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What is the difference between moving-horizon DP (dynamic programming) and MPC (modelbased predictive control)? In both cases, the system input at time $t$ is determined by solving a finite-time optimization problem.

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Moving Horizon Dynamic Programming has two applications this field: Moving Horizon Estimators (MHE) and Controllers.

MHE are optimal observers that can be either linear or nonlinear. If you consider the MHE with unconstrained linear dynamics and quadratic cost its estimation of the current state $x_{t \mid t}$ is equivalent to the Kalman Filter.

Meanwhile, MHC is MHE control counterpart. It is just the name previously used for Model Predictive Control. Which, like the MHE in the unconstrained linear dynamics with quadratic cost is equivalent to the Linear Quadratic Regulator.

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