Modeling and control of electromechanical systems

The material presented in the these notes covers the sessions Modelling of electromechanical systems, Passive control theory I and Passive control theory II of the II EURON/GEOPLEX Summer School on Modelling and Control of Complex Dynamical Systems.We start with a general description of what an electromechanical system is from a network modelling point of view. Next, a general formulation in terms of PHDS is introduced, and some of the previous electromechanical systems are rewritten in this formalism. Power converters, which are variable structure systems (VSS), can also be given a PHDS form.We conclude the modelling part of these lectures with a rather complex example, showing the interconnection of subsystems from several domains, namely an arrangement to temporally store the surplus energy in a section of a metropolitan transportation system based on dc motor vehicles, using either arrays of supercapacitors or an electric poweredflywheel. The second part of the lectures addresses control of PHD systems. We first present the idea of control as power connection of a plant and a controller. Next we discuss how to circumvent this obstacle and present the basic ideas of Interconnection and Damping Assignment (IDA) passivity-based control of PHD systems.

Matching stages of heavy-ion collision models

Heavy-ion reactions and other collective dynamical processes are frequently described by different theoretical approaches for the different stages of the process, like initial equilibration stage, intermediate locally equilibrated fluid dynamical stage, and final freeze-out stage. For the last stage, the best known is the Cooper-Frye description used to generate the phase space distribution of emitted, noninteracting particles from a fluid dynamical expansion or explosion, assuming a final ideal gas distribution, or (less frequently) an out-of-equilibrium distribution. In this work we do not want to replace the Cooper-Frye description, but rather clarify the ways of using it and how to choose the parameters of the distribution and, eventually, how to choose the form of the phase space distribution used in the Cooper-Frye formula. Moreover, the Cooper-Frye formula is used in connection with the freeze-out problem, while the discussion of transition between different stages of the collision is applicable to other transitions also. More recently, hadronization and molecular dynamics models have been matched to the end of a fluid dynamical stage to describe hadronization and freeze-out. The stages of the model description can be matched to each other on space-time hypersurfaces (just like through the frequently used freeze-out hypersurface). This work presents a generalized description of how to match the stages of the description of a reaction to each other, extending the methodology used at freeze-out, in simple covariant form which is easily applicable in its simplest version for most applications.

Modeling morphogen gradient formation from arbitrary realistically shaped sources.

Much of the analytical modeling of morphogen profiles is based on simplistic scenarios, where the source is abstracted to be point-like and fixed in time, and where only the steady state solution of the morphogen gradient in one dimension is considered. Here we develop a general formalism allowing to model diffusive gradient formation from an arbitrary source. This mathematical framework, based on the Green's function method, applies to various diffusion problems. In this paper, we illustrate our theory with the explicit example of the Bicoid gradient establishment in Drosophila embryos. The gradient formation arises by protein translation from a mRNA distribution followed by morphogen diffusion with linear degradation. We investigate quantitatively the influence of spatial extension and time evolution of the source on the morphogen profile. For different biologically meaningful cases, we obtain explicit analytical expressions for both the steady state and time-dependent 1D problems. We show that extended sources, whether of finite size or normally distributed, give rise to more realistic gradients compared to a single point-source at the origin. Furthermore, the steady state solutions are fully compatible with a decreasing exponential behavior of the profile. We also consider the case of a dynamic source (e.g. bicoid mRNA diffusion) for which a protein profile similar to the ones obtained from static sources can be achieved.

Implicit methods for qualitative modeling of gene regulatory networks.

Advancements in high-throughput technologies to measure increasingly complex biological phenomena at the genomic level are rapidly changing the face of biological research from the single-gene single-protein experimental approach to studying the behavior of a gene in the context of the entire genome (and proteome). This shift in research methodologies has resulted in a new field of network biology that deals with modeling cellular behavior in terms of network structures such as signaling pathways and gene regulatory networks. In these networks, different biological entities such as genes, proteins, and metabolites interact with each other, giving rise to a dynamical system. Even though there exists a mature field of dynamical systems theory to model such network structures, some technical challenges are unique to biology such as the inability to measure precise kinetic information on gene-gene or gene-protein interactions and the need to model increasingly large networks comprising thousands of nodes. These challenges have renewed interest in developing new computational techniques for modeling complex biological systems. This chapter presents a modeling framework based on Boolean algebra and finite-state machines that are reminiscent of the approach used for digital circuit synthesis and simulation in the field of very-large-scale integration (VLSI). The proposed formalism enables a common mathematical framework to develop computational techniques for modeling different aspects of the regulatory networks such as steady-state behavior, stochasticity, and gene perturbation experiments.

An Improved Hybrid Model for the Generic Hoist Scheduling Problem

Peer-reviewed

Multi-scale systems analysis of plant development : beyond cellular signaling and tissue morphodynamics

Les plantes sont essentielles pour les sociétés humaines. Notre alimentation quotidienne, les matériaux de constructions et les sources énergétiques dérivent de la biomasse végétale. En revanche, la compréhension des multiples aspects développementaux des plantes est encore peu exploitée et représente un sujet de recherche majeur pour la science. L'émergence des technologies à haut débit pour le séquençage de génome à grande échelle ou l'imagerie de haute résolution permet à présent de produire des quantités énormes d'information. L'analyse informatique est une façon d'intégrer ces données et de réduire la complexité apparente vers une échelle d'abstraction appropriée, dont la finalité est de fournir des perspectives de recherches ciblées. Ceci représente la raison première de cette thèse. En d'autres termes, nous appliquons des méthodes descriptives et prédictives combinées à des simulations numériques afin d'apporter des solutions originales à des problèmes relatifs à la morphogénèse à l'échelle de la cellule et de l'organe. Nous nous sommes fixés parmi les objectifs principaux de cette thèse d'élucider de quelle manière l'interaction croisée des phytohormones auxine et brassinosteroïdes (BRs) détermine la croissance de la cellule dans la racine du méristème apical d'Arabidopsis thaliana, l'organisme modèle de référence pour les études moléculaires en plantes. Pour reconstruire le réseau de signalement cellulaire, nous avons extrait de la littérature les informations pertinentes concernant les relations entre les protéines impliquées dans la transduction des signaux hormonaux. Le réseau a ensuite été modélisé en utilisant un formalisme logique et qualitatif pour pallier l'absence de données quantitatives. Tout d'abord, Les résultats ont permis de confirmer que l'auxine et les BRs agissent en synergie pour contrôler la croissance de la cellule, puis, d'expliquer des observations phénotypiques paradoxales et au final, de mettre à jour une interaction clef entre deux protéines dans la maintenance du méristème de la racine. Une étude ultérieure chez la plante modèle Brachypodium dystachion (Brachypo- dium) a révélé l'ajustement du réseau d'interaction croisée entre auxine et éthylène par rapport à Arabidopsis. Chez ce dernier, interférer avec la biosynthèse de l'auxine mène à la formation d'une racine courte. Néanmoins, nous avons isolé chez Brachypodium un mutant hypomorphique dans la biosynthèse de l'auxine qui affiche une racine plus longue. Nous avons alors conduit une analyse morphométrique qui a confirmé que des cellules plus anisotropique (plus fines et longues) sont à l'origine de ce phénotype racinaire. Des analyses plus approfondies ont démontré que la différence phénotypique entre Brachypodium et Arabidopsis s'explique par une inversion de la fonction régulatrice dans la relation entre le réseau de signalisation par l'éthylène et la biosynthèse de l'auxine. L'analyse morphométrique utilisée dans l'étude précédente exploite le pipeline de traitement d'image de notre méthode d'histologie quantitative. Pendant la croissance secondaire, la symétrie bilatérale de l'hypocotyle est remplacée par une symétrie radiale et une organisation concentrique des tissus constitutifs. Ces tissus sont initialement composés d'une douzaine de cellules mais peuvent aisément atteindre des dizaines de milliers dans les derniers stades du développement. Cette échelle dépasse largement le seuil d'investigation par les moyens dits 'traditionnels' comme l'imagerie directe de tissus en profondeur. L'étude de ce système pendant cette phase de développement ne peut se faire qu'en réalisant des coupes fines de l'organe, ce qui empêche une compréhension des phénomènes cellulaires dynamiques sous-jacents. Nous y avons remédié en proposant une stratégie originale nommée, histologie quantitative. De fait, nous avons extrait l'information contenue dans des images de très haute résolution de sections transverses d'hypocotyles en utilisant un pipeline d'analyse et de segmentation d'image à grande échelle. Nous l'avons ensuite combiné avec un algorithme de reconnaissance automatique des cellules. Cet outil nous a permis de réaliser une description quantitative de la progression de la croissance secondaire révélant des schémas développementales non-apparents avec une inspection visuelle classique. La formation de pôle de phloèmes en structure répétée et espacée entre eux d'une longueur constante illustre les bénéfices de notre approche. Par ailleurs, l'exploitation approfondie de ces résultats a montré un changement de croissance anisotropique des cellules du cambium et du phloème qui semble en phase avec l'expansion du xylème. Combinant des outils génétiques et de la modélisation biomécanique, nous avons démontré que seule la croissance plus rapide des tissus internes peut produire une réorientation de l'axe de croissance anisotropique des tissus périphériques. Cette prédiction a été confirmée par le calcul du ratio des taux de croissance du xylème et du phloème au cours de développement secondaire ; des ratios élevés sont effectivement observés et concomitant à l'établissement progressif et tangentiel du cambium. Ces résultats suggèrent un mécanisme d'auto-organisation établi par un gradient de division méristématique qui génèrent une distribution de contraintes mécaniques. Ceci réoriente la croissance anisotropique des tissus périphériques pour supporter la croissance secondaire. - Plants are essential for human society, because our daily food, construction materials and sustainable energy are derived from plant biomass. Yet, despite this importance, the multiple developmental aspects of plants are still poorly understood and represent a major challenge for science. With the emergence of high throughput devices for genome sequencing and high-resolution imaging, data has never been so easy to collect, generating huge amounts of information. Computational analysis is one way to integrate those data and to decrease the apparent complexity towards an appropriate scale of abstraction with the aim to eventually provide new answers and direct further research perspectives. This is the motivation behind this thesis work, i.e. the application of descriptive and predictive analytics combined with computational modeling to answer problems that revolve around morphogenesis at the subcellular and organ scale. One of the goals of this thesis is to elucidate how the auxin-brassinosteroid phytohormone interaction determines the cell growth in the root apical meristem of Arabidopsis thaliana (Arabidopsis), the plant model of reference for molecular studies. The pertinent information about signaling protein relationships was obtained through the literature to reconstruct the entire hormonal crosstalk. Due to a lack of quantitative information, we employed a qualitative modeling formalism. This work permitted to confirm the synergistic effect of the hormonal crosstalk on cell elongation, to explain some of our paradoxical mutant phenotypes and to predict a novel interaction between the BREVIS RADIX (BRX) protein and the transcription factor MONOPTEROS (MP),which turned out to be critical for the maintenance of the root meristem. On the same subcellular scale, another study in the monocot model Brachypodium dystachion (Brachypodium) revealed an alternative wiring of auxin-ethylene crosstalk as compared to Arabidopsis. In the latter, increasing interference with auxin biosynthesis results in progressively shorter roots. By contrast, a hypomorphic Brachypodium mutant isolated in this study in an enzyme of the auxin biosynthesis pathway displayed a dramatically longer seminal root. Our morphometric analysis confirmed that more anisotropic cells (thinner and longer) are principally responsible for the mutant root phenotype. Further characterization pointed towards an inverted regulatory logic in the relation between ethylene signaling and auxin biosynthesis in Brachypodium as compared to Arabidopsis, which explains the phenotypic discrepancy. Finally, the morphometric analysis of hypocotyl secondary growth that we applied in this study was performed with the image-processing pipeline of our quantitative histology method. During its secondary growth, the hypocotyl reorganizes its primary bilateral symmetry to a radial symmetry of highly specialized tissues comprising several thousand cells, starting with a few dozens. However, such a scale only permits observations in thin cross-sections, severely hampering a comprehensive analysis of the morphodynamics involved. Our quantitative histology strategy overcomes this limitation. We acquired hypocotyl cross-sections from tiled high-resolution images and extracted their information content using custom high-throughput image processing and segmentation. Coupled with an automated cell type recognition algorithm, it allows precise quantitative characterization of vascular development and reveals developmental patterns that were not evident from visual inspection, for example the steady interspace distance of the phloem poles. Further analyses indicated a change in growth anisotropy of cambial and phloem cells, which appeared in phase with the expansion of xylem. Combining genetic tools and computational modeling, we showed that the reorientation of growth anisotropy axis of peripheral tissue layers only occurs when the growth rate of central tissue is higher than the peripheral one. This was confirmed by the calculation of the ratio of the growth rate xylem to phloem throughout secondary growth. High ratios are indeed observed and concomitant with the homogenization of cambium anisotropy. These results suggest a self-organization mechanism, promoted by a gradient of division in the cambium that generates a pattern of mechanical constraints. This, in turn, reorients the growth anisotropy of peripheral tissues to sustain the secondary growth.

Energy-based modelling and simulation of the interconnection of a back-to-back converter and a doubly-fed induction machine

This paper describes the port interconnection of two subsystems: a power electronics subsystem (a back-to-back AC/CA converter (B2B), coupled to a phase of the power grid), and an electromechanical subsystem (a doubly-fed induction machine (DFIM). The B2B is a variable structure system (VSS), due to presence of control-actuated switches: however, from a modelling simulation, as well as a control-design, point of view, it is sensible to consider modulated transformers (MTF in the bond graph language) instead of the pairs of complementary switches. The port-Hamiltonian models of both subsystems are presented and, using a power-preserving interconnection, the Hamiltonian description of the whole system is obtained; detailed bond graphs of all subsystems and the complete system are also provided. Using passivity-based controllers computed in the Hamiltonian formalism for both subsystems, the whole model is simulated; simulations are run to rest the correctness and efficiency of the Hamiltonian network modelling approach used in this work.

Symmetries of the free Schrödinger equation in the non-commutative plane

We study all the symmetries of the free Schr odinger equation in the non-commu- tative plane. These symmetry transformations form an infinite-dimensional Weyl algebra that appears naturally from a two-dimensional Heisenberg algebra generated by Galilean boosts and momenta. These infinite high symmetries could be useful for constructing non-relativistic interacting higher spin theories. A finite-dimensional subalgebra is given by the Schröodinger algebra which, besides the Galilei generators, contains also the dilatation and the expansion. We consider the quantization of the symmetry generators in both the reduced and extended phase spaces, and discuss the relation between both approaches.

Symmetries of the free Schrödinger equation in the non-commutative plane

We study all the symmetries of the free Schrödinger equation in the non-commu- tative plane. These symmetry transformations form an infinite-dimensional Weyl algebra that appears naturally from a two-dimensional Heisenberg algebra generated by Galilean boosts and momenta. These infinite high symmetries could be useful for constructing non-relativistic interacting higher spin theories. A finite-dimensional subalgebra is given by the Schröodinger algebra which, besides the Galilei generators, contains also the dilatation and the expansion. We consider the quantization of the symmetry generators in both the reduced and extended phase spaces, and discuss the relation between both approaches.

Steady-state global optimization of metabolic non-linear dynamic models through recasting into power-law canonical models

Background: Design of newly engineered microbial strains for biotechnological purposes would greatly benefit from the development of realistic mathematical models for the processes to be optimized. Such models can then be analyzed and, with the development and application of appropriate optimization techniques, one could identify the modifications that need to be made to the organism in order to achieve the desired biotechnological goal. As appropriate models to perform such an analysis are necessarily non-linear and typically non-convex, finding their global optimum is a challenging task. Canonical modeling techniques, such as Generalized Mass Action (GMA) models based on the power-law formalism, offer a possible solution to this problem because they have a mathematical structure that enables the development of specific algorithms for global optimization. Results: Based on the GMA canonical representation, we have developed in previous works a highly efficient optimization algorithm and a set of related strategies for understanding the evolution of adaptive responses in cellular metabolism. Here, we explore the possibility of recasting kinetic non-linear models into an equivalent GMA model, so that global optimization on the recast GMA model can be performed. With this technique, optimization is greatly facilitated and the results are transposable to the original non-linear problem. This procedure is straightforward for a particular class of non-linear models known as Saturable and Cooperative (SC) models that extend the power-law formalism to deal with saturation and cooperativity. Conclusions: Our results show that recasting non-linear kinetic models into GMA models is indeed an appropriate strategy that helps overcoming some of the numerical difficulties that arise during the global optimization task.

Identifying the preferred subset of enzymatic profiles in nonlinear kinetic metabolic models via multiobjective global optimization and pareto filters

Optimization models in metabolic engineering and systems biology focus typically on optimizing a unique criterion, usually the synthesis rate of a metabolite of interest or the rate of growth. Connectivity and non-linear regulatory effects, however, make it necessary to consider multiple objectives in order to identify useful strategies that balance out different metabolic issues. This is a fundamental aspect, as optimization of maximum yield in a given condition may involve unrealistic values in other key processes. Due to the difficulties associated with detailed non-linear models, analysis using stoichiometric descriptions and linear optimization methods have become rather popular in systems biology. However, despite being useful, these approaches fail in capturing the intrinsic nonlinear nature of the underlying metabolic systems and the regulatory signals involved. Targeting more complex biological systems requires the application of global optimization methods to non-linear representations. In this work we address the multi-objective global optimization of metabolic networks that are described by a special class of models based on the power-law formalism: the generalized mass action (GMA) representation. Our goal is to develop global optimization methods capable of efficiently dealing with several biological criteria simultaneously. In order to overcome the numerical difficulties of dealing with multiple criteria in the optimization, we propose a heuristic approach based on the epsilon constraint method that reduces the computational burden of generating a set of Pareto optimal alternatives, each achieving a unique combination of objectives values. To facilitate the post-optimal analysis of these solutions and narrow down their number prior to being tested in the laboratory, we explore the use of Pareto filters that identify the preferred subset of enzymatic profiles. We demonstrate the usefulness of our approach by means of a case study that optimizes the ethanol production in the fermentation of Saccharomyces cerevisiae.

Complexation to macromolecules with a large number of sites

This paper presents an approach based on the saddle-point approximation to study the equilibrium interactions between small molecules and macromolecules with a large number of sites. For this case, the application of the Darwin–Fowler method results in very simple expressions for the stoichiometric equilibrium constants and their corresponding free energies in terms of integrals of the binding curve plus a correction term which depends on the first derivatives of the binding curve in the points corresponding to an integer value of the mean occupation number. These expressions are simplified when the number of sites tends to infinity, providing an interpretation of the binding curve in terms of the stoichiometric stability constants. The formalism presented is applied to some simple complexation models, obtaining good values for the free energies involved. When heterogeneous complexation is assumed, simple expressions are obtained to relate the macroscopic description of the binding, given by the stoichiomeric constants, with the microscopic description in terms of the intrinsic stability constants or the affinity spectrum. © 1999 American Institute of Physics.

Ehrenfest dynamics is purity non-preserving: a necessary ingredient for decoherence

We discuss the evolution of purity in mixed quantum/classical approaches to electronic nonadiabatic dynamics in the context of the Ehrenfest model. As it is impossible to exactly determine initial conditions for a realistic system, we choose to work in the statistical Ehrenfest formalism that we introduced in Alonso et al. [J. Phys. A: Math. Theor. 44, 396004 (2011)10.1088/1751-8113/44/39/395004]. From it, we develop a new framework to determine exactly the change in the purity of the quantum subsystem along with the evolution of a statistical Ehrenfest system. In a simple case, we verify how and to which extent Ehrenfest statistical dynamics makes a system with more than one classical trajectory, and an initial quantum pure state become a quantum mixed one. We prove this numerically showing how the evolution of purity depends on time, on the dimension of the quantum state space D, and on the number of classical trajectories N of the initial distribution. The results in this work open new perspectives for studying decoherence with Ehrenfest dynamics.

Moment Operators of Observables in Quantum Mechanics, with Applications to Quantization and Homodyne Detection

The questions studied in this thesis are centered around the moment operators of a quantum observable, the latter being represented by a normalized positive operator measure. The moment operators of an observable are physically relevant, in the sense that these operators give, as averages, the moments of the outcome statistics for the measurement of the observable. The main questions under consideration in this work arise from the fact that, unlike a projection valued observable of the von Neumann formulation, a general positive operator measure cannot be characterized by its first moment operator. The possibility of characterizing certain observables by also involving higher moment operators is investigated and utilized in three different cases: a characterization of projection valued measures among all the observables is given, a quantization scheme for unbounded classical variables using translation covariant phase space operator measures is presented, and, finally, a mathematically rigorous description is obtained for the measurements of rotated quadratures and phase space observables via the high amplitude limit in the balanced homodyne and eight-port homodyne detectors, respectively. In addition, the structure of the covariant phase space operator measures, which is essential for the above quantization, is analyzed in detail in the context of a (not necessarily unimodular) locally compact group as the phase space.

Incompressibility of neutron-rich matter

The saturation properties of neutron-rich matter are investigated in a relativistic mean-field formalism using two accurately calibrated models: NL3 and FSUGold. The saturation properties density, binding energy per nucleon, and incompressibility coefficient are calculated as a function of the neutron-proton asymmetry α≡(N-Z)/A to all orders in α. Good agreement (at the 10% level or better) is found between these numerical calculations and analytic expansions that are given in terms of a handful of bulk parameters determined at saturation density. Using insights developed from the analytic approach and a general expression for the incompressibility coefficient of infinite neutron-rich matter, i.e., K0(α)=K0+Kτα2+ , we construct a hybrid model with values for K0 and Kτ as suggested by recent experimental findings. Whereas the hybrid model provides a better description of the measured distribution of isoscalar monopole strength in the Sn isotopes relative to both NL3 and FSUGold, it significantly underestimates the distribution of strength in 208Pb. Thus, we conclude that the incompressibility coefficient of neutron-rich matter remains an important open problem.