Section: New Results

New Techniques for Robust Control of Linear Infinite-Dimensional Systems

Robust stabilization of a flexible rod moving in rotation and translation

We develop a hierarchy of models for a flexible rod moving in rotation and translation from a nonlinear partial differential model (generalization of the Euler-Bernoulli equation) to a linear partial differential equation and finite-dimensional models via approximations. We study the stability of those models as well as their robust stabilizations. This work is an extension of the results obtained in [61] . This work will be pursued within the framework of a CIFRE PhD thesis developed in collaboration with SAGEM (2015).

Noncommutative geometric approach to infinite-dimensional systems theory:

This new field of research aims at showing that noncommutative geometric structures such as connections and curvatures exist on internally stabilizable infinite-dimensional linear systems and on their stabilizing controllers. To see this new geometry, using the noncommutative geometry developed by Connes, we have to replace the standard differential calculus by the quantized differential calculus and classical vector bundles by projective modules. In [50] , we give an explicit description of the connections on an internally stabilizable system and on its stabilizing controllers in terms of the projectors of the closed-loop system classically used in robust control. Their curvatures are explicitly computed. These connections aim at studying the variations of the signals in the closed-loop system in response to a disturbance or a change of the reference. The study of these connections are useful to understand how techniques of (noncommutative) differential geometry can be used in the study of $H^{\infty }$ control theory.

A fractional ideal approach to the robust regulation problem

We show how fractional ideal techniques developed in [8] can be used to obtain a general formulation of the internal model principle for stabilizable infinite-dimensional SISO plants which do not necessarily admit coprime factorization. This result is then used to obtain necessary and sufficient conditions for the robust regulation problem. In particular, we find again all the standard results obtained in the literature.

Robust control as an application to the homological perturbation lemma:

Within the lattice approach to transfer matrices developed in [8] , we have recently shown how standard results on robust control can be obtained in a unified way and generalized when interpreted as a particular case of the so-called Homological Perturbation Lemma. This lemma plays a significant role in algebraic topology, homological algebra, algebraic and differential geometry, computer algebra, $...$ Our results show that it is also central to robust control theory for infinite-dimensional linear systems.