Nonlinear oscillator design based on phase reduction method for closed‐loop system

Abstract

Summary To widely utilize the synchronization properties of nonlinear oscillators in the control and signal processing engineering field, a common design method for a nonlinear oscillator model would be helpful. The output response of a nonlinear oscillator excited by a periodic signal implicitly includes the information of both the instantaneous phase and the amplitude of the input signal. Explicitly expressing the dynamics of both phase and amplitude with the phase reduction method, and organizing them by the number of events in one period, could enable the design of a nonlinear oscillator model in the same manner that can synchronize with an arbitrary cyclic phenomenon in a broad sense. Assuming the cyclic phenomenon is a single periodic signal, the property of the generalized nonlinear oscillator model has the role as a common‐dimensional observer to attain the pseudo‐property of both iso‐damping and flat‐gain. Thus, this article determines the essential role of the nonlinear oscillator inserted in the closed‐loop system through investigating the input–output characteristic of the generalized nonlinear oscillator model. As a result of the numerical simulation using a simple vibration excitation and suppression system, it was found that using the generalized nonlinear oscillator model enables us to easily build the closed‐loop system including the pseudo‐property of both iso‐damping and flat‐gain.

Relation of nonlinear oscillator design based on phase reduction method and fractional derivative

“…As presented above, assuming that the cyclic phenomenon is a single periodic signal, the generalized nonlinear oscillator model has the role as a common dimensional observer to attain the pseudo‐property of iso‐damping and flat‐gain. More design examples and demonstrations are described in Hongu and Iba 40 …”

“…Hence, we attempted to establish a common design method for a nonlinear oscillator based on the phase‐reduction method 40 as follows. …”

“…Next, the input‐output characteristics of the generalized nonlinear oscillator model are discussed. We reuse the content described in Hongu and Iba 40 here. Finally, it is verified that utilizing fractional calculus enables us to improve the estimation accuracy of the phase and amplitude of the signal using the generalized nonlinear oscillator model.…”