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Stability of networks with distributed and nonlinear elements. I
Author(s) -
Schwyn P. J.,
Bickart T. A.
Publication year - 1972
Publication title -
networks
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.977
H-Index - 64
eISSN - 1097-0037
pISSN - 0028-3045
DOI - 10.1002/net.3230020105
Subject(s) - mathematics , bounded function , nonlinear system , stability (learning theory) , mathematical analysis , matrix (chemical analysis) , differential equation , function (biology) , topology (electrical circuits) , combinatorics , computer science , physics , materials science , quantum mechanics , machine learning , evolutionary biology , composite material , biology
This paper, in two parts, considers the stability of electrical networks consisting of transmission lines interconnected with lumped linear and memoryless nonlinear elements. Let x t be the state of such a network represented as a point in the space C([‐∞, 0], E n ) of bounded continuous functions mapping the interval [‐∞, 0] into E n and let x(t) = x t (0). Then large classes of such networks may be described by an integro‐differential equation of the formwhere A is a real n × n matrix, G(τ) is a real n x n matrix‐valued function of bounded variation on [‐∞,0], and E is a nonlinear function mapping E n into E n . The integral is of the Riemann‐Stieltjes type. A functional is defined on a subset of C and used to obtain stability theorems for the integro‐differential equation using Liapunov stability theory as extended to functional differential equations. Then, these theorems are applied to networks with transmission lines, with the result being stability criteria which depend only on the lumped elements and the total variation of the step response of the transmission lines with suitably chosen terminations. The total variations associated with uniform, distortionless LC lines and nonuniform RC lines are presented. For networks with RC lines the stability criteria are simplified such that they are functions only of the lumped elements and the total resistances of the RC lines.