Designing Nonlinear Zero Dynamics to Reject Periodic Waveforms

In linear systems, designers use zeros for loop-shaping and the attenuation of disturbances. For example, imaginary-axis zeros can be used to reject sinusoids at a specific frequency such as a 50 Hz or 60 Hz power line interference. If the disturbance is nonsinusoidal, repetitive control methods may be used but additional controller states for each harmonic are required. Thus, rejecting periodic disturbance that are nonsinusoidal with low controller order remains an open problem. In this paper, we constructively design low-order nonlinear dynamics that reject a nonsinusoidal disturbance. We identify a particular case of the nonlinear regulator theory wherein the Byrnes–Isidori PDE has a simple solution. This case leads to a constructive procedure for designing nonlinear zero dynamics in systems with input disturbances. The constructive procedure is accessible to designers who do not have experience with the nonlinear geometric control theory.