Integrity of systems under decentralized integral control

A decentralized controller that maintains closed loop stability, when the individual controllers fail or are taken out of service, provides fault tolerance and flexibility in operation. Recently, necessary and sufficient conditions [Gündes, Kabuli, 2001. IEEE Transactions on Automatic Control, 46(2), 296-301] for the existence of a block decentralized controller with integral action for a system to possess integrity against controller failures were proposed. In this paper, these conditions are equivalently expressed using the well-known notions of Niederlinski index (NI) and block relative gain (BRG). The alternate representation implies that under minor assumptions, the available necessary conditions based on NI and BRG are actually both necessary and sufficient. We also show that confirming the existence of a block decentralized controller with integral action such that the system has integrity is NP-hard.