Tuning of PID controller for tractor power take-off shaft rotational speed using particle swarm optimization method
DOI:
https://doi.org/10.31210/spi2026.29.02.30Keywords:
PID controller, particle swarm optimization, power take-off, tractor, Hurwitz criterion, stability, robustnessAbstract
The paper investigates the application of the Particle Swarm Optimization algorithm for automatic tuning of the PID controller coefficients for tractor power take-off shaft rotational speed control. A generalized mathematical model of the system with parameters typical for medium-class tractors was developed, with engine gain Ke = 50 N·m/unit, engine time constant Te = 0.3 s, reduced moment of inertia J = 2.5 kg·m², viscous friction coefficient b = 0.8 N·m·s/rad, PTO gear ratio i = 3.5 and transmission efficiency η = 0.92. Two approaches to automatic PID controller tuning were investigated: the Ziegler-Nichols method and the Particle Swarm Optimization method, the latter running a swarm of 30 particles over 80 iterations and converging after about 20. The Integral of Time-weighted Absolute Error (ITAE) was selected as the objective function for the particle swarm optimization. Unlike other integral criteria, the ITAE criterion, due to the incorporation of the time parameter, penalizes errors that persist over extended periods. This corresponds to the specifics of PTO control, where a short-term drop in rotational speed during a sudden load change is inevitable and acceptable, whereas prolonged oscillations in rotational speed lead to uneven processing, equipment wear, and excessive fuel consumption. The stability of the closed-loop system was proven using the Routh-Hurwitz criterion with a complete derivation of the characteristic polynomial, from which an explicit upper bound on the integral gain was obtained; both tunings satisfy the criterion, yet the determinants of the optimized controller exceed those of the Ziegler-Nichols tuning by several orders of magnitude (Δ₂ = 42 273 against 102.6). Robustness analysis was performed using the statistical Monte Carlo method under simultaneous ±30 % variation of the plant parameters (J, b, Ke, Te) across 100 realizations. The results showed that the PSO-optimized controller (Kp = 3.63, Ki = 15.00, Kd = 0.25) provides significantly lower overshoot (12.4 % against 97.1 %) and a substantially larger phase margin (79.0° against 33.8°) compared to the Ziegler-Nichols method, cuts the rise time from 0.248 s to 0.029 s at the reference speed of 540 rpm and the ITAE from 16.06 to 0.177, and maintains stability across all realizations of the Monte Carlo robustness analysis.
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