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

par Ayman Aljarbouh

Projet de thèse en Informatique

Sous la direction de Benoît Caillaud.


  • Résumé

    The theme of this dissertation is to deal with Zeno behavior 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 behavior of the system, and discrete event dynamics such as finite state machines (synchronous or data-flow programs) that describe the discrete behavior of the system. However, due to the complex interaction between the continuous and discrete dynamics, designers should pay special attention when modeling hybrid systems. In fact, realistic models of hybrid systems almost always necessitate part of the hybrid system’s physical behavior to be abstracted by means of “ideal equations” such as reset and conditional constraints that typically lead to discontinuities in physical signals. The term modeling abstraction is thus designated to any mechanism that enables concrete physical behavior 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 modeling abstraction mechanism may result in hybrid models that exhibit Zeno behavior. Formally, we define Zeno behavior as an infinite sequence of discrete events occurring in a finite amount of time. Basically, the presence of Zeno behavior indicates that the hybrid system’s model incompletely describes the real physical behavior of the system being modeled. If we consider the standard semantics1 of executions of hybrid systems models, the problem can best be described as follows: the physical system keeps evolving contentiously 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of the hybrid system model makes the solution of the system reaches a (Zeno limit) point in time at which the model is no longer valid.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 modeling abstraction mechanism incompletely describes the complex interaction between the continuous and discrete dynamics of the hybrid system being modeled. That is, Zeno behavior can be seen as a modeling artifact that never can occur in reality. Analytically, we distinguish between two different types of Zeno behavior 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 behavior 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 it geometric-Zeno. In models that exhibit geometric-Zeno, an accumulation of an infinite number of mode switchings occurs in finite time. Geometric-Zeno behavior 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 behavior 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, terminates with an error message, or becomes numerically inaccurate, 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 con- forming to the defined hybrid semantic model. 3. Design a language capable of expressing all models elements and components con- forming 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 modeling 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), 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 modeling 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 behavior 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 behavior 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 attributed to proposing 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 behavior 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 attributed to proposing 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 past Zeno. The issue of existence and uniqueness of solution past 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 past 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 past 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.

  • Titre traduit

    Accelerated Simulation of Hybrid Systems : method combining static analysis and run-time execution analysis.


  • Résumé

    The theme of this dissertation is to deal with Zeno behavior 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 behavior of the system, and discrete event dynamics such as finite state machines (synchronous or data-flow programs) that describe the discrete behavior of the system. However, due to the complex interaction between the continuous and discrete dynamics, designers should pay special attention when modeling hybrid systems. In fact, realistic models of hybrid systems almost always necessitate part of the hybrid system’s physical behavior to be abstracted by means of “ideal equations” such as reset and conditional constraints that typically lead to discontinuities in physical signals. The term modeling abstraction is thus designated to any mechanism that enables concrete physical behavior 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 modeling abstraction mechanism may result in hybrid models that exhibit Zeno behavior. Formally, we define Zeno behavior as an infinite sequence of discrete events occurring in a finite amount of time. Basically, the presence of Zeno behavior indicates that the hybrid system’s model incompletely describes the real physical behavior of the system being modeled. If we consider the standard semantics1 of executions of hybrid systems models, the problem can best be described as follows: the physical system keeps evolving contentiously 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 modeling abstraction mechanism incompletely describes the complex interaction between the continuous and discrete dynamics of the hybrid system being modeled. That is, Zeno behavior can be seen as a modeling artifact that never can occur in reality. Analytically, we distinguish between two different types of Zeno behavior 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 behavior 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 it geometric-Zeno. In models that exhibit geometric-Zeno, an accumulation of an infinite number of mode switchings occurs in finite time. Geometric-Zeno behavior 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 behavior 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, terminates with an error message, or becomes numerically inaccurate, 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 con- forming to the defined hybrid semantic model. 3. Design a language capable of expressing all models elements and components con- forming 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 modeling 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), 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 modeling 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 behavior 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 behavior 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 attributed to proposing 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 behavior 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 attributed to proposing 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 past Zeno. The issue of existence and uniqueness of solution past 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 past 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 past 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.