Abstract
Adaptive oscillators (AOs) are nonlinear oscillators with plastic states that encode information. Here, an analog implementation of a four-state adaptive oscillator, including design, fabrication, and verification through hardware measurement, is presented. The result is an oscillator that can learn the frequency and amplitude of an external stimulus over a large range. Notably, the adaptive oscillator learns parameters of external stimuli through its ability to completely synchronize without using any pre- or post-processing methods. Previously, Hopf oscillators have been built as two-state (a regular Hopf oscillator) and three-state (a Hopf oscillator with adaptive frequency) systems via VLSI and FPGA designs. Building on these important implementations, a continuous-time, analog circuit implementation of a Hopf oscillator with adaptive frequency and amplitude is achieved. The hardware measurements and SPICE simulation show good agreement. To demonstrate some of its functionality, the circuit’s response to several complex waveforms, including the response of a square wave, a sawtooth wave, strain gauge data of an impact of a nonlinear beam, and audio data of a noisy microphone recording, are reported. By learning both the frequency and amplitude, this circuit could be used to enhance applications of AOs for robotic gait, clock oscillators, analog frequency analyzers, and energy harvesting.
Highlights
Adaptive oscillators are similar to phase-locked loops, except they have an additional, direct injection of the external forcing on the oscillator itself
The Hopf oscillator is a nonlinear oscillator described by the following ordinary differential equations (ODEs) [27]: x_ 1⁄4 ðm À ðx2 þ y2ÞÞx À o0y þ kða sinðOt þ 0ÞÞ ð1Þ
The Hopf Adaptive Frequency Oscillator (HAFO) may learn the frequency of an external stimuli; this implies that the HAFO is a three-state nonlinear oscillator with dynamical plasticity
Summary
Adaptive oscillators are similar to phase-locked loops, except they have an additional, direct injection of the external forcing on the oscillator itself. The resulting system provides the dynamics of 1) A two-state regular Hopf oscillator, 2) A three-state AO that can learn the frequency of an input sinusoid, and 3) A four-state AO that can learn the frequency and amplitude of an input sinusoid and extends the work of [1, 2] These schemes are similar to AOs that have previously improved phase-locked loop designs through Lyapunov treatments [26]. We show a topology that extends the number of adaptive states previously implemented and show direct injection of a forcing signal This AO is similar to the phase-locked loop Lyapunov design [26]; in the Lyapunov design, the forcing frequency needed to be directly inputted into the system. This treatment results in 1) the addition of a fourth, plastic state and 2) three systems that inform circuit designs based on state variable networks
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