In several applications, the main objective of the controllers is to ensure some process variable to track (and/or reject) a sinusoidal reference (disturbance) signal. To that end, two control approaches are defined: those based on the sinusoidal signals, and those based on the envelope (amplitude and phase) of these signals. The former one, which we name direct approach, corresponds to the classical architectures used in control engineering. This approach offers a broad range of methods to design linear controllers with guarantees of stability and performance. In general, envelope-based approaches are nonlinear and do not provide those guarantees. However, they allow obtaining controllers with a bandwidth much smaller than it would have in the direct approach. In this paper, we use time-varying complex phasors to describe the envelopes of the signals in the system. Then, we show that with a suitable reformulation, the system remains linear. Hence, links between these approaches are established under the assumption that a phasor can be instantaneously defined from a modulated signal (ideality). We propose thus two methods to design a phasor-based controller: one considering the ideal case and another where nonidealities are taken into account. Numerical examples, based on the operation of MEMS gyroscopes, show the effectiveness of these methods.
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