Analysis and control of nonlinear differential-algebraic systems

Abstract

In this Thesis the topics of integration, analysis and control of nonlinear differential-algebraic systems are studied using notions and tools from classical control theory. Typical problems arising when differential-algebraic systems are numerically integrated include inconsistent initial conditions, round-off errors and constraint drift. Constraint stabilization methods represent an efficient solution to these issues. It is observed, however, that the application to nonlinear systems of constraint stabilization methods which rely on a linear feedback mechanism may result in trajectories with finite escape time. To overcome this problem we propose a method based on a nonlinear stabilization mechanism which guarantees the global existence and convergence of the solutions. Discretization schemes, which preserve the properties of the method, are also presented. Classical approaches to the stability analysis and control of nonlinear differential-algebraic systems rely on the calculation of the underlying unconstrained dynamics to which classical results can be applied. Different from these approaches, an alternative which allows studying the stability properties of the equilibrium points directly in the differential-algebraic formulation of the system is proposed. In particular, sufficient stability conditions relying on matrix inequalities are established via Lyapunov Direct Method. In addition, a novel interpretation of differential-algebraic systems as feedback interconnection of a purely differential system and an algebraic system allows reducing the stability analysis to a small-gain-like condition. The stability analysis for constrained mechanical systems, the stabilization problem for a class of Lipschitz differential-algebraic systems and the control problem for an air suspension system, along with several numerical examples, are used to illustrate the theory.