On Computing the Real Decentralized Fixed Mode Radius

This paper considers 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, but this gives a binary answer with no information about robustness. Measures have been developed to further determine how far a system is from having a fixed mode, but these involve intractable optimizations and have thus not been applied to the large-scale systems for which they would be able to provide valuable sensitivity information. Recent work has addressed the decentralized assignability measure of Vaz and Davison from 1988, or complex Decentralized Fixed Mode (DFM) radius, which captures the smallest complex perturbation of the state-space matrices which would result in a fixed mode. This involved a minimization over a particular singular value of a matrix variable, as well as over the power set of the subsystems. Scalable methods were developed to compute close upper bounds for this metric, along with methods to compute lower bounds. In this paper we address the more realistic and less conservative measure of the smallest real perturbation of the state-space matrices which results in a fixed mode, or real DFM radius, which was developed by Lam and Davison. This involves two difficult minimizations similar to the ones encountered for the complex DFM radius, along with a non-concave maximization over an additional parameter. We adapt and apply the tools developed previously while further addressing the maximization difficulty, developing scalable methods of approximating the real DFM radius, and further discussing methods of obtaining upper and lower bounds.