Consistent Fixed Points and Negative Gain

Abstract

We discuss the stabilization properties of networks that are composed of “ displacement elements”. Each displacement element is defined by an integer K, called the displacement of the element, an input variable x, and an output variable y, where the values of x and y are non-negative integers. An execution step of this element assigns to y the maximum of 0 and K + x. The objective of our discussion is to demonstrate that two principles play an important role in ensuring that a network N is stabilizing, i. e. starting from any global state, network N is guaranteed to reach a global fixed point. Specifically, the principle of consistent fixed points is analogous to the requirement that a control system be free from self-oscillations. And the principle of negative gain is analogous to the requirement that the feedback loop of a sum of displacements along every directed loop in network N is negative.