Consistency of a time-stepping method for a class of piecewise-linear networks

In this brief, we will study the computation of transient solutions of a class of piecewise-linear (PL) circuits. The network models will be so-called linear complementarity systems, which can be seen as dynamical extensions of the PL modeling structure. In particular, the numerical simulation will be based on a time-stepping method using the well-known backward Euler scheme. It will be demonstrated, by means of an example, that this widely applied time-stepping method does not necessarily produce useful output for arbitrary linear dynamical systems with ideal diode characteristics. Next the consistency of the method will be proven for PL networks that can be realized by linear passive circuit elements and ideal diodes by showing that the approximations generated by the method converge to the true solution of the system in a suitable sense. To give such a consistency proof, a fundamental framework developed previously is indispensable as it proposes a precise definition of a "solution" of a linear complementarity system and provides conditions under which solutions exist and are unique.