On multivariable stability in the grain space

General existence conditions under which the stability of the individual loops of a multivariable system in a given compact and bounded gain space imply the global asymptotic stability of the multivariable system in the same space are very useful in practice, and they essentially indicate when in principle it is possible to obtain stable closed loop performance of a multivariable system by tuning every loop separately. Such conditions are particularly useful for fault tolerant control of multivariable systems. The authors give the required necessary and sufficient conditions and effectively unify all previous work on diagonal stabilizability which includes as special cases the diagonal dominance, the H-matrix, the GKK-matrix, and the decentralized integral controllability (DIC) conditions reported previously in the literature. Some application examples are included