Complexity of checking the existence of a stabilizing decentralized controller

Given an interconnected system, this paper is concerned with the time complexity of verifying if any given unrepeated mode of the system is a decentralized fixed mode (DFM). It is shown that checking the decentralized fixedness of any distinct mode is tantamount to testing the strong connectivity of a digraph formed based on the system. It is subsequently proved that the time complexity of this decision problem using the proposed approach is the same as the complexity of matrix multiplication. This work concludes that the identification of distinct decentralized fixed modes (by means of a deterministic algorithm, rather than a randomized one) is computationally very easy, although the existing algorithms for solving this problem would wrongly imply that it is cumbersome. This paper provides not only a complexity analysis, but also an efficient algorithm for tackling the underlying problem.