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High Performance RF and Basdband Analog-to-Digital Interface for Multi-standard/Wideband ApplicationsZhang, Heng 2010 December 1900 (has links)
The prevalence of wireless standards and the introduction of dynamic
standards/applications, such as software-defined radio, necessitate the next generation
wireless devices that integrate multiple standards in a single chip-set to support a variety
of services. To reduce the cost and area of such multi-standard handheld devices,
reconfigurability is desirable, and the hardware should be shared/reused as much as
possible. This research proposes several novel circuit topologies that can meet various
specifications with minimum cost, which are suited for multi-standard applications. This
doctoral study has two separate contributions: 1. The low noise amplifier (LNA) for the
RF front-end; and 2. The analog-to-digital converter (ADC).
The first part of this dissertation focuses on LNA noise reduction and linearization
techniques where two novel LNAs are designed, taped out, and measured. The first LNA,
implemented in TSMC (Taiwan Semiconductor Manufacturing Company) 0.35Cm
CMOS (Complementary metal-oxide-semiconductor) process, strategically combined an
inductor connected at the gate of the cascode transistor and the capacitive cross-coupling
to reduce the noise and nonlinearity contributions of the cascode transistors. The proposed technique reduces LNA NF by 0.35 dB at 2.2 GHz and increases its IIP3 and
voltage gain by 2.35 dBm and 2dB respectively, without a compromise on power
consumption. The second LNA, implemented in UMC (United Microelectronics
Corporation) 0.13Cm CMOS process, features a practical linearization technique for
high-frequency wideband applications using an active nonlinear resistor, which obtains a
robust linearity improvement over process and temperature variations. The proposed
linearization method is experimentally demonstrated to improve the IIP3 by 3.5 to 9 dB
over a 2.5–10 GHz frequency range. A comparison of measurement results with the prior
published state-of-art Ultra-Wideband (UWB) LNAs shows that the proposed linearized
UWB LNA achieves excellent linearity with much less power than previously published
works.
The second part of this dissertation developed a reconfigurable ADC for multistandard
receiver and video processors. Typical ADCs are power optimized for only one
operating speed, while a reconfigurable ADC can scale its power at different speeds,
enabling minimal power consumption over a broad range of sampling rates. A novel
ADC architecture is proposed for programming the sampling rate with constant biasing
current and single clock. The ADC was designed and fabricated using UMC 90nm
CMOS process and featured good power scalability and simplified system design. The
programmable speed range covers all the video formats and most of the wireless
communication standards, while achieving comparable Figure-of-Merit with customized
ADCs at each performance node. Since bias current is kept constant, the reconfigurable
ADC is more robust and reliable than the previous published works.
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Inférence doublement robuste en présence de données imputées dans les enquêtesPicard, Frédéric 02 1900 (has links)
L'imputation est souvent utilisée dans les enquêtes pour traiter la non-réponse partielle. Il est bien connu que traiter les
valeurs imputées comme des valeurs observées entraîne une
sous-estimation importante de la variance des estimateurs
ponctuels. Pour remédier à ce problème, plusieurs méthodes
d'estimation de la variance ont été proposées dans la littérature,
dont des méthodes adaptées de rééchantillonnage telles que le
Bootstrap et le Jackknife. Nous définissons le concept de
double-robustesse pour l'estimation ponctuelle et de variance
sous l'approche par modèle de non-réponse et l'approche par modèle
d'imputation. Nous mettons l'emphase sur l'estimation de la
variance à l'aide du Jackknife qui est souvent utilisé dans la
pratique. Nous étudions les propriétés de différents estimateurs
de la variance à l'aide du Jackknife pour l'imputation par la
régression déterministe ainsi qu'aléatoire. Nous nous penchons
d'abord sur le cas de l'échantillon aléatoire simple. Les cas de
l'échantillonnage stratifié et à probabilités inégales seront
aussi étudiés. Une étude de simulation compare plusieurs méthodes
d'estimation de variance à l'aide du Jackknife en terme de biais
et de stabilité relative quand la fraction de sondage n'est pas
négligeable. Finalement, nous établissons la normalité
asymptotique des estimateurs imputés pour l'imputation par
régression déterministe et aléatoire. / Imputation is often used in surveys to treat item nonresponse. It
is well known that treating the imputed values as observed values
may lead to substantial underestimation of the variance of the
point estimators. To overcome the problem, a number of variance
estimation methods have been proposed in the literature, including
appropriate versions of resampling methods such as the jackknife
and the bootstrap. We define the concept of doubly robust point
and variance estimation under the so-called nonresponse and
imputation model approaches. We focus on jackknife variance
estimation, which is widely used in practice. We study the
properties of several jackknife variance estimators under both
deterministic and random regression imputation. We first consider
the case of simple random sampling without replacement. The case
of stratified simple random sampling and unequal probability
sampling is also considered. A limited simulation study compares
various jackknife variance estimators in terms of bias and
relative stability when the sampling fraction is not negligible.
Finally, the asymptotic normality of imputed estimator is
established under both deterministic and random regression
imputation.
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[en] A NEW APPROACH FOR ESTIMATING THE COEFFICIENTS OF SCALABILITY ASSOCIATED WITH NON PARAMETRIC ITEM RESPONSE THEORY / [pt] UMA NOVA ABORDAGEM PARA A ESTIMAÇÃO DOS COEFICIENTES DE ESCALONABILIDADE ASSOCIADOS À TEORIA DE RESPOSTA AO ITEM NÃO PARAMÉTRICAMARCIA SANTOS ANDRADE 10 April 2014 (has links)
[pt] A finalidade desta tese é propor estimadores pontuais para os coeficientes de
escalonabilidade associados à Teoria de Resposta ao Item não Paramétrica
(TRIN), a saber: Hij, Hi e H, e seus respectivos estimadores da variância, baseados
na abordagem da amostragem de populações finitas. Com o objetivo de investigar
empiricamente a qualidade destes estimadores são consideradas as populações de
referência que são formadas pelos alunos que frequentavam o 9° ano do Ensino
Fundamental, na rede pública, em áreas urbanas dos Estados de Roraima e do Rio
de Janeiro, que participaram da Prova Brasil 2007. As respostas obtidas destes
alunos a um conjunto de 10 itens dicotomizados que mensuram o capital
econômico da sua família foram usadas na construção dos coeficientes de
escalonabilidade do Modelo de Homogeneidade Monótona da TRIN. Repetidas
amostras foram selecionadas de cada população de referência empregando dois
planos amostrais: AC1S (amostragem por conglomerados em único estágio) e
AC2-SAEB (com seleção de escolas e turmas, estratificação e sorteio das unidades
do primeiro estágio com probabilidade proporcional a uma medida de tamanho da
escola. A estimação pontual é baseada no Modelo de Superpopulação. Duas
técnicas foram tratadas para a estimação da variância: método do Conglomerado
Primário e Delete - 1 Jackknife. As medidas usuais: vício relativo, erro relativo
médio, intervalo de confiança e efeito do plano amostral são usadas para a
avaliação da qualidade dos estimadores em termos das propriedades de vício e
precisão. O estudo assinala que os estimadores pontuais apresentam boas
propriedades e, além disto, o estimador da variância corrigido pelo fator de
correção de população finita é o mais apropriado em termos de vício e precisão. O
plano amostral complexo adotado teve impacto na estimação pontual e da
variância dos estimadores dos coeficientes de escalonabilidade. / [en] The purpose of this thesis is to propose estimators for the coefficients of
scalability associated with Non Parametric Item Response Theory (NIRT),
namely: Hij, Hi and H, and their variance estimators, based on the approach of
sampling finite populations. To investigate empirically the quality of these
estimators are considered the reference populations that are formed by students
attending the 9th year of elementary school, in public, in urban areas of the states
of Roraima and Rio de Janeiro, who participated Prova Brasil 2007. The
responses of students to a set of 10 dichotomized items that measure the economic
status of their families were used in the construction of the coefficients of
scalability of the Homogeneity Model. Repeated samples were selected from each
reference population using two sampling plans: AC1S (cluster sampling single
stage) and AC2-SAEB (with selecting schools and classes, stratification and draw
units of the first stage with probability proportional to a measure of school size).
The point estimate is based on the approach of the Model Overpopulation. Two
techniques were treated to estimate the variance: Ultimate Cluster method and
Delete - 1 Jackknife. The usual measures: relative bias, mean relative error,
confidence intervals and effect of the sampling plan is used to assess the quality of
the estimators in terms of the properties of bias and accuracy. The study notes that
the estimators have good properties and, in addition, the estimator of the variance
corrected by the correction factor for finite population is the most appropriate in
terms of accuracy and bias. The complex sampling (AC2-SAEB) impacted the
point estimate and variance of the estimators of the coefficients of scalability.
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Linéarisation des convertisseurs analogique-numérique pour l’amélioration des performances de dynamiques instantanées des numériseurs radioélectriques / Analog-to-digital converter linearization for improving digital radio receiver dynamic rangesMinger, Bryce 18 May 2017 (has links)
Le convertisseur analogique-numérique (ADC), fait fonction d’interface entre les domaines de représentation analogique et numérique des systèmes mixtes de traitement du signal.Il est un élément central en cela que ses performances circonscrivent celles des traitements numériques qui lui succèdent et a fortiori celles de son dispositif hôte. C’est notamment le casdes récepteurs radioélectriques numériques à large bande instantanée. De fait, ces systèmes voient leurs performances de dynamiques instantanées monotonale (DTDR) et bitonale (STDR)– i.e. leur capacité à traiter simultanément des composantes de faible puissance en présence d’une ou plusieurs autres composantes de plus forte puissance – limitées par la linéarité de leur ADC.Ce dernier caractère est quantifié par les performances de dynamique sans raies parasites (SFDR)et distorsion d’intermodulation (IMD) d’un ADC.Les critères de DTDR et de STDR sont essentiels pour les récepteurs radios numériques de guerre électronique conçus pour le traitement des signaux de radiocommunications. En effet, ces dispositifs sont employés à l’établissement de la situation tactique de l’environnement électromagnétique à des fins de support de manoeuvres militaires. La fidélité de la représentation numérique du signal analogique reçu est donc critique. Ainsi, cette thèse vise à étudier la linéarisation des ADC, i.e. l’augmentation des SFDR et IMD, en vue de l’amélioration des dynamiques instantanées de ces récepteurs.Dans ce manuscrit, nous traitons cette problématique selon deux axes différents. Le premier consiste à corriger les distorsions introduites par un ADC au moyen de tables de correspondances(LUT) pré-remplies. À cette fin, nous proposons un algorithme de remplissage de LUT procédant d’une méthode de la littérature par la réduction de moitié du nombre de coefficients à déterminer pour estimer la non-linéarité intégrale (INL) d’un ADC. Sur la base de cette nouvelle méthode,nous développons une approche de correction des non-linéarités dynamiques introduites par un ADC reposant sur une paire de LUT statiques et présentons un exemple d’algorithme permettant de l’opérer. Le second axe du manuscrit repose sur la modélisation comportementale de l’ADC par les séries de Volterra à temps discrets et leurs dérivés. En premier lieu, nous considérons les trois problématiques fondamentales de cette approche de linéarisation : la modélisation ;l’identification de modèle ; et l’inversion de modèle. Puis, nous définissons trois solutions de linéarisation d’ADC aveugles. Enfin, nous analysons l’implémentation sur circuits à réseaux logiques programmables (FPGA) de l’un de ces algorithmes afin d’évaluer la pertinence d’uneopération en temps-réel des échantillons de sortie d’un ADC échantillonnant à une fréquence d’environ 400 MHz. / The analog-to-digital converter (ADC) is a central component of mixed signal systems as the interface between the analog and digital representation spaces. Its performance bounds that of the device it is integrated in. Indeed, ADC linearity is essential for maintaining in the digital space the reliability of its input signal and then that of the information it carries.Wideband digital radio receivers are particularly sensitive to ADC non-linearities. Single-tone and dual-tone dynamic range (respectively STDR and DTDR) of such systems – i.e. the abilityto process simultaneously signal components with high power ratio – are limited by the spurious free dynamic range (SFDR) and intermodulation distortion (IMD) of their internal ADC.DTDR et de STDR are key metrics for electronic warfare wideband digital radio receivers developed for radiocommunication signal processing. As a matter of fact, these equipments are employed for analyzing the tactical situation of the radiofrequency spectrum in order to support military maneuvers. Hence, signal integrity is critical. This thesis deals with the ADC linearization issue in this context. Thus, it aims to study techniques for increasing ADC SFDRand IMD for the purpose of improving dynamic ranges of electronic warfare wideband digitalr eceivers.In this dissertation, the problematic of ADC linearization is approached in two different ways.On the one hand, we consider distortion compensation using pre-filled look-up tables (LUT). Wepropose an algorithm for filling LUTs that stems from an existing method by halving the numberof coefficients required for the integral non-linearity (INL) estimation. Then, based on this new method, we develop an approach for correcting ADC dynamic non-linearities using a couple ofstatic LUTs and we present an example of algorithm for operating this method. On the other hand,we study linearization solutions that rely on behavioural modelling of ADCs using discrete-time Volterra series and its derivatives. First, we address the three fundamental issues of this approach:modelling ; model identification ; and model inversion. Then, we propose three blind linearization algorithms. Finally, we consider the implementation on field programmable gate array (FPGA) of one of them for the purpose of evaluating the relevance of real-time linearization of an ADC sampling at about 400 MHz.
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Analysis of Bolted Connections in Creo Simulate - Theory, Software Functionality and Application Examples / Analyse von Schraubenverbindungen mit Creo Simulate - Theorie, Softwarefunktionalität und AnwendungsbeispieleJakel, Roland 25 June 2013 (has links) (PDF)
Die Präsentation stellt kurz die Grundlagen der Berechnung von Schraubenverbindungen in Anlehnung an die VDI-Richtlinie 2230 Teil 1 dar. Auch die vier FEM-Modellklassen, die die VDI 2230 Teil 2 (Entwurf) zur Berechnung von Mehrschraubenverbindungen vorschlägt, werden behandelt, und die in Creo Simulate vorhandenen Softwarefeatures zu deren Umsetzung vorgestellt.
Es folgt eine Darstellung, was bei der Linearisierung von Schraubenverbindungen zur vereinfachten Berechnung zu beachten ist, und wieso bei der Berechnung im FEM-System dann nicht notwendigerweise eine Vorspannung benötigt wird.
Ausführlich wird das neue Schraubenfeature in Creo Simulate betrachtet, das eine weitgehend automatisierte Modellierung und Berechnung von Standardverschraubungen erlaubt. Weitere Features, wie die neuen Vorspannelemente, werden erläutert, sowie auch die Grenzen der Software aufgezeigt.
Abschließend werden zwei anspruchsvolle Anwendungsbeispiele vorgestellt: Eine zentrisch belastete Verschraubung mit Berücksichtigung von Elasto-Plastizität und einer komplexen Lasthistorie (Anziehen durch Anzugsmoment, Setzeffekte, Entfall des Torsionsmomentes durch das Anziehen, Betriebskraft) sowie eine exzentrisch belastete Verschraubung, die wegen eines relativ dünnen Flansches starke Biegezusatzbeanspruchungen erfährt. / The presentation shows the foundations of bolt analysis according to VDI-guideline 2230 part 1. In addition, the four FEM model classes proposed in VDI 2230 part 2 (draft) are described, as well as the features available in Creo Simulate to realize these model classes.
Next, the presentation shows the requirements for linearizing bolted connections, and why in a FEM analysis with a linearized connection no preload is necessary.
The new fastener feature introduced in Creo Simulate is explained in detail. This feature allows the automated modeling and analysis of bolted connections having standard geometry. Further software features, like pretension elements, as well as the current software limitations are shown.
Finally, two advanced application examples are shown: A centrically loaded bolted connection taking into account elasto-plasticity and a complex load history (tightening torque, embedding, removal of tightening stress, operational load), and an eccentrically loaded flange connection, which is subjected to high additional bending loads because the flange is relatively thin.
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鋼製橋脚ー基礎ー地盤連成系の大地震時挙動Usami, Tsutomu, 葛西, 昭, Kasai, Akira, 河村, 康文, Kawamura, Yasufumi, 宇佐美, 勉 03 1900 (has links)
No description available.
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Novel Sub-Optimal And Particle Filtering Strategies For Identification Of Nonlinear Structural Dynamical SystemsGhosh, Shuvajyoti 01 1900 (has links)
Development of dynamic state estimation techniques and their applications in problems of identification in structural engineering have been taken up. The thrust of the study has been the identification of structural systems that exhibit nonlinear behavior, mainly in the form of constitutive and geometric nonlinearities. Methods encompassing both linearization based strategies and those involving nonlinear filtering have been explored.
The applications of derivative-free locally transversal linearization (LTL) and multi-step transversal linearization (MTrL) schemes for developing newer forms of the extended Kalman filter (EKF) algorithm have been explored. Apart from the inherent advantages of these methods in avoiding gradient calculations, the study also demonstrates their superior numerical accuracy and considerably less sensitivity to the choice of step sizes. The range of numerical illustrations covers SDOF as well as MDOF oscillators with time-invariant parameters and those with discontinuous temporal variations.
A new form of the sequential importance sampling (SIS) filter is developed which explores the scope of the existing SIS filters to cover nonlinear measurement equations and more general forms of noise involving multiplicative and (or) Gaussian/ non-Gaussian noises. The formulation of this method involves Ito-Taylor’s expansions of the nonlinear functions in the measurement equation and the development of the ideal ispdf while accounting for the non-Gaussian terms appearing in the governing equation. Numerical illustrations on parameter identification of a few nonlinear oscillators and a geometrically nonlinear Euler–Bernoulli beam reveal a remarkably improved performance of the proposed methods over one of the best known algorithms, i.e. the unscented particle filter.
The study demonstrates the applicability of diverse range of mathematical tools including Magnus’ functional expansions, theory of SDE-s, Ito-Taylor’s expansions and simulation and characterization of the non-Gaussian random variables to the problem of nonlinear structural system identification.
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Études combinatoires sur les permutations et partitions d'ensemble / Combinatorial studies on set partitions and permutationsKasraoui, Anisse 12 March 2009 (has links)
Cette thèse regroupe plusieurs travaux de combinatoire énumérative sur les permutations et permutations d'ensemble. Elle comporte 4 parties.Dans la première partie, nous répondons aux conjectures de Steingrimsson sur les partitions ordonnées d'ensemble. Plus précisément, nous montrons que les statistiques de Steingrimsson sur les partitions ordonnées d'ensemble ont la distribution euler-mahonienne. Dans la deuxième partie, nous introduisons et étudions une nouvelle classe de statistiques sur les mots : les statistiques "maj-inv". Ces dernières sont des interpolations graphiques des célèbres statistiques "indice majeur" et "nombre d'inversions". Dans la troisième partie, nous montrons que la distribution conjointe des statistiques"nombre de croisements" et "nombre d'imbrications" sur les partitions d'ensemble est symétrique. Nous étendrons aussi ce dernier résultat dans le cadre beaucoup plus large des 01-remplissages de "polyominoes lunaires".La quatrième et dernière partie est consacrée à l'étude combinatoire des q-polynômes de Laguerre d'Al-Salam-Chihara. Nous donnerons une interprétation combinatoire de la suite de moments et des coefficients de linéarisations de ces polynômes. / This thesis consists of four chapters, each on a different topic in enumerative combinatorics, all related in some way to the enumeration of permutations or set partitions. In the first chapter, we prove and generalize Steingrimsson's conjectures on Euler-Mahonian statistics on ordered set partitions. In the second chapter, we introduce and study a new class of statistics on words: the "maj-inv" statistics. These are graphical interpolation of the well-known "major index" and "inversion number".In the third chapter, we show that the joint distribution of the numbers of crossings and nestings on set partitions is symmetric. We also put this result in the larger context of enumeration of increasing and decreasing chains in 01-fillings of moon polyominoes.In the last chapter, we decribe various aspects of the Al-Salam-Chihara q-Laguerre polynomials. These include combinatorial descriptions of the polynomials, the moments, the orthogonality relation and a combinatorial interpretation of the linearization coefficients.
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Inférence doublement robuste en présence de données imputées dans les enquêtesPicard, Frédéric 02 1900 (has links)
L'imputation est souvent utilisée dans les enquêtes pour traiter la non-réponse partielle. Il est bien connu que traiter les
valeurs imputées comme des valeurs observées entraîne une
sous-estimation importante de la variance des estimateurs
ponctuels. Pour remédier à ce problème, plusieurs méthodes
d'estimation de la variance ont été proposées dans la littérature,
dont des méthodes adaptées de rééchantillonnage telles que le
Bootstrap et le Jackknife. Nous définissons le concept de
double-robustesse pour l'estimation ponctuelle et de variance
sous l'approche par modèle de non-réponse et l'approche par modèle
d'imputation. Nous mettons l'emphase sur l'estimation de la
variance à l'aide du Jackknife qui est souvent utilisé dans la
pratique. Nous étudions les propriétés de différents estimateurs
de la variance à l'aide du Jackknife pour l'imputation par la
régression déterministe ainsi qu'aléatoire. Nous nous penchons
d'abord sur le cas de l'échantillon aléatoire simple. Les cas de
l'échantillonnage stratifié et à probabilités inégales seront
aussi étudiés. Une étude de simulation compare plusieurs méthodes
d'estimation de variance à l'aide du Jackknife en terme de biais
et de stabilité relative quand la fraction de sondage n'est pas
négligeable. Finalement, nous établissons la normalité
asymptotique des estimateurs imputés pour l'imputation par
régression déterministe et aléatoire. / Imputation is often used in surveys to treat item nonresponse. It
is well known that treating the imputed values as observed values
may lead to substantial underestimation of the variance of the
point estimators. To overcome the problem, a number of variance
estimation methods have been proposed in the literature, including
appropriate versions of resampling methods such as the jackknife
and the bootstrap. We define the concept of doubly robust point
and variance estimation under the so-called nonresponse and
imputation model approaches. We focus on jackknife variance
estimation, which is widely used in practice. We study the
properties of several jackknife variance estimators under both
deterministic and random regression imputation. We first consider
the case of simple random sampling without replacement. The case
of stratified simple random sampling and unequal probability
sampling is also considered. A limited simulation study compares
various jackknife variance estimators in terms of bias and
relative stability when the sampling fraction is not negligible.
Finally, the asymptotic normality of imputed estimator is
established under both deterministic and random regression
imputation.
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Simulation of Piecewise Smooth Differential Algebraic Equations with Application to Gas NetworksStreubel, Tom 10 June 2022 (has links)
Zuweilen wird gefördertes Erdgas als eine Brückentechnologie noch eine Weile erhalten bleiben, aber unsere Gasnetzinfrastruktur hat auch in einer Ära post-fossiler Brennstoffe eine Zukunft, um Klima-neutral erzeugtes Methan, Ammoniak oder Wasserstoff zu transportieren.
Damit die Dispatcher der Zukunft, in einer sich fortwährend dynamisierenden Marktsituation, mit sich beständig wechselnden Kleinstanbietern, auch weiterhin einen sicheren Gasnetzbetrieb ermöglichen und garantieren können, werden sie auf moderne, schnelle Simulations- sowie performante Optimierungstechnologie angewiesen sein. Der Schlüssel dazu liegt in einem besseren Verständnis zur numerischen Behandlung nicht differenzierbarer Funktionen und diese Arbeit möchte einen Beitrag hierzu leisten.
Wir werden stückweise differenzierbare Funktionen in sog. Abs-Normalen Form betrachten.
Durch einen Prozess, der Abs-Linearisierung genannt wird, können wir stückweise lineare Approximationsmodelle erster Ordnung, mittels Techniken der algorithmischen Differentiation erzeugen.
Jene Modelle können über Matrizen und Vektoren mittels gängiger Software-Bibliotheken der numerischen linearen Algebra auf Computersystemen ausgedrückt, gespeichert und behandelt werden.
Über die Generalisierung der Formel von Faà di Bruno können auch Splinefunktionen höherer Ordnung generiert werden, was wiederum zu Annäherungsmodellen mit besserer Güte führt.
Darauf aufbauend lassen sich gemischte Taylor-Kollokationsmethoden, darunter die mit Ordnung zwei konvergente generalisierte Trapezmethode, zur Integration von Gasnetzen, in Form von nicht glatten Algebro-Differentialgleichungssystemen, definieren.
Numerische Experimente demonstrieren das Potential.
Da solche implizite Integratoren auch nicht lineare und in unserem Falle zugleich auch stückweise differenzierbare Gleichungssysteme erzeugen, die es als Unterproblem zu lösen gilt, werden wir uns auch die stückweise differenzierbare, sowie die stückweise lineare Newtonmethode betrachten. / As of yet natural gas will remain as a bridging technology, but our gas grid infrastructure does have a future in a post-fossil fuel era for the transportation of carbon-free produced methane, ammonia or hydrogen.
In order for future dispatchers to continue to enable and guarantee safe gas network operations in a continuously changing market situation with constantly switching micro-suppliers, they will be dependent on modern, fast simulation as well as high-performant optimization technology. The key to such a technology resides in a better understanding of the numerical treatment of non-differentiable functions and this work aims to contribute here.
We will consider piecewise differentiable functions in so-called abs-normal form.
Through a process called abs-linearization, we can generate piecewise linear approximation models of order one, using techniques of algorithmic differentiation.
Those models can be expressed, stored and treated numerically as matrices and vectors via common software libraries of numerical linear algebra.
Generalizing the Faà di Bruno's formula yields higher order spline functions, which in turn leads to even higher order approximation models.
Based on this, mixed Taylor-Collocation methods, including the generalized trapezoidal method converging with an order of two, can be defined for the integration of gas networks represented in terms of non-smooth system of differential algebraic equations.
Numerical experiments will demonstrate the potential.
Since those implicit integrators do generate non-linear and, in our case, piecewise differentiable systems of equations as sub-problems, it will be necessary to consider the piecewise differentiable, as well as the piecewise linear Newton method in advance.
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