Aiming at addressing the challenges of manual adjustment in drug dosage during multistage sub-process rare earth extraction production, as well as the significant fluctuations in production index caused by such manual adjustments, a distributed model predictive control method based on the Laguerre function is proposed. Firstly, a distributed state-space equation with series structure is established based on the extraction mechanism of multi-rare earth components, followed by the development of an augmented state-space model with embedded integrators. Secondly, a non-iterative hierarchical optimization function with input constraints is constructed to solve the distributed drug dose objective optimization function, taking advantage of the fact that the sub-process of multi-component rare earth extraction is only coupled with its upstream sub-process. Thirdly, while higher control accuracy can be achieved by using a larger prediction time domain, it also leads to significantly increased computational workload. To address this issue, the Laguerre function is utilized to represent control increment and transform the original objective function into one containing Laguerre coefficients for solution through quadratic programming. This approach ensures that computational workload remains independent of prediction time domain scope but is solely related to the number of coefficients in Laguerre function and does not change accordingly. Simulation results demonstrate the effectiveness of the proposed method.