## 3 Control Strategy
### Open-Loop Behavioral Interventions Analysis: Tuning of MPC  
The proper selection of controller parameters (tuning) can be estimated by analyzing the open-loop step response of the proposed behavioral interventions. The choice of suitable values for these parameters is important because they affect not only the controller's performance but also the computational complexity of the MPC algorithm, which solves an optimization problem online at each time step. Analyzing an open-loop step response is important because it provides information about the dynamic behavior of a system. It allows us to observe the system's response to a sudden change in its input and measure its performance in terms of characteristics such as rise time, settling time, overshoot, and steady-state error. This information is crucial for obtaining specifications of the system (human behavior) and tuning the controller parameters to achieve desired performance. By analyzing the open-loop step response, we can gain insights into the system's stability, transient response, and steady-state behavior, which are all essential for designing and implementing control systems. Given the aim of reaching 10,000 steps daily, our approach to conducting the open-loop test involves setting a target of 1500 steps per day for the model and observing the point at which the system attains stability, as indicated in Figure 6. Based on the system's stabilization from day 20 onwards, calculations are made for the dynamic specifications of the performed step response. From this, some specifications and parameters of the system are calculated: percent overshoot (%OS), damping ratio (\(\zeta\)), natural frequency (\(\omega_{n}\)), and rise time (\(T_{r}\)). Rise time is the time required for the system output to reach and maintain a percentage of its final value, typically 90%. It is an important characteristic of a system's response because it indicates how quickly the system can respond to changes in its input. A shorter rise time indicates a faster response and better performance. See Equations (12)-(15). \[\%\text{OS}=\frac{2763-2276}{2276-776.1}\times 100=32.46\%, \tag{12}\]  
\[\zeta=\frac{-ln(\%OS/100)}{\sqrt{\pi^{2}+(ln(\%OS/100))^{2}}}=0.337, \tag{13}\]  
\[\omega_{n}=\frac{\pi}{T_{p}\sqrt{1-\zeta^{2}}}=0.67\ \ \ \ \ (rad/day), \tag{14}\]  
Figure 6: “Step response” refers to the response of a system to a step input of the Social Cognitive Theory (SCT) model in an open loop configuration. On the other hand, “steps” refers to the number of steps in the output signal (\(\eta_{4}\)). It is a measure of the output of the system. \[T_{r}=\frac{\pi-cos^{-1}(\zeta)}{\omega_{n}\sqrt{1-\zeta^{2}}}=1.99\ \ \ \ \ (days). \tag{15}\]  
The sampling time parameter (\(T_{s}\)) is the rate at which data are sampled. This time is determined based on two considerations: limiting the number of samples to 20 during the rise time to avoid computational overload, and following [PERSON]'s theorem, which requires sampling a periodic signal at more than twice the highest frequency component of the signal [24]. See Equations (16) and (17). \[ \begin{split} T_{s}&>\frac{T_{r}}{20}\\ T_{s}&>\frac{1.99}{20}\\ T_{s}&>0.099\ \ \ \ \ (days),\end{split}  \tag{16}\]  
\[ \begin{split}\omega_{s}&>2\omega_{n}\\ \frac{2\pi}{T_{s}}&>2\omega_{n}\\ T_{s}&<\frac{\pi}{\omega_{n}}\\ T_{s}&<4.69\ \ \ \ \ (days).\end{split}  \tag{17}\]  
Thus, given \(0.099<T_{s}<4.69\), a value of \(T_{s}=1\) is recommended. Selecting a prediction horizon that covers its significant dynamics is crucial to predicting a system's future behavior accurately. A recommended approach is to choose a value that is high enough to capture the relevant dynamics but not too high to introduce unnecessary delay [13]. Based on the calculation of the sampling time, it is recommended to use a prediction horizon of 20 to 40 samples in order to cover the transient response of the open-loop system. A prediction horizon parameter value of 30 was selected. Choosing a very large control horizon only increases computational complexity.