## 4 Application of Control Parameter Optimization on Five-Story Benchmark Structure  
To validate the PSO-H optimization on the control weighting vector, the five-story Kajima-Shizouka building was used as a benchmark structure (Figure 3). All simulations were executed using MATLAB. The structure was modeled as a lumped mass system, similar to that proposed by [32], which is based on the actual structure used in the study conducted in [33]. The structural properties are given in Table 1, yielding five natural frequencies of 1.00, 2.82, 4.49, 5.80, and 6.77 Hz. The damping in the structure was modeled as a 5% damping ratio based on Rayleigh damping that is both mass-proportional and stiffness-proportional [34]. It is assumed that only the horizontal degrees-of-freedom is measurable and controllable. Each floor is assumed to include an installed transducer, which measures inter-story displacement, and an ideal actuator, which is capable of supplying the demanded control force. In other words, the actuator block in Figure 2 is set to a value of one and \(u(t_{c})\) equals \(f(t_{c})\). The base-excited structural system is modeled in continuous time as an \(m\) degree-of-freedom, linear time-invariant, lumped mass shear structure. This can be generalized through \(m\) equations of motion [34]:  
\[M\ddot{y}(t)+C_{d}\,y(t)+K_{s}y(t)=-M\ddot{y}_{g}(t)+lf\,(t), \tag{9}\]  
where \(M\), \(C_{d}\), and \(K_{s}\in\mathbb{R}^{m\times m}\) are the mass, damping and stiffness matrices, respectively. The displacement vector relative to the base of the structure is \(y\in\mathbb{R}^{m}\), \(\ddot{y}_{g}\) is the ground acceleration, and \(l\in\mathbb{R}^{m}\) is the ground acceleration influence vector, where each term takes \begin{table}
\begin{tabular}{c c c} \hline \hline
**Floor** & **Seismic Mass (kg)** & **Inter-Story Stiffness (kN/m)** \\ \hline
1 & \(215.2\times 10^{3}\) & \(147\times 10^{3}\) \\
2 & \(209.2\times 10^{3}\) & \(113\times 10^{3}\) \\
3 & \(207.0\times 10^{3}\) & \(99\times 10^{3}\) \\
4 & \(204.8\times 10^{3}\) & \(89\times 10^{3}\) \\
5 & \(266.1\times 10^{3}\) & \(84\times 10^{3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Benchmark structure properties. Figure 3: Five-Story Kajima-Shizouka building used as a benchmark structure. a value of one. The locations of the actuators are described by the matrix \(l\ e\ \mathbb{R}^{mxp}\) and \(f\ e\ \mathbb{R}^{p}\) is a vector of control forces given that \(p\) is the number of input control forces. The variable \(t\) represents continuous time. By setting the control force, \(f\), equal to zero, the uncontrolled response of the structure to the base excitation is approximated using the integration method proposed by [PERSON] [34]. In this simulation, the control frequency was set to 100 Hz and the simulation time-step was 488 usec. It is assumed that the control force is held constant in between control time-steps. To introduce the weighted-sum control algorithm, it is assumed that each floor is outfitted with a bank of bandpass filters that are optimized to fully capture the displacement signal of that floor. Previous parametric studies have shown that a sensor with 11 filters, each having a center frequency every 0.7 Hz and a passband of 0.5 Hz, is optimal for representing the response of the structure to seismic base excitation [30]. The dynamics of each filter is modeled using Equations (5) and (6). It is assumed that every unit in the filter bank is able to broadcast the peaks from its filtered signal to the controller on each floor and therefore every filter on every floor in the filter bank has a weighted connection with every controller on every floor. For the five-story structure, this resulted in a weight matrix, \(W\), that was 55 \(\times\) 5 elements. The 55 rows in this matrix were derived from five sensors, each having eleven filters, and the five columns were due to the connections with the five actuators (Figure 4).