Model predictive control

Definition

Let $f \colon \R^n \times \R^m \to \R^n$ define the discrete-time control system $x_{k+1} = f(x_k, u_k)$. Let $\mathbb{X} \subseteq \R^n$, $\mathbb{U} \subseteq \R^m$ and $\mathbb{X}_f \subseteq \mathbb{X}$ be constraint sets. Let $\ell \colon \R^n \times \R^m \to \R$ be a stage cost, let $V_f \colon \R^n \to \R$ be a terminal cost, and let $N \ge 1$ be the horizon. For $x \in \R^n$, let $P_N(x)$ be the problem: minimise $$\sum_{i=0}^{N-1} \ell(z_i, v_i) + V_f(z_N)$$ over $v_0, \dots, v_{N-1} \in \mathbb{U}$, subject to $z_0 = x$, $z_{i+1} = f(z_i, v_i)$, $z_i \in \mathbb{X}$ for $i < N$, and $z_N \in \mathbb{X}_f$. Assume that $P_N(x)$ has a minimiser $(v_0^*(x), \dots, v_{N-1}^*(x))$ for each $x$ in a set $\mathcal{X}_N$, and fix a choice of it. The model predictive control law is $\kappa_N(x) = v_0^*(x)$ for $x \in \mathcal{X}_N$. Model predictive control operates the system in closed loop by $u_k = \kappa_N(x_k)$: at each time $k$ it solves $P_N(x_k)$ and applies only the first control.