## 4 Results Analysis  
It is essential to use accurate system parameters when designing and testing a control system to ensure that it performs as expected and achieves the desired control objectives. Thus, an identification procedure with the data from the \"hypothetical model\" was performed to obtain a linear state-space variable model. This discrete model was then used to design an MPC controller that considers the actual system dynamics. However, to test the controller the \"hypothetical model\" was used. This led to a difference between the identified parameters used for the MPC design and the system's parameters used for simulation, similar to what may occur in a real scenario. The proposed controller was tested and the following results were obtained (Figure 10). The results from day 30 to day 120 are shown. During the first month, an initial desired behavior of 1000 daily steps was set. \begin{table}
\begin{tabular}{c c c c} \hline
**Parameter** & **Min** & **Max** & **Unit** \\ \hline \(u_{8}\) & 0 & 15,000 & (steps/day) \\ \(u_{9}\) & 0 & 5000 & (points/day) \\ \hline \(\Delta u_{8}\) & \multicolumn{2}{c}{unconstrained} & (steps/day) \\ \(\Delta u_{9}\) & \multicolumn{2}{c}{unconstrained} & (points/day) \\ \hline \end{tabular}
\end{table} 
Table 4: MPC constraints. Goals (\(u_{8}\)) and expected points (\(u_{9}\)) are the manipulated variables (MV). \(\Delta u_{i}\) refers to the change in the signal \(i\) between two consecutive time instances. Figure 10: Step tracking. This figure describes the ability of a controller to track or follow a step input signal. A step input signal is a signal that abruptly changes from one constant value to another, often used to represent a change in the desired setpoint of a controlled system. In this case, the desired behavior or setpoint was a change in daily steps from 1000 to 10,000. The figure shows that the control system was able to achieve this desired behavior in less than 20 days. The settling time for this test was 10 days and the percentage overshoot was 17%. \begin{table}
\begin{tabular}{c c c c} \hline
**Parameter** & **Weights** & **Unit** \\ \hline \(u_{8}\) & 0 & & \\ \(u_{9}\) & 1 & & dimensionless \\ \(\Delta u_{8}\Delta u_{9}\) & 0.1 & & \\ \hline Output Variables & \(\eta_{2}=0\) & \(\eta_{3}=0\) & \(\eta_{4}=1\) & \(\eta_{5}=0\) & dimensionless \\ \hline \end{tabular}
\end{table} 
Table 5: MPC weights. Taking into account the economic factor which is critical for the project, the controller will try to penalize abrupt values of offered reward points (\(u_{9}\)) for which the weight of 1 is assigned. The weight of the daily step goal (\(u_{8}\)) was set to zero. The weights assigned to \(\Delta u_{8}\) and \(\Delta u_{9}\) are both 0.1, indicating that those variables are not considered significant. The weight of the output variable \(\eta_{4}\) is also equal to 1 since that variable, which corresponds to the behavior, is the one to be regulated, leaving the other outputs to vary freely (other output weight = 0). The response of the manipulated variables (i.e., goals (\(u_{8}\)) and expected points (\(u_{9}\))) is shown in Figure 11, demonstrating the successful achievement of the constraints considered in the MPC design. Since the weight of the expected points was set to 1 (the highest level of importance), the signal tends to decrease over time. On the other hand, the weight of the goals was set to 0, resulting in an increase and stabilization of the signal at a steady-state value of around 10,000 steps. In Figures 12 and 13, the other input variables (granted points (\(u_{10}\)) and goal attainment (\(u_{11}\))) are shown. It is noticeable that the number of granted points in Figure 12 was lower than the number of expected points in Figure 11. Meanwhile, the goal attainment signal in Figure 13 shows that, for the majority of the experiment, the number of daily steps taken surpassed the proposed goal. In Figure 14, the steady state of the goal attainment signal is shown.