Stability of third order conewise linear systems

Abstract

A conewise linear, time-invariant system is a piecewise linear system in which the state-space is a union of polyhedral cones. Each cone has its own dynamics so that a multi-modal system results. We focus our attention to global asymptotic stability so that each mode (or subsystem) is autonomous. i.e., driven only by initial states. Conewise linear systems are of great relevance from both practical and theoretical point of views as they represent an immediate extension of linear, time-invariant systems. A clean and complete necessary and sufficient condition for stability of this class of systems has been obtained only when the cones are planar, that is only when the state space is R2. This thesis is devoted to the case of state-space being R3, although occasionally we also consider the general case Rn. We aim to determine conditions for stability exploring the geometry of the modes. Thus our results do not make use of a Lyapunov function based approach for stability analysis. We first consider an individual mode and determine whether a cone with a given dynamics can be classified as a sink, source, or transitive from one or two borders. It turns out that the classification not only depends on the geometry of the eigenvectors and the geometry of the cone but also on entries of the A-matrix that defines the dynamics. Under suitable assumptions on the configuration of the eigenvectors relative to the cone, we manage to obtain relatively clean charecterizations for transitive modes. Combining this with a complete characterization of sinks and sources, we use some tools from graph theory and obtain an interesting sufficient condition for stability of a conewise system composed of transitive modes, sources, and sinks. Finally, we apply our results to study the stability of a linear RC electrical network containing diodes.