Laurent Fribourg

Laurent Fribourg, Bertrand Revol, Romain Soulat

We consider here systems with piecewise linear dynamics that are periodically sampled with a given period τ . At each sampling time, the mode of the system, i.e., the parameters of the linear dynamics, can be switched, according to a switching rule. Such systems can be modelled as a special form of hybrid automata, called "switched systems", that are automata with an infinite real state space. The problem is to find a switching rule that guarantees the system to still be in a given area V at the next sampling time, and so on indefinitely. In this paper, we will consider two approaches: the indirect one that abstracts the system under the form of a finite discrete event system, and the direct one that works on the continuous state space. Our methods rely on previous works, but we specialize them to a simplified context (linearity, periodic switching instants, absence of control input), which is motivated by the features of a focused case study: a DC-DC boost converter built by electronics laboratory SATIE (ENS Cachan). Our enhanced methods allow us to treat successfully this real-life example.

Daniel Martin Xavier, Ludovic Chamoin, Jawher Jerray et al.

The early stopping strategy consists in stopping the training process of a neural network (NN) on a set $S$ of input data before training error is minimal. The advantage is that the NN then retains good generalization properties, i.e. it gives good predictions on data outside $S$, and a good estimate of the statistical error (``population loss'') is obtained. We give here an analytical estimation of the optimal stopping time involving basically the initial training error vector and the eigenvalues of the ``neural tangent kernel''. This yields an upper bound on the population loss which is well-suited to the underparameterized context (where the number of parameters is moderate compared with the number of data). Our method is illustrated on the example of an NN simulating the MPC control of a Van der Pol oscillator.

Étienne André, Emmanuel Coquard, Laurent Fribourg et al.

The next generation of space systems will have to achieve more and more complex missions. In order to master the development cost and duration of such systems, an alternative to a manual design is to automatically synthesize the main parameters of the system. In this paper, we present an approach for the specific case of the scheduling of the flight control of a space launcher. The approach requires two successive steps: (1) the formalization of the problem to be solved in a parametric formal model and (2) the synthesis of the model parameters with a tool. We first describe the problem of the scheduling of a launcher flight control, then we show how this problem can be formalized with parametric stopwatch automata; we then present the results computed by the parametric timed model checker IMITATOR. We enhance our model by taking into consideration the time for switching context, and we compare the results to those obtained by other tools classically used in scheduling.

Étienne André, Emmanuel Coquard, Laurent Fribourg et al.

The next generation of space systems will have to achieve more and more complex missions. In order to master the development cost and duration of such systems, an alternative to a manual design is to automatically synthesize the main parameters of the system. In this paper, we present an approach on the specific case of the scheduling of the flight control of a space launcher. The approach requires two successive steps: (1) the formalization of the problem to be solved in a parametric formal model and (2) the synthesis of the model parameters with a tool. We first describe the problematic of the scheduling of a launcher flight control, then we show how this problematic can be formalized with parametric stopwatch automata; we then present the results computed by IMITATOR. We compare the results to the ones obtained by other tools classically used in scheduling.

Gilles Feld, Laurent Fribourg, Denis Labrousse et al.

High-power converters based on elementary switching cells are more and more used in the industry of power electronics owing to various advantages such as lower voltage stress and reduced power loss. However, the complexity of controlling such converters is a major challenge that the power manufacturing industry has to face with. The synthesis of industrial switching controllers relies today on heuristic rules and empiric simulation. The state of the system is not guaranteed to stay within the limits that are admissible for its correct electrical behavior. We show here how to apply a formal method in order to synthesize a correct-by-design control that guarantees that the power converter will always stay within a predefined safe zone of variations for its input parameters. The method is applied in order to synthesize a correct-by-design control for 5-level and 7-level power converters with a flying capacitor topology. We check the validity of our approach by numerical simulations for 5 and 7 levels. We also perform physical experimentations using a prototype built by SATIE laboratory for 5 levels.