On the Exponentiation of Interval Matrices

The numerical computation of the exponentiation of a real matrix has been intensively studied. The main objective of a good numerical method is to deal with round-off errors and computational cost. The situation is more complicated when dealing with interval matrices exponentiation: Indeed, the main problem will now be the dependency loss of the different occurrences of the variables due to interval evaluation, which may lead to so wide enclosures that they are useless. In this paper, the problem of computing a sharp enclosure of the interval matrix exponential is proved to be NP-hard. Then the scaling and squaring method is adapted to interval matrices and shown to drastically reduce the dependency loss w.r.t. the interval evaluation of the Taylor series

Evaluation of the effectiveness of the interval computation method to simulate the dynamic behavior of subdefinite system: application on an active suspension system

International audienceA new design approach based on methods by intervals adapted to the integration of the simulation step at the earliest stage of preliminary design for dynamic systems is proposed in this study. The main idea consists on using the interval computation method to make a simulation by intervals in order to minimize the number of simulations which allow obtaining a set of solutions instead of a single one. These intervals represent the domains of possible values for the design parameters of the subdefinite system. So the parameterized model of the system is solved by interval. This avoids launching n simulations with n values for each design parameter. The proposed method is evaluated by several tests on a scalable numerical example. It has been applied to solve parameterized differential equations of a Macpher-son suspension system and to study its dynamic behavior in its passive and active form. The dynamic model of the active suspension is nonlinear but linearisable. It is transformed into a parameterized state equation by intervals. The solution to this state equation is given in the form of a matrix exponential. Three digital implementations of exponential have been tested to obtain convergent results. Simulations results are presented and discussed

Rigorous numerics in floquet theory: Computing stable and unstable bundles of periodic orbits

In this paper, a rigorous method to compute Floquet normal forms of fundamental matrix solutions of nonautonomous linear differential equations with periodic coefficients is introduced. The Floquet normal form of a fundamental matrix solution F(t) is a canonical decomposition of the form F(t) = Q(t)eRt, where Q(t) is a real periodic matrix and R is a constant matrix. To rigorously compute the Floquet normal form, the idea is to use the regularity of Q(t) and to simultaneously solve for R and Q(t) with the contraction mapping theorem in a Banach space of rapidly decaying coefficients. The explicit knowledge of R and Q can then be used to construct, in a rigorous computer-assisted way, stable and unstable bundles of periodic orbits of vector fields. The new proposed method does not require rigorous numerical integration of the ODE

Imprecise Continuous-Time Markov Chains

Continuous-time Markov chains are mathematical models that are used to describe the state-evolution of dynamical systems under stochastic uncertainty, and have found widespread applications in various fields. In order to make these models computationally tractable, they rely on a number of assumptions that may not be realistic for the domain of application; in particular, the ability to provide exact numerical parameter assessments, and the applicability of time-homogeneity and the eponymous Markov property. In this work, we extend these models to imprecise continuous-time Markov chains (ICTMC's), which are a robust generalisation that relaxes these assumptions while remaining computationally tractable. More technically, an ICTMC is a set of "precise" continuous-time finite-state stochastic processes, and rather than computing expected values of functions, we seek to compute lower expectations, which are tight lower bounds on the expectations that correspond to such a set of "precise" models. Note that, in contrast to e.g. Bayesian methods, all the elements of such a set are treated on equal grounds; we do not consider a distribution over this set. The first part of this paper develops a formalism for describing continuous-time finite-state stochastic processes that does not require the aforementioned simplifying assumptions. Next, this formalism is used to characterise ICTMC's and to investigate their properties. The concept of lower expectation is then given an alternative operator-theoretic characterisation, by means of a lower transition operator, and the properties of this operator are investigated as well. Finally, we use this lower transition operator to derive tractable algorithms (with polynomial runtime complexity w.r.t. the maximum numerical error) for computing the lower expectation of functions that depend on the state at any finite number of time points

Métodos lineares-quadráticos para sistemas intervalares : aplicações em controle amostrado

Orientador: Matheus SouzaDissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Elétrica e de ComputaçãoResumo: Robustez é uma das muitas propriedades que um sistema de controle deve apresentar, juntamente com desempenho. Neste trabalho, o objetivo é controlar um sistema linear com incertezas intervalares usando um sinal de controle amostrado a fim de minimizar um critério de desempenho quadrático. Como o desempenho depende dos parâmetros incertos do sistema, buscamos minimizar um limitante superior para o custo. Para alcançar esse objetivo, adaptamos resultados clássicos existentes na literatura para lidar com este problema de controle robusto. Os principais resultados alcançados nesta dissertação são condições de análise e de projeto de controladores robustos para sistemas intervalares baseadas em desigualdades matriciais lineares (LMIs). Estas condições consideram critérios de desempenho clássicos e foram desenvolvidas para sistemas a tempo contínuo e a tempo discreto, além de sistemas de controle amostrado. As condições desenvolvidas permanecem válidas mesmo para sistemas com incertezas variantes no tempo. Embora nossas condições sejam mais conservadoras quando comparadas à abordagem politópica, precisam apenas de um número constante de restrições e um número polinomial de variáveis de decisão, o que permite resolvê-las de modo mais eficiente mesmo para sistemas grandes ou com muitas entradas incertas. Por fim, apresentamos exemplos numéricos para apontar as principais características dos métodos propostosAbstract: Robustness is one of many properties that a control system should present, together with performance. In this thesis, we aim to control a linear system with interval uncertainties using a sampled control signal in order to minimize a quadratic performance criterion. As the performance depends on the uncertain parameters of the system, we consider minimizing a higher limit for the cost. To achieve this goal, we adapt classical results in the literature to deal with this robust control problem. The main results obtained in this thesis are robust control analysis and design conditions for interval systems based on linear matrix inequalities (LMIs). These conditions consider classic performance indices and were developed for continuous and discrete-time systems, as well as sampled-data systems. The devised conditions remain valid even for time-varying uncertain systems. Although our conditions are more conservative when compared with the polytopic approach, they need only a constant number of constraints and a polynomial number of variables, which allows them to be solved more efficiently even for large systems with many uncertain inputs. Finally, we present numerical examples to point out the main characteristics of the proposed methodsMestradoAutomaçãoMestre em Engenharia ElétricaCAPE

Contribution à la prise en compte d'exigences dynamiques en conception préliminaire de systèmes complexes

Cette thèse traite de problématique de dimensionnement d'un système technique complexe. L'objectif est de proposer et d'outiller un processus de conception selon lequel le dimensionnement statique de l'architecture initiale d'un système satisfait dès le début les exigences statiques et dynamiques sans nécessité de redimensionnement. Ainsi, nous avons proposé une nouvelle démarche de conception dans laquelle la prise en compte des exigences statiques et dynamiques est effectuée de maniéré simultanée et globale dans la phase de conception préliminaire. Cette démarche se base sur les exigences pour déterminer les solutions admissibles et utilise des méthodes de résolution ensemblistes telles que la méthode de calcul par intervalle et la méthode de propagation par contraintes. En effet, les variables de conception sont exprimées par intervalles et les exigences statiques et dynamiques sont implémentées dans un même modèle NCSP. Les exigences dynamiques sont plus difficiles à intégrer. Il s'agit des exigences fonctionnelles du système, de la résonance et des critères de stabilité, de commandabilité et de transmittance. Dans un premier temps, nous avons réussi à intégrer le comportement dynamique d'un système technique sous forme d'équation différentielle ordinaire par intervalles et dans un deuxième temps, nous avons traduit les exigences dynamiques sous forme de contraintes algébriques définies par un ensemble d'équations et inéquations. La solution générée représente les valeurs admissibles des variables de conception satisfaisant simultanément les exigences statiques et dynamiques imposées. Ce couplage entre le dimensionnement statique et dynamique dans l'approche de conception proposée permet d'éviter le sur-dimensionnement puisque les exigences dynamiques interviennent dans le choix des coefficients de sécurité, et d'éviter les boucles de redimensionnement en cas d'échec ce qui permet de gagner en temps de calcul et de réduire le coût de conception. La démarche de conception proposée est validée par application sur le cas de dimensionnement d'un système de suspension active MacPherson.This thesis deals with design problems of a complex technical system. The objective is to find a design process which the static design of the initial architecture of a system meets from the first static and dynamic requirements with no need to resize it. Thus, we propose a new design approach which the consideration of static and dynamic requirements is done simultaneously and globally in the preliminary design phase. This approach is based on the requirements to determine admissible solutions and uses set-based methods such as interval computation and constraint propagation. Indeed, the design variables are expressed by intervals and the static and dynamic requirements are implemented in a NCSP model. The dynamic requirements are more difficult to integrate. They represent the functional requirements of the system, the resonance and stability criteria, controllability and transmittance. On the one hand, we succeed to integrate the dynamic behavior of a technical system in the form of ordinary differential equation by intervals. On the other hand, we formalize the dynamic requirements in the form of algebraic constraints defined by a set of equations and inequalities. The generated solution is the set of acceptable values of design variables satisfying simultaneously static and dynamic requirements. This coupling between the static and dynamic sizing steps in the proposed design approach avoids over- sizing of the system as the dynamic requirements involved in the choice of safety factors. Il also avoid resizing loops in case of failure, which saves significant computation time and reduce the cost of design. The proposed design approach is applied on the sizing of a MacPherson active suspension system.CHATENAY MALABRY-Ecole centrale (920192301) / SudocSudocFranceF