Piecewise-linear Networks Research Articles

A Note on the Generalised Katzenelson’s Method for Piecewise Linear Resistive Networks

Ohtsuki, Fijisawa, and Kumagai [SIAM J. Math. Anal., 8 (1977), pp. 69–99] proposed a generalisation of Katzenelson’s method that is useful for a large class of piecewise linear resistive networks. In this note, errors are pointed out and the correct proof is presented.

Necessary and sufficient condition for uniqueness of solutions of certain piecewise linear resistive networks containing transistors and diodes

Equations of the form F(x)=f(x) + Ax = y where f(x) = [f1(x1)f2(x2)…fn(xn)]T, A is a P0 matrix and for all i = 1, 2,…, n, fi(xi), are monotonic and piecewise linear but not necessarily strictly monotonic, are studied. Such networks are shown to have a unique solution if the Jacobian determinant has the same sign in all the regions. This is much more general than several sufficient conditions available in the papers by Sandberg and Willson and by Chien. Furthermore it is not possible to improve this any further as the condition is both necessary and sufficient A new sufficient condition is proposed to quickly check the sign condition on the Jacobians.

A simple algorithm for finding all solutions of piecewise-linear networks

An efficient algorithm is presented for the searching of all solutions of a piecewise-linear (PWL) resistive network. Let P/sub /n be the number of largest segments of PWL resistors, and P/sub /n/sub -1/ the second largest. The algorithm presented is better than the brute-force method by a factor of P/sub /n*P/sub /n/sub -1/ and the Chua and Ying method, the most efficient method at the present time, by a factor of at least P/sub /n/sub -1/. The algorithm is based on a PWL network theorem, which is also introduced.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Uniqueness of solutions of piecewise linear resistive networks with two terminal resistors

Consider a network comprising two terminal linear, piecewise linear resistors and independent sources. Contrary to widely held belief, it is shown that the determinant sign condition (i.e. the Jacobian determinant is nonzero and has the same sign in all the regions) is both necessary as well as sufficient to guarantee uniqueness of solutions of these networks. For this purpose we do not require the existence of specific hybrid matrix.

D.C. ANALYSIS OF PIECEWISE LINEAR NETWORKS USING THE GAUSS‐SEIDEL METHOD

Analysis of piecewise linear resistive networks, described by nodal equations, using the Gauss‐Seidel method is presented in the paper. The conditions which guarantee the existence of a unique solution and convergence of the iterative technique are formulated. An algorithm is described and a numerical example is demonstrated.

Piecewise linear resistive networks with unique solutions

New conditions for the existence of unique solutions of piecewise linear resistive networks are presented. They require that certain submatrices of unbounded regions are all positive definite or all negative definite. The applicability of these conditions can be enhanced by using a suitable transformation.

Synthesis of resistive n-port section-wise piecewise-linear networks

A nonlinear <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -port described by an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -dimensional sectionwise piecewise-linear (SPL) function is shown to be realizable by a circuit composed of a nonlinear converter and several one-ports. Also, a decomposition of the required converter is presented which is based on nonenergic nonlinear subnetworks.

Synthesis of piecewise-linear networks

Systematic methods for synthesising nonlinear networks having a prescribed scalar or multidimensional piecewise-linear function are described. In the scalar case, precision active-circuit building blocks using operational amplifiers, transistors, diodes, and resistors are given for realising piecewise-linear drivingpoint and transfer characteristic plots. In the multidimensional case, methods are given for realising a multiterminal nonlinear network having a multidimensional piecewise-linear transfer function. Finally, these methods are generalised for synthesising nonlinear n-ports having a prescribed multidimensional piecewiselinear driving-point function. Although most of the basic building blocks are grounded active networks, convertor networks are developed for transforming such grounded networks into floating networks having the same properties. By slight modifications of these convertor networks, other useful conversion properties are also presented

Simplex analysis of grounded concave piecewise‐linear resistive networks

AbstractThe physical properties of the networks under consideration may be advantageously used for saving time and storage in analysing nonlinear networks. The algorithm proposed in the present paper represents an attempt in this direction. It concerns a particular class of piecewise linear resistive networks and it is based on a modification of the simplex as originally proposed by Dantzig.The algorithm requires: The analysis of the linear network obtained by the given one by replacing the nonlinear resistances by linear ones. The application of the modified simplex. This second step requires a number of arithmetical operations less than that necessary for analysing a linear resistive network having a number of nodes equal to the total number of break‐points of the nonlinear resistances.

A sparse matrix method for analysis of piecewise-linear resistive networks

Nonlinear resistive networks, which can be characterized by the equation <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f(x) =y</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f(\cdot)</tex> is a continuous piecewise linear mapping of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R^{n}</tex> into itself, are discussed. <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</tex> is a point in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R^{n}</tex> and represents a set of chosen network variables and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">y</tex> is an arbitrary point in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R^{n}</tex> and represents the input to the network. New theorems on the existence of solutions together with a convergent method for obtaining at least one of the solutions are given. Also dealt with is an efficient computational algorithm which is especially suited for analysis of large piecewise-linear networks. The effectiveness of the method in terms of the amount of computation and data handling and storage is demonstrated.