On a Hankel-based Measure of Decentralized Controllability and Observability

This paper discusses the controllability of linear time-invariant (LTI) systems with decentralized controllers. Whether an LTI system is controllable (by LTI controllers) with respect to a given information structure can be determined by testing for fixed modes. Measures have been developed to further determine how far a system is from having a fixed mode, but these measures cannot actually be computed in most cases. We thus seek an easily computable, non-binary measure of controllability for LTI systems with decentralized controllers of arbitrary information structure. We develop a measure based on the Hankel operator, that combines the controllability gramian, the observability gramian, and a version of the cross-gramian which incorporates the information structure.