On the stability robustness in a closed-loop interconnection under a class of operations performed on the controller: a novel characterization for static controllers

A closed-loop stable interconnection of two linear time-invariant finite-dimensional systems (one the plant, the other the controller) is considered. We analyze the preservation of the stability in the closed-loop interconnection whenever the input and output signals of the controller are multiplied by time-variant gains, one the reciprocal of the other, and in addition the function that represents those gains belongs to a specific class of functions. An important consequence of that analysis (and the main motivation in considering the aforementioned seemingly artificial robust stability setting) is the new characterization for output static stabilizing controllers we present in this communication. Moreover, a new technical tool is also presented: the principal function of the closed-loop stable interconnection. This function provides with information relative to the property of being static, of a stabilizing controller.