In this paper, we give a stronger version of the notion of behavioural controlled-invariance introduced in a previous paperin the context of regular partial interconnections. In such interconnections, the variables are divided into two sets: the variables to-be-controlled and the variables on which it is allowed to enforce restrictions (control variables); moreover, regularity means that the restrictions of the controller do not overlap with the ones already implied by the laws of the original behaviour. A complete characterisation of strong controlled-invariance for nD behaviours is derived making use of a special controller behaviour known as the canonical controller.