Decomposition Functions for Interconnected Mixed Monotone Systems

A dynamical system is mixed monotone when there exists a related decomposition function that separates the system dynamics into cooperative and competitive state interactions. Such a decomposition enables, e.g., efficient computation of robust reachable sets and forward invariant sets, but obtaining a decomposition function can be challenging. In this letter, we present a method for obtaining a decomposition function for a system that can be represented as an interconnection of subsystems with known decomposition functions. We further extend this approach using tools from interval reachability analysis to accommodate systems with outputs and we provide also conditions for when the system’s unique tight decomposition function is obtained via this approach. We demonstrate this methodology for computing decomposition functions with an example of a 3-dimensional unicycle model and with a case study of a 7-dimensional nonlinear spacecraft system defined as an interconnection of subsystems and feedback controllers. Reachable sets for the systems are then computed using their decomposition functions and the standard tools from mixed monotone systems theory.