Polynomial optimization methods for determining lower bounds on decentralized assignability

This work considers the determination of non-binary measures of controllability or robustness with respect to decentralized controllers. A measure developed by Vaz and Davison in 1988 nicely captures the distance from a plant to the closest one with a fixed mode, and ties it to eigenvalue assignability; that is, how much effort is at most required to move the modes a given amount with the prescribed information structure. This metric is intractable to compute, but recent work has been very successful in finding close upper bounds. Finding lower bounds is not only important for providing guarantees on where the true metric lies, but is typically more important since determining whether the metric is bounded away from zero corresponds to whether the system can be controlled at all. This paper will address these lower bounds, in particular by using the Courant-Fischer formulation of singular values, we will formulate our problem as a polynomial optimization problem, for which we can then use Sum-of-Squares (SOS) techniques to find a lower bound.