The inversion problem of recovering the Hamiltonian and dipole moment is considered in a quantum control framework. The inversion process uses as inputs some measurable quantities (observables) for each admissible control; however the implementation of the control is noisy (the perturbations are additive constants in a countable set of values) and therefore the data available is only in the form of the law of the measured observable. Nevertheless it is proved that the inversion process still has unique solutions (up to some phase factors). Numerical illustrations support the theoretical results.