The problem of robust output feedback control for linear systems with unknown inputs based on hierarchical super twisting (HST) observation and identification is considered. Theoretically, the finite time convergence of HST observation and identification errors allows the compensation of the matched uncertainties. The accuracy of the observation, the identification, and the compensation is discussed in terms of upper bounds for deterministic noise, sampling step, and execution time. The LQ control for the nominal system is considered as a case study. Two methods to be compared are proposed: compensation via the identification of the uncertainties, which allows to avoid the chattering, and compensation via integral sliding mode control. Based on this comparison, the designer could select the more adequate controller for the system. Numerical simulations and experimental results are given for an inverted rotary pendulum system.