## 4 Application of Control Parameter Optimization on Five-Story Benchmark Structure
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Control Scenario** & **Floor** & & & & & \\ \hline \multirow{2}{*}{\(J_{1}\)} & Weighted-sum & 0.359 & 0.425 & 0.468 & 0.452 & 0.385 \\  & Weighted-sum–220 & **0.347** & 0.413 & 0.451 & 0.401 & 0.347 \\  & LQR & 0.415 & **0.350** & **0.318** & **0.306** & **0.305** \\ \hline \multirow{2}{*}{\(J_{2}\)} & Weighted-sum & 0.305 & 0.300 & 0.320 & 0.341 & 0.392 \\  & Weighted-sum–220 & 0.293 & 0.288 & 0.306 & 0.298 & 0.328 \\  & LQR & **0.237** & **0.199** & **0.195** & **0.200** & **0.204** \\ \hline \multirow{2}{*}{\(J_{3}\)} & Weighted-sum & 0.841 & 0.815 & 0.789 & 0.803 & 0.679 \\  & Weighted-sum–220 & **0.757** & **0.787** & **0.712** & **0.750** & **0.661** \\  & LQR & 1.024 & 0.932 & 0.868 & 0.918 & 0.703 \\ \hline \multirow{2}{*}{\(J_{4}\)} & Weighted-sum & 0.743 & 0.687 & 0.693 & 0.634 & 0.564 \\  & Weighted-sum–220 & 0.741 & 0.686 & 0.666 & 0.610 & 0.533 \\  & LQR & **0.701** & **0.581** & **0.518** & **0.468** & **0.413** \\ \hline \multirow{2}{*}{\(J_{5}\)} & Weighted-sum & 0.055 & **0.045** & 0.081 & **0.064** & 0.152 \\  & Weighted-sum–220 & 0.077 & 0.102 & **0.075** & 0.066 & **0.145** \\  & LQR & **0.052** & 0.077 & 0.119 & 0.153 & 0.178 \\ \hline \multirow{2}{*}{\(J_{6}\)} & Weighted-sum: LQR & & & 1.284 & & \\  & Weighted-sum–220: & & & 0.806 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Cost functions \(J_{1}\) through \(J_{6}\) for El Centro earthquake. Bold indicates the cost function that yields the minimum cost function for each floor. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Control Scenario** & **Floor** & & & & & \\ \hline \multirow{2}{*}{\(J_{1}\)} & Weighted-sum & 0.541 & 0.497 & 0.566 & 0.483 & 0.511 \\  & Weighted-sum–220 & 0.526 & 0.464 & 0.565 & 0.438 & 0.458 \\  & LQR & **0.392** & **0.318** & **0.325** & **0.287** & **0.262** \\ \hline \multirow{2}{*}{\(J_{2}\)} & Weighted-sum & 0.510 & 0.510 & 0.539 & 0.518 & 0.521 \\  & Weighted-sum–220 & 0.499 & 0.491 & 0.518 & 0.469 & 0.458 \\  & LQR & **0.348** & **0.295** & **0.285** & **0.276** & **0.261** \\ \hline \multirow{2}{*}{\(J_{3}\)} & Weighted-sum & 0.810 & 0.858 & 0.914 & 0.924 & 0.669 \\  & Weighted-sum–220 & 0.090 & 0.854 & 0.909 & 0.753 & 0.634 \\  & LQR & **0.757** & **0.727** & **0.659** & **0.533** & **0.412** \\ \hline \multirow{2}{*}{\(J_{4}\)} & Weighted-sum & 0.768 & 0.650 & 0.776 & 0.843 & 0.707 \\  & Weighted-sum–220 & 0.760 & 0.637 & 0.754 & 0.856 & 0.684 \\  & LQR & **0.542** & **0.488** & **0.522** & **0.556** & **0.451** \\ \hline \multirow{2}{*}{\(J_{5}\)} & Weighted-sum & 0.067 & **0.048** & **0.087** & **0.080** & 0.245 \\  & Weighted-sum–220 & 0.090 & 0.097 & 0.102 & 0.097 & **0.202** \\ \cline{1-1}  & LQR & **0.042** & 0.072 & 0.105 & 0.141 & 0.174 \\ \hline \multirow{2}{*}{\(J_{6}\)} & Weighted-sum: LQR & & & 1.044 & & \\ \cline{1-1}  & Weighted-sum–220: & & & 0.525 & & \\ \cline{1-1}  & LQR & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Cost functions \(J_{1}\) through \(J_{6}\) for Kobe earthquake. Bold indicates the cost function that yields the minimum cost function for each floor. to the other Kobe and Loma Prieta earthquakes, but it is significantly less effective at reducing the acceleration measures (i.e., \(J_{3}\) and \(J_{4}\)) when compared to the LQR controller (Tables 3 and 4). In application, retraining the weight matrix for the characteristics of the anticipated seismic response could improve the performance of the control algorithm. For all control scenarios, the controllers were more effective at reducing displacement than acceleration, which is typically a trade-off when controlling structures during seismic events. For all earthquakes, the weighted-sum control algorithm did place minimal demand on the actuators, as denoted by the \(J_{5}\) cost function, and this will contribute to its decrease in control effectiveness.