Single-tone narrowband

Adaptive notch control targets one sinusoidal component with a quadrature pair.

The controller synthesizes cosine and sine references at the tracked frequency, then adapts two weights to cancel the measured component after accounting for secondary-path delay or compensation filtering. It is the narrowband choice for a dominant rotating-order or single-frequency disturbance.

Core Equations

The synthesized references are:

\[x_0(n) = A \cos(k \omega_0 n)\] \[x_1(n) = A \sin(k \omega_0 n)\]

The canceling signal is the weighted combination:

\[y(n) = w_0(n) x_0(n) + w_1(n) x_1(n)\]

and the update follows the same ANC sign convention used elsewhere:

\[w_i(n+1) = w_i(n) - \mu e(n) x'_i(n)\]

where $x’_i(n)$ is either delay-compensated or filtered through the secondary-path estimate.

Benchmark Interpretation

The primary benchmark shows the notch controller converging quickly in the single-tone scenario and delivering a solid narrowband attenuation result. The scenario comparison also shows the limit of the method: once the disturbance drifts or stops being a single stable tone, a fixed-frequency notch section is no longer well matched.

Use the right model class

Adaptive notch is excellent when the disturbance really is one tracked sinusoid. It is not a broadband controller and should not be evaluated as if it were one.

Tuning Notes

The tracked frequency matters more than controller order because the controller only has two adaptive weights.
Delay compensation or secondary-path compensation is what keeps the cosine and sine channels aligned with the physical plant.
When the disturbance contains several harmonics instead of one, move to the multi-tone controller instead of trying to stretch a single notch section too far.

Using It From C++

#include "anc/adaptive_notch_controller.h"

anc::AdaptiveNotchConfig config;
config.frequencyRadians = 2.0 * 3.14159265358979323846 / 64.0;
config.amplitude = 1.0;
config.stepSize = 0.02;
config.delaySamples = 1;
config.secondaryEstimate = {0.0, 0.12, 0.38, 0.21, -0.05};

anc::AdaptiveNotchController controller(config);

for (double error : errorSignal) {
  const double control = controller.synthesize();
  sendToActuator(control);
  controller.adapt(error);
}

CLI Reproduction

build\Debug\anc_cli.exe `
  --algorithm notch `
  --summary analysis_output\raw\notch_single_tone_summary.csv `
  --timeseries analysis_output\raw\notch_single_tone_timeseries.csv `
  --period-samples 64 `
  --delay-samples 1 `
  --step-size 0.02