Consensus Controllability for Node and Edge Multiagent Systems

The multiagent controllability of both node and edge dynamics depends on the communication topology, leader selection, as well as the weight adjustments. This article deals with the controllability relationship between node dynamics and edge dynamics using an edge dynamics model that describes the states of nodes and edges. According to the number of nodes <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> and edges <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>, topology graphs are divided into three categories: <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N&gt;M$</tex-math></inline-formula>, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N=M$</tex-math></inline-formula>, and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N&lt; M$</tex-math></inline-formula>. Sufficient and necessary conditions are derived for determining the controllability relationship between node and edge dynamics, under which a large number of general, special, and mixed topology graphs are analyzed and verified. In cases where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N&gt;M$</tex-math></inline-formula>, the uncontrollable node and edge dynamics are converted to a controllable situation by adding leaders or adjusting weights. Finally, algorithms are provided to validate the effectiveness of the outcomes.