Uncertainty in Sampled Systems

The recently obtained evidence of the need for a positive real element in an adaptive system leaves us with a disturbing gap in adaptive control theory. It is a fact that in some applications adaptive controllers are performing well in practice. How can these systems behave well in practical situations which must contain modeling error? This paper introduces a preliminary result which indicates that it may be possible to maintain the needed positive real system in the presence of modeling error. The result shows that if a continuous-time system with large high frequency uncertainty is treated appropriately with antialiasing filters and sampled slowly enough, the resulting discrete-time system may contain very little uncertainty. With small enough uncertainty in the plant, a positive real system in the adaptive loop is possible.