The waterbed effect on quasiperiodic disturbance observer: Avoidance of sensitivity tradeoff with time delays

In linear time-invariant systems, the sensitivity function to disturbances is designed under a sensitivity tradeoff known as the waterbed effect. To compensate for a quasiperiodic disturbance, a quasiperiodic disturbance observer using time delays was proposed. Its sensitivity function avoids the sensitivity tradeoff, achieving wideband harmonic suppression without amplifying aperiodic disturbances or shifting harmonic suppression frequencies. However, its open-loop transfer function is not rational and does not satisfy the assumptions of existing Bode sensitivity integrals due to its time delays. This paper provides Bode-like sensitivity integrals for the quasiperiodic disturbance observer in both continuous-time and discrete-time representations and clarifies the avoided sensitivity tradeoff with time delays.