Stability of networks with distributed and nonlinear elements. I
P. J. Schwyn, Theodore A. Bickart · Networks · 1972
Abstract This paper, in two parts, considers the stability of electrical networks consisting of transmission lines interconnected with lumped linear and memoryless nonlinear elements. Let xt be the state of such a network represented as a point in the space C([‐∞, 0], En) of bounded continuous functions mapping the interval [‐∞, 0] into En and let x(t) = xt (0). Then large classes of such networks may be described by an integro‐differential equation of the form magnified image where 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 En into En. 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.