Variable-Order Fuzzy Fractional PID Control in a Reduced Quadrotor Altitude Benchmark: Paired Monte Carlo Evaluation and Mechanism-Oriented Analysis

This paper compares fixed-order (FO) and variable-order (VO) fuzzy fractional PID control in a reduced numerical benchmark for quadrotor altitude tracking. The study asks whether online order scheduling changes paired performance when the fuzzy gain structure, gain channels, fractional approximation, plant, reference, random realizations, and evaluation protocol are fixed. The benchmark comprises vertical double-integrator dynamics, a first-order actuator lag, and asymmetric clipping of an equivalent net-thrust command; it is not full-vehicle or hardware validation. Both controllers use the same two-input, five-output Sugeno structure. FO is optimized first, and VO then optimizes four scheduling parameters while retaining the FO gain channels. Evaluation uses a six-segment reference, a composite index, 100 pre-specified paired Monte Carlo seeds, paired effect estimates, two counterfactual order variants, and an existing stronger-disturbance test without retraining. At baseline, VO reduces mean J by 3.97%, ITAE by 7.06%, positive-upper-limit near-saturation exposure by 43.36%, and RMS input by 3.18% while increasing the total in-band ratio by 6.80%. Overshoot and robust settling time show no statistically supported improvement. The counterfactual results are more consistent with state-dependent variation than a fixed-order shift alone but do not form an additive causal decomposition. Within this benchmark, the results support a measurable role for online order scheduling but do not establish full-vehicle or hardware performance.