Formal computational methods for control theory

Robertz, Daniel; Plesken, Wilhelm (Thesis advisor)

Aachen : Publikationsserver der RWTH Aachen University (2006)
Dissertation / PhD Thesis

Aachen, Techn. Hochsch., Diss., 2006

Abstract

This thesis treats structural properties of control systems, e.g. controllability and parametrizability of their behavior, from an algebraic point of view. It contributes the following formal computational methods. Janet's algorithm is extended to Ore algebras which are relevant for system theoretic applications. The generalized Hilbert series is introduced, which enumerates a vector space basis of a finitely presented module over an Ore algebra. A method for linearizing differential equations that is independent of any chosen trajectory is presented. This generic linearization results in a system of linear differential equations with non-constant coefficients which are subject to the original nonlinear equations. Therefore, a computational way for dealing with these equations is explained in the framework of jet calculus and differential rings. The algebraic approach to systems theory which is employed in this thesis associates with every linear system a module over a ring which is chosen in accordance with the type of the given equations (e.g. ordinary or partial differential equations, difference equations, retarded differential equations, etc.). The precision in which structural properties of the solution space of the linear system are represented by the module depends on the choice of the space of admissible functions. A faithful correspondence of homological conditions holds for function spaces which are injective cogenerators. In this thesis an injective cogenerator for every Ore algebra which is relevant for the applications to systems theory is presented. The possibility to parametrize the solution spaces of linear systems is investigated more closely. An extension of the established theory to linear systems which are not completely controllable is explained and a method for computing flat outputs of a certain class of linear systems over Weyl algebras is given. The presented theory and formal methods are illustrated on mechanical and chemical engineering systems.