Comparative simulation of fractional-order PD sliding mode and fuzzy logic controllers for a second-order discrete-time nonlinear system

Abstract

Tight output regulation in power converters and electric drive systems requires a control strategy that simultaneously minimizes tracking error and maintains smooth actuation-two objectives that are intrinsically in tension for nonlinear, parameter-varying plants. Despite the widespread deployment of fractional-order PD sliding mode control (FOPD-SMC) and Mamdani fuzzy logic control (FLC) in this domain, no prior study has placed them in a direct, metric-identical comparison on a common plant. The present work closes this gap by implementing both controllers on the same second-order discrete-time nonlinear plant-representative of DC-DC converter output dynamics and motor-drive input-output behavior and evaluating them under a composite reference that combines sinusoidally-modulated ramps with step transitions, scored by the integral of squared error (ISE) and integral of absolute error (IAE). FOPD-SMC achieves ISE= 1.639 × 10-2 and IAE= 3.345 × 10-2, outperforming FLC by 87.6% and 50.4%, respectively; the advantage originates from the non-integer memory embedded in the sliding surface via the Gr¨unwald-Letnikov operator and from the explicit decomposition of the control law into nominal-tracking and robustness components. FLC, conversely, produces a chattering-free, continuously varying control signal a structural consequence of smooth Gaussian membership functions and linguistic rule aggregation, at the cost of a mean absolute tracking error twice that of FOPD-SMC. These findings establish a quantitative selection criterion: FOPD-SMC is recommended when tight voltage or current regulation is the primary objective, while FLC is preferred where smooth torque delivery and reduced actuator stress outweigh marginal gains in tracking accuracy.