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Error-free approximation of explicit linear MPC through lattice piecewise affine expression

In this paper, the disjunctive and conjunctive lattice piecewise affine (PWA) approximations of explicit linear model predictive control (MPC) are proposed. The training data are generated uniformly in the domain of interest, consisting of the state samples and corresponding affine control laws, based on which the lattice PWA approximations are constructed. Re-sampling of data is also proposed to guarantee that the lattice PWA approximations are identical to explicit MPC control law in the unique order (UO) regions containing the sample points as interior points. Additionally, under mild assumptions, the equivalence of the two lattice PWA approximations guarantees that the approximations are error-free in the domain of interest. The algorithms for deriving statistically error-free approximation to the explicit linear MPC are proposed and the complexity of the entire procedure is analyzed, which is polynomial with respect to the number of samples. The performance of the proposed approximation strategy is tested through two simulation examples, and the result shows that with a moderate number of sample points, we can construct lattice PWA approximations that are equivalent to optimal control law of the explicit linear MPC.