Piecewise Smooth Dynamical Systems: Analysis, Numerics and Applications

University of Bristol, 13th-16th September 2004

Abstracts

Invited speakers

Vladimir Babitsky: Excitation and control of micro-vibro-impact mode of ill defined time-varying nonlinear dynamical structure for ultrasonic applications

It has been shown that resonant excitation of the high-frequency nonlinear mode of the tool-workpiece interaction is the most effective method of ultrasonic influence on the dynamic characteristics of machining. This repetitive influence imparts some unique properties to metal cutting where the interaction between the workpiece and the cutting tool is transformed into a micro-vibro-impact process. The investigation of micro-vibro-impact mode needs an analysis of the boundary problem for a rod structure with the variable cross section and nonlinear boundary conditions. The boundary conditions reflects a complex process of tool-workpiece interaction during the cutting. This describes both elasto-plastic deformation of the material and following friction processes.

The specific of such machining system, which is ill-defined mathematically, transforms the dynamical system of a machine in a processing regime into a nonlinear system with poorly predictable response to the excitation. Excitation, stabilisation and control of a nonlinear mode at the top intensity in such a system is an engineering challenge and needs a new method of adaptive control for its realisation. To make the system adaptable to unpredictable variations of cutting conditions the new method for excitation and stabilisation of ultrasonic vibration was proposed, known as autoresonance. In an autoresonant system the amplified signal obtained from the performance sensor is fed to the piezo transducer by means of a positive feedback. This leads to dynamic instability of the acousto-mechanical system, which is controlled by intelligent tracing of the optimal relationship between phase shifting and limitation in the feedback circuitry.

Effectiveness of the autoresonant control relies on a specific topology of amplitude-phase characteristics of the vibration systems. It is shown that these characteristics retain robust single-valued and gently sloping bell-type configurations for many cases of practical importance regardless of Q-factor and nonlinearity of the vibration system. The autoresonant control provides the possibility of self-tuning and self-adaptation mechanisms for the system to keep the nonlinear resonant mode of oscillation under unpredictable variation of load and parameters. Dynamics of the autoresonant ultrasonic machine was investigated and the results of analysis, design and experimentation are presented. The opportunity of application of the robust and high quality nonlinear resonant system under wide deviation of processing loads results in essential increase of machine productivity, efficiency and improvement of design.

Soumitro Banerjee: Different types of border collision bifurcations in piecewise smooth maps

Chris Budd: Simple and complex impact oscillators and their bifurcations

The simple impact oscillator,comprising a smooth system impacting with an obstacle is one of the oldest non smooth dynamical systems, and yet has very rich dynamics which is still being investigated. In this talk I will give and introduction to the theory of the simple impact oscillator and will also show how theory and experiment agree. I will also give a simple introduction to the grazing bifurcation and the period adding route to chaos. We will then look at systems of impact oscillators (such as Newtons cradle) and show that these have special dynamical behaviour driven by the rare effect of simultaneous multiple collisions.

Harry Dankowicz: Discontinuity-driven design and control of an impact microactuator

Impact microactuators rely on repeated collisions to generate gross displacements of a microelectromechanical machine element without the need for large applied forces. Their design and control relies on an understanding of the critical transition between non-impacting and impacting long-term system dynamics and the associated changes in system behavior,known as grazing bifurcations. Here, a theoretical normal-form analysis is presented that predicts the character of such transitions from a set of conditions that are computable in terms of system properties at grazing. The analysis also suggests opportunities for using passive design or active control to regulate the system response near grazing. The theoretical analysis is validated against numerical simulations of an experimentally realized impact microactuator.

Fabio Dercole: Border collision bifurcations in the evolution of mutualistic interactions

he talk describes the slow evolution of two adaptive traits that regulate the interactions between two mutualistic populations (e.g. a flowering plant and its insect pollinator). For frozen values of the traits, the two populations can either coexist or go extinct. The values of the traits for which populations extinction is guaranteed are therefore of no interest from an evolutionary point of view. In other words, the evolutionary dynamics must be studied only in a viable subset of trait space, which is bounded due to the physiological cost of extreme trait values. Thus, evolutionary dynamics experience so-called border collision bifurcations, when a system invariant in trait space hits the border of the viable subset. The unfolding of standard and border collision bifurcations with respect to two parameters of biological interest is presented. The algebraic and boundary-value problems characterizing the border collision bifurcations are described together with some details concerning their computation.

Enric Fossas: SMC applications in Power Electronics

Power converters are widely used in applications where it is desired to obtain a totally regulated electric signal from a non regulated one, keeping optimum energy efficiency in the conversion. These converters can be linear or switched, the later being the most common due to their better energy efficiency. As will be seen in this presentation, switching converters can be modelled as variable structure systems. They therefore constitute a natural field of application of Sliding Mode Control techniques. The most usual conversion types, namely DC-DC, DC-AC and AC-DC, will be considered here. SMC controllers will be designed and several aspects involving the electronic implementation of the controllers will be discussed.

Ian Hiskens: Shooting Methods for Locating Grazing Phenomena in Hybrid Systems

Hybrid systems are typified by strong coupling between continuous dynamics and discrete events. For such systems, event triggering generally has a significant influence over subsequent system behaviour. Therefore it is important to identify situations where a small change in parameter values alters the event triggering pattern. The bounding case, which separates regions of (generally) quite different dynamic behaviour, is referred to as grazing. At a grazing point, the system trajectory makes tangential contact with an event triggering hypersurface. The talk will present conditions governing grazing points. Both transient and periodic behaviour will be considered. The resulting boundary value problems are solved using shooting methods. The approach is applicable for general nonlinear hybrid systems. Examples will be drawn from power electronics, power systems and robotics, all of which involve intrinsic interactions between continuous dynamics and discrete events.

Karl Johansson: Dynamical properties of hybrid automata

Many technical systems involve both analogue and logic components. Hybrid system is a generic term for such systems, where the time-driven and event-driven dynamics are interacting. These systems arise naturally in many applications of automatic control, for example when a physical plant is controlled by a finite set of controls. Motivated by some application examples, we will in this seminar introduce a mathematical model for hybrid systems denoted hybrid automaton. Based on this model, we will discuss the verification problem for hybrid systems as well as some of their dynamical properties.

Mikael Johansson: Computational Analysis of Piecewise Linear Systems

Piecewise linear systems have a broad applicability in a wide range of engineering sciences; systems operating in different modes or subject to physical constraints are naturally modeled as piecewise linear; some of the most common nonlinearities in control systems, such as relays and saturations are piecewise linear, etc. Demanding applications, a shortage of techniques for dealing with hybrid systems, and the emergence of new computational tools have sparked a strong interest for piecewise linear systems in the control community. In this talk, we will survey some recent results on piecewise linear systems developed in the control community. We cover existing results and outstanding research issues, trying to balance basic theory, applications and computational tools.

Yuri Kuznetsov: Continuation of sliding bifurcations in SlideCont 2.0

The latest version of SlideCont, an AUTO97 driver for bifurcation analysis of discontiuous piecewise-smooth autonomous Filippov systems, will be described. The software allows for detection and continuation of codimension-1 bifurcation of Filippov systems in which some sliding on the discontinuity boundary is critically involved. Both local and global bifurcations are supported. Results obtained with SlideCont 2.0 will be presented. Future software developments will be discussed

Manfred Morari: Hybrid Systems - A Control Engineering Perspective

Hybrid systems - loosely defined as systems comprised of continuous and discrete/switched components - are prevalent in all domains of engineering. Over the last few years this system class has attracted much attention and various tools have emerged for studying and affecting its behavior. In this presentation I will describe a recently developed approach for modeling, analysis and controller synthesis that is built on mixed integer mathematical programming. I will illustrate the merits of the technique on a wide range of examples from the automotive, the electrical power and the biomedical domains.

I will start by describing a new framework for modeling, analyzing and controlling systems whose behavior is governed by interdependent physical laws, logic rules, and operating constraints, denoted as Mixed Logical Dynamical (MLD) systems. They are described by linear dynamic equations subject to linear inequalities involving real and integer variables. MLD models are equivalent to various other system descriptions like Piece-Wise Affine (PWA) systems and Linear Complementarity (LC) systems. They have the advantage, however, that all problems of system analysis (like controllability, observability, stability and verification) and all problems of synthesis (like controller design and filter design) can be readily expressed as mixed integer linear or quadratic programs, for which many commercial software packages exist.

In the second part of the talk I recall some concepts of mathematical programming and show their connections with optimal control. In particular, I point out that finite time optimal control problems with constraints can be expressed as mathematical programs that depend on the initial state as a parameter, so called multi-parametric programs. "Solving" a multi-parametric program is synonymous with finding the solution of the mathematical program as an explicit function of the parameter. In the control context, solving the multi-parametric program is synonymous with finding the optimal state feedback controller. I will review the various algorithms that have emerged for the solution of multi-parametric (mixed integer) linear and quadratic programs and describe the broad range of controller synthesis problems that can be addressed with these new tools.

In the final part of the presentation I will discuss in detail some practical applications that have been tackled with these new tools: Traction control for automobiles (Ford), optimal control of co-generation power plants, the control of voltage collapse in power grids, Direct Torque Control of electrical machines (all with ABB), electronic throttle control (with Ford and U. of Zagreb), and Driver Assistance Systems (with Daimler-Chrysler).

Arne Nordmark: Grazing bifurcations of non-hyperbolic limit cycles in impacting systems

Karl Popp: Modelling and Control of Friction Induced Vibrations

Friction induced vibrations in technical applications are usually unwanted, as they create noise, diminish accuracy and increase wear. This lecture intends to give insight in basic excitation mechanisms of friction induced vibrations and to show possible ways of avoidance. Excitation mechanisms under investigation are a friction characteristic decreasing with increasing relative velocity, fluctuating normal forces, nonconservative restoring forces and sprag-slip. They are treated using mechanical models up to 2 degrees of freedom (DOF). Analytical and numerical stability analysis as well as numerical time step integration is used to show the influence of model parameters on the excitation of friction induced vibrations. In the second part of the lecture measures to avoid friction induced vibrations are presented and explained. Such measures are an increase of external damping, additional external excitation as well as active and passive vibration control.

Lawrence Virgin: Piecewise smooth systems: Interrogating the robustness of behavior using perturbations

The stability of solutions for dynamical systems is assessed via the behavior of small perturbations. Specifically the eigenvalues of the Jacobian determine the extent to which a solution persists, and, under changing conditions, whether stability is lost. Furthermore, if perturbations are sufficiently large then a remote solution may be picked up. Thus, both local and global stability characteristics can be interrogated by disturbing a system in equilibrium or undergoing steady-state motion. This talk will discuss this type of situation from a primarily experimental perspective in which a number of non-smooth mechanical oscillators are used as practical examples.

Marian Wiercigroch: Piecewise-Smooth Models in Engineering Dynamics

James Yorke: Border collisions and other weird situations that frequently occur

Zhanybai Zhusubalyiev: Torus Birth Bifurcations in Piecewice-Smooth Dynamical Systems

The talk is devoted to the discussion of the new type of border-collision bifurcation that can occur in multidimensional piecewise-smooth dynamical system displaying a quasiperiodic rote to chaos. We demonstrate how a two-dimensional torus can arise from a periodic orbit through a bifurcation in which two complex-conjugated eigenvalues jump abruptly from the inside to the outside of the unit circle. The torus may be ergodic or resonant. We also consider the birth of a torus via a subcritical Neimark-Sacker bifurcation. Special emphasis is given to the torus destruction through homoclinic and heteroclinic bifurcations in piecewise-smooth systems.

Contributed posters

U. Aeberhard, M. Payr. Ch. Glocker: Theoretical and Experimental Treatment of Impacts (SICONOS)

Excited Multicontact Collision
Considering two kinds of excitations during impact of a finite DOF multibody systems: Kinetic and kinematic excitations. Kinematic excitations are the explicit dependencies in time of the constraints, defining the admissible displacements and velocities. Energetic considerations are important in the treatment of perfect collisions, i.e. as a parameter in impact laws. If a system is kinematically excited, energy has to be defined regarding a reference velocity in order to physically describe the energy budget during collision. A possible reference velocity got vanishing contact velocities in all closed contacts. In case of non-uniform kinematical excitations there is no such velocity. Being unable to define an energy in such cases, the contact work, which can be defined in contact coordinates and therefore is always definable, will be used. Some properties of the contact work, e.g. non-uniqueness, are shown in case of non-uniform kinematically excited systems. Kinetical excitations are external forces with impulsive character, acting during a collision. It is shown in an example, that if only the impulse of such an excitation is given, nothing can be said about the impact intensity and hereby energy. Out of this it will be postulated, that kinetically unexcited systems do not increase the contact work during collision.

Experimental Treatment of Impacts:
To validate the general impact law experiments on multibody collisions are necessary. Interesting impact configurations are chains of balls (Newton's Cradle) as well as longitudinal bars of different length and forked bars. The experiments with longitudinal bars will show the influence of wave effects on the topology of multibody collisions (Lagrange diagram). The test setup consist of the impacting bodies, equipped with a steel scale tape for a exposed linear encoder, hanging on threads. The velocity of the bodies are obtained applying a finite difference scheme on the position vector sampled by the exposed linear encoder. This is the base to identify the Fremont Matrix of the tested impact configuration. A test circuit applying binary coding of the index set H of closed contacts is used to measure the contact time of each couple of colliding bodies. It is compared with the theoretical results of Hertz theory and the Lagrange diagram.

Carles Battle, Enric Fossas, Ivan Merillas, Alicia Miralles: Generalized discontinuous conduction modes in the complementarity formalism (SICONOS & BCANM)

Using a complementarity description of dc-dc power converters we show that, for each position of the switches, the dynamics is given by a linear complementarity problem to which standard techniques can be applied. For systems with a single diode, an analytical condition for the presence of generalized discontinuous conduction modes (GDCM), characterized by a reduction of the dimension of the effective dynamics, can be proved. This result is used to identify the GDCM for the switch configurations of the boost and Cuk converters. Simulation results, showing a variety of behaviours, such as persistent or re-entering GDCM, are presented.

A Brasiello, E Mancusi, L Russo and S Crescitelli: Hybrid system approach to study the dynamics of a controlled reverse flow reactor II. Continuation of limit cycles and bifurcations (SICONOS & BCANM)

A hybrid system approach is adopted to study the dynamic behavior of a controlled reverse flow reactor: a feedback control law dictates the occurrence of flow inversion. We have found a very complex dynamic behavior when a control parameter is varied: Zeno phenomena are coexisting with other regimes like limit cycles and quasi-periodic solutions. We characterize these limit cycles and their bifurcations with the application of continuation technique to the impact map of the system. Several local bifurcations of limit cycles are detected: pitchfork, Neimark- Sacker and period-doubling bifurcations. Symmetric and asymmetric periodic and quasi-periodic solutions are connected to the symmetry of the hybrid system. The coexistence of these regimes with Zeno state is also discussed as the control parameter is varied: beyond a critical value only the Zeno state exists. The stability and the bifurcations of the limit cycles are studied with the systematic computation of the Floquet multipliers. In spite of this hybrid system is spatially extended (an infinite dimensional system), few Floquet multipliers determine the system dynamics in wide range of the control parameter. However, near the critical value that signs the end of the coexistence region, the dynamics seems to be fully infinite dimensional.

Kanat Camlibel: Controllability of complementarity systems (SICONOS)

This paper deals with the controllability problem of a class of piecewuse linear systems, known as linear complementarity systems. It is well-known that checking certain controllability properties of very simple piecewise linear systems are undecidable problems. However, we establish algebraic necessary and sufficient conditions for the controllability of linear conmplementarity systems. Our treatment makes use of the ideas and techniques from geometric control theory together with mathematical programming.

Victoriano Carmona, Emilio Freire, Enrique Ponce and Francisco Torres: Can the continuous matching of two stable linear systems give rise to a unstable system? (BCANM)

The seemingly straightforward stability issue in three-dimensional homogeneous continuous piecewise linear systems with two linear zones is considered. The only equilibrium at the origin, being in the separation plane of the linear zones, possesses two linearization matrices. For the important case where both matrices have complex eigenvalues with the whole spectrum in the left half plane, the possible counter-intuitive instability of the origin is proved.

Gabor Csernak: Nonlinear Stability Analysis of a Simple Coulomb-friction Oscillator (BCANM)

A harmonically excited dry friction oscillator is examined analytically and numerically. We search for non-sticking 2*pi/Omega-periodic solutions, where Omega is the excitation frequency. Exploiting that there are only two turnarounds during each cycle, we prove that the two phases of the motion are of equal length and the motion is symmetric in the coordinate at almost every value of the excitation frequency. However, innumerable asymmetric, linearly marginally stable solutions can be found at frequencies Omega = 1/(2n), and sticking solutions appear at \Omega = 1/(2n+1). The non-linear stability of the asymmetric solutions is examined using a normal form technique.

Max Demenkov: Algorithms for parametric continuation of solutions for nonsmooth systems and their parallel implementation (BCANM)

I will discuss one-parameter continuation algorithms for tracing steady-state solutions of continuous nonsmooth dynamical systems. The algorithms require nonsmooth equation solver as a part. Parallel realization of these algorithms on a Linux HPC cluster using Matlab/C++ & MPI will be considered.

Apostolos Doris: Switching observer design for an experimental piecewise linear beam system (SICONOS)

This project is within the scope of the European SICONOS project on analysis and control of non-smooth dynamical systems. It involves an experimental study of a switching observer design strategy for a class of piece-wise linear systems by application to an elastic beam with a one-sided support.

The beam system consists of a steel beam, which is clamped on two sides and is supported at a discrete location by a one-sided linear spring. Due to the one-sided spring the beam has two different dynamics regimes.

Focus is on the implementation of a model-based switching observer on the experimental beam system and on the comparison of the observer predictions with experimental measurements. To achieve that, the observer incorporates a 3DOF model of the experimental system, which is based on a finite element model of the beam and is reduced using the Rubin reduction method.

According to the observer design strategy global asymptotic stability of the state estimation error can be achieved. Furthermore, the designed observer does not require information on the currently active dynamic regime of the piece-wise linear system.

Comparison of experimental results with observer predictions confirms the stability of the implemented observer and indicates a satisfactory observer performance in practice

Raoul Dzonou: Convergence result in vibro-impact system with non trivial mass matrix: the inelastic case(SICONOS & BCANM)

We consider a mechanical system with a finite number of degrees of freedom submitted to perfect unilateral constraints. J.J. Moreau gave a formulation of the dynamic as a measure differential inclusion. In the case of the trivial mass matrix, M. Schatzman and L. Paoli proved an existence result of such problem by solving a second order differential inclusion with a penalisation technique; M. Marques and M. Mabrouk in the same case have solve the problem as a "sweeping process" problem. Following this last formulation, we present here a model with a non tivial mass matrix and we report on the current state of proof of existence of solution by using a time discretisation scheme.

Fabiola Angulo Garcia: Rich dynamics, chaos and control of chaos in a zero average dynamics buck converter (SICONOS & BCANM)

In this poster the dynamics of a fixed frequency switching controller for a power converter is considered. A centered Pulse Width Modulator implements the control action. The output, a first order differential equation in the error, is parametrised by the time constant. The proposed duty cycle guarantees output Zero Averaged Dynamics in each PWM-period, this resulting in a very rich dynamics which, when stable, results in a robust behaviour and an excellent performance. The output time constant is used as bifurcation parameter, yielding to chaos and unstable periodic orbits. Lyapunov and Floquet exponents have been computed and chaos has been controlled through Time Delay AutoSynchronization techniques.

Ch. Glocker: A Converter Model in Complementarity Framework (SICONOS)

Idealized modeling of diodes, relays and switches in the framework of linear complementarity is introduced. Within the charge approach, the classical electromechanical analogy is extended to passively and actively switching components in electrical circuits. The associated branch relations are expressed in terms of set-valued functions, which allow to formulate the circuit's dynamic behavior as a differential inclusion. This approach is demonstrated by the example of the DC-DC buck converter. A difference scheme, known in mechanics as time stepping, is applied for numerical approximation of the evolution problem. The discretized inclusions are formulated as a linear complementarity problem in standard form, which implicitly takes care of all switching events by its solution. State reduction, which requires manipulation of the set-valued branch relations in order to obtain a minimal model, is performed on the example of the buck converter.

Tomas Hanus: Bifurcations of Filippov systems with sliding (BCANM)

Filippov systems were shown to exhibit sliding bifurcations. The sliding bifurcations occur when the system trajectory interacts with regions on the discontinuity set where sliding is possible. Using the classical approach of topological equivalence, these bifurcations are defined, classified and their normal forms are derived.

Galyna Kriukova: Canonical forms and hyperbolicity of two-dimentional piecewise linear dynamical systems (BCANM)

We construct canonical forms for some classes of piecewise linear mapping of the plane. Using these forms we investigate a behavior of such systems, in particular we find some sufficient conditions of hyperbolicity of such systems.

Stefano Lenci: A numerical study of the non-smooth dynamics of an impacting inverted pendulum (BCANM)

A systematic numerical investigation of the nonlinear dynamics of an inverted pendulum between lateral rebounding barriers is performed. The attractor scenario is determined and illustrated by bifurcation diagrams. Attention is focused on local and global, classical and non-classical bifurcations that lead to the attractors-basins metamorphoses, and on the role played by invariant manifolds of appropriate saddles. The fractal nature of the basins of attraction appearing at a certain critical threshold is highlighted, and attention is then focused on the onset of the scattered cross-well chaotic attractor.

Max Little: Piecewise smooth non-linear models of vocal chord oscillation (BCANM)

There exist several non-linear models of vocal chord vibration of varying dimensions, often involving piecewise smooth regions and discontinuous impact events. Numerically simulating these models is possible, but stability and consistency analysis is not a simple matter. Furthermore, finding bifurcations is a complex problem. One goal of my research therefore is to find low-dimensional counterparts that capture all the important dynamics in an analytically and computationally tractable way. The aim of this presentation will be to describe the results of some of the candidate approaches that I have taken, which include piecewise state-space region analysis, finding topological regions where the dynamics are locally smooth, and fitting discrete threshold autoregression models that approximate the dynamics successfully.

Qi-shao Lu: Grazing Bifurcation and Global Dynamics in Rub-impact Rotor Systems (BCANM)

In large high-speed rotating machines, such as turbines, generators and engines, rub-impact events between the rotor and the stator have drawn the attentions of engineers and other authors [1,2]. Due to the non-smoothness, the rub-impact rotor system experiences complex bifurcation processes from stable periodic motions to chaotic rub-impact motions. Grazing impact is one of typical phenomena of non-smooth dynamic systems. Grazing impacts of non-smooth systems generally refer to the uncertain motions in the phase space when the trajectories touch the constraint surface with a zero normal velocity, and they can be described mathematically by discrete maps with square root singularities and treated by the theory of dynamical systems.

Enlightened by the work for impact oscillators of Nordmark et al [3-6], this paper focuses the attention on rub-impact rotors of two degrees of freedom. Taking account of both the rotation and rub-impact features of rotor systems, a general method to construct the rub-impact Poincaré map for rotor systems is proposed through an appropriate composition of discrete maps. By means of time reversing and grazing singularity analysis, an approximate analytical expression of the rub-impact Poincaré map during grazing is derived for further dynamical analysis. Some theoretical results for single impact periodic motions near grazing, including the stability, bifurcation and the variation of trapping regions with parameters, are obtained by detailed analysis of the local rub-impact Poincaré map. Numerical results also reveal complex bifurcation processes and global dynamics near grazing in rub-impact rotor systems, such as period-adding bifurcation, co-existing multiple attractors and chaos.

[1] F.K. Choy, J. Padovan, Nonlinear transient analysis of rotor-casing rub events, J. Sound Vibr., 113(1987), 529-545.
[2] G.-X. Li, M.P. Paidoussis, Impact phenomena of rotor-casing dynamical systems, Nonlinear Dynamics, 5(1994), 53-70.
[3] A.B. Nordmark, Non-periodic motion caused by grazing incidence in an impact oscillator, J. Sound Vibr., 145(1991), 279-297.
[4] M.H. Fredriksson, A.B. Nordmark, Bifurcations caused by grazing incidence in many degrees of freedom impact oscillators, Proc. R. Soc. London, Series A, 453(1997), 1261-1276.
[5] A.P. Ivanov, Stabilization of an impact oscillator near grazing incidence owing to resonance, J. Sound Vibr., 162(1993), 562-565.
[6] J. Molenaar, J.G. de Weger, W. van de Water, Mappings of grazing- impact oscillators, Nonlineaity , 14(2001), 301-321.

E Mancusi, L Russo, M di Bernardo and S Crescitelli: Hybrid system approach to study the dynamics of a controlled reverse flow reactor I. Zeno executions and regularization (SICONOS & BCANM)

Recent and extended studies have shown that reverse flow reactors can improve many catalytic processes (Matros and Bunimovich 1996). These catalytic fixed bed reactors are operated by periodically inverting the flow direction. Particularly promising is the application to the catalytic combustion of VOCs for which an autothermal operation can be carried out even in the case of low feed concentrations. The extinction of the reaction and the hot spots formation are two problems related to the conduction of these reactors. To overcome these problems a feedback controller can be adopted (Barresi and Vanni 2002): the flow direction is inverted when the temperature at the first layer of catalyst falls below a fixed value. The control law regulates the occurrence of discrete events (the inversions of the flow direction), while the continuous dynamics between two successive inversions is described by partial differential equations: the mathematical model is a hybrid spatial extended system. We analyze the dynamics of the controlled reverse flow reactor with the above mentioned control policy. In particular, a typical feature of hybrid system dynamics, Zeno and quasi-Zeno executions (Zhang et al. 2001; Mancusi et al. 2004) are discussed. Model modifications are considered to regularize this dynamical behaviour.

Matros, Yu. Sh., and G. A. Bunimovich, Reverse Flow Operation in Fixed Bed Catalytic Reactors, Catal. Rev. -Sci. Eng., 38, 1 (1996).
Barresi, A. A., and M. Vanni, Control of Catalytic Combustors with Periodical Flow Reversal, AIChE J., 48, 648 (2002).
Mancusi E., Russo L., di Bernardo M. and S.Crescitelli, A controlled reverse flow reactor: a hybrid system approach, submitted to AIChE J. (2004).
Zhang, J., K. H. Johansson, J. Lygeros and S. Sastry, Zeno Hybrid Systems, Int. J. Robust Nonlinear Control, 11, 435 (2001).

Iryna Matskiv: Synchronization and clustering in the ensembles of piecewise linear and smooth maps (BCANM)

We investigate full and different types of partial synchronization in the three types of coupled maps systems. As individual elements we choose piecewise linear or smooth maps, and compare results obtained in both cases. We delineate regions of transverse stability for different synchronized regimes in the parameter space. We show, that considered coupling structures of the systems provide for wide parameter regions of stability for fully and partially synchronized states. Parameter regions of weak and asyptotic stability and instability are obtained in the case of full synchronization. We analyse qualitatively different transition from desynchronization to synchronization in one of the considered models for piece wise and smooth cases.

Gerard Olivar Tost: Non-smooth Bifurcations in a planar model of a DC-DC converter (SICONOS & BCANM)

DC-DC Power Electronic Converters are usually modelled by switching systems, that means discontinuous systems of Filippov type. These systems are piecewise linear, and simulation is straightforward since analytical solutions are available in each linear topology. Recently, it has been shown that they can undergo some nonsmooth bifurcations and transitions. In this presentation we show that a DC-DC Buck converter can undergo several nonsmooth Hopf transitions.

Anastasiya Panchuk: Partial synchronization in the turbulent phase (BCANM)

We investigate partial synchronization phenomenon appearing in a system of globally coupled maps. We analyze the structure of clustering zones for small coupling parameter values and the prerequisites for periodic and chaotic attractors being formed on cluster manifolds. We obtain the relationship between transverse and longitudinal Lyapunov numbers for a single trajectory. We also derive the necessary and sufficient conditions for its transverse stability.

Pavel Pokorny: Piecewise continuous models of biological systems (BCANM)

An important class of piecewise smooth dynamical systems can be put in the form

dx/dt = u(x) if f(x,t)=1
dx/dt = v(x) if f(x,t)=0.

f(x,t) = 1 for 0<=t<p
f(x,t) = 0 for p<=t<2p.

dx/dt = (u(x)+v(x))/2.

Miss Nineb Sheherazade: Siconos Numerics and Applications (SICONOS)

Presentation of the various methods implemented in the WP2 Siconos Software. Illustration on tensegrity structures and granular materials.

Bart De Schutter: Model predictive control for max-min-plus-scaling systems (SICONOS)

Continuous piecewise-affine systems can be shown to be equivalent to max-min-plus-scaling systems (i.e., systems that can be modeled using maximization, minimization, addition and scalar multiplication). We use this equivalence to derive non-integer programming algorithms for model predictive control based on canonical forms for max-min-plus-scaling functions. We consider both the deterministic model predictive control case and the case with bounded perturbations.

David Wagg: Periodic sticking motion and multi-sliding in a two-degree of freedom impact oscillator (SICONOS & BCANM)

In mechanical systems with vibration and impact, chatter and sticking are phenomena which have been observed for a wide range of parameter values. These systems exhibit a rich variety of periodic motions and periodic sticking motions can be found for particular parameter values in both single and multi-degree-of-freedom systems. In this study we consider the dynamics of periodic sticking motions and focus particularly on multi-sliding events and the region in state space where sticking motions exist.