A necessary and sufficient frequency domain criterion for the passivity of SISO sampled-data systems

We present a frequency domain solution to the sampled-data passivity problem. Our analysis is exact in the sense that we take into account the intersample behavior of the system. We use frequency response (FR) operators to first obtain necessary conditions on the sampling rate T and the relative degree of the open-loop transfer function G₁₁ for achieving a passive continuous-time closed-loop system. Then, assuming passivity of G₁₁ and closed-loop stability, we derive a necessary and sufficient condition for discrete-time controllers that render a passive closed-loop system. We apply the obtained results to the problem of stability of haptic systems.