jeudi 19 octobre , 14h en salle Minquiers
Structural Analysis of Multi-Mode DAE Systems
Differential Algebraic Equation (DAE) systems constitute the mathematical model supporting physical modeling languages such as Modelica, VHDL-AMS, or Simscape. Unlike ODEs, they exhibit subtle issues because of their implicit latent equations and related differentiation index. Multi-mode DAE (mDAE) systems are much harder to deal with, not only because of their mode-dependent dynamics, but essentially because of the events and resets occurring at mode transitions. Unfortunately, the large literature devoted to the numerical analysis of DAEs does not cover the multi-mode case. It typically says nothing about mode changes. This lack of foundations cause numerous difficulties to the existing modeling tools. Some models are well handled, others are not, with no clear boundary between the two classes. In this talk we develop a comprehensive mathematical approach to the structural analysis of mDAE systems which properly extends the usual analysis of DAE systems. We define a constructive semantics based on nonstandard analysis and show how to produce execution schemes in a systematic way.
Exposés des semaines suivantes
jeudi 2 Novembre, 14h en salle Aurigny
jeudi 9 Novembre, 14h en salle Aurigny
Euler’s method applied to the control of switched systems
Hybrid systems are a powerful formalism for modeling and reasoning about cyber-physical systems. They mix the continuous and discrete natures of the evolution of computerized systems. Switched systems are a special kind of hybrid systems, with restricted discrete behaviours: those systems only have finitely many different modes of (continuous) evolution, with isolated switches between modes. Such systems provide a good balance between expressiveness and controllability, and are thus in widespread use in large branches of industry such as power electronics and automotive control. The control law for a switched system defines the way of selecting the modes during the run of the system. Controllability is the problem of (automatically) synthezing a control law in order to satisfy a desired property, such as safety (maintaining the variables within a given zone) or stabilisation (confinement of the variables in a close neighborhood around an objective point).In order to compute the control of a switched system, we need to compute the solutions of the differential equations governing the modes. Euler’s method is the most basic technique for approximating such solutions. We present here an estimation of the Euler’s method local error, using the notion of “one-sided Lispchitz constant’’ for modes. This yields a general synthesis approach which can encompass uncertain parameters, local information and stochastic noise. We will sketch out how the approach relates with other symbolic methods based on interval arithmetic and Lyapunov functions. We will also present some applicative examples which illustrate its advantages and limitations.
jeudi 14 Décembre, 14h en salle Aurigny
Shuvra Bhattacharyya, University of Maryland