Controllability of Multilayer Snapback Networks

Abstract

This article studies the controllability of multi-input/multioutput linear time-invariant (LTI) systems in a snapback interlayer coupling framework. Several necessary and sufficient conditions for the controllability of the three-layer snapback networks are established. In addition, controllability conditions about the superposition of the three-layer networks are obtained. These conditions are related to smaller scale factor networks, showing the effects of the interlayer coupling frameworks, intralayer network topologies, node dynamics, inner interactions, and external control inputs on the controllability of the snapback networks. Moreover, controllability conditions of the three-layer snapback networks are extended to the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$M$</tex-math></inline-formula> -layer setting. Finally, some examples are presented to illustrate the effectiveness of the controllability conditions.