# Real-Time Model Predictive Control of Human Bodyweight Based on Energy Intake
## 3 Results and Discussion
### Time-Invariant SISO TF Models
\begin{table}
\begin{tabular}{c c c c c} \hline  & **[1 1 0]** & **[1 2 0]** & **[2 1 0]** & **[2 2 0]** \\ \hline \(YIC\) & \(-12\pm 2\) & \(-5\pm 3\) & \(-6\pm 4\) & \(-1\pm 5\) \\ \(R_{T}^{2}\) & \(0.99\pm 0.01\) & \(0.99\pm 0.01\) & \(0.93\pm 0.24\) & \(0.83\pm 0.34\) \\ \hline  & **[1 1 1]** & **[1 2 1]** & **[2 1 1]** & **[2 2 1]** \\ \hline \(YIC\) & \(-11\pm 2\) & \(-6\pm 2\) & \(-1\pm 6\) & \(2\pm 6\) \\ \(R_{T}^{2}\) & \(0.99\pm 0.01\) & \(0.99\pm 0.01\) & \(0.60\pm 0.47\) & \(0.53\pm 0.49\) \\ \hline  & **[1 1 2]** & **[1 2 2]** & **[2 1 2]** & **[2 2 2]** \\ \hline \(YIC\) & \(-9\pm 1\) & \(-6\pm 2\) & \(5\pm 3\) & \(7\pm 6\) \\ \(R_{T}^{2}\) & \(0.97\pm 0.01\) & \(0.98\pm 0.01\) & \(0.09\pm 0.28\) & \(0.21\pm 0.39\) \\ \hline  & **[1 1 3]** & **[1 2 3]** & **[2 1 3]** & **[2 2 3]** \\ \hline \(YIC\) & \(-8\pm 1\) & \(-4\pm 2\) & \(5\pm 2\) & \(5\pm 4\) \\ \(R_{T}^{2}\) & \(0.96\pm 0.02\) & \(0.96\pm 0.02\) & \(0.03\pm 0.17\) & \(0.24\pm 0.36\) \\ \hline  & **[1 1 4]** & **[1 2 4]** & **[2 1 4]** & **[2 2 4]** \\ \hline \(YIC\) & \(-6\pm 2\) & \(1\pm 4\) & \(6\pm 2\) & \(6\pm 4\) \\ \(R_{T}^{2}\) & \(0.90\pm 0.16\) & \(0.84\pm 0.27\) & \(0.03\pm 0.14\) & \(0.18\pm 0.30\) \\ \hline  & **[1 1 5]** & **[1 2 5]** & **[2 1 5]** & **[2 2 5]** \\ \hline \(YIC\) & \(-2\pm 4\) & \(1\pm 3\) & \(4\pm 3\) & \(7\pm 5\) \\ \(R_{T}^{2}\) & \(0.75\pm 0.34\) & \(0.72\pm 0.37\) & \(0.15\pm 0.31\) & \(0.19\pm 0.35\) \\ \hline \end{tabular}
\end{table} 
Table 1: Average Young information criterion (\(YIC\)) and coefficient of determination, \(R_{T}^{2}\), together with their standard deviation, obtained during the system identification process for each of the 32 sub-datasets created from the Minnesota starvation excel spreadsheet used in the work of [31]. [\(n_{a}\)\(n_{b}\)\(\delta\)] indicates the different combinations (24 in total) of time –invariant first and second order models: number of \(a\)-parameters (\(n_{a}\)) ranging from 1 to 2 (Equation (2)), number of \(b\)-parameters (\(n_{b}\)) ranging from 1 to 2 (Equation (3)) and delays (\(\delta\)) ranging from 0 to 5 samples. \begin{table}
\begin{tabular}{c c c c} \hline
**[1 1 0]** & **[1 2 0]** & **[2 1 0]** & **[2 2 0]** \\ \hline \(0.999\pm 0.001\) & \(0.999\pm 0.001\) & \(0.999\pm 0.001\) & \(0.999\pm 0.001\) \\ \hline \end{tabular}
\end{table} 
Table 2: Mean and standard deviation of the Pearson correlation coefficient obtained when comparing the first and second orders time-invariant models with no delay ([\(n_{a}\)\(n_{b}\)\(0\)]) output and each one of the weight curves collected for the 32 participants in the experiment. estimating the impact of bodyweight of a given energy intake. When the \(a\)- and \(b\)-parameters from the individual estimations are averaged together, the following model is obtained,  
\[BW_{t}=a_{1}\cdot BW_{t-1}+b_{0}\cdot EI_{t}=(0.92\pm 0.03)\cdot BW_{t-1}+(9\pm 3) \cdot 10^{-4}\cdot EI_{t} \tag{20}\]  
where \(BW\) is bodyweight in kg, \(EI\) is the energy intake in kcal day\({}^{-1}\) and \(t\) is the time sample in weeks. It can be seen that the \(b\)-parameter exhibits more variability than the \(a\)-parameter. This is expected, taking into account the biological process under study. Translating the model to the biological process, it states that the bodyweight at time \(t\) is the bodyweight of the time sample before plus a contribution of a certain percentage of the energy intake at time sample \(t\). Therefore, on one hand it is expected that the \(a\)-parameter should be close to one and should not show too much variability between participants. On the other hand, the \(b\)-parameter should resemble energy efficiency and thus it is expected to vary between the different individuals. From Equation (20), it can be the seen that the results matched perfectly this hypothesis. Running this average model in all datasets results in an average coefficient of determination of \(R_{T}^{2}=0.3\pm 0.2\). Therefore, by defining an average model in which an average \(b\)-parameter value is set, the performance of the model drops drastically.