K-theory of geometric modules with fibred control

Controlled algebra plays a central role in many recent advances in geometric topology. This paper studies the iteration construction that was present from the very origins of the theory but started being exploited only recently. We develop the general framework for fibred control, prove localization theorems required for fibrewise excision, and then prove several versions of fibrewise excision theorems. However, we also demonstrate how the standard tools break down in the presence of new equivariant phenomena which require advanced localization methods.