This paper introduces a robust method for performing active actuator fault detection and diagnostics (FDD) in heating ventilation and air conditioning (HVAC) systems. The proposed actuator FDD strategy, for testing whether an actuator is stuck in a given position, is designed on using an invariant hypothesis testing approach and is an improvement of a previous strategy that employed an adaptive detection strategy. The parameter-invariant detector is formulated to provide a constant detection performance, invariant to unknown building parameters, and it is described how this approach can replace the adaptive detector in the previous work. A closed-loop experimental HVAC testbed at the KTH Royal Institute of Technology campus in Stockholm, Sweden is introduced and employed to evaluate the parameter-invariant detector.