Finite Horizon Optimal Control MPC

Topics: Model Predictive Control

Problem Definition

The Finite Horizon Optimal Control MPC is a special case of the General MPC Formulation where the cost is written as a function of the stage costs and terminal cost typically used in optimal control problems.

At time step $k$, the controller solves the following optimization problem

$$\begin{equation} \begin{aligned} \min_{\{u_0,\ldots,u_{N-1}\}} \quad & \sum_{i=0}^{N-1} \ell(x_{i},u_{i}) + V_f(x_{N}) \\ \text{subject to} \quad & x_{i+1} = f(x_{i},u_{i}), \qquad i=0,\ldots,N-1, \\ & x_0 = x_{\mathrm{current}}, \\ & x_{i} \in \mathcal{X}, \qquad i=1,\ldots,N, \\ & u_{i} \in \mathcal{U}, \qquad i=0,\ldots,N-1, \\ & x_{N} \in \mathcal{X}_f. \end{aligned} \end{equation}$$

where

• $N$ is the prediction horizon
• $l(x,u)$ is the stage cost
• $V_f(x)$ is the terminal cost
• $\mathcal{X}$ is the set of allowed states
• $\mathcal{U}$ is the set of allowed input
• $\mathcal{X}_f$ is the terminal constraint set