Novel methods for estimating bandwidth and stability margins of pilot-in-loop systems

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Bachelder, E.
Aponso, B.
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A standard metric to estimate system bandwidth is the pilot cutoff frequency, which is the frequency at which the cumulative power ratio of the pilot’s control response equals 0.5. Implicit with using the pilot cutoff frequency is that the vehicle output being tracked is approximately the integral of the control input (i.e., rate-commanded). However, errors in estimation will occur when the technique is applied to systems whose tracked outputs differ significantly from this assumption. Furthermore, the type of disturbance spectrum impinging on the tracking task has a strong influence. The effect of these factors on pilot cutoff frequency is examined theoretically, and a method for transforming the cumulative control power ratio is developed that enables the transformed ratio to be applied to any tracked state. A method for determining effective time delay, and effective phase and gain margin from the slope of the transformed cumulative power ratio is also developed. Assuming knowledge of the disturbance spectrum and vehicle dynamics, two techniques are offered to estimate system bandwidth and time delay using: 1) A cutoff frequency (dependent on the forcing function) using the transformed stick response cumulative power ratio; 2) Iteration on the crossover frequency and time delay parameters in the closed-loop Crossover Model until a best match is found between the transformed cumulative power ratios of the modeled and observed stick response. The latter approach does not require that the forcing function contain power extending to or beyond crossover. The development demonstrates that bounding the upper frequency of the computed control power is a critical step of the estimation process, as this reduces the effect of uncorrelated high frequency content arising from sources such as the neuromuscular mode and harmonics of pulse-like control on the estimates. A unique bi-directional spatial filter that allows the frequency and slope from cumulative power ratios to be continuously analyzed when using discrete spectra forcing functions (such as sum-of-sines) is developed. The filter also improves estimation when the forcing spectrum is continuous. A new system bandwidth estimation method that uses the vehicle output cumulative power ratio is proposed, which unlike the cumulative stick power approach does not require an assumption about or measurement of the vehicle dynamics. This technique transforms the output by simple differentiation, allowing similar application of the stick power methods (cumulative power ratio cutoff and model matching). Finally, effective time delay and crossover frequency are estimated using the ideal Crossover Model by matching the observed system time response. The novelty introduced here is the that the effective stability margin arising from these two effective parameters closely coincides with the actual system stability margins (phase and gain), irrespective of the differences between the idealized and actual dynamics. This allows the accuracy of any bandwidth estimate to be assessed - establishing the actual bandwidth associated with human-in-loop operation has heretofore proven elusive. The technique lends itself to both manual and automated systems and will be useful for assessing handling qualities. Pilot data from a simulation tracking experiment is used to demonstrate the efficacy of these various estimation techniques.