Section: Application Domains

Control theory

Many problems in control theory have been studied using general exact polynomial solvers in the past. One can cite the famous Routh-Hurwitz criterion (late 19th century) for the stability of a linear time invariant (LTI) control system and its relation with Sturm sequences and Cauchy index. However most of the strategies used were involving mostly tools for univariate polynomials and then tried to tackle multivariate problems recursively with respect to the variables. More recent work are using a mix of symbolic/numeric strategies, using semi-definite programming for classes of optimization problems or homotopy methods for some algebraic problems, but still very few practical experiments are currently involving certified algebraic using general solvers for polynomial equations.

Our work in control theory is a recent activity and it is done in collaboration with a group of specialists, the GAIA team, Inria Lille-Nord Europe. We started with a well-known problem, the study of the stability of differential delay systems and multidimensional systems with an important observation: with a correct modelization, some recent algebraic methods, derived from our work in algorithmic geometry and shared with applications in robotics, now allow some previously impossible computations and lead to a better understanding of the problems to be solved [37], [36]. The field is porous to computer algebra since one finds for a long time algebraic criteria of all kinds but the technology seems blocked on a recursive use of one-variable methods, whereas our approach involves the direct processing of problems into a larger number of variables or variants.

The structural stability of $n$-D discrete linear systems (with $n\ge 2$) is a good source of problems of several kinds ranging from solving univariate polynomials to studying algebraic systems depending on parameters. For example, we have shown that the standard characterization of the structural stability of a multivariate rational transfer function (namely, the denominator of the transfer function does not have solutions in the unit polydisc of ${ℂ}^n$) is equivalent to deciding whether or not a certain system of polynomial equations has real solutions. The use state-of-the-art computer algebra algorithms to check this last condition, and thus the structural stability of multidimensional systems has been validated in several situations from toy examples with parameters to state-of-the-art examples involving the resolution of bivariate systems.