Product Form of Projection-Based Model Reduction and its Application to Multi-Agent Systems

Orthogonal projection-based reduced order models (PROM) are the output of widely-used model reduction methods. In this work, a novel product form is derived for the reduction error system of these reduced models, and it is shown that any such PROM can be obtained from a sequence of 1-dimensional projection reductions. Investigating the error system product form, we then define interface-invariant PROMs, model order reductions with projection-invariant input and output matrices, and it is shown that for such PROMs the error product systems are strictly proper. Furthermore, exploiting this structure, an analytic $\mathcal{H}_{\infty}$ reduction error bound is obtained and an $\mathcal{H}_{\infty}$ bound optimization problem is defined. Interface-invariant reduced models are natural to graph-based model reduction of multi-agent systems where subsets of agents function as the input and output of the system. In the second part of this study, graph contractions are used as a constructive solution approach to the $\mathcal{H}_{\infty}$ bound optimization problem for multi-agent systems. Edge-based contractions are then utilized in a greedy-edge reduction algorithm and are demonstrated for the model reduction of a first-order Laplacian controlled consensus protocol.