Obstructions to Genericity in Study of Parametric Problems in Control Theory

TL;DR

This paper develops algorithms using Groebner bases to analyze parametric systems in control theory, revealing obstructions to genericity and providing complete solutions to complex classical problems.

Contribution

It introduces an optimized algorithm for identifying parameter values affecting system properties, with practical implementation and application to classical control problems.

Findings

Algorithm reveals parameter values where system properties change

Complete solution to the 'two pendula on a cart' problem including friction

Demonstrates practical use of Groebner bases in non-commutative algebra

Abstract

We investigate systems of equations, involving parameters from the point of view of both control theory and computer algebra. The equations might involve linear operators such as partial (q-)differentiation, (q-)shift, (q-)difference as well as more complicated ones, which act trivially on the parameters. Such a system can be identified algebraically with a certain left module over a non-commutative algebra, where the operators commute with the parameters. We develop, implement and use in practice the algorithm for revealing all the expressions in parameters, for which e.g. homological properties of a system differ from the generic properties. We use Groebner bases and Groebner basics in rings of solvable type as main tools. In particular, we demonstrate an optimized algorithm for computing the left inverse of a matrix over a ring of solvable type. We illustrate the article with interesting examples. In particular, we provide a complete solution to the "two pendula, mounted on a cart" problem from the classical book of Polderman and Willems, including the case, where the friction at the joints is essential . To the best of our knowledge, the latter example has not been solved before in a complete way.