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Fuel cells are electrochemical devices that convert the chemical energy of a gaseous fuel directly into electricity. They are widely regarded as potential future stationary and mobile power sources. The response of a fuel-cell system depends on the air and hydrogen feed, flow and pressure regulation, and heat and water management. In this paper, the study is concentrated on the air subsystem that feeds the fuel-cell cathode with oxygen. Proceeding from a fourth-order model representing the air subsystem of a proton exchange membrane (PEM) fuel cell, a reduced third-order model is presented. Simulations show that the relative error caused by this reduction does not exceed 5%. Experimental validation has been done on a 33-kW PEM fuel cell, for both fourth- and reduced third-order models with less than 5% relative error. Additionally, a higher order sliding-mode supertwisting algorithm, with a well-known heuristic modification using variable gains, has been designed and validated experimentally to control a permanent-magnet synchronous motor that drives a volumetric compressor (double screw) designed to feed the 33-kW fuel cell with air.