This master thesis treats the modelling of a common-rail direct fuel injection system where pressure generation is decoupled from the injection process. It has been shown that the fuel pressure plays a vital role for the general performance of the engine, affecting both emissions and efficiency, and it is carefully regulated to achieve optimal performance at different operating points. In an attempt to facilitate the development of the responsible control algorithms, a simulation framework has been requested. A model describing the complete work cycle of the high-pressure fuel system is developed and implemented in a Simulink environment. It is to a large extent based on the underlying physics and constructed in a modular manner, which allows for different engine configurations to be simulated. The modelled pressure signal is compared to experimental data at different operating points with promising results in capturing the transient behaviour from a low-level perspective. Additionally, it manages to replicate some of the pressure oscillations which has been observed in the real system and it shows good response to changes in the input signals. However, there are some areas which are subject to improvement since capturing the static pressure levels over longer drive cycles has proved to be a difficult task. Overall, the developed model serves as a starting point for future development and validation of control algorithms.

Cette thèse vise à concevoir des estimateurs non-asymptotiques et robustes pour les systèmes linéaires d’ordre fractionnaire dans un environnement bruité. Elle traite une classe des systèmes linéaires d’ordre fractionnaire modélisée par la dite pseudo représentation d’état avec des conditions initiales inconnues. Elle suppose également que les systèmes étudiés ici peuvent être transformés sous la forme canonique de Brunovsky. Pour estimer le pseudo-état, la forme précédente est transformée en une équation différentielle linéaire d’ordre fractionnaire en prenant en compte les valeurs initiales des dérivées fractionnaires séquentielles de la sortie. Ensuite, en utilisant la méthode des fonctions modulatrices, les valeurs initiales précédentes et les dérivées fractionnaires avec des ordres commensurables de la sortie sont données par des formules algébriques avec des intégrales à l’aide d’une méthode récursive. Ainsi, ces formules sont utilisés pour calculer le pseudo-état dans le cas continu sans bruit. En outre, elle fournit un algorithme pour construire les fonctions modulatrices requises à l’accomplissement de l’estimation. Deuxièmement, inspiré par la méthode des fonctions modulatrices développée pour l’estimation de pseudo-état, cette méthode algébrique basée sur un opérateur est introduite pour estimer la dérivée fractionnée avec un ordre arbitraire fractionnaire de la sortie pour les systèmes considérés. Cet opérateur sert à annuler les valeurs initiales non désirées, puis permet d’estimer la dérivée fractionnaire souhaitée par une nouvelle formule algébrique à l’aide d’une méthode récursive. Troisièmement, l’estimateur du pseudo-état et le différenciateur d’ordre fractionnaire obtenus précédemment sont étudiés respectivement dans le cas discret et bruité. Chacun d’entre eux contient une erreur numérique due à la méthode d’intégration numérique utilisée et au bruit. En particulier, elle fournit une analyse pour diminuer la contribution du bruit au moyen d’une d’erreur bornée qui permet de sélectionner les degrés optimaux des fonctions de modulation à chaque instant. Ensuite, des exemples numériques sont donnés pour mettre en évidence la précision, la robustesse et la propriété non-asymptotique des estimateurs proposés. En outre, les comparaisons avec certaines méthodes existantes et avec un nouvel observateur d’ordre fractionnaire de typeH1sont montrées. Enfin, elle donne des conclusions / This thesis aims to design non-asymptotic and robust estimators for a class of fractional order linear systems in noisy environment. It deals with a class of commensurate fractional order linear systems modeled by the so-called pseudo-state space representation with unknown initial conditions. It also assumed that linear systems under study can be transformed into the Brunovsky’s observable canonical form. Firstly, the pseudo-state of the considered systems is estimated. For this purpose, the Brunovsky’s observable canonical form is transformed into a fractional order linear differential equation involving the initial values of the fractional sequential derivatives of the output. Then, using the modulating functions method, the former initial values and the fractional derivatives with commensurate orders of the output are given by algebraic integral formulae in a recursive way. Thereby, they are used to calculate the pseudo-state in the continuous noise-free case. Moreover, to perform this estimation, it provides an algorithm to build the required modulating functions. Secondly, inspired by the modulating functions method developed for pseudo-state estimation, an operator based algebraic method is introduced to estimate the fractional derivative with an arbitrary fractional order of the output. This operator is applied to cancel the former initial values and then enables to estimate the desired fractional derivative by a new algebraic formula using a recursive way. Thirdly, the pseudo-state estimator and the fractional order differentiator are studied in discrete noisy case. Each of them contains a numerical error due to the used numerical integration method, and the noise error contribution due to a class of stochastic processes. In particular, it provides ananalysis to decrease noise contribution by means of an error bound that enables to select the optimal degrees of the modulating functions at each instant. Then, several numerical examples are given to highlight the accuracy, the robustness and the non-asymptotic property of the proposed estimators. Moreover, the comparisons to some existing methods and a new fractional orderH1-like observer are shown. Finally, conclusions are outlined with some perspectives

Periodically correlated time series generally exist in several fields including hydrology, climatology, economics and finance, and are commonly modelled using periodic autoregressive (PAR) model. For a time series with stochastic periodic trend, for which a unit root is expected, a periodically integrated autoregressive PIAR model with periodic and/or seasonal unit root has been shown to be a satisfactory model. The existing theory used the multivariate methodology to study PIAR models. However, this theory is convoluted, majority of it only developed for quarterly time series and its generalisation to time series with larger number of periods is quite cumbersome. This thesis studies the existing theory and highlights its restrictions and flaws. It provides a coherent presentation of the steps for analysing PAR and PIAR models for different number of periods. It presents the different unit roots representations and compares the performance of different unit root tests available in literature. The restrictions of existing studies gave us the impetus to develop a unified theory that gives a clear understanding of the integration and unit roots in the periodic models. This theory is based on the spectral information of the multi-companion matrix of the periodic models. It is more general than the existing theory, since it can be applied to any number of periods whereas the existing methods are developed for quarterly time series. Using the multi-companion method, we specify and estimate the periodic models without the need to extract complicated restrictions on the model parameters corresponding to the unit roots, as required by NLS method. The multi-companion estimation method performed well and its performance is equivalent to the NLS estimation method that has been used in the literature. Analysing integrated multivariate models is a problematic issue in time series. The multi-companion theory provides a more general approach than the error correction method that is commonly used to analyse such time series. A modified state state representation for the seasonal periodically integrated autoregressive (SPIAR) model with periodic and seasonal unit roots is presented. Also an alternative state space representations from which the state space representations of PAR, PIAR and the seasonal periodic autoregressive (SPAR) models can be directly obtained is proposed. The seasons of the parameters in these representations have been clearly specified, which guarantees correct estimated parameters. Kalman filter have been used to estimate the parameters of these models and better estimation results are obtained when the initial values were estimated rather than when they were given.

Les études menées permettent de prendre en main un système depuis l’identification jusqu’à la commande robuste des systèmes non entiers. Les principes de la platitude permettent de parvenir à la planification de trajectoire à condition de connaître le modèle du système, d’où l’intérêt de l’identification des paramètres du système. Les principaux travaux de cette thèse concernent l’identification de système par modèles non entiers, la génération et la poursuite robuste de trajectoire par l’application des principes de la platitude aux systèmes non entiers.Le chapitre 1 rappelle les définitions et propriétés de l’opérateur non entier ainsi que les diverses méthodes de représentation d’un système non entier. Le théorème de stabilité est également remémoré. Les algèbres sur les polynômes non entiers et sur les matrices polynômiales non entières sont introduites pour l’extension de la platitude aux systèmes non entiers.Le chapitre 2 porte sur l’identification par modèle non entier. Après un état de l’art sur les méthodes d’identification par modèle non entier, deux contextes sont étudiés : en présence de bruit blanc et en présence de bruit coloré. Dans chaque cas, deux estimateurs optimaux (sur la variance et le biais) sont propos´es : l’un, en supposant une structure du modèle connue et d’ordres de dérivation fixés, et l’autre en combinant des techniques de programmation non linéaire qui optimise à la fois les coefficients et les ordres de dérivation.Le chapitre 3 établit l’extension des principes de la platitude aux systèmes non entiers.La platitude des systèmes non entiers linéaires en proposant différentes approches telles que les fonctions de transfert et la pseudo-représentation d’état par matrices polynômiales est étudiée.La robustesse du suivi de trajectoire est abordée par la commande CRONE. Des exemples de simulations illustrent les développements théoriques de la platitude au travers de la diffusion thermique sur un barreau métallique.Enfin, le chapitre 4 est consacré à la validation des contributions en identification, en planification de trajectoire et en poursuite robuste sur un système non entier réel : un barreau métallique est soumis à un flux de chaleur. / The general theme of the work enables to handle a system, from identification to robust control. Flatness principles tackle path planning unless knowing the system model, hence the system parameter identification necessity. The principal contribution of this thesis deal with system identification by non integer models and with robust path tracking by the use of flatness principles for fractional models.Chapter 1 recalls the definitions and properties of a fractional operator and also the various representation methods of a fractional system. The stability theorem is also brought to mind. Fractional polynomial and fractional polynomial matrice algebras are introduced for the extension of flatness principles for fractional systems.Chapter 2 is about non integer model identification. After a state of the art on system identification by non integer model. Two contexts are considered : in presence of white noise and of colored noise. In each situation, two optimal (in variance and bias sense) estimators are put forward : one, when considering a known model structure with fixed differentiating orders, and another one by combining nonlinear programming technics for the optimization of coefficients and differentiating orders.Chapter 3 establishes the extension of flatness principles to fractional systems. Flatness of linear fractional systems are studied while considering different approaches such as transfer functions or pseudo-state-space representations with polynomial matrices. Path tracking robustness is ensured with CRONE control. Simulation examples display theoretical developments on flatness through thermal diffusion on a metallic rod. Finally, Chapter 4 is devoted to validate the contributions to system identification, to trajectory planning and to robust path tracking on a real fractional system : a metallic rod submitted to a heat flux.

In this thesis, we explore the potential of financing a fast charging system with energy storage by delivering ancillary services from the energy storage in an optimal way. Specifically, a system delivering frequency regulation services FCR-D Up and FCR-D Down in combination with energy arbitrage trading is considered. An optimization model is developed that could be implemented operationally and then used in a Monte-Carlo simulation to estimate the net present value of the system for four identified cases at three different energy market price scenarios. The main modeling approach is to formulate the system as a state-space model serving as the foundation for model predictive control, with the delay between decision and delivery of the frequency regulation services incorporated as a part of the system state. The optimization of the system is implemented using a dynamic programming approach with a time horizon of 48h, where the choice of admissible controls is optimized for computational efficiency. The result shows that the system could profitable under optimal operation, but it is heavily dependent on the size of the grid connection, future price levels for ancillary services, and the nature of fast-charging demand. As such, the business case and profitability should be evaluated with a specific use case in mind. The developed model showed relatively good computational efficiency for operational implementations with a run time for one iteration of the optimization problem of 15 seconds. The model could therefore be used as the foundation for future research within the specific field and for similar control problems considering delayed controls and stochastic demand. Several proposed improvements and suggested areas of future research are proposed. / I den här uppsatsen utforskar vi huruvida det är finansiellt lönsamt att leverera snabbladdning från ett energilager samtidigt som energilagret används för att leverera systemtjänster på ett optimalt sätt. Mer specifikt undersöks ett potentiellt system som levererar frekvensregleringstjänsterna FCR-D Up och FCR-D Down samt energiarbitragehandel. Vi utvecklar en optimeringsmodell som kan implementeras i ett fysiskt system och använder sedan modellen i en Monte-Carlo-simulering för att estimera nuvärdet av fyra olika systemkonfigurationer för tre olika prisscenarion. Den huvudsakliga modelleringsmetoden är att formulera systemet som en tillstånds-rum modell, som sedan används som grund för modellprediktiv styrning, där fördröjningen mellan beslut och leverans av frekvensregleringstjänster inkluderas som en del av systemets tillstånd. Optimeringen av systemet implementeras med en dynamisk programmeringsmetodik med en tidsram på 48 timmar, där valet av tillåtna kontroller optimeras för beräkningseffektivitet. Resultatet visar att systemet kan vara lönsamt under optimal drift, men det är starkt beroende av storleken på nätanslutningen, framtida prisnivåer för systemtjänster och typen av snabbladdningsbehovet. Därför bör lönsamheten utvärderas för varje specifikt fall. Den utvecklade modellen visade relativt god beräkningseffektivitet för praktiskt implementation med en körtid för en enskilt iteration på 15 sekunder. Modellen kan därför användas som grund för framtida forskning inom området och för liknande problem inom optimal styrteori som involverar fördröjda kontroller och stokastisk efterfrågan. Flera föreslagna förbättringar och områden för framtida forskning föreslås.

Les études menées permettent d’étendre la méthodologie de commande H∞ aux modèles décrits par des équations différentielles faisant intervenir des ordres de dérivation non entiers. Deux approches sont proposées. La première consiste à réécrire le modèle non entier comme un modèle entier incertain afin de pouvoir utiliser les méthodes de commande H∞ développées pour les modèles entiers. La seconde approche consiste à développer des conditions LMI spécifiques aux modèles non entiers à partir de leur pseudo représentation d’état. Ces deux approches sont appliquées à l’isolation vibratoire d’un pont. / The general theme of the work enables to extend H∞ control methodology to fractional order models. Two approaches are proposed. The first one consists in rewriting the fractional order model as an uncertain integer order model in order to use existing H∞ control methods for integer order models. The second approach consists in developing specific LMI conditions for fractional order models based on their pseudo state space representation. These two approaches are applied to the vibratory isolation of a bridge.

With the advances in the telecommunications industry, there is a need to solve the in-band full-duplex (IBFD) problem for antenna systems. One premise for solving the IBFD problem is to have strong isolation between transmitter and receiver antennas in an antenna system. To increase isolation, antenna engineers are dependent on simulation software to calculate the isolation between the antennas, i.e., the mutual coupling. Full-wave simulations that accurately calculate the mutual coupling between antennas are timeconsuming, and there is a need to reduce the required time. In this thesis, we investigate how implemented data-driven interpolation methods can be used to reduce the simulation times when applied to frequency domain solvers. Here, we benchmark the four different interpolation methods vector fitting, the Loewner framework, Cauchy interpolation, and a modified version of Nevanlinna-Pick interpolation. These four interpolation methods are benchmarked on seven different antenna frequency responses, to investigate their performance in terms of how many interpolation points they require to reach a certain root mean squared error (RMSE) tolerance. We also benchmark different frequency sampling algorithms together with the interpolation methods. Here, we have predetermined frequency sampling algorithms such as linear frequency sampling distribution, and Chebyshevbased frequency sampling distributions. We also benchmark two kinds of adaptive frequency sampling algorithms. The first type is compatible with all of the four interpolation methods, and it selects the next frequency sample by analyzing the dynamics of the previously generated interpolant. The second adaptive frequency sampling algorithm is solely for the modified NevanlinnaPick interpolation method, and it is based on the free parameter in NevanlinnaPick interpolation. From the benchmark results, two interpolation methods successfully decrease the RMSE as a function of the number of interpolation points used, namely, vector fitting and the Loewner framework. Here, the Loewner framework performs slightly better than vector fitting. The benchmark results also show that vector fitting is less dependent on which frequency sampling algorithm is used, while the Loewner framework is more dependent on the frequency sampling algorithm. For the Loewner framework, Chebyshev-based frequency sampling distributions proved to yield the best performance. / Med de snabba utvecklingarna i telekomindustrin så har det uppstått ett behov av att lösa det så kallad i-band full-duplex (IBFD) problemet. En premiss för att lösa IBFD-problemet är att framgångsrikt isolera transmissionsantennen från mottagarantennen inom ett antennsystem. För att öka isolationen mellan antennerna måste antenningenjörer använda sig av simulationsmjukvara för att beräkna isoleringen (den ömsesidiga kopplingen mellan antennerna). Full-wave-simuleringar som noggrant beräknar den ömsesidga kopplingen är tidskrävande. Det finns därför ett behov av att minska simulationstiderna. I denna avhandling kommer vi att undersöka hur våra implementerade och datadrivna interpoleringsmetoder kan vara till hjälp för att minska de tidskrävande simuleringstiderna, när de används på frekvensdomänslösare. Här prestandajämför vi de fyra interpoleringsmetoderna vector fitting, Loewner ramverket, Cauchy interpolering, och modifierad Nevanlinna-Pick interpolering. Dessa fyra interpoleringsmetoder är prestandajämförda på sju olika antennsystemsvar, med avseende på hur många interpoleringspunkter de behöver för att nå en viss root mean squared error (RMSE)-tolerans. Vi prestandajämför också olika frekvenssamplingsalgoritmer tillsammas med interpoleringsmetoderna. Här använder vi oss av förbestämda frekvenssamplingsdistributioner så som linjär samplingsdistribution och Chebyshevbaserade samplingsdistributioner. Vi använder oss också av två olika sorters adaptiv frekvenssamplingsalgoritmer. Den första sortens adaptiv frekvenssamplingsalgoritm är kompatibel med alla de fyra interpoleringsmetoderna, och den väljer nästa frekvenspunkt genom att analysera den föregående interpolantens dynamik. Den andra adaptiva frekvenssamplingsalgoritmen är enbart till den modifierade Nevanlinna-Pick interpoleringsalgoritmen, och den baserar sitt val av nästa frekvenspunkt genom att använda sig av den fria parametern i Nevanlinna-Pick interpolering. Från resultaten av prestandajämförelsen ser vi att två interpoleringsmetoder framgångsrikt lyckas minska medelvärdetsfelet som en funktion av antalet interpoleringspunkter som används. Dessa två metoder är vector fitting och Loewner ramverket. Här så presterar Loewner ramverket aningen bättre än vad vector fitting gör. Prestandajämförelsen visar också att vector fitting inte är lika beroende av vilken frekvenssamplingsalgoritm som används, medan Loewner ramverket är mer beroende på vilken frekvenssamplingsalgoritm som används. För Loewner ramverket så visade det sig att Chebyshev-baserade frekvenssamplingsalgoritmer presterade bättre.