Frequency domain interpretations of the invariance condition of sliding mode control theory

The sliding-surface design problem for unmatched uncertain systems is considered. In the literature, it has been shown that if a system with unmatched uncertainties satisfies an invariance condition, then there exists a linear switching surface such that the sliding-mode dynamics restricted to the switching surface are not only stable but completely invariant to unmatched uncertainties. Frequency domain interpretations of the invariance condition of the sliding-mode control theory are given. It is shown that a certain H_infin-norm bound constraint should be satisfied in order to guarantee the invariance condition. It is also shown that the invariance condition is guaranteed for any unmatched uncertainties satisfying a certain minimum phase condition. These results can be used to efficiently design a sliding surface. Finally, the effectiveness of the results is verified via some numerical design examples.