Passivity degradation under the discretization with the zero-order hold and the ideal sampler

Quantitative study is made on how much passivity is lost when a continuous-time nonlinear system is discretized for digital implementation of a controller. The standard discretization with the zero-order hold and the ideal sampler is mainly considered though other types of discretization are also considered. The result can be used for the choice of the sampling period. Namely, the sampling period has to be small when the original continuous-time system is oscillatory. The general input-output framework is used for the discussion.