Safety control constitutes an important aspect of hybrid systems and control. Invariance controllers guarantee that a system can stay within a given safe set for all future time. While abstraction-based approach to control synthesis takes the advantages of formal methods in automatic synthesis, abstractions often introduce spurious transitions that can lead to failure of controller synthesis for a given specification, even though the original system can be controlled to satisfy this specification. For discrete-time switched systems, this paper presents an interval arithmetic-based approach for invariance control synthesis. The main synthesis algorithms rely on partition refinement techniques and backward reachable set computation using interval analysis. The use of rigorous numerics allow us to prove formal guarantees of finding an invariance controller via abstraction refinement, provided that a robustly invariant condition is satisfied for the original switched system. The results are illustrated with polynomial dynamics.