Absolute stability of a system of nonlinear networks interconnected by lossless transmission lines

This paper considers the problem of global asymptotic stability of a system S of nonlinear networks I (i), j = 1, - ,n, interconnected by lossless transmission lines. Each 0(i) consists of a linear time-invariant lumped-parameter multiport network with a nonlinear element (resistor or capacitor) in parallel or (resistor or inductor) in series with each of its output ports. The voltage-current, voltage-charge, and current-flux linkage relationships for the nonlinear elements are assumed to lie in a sector. The transmission lines introduce time delays in the overall system as well as loading effects at the terminals of the networks. On the assumption that the linear system obtained by deleting (appropriately shorting or opening) the nonlinear elements is asymptotically stable by satisfying, for example, Brayton's conditions, this paper develops a frequency-domain condition that guarantees global asymptotic stability of the system S. This result is achieved by suitably modifying and extending the result of Popov and Halanay.