Formal Definition of a Coupled System

Question

I'm working through a paper on interconnected systems, and the main result relies on the assumption that the systems are weakly coupled. Intuitively, I understand what weak coupling means, but I am looking for a formal definition for coupled systems--hopefully one that will allow me to determine to what extent an interconnected system is weakly coupled. Both providing a definition and any relevant sections from papers/articles/textbooks would be helpful.

Answer 1

Often, in interconnected systems containing linear subsystems, diagonal dominance is considered as the condition for "weakly interconnected". To be more precise, the effect of other subsystems is quantifiably less than the effect of each subsystem's own dynamics.

Some state space level definitions of interconnected systems are outlined in following papers :

http://www.cba.mit.edu/events/03.11.ASE/docs/Dandrea.pdf http://www-bcf.usc.edu/~ioannou/2003update/d8.pdf

A simple example is as follows:

$ \begin{eqnarray*} \frac{d}{dt}\left[\begin{array}{c} x_{1}\\ x_{2} \end{array}\right] & = & \left[\begin{array}{cc} A_{11} & A_{12}\\ A_{21} & A_{22} \end{array}\right]\left[\begin{array}{c} x_{1}\\ x_{2} \end{array}\right] \end{eqnarray*} $

The system is weakly interconnected when the diagonal terms in the $A$ matrix dominate non-diagonal terms. I hope this is helpful.