The basal metabolic rate characterizes the energy consumption of the human body at rest. It can be estimated by using respiratory gas exchange analyzers and indirect calorimetry. During cycling this value could vary due to several adaptation processes such as an increase in breathing and blood circulation, controlling body temperature, among others. In this paper we propose to reconstruct the instantaneous value of this basal metabolic rate, referred here as basal power (measured in Watts), by using a robust discrete-time proportional integral (PI) observer. The observer design is based on a solution of linear matrix inequalities and uses an uncertain linear parameter varying model of the gas exchange dynamics. The proposed methodology allows the reconstruction of the basal power while making the respiratory gas exchange estimation robust to bounded uncertainties and disturbances. The PI observer has been validated in simulation.