Theory behind MPC
MPC is based on iterative, finite horizon optimization of a plant model. At time t the current plant state is sampled and a cost minimizing control strategy is computed (via a numerical minimization algorithm) for a relatively short time horizon in the future: ![[t,t+T]](https://upload.wikimedia.org/math/6/4/8/648573482abe10af9ae83915e5751af8.png)
. Specifically, an online or on-the-fly calculation is used to explore state trajectories that emanate from the current state and find (via the solution of Euler-Lagrange equations) a cost-minimizing control strategy until time 
. Only the first step of the control strategy is implemented, then the plant state is sampled again and the calculations are repeated starting from the current state, yielding a new control and new predicted state path. The prediction horizon keeps being shifted forward and for this reason MPC is also called receding horizon control.
. Specifically, an online or on-the-fly calculation is used to explore state trajectories that emanate from the current state and find (via the solution of Euler-Lagrange equations) a cost-minimizing control strategy until time . Only the first step of the control strategy is implemented, then the plant state is sampled again and the calculations are repeated starting from the current state, yielding a new control and new predicted state path. The prediction horizon keeps being shifted forward and for this reason MPC is also called receding horizon control.
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