Decentralized integral controllability (DIC) is an important property of industrial and process multivariable control systems. In this paper, some of the previously proposed conditions for the study of the DIC of certain systems are extended to treat systems with correlated and independent uncertainty. These conditions are based on the concept of the stability of Polytopic and Interval family of matrices, and evaluation of the Lyapunov condition only at special vertices of these families. By comparison with previous sufficient conditions, and known necessary conditions, it is demonstrated that the newly proposed criteria are significantly less conservative, especially in the case of correlated uncertainties. (c) 2013 Elsevier Ltd. All rights reserved.