Controller-Conditioned Empirical Frequency Responses of Scalar Polynomial Systems under Feedback

A nonlinear system has no input-independent transfer function, yet ratios of measured spectra are computed for such systems. For the scalar polynomial class ẋ = Σk ak xk + u under digital PID feedback, the controller-conditioned empirical frequency response, called the attainable frequency response (AFR), is defined as an experiment-dependent object with a valid-frequency mask. Inverted with the empirical controller ratio of the record, it equals the direct ratio X/U. Inverted with the known controller, it differs from X/U by an exact end term of the finite record. Its sensitivity δH/[H(1 − H)] amplifies errors that do not pass through the loop in the indirect computation, while noise inside the loop does not show this amplification, to first order for an LTI plant on a periodic record and in simulation for the x2 plant. The direct ratio X/U does not show it in the cases examined. Single non-periodic records give estimates dominated by the transient term, except for the two records tested that start and end at rest. Removing that term with the local polynomial method recovers known LTI plants from the same records to 1.4 × 10−3. With a periodic reference the loop reaches steady state within five periods and the response of ẋ = x2 + u departs from the linearization 1/s at low frequency by an amount that grows with the reference level, up to 310% at 1 Hz for |x| ≤ 8. On the independent references tested with a high-gain PID (Kp = 500, Kd = 0.2, Ki = 750), the linearization inside the loop predicts the nonlinear loop to 1% for |x| ≤ 5, the closed-loop best linear approximation and a stepped sine recover it, and the one-record response does not.