Two quantum control systems that are driven by an external field are said to be input–output equivalent if, for any control field, the measured value of a given observable is the same. Equivalent models cannot be distinguished by experiments involving state evolutions and measurements. In this paper, we characterize the equivalent models of networks of spin driven by electro-magnetic fields for which the expectation value of the total magnetization is measured. Extending previous results and definitions that only dealt with the case of a single measurement, we describe the class of equivalent models under a sequence of Von Neumann measurements. The results are motivated by the problem of parameter identification for Heisenberg spin systems modeling molecular magnets.