In this article, we develop a predictive tracking controller for a nonlinear multi-compartment lung mechanics model. Specifically, for a given clinically plausible reference volume pattern, we use model predictive control to design a tracking controller that accounts for amplitude and rate input constraints. The predictive control law is derived by minimising a quadratic performance criterion involving a prediction of the system response over a prescribed time step. The proposed tracking control framework is applied to a two-compartment lung mechanics model with nonlinear lung compliance parameters.