The First Five Years of a Phase Theory for Complex Systems and Networks

In this paper, we review the development of a phase theory for systems and networks in its first five years, represented by a trilogy: Matrix phases and their properties; The MIMO LTI system phase response, its physical interpretations, the small phase theorem, and the sectored real lemma; The synchronization of a multi-agent network using phase alignment. Towards the end, we also summarize a list of ongoing research on the phase theory and speculate what will happen in the next five years.