## 2 Model Predictive Control  
With the increasing complexity of industrial plants and the search for better systems performance, the development of new controllers is increasingly important. Among the advanced control techniques suitable for industrial applications is model predictive control (MPC) [48; 49; 50; 51]. The concept of prediction is increasingly used in several sectors beyond the control of processes themselves: maintenance, cost forecasting, early detection of malfunctions, etc. This concept has been increasingly complementary to feedback, the action is more related to reaction than to anticipation or prediction. The feed-forward control is naturally encompassed by the concept of prediction. The basic idea of predictive control is simple: based on a model of the process, the behavior of the process is predicted for different actions (controls) [52]. From the optimization of a functional, dependent on these actions, the optimal action to be applied to the process is obtained [53]. MPC is not a specific control strategy, but is the name given to a set of control methods that were developed considering the concept of prediction and obtaining the control signal, by minimizing a certain objective function [54]. It is advanced process control (APC) with less variation in process variables [55]. This function considers future error and constraints on process and/or control variables [56]. An overall operation of this control method can be observed in Figure 1.  
Figure 1: The benefits of the MPC enabling optimization. A major advantage of linear MPC compared to non-linear is the associated optimization problem and simpler to solve. However, when processes have a medium or severe degree of non-linearity, when there is a range of operations and variables, or when the processes undergo continuous transitions in their operation, a nonlinear model of control design must be considered, so that it allows maintaining stability and the desired performance for the closed loop system [57]. The MPC rather than a specific controller is a methodology for the calculation of control actions. It is also a comprehensible methodology, which in a way tries to reproduce the way an expert operator would operate in the control of a certain process [48]. To achieve a higher quality control, this same operator would repeat all the calculations whenever it has updated information, being new measures of the status of the process or updated knowledge about the behavior of the process (new information of the model). In the MPC, this concept is called the receding horizon, resulting in the resolution of a different optimization (minimization) problem in each sampling period, since new information is incorporated in the dynamic evolution of the process [55]. The concept of a receding horizon can be seen in Figure 2.  
This example allows understanding that the first controls performed manually by operators who knew the process well could have been included in the area of model-based predictive control. In short, it is a very intuitive methodology to address the control of a process and this has influenced its dissemination at an industrial level [58]. Based on the process model, the predictor is responsible for calculating, for each instant \(t\), the predictions of the dynamic evolution of the process [\(y(t+1\,|\,t)\),..., and \(y(t+N\,|\,t)\)]\({}^{2}\) throughout the horizon of prediction \(N\), from the dynamic information available up to that moment (measurements of the process variables and inputs passed up to the current time \(t\)) and a postulated or future control law [\(u(t\,|\,t)\),..., \(u(t+N\,|\,t)\)], along the prediction horizon, as shown in Figure 3[48]. Future control actions are calculated in a way that minimizes a certain cost function. Thus, the cost function assigns a value to each prediction and therefore to each postulated control law. This value tries to show the degree of compliance with the static and dynamic specifications compatible with possible operating restrictions.