In the paper, we study the stability and performance robustness of composite feedback systems under decentralised control. It is shown that, if the perturbed system transfer matrix is a compositeH-matrix for everyson the Nyquist contour, then stabilisation of the diagonal blocks of the composite system implies stabilisation of the overall system, provided the number of unstable poles of the composite system coincide with those of the diagonal blocks. This is then used to define decentralised controllers for robust stability. When restricted to regular multivariable systems, the new criterion leads to an interesting and simple condition on the peak magnification factorMpof the closed-loop system.