This article deals with fault tolerant multisensor control schemes for systems with linear dynamics. Positive invariance is a common analysis and control design tool for systems affected by bounded constraints and disturbances. This article revisits the construction of epsilon-approximations of minimal robust positive invariant sets for linear systems upon contractive set-iterations. The cases of switching between different sets of disturbances and the inclusion of a predefined region of the state space are treated in detail. All these results are used in multisensor control schemes which have to deal with specific problems originated by the switching between different estimators and by the presence of faults in some of the sensors. The construction of positive invariant sets for different operating regimes provides, in this context, effective fault detection information. Within the same framework, global stability of the switching strategies can be assured if the invariant sets topology allows the exclusive selection of estimates obtained from healthy sensors.