Receding horizon — control moves $u^$ over the horizon $M$ steer the predicted output $\hat{y}$ toward the target $y^$ over the prediction horizon $P$, with the disturbance $d$ held constant.
Model Predictive Control (MPC) is a control strategy that repeatedly optimises a sequence of control moves over a receding horizon using a predictive model, applying only the first move before re-optimising at the next step.
1. Train Prediction Model
With sufficient lookback $L$:
$$ \hat{y}{t+1} = f\left(u{(t-L):t}, \, d_{(t-L):t}\right) $$
2. Set Target Trajectory
$$ \mathbf{Y}^* = [y_{t+1}^, \dots, y_{t+P}^] $$
3. Select Candidate Control Trajectory
Held constant after the control horizon $M$:
$$ \mathbf{U} = [u_t, \dots, u_{t+M-1}] $$
4. Freeze Disturbance Trajectory
Held constant over the prediction horizon $P$:
$$ \mathbf{D} = [d_t, \dots, d_t] $$
5. Predict Target Trajectory
$$ \hat{\mathbf{Y}}(\mathbf{U}) = [\hat{y}{t+1}, \dots, \hat{y}{t+P}] = F(\mathbf{U}) $$
6. Correct Target Bias (optional)
$$ \hat{\mathbf{Y}} \leftarrow \hat{\mathbf{Y}} + b, \qquad b = y_t - \hat{y}_t $$
7. Compute Cost Function