Performance Bounds for Optimal and Robust Feedback Control in Networks

Abstract

Important complex networks, including critical infrastructure and emerging industrial automation systems, are becoming increasingly intricate webs of interacting feedback control loops. A fundamental concern is to quantify the control properties and performance limitations of the network as a function of the structure of its dynamics and control architecture. We study performance bounds for networks in terms of both optimal and robust feedback control costs as a function of the system dynamics and actuator structure. For unstable network dynamics, we demonstrate a tradeoff between feedback control performance and the number of control inputs, in particular, showing that optimal cost can increase exponentially with the number of unstable nodes in the network. Likewise, we demonstrate a tradeoff between robustness and the numbers of control and adversarial inputs, which shows that either an increase in the number of unstable nodes or the number of adversarial inputs can make optimal cost increase exponentially or become infinite. We also derive a bound on the performance of the worst-case actuator subset for open-loop stable networks, providing insight into dynamics properties that affect the potential efficacy of actuator selection in terms of measures for how close the open-loop dynamics are to instability. However, we show that such a bound is generally not guaranteed even for open-loop stable networks in terms of robustness to adversarial inputs. When the open-loop network dynamics has a finite <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$H_\infty$</tex-math></inline-formula> norm with respect to the adversarial inputs, we find an analogous performance bound of the worst case actuator set. We illustrate our results with numerical experiments that analyze performance in regular and random networks.