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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 4-th order model representing the air subsystem of a PEM fuel cell, a reduced 3-rd order model is presented. Simulations show that the relative error caused by this reduction doesn't exceed 0.5%. Experimental validation has been done on a 33 kW PEM fuel cell, for both 4-th and reduced 3-rd order models with less than 5% relative error. Additionally, a higher order sliding mode, super-twisting algorithm, with adaptive parameters has been designed and validated experimentally to control a permanent magnet synchronous motor (PMSM) that drives a volumetric compressor (double screw) designed to feed the 33 kW fuel cell with air.