In this paper, a PID controller is compared to an extended moving horizon estimator coupled with a model predictive control approach for the problem of dosing propofol and remifentanil during the induction of anesthesia. For the PID controller, taken from the literature, a fixed ratio is considered between propofol and remifentanil flow rates and an anti-windup strategy is used to prevent integration wind-up. The optimal control approach uses an MHE to estimate both the states and the pharmacodynamic parameters of the system, followed by a non-linear model predictive controller to compute the optimal drug rates according to the model. Both controllers are tuned using the same criterion, and are compared by simulating 500 uncertain patient models for the induction phase.