This paper presents an optimal control scheme for a real-time feedback control rate-monotonic scheduling (FC-RMS) system. We consider two-version tasks composed of a mandatory and an optional part to be scheduled according to the FC-RMS. In FC-RMS, the controller provides a feedback strategy for deciding about the execution or rejection of the optional sub-tasks. By modeling the task execution times as random variables, we first find the statistical model of FC-RMS and then we design a pure optimal controller and an optimal controller with feedforward integral compensation. The comparison of these two schemes with common Proportional-Integral-Derivative (PID) controller highlights the benefit of the optimal scheme with integral compensation. The results are demonstrated through the real implementation of FC-RMS on RT-Linux.