Phase Margin Degradation from Digital Filter Latency in Control Loops
Digital filter latency introduces pure time delay into feedback loops, eroding phase margin at crossover frequencies by 360 degrees per cycle of propagation delay.
Sensitivity Function
Disturbance Transfer
Closed-loop feedback structures alter the relationship between external disturbances and output tracking through frequency-dependent dynamic attenuation. The sensitivity function defines the complex frequency response mapping disturbance inputs directly onto system errors within an active feedback control loop. Represented mathematically as the reciprocal of one plus the open-loop transfer function, it quantifies feedback efficiency and disturbance rejection capability.
The relationship governs linear time-invariant systems, losing direct predictive utility when loop elements operate in non-linear or saturated regimes.
Spectral Profile
At low frequencies within the control bandwidth, open-loop gain remains large, driving the value of the function toward zero and suppressing disturbances effectively. Near the loop crossover frequency, open-loop gain approaches unity while phase lag nears one hundred and eighty degrees, causing the denominator to shrink. This interaction forces the magnitude of the function to rise above unity, creating an unavoidable amplification zone where disturbances are exaggerated rather than suppressed.
At very high frequencies beyond the loop bandwidth, open-loop gain drops to zero, and the function asymptotically approaches unity, leaving high-frequency disturbances unattenuated.
Stability Indices
The peak magnitude of this response provides a metrological indicator of relative loop stability and robustness against unmodeled dynamics. A high peak value indicates poor gain and phase margins, warning of pronounced resonance, long settling times, and sensitivity to physical parameter variations. Conservative industrial motion and process controllers target peak values between one point three and two point zero, corresponding to roughly six decibels of maximum disturbance amplification.
Higher peak values leave the control loop vulnerable to limit-cycle oscillations when physical plant parameters drift because of thermal expansion or mechanical wear.
Measurement Verification
Dynamic signal analyzers measure this transfer function in physical hardware by injecting wideband noise or swept-sine perturbations into the loop error junction. The analyzer records the ratio of the resulting error signal to the injected disturbance signal across the target operational bandwidth. Qualification protocols verify that the measured peak remains below specified thresholds across all operating envelopes, including variations in supply voltage and ambient temperature.
Production acceptance documents must include swept plots of the function alongside calculated stability margins to validate control law robustness before field deployment. Measurements showing unexpected resonance peaks indicate unmodeled physical compliance or excessive sensor filter delays.