An Intuiative Formulation of Model Predictive Controls

Feb 3, 2026

The problem of MPC can be boiled down to the following formulation

\min_{\vec u} \left(\sum_{t\in[1,n]}j(\vec u_t, \vec x_t) = J(\mathcal U, \mathcal X)\right)\\ \vec x_{t+1} = f(\vec x_t, \vec u_t)\\ \vec x\in [\vec x_{min}, \vec x_{max}]\\ \vec u\in [\vec u_{min}, \vec u_{max}]\\ x_0\text{ given (current sensor values)}

Where

x is the state
u is the action
J is the cost function
f is the system dynamics (the model in question), namely given the current state and some action, what is the state at the next timestep

Note that I am currently abusing notation quite significantly. In particular, we imagine $\mathcal U$ and $\mathcal X$ to be the joined vectors of actions and states over the entire prediction horizon, i.e. $\mathcal U = \{\vec u_1, \vec u_2, \ldots, \vec u_n\}\\ \mathcal X = \{\vec x_1, \vec x_2, \ldots, \vec x_n\}$

As a (not so arbitrary) example, we can imagine x could be the state of a car, u is the torque commands provided by the driver, f is the car dynamics, and J is some cost function that penalizes deviation from a desired trajectory.

Simplifying the formulation

We can observe pretty easily that $x_1=f(x_0,u_0)\\ x_2=f(x_1,u_1)=f(f(x_0,u_0),u_1)\\ \vdots$ Specifically, we can find that $\mathcal X$ is entirely determined by $x_0$ and $\mathcal U$ . Thus, we can simplify $\mathcal X$ to be a function of $x_0$ and $\mathcal U$ , i.e. $\mathcal X_f(x_0, \mathcal U) = \{x_0, f(x_0, u_0), \dots f(f(x_0, u_0), u_1) \}$ Then, we can write the cost as $\hat J(\mathcal U) = J(\mathcal U, \mathcal X_f(x_0, \mathcal U))$ and thus reduce the problem to an optimization over just $\mathcal U$ .