Event-triggered control for piecewise affine discrete-time systems

In the present work, we study the problems of stability analysis of piecewise-affine (PWA) discrete-time systems, and trigger-function design for discrete-time event-triggered control systems. We propose a representation for piecewise-affine systems in terms of ramp functions, and we rely on Lyapunov theory for the stability analysis. The proposed implicit piecewise-affine representation prevents the shortcomings of the existing stability analysis approaches of PWA systems. Namely, the need to enumerate regions and allowed transitions of the explicit representations. In this context, we can emphasize two benefits of the proposed approach: first, it makes possible the analysis of uncertainty in the partition and, thus, the transitions. Secondly, it enables the analysis of event-triggered control systems for the class of PWA systems since, for ETC, the transitions cannot be determined as a function of the state variables. The proposed representation, on the other hand, implicitly encodes the partition and the transitions. The stability analysis is performed with Lyapunov theory techniques. We then present conditions for exponential stability. Thanks to the implicit representation, the use of piecewise quadratic Lyapunov functions candidates becomes simple. These conditions can be solved numerically using a linear matrix inequality formulation. The numerical analysis exploits quadratic expressions that describe ramp functions to verify the positiveness of extended quadratic forms. For ETC, a piecewise quadratic trigger function defines the event generator. We find suitable parameters for the trigger function with an optimization procedure. As a result, this function uses the information on the partition to reduce the number of events, achieving better results than the standard quadratic trigger functions found in the literature. We provide numerical examples to illustrate the application of the proposed representation and methods.Ce manuscrit présente des résultats sur l’analyse de stabilité des systèmes affines par morceaux en temps discret et sur le projet de fonctions de déclenchement pour des stratégies de commande par événements. Nous proposons une représentation pour des systèmes affines par morceaux et l’on utilise la théorie de stabilité de Lyapunov pour effectuer l’analyse de stabilité globale de l’origine. La nouvelle représentation implicite que nous proposons rend plus simple l’analyse de stabilité car elle évite l’énumération des régions et des transitions entre régions tel que c’est fait dans le cas des représentations explicites. Dans ce contexte nous pouvons souligner deux avantages principaux, à savoir I) la possibilité de traiter des incertitudes dans la partition qui définit le système et, par conséquent des incertitudes dans les transitions, II) l’analyse des stratégies de commande par événements pour des systèmes affines par morceaux. En effet, dans ces stratégies les transitions ne peuvent pas être définies comme des fonctions des variables d’état. La théorie de stabilité de Lyapunov est utilisée pour établir des conditions pour la stabilité exponentielle de l’origine. Grâce à la représentation implicite des partitions nous utilisons des fonctions de Lyapunov quadratique par morceaux. Ces conditions sont données par des inégalités dont la solution numérique est possible avec une formulation par des inégalités matricielles linéaires. Ces formulations numériques se basent sur des expressions quadratiques décrivant des fonctions rampe. Pour des stratégies par événement, une fonctions quadratique par morceaux est utilisée pour le générateur d’événements. Nous calculons les paramètres de ces fonctions de déclenchement a partir de solutions de problèmes d’optimisation. Cette fonction de déclenchement quadratique par morceaux permet de réduire le nombre de d’événementsen comparaison avec les fonctions quadratiques utilisées dans la littérature. Nous utilisons des exemples numériques pour illustrer les méthodes proposées.No presente trabalho, são estudados os problemas de análise de estabilidade de sistemas afins por partes e o projeto da função de disparo para sistemas de controle baseado em eventos em tempo discreto. É proposta uma representação para sistemas afins por partes em termos de funções rampa, e é utilizada a teoria de Lyapunov para a análise de estabilidade. A representação afim por partes implícita proposta evita algumas das deficiências das abordagens de análise de estabilidade de sistemas afins por partes existentes. Em particular, a necessidade de anumerar regiões e transições admissíveis das representações explícitas. Neste contexto, dois benefícios da abordagem proposta podem ser enfatizados: primeiro, ela torna possível a análise de incertezas na partição, e, assim, nas transições. Segundo, ela permite a análise de sistemas de controle baseado em eventos para a classe de sistemas afins por partes, já que, para o controle baseado em eventos, as transições não podem ser determinadas como uma função das variáveis de estado. A representação proposta, por outro lado, codifica implicitamente a partição e as transições. A análise de estabilidade é realizada com técnicas da teoria de Lyapunov. Condi- ções para a estabilidade exponencial são então apresentadas. Graças à representação implícita, o uso de funções candidatas de Lyapunov se torna simples. Essas condições podem ser resolvidas numéricamente usando uma formulação de desigualdades matriciais lineares. A análise numérica explora expressões quadráticas que descrevem funções de rampa para verificar a postivividade de formas quadráticas extendidas. Para o controle baseado em eventos, uma função de disparo quadrática por partes define o gerador de eventos. Parâmetros adequados para a função de disparo sãoencontrados com um procedimento de otimização. Como resultado, esta função usa informação da partição para reduzir o número de eventos, obtendo resultados melhores do que as funções de disparo quadráticas encontradas na literatura. Exemplos numéricos são fornecidos para ilustrar a aplicação da representação e mé- todos propostos

Currency recognition using a smartphone: Comparison between color SIFT and gray scale SIFT algorithms

AbstractBanknote recognition means classifying the currency (coin and paper) to the correct class. In this paper, we developed a dataset for Jordanian currency. After that we applied automatic mobile recognition system using a smartphone on the dataset using scale-invariant feature transform (SIFT) algorithm. This is the first attempt, to the best of the authors knowledge, to recognize both coins and paper banknotes on a smartphone using SIFT algorithm. SIFT has been developed to be the most robust and efficient local invariant feature descriptor. Color provides significant information and important values in the object description process and matching tasks. Many objects cannot be classified correctly without their color features. We compared between two approaches colored local invariant feature descriptor (color SIFT approach) and gray image local invariant feature descriptor (gray SIFT approach). The evaluation results show that the color SIFT approach outperforms the gray SIFT approach in terms of processing time and accuracy

Model-based Calibration of Engine Control Units Using Gaussian Process Regression

Reducing the number of tests on vehicles is one of the most important requirements for increasing cost efficiency in the calibration process of engine control units (ECU). Here, employing virtual vehicles for a model-based calibration of ECUs is essential. Modelling components for virtual vehicles can be a tedious and time-consuming task. In this context, data-based modelling techniques can be an attractive alternative to physical models to increase efficiency in the modelling process. Data-based models can incorporate unknown nonlinearities encoded in the sampled data, resulting in more accurate models in practice. In combination with automated measurement, data-based modelling can help to significantly accelerate the calibration process. Furthermore, the fast simulation speed of the resulting models allows their implementation into real-time simulation environments, such as Hardware-in-the-Loop (HiL) systems, and thus enables a model-based calibration of the related ECU software function. However, generating appropriate data for learning dynamic models, i.e., the transient Design of Experiments (DoE), is not straightforward, since system boundaries and permissible excitation frequencies are not known beforehand. Thus the training data of the system measurement will be inconsistent and the main challenge of the identification process is to deal with this data to achieve a globally valid model. Furthermore, when dealing with dynamic systems in an automotive context, the Engine Control Unit typically changes operating modes while driving. Thus nonlinearities and changes of physical structures appear, which need to be considered in the model. In this thesis, a modelling system called the Local Gaussian Process Regression (LGPR), is used and adapted in order to receive a flexible modelling approach, which allows an iterative modelling process and obtains robust and globally valid dynamic models. The adapted LGPR approach is employed for the ECU calibration of dynamical automotive systems, which is critical regarding system excitation. Using LGPR, it is possible to measure the system iteratively while exploring the relevant state-space regions and improving the quality of the model step by step. The results show that LGPR is beneficial for iterative modelling of dynamical systems. Compared to the traditional Gaussian Process Regression (GPR) modelling approach, LGPR yields better results regarding the variable system dynamics