Constructive Commutative Algebra in Nonlinear Control Theory

Sammanfattning: This thesis consists of two parts. The first part is a short review of those results from commutative algebra, algebraic geometry and differential algebra that are needed in this work. The emphasis is on constructive methods. The second part contains applications of these methods to topics in control theory, mainly nonlinear systems.When studying nonlinear control systems it is common to consider C00 affine systems, with differential geometry as a mathematical framework. If instead we regard systems where all nonlinearities are polynomial the most natura] tool is commutative algebra. One of the most important constructive methods in commutative algebra is Gröbner bases, which are used for several different purposes in this thesis.Conversion from state space realization to input-output form is one of the problems solved. Furthermore, transformations between different realizations of the same input-output equation can be found with the help of elimination theory. Testing algebraic observability is also possible to do with Gröbner bases.In Lyapunov theory the classical problem of finding the critical level of a local Lyapunov function of a system is addressed. Some variants of this problem are used to analyze the stability and performance robustness of a system subject to structured uncertainty. Similarly, it is possible to make the optimal choice of parameters in a family of local Lyapunov functions for a given system.A problem that has not been solved entirely constructively is the question of when a polynomial input-output differential equation has a rational state space realization. This problem is showed to be strictly harder than the one of determining unirationality of hypersurfaces, which is an open question in algebraic geometry.