Piecewise-Linear Theory of Nonlinear Networks

This paper deals with nonlinear networks which can be characterized by the equation ${\bf f}( {\bf x} ) = {\bf y}$, where $f( \cdot )$ is a continuous piecewise-linear mapping of $R^n $ into itself. ${\bf x}$ is a point in $R^n $ and represents a set of chosen network variables, and ${\bf y}$ is an arbitrary point in $R^n $ and represents the input to the network. The Lipschitz condition and global homeomorphism are studied in detail. Two theorems on sufficient conditions for the existence of a unique solution of the equation for all ${\bf y} \in R^n $ in terms of the constant Jacobian matrices are derived. The theorems turn out to be pertinent in the numerical computation of general nonlinear resistive networks based on the piecewise-linear analysis. A comprehensive study of the Katzenelson’s algorithm applied to general networks is carried out, and conditions under which the method converges are obtained. Special attention is given to the problem of boundary crossing of a solution curve.