For economic reasons, above a certain water speed, it is desirable for current turbines to maintain a constant power output. This requires a control strategy to shed power. Here, such a strategy is evaluated through simulation and laboratory experiment for a helically bladed turbine with four blades and a straight-bladed turbine with two blades. These are contrasting cases because hydrodynamic torque produced by the straight-bladed turbine has a substantially more azimuthal phase variability than the helically bladed turbine. For practical implementation, a control algorithm is desired that requires only a time-average characteristic performance curve and estimates of angular velocity and control torque. Additionally, the transition between power maximizing and power tracking regimes should be smooth and automatic. The controller can be further constrained to only apply a resistive torque to the turbine. A control strategy satisfying these constraints is shown experimentally to achieve these objectives for both types of turbines at a variety of power set points. The power-tracking error (<3%) is primarily at the blade passage frequency for the straight-bladed turbine and at the rate of turbine rotation for the helical turbine. While partially a consequence of how the generator drive is tuned, comparison to simulation indicates that perfect power-tracking is not generally possible even under ideal conditions for a fixed-pitch, cross-flow turbine using purely resistive torque control.