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Describing function

In control systems theory, the describing function (DF) method, developed by Nikolay Mitrofanovich Krylov and Nikolay Bogoliubov in the 1930s, and extended by Ralph Kochenburger is an approximate procedure for analyzing certain nonlinear control problems. It is based on quasi-linearization, which is the approximation of the non-linear system under investigation by a linear time-invariant (LTI) transfer function that depends on the amplitude of the input waveform. By definition, a transfer function of a true LTI system cannot depend on the amplitude of the input function because an LTI system is linear. Thus, this dependence on amplitude generates a family of linear systems that are combined in an attempt to capture salient features of the non-linear system behavior. The describing function is one of the few widely applicable methods for designing nonlinear systems, and is very widely used as a standard mathematical tool for analyzing limit cycles in closed-loop controllers, such as

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Les systèmes non linéaires sont analysés en vue d'établir des lois de commande. On présente la stabilité au sens de Lyapunov, ainsi que des méthodes de commande géométrique (linéarisation exacte). Divers exemples illustrent la théorie (exercices papier crayon et simulations à l'ordinateur).

This class teaches the theory of linear time-invariant (LTI) systems. These systems serve both as models of physical reality (such as the wireless channel) and as engineered systems (such as electrical circuits, filters and control strategies).

Cours introductif à la commande des systèmes dynamiques. On part de quatre exemples concrets et on introduit au fur et à mesure un haut niveau d'abstraction permettant de résoudre de manière unifiée les problèmes d'asservissement et de régulation, en particulier les questions de stabilité.

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Modélisation de matériaux supraconducteurs

Joseph Duron

This thesis is focused on the modelling of superconducting materials, and in particular, high-temperature superconducting materials. This work is divided in two parts: first, the dynamic magnetic field mapping obtained by measurements is used in order to reconstruct the dynamic current distribution inside a tape assuming a model describing its global behavior. In the second part, superconductors in over-critical regime have been both electrically and thermally modelized with a finite element method. Calculations have been applied on a current limiter and compared to a constructed and measured device at the University of Geneva. The first chapter is an introduction to high-temperature superconducting materials, where the main superconducting physical properties used are presented. Next, we introduce the different existing models which can be used for describing the behavior of a superconducting tape. With the help of the Bean model, a novel method to evaluate the current distribution in a mono-filamentary superconducting tape from surface magnetic field measurements is proposed. The obtained results are compared with more precise, but more complex, methods like finite element modeling, which will be used for the second part of this work. The last part is focused on the modelization of superconducting material in overcritical regime. To achieve this goal, we need to introduce a different expression than the usual Ec(J/Jc)n power-law for describing the electric behavior of the material in a much wider current range covering over-critical excursion. The original proposition we made is able to fit the measurements made on YBCO tapes, allowing to describe the transition from the superconducting to the normal state. Then, to taking into account the thermal phenomena, the temperature dependence of the electrical parameters has been introduced in order to solve a coupled electromagnetic and thermal problem: solving a time step of the electromagnetic part leads to the knowledge of the local losses. These losses are injected inside the thermal part. The resulting temperature computation is used to modify the electrical parameters for the next time step. Finally, this method has been applied in order to simulate a superconducting current limiter. The global behavior of the device can be reproduced by the implemented numerical model, which also allows the study of the local variables, like current density distribution or temperature profile. The obtained results can be used to optimize a device according to specific criteria. In particular, we have proposed geometric modifications for avoiding possible local thermal runaway, which can lead to the destruction of the device. This proposition has beeing inspired the new design, which is now tested.

EPFL2006

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A Data-Driven Frequency-Domain Approach for Robust Controller Design via Convex Optimization

Achille Nicoletti

The objective of this dissertation is to develop data-driven frequency-domain methods for designing robust controllers through the use of convex optimization algorithms. Many of today's industrial processes are becoming more complex, and modeling accurate physical models for these plants using first principles may be impossible. With the increased developments in the computing world, large amounts of measured data can be easily collected and stored for processing purposes. Data can also be collected and used in an on-line fashion. Thus it would be very sensible to make full use of this data for controller design, performance evaluation, and stability analysis. The design methods imposed in this work ensure that the dynamics of a system are captured in an experiment and avoids the problem of unmodeled dynamics associated with parametric models. The devised methods consider robust designs for both linear-time-invariant (LTI) single-input-single-output (SISO) systems and certain classes of nonlinear systems. In this dissertation, a data-driven approach using the frequency response function of a system is proposed for designing robust controllers with H∞ performance. Necessary and sufficient conditions are derived for obtaining H∞ performance while guaranteeing the closed-loop stability of a system. A convex optimization algorithm is implemented to obtain the controller parameters which ensure system robustness; the controller is robust with respect to the frequency-dependent uncertainties of the frequency response function. For a certain class of nonlinearities, the proposed method can be used to obtain a best-linear-approximation with an associated frequency dependent uncertainty to guarantee the stability and performance for the underlying linear system that is subject to nonlinear distortions. The concepts behind these design methods are then used to devise necessary and sufficient conditions for ensuring the closed-loop stability of systems with sector-bounded nonlinearities. The conditions are simple convex feasibility constraints which can be used to stabilize systems with multi-model uncertainty. Additionally, a method is proposed for obtaining H∞ performance for an approximate model (i.e., describing function) of a sector-bounded nonlinearity. This work also proposes several data-driven methods for designing robust fixed-structure controllers with H∞ performance. One method considers the solution to a non-convex problem, while another method convexifies the problem and implements an iterative algorithm to obtain the local solution (which can also consider H2 performance). The effectiveness of the proposed method(s) is illustrated by considering several case studies that require robust controllers for achieving the desired performance. The main applicative work in this dissertation is with respect to a power converter control system at the European Organization for Nuclear Research (CERN) (which is used to control the current in a magnet to produce the desired field in controlling particle trajectories in accelerators). The proposed design methods are implemented in order to satisfy the challenging performance specifications set by the application while guaranteeing the system stability and robustness using data-driven design strategies.

EPFL2018

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Effect of Fe(II)/Fe(III) species, pH, irradiance and bacterial presence on viral inactivation in wastewater by the photo-Fenton process: Kinetic modeling and mechanistic interpretation

Michaël Bensimon, Anna Carratala Ripolles, Stefanos Giannakis, Siting Liu, César Pulgarin

Advanced Oxidation Processes and in particular photo-Fenton, represent promising strategies of pathogen inactivation in wastewater effluents. Nevertheless, its full potential is not yet unlocked, as the efficacy of photo-Fenton against viruses has not been deeply explored. In this work, we characterize the effect of major parameters (Fe species and concentration, solar irradiance, pH and microbial competition) on the inactivation of MS2 Coliphage by the photo-Fenton process. The use of Fe(II) salts, under any combination of H2O2 concentration, sunlight irradiance or starting pH (6–8), induced a faster inactivation compared to their Fe(III) counterparts. Moreover, ICP-MS analyses revealed that starting with Fe(II) resulted to higher amount of iron in solution longer than Fe(III), which led to higher inactivation kinetics. Even so, a 4-log MS2 inactivation was achieved upon exposure to 600 W/m2 for 30 min in presence of Fe(III) and H2O2 (1:1 ratio). Furthermore, the inactivation of MS2 was only slightly decreased in presence of the bacterial host, suggesting a low competition for the oxidants in the bulk. The enhancement of iron solubilization through its complexation by organic matter present in wastewater was also investigated, observing an efficient viral inactivation despite the presence of reactive oxygen species (ROS) scavengers. The present data have been used to propose a simple model describing MS2 photo-Fenton inactivation in wastewater. Finally, the pathway describing the photo-Fenton-induced MS2 inactivation in wastewater was proposed, in presence or absence of bacteria.

Elsevier2017

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Théorie du contrôle

En mathématiques et en sciences de l'ingénieur, la théorie du contrôle a comme objet l'étude du comportement de systèmes dynamiques paramétrés en fonction des trajectoires de leurs paramètres.

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