Conewise Linear Systems: Non-Zenoness and Observability

Conewise linear systems are dynamical systems in which the state space is partitioned into a finite number of nonoverlapping polyhedral cones on each of which the dynamics of the system is described by a linear differential equation. This class of dynamical systems represents a large number of piecewise linear systems, most notably, linear complementarity systems with the P-property and their generalizations to affine variational systems, which have many applications in engineering systems and dynamic optimization. The challenges of dealing with this type of hybrid system are due to two major characteristics: mode switchings are triggered by state evolution, and states are constrained in each mode. In this paper, we first establish the absence of Zeno states in such a system. Based on this fundamental result, we then investigate and relate several state observability notions: short-time and $T$-time (or finite-time) local/global observability. For the short-time observability notions, constructive, finitely verifiable algebraic (both sufficient and necessary) conditions are derived. Due to their long-time mode-transitional behavior, which is very difficult to predict, only partial results are obtained for the $T$-time observable states. Nevertheless, we completely resolve the $T$-time local observability for the bimodal conewise linear system, for finite $T$, and provide numerical examples to illustrate the difficulty associated with the long-time observability.