Algebraic Methods for Modeling and Design in Control

Abstract: Computer algebra or symbolic computation has been recognized as an important tool in many engineering disciplines and continues to find new fields of application. In this thesis we investigate a number of constructive methods implemented as parts of computer algebra packages and utilize them both theoretically and practically in connection with problems in modeling and design of control systems.A large class of systems can be described by a set of polynomial differential equations and there are different approaches to analyze systems of this kind. In this thesis we utilize mathematical tools from commutative and differential algebra such as Gröbner bases and characteristic sets to study input-output relations of systems given in state space form.We show that despite the fact that the state space representation of a system can be described by a finitely generated differential ideal, this is generally not true for the corresponding differential ideal of input-output relations. However, the input-output relations up to a fixed order can be computed and represented using non-differential tools. Using characteristic sets we also show how the above problem of finite representation of the input-output relations can be resolved.Many problems in control theory can be reduced to finding solutions of a system of polynomial equations, inequations and inequalities, a so called real polynomial system. The cylindrical algebraic decomposition method, which was invented in the mid seventies is an algorithm that can be utilized to find solutions to such systems. The extension of real polynomial systems to expressions involving Boolean operators and quantifiers is called the first-order theory of real closed fields. There are algorithms to perform quantifier elimination in such expressions, i.e., to derive equivalent expressions without any quantified variables.We show how these algorithms can be used to solve problems in control such as choosing parameters in a controller to stabilize an unstable system with parameter uncertainties; feedback design of linear, time-invariant systems; computation of bounds on static nonlinearities in feedback systems that ensure stability; computation of (asymptotically stable) equilibrium points for nonlinear systems subjected to constraints on the control and state variables; and curve following in the state space of a nonlinear system.