For a companion robot that follows a person as an assistant, predicting human walking is important to produce a proactive movement that is helpful to maintain an appropriate area decided by the human personal space. However, fully trusting the prediction may result in obstructing human walking because it is not always accurate. Hence, we consider the estimation of uncertainty (i.e., entropy) of the prediction to enable the robot to move without causing overconfident motion and without being late for the person it follows. To consider this uncertainty of the prediction to the controller, we introduce a reliability value that changes based on the entropy of the prediction. This value expresses the extent the controller should trust the prediction result, and it affects the cost function of our controller. We propose an uncertainty-aware robot controller based on nonlinear model predictive control to realize natural human-followings. We found that our uncertainty-aware control system can produce an appropriate robot movement, such as not obstructing the human walking and avoiding delay, in both simulations using actual human walking data and real-robot experiments.