Numerical methods for the investigation of stabilizability of constrained systems

The systematic stabilization of dynamical systems based on a suitable controller is a central task in automatic control. Obviously, this task can only be solved if the underlying system is indeed stabilizable, i. e., if the existence of a stabilizing control law can be guaranteed. For linear unconstrained systems, simple algebraic criteria for the verification of stabilizability are available. In contrast, for nonlinear systems with state and input constraints, such criteria are missing. The thesis deals with numerical methods for the rigorous investigation of stabilizability of constrained systems. In particular, we present and extend procedures for the computation of stabilizable sets. In this context, the accuracy of the evaluated sets w. r. t. the largest stabilizable set (LSS) is of central interest. To measure accuracy, we rigorously compute inner and outer approximations of the LSS using reachability analysis. With regard to the initially mentioned task, the identification of stabilizable states only marks the first step towards the stabilization of dynamical systems. Consequently, the second part of the thesis is aimed at the design of stabilizing control laws. Based on the evaluated stabilizable sets, we present efficient model predictive and time-optimal control schemes suitable for nonlinear, bilinear, and linear systems, respectively. The last part of the thesis concentrates on the exemplary application of the developed methods to some selected systems with practical relevance. Inter alia, we provide control concepts for blood glucose regulation and the operation of a chemical reactor. It is remarkable that, for both applications, we are able to guarantee that at least 97% of all stabilizable states will indeed be stabilized by the designed controllers.