controlled $k$ fusion frame

In this manuscript, we study the relation between K-fusion frame and its local components which leads to the definition of a $C$-controlled $K$-fusion frames, also we extend a theory based on K-fusion frames on Hilbert spaces, which prepares exactly the frameworks not only to model new frames on Hilbert spaces but also for deriving robust operators. In particular, we define the analysis, synthesis and frame operator for $C$-controlled $K$-fusion frames, which even yield a reconstruction formula. Also, we define dual of $C$-controlled $K$-fusion frames and study some basic properties and perturbation of them.