Steady-state analysis of piecewise-linear dynamic systems

Piecewise-linear (PWL) systems form a subclass of the general class of nonlinear systems. However, because PWL functions are not differentiable everywhere, results derived for general differentiable systems do not generally apply to PWL systems. In this paper, the properties of the solutions of continuous PWL dynamic systems are investigated in detail and theorems on the continuity and differentiability of the solutions with respect to initial conditions are derived. The results obtained are applied in the study of the convergence properties of steadystate algorithms when applied to continuous PWL dynamic systems. It is found that under fairly mild conditions the convergence is of order two, which is the same order of convergence of the algorithms when applied to differentiable dynamic systems. It is also found, however, that there are special cases when the order of convergence may be reduced to one.