On robust stability of disturbance observer for sampled-data systems under fast sampling: An almost necessary and sufficient condition

Despite increased interest on discrete-time disturbance observer (DOB) in both theory and application, study of robust stabilization via the DOB does not seem to be mature. This is because most of the existing studies on robust stability are based on the small-gain theorem, so that the results are just sufficient conditions and the amount of uncertainty that the DOB-based control system can tolerate is conservative. In this paper, motivated by a recent work on the continuous-time case, we present an almost necessary and sufficient condition for robust stability of the sampled-data system controlled by a discrete-time DOB under fast sampling. In particular, our study clarifies the phenomenon that the sampling process can hamper stability of the DOB-controlled systems by generating additional (and possibly nonminimum-phase) zeros, and explains why the blind discretization of the continuous-time DOB controller using fast sampling may fail. For the robust stabilization against both these extra zeros and plant uncertainty, we also propose a new design methodology for the nominal model and the Q-filter. It turns out that arbitrarily large uncertainty can be compensated by appropriately designing the Q-filters and by fast sampling. A benchmark problem is revisited to illustrate the validity of the proposed analysis and design method.