Stability of planar piecewise linear systems :a geometric approach

Abstract

This thesis focuses on the stability analysis of piecewise linear systems. Such systems consist of linear subsystems, each of which is active in a particular region of the state-space. Many practical and theoretical systems can be modelled as piecewise linear systems. Despite their simple structure, analysis of piecewise linear systems can be rather complex. For instance, most of the results for stability can be based on a Lyapunov approach. However, a major drawback of applying this method is that, it usually only provides su cient conditions for stability. A geometric approach will be used to derive new stability criteria for planar piecewise linear systems. Any planar piecewise linear (multi-modal) system is shown to be globally asymptotically stable just in case each linear mode satis es certain conditions that solely depend on how its eigenvectors stand relative to the cone on which it is de ned. The stability conditions are in terms of the eigenvalues, eigenvectors, and the cone. The improvements on the known stability conditions are the following: i) The condition is directly in terms of the \givens" of the problem. ii) Non-transitive modes are identi ed. iii) Initial states and their trajectories are classi ed (basins of attraction and repulsion are indicated). iv) The known condition for bimodal systems is obtained as an easy corollary of the main result. Additionally, using our result on stability, we design a hybrid controller for a class of second order LTI systems that do not admit a static output feedback controller. The e ectiveness of the proposed controller is illustrated on a magnetic levitation system.