A. Alavian and M.C. Rotkowitz
Enhanced Approximation of the Decentralized Assignability Measure by Subgradient Methods
Proceedings of the 22nd International Symposium on Mathematical Theory of Networks and Systems,
pp. 511-514, July 2016.

Abstract

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, 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, in particular the decentralized assignability measure of Vaz and Davison in 1988, 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.

This problem has been addressed by utilizing modern optimization techniques that tackle the decentralized assignability measure. The main difficulties which have previously precluded its widespread use, are that it involves the minimization of the n-th singular value of a matrix, which must further be minimized over a power set of the subsystems. A recently developed algorithm uses the nuclear norm in place of the singular value, then employs Alternating Direction Method of Multipliers (ADMM) to decouple those variables that cause further non-convexity. In this paper, we aim to improve this algorithm by using its solution as a starting point for a tuning method that utilizes subgradient methods to directly target the n-th singular value, which is the original objective of the decentralized assignability measure.