Efficient MPC Optimization using Pontryagin's Minimum Principle

A method of solving the online optimization in model predictive control (MPC) of input-constrained linear systems is described. Using Pontryagin's Minimum Principle, the matrix factorizations performed by general purpose quadratic programming (QP) solvers are replaced by recursions of state and co-state variables over the MPC prediction horizon. This allows for the derivation of solvers with computational complexity per iteration that depends only linearly on the length of the prediction horizon. Parameterizing predicted input and state variables in terms of the terminal predicted state results in low computational complexity but can lead to numerical sensitivity in predictions. To avoid ill-conditioning an alternative parameterization is derived using Riccati recursions. Comparisons are drawn with the multiparametric QP solution, and the computational savings are demonstrated over generic QP solvers