Section: Research Program
Applications to control theory and signal processing
Robust stability analysis and stabilization problems for functional systems
Our expertise in the computer algebra aspects to stability and stabilization problems for multidimensional systems and for differential constant/distributed/varying delay systems [5], [6], [48], [47] will further be developed.
Computation of Lyapunov functions for homogeneous dynamical systems
We shall investigate the possibility to develop a computer algebra package for the design of Lyapunov functions for homogeneous dynamical systems based on an effective study of the differential algebra of generalized forms, i.e. of Puiseux polynomials in signed powers [106], [107].
Robust stability analysis of differential time-delay systems
The symbolic-numeric study of the robust stability of a differential
constant time-delay system with respect to the delay
Moreover, in collaboration with Mouze (Centrale Lille, France), we
want to develop an effective study of the ring
Stabilization problems for functional systems
We want to use the results developed
on the module-theoretic aspects of the ring of multivariate rational
functions without poles in the closed unit polydisc of
Finally, the noncommutative geometric approach to robust problems for infinite-dimensional linear systems (e.g. differential time-delay or PD systems) [54], initiated in [92], will be further studied based on the mathematical concepts and methods introduced by (the Fields medalist) Connes [53]. We particularly want to investigate generalizations of Nyquist's theorem to infinite-dimensional systems
based on index theory (pairing of
Parameter estimation for linear & nonlinear functional systems
Linear functional systems
Our expertise on algebraic parameter estimation problem, coming from the former Non-A project-team, will be further developed. Following [65], this problem consists in estimating a set
A first aim in this direction is to develop to a greater extent our recent work [108] that shows how the above approach can cover wider classes of signals such as holonomic signals (e.g. signals decomposed into orthogonal polynomial bases, special functions, possibly wavelets).
Moreover, [94] explains how larger classes of structured
perturbations
Furthermore, as
an alternative to passing forth and backwards from the time domain
to the operational (Laplace/frequency) domain by means of Laplace transform and its
inverse as done in the standard algebraic parameter
estimation method [65], [108], we aim to develop a direct time domain approach
based on calculus on rings of integro-differential operators as
described by the following picture (
The direct computation will be handled by means of the effective methods of rings of integro-differential operators described in the above sections.
Nonlinear functional systems
For nonlinear control systems, the approach to the parameter estimation problem, recently proposed in [3] and based on the computation of integro-differential input-output equations, will be further developed based on the integration of fractions [2]. Such a representation better suits a numerical estimation of the parameters as shown in [3].
In [29], we have recently initiated an extension of the results developed in [3] to handle integro-differential equations such as Volterra-Kostitzin's equation. This general approach, based on an extension of the input-output ideal method for ordinary differential equations to the integro-differential ones, will be further developed based on the effective elimination theory for systems of integro-differential equations. An important advantage of this approach is that not only it solves the identifiability theoretical question but it also prepares a further parameter estimation step [57].