Model predictive control

The Model Predictive Control (MPC) is an advanced control method, which is able to control multi-inputs and multi-output systems. It uses an internal model of the process to be controlled, enabling it to predict its behavior over a certain horizon length, and to optimize its control inputs on this basis. The implementation of an MPC control involves the real-time resolution of an optimization problem on the basis of a criterion defining the control objectives, such as setpoint tracking. The foundation of this control method is attributed to J. Richalet, in 1978 [1].

Formulation

The model predictive control formulation envolves a quadratic cost function with constraints [2], such as:

\[\text{min}~ \sum_{i=1}^N \tilde{x}_i^TQ \tilde{x}_i + \tilde{u}_i^T R \tilde{u}_i\]

\[\text{subject to:} \\ \hat{x}_{i+1} = f(\hat{x}_i, u_i) \\ \tilde{x}_i = \hat{x}_i - x_i^r \\ \tilde{u}_i = u_i - u_i^r \\ \hat{x}_0 = \bar{x}(t_k) \\ x_{min} \leq \hat{x}_i \leq x_{max} \\ u_{min} \leq u_i \leq u_{max}\]

where $\hat{x}$ is the state prediction according the model of the dynamical system, $u$is the input control, $\tilde{x}$ is the state deviation from the state reference, $\tilde{u}$ is the input control deviation from the input control reference, $\bar{x}$ is the state initialization, $x_{min}$ and $x_{max}$ is the state constraints and umin and umax is the input control constraints.

Terminal ingredients

For the moment, the terminal ingredients cannot be added.

References

[1] Richalet, J., Rault, A., Testud, J. L., & Papon, J. (1978). Model predictive heuristic control: Applications to industrial processes. Automatica, 14(5), 413-428.

[2] Mayne, D. Q., Rawlings, J. B., Rao, C. V., & Scokaert, P. O. (2000). Constrained model predictive control: Stability and optimality. Automatica, 36(6), 789-814.