Necessary and sufficient conditions for the global invertibility of certain nonlinear operators that arise in the analysis of networks

Author

Sandberg, Irwin W.

Volume

Issue

fYear

1971

fDate

3/1/1971 12:00:00 AM

Firstpage

260

Lastpage

263

Abstract

For a large class of operators of the form $[F(\\cdot)+A],$ in which $A$ is a not necessarily nonsingular real $n \\times n$ matrix, and $F(\\cdot)$ is a diagonal strictly monotone-increasing mapping of the set of all real $n$ vectors $E^{n}$ onto an open subset of $E^{n}$ , we give necessary and sufficient conditions under which $F(\\cdot)+A]$ possesses a global inverse on $E^{n}$ . Operators of the type $[F(\\cdot)+A]$ frequently arise in the analysis of nonlinear networks and are encountered in other areas as well. In particular, for $A$ the short-circuit conductance matrix of a resistance network, and $F(x)$ the transpose of $(f_{1}(x_{1}), f_{2}(x_{2}), \\cdots , f_{n}(x_{n}))$ for all $x \\in E^{n}$ in which the $f_{j}(\\cdot)$ are the usual exponential diode functions, we give a complete solution to the problem of determining whether or not $[F(\\cdot)+A]$ possesses an inverse on $E^{n}$ .

Keywords

Matrix methods; Nonlinear network analysis & design; Nonlinear networks; Transient analysis; Computer networks; Computer simulation; Differential equations; Nonlinear equations; Resistors; Semiconductor diodes; Sufficient conditions; Telephony;

fLanguage

English

Journal_Title

Circuit Theory, IEEE Transactions on

Publisher

ieee

ISSN

0018-9324

Type

jour

DOI

10.1109/TCT.1971.1083269

Filename

1083269

Link To Document

https://search.isc.ac/dl/search/defaultta.aspx?DTC=49&DC=1168405