Robust Simulation for Hybrid Systems: Chattering Path Avoidance

The sliding mode approach is recognized as an efficient tool for treating the chattering behavior in hybrid systems. However, the amplitude of chattering, by its nature, is proportional to magnitude of discontinuous control. A possible scenario is that the solution trajectories may successively enter and exit as well as slide on switching mani-folds of different dimensions. Naturally, this arises in dynamical systems and control applications whenever there are multiple discontinuous control variables. The main contribution of this paper is to provide a robust computational framework for the most general way to extend a flow map on the intersection of p intersected (n--1)-dimensional switching manifolds in at least p dimensions. We explore a new formulation to which we can define unique solutions for such particular behavior in hybrid systems and investigate its efficient computation/simulation. We illustrate the concepts with examples throughout the paper.Comment: The 56th Conference on Simulation and Modelling (SIMS 56), Oct 2015, Link\"oping, Sweden. 2015, Link\"oping University Pres

On the Regularization of Chattering Executions in Real Time Simulation of Hybrid Systems

International audienceIn this paper we present a new method to perform the higher order sliding modes analysis of trajectories of hybrid systems with chattering behavior. This method improves our previous work [AC15] as it modifies numerical simulation algorithms to make them compute the higher order terms of the normal unit vectors of the systems dynamics whenever the first order sliding mode theory cannot be applied. Such modification does not affect the generality of our previous contribution in [AC15]. Our algorithm is general enough to handle both chattering on a single (n−1) switching manifold (i.e. chattering between two dynamics) as well as chattering on the intersection of finitely many intersected (n−1) switching manifolds. In this last case, we show by a special hierarchical application of convex combinations, that unique solutions can be found in general cases when the switching function takes the form of finitely many intersecting manifolds so that an efficient numerical treatment of the sliding motion constrained on the entire discontinuity region (including the switching intersection) is guaranteed. Illustrations of the techniques developed in this article are given on representative examples

Microgrid energy management system for smart home using multi-agent system

This paper proposes a multi-agent system for energy management in a microgrid for smart home applications, the microgrid comprises a photovoltaic source, battery energy storage, electrical loads, and an energy management system (EMS) based on smart agents. The microgrid can be connected to the grid or operating in island mode. All distributed sources are implemented using MATLAB/Simulink to simulate a dynamic model of each electrical component. The agent proposed can interact with each other to find the best strategy for energy management using the java agent development framework (JADE) simulator. Furthermore, the proposed agent framework is also validated through a different case study, the efficiency of the proposed approach to schedule local resources and energy management for microgrid is analyzed. The simulation results verify the efficacy of the proposed approach using Simulink/JADE co-simulation

Fuzzy logic-based controller of the bidirectional direct current to direct current converter in microgrid

Microgrids are small-scale power networks that include renewable energy sources, load, energy storage systems, and energy management systems (EMS). Lithium-ion batteries are the most used battery for energy storage in microgrids due to their advantages over other types of batteries. However, to protect the battery from the explosion and to manage to charge and discharge based on state-of-charge (SoC) value, this type of battery requires the use of an energy management system. The main objective of this paper is to propose an intelligent control strategy for energy management in the microgrid to control the charge and discharge of Li-ion batteries to stabilize the system and reduce the cost of electricity due to the high cost of grid electricity. The proposed technique is based on a fuzzy logic controller (FLC) for voltage control. The FLC is based on the measured voltage of the direct current (DC) bus and the fixed reference voltage to generate buck/boost converter signal control. The proposed technique has been simulated and tested using MATLAB/Simulink software which illustrates the tracking of desired power and DC bus voltage regulation. The simulation results confirm that the proposed systems can diminish the deviations of the system's voltage

Microgrid energy management and monitoring systems: A comprehensive review

Microgrid (MG) technologies offer users attractive characteristics such as enhanced power quality, stability, sustainability, and environmentally friendly energy through a control and Energy Management System (EMS). Microgrids are enabled by integrating such distributed energy sources into the utility grid. The microgrid concept is proposed to create a self-contained system composed of distributed energy resources capable of operating in an isolated mode during grid disruptions. With the Internet of Things (IoT) daily technological advancements and updates, intelligent microgrids, the critical components of the future smart grid, are integrating an increasing number of IoT architectures and technologies for applications aimed at developing, controlling, monitoring, and protecting microgrids. Microgrids are composed of various distributed generators (DG), which may include renewable and non-renewable energy sources. As a result, a proper control strategy and monitoring system must guarantee that MG power is transferred efficiently to sensitive loads and the primary grid. This paper evaluates MG control strategies in detail and classifies them according to their level of protection, energy conversion, integration, benefits, and drawbacks. This paper also shows the role of the IoT and monitoring systems for energy management and data analysis in the microgrid. Additionally, this analysis highlights numerous elements, obstacles, and issues regarding the long-term development of MG control technologies in next-generation intelligent grid applications. This paper can be used as a reference for all new microgrid energy management and monitoring research

Simulation Accélérée des Systèmes Hybrides : méthode combinant analyse statique et analyse à l’exécution.

The theme of this dissertation is to deal with Zeno behaviour of hybrid systems from a simulation perspective.Hybrid systems can be defined as dynamical systems in which continuous and discrete dynamics interact with each other. Such systems exist in a large number of technological systems where the physical continuous evolution of the system is combined with embedded control actions. The mathematical models of hybrid systems consist typically of continuous time dynamics usually described by differential equations describing the continuous behaviour of the system, and discrete event dynamics such as finite state machines (synchronous or data-flow programs) that describe the discrete behaviour of the system.However, due to the complex interaction between the continuous and discrete dynamics, designers should pay special attention when modelling hybrid systems. In fact, realistic models of hybrid systems almost always necessitate part of the hybrid system’s physical behaviour to be abstracted by means of “ideal equations” such as reset and conditional constraints that typically lead to discontinuities in physical signals. The term modelling abstraction is thus designated to any mechanism that enables concrete physical behaviour to be “hidden” by considering idealized models. Intuitively, because of such abstraction, the model jumps over instants corresponding to the violation of abstraction mechanisms. Such modelling abstraction mechanism may result in hybrid models that exhibit Zeno behaviour. Formally, we define Zeno behaviour as an infinite sequence of discrete events occurring in a finite amount of time. Basically, the presence of Zeno behaviour indicates that the hybrid system’s model incompletely describes the real physical behaviour of the system being modelled. If we consider the standard semantics of executions of hybrid systems models, the problem can best be described as follows: the physical system keeps evolving continuously beyond a certain point, but the model’s continuous evolution is undefined beyond that point, because of the infinity of the discrete transitions or mode switchings. Such inherent limitation of the hybrid system model makes the solution of the system reaches a (Zeno limit) point in time at which the model is no longer valid. This is due the fact that the modelling abstraction mechanism incompletely describes the complex interaction between the continuous and discrete dynamics of the hybrid system being modelled. That is, Zeno behaviour can be seen as a modelling artifact that can never occur in reality.Analytically, we distinguish between two different types of Zeno behaviour in hybrid systems: i) chattering-Zeno, and ii) genuinely-Zeno. In models that exhibit chattering- Zeno, the system infinitely moves back and forth between modes in a discrete fashion with infinitesimal time spent between the successive mode switchings. Any Zeno behaviour that is not chattering-Zeno can be classified as genuinely-Zeno. In this dissertation we focus on both chattering-Zeno and a particular type of genuinely-Zeno which we call geometric-Zeno. In models that exhibit geometric-Zeno, an accumulation of an infinite number of mode switchings occurs in finite time. Geometric-Zeno behaviour leads the solution of the system to converge to a Zeno limit point according to a geometric series. Roughly speaking, in geometric-Zeno models discrete events occur at an increasingly smaller distance in time, converging against a limit point.Zeno behaviour is highly challenging from a simulation perspective. In fact, although chattering-Zeno and geometric-Zeno are analytically different, the effect of these two types of Zeno during the numerical simulation is similar: the simulation process effectively stalls, halts and terminates with an error message, or becomes numerically incorrect and produces faulty results, as infinitely many discrete transitions would need to be simulated.This dissertation takes the perspective that models of hybrid systems are executable programs written in programming languages having a hybrid system semantics. Basically, defining a proper hybrid semantic model is the first step of developing a simulation framework for hybrid systems. This step is mandatory even before designing the language or the simulator. The development of a hybrid simulation framework typically include the following steps:1. Properly define a hybrid semantic model that can account for the expected properties of the hybrid systems under simulation.2. Design and develop a simulator that could approximate the model dynamics conforming to the defined hybrid semantic model.3. Design a language capable of expressing all models elements and components conforming to the hybrid semantic model. Type-checking must be included in this step to prove statically the semantic validity of the simulated models.4. Design a compiler for the language. The compiler should be capable of performing static checks of models and also rejecting models that are invalid.Many modelling and simulation tools for hybrid systems have been developed in the past years. They can be classified into two categories: those who put special attention on defining models rigorously, such as for instance SpaceEx, Ptolemy (based on the super-dense time semantics in), and Zélus (based on the non-standard semantics); and those who use informal approach for model definition such as Simulink, Modelica language and its associated tools. All these modelling and simulation tools share the same approach of hybrid model execution alternating between continuous evolution and sequences of discrete switchings as defined by the notion of hybrid automata. For all of these tools, the operational semantics of continuous dynamics (differential equations) is not included in the core semantic model: numerical solvers execute the continuous behaviour by advancing time and computing the values of physical continuous variables. None of the above tools have a Zeno-free semantic model. They all rely on analyzing the solver output at each integration time step, with the solver behaviour at the Zeno-limit point being usually unspecified.In this dissertation we focus on the first two steps above. In particular, we focus on proposing a rigorous Zeno-free computational framework for hybrid semantic model design, and how this Zeno-free computational framework can be implemented in the design of hybrid systems simulators.The first part of our contribution is to propose non-standard semantics for Zeno executions of hybrid systems models. This is based on interpreting Zeno executions in a non-standard densely ordered hybrid time domain. The advantages of using non- standard semantics in the analysis of Zeno behaviour is that it allows for solutions of Zeno hybrid models to be well-defined beyond the Zeno limit points, as well as representing the complex interaction between continuous and discrete dynamics in a concrete way:1. The continuous dynamics of the hybrid system is reduced to the recurrence equation that represents the infinite iteration of infinitesimal discrete changes with infinitesimal duration. Therefore, we can handle the hybrid dynamics based only on fully discrete paradigm.2. The representation of dynamics based on non-standard analysis is complete and the exact limit point of discrete change, like chattering-Zeno and geometric-Zeno limit points, can be handled.The second part of our contribution is to propose a rigorous Zeno-free computational framework for hybrid semantic model design and implementation. The key idea in our proposed computational framework is to perform Zeno detection and avoidance by using behavioral analysis of the system, instead of only considering zero-crossings in a hybrid time domain. The behavioral analysis technique we propose for treating Zeno is based on analyzing both types of Zeno systematically. We provide formal conditions on when the simulated models of hybrid systems display chattering-Zeno and geometric-Zeno executions. We also provide methods for carrying solutions beyond Zeno. The issue of existence and uniqueness of solution beyond Zeno is also studied in this dissertation. Our Zeno-free computational framework allows sacrificing the notion of Zeno-freeness as: i) the decision on whether or not the Zeno limiting state is chattering-Zeno or geometric-Zeno is based on formal conditions explicitly defined and provided to the hybrid simulator’s solver, and ii) the correct notion of solution beyond Zeno is well defined and established in our framework.Our approach also supports mixing compile-time transformations of hybrid programs (i.e. generating what is necessary for Zeno detection and avoidance), the decision, in run-time, for the urgent transition from pre-Zeno to post-Zeno state (based on formal conditions for the existence of Zeno), and the computation, in run-time, of the system dynamics beyond Zeno.Examples of hybrid systems with domains, guard sets, vector fields, and reset maps, illustrating the use of the methods proposed in this dissertation, are also provided.Cette thèse de doctorat porte sur la modélisation et la simulation de systèmes hybridescomportant des phénomènes Zénon.Les systèmes hybrides peuvent être définis comme des systèmes dynamiques danslesquels les dynamiques en temps continu et les dynamiques en temps discret inter-agissent les unes avec les autres. De tels systèmes existent dans un grand nombred’applications technologiques où l’évolution de la partie physique du système, le plussouvent modélisée par un système dynamique en temps continu, est combinée avec desactions de contrôle intégrées, modélisée par un système dynamique en temps discret. Lesmodèles mathématiques des systèmes hybrides consistent généralement en dynamiquesde temps continu habituellement décrites par des équations différentielles qui décriventle comportement continu du système, et des dynamiques d’événements discrets telles queles machines à états finis, qui décrivent le comportement discret du système.Cependant, en raison de l’interaction complexe entre les dynamiques continues etles dynamiques discrètes des systèmes hybrides, les concepteurs des systèmes complexestechnologiques devraient accorder une attention particulière lors de la modélisation dessystèmes hybrides. En fait, les modèles réalistes de systèmes hybrides nécessitent presquetoujours l’abstraction d’une partie du comportement physique du système modélisé.Cette abstraction se fait au moyen d’équations idéales telles que la ré-initialisation et lescontraintes conditionnelles qui conduisent généralement à des discontinuités dans les sig-naux physiques du modèle. Le termeabstraction de modélisationest donc désigné pourtout mécanisme qui permet de “cacher” un comportement physique concret en consid-érant des modèles idéalisés. Les modèles ainsi produits peuvent présenter des comporte-ments Zénon, c’est à dire une séquence infinie d’événements discrets se produisant dansun intervalle de temps fini. Fondamentalement, la présence du comportement Zénonindique que le modèle décrive de manière incomplète le comportement physique réel dusystème hybride. Ce comportement peut être considéré donc comme un problème demodélisation.Nous distinguons deux types de comportement Zénon dans les systèmes hybrides:1) chattering-Zénon, et 2) genuinely-Zénon. Dans les modèles qui présentent de com-portement chattering-Zénon, le système évolue de façon infinie entre ses états discrets,selon un alternance de transitions de modes et de dynamiques continues différentes àune fréquence infinie. Tout comportement Zénon qui n’est pas chattering-Zénon peutêtre classifié comme un comportement genuinely-Zénon. Dans cette thèse nous étudions,d’une manière systématique et analytique, le comportement chattering-Zénon et aussi untype particulier de comportement genuinely-Zénon appelé genuinely-Zénon géométrique.Dans les modèles qui présentent de comportement genuinely-Zénon géométrique, une ac-cumulation d’un nombre infini de commutations entre les états discrets — du systèmehybride — se produit en temps fini. Le comportement genuinely-Zénon géométriqueamène la solution du système à converger vers un point limite Zénon selon une sériegéométrique, c’est à dire, dans les modèles qui présentent de comportement genuinely-Zénon géométrique, les événements discrets se produisent à une période de plus en plusfaible, convergeant vers une date limite.La simulation des comportements Zénon — des systèmes hybrides — pose une dif-ficulté majeure: la simulation devient numériquement incorrect; le simulateur devientincapable d’avancer la simulation au delà du point limite Zénon, à cause de l’infinité descommutations discrètes en temps fini.Dans cette thèse, nous considérons les modèles de systèmes hybrides comme des pro-grammes exécutables écrits par des langages de programmation ayant des sémantiqueshybrides. Fondamentalement, la définition d’un modèle sémantique hybride appropriéest la première étape vers un développement d’un environnement propre de simulationpour les systèmes hybrides. Le développement d’un environnement de simulation hybrideconsiste généralement à suivre les étapes suivantes:1. Définir correctement un modèle sémantique hybride, qui peut considérer les pro-priétés de comportement continue/discret des systèmes hybrides.2. Concevoir et développer un simulateur capable à calculer les dynamiques et lessolutions des modèles des systèmes hybrides conformément au modèle sémantiquehybride défini.3. Concevoir et développer un langage de programmation capable d’exprimer tous leséléments et composants des modèles hybrides conformément au modèle sémantiquehybride défini. La vérification de code doit être incluse dans cette étape pourprouver de manière statique la validité sémantique des modèles simulés.4. Concevoir et développer un compilateur pour le langage de programmation développé.Le compilateur devrait être capable d’effectuer une vérification statique efficace desmodèles, et rejeter les modèles invalides.De nombreux outils de modélisation et de simulation pour les systèmes hybrides ont étédéveloppés ces dernières années. Ils peuvent être classés en deux catégories: ceux quiaccordent une attention particulière à une définition rigoureuse des modèles, comme parexemple SpaceEx [44], Ptolemy [27], et Zélus [57], et ceux qui utilisent une approcheinformelle pour la définition des modèles, comme Simulink1, le langage Modelica [55]et ses outils associés. Tous ces outils partagent la même approche de l’exécution dumodèle hybride alternant entre l’évolution continue et les séquences de commutationsdiscrètes similaire à l’approche définie par la notion d’automate hybride [30]. Pour tousces outils, la sémantique opérationnelle de la dynamique continue (équations différen-tielles) n’est pas incluse dans le modèle sémantique: les solveurs numériques exécutentle comportement continu en faisant progresser le temps et en calculant les valeurs desvariables continues physiques. Aucun de ces outils n’a un modèle sémantique qui permetde détecter le comportement Zénon et de l’éliminer. Ils comptent tous sur l’analyse dela sortie du solveur à chaque pas d’intégration numérique, sans aucun moyen de spécifierle comportement du solveur au point limite Zénon.Dans cette thèse, nous proposons une solution à ce problème. En particulier, nousproposons une méthode combinant un analyse statique et un analyse à l’execution pourdétecter et éliminer le comportement Zénon lors de la simulation des modèles des sys-tèmes hybrides. Nous montrons aussi comment notre méthode peut être utilisée dans ledéveloppement d’outils de modélisation et de simulation qui produisent une simulationcorrecte éliminatoire de comportement Zénon.La première partie de notre contribution est destinée à proposer de sémantiquenon-standard pour les exécutions Zénon des automates hybrides. Ceci est basé surl’interprétation des exécutions Zénon dans un domaine de temps hybride non-standard.L’ avantage de l’utilisation de la sémantique non-standard dans l’analyse du comporte-ment Zénon c’est que l’analyse non-standard permet aux solutions des modèles ayantun comportement Zénon d’être bien définies au-delà des points limites Zénon, ainsi quede représenter l’interaction complexe entre les dynamiques continues et les dynamiquesdiscrètes de manière concrète:1. Les dynamiques continues du système hybride sont réduites à des équations récur-rentes qui représentent rigoureusement l’itération infinie des commutations dis-crètes en durée infinitésimale. Par conséquent, nous pouvons gérer les dynamiqueshybrides en appuyant seulement sur un paradigme complètement discret.2. La représentation non-standard des dynamiques hybrides est complète, qui permetde traiter les points limites Zénon.La deuxième partie de notre contribution est attribuée à proposer une techniquede calcul éliminatoire de comportement Zénon. L’idée clé dans notre technique estd’effectuer la détection et l’élimination de comportement Zénon en utilisant l’analysecomportementale du système, au lieu de seulement considérer le nombre des passages àzéro, comme ce qui est le cas désormais dans tous les outils de modélisation et de simu-lation développés pour les systèmes hybrides. La technique d’analyse comportementaleque nous proposons pour le traitement de comportement Zénon est basée sur un anal-yse systématique des deux types de Zénon. Nous proposons des conditions formellespour bien distinguer si les modèles hybrides simulés présentent ou non de comporte-ment Zénon. Nous proposons également des méthodes pour une définition correcte dessolutions au delà des points limites Zénon. La question de l’existence et l’unicité de lasolution au delà du point limite Zénon est également bien étudiée dans cette thèse. Nosméthodes proposées dans cette thèse permettent de sacrifier la notion de Zénon-free: 1)la décision algorithmique est basé sur des conditions formelles explicitement définies etfournies au simulateur hybride, et 2) la notion correcte de solution au delà du pointlimite Zénon est bien définie et établie dans notre technique proposée.Des exemples de systèmes hybrides, illustrant l’utilisation des méthodes proposéesdans cette thèse, sont également présentés

Hybrid Sliding Mode Control of Full-Car Semi-Active Suspension Systems

With the advance in technology in driving vehicles, there is currently more emphasis on developing advanced control systems for better road handling stability and ride comfort. However, one of the challenging problems in the design and implementation of intelligent suspension systems is that there is currently no solution supporting the export of generic suspension models and control components for integration into embedded Electronic Control Units (ECUs). This significantly limits the usage of embedded suspension components in automotive production code software as it requires very high efforts in implementation, manual testing, and fulfilling coding requirements. This paper introduces a new dynamic model of full-car suspension system with semi-active suspension behavior and provides a hybrid sliding mode approach for control of full-car suspension dynamics such that the road handling stability and ride comfort characteristics are ensured. The semi-active suspension model and the hybrid sliding mode controller are implemented as Functional Mock-Up Units (FMUs) conforming to the Functional Mock-Up Interface for embedded systems (eFMI) and are calibrated with a set experimental tests using a 1/5 Soben-car test bench. The methods and prototype implementation proposed in this paper allow both model and controller re-usability and provide a generic way of integrating models and control software into embedded ECUs

Chattering-Free Simulation of Hybrid Dynamical Systems with the Functional Mock-Up Interface 2.0

International audienceThe numerical simulation of non-smooth hybrid systems exhibiting chattering behavior requires high computational costs. In the worst case, the simulation appears to come to a halt, since infinitely many discrete transitions would need to be simulated. In this paper we present an FMI-based framework and prototypical implementation for robust and reliable detection and elimination “On the Fly” of chattering behavior in run-time simulation of non-smooth hybrid systems. The main benefit of the developed framework is that it establishes solvability requirements and theorems for simulating hybrid systems while performing the chattering path avoidance internally in the master algorithm of the interface. This increases the efficiency of the chattering-free simulation as no enumeration of modes is required during the chattering detection and elimination process. The developed FMI-based framework can generate a chattering-free simulation for any generic chattering Functional Mockup Unit (FMU) conforming to the FMI standard v2.0 Specification for model exchange

Non-Standard Analysis for Regularization of Geometric-Zeno Behaviour in Hybrid Systems

Geometric-Zeno behaviour is a highly challenging problem in the analysis (including simulation) of hybrid systems. Geometric-Zeno can be defined as an infinite number of discrete mode switches in a finite time interval. Typically, for hybrid models exhibiting geometric-Zeno, the numerical simulation either halts or produces false results, because an infinite number of discrete events occur in a given simulation time-step. In this paper, we provide formal methods for regularization of geometric-Zeno behaviour by using a non-standard analysis. In particular, we provide formal conditions for the existence of geometric-Zeno in hybrid systems, and we propose methods to allow geometric-Zeno executions to be continued beyond geometric-Zeno limit points. The concepts are illustrated with a case study throughout the paper