Section: New Results

Control

Digital sliding mode control

Participants : Vincent Acary, Bernard Brogliato.

The problem of digital sliding mode controllers is a long-standing issue not yet satisfactorily solved. We propose in ideas which are inspired from the numerical methods of contact mechanics [2] and which permit a) to suppress the numerical chattering, b) to obtain a smooth stabilization on the sliding surfaces. The work is continued together with Yury Orlov in more general cases where the system is acted upon by disturbances and a disturbance estimation is added [19] .

Discrete-time discontinuous systems

Participants : Vincent Acary, Bernard Brogliato, Carmina Georgescu, Scott Greenhalg, Thorsten Schindler.

We focus on some classes of discontinuous dynamical systems like relay systems, linear complementarity systems. The objectives are to show that the time-stepping numerical schemes like Moreau's algorithm can be used to successfully simulate such systems (like in the case of biological systems like gene networks), and also to study the properties of these schemes for finite nonzero time steps (like preservation of disssipativity properties). Results are in [23] , [35] . Further work deals with timestepping schemes for nonsmooth dynamical systems. So far, these schemes are locally of order one both in smooth and nonsmooth segments. This is inefficient for applications with few events like circuit breakers. To consistently improve the behavior during smooth episodes, the traditional schemes are being embedded in time discontinuous Galerkin methods. After establishing the correct mathematical setting, a Petrov-Galerkin distributional differential inclusion is outlined. The bouncing ball example illustrates its capabilities.