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.
In this paper, we address this problem by utilizing modern optimization techniques to 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. We will propose three methods to address its computation. First, we will discuss a relaxed convex problem, using the nuclear norm in place of the singular value, and expressing the power set minimization as binary constraints which can be relaxed to the hypercube. Our second algorithm simply entails rounding when the first method fails to reach a corner of the hypercube. Our final algorithm is developed using the Alternating Direction Method of Multipliers (ADMM), and is shown to decouple the effects of the binary variables, such that they can be optimized directly with per-iteration computations scaling linearly, rather than exponentially, with the number of subsystems. This final method is shown to produce results which closely track the assignability measure across a variety of fixed mode types.