New tools for nonsmooth stability analysis and observer design

In this dissertation, we study two problems of interest in nonsmooth control theory. The problems are the development of nonsmooth Lyapunov stability theory and the development of a state estimator for nonlinear systems with quantized outputs. We first motivate the need for these tools by giving an overview of smooth dynamic and control systems. The overview identifies salient differences between smooth and nonsmooth systems. Background information on smooth and nonsmooth systems is the content of the first chapter.;In chapter two we develop nonsmooth Lyapunov stability theory for a class of nonsmooth Lipschitz continuous Lyapunov functions and absolutely continuous state trajectories. The primary use of these results is in analyzing the stability of equilibria of differential equations with discontinuous right hand side. Such equations arise in nonsmooth dynamic systems and variable structure control. Generalizations of theorems for stability, asymptotic stability, and Lasalle's theorems are made for differential equations with discontinuous right-hand sides. We observe that modifications to the smooth theory are needed to obtain a general nonsmooth theory with the use of Filippov's differential inclusion and Clarke's generalized gradient. One feature of the theorems is that although the derivatives of both the trajectories and the Lyapunov functions may not exist everywhere, the stability conditions in the nonsmooth theorems are computable. The use of nonsmooth Lyapunov functions is motivated by several examples, including the analysis of a harmonic oscillator with Coulomb friction and a stable autonomous nonsmooth dynamic system for which no continuously differentiable time independent Lyapunov function exists.;In chapter three we develop an observer for nonlinear systems with quantized outputs. The observer is a recursive algorithm based on the recursive intersection of sets: each measurement defines a set in state space consistent with the measurement. By recursive intersection, each measurement is used to refine our knowledge of the state. We develop the necessary data structures and procedures to implement the algorithm numerically. Comparisons are drawn between the proposed observer, the Kalman filter, and the equations of nonlinear filtering. Estimates are given for the error due to the triangulation of the set of consistent states and the computational complexity of the numerical implementation of our observer. Finally, the algorithm is applied to two example systems. The first example is applying the observer to a general linear system, proving that the observer generates the correct state estimate in this simple case. The second example is an application of the observer to a nonlinear oscillator.;In chapter four we suggest directions for future research and ways in which the research presented in this dissertation could be applied to active areas of nonsmooth control theory.