On the Stability Regions of Nonlinear Control Systems

Abstract

In this thesis, we report advances to the theory of stabilization of control systems. A first contribution of this thesis is to utilize the concept of stability regions to derive control laws to stabilize nonlinear systems with stochastic disturbances and subject to lack of availability of state measurements. In this direction, the probability of destabilizing behavior is approximated that incorporates the effect of the observation error. In a second contribution, we determine, for any given input-constrained nonlinear control system, the largest region from which stabilization can be achieved, termed the null controllable region (NCR). Indeed, a key result of this thesis is to propose a control law that induces stabilization from the entire NCR. In this direction, two methods to construct NCRs are proposed using completely different principles. In one major contribution, the NCR is given by special smooth optimal trajectories integrated in reverse-time using Pontryagin's Maximum Principle. In another major contribution, this same region is given by iteratively expanding an initial Lyapunov-based stability region by applying a reachability test to boundary cells. A further contribution of this thesis is to use the geometry of a collection of NCRs, corresponding to modulated input constraints, to construct a control law that guarantees the stabilization of the closed-loop system. In this direction, special modifications are introduced in order to apply this technique to a chemical reactor example. In a final contribution, knowledge of the NCR is used to enable the formulation of a simple finite difference scheme to solve otherwise intractable Hamilton-Jacobi-Bellman equations in order to generate Lyapunov functions with maximal regions of attraction.