For the multi-stage sub-process rare earth extraction production, manual adjustment of medication at each stage based on experience can lead to problems such as manual cyclic medication and large fluctuations in production indicators. In response to this, a distributed model predictive control method based on Laguerre functions is proposed to regulate medication dosage. Firstly, a distributed state space equation with a serial structure is established based on the extraction mechanism of multi-rare earth components, which is then expanded into an augmented state space model embedded with an integrator. Secondly, taking advantage of the fact that each level of sub-process in the extraction of multiple rare earth components is only coupled with its upstream sub-process, a non-iterative recursive solution method for the distributed medication dosage target optimization function with input constraints is constructed. Furthermore, since using a larger prediction horizon can achieve higher control accuracy but drastically increase computational complexity,Laguerre functions are employed to represent control increments. The original objective function is transformed into one containing Laguerre coefficients, and solved using quadratic programming. This method ensures that the computational complexity is independent of the prediction horizon and only related to the number of Laguerre function coefficients,thus not changing with the range of the prediction horizon. Simulation experiments demonstrate the effectiveness of the proposed method.