Modal Barriers to Controllability in Networks with Linearly-Coupled Homogeneous Subsystems
This work provides theoretical insights for engineers designing built networks (e.g., power grids) where subsystem interfaces impose fundamental controllability barriers, distinguishing them from multi-agent systems where protocols can be designed.
The paper examines controllability in networks of homogeneous linear subsystems with linear couplings, identifying network-invariant modes that inherently limit controllability unless a sufficient fraction of subsystems are actuated. It shows that special-repeat modes do not affect controllability.
The controllability of networks comprising homogeneous multi-input multi-output linear subsystems with linear couplings among them is examined, from a modal perspective. The eigenvalues of the network model are classified into two groups: 1) network-invariant modes, which have very high multiplicity regardless of the network's topology; and 2) special-repeat modes, which are repeated for only particular network topologies and have bounded multiplicity. Characterizations of both types of modes are obtained, in part by drawing on decentralized-fixed-mode and generalized-eigenvalue concepts. We demonstrate that network-invariant modes necessarily prevent controllability unless a sufficient fraction of the subsystems are actuated, both in the network as a whole and in any weakly-connected partition. In contrast, the multiplicities of special-repeat modes have no influence on controllability. Our analysis highlights a distinction between built networks where subsystem interfaces may be unavoidable barriers to controllability, and multi-agent systems where protocols can be designed to ensure controllability.