Section: Application Domains

Control theory

Certain problems studied in mathematical systems theory and control theory can be better understood and finely studied by means of algebraic structures and methods. Hence, the rich interplay between algebra, computer algebra, and control theory has a long history. For instance, the first main paper on Gröbner bases written by their creators, Buchberger, was published in Bose's book [46] on control theory of multidimensional systems. Moreover, the differential algebra approach to nonlinear control theory (see [72], [73] and the references therein) was a major motivation for the algorithmic study of differential algebra [47], [76]. Finally, the behaviour approach to linear systems theory [141], [119] advocates for an algorithmic study of algebraic analysis (see Section 2.2.4). More generally, control theory is porous to computer algebra since one finds algebraic criteria of all kinds in the literature even if the control theory community has a very few knowledge in computer algebra.

OURAGAN has a strong interest in the computer algebra aspects of mathematical systems theory and control theory related to both functional and polynomial systems, particularly in the direction of robust stability analysis and robust stabilization problems for multidimensional systems [46], [119] and infinite-dimensional systems [66] (such as, e.g., differential time-delay systems).

Let us shortly state a few points of our recent interests in this direction.

In control theory, stability analysis of linear time-invariant control systems is based on the famous Routh-Hurwitz criterion (late 19th century) and its relation with Sturm sequences and Cauchy index. Thus, stability tests were only involving tools for univariate polynomials [102]. While extending those tests to multidimensional systems or differential time-delay systems, one had to tackle multivariate problems recursively with respect to the variables [46]. Recent works use a mix of symbolic/numeric strategies, Linear Matrix Inequalities (LMI), sums of squares, etc. But still very few practical experiments are currently involving certified algebraic computations based on general solvers for polynomial equations. We have recently started to study certified stability tests for multidimensional systems or differential time-delay 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 $-$ can now handle previously impossible computations and lead to a better understanding of the problems to be solved [52], [54], [55]. The previous approaches seem to be blocked on a recursive use of one-variable methods, whereas our approach involves the direct processing of the problem for a larger number of variables.

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 instance, we show [53], [54], [55] 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, e.g., the resolution of bivariate systems [51], [50].

The rich interplay between control theory, algebra, and computer algebra is also well illustrated with our recent work on robust stabilization problems for multidimensional and finite/infinite-dimensional systems [48], [123], [129], [132], [130], [131].