On the Dynamic Analysis of Piecewise Linear Networks

Piecewise linear (PL) modelling is often used to approximate the behavior of nonlinear circuits. One of the possible PL modelling methodologies is based on the linear complementarity problem, and this approach has already been used extensively in the circuits and systems community for static networks. In this paper the object of study will be dynamic electrical circuits that can be recast as linear complementarity systems, i.e., as interconnections of linear time-invariant differential equations and complementarity conditions (ideal diode characteristics). A mathematically precise framework is developed that formalizes the mixed discrete and continuous behavior of these switched networks. Within this framework the fundamental question of well-posedness (existence and uniqueness of solution trajectories given an initial condition) is studied and additional properties of the behavior are derived. For instance, a full characterization is presented of the inconsistent states. Keywords— circuit analysis, piecewise linear networks, switched circuits, linear complementarity problem, passivity.