Section: New Results

Analysis and Control of Set-Valued Systems

Participants : Bernard Brogliato, Christophe Prieur, Alexandre Vieira.

This work [5] continues our previous results in [33], to the case when exogeneous terms are present in both the unilateral constraint, and in the dynamics. A suitable change of state variables allows one to recast the dynamics in a format that is close to the autonomous case, so that the well-posedness issues (existence and uniqueness of solutions) is shown (see the preprint [20] for a complete analysis, which in fact differs only slightly from the original one in [33]). The link with switching DAEs is made.

This work [10] concerns the robust control of linear time-invariant systems, subjected to nonlinear varying state dependent disturbances as well as parameter uncertainties. A specific set-valued class of sliding-mode controllers is designed, and its discretization (with the implicit method introduced in [32], [34]) is analysed. One difficulty is that the parameter uncertainties, as well as the discretization, create unmatched disturbances. Stability and convergence results are proved. Let us mention also [9] that corrects a slight mistake in [80]. In the same way it is worth citing [14], [14] which continues the analysis of the implicit discretization of set-valued systems, this time oriented towards the consistency of time-discretizations for homogeneous systems, with one discontinuity at zero (sometimes called quasi-continuous, strangely enough).

In [13] we continue our previous works on well-posedness and stabilization/control of a class of set-valued systems, that take the form of evolution variational inequalities. Dissipativity is then a key property. Regulation with state and output feedback, viability issues, are solved, with absolutely continuous and bounded variations solutions. Applications are in power converters.

The quadratic and minimum time optimal control of LCS as in (6) is tackled in [24], [25]. This work relies on the seminal results by Guo and ye (SIAM 2016), and aims at particularizing their results for LCS, so that they become numerically tractable and one can compute optimal controllers and optimal trajectories. The basic idea is to take advantage of the complementarity, to construct linear complementarity problems in the Pontryagin's necessary conditions which can then be integrated numerically, without having to guess a priori the switching instants (the optimal controller can eb discontinuous and the optimal trajectories can visit several modes of the complementarity conditions).