Aspects of the invention disclosed herein generally provide a heat engine system, a turbopump system, and methods for lubricating a turbopump while generating energy. The systems and methods provide proper lubrication and cooling to turbomachinery components by controlling pressures applied to a thrust bearing in the turbopump. The applied pressure on the thrust bearing may be controlled by a turbopump back-pressure regulator valve adjusted to maintain proper pressures within bearing pockets disposed on two opposing surfaces of the thrust bearing. Pocket pressure ratios, such as a turbine-side pocket pressure ratio (P1) and a pump-side pocket pressure ratio (P2), may be monitored and adjusted by a process control system. In order to prevent damage to the thrust bearing, the systems and methods may utilize advanced control theory of sliding mode, the multi-variables of the pocket pressure ratios P1 and P2, and regulating the bearing fluid to maintain a supercritical state.