i.e., systems that are characterized by a lack of differentiability, either of the mappings in their formulations, or of their solutions with respect to time. In mechanics, the main instances of nonsmooth dynamical systems are multibody systems with Signorini unilateral contact, set-valued (Coulomb-like) friction and impacts. In Electronics, examples are found in switched electrical circuits with ideal components (diodes, switches, transistors). In Control, nonsmooth systems arise in the sliding mode control theory and in optimal control. A lot of examples can also be found in cyber-physical systems (hybrid systems), in transportation sciences, in mathematical biology or in finance.

Nonsmooth dynamics concerns the study of the time evolution of systems that are not smooth in the mathematical sense, i.e., systems that are characterized by a lack of differentiability, either of the mappings in their formulations, or of their solutions with respect to time.
The class of nonsmooth dynamical systems recovers a large variety of dynamical systems that arise in many applications. The term “nonsmooth”, as the term “nonlinear”, does not precisely define the scope of the systems we are interested in but, and most importantly, they are characterized by the mathematical and numerical properties that they share. To give more insight of what are nonsmooth dynamical systems, we give in the sequel a very brief introduction of their salient features. For more details, we refer to , , , , , .

As a first illustration, let us consider a linear finite-dimensional system described by its state

subjected to a set of

If the constraints are physical constraints, a standard modeling approach is to augment the dynamics in () by an input vector

which models the one-sided effect of the inequality constraints. The notation

where

leading to a general definition of LCS as

The complementarity condition, illustrated in Figure is the archetype of a nonsmooth graph that we extensively use in nonsmooth dynamics. The mapping

This function is convex, proper and can be sub-differentiated . The definition of the subdifferential of a convex function

A basic result of convex analysis reads as

that gives a first functional meaning to the set-valued mapping

It is easy to check that

Finally, the definition of the normal cone yields a variational inequality:

The relations () and () allow one to formulate the complementarity system with

The mathematical concept of solutions depends strongly on the nature of the matrix quadruplet

admits a unique solution

is a standard ODE with a Lipschitz right-hand side with a

that admits a solution that is absolutely continuous if

A lot of variants can be derived from the basic form of linear complementarity systems, by changing the form of the dynamics including nonlinear terms or by changing the complementarity relation by other multivalued maps. In particular the nonnegative orthant may be replaced by any convex closed cone

where

In Figure , we illustrate some other basic maps that can used for defining the relation between

Using again convex analysis, the multivalued sign function may be formulated as an inclusion into a normal cone as

More generally, any system of the type,

can reformulated in terms of the following set-valued system

The system () appears in a lot of applications; among them, we can cite the sliding mode control, electrical circuits with relay and Zener diodes , or mechanical systems with friction .

Though this class of systems seems to be rather specific, it includes as well more general dynamical systems such as piecewise smooth systems and discontinuous ordinary differential equations. Indeed, the system () for scalars

One of the most well-known mathematical framework to deal with such systems is the Filippov theory that embed the discontinuous differential equations into a differential inclusion. In the case of a single discontinuity surface given in our example by

Generally, the nonsmooth dynamical systems we propose to study mainly concern systems that possess the following features:

The nonsmooth dynamical systems we are dealing with, have a nonempty intersection with hybrid systems and cyber-physical systems, as it is briefly discussed in Sect. . Like in hybrid systems, nonsmooth dynamical systems define continuous–time dynamics that can be identified to modes separated by guards, defined by the constraints. However, the strong mathematical structure of nonsmooth dynamical systems allows us to state results on the following points:

These latter properties, that are common for smooth nonlinear dynamical systems, distinguish the nonsmooth dynamical systems from the very general definition of hybrid or cyber-physical systems , . Indeed, it is difficult to give a precise mathematical concept of solutions for hybrid systems since the general definition of hybrid automata is usually too loose.

To conclude this brief exposition of nonsmooth dynamical systems, let us recall an important fact related to numerical methods. Beyond their intrinsic mathematical interest, and the fact that they model real physical systems, using nonsmooth dynamical systems as a model is interesting, because it exists a large set of robust and efficient numerical techniques to simulate them. Without entering into deeper details, let us give two examples of these techniques:

In the Axis 2 of the research program (see Sect. ), we propose to perform new research on the geometric time-integration schemes of nonsmooth dynamical systems, to develop new integration schemes for Boundary Value Problem (BVP), and to work on specific methods for two time-discretized problems: the Mathematical Program with Equilibrium Constraints (MPEC) for optimal control and Second Order Cone Complementarity Problems (SOCCP) for discrete frictional contact systems.

In this section, we develop our scientific program. In the framework of nonsmooth dynamical systems, the activities of the project–team will be on focused on the following research axes:

These research axes will be developed with a strong emphasis on the software development and the industrial transfer.

This axis is dedicated to the modeling and the mathematical analysis of nonsmooth dynamical systems. It consists of four main directions. Two directions are in the continuation of BIPOP activities: 1) multibody vibro-impact systems (Sect. ) and 2) excitable systems (Sect. ). Two directions are completely new with respect to BIPOP: 3) Nonsmooth geomechanics and natural hazards assessment (Sect. ) and 4) Cyber-physical systems (hybrid systems) (Sect. ).

Participants: B. Brogliato, F. Bourrier, G. James, V. Acary

As a continuation of the work in the BIPOP team, our software code, Siconos (see Sect. ) will be our favorite software platform for the integration of these new modeling results.

Participants: A. Tonnelier, G. James

An excitable system elicits a strong response when the applied perturbation is greater than a threshold , , , . This property has been clearly identified in numerous natural and physical systems. In mechanical systems, non-monotonic friction law (of spinodal-type) leads to excitability. Similar behavior may be found in electrical systems such as active compounds of neuristor type. Models of excitable systems incorporate strong non-linearities that can be captured by non-smooth dynamical systems. Two properties are deeply associated with excitable systems: oscillations and propagation of nonlinear waves (autowaves in coupled excitable systems). We aim at understanding these two dynamical states in excitable systems through theoretical analysis and numerical simulations. Specifically we plan to study:

Participants: F. Bourrier, B. Brogliato, G. James, V. Acary

More generally, our collaboration with INRAe opens new long term perspectives on granular flows applications such as debris and mud flows, granular avalanches and the design of structural protections. The numerical methods that go with these new modeling approaches will be implemented in our software code, Siconos (see Sect.)

Participants: V. Acary, B. Brogliato, C. Prieur, A. Tonnelier

Nonsmooth systems have a non-empty intersection with hybrid systems and cyber–physical systems. However, nonsmooth systems enjoy strong mathematical properties (concept of solutions, existence and uniqueness) and efficient numerical tools. This is often the result of the fact that nonsmooth dynamical systems are models of physical systems, and then, take advantage of their intrinsic property (conservation or dissipation of energy, passivity, stability).
A standard example is a circuit with

Research actions:
There are two main actions in this research direction that will be implemented in the framework of the Inria Project Lab (IPL “ Modeliscale”, see for partners and details of the research program):

Structural analysis of multimode DAE : When a hybrid system is described by a Differential Algebraic Equation (DAE) with different differential indices in each continuous mode, the structural analysis has to be completely rethought. In particular, the re-initialization rule, when a switching occurs from a mode to another one, has to be consistently designed. We propose in this action to use our knowledge in complementarity and (distribution) differential inclusions to design consistent re-initialization rule for systems with nonuniform relative degree vector

Cyber–physical in hybrid systems modeling languages :
Nowadays, some hybrid modeling languages and tools are widely used to describe and to simulate hybrid systems (modelica, simulink, and see for references therein). Nevertheless, the compilers and the simulation engines behind these languages and tools suffer from several serious weaknesses (failure, weird output or huge sensitivity to simulation parameters), especially when some components, that are standard in nonsmooth dynamics, are introduced (piecewise smooth characteristic, unilateral constraints and complementarity condition, relay characteristic, saturation, dead zone, ...).
One of the main reasons is the fact that most of the compilers reduce the hybrid system to a set of smooth modes modeled by differential algebraic equations and some guards and reinitialization rules between these modes. Sliding mode and Zeno–behaviour are really harsh for hybrid systems and relatively simple for nonsmooth systems.
With B. Caillaud (Inria HYCOMES) and M. Pouzet (Inria PARKAS), we propose to improve this situation by implementing a module able to identify/describe nonsmooth elements and to efficiently handle them with siconos as the simulation engine. They have already carried out a first implementation in , a synchronous language for hybrid system. Removing the weaknesses related to the nonsmoothness of solutions should improve hybrid systems towards robustness and certification.

A general solver for piecewise smooth systems This direction is the continuation of the promising result on modeling and the simulation of piecewise smooth systems .
As for general hybrid automata, the notion or concept of solutions is not rigorously defined from the mathematical point of view. For piecewise smooth systems, multiplicity of solutions can happen and sliding solutions are common. The objective is to recast general piecewise smooth systems in the framework of differential inclusions with Aizerman–Pyatnitskii extension , . This operation provides a precise meaning to the concept of solutions. Starting from this point, the goal is to design and study an efficient numerical solver (time–integration scheme and optimization solver) based on an equivalent formulation as mixed complementarity systems of differential variational inequalities. We are currently discussing the issues in the mathematical analysis.
The goal is to prove the convergence of the time–stepping scheme to get an existence theorem. With this work, we should also be able to discuss the general Lyapunov stability of stationary points of piecewise smooth systems.

This axis is dedicated to the numerical methods and simulation for nonsmooth dynamical systems. As we mentioned in the introduction, the standard numerical methods have been largely improved in terms of accuracy and dissipation properties in the last decade. Nevertheless, the question of the geometric time–integration techniques remains largely open. It constitutes the objective of the first research direction in Sect. . Beside the standard IVP, the question of normal mode analysis for nonsmooth systems is also a research topic that emerged in the recent years. More generally, the goal of the second research direction (Sect. ) is to develop numerical methods to solve boundary value problems in the nonsmooth framework. This will serve as a basis for the computation of the stability and numerical continuation of invariants. Finally, once the time-integration method is chosen, it remains to solve the one-step nonsmooth problem, which is, most of time, a numerical optimization problem. In Sect. , we propose to study two specific problems with a lot of applications: the Mathematical Program with Equilibrium Constraints (MPEC) for optimal control, and Second Order Cone Complementarity Problems (SOCCP) for discrete frictional contact systems. After some possible prototypes in scripting languages (Python and Matlab), we will be attentive that all these developments of numerical methods will be integrated in Siconos.

Participants: V. Acary, B. Brogliato, G. James, F. Pérignon

The objective of this research item is to continue to improve classical time–stepping schemes for nonsmooth systems to ensure some qualitative properties in discrete-time. In particular, the following points will be developed

Participants: G. James, V. Acary, A. Tonnelier, F. Pérignon,

By invariants, we mean equilibria, periodic solutions, limit cycles or waves. Our preliminary work on this subject raised the following research perspectives:

Computation of periodic solutions of discrete mechanical systems . The modal analysis, i.e., a spectral decomposition of the problem into linear normal modes is one of the basic tools for mechanical engineers to study dynamic response and resonance phenomena of an elastic structure.
Since several years, the concept of nonlinear normal modes , that is closely related to the computation of quasi-periodic solutions that live in a nonlinear manifold, has emerged as the nonlinear extension of the modal analysis.
One of the fundamental question is: what remains valid if we add unilateral contact conditions ?
The computation of nonsmooth modes amounts to computing periodic solutions, performing the parametric continuation of solution branches and studying the stability of these branches.

This calls for time integration schemes for IVP an BVP that satisfy some geometric criteria: conservation of energy, reduced numerical dispersion, symplecticity as we described before. Though the question of conservation of energy for unilateral contact has been discussed in , the other questions remain open. For the shooting technique and the study of stability, we need to compute the Jacobian matrix of the flow with respect to initial conditions, the so-called saltation matrix , for nonsmooth flows. The eigenvalues of this matrix are the Floquet multipliers that give some information on the stability of the periodic solutions. The question of an efficient computation of this matrix is also an open question. For the continuation, the question is also largely open since the continuity of the solutions with respect to the parameters is not ensured.

Participants: V. Acary, M. Brémond, F. Pérignon, B. Brogliato, C. Prieur

Participants: B. Brogliato, C. Prieur, V. Acary, F. Miranda-Villatoro

This last axis is dedicated to the automatic control of nonsmooth dynamical systems, or the nonsmooth control of smooth systems. The first item concerns the discrete-time sliding mode control for which significant results on the implicit implementation have been obtained in the BIPOP team. The idea is to pursue this research towards state observers and differentiators (Sect ). The second direction concerns the optimal control which brings of nonsmoothness in their solution and their formulation. After the preliminary work in BIPOP on the quadratic optimal control of Linear Complementarity systems(LCS), we propose to go further to the minimal time problem, to impacting systems and optimal control with state constraints (Sect. ). In Sect , the objective is to study the control of nonsmooth systems that contain unilateral constraint, impact and friction. The targeted systems are cable–driven systems, multi-body systems with clearances and granular materials. In Sect , we will continue our work on the higher order Moreau sweeping process. Up to now, the work of BIPOP was restricted to finite-dimensional systems. In Sect , we propose to extend our approach to the control of elastic structures subjected to contact unilateral constraints.

It is noteworthy that most of the problems listed below, will make strong use of the numerical tools analyzed in Axis 2, and of the Modeling analysis of Axis 1. For instance all optimal control problems yield BVPs. Control of granular materials will undoubtedly use models and numerical simulation developed in Axis 1 and 2. And so on. It has to be stressed that the type of nonsmooth models we are working with, deserve specific numerical algorithms which cannot be found in commercial software packages. One of the goals is to continue to extend our software package Siconos, and in particular the siconos/control toolbox with these developments.

Nonsmooth dynamical systems arise in a lot of application fields. We briefly expose here some applications that have been treated in the BIPOP team and that we will continue in the TRIPOP team, as a validation for the research axes and also in terms of transfer. In mechanics, the main instances of nonsmooth dynamical systems are multibody systems with Signorini's unilateral contact, set-valued (Coulomb-like) friction and impacts, or in continuum mechanics, ideal plasticity, fracture or damage. Some illustrations are given in Figure (a-f). Other instances of nonsmooth dynamical systems can also be found in electrical circuits with ideal components (see Figure (g)) and in control theory, mainly with sliding mode control and variable structure systems (see Figure (h)). More generally, every time a piecewise, possibly set–valued, model of systems is invoked, we end up with a nonsmooth system. This is the case, for instance, for hybrid systems in nonlinear control or for piecewise linear modeling of gene regulatory networks in mathematical biology (see Figure (i)). Another common example of nonsmooth dynamics is also found when the vector field of a dynamical system is defined as a solution of an optimization problem under constraints, or a variational inequality. Examples of this kind are found in the optimal control theory, in dynamic Nash equilibrium or in the theory of dynamic flows over networks.

Granular chains made of aligned beads interacting by contact (e.g. Newton's cradle) are widely studied in the context of impact dynamics and acoustic metamaterials. In order to describe the response of such systems to impacts or vibrations, it is important to analyze different wave effects such as the propagation of compression waves (solitary waves or fronts) or localized oscillations (traveling breathers), or the scattering of vibrations through the chain. Such phenomena are strongly influenced by contact nonlinearities (Hertz force), spatial inhomogeneities and dissipation.

In the works , , we analyze nonlinear oscillations in chains of impact oscillators, such as localized oscillations (breathers). In , the oscillations are obtained by continuation from the limit of vanishing coupling. In , a multiscale expansion is used to approximate breather waves generated by impacts in granular chains with local potentials.

In the works , , , , we investigate the propagation of impacts in chains of beads with contact damping, using both analytical and numerical approaches. In , we show that the Kuwabara-Kono (KK) viscoelastic contact law is equivalent at leading order to a numerical dissipation for some classical first-order time-integration schemes. This result allows to simulate KK dissipation using numerical dissipation, without incorporating the nonsmooth terms of the KK model in dynamical equations. Conversely, it shows that studying the effects of KK dissipation on wave profiles allows to determine the effects of numerical dissipation. In , we show that granular chains with KK dissipation sustain traveling compression fronts that do not propagate steadily in nondissipative chains. The wave profiles are approximated using a special kind of multiscale expansion adapted to the strongly nonlinear contact forces, leading to a logarithmic KdV-Burgers equation (or a logarithmic Burgers equation for overdamped fronts).

In , , we derive an analytical approximation of the restitution coefficient for two-bead viscoelastic impacts under constant external loading. This allows in particular to approximate the critical damping above which beads do not detach after impact.

The overexploitation of natural resources questions the long-term sustainability of our society. Recently, a simple nature-society interrelations model, called the HANDY model, has been proposed by Montesharrei

(2014) to address this concern with a special emphasis on the role of the stratification of the society. We analyse the dynamics of this model and we explore the influence of two parameters: the nature depletion rate and the inequality factor. We characterize the asymptotic states of the system through a bifurcation analysis and we derive several quantitative results on the trajectories. We examine the possible collapses, sustainable equilibria and oscillations. We show that some collapses are irreversible and, depending on the wealth production factor, a bistability regime between a sustainable equilibrium and cycles of collapse-and-regeneration can be obtained. We discuss the possible policies to avoid dramatic scenarios. Part of this work was the subject of Camille Julliard's master's internship and results are presented in an article in preparation.

In

, we study differential algebraic equations with constraints defined in a piecewise manner using a conditional statement. Such models classically appear in systems where constraints can evolve in a very small time frame compared to the observed time scale. The use of conditional statements or hybrid automata are a powerful way to describe such systems and are, in general, well suited to simulation with event driven numerical schemes. However, such methods are often subject to chattering at mode switch in presence of sliding modes, or can result in Zeno behaviours. In contrast, the representation of such systems using differential inclusions and method from non-smooth dynamics are often closer to the physical theory but may be harder to interpret. Associated time-stepping numerical methods have been extensively used in mechanical modelling with success and then extended to other fields such as electronics and system biology. In a similar manner to the previous application of non-smooth methods to the simulation of piecewise linear ODEs, non-smooth event-capturing numerical scheme are applied to piecewise linear DAEs. In particular, the study of a 2-D dynamical system of index-2 with a switching constraint using set-valued operators, is presented. In

and in

, we study classes of differential-algebraic linear complementarity systems which possess some passivity properties. Specifically, the implicit Euler time-discretisation is analysed: the one-step nonsmooth problem's wellposedness is shown, then the convergence of the discrete solutions is studied. Examples are in circuits with nonsmooth components (like ideal diodes).

This scientific theme concerns the numerical analysis of mechanical systems with bilateral and unilateral constraints, with or without friction . They form a particular class of dynamical systems whose simulation requires the development of specific methods for analysis and dedicated simulators .

In

, we review several formulations of the discrete frictional contact problem that arises in space and time discretized mechanical systems with unilateral contact and three-dimensional Coulomb’s friction. Most of these formulations are well–known concepts in the optimization community, or more generally, in the mathematical programming community. To cite a few, the discrete frictional contact problem can be formulated as variational inequalities, generalized or semi–smooth equations, second–order cone complementarity problems, or as optimization problems such as quadratic programming problems over second-order cones. Thanks to these multiple formulations, various numerical methods emerge naturally for solving the problem. We review the main numerical techniques that are well-known in the literature and we also propose new applications of methods such as the fixed point and extra-gradient methods with self-adaptive step rules for variational inequalities or the proximal point algorithm for generalized equations. All these numerical techniques are compared over a large set of test examples using performance profiles. One of the main conclusion is that there is no universal solver. Nevertheless, we are able to give some hints to choose a solver with respect to the main characteristics of the set of tests.

Recently, new developments have been carried out on two new applications of well-known numerical methods in Optimization:

Rockfall propagation models are routinely used for the quantitative assessment of rockfall hazard. Their capacities and limitations remain difficult to assess due to the limited amount of exhaustive experimental data at the slope scale.

Complementary with the research held in , the predictive capabilities of block propagation models after a preliminary calibration phase is investigated. It is focused on models integrating the shape of blocks since, despite their sound physical bases, they remain less used than lumped-mass approaches due to their more recent popularisation. We first performed an expert-based calibration based on the use of the 2D model and, second, evaluated the predictive capabilities of the calibrated model in 2D and in 3D using the remaining part of the experimental results. The calibrated model simulations predict the main characteristics of the propagation : after a calibration phase on sufficient amount of soil types, the model may be used in a predictive manner. The adequacy between 2D and 3D simulations also favors applicability of the model since easier and faster calibrations based on 2D simulations only can be envisaged. As classically observed for block propagation models, the model is not sufficient to predict the details of the velocity and stopping points but provides accurate prediction of the global ranges of these quantities, in particular of the extreme values. To lift these limitations in terms of predictive capabilities, more advanced calibration procedures based on optimization techniques can constitute a promising perspective as it is studied in .

Standard finite element discretization for cable structures suffer from several drawbacks. The first one is related to the mechanical assumption that the cable can not support compression. Standard formulations do not take into account this assumption. The second drawback comes from the high stiffness of the cable model when we deal with large lengths with high Young modulus such as cable ropeways installations. In this context, standard finite element applications cannot avoid compressive solutions and have huge difficulties to converge. In a forthcoming paper, we propose to a formulation based on a piecewise linear modeling of the cable constitutive behavior where the elasticity in compression is canceled. Furthermore, a dimensional analysis help us to formulate a problem that is well-balanced and the conditioning of the problem is diminished. The finite element discretization of this problem yields a robust method where convergence is observed with the number of elements and the nonlinear solver based on nonsmooth Newton strategy is converging up to tight tolerances. The convergence with the number of element allows one to refine the mesh as much as we want that will be of utmost importance for applications with contact and friction. Indeed, a fine discretization with respect to the whole length of the cable will be possible in the contact zone. This work has been the object of the following publication . In , consistent reduced order modeling has been proposed and comparisons with the model developped in has been performed.

The implicit method for the time-discretization of set-valued sliding-mode controllers was introduced in , . The backstepping approach is used in to design a continuous-time and a discrete-time nested set-valued controller that is able to reject unmatched disturbances (a problem that is known to be tough in the sliding-mdoe control community). In , we continue the analysis of the implicit discretization of set-valued systems, this time oriented towards the consistency of time-discretizations for homogeneous systems, with one discontinuity at zero (sometimes called quasi-continuous, strangely enough). The discrete-time analysis of the twisting and the super-twisting algorithms are tackled in , . A tutorial/survey of the implicit Euler method applied to set-valued sliding-mode control can be found in . In , we propose an upgrade of a linear controller to a set-valued sliding-mode controller, and we study the discrete-time implementation. Experimental validations are made on a laboratory setup available at INRIA Lille (an inverted rotary pendulum). In , the sliding-mode control of a nonsmooth hydraulic actuator is analysed. The implicit Euler discretisation is carefully studied.

Lur'e systems are very popular in the Automatic Control field since their introduction by Lur'e in 1944. In we propose a very complete survey/tutorial on the set-valued version of such dynamical systems (in finite dimension) which mainly consist of the negative feedback interconnection of an ODE with a maximal monotone set-valued operator. The first studies can be traced back to Yakubpovich in 1963 who analysed the stability of a linear time invariant system with positive real constraints, in negative feedback connection with a hysteresis operator. About 600 references are analysed from the point of view of the mathematical formalisms (Moreau's sweeping process, evolution variational inequalities, projected dynamical systems, complementarity dynamical systems, maximal monotone differential inclusions, differential variational inequalities), the relationships between these formalisms, the numerous fields of application, the well-posedness issues (existence, uniqueness and continuous dependence of solutions), and the stability issues (generalized equations for fixed points, Lyapunov stability, invariance principles). In , we propose a pedagogical survey about the use of passivity (in Willems' sense) in the well-posedness analysis of set-valued Lur'e systems.

The quadratic and minimum time optimal control of LCS as in () is tackled in , . This work relies on the seminal results by Guo and ye (SIAM 2016), and aims at particularizing their results for LCS, so that they become numerically tractable and one can compute optimal controllers and optimal trajectories. The basic idea is to take advantage of the complementarity, to construct linear complementarity problems in the Pontryagin's necessary conditions which can then be integrated numerically, without having to guess a priori the switching instants (the optimal controller can eb discontinuous and the optimal trajectories can visit several modes of the complementarity conditions). In we study the optimal control of complementarity Lagrangian systems. The analysis is based on the transformation of the system with impacts, into a system without impacts, using an equivalent Filippov's differential inclusion with absolutely continuous solutions. The difficulty is in the correct design of the sliding surface so that the post-impact velocity is computed corectly.

It is well-known that linear complementarity systems can undergo bifurcations at which equilibria, limit cycle, or chaotic solutions can appear, dissappear or change stability. From a control viewpoint, it is important to know the range of parameters for which such changes take place. The work addresses this issue by proposing a novel notion of equivalence between LCPs (linear complementarity problems) that permits to make a classification of steady state bifurcations in dynamic LCS. The proposed approach takes advantage of the geometric structure of the problem and allows to closely mimic the bifurcation theory of smooth maps. This type of results allows us, for instance, to design LCSs with asymptotic behaviors showing multiple steady states, as is the case in negative resistance circuits. In addition, a full classification of bifurcations for the planar LCP is provided.

The third edition of the book has been released . Also a short proof of equivalence of so-called side conditions for strictly positive real (SPR) transfer functions is done in , closing a long debate in the Automatic Control community about the frequency-domain characterization of SPR transfer matrices. An erratum/addendum for the third edition of our monograph on dissipative systems, is available at .

In

,

, we study the derivation of approximations of the coefficient of restitution associated with a class of nonlinear spring/dashpot contact/impact models (the most well-known being the so-called Simon-Hunt-Crossley, and Kuwabara-Kono models). The originality of our work (compared to previous results obtained by authors like Poschel and Brilliantov) is that the external load which acts on the system during the collision, is taken into account. This load can influence the restitution coefficient. The approximated expressions are obtained by careful analysis of suitable expansions. A piece of code which allows one to calculate the coefficients has been developed.

This action started in 2001 with the post-doc of V. Acary co–supported by Schneider Electric and CNRS. With some brief interruptions, this action is still active and should further continue. It concerns mainly the simulation and modeling of multi–body systems with contact, friction and impacts with the application for the virtual prototyping of electrical circuit breakers.

During these years, various forms of collaborations have been held. Two PhD thesis have been granted by Schneider Electric (D.E. Taha and N. Akhakdar) accompanied with research contracts between INRIA and Schneider Electric. Schneider Electric participated also the ANR project Saladyn as a main partner.

Without going into deep details of the various actions over the years, the major success of this collaboration is the statistical tolerance analysis of the functional requirements of the circuit breakers with respect to clearance in joints and geometrical tolerances on the parts. Starting from the geometrical descriptions (CAD files) of a mechanism with prescribed tolerances on the manufacturing process, we perform worst-case analysis and Monte–Carlo simulations of the circuit breaker with Siconos and we record the variations in the functional requirements. The difficulty in such simulations are the modeling of contact with friction that models the joints with clearances. The results of these analysis enable Schneider Electric to define the manufacturing precision that has a huge impact of the production cost (Schneider Electric produces several millions of C60-type circuit breaker per year). Note that it is not possible to perform such simulations with the existing software codes of the market.

At the beginning, our interlocutor at Schneider Electric was the innovation (R&D) department. Now, we are working and discussing with the business unit, Division Power and Dinnov (M. Abadie, E. Boumediene, X. Herreros) in charge of designing and producing the circuit–breakers. The targeted users are the R&D engineers of Schneider Electric that use simulation tools for designing new models or improving existing circuit breakers. This collaboration continues with new modeling and simulation challenges (flexible parts, multiple impact laws) with the CIFRE PhD of Rami Sayoud.

We have started with STRMTG a research contract about modelling, simulation and control of cable-transport systems. In such systems, the question of the coupling between the nonlinear dynamics of cables and their supports with unilateral contact and friction appears now to be determinant in order to increase the performances of the cableway systems, especially for urban transportation systems.

Inria Project Lab (IPL): , Languages and Compilation for Cyber-Physical System Design.

The project gathers researchers from three Inria teams, and from three other research labs in Grenoble and Paris area (see table ).

The main objective of ModeliScale is to advance modeling technologies (languages, compile-time analyses, simulation techniques) for CPS combining physical interactions, communication layers and software components. We believe that mastering CPS comprising thousands to millions of components requires radical changes of paradigms. For instance, modeling techniques must be revised, especially when physics is involved. Modeling languages must be enhanced to cope with larger models. This can only be done by combining new compilation techniques (to master the structural complexity of models) with new mathematical tools (new numerical methods, in particular).

ModeliScale gathers a broad scope of experts in programming language design and compilation (reactive synchronous programming), numerical solvers (nonsmooth dynamical systems) and hybrid systems modeling and analysis (guaranteed simulation, verification). The research program is carried out in close cooperation with the Modelica community as well as industrial partners, namely, Dassault Systèmes as a Modelica/FMI tool vendor, and EDF and Engie as end users.

B. Brogliato coordinates the ANR project Digitslid (PRC, ANR-18-CE40-0008-01), Differentiateurs et commandes homogenes par modes glissants en temps discret: l'approche implicite. Partners: LS2N (Ecole Centrale de Nantes), INRIA Lille Nord Europe (team Non-A-Post), and Tripop. October 2018-September 2021. 12 participants overall (3 post-doc students recruited by the project, 3 Ph.D. students supported by other means). Total financial support by the ANR: 338 362 euros (100 762 for Tripop, 18 months of post-doc).

The ModeliScale FUI focuses on the modeling, simulation and analysis of large cyber-physical systems. It federates the research activities of several teams, covering a broad spectrum of topics, namely hybrid systems modeling & verification, numerical analysis, programming language design and automatic control. Our research agenda includes the following tracks:

This project deals with the modeling and control of overhead cranes, in collaboration with Schneider Electric. It involves M.R. Mojallizadeh (two-year contract funded by the IRT), B. Brogliato and C. Prieur. The major objective is to better understand how to minimize sway effects that may be created during operation. The major issues for modeling are how to choose in the set of finite-dimensional (pendulum-like), infinite-dimensional (string or wave equations), and finite-element-method based, models. One major issue for feedback control, concerns the choice of the measured cable angles.

Self-sustained Oscillations in Nonsmooth Dynamical Systems. October 2021 - September 2024. INRIA Grenoble TRIPOP team (F. Miranda-Villatoro, B. Brogliato) and Gipsa-Lab UGA (F. Ferrante). Coordinated by F. Miranda-Villatoro. The SsONDS project aims at developing theoretical methods for the analysis, design, and control of systems with robust self-sustained oscillations in environments with uncertainties (as for instance, lack of knowledge of certain parameters of the model, or the presence of external disturbances). Potential applications include the design of central pattern generators (CPGs) for motion control, and mathematical analysis of models from computational biology. Total fundings for TRIPOP = 99 keuros.

The project aims to develop and test an innovative structure for protection against natural hazards. It is funded by the Auvergne Rhône-Alpes region as part of the R&D operation BOOSTER 2019. The partnerships (GEOLITHE INNOV, GEOLITHE, MYOTIS, INRIA and INRAE) and the operational solutions and tools developed as part of the "Smart-Protect" project will constitute major advances in the methods and means for the natural risk management, both nationally and internationally. GEOLITHE INNOV is leader of the SMART-PROTECT collaborative project. The financial support for INRIA is devoted to the post-doc of Nicholas Collins Craft for the study and the development of cohesive zone model for fracture mechanics simulation.

The OCIRN Project is supported and accompanied by the Auvergne Rhône-Alpes Region. The partners of the project are Géolithe, CAN, INRIA, department of Isère, Halias Technologies and INDURA cluster. The general ambition of the OCIRN project is to support the development of the natural gravity hazards sector in the development and integration of new digital practices. Natural gravity hazards are a growing concern in the context of global warming generating an increase in the frequency and intensity of events, combined with the reduction of societal and economic tolerance of these risks. A functional ambition of the project is to contribute to the integrated and reasoned management of natural gravitational risks, coordinated with the projects of development of the territories, in order to allow important progress in the reduction of the risks, the continuity of service of the installations and the optimization of the operations of mitigation and protection. The OCIRN project aims at 3 major objectives:

These three objectives are addressed through access to shared tools on a scalable and collaborative digital platform made available to the sector. In addition, related training, data collection and processing services will be set up.