In this paper, an actuator-fault detection and isolation (FDI) approach is proposed. The FDI approach is based on a bank of interval observers, each of which is designed to match a healthy or faulty system mode. To ensure reliable FDI for all considered actuator faults, a collection of invariant set-based FDI conditions are established for the proposed technique. Under these guaranteed FDI conditions, all the considered faults can be detected and isolated during the transition induced by fault occurrences. Comparing with the existing set-based FDI approaches, the advantage of the proposed technique consists in that it combines the advantages of interval observers in the transient-state functioning and the advantages of invariant sets in the steady-state functioning. This paper is completed with the study of a continuous stirred-tank reactor (CSTR), which illustrates the effectiveness of the proposed method.