## 4 Application of Control Parameter Optimization on Five-Story Benchmark Structure
To confirm convergence of the algorithm, the particles are trained until the best solution, \(g\), does not change for 50 consecutive iterations. This does not ensure locating a global minimum, but with an adequate number of particles and with executing the homing feature, the algorithm does locate a competitive local minimum. Using these metrics, the weight matrix is trained using the PSO-H algorithm, resulting in the cost functions shown in Table 2, denoted as Weighted-sum in the table. A sample time history response for the drift is shown in Figure 6. To ensure that the weight matrix is not over-trained to the El Centro record, it is also validated using the 1995 Kobe (NS JMA) and 1989 Loma Prieta (CORRALITOS) earthquake ground acceleration records (Figure 5). These results are shown in Tables 3 and 4. As a comparison to the weighted-control algorithm, a traditional full-state feedback linear quadratic regulator (LQR) [37] was also considered. This controller assumes that all necessary states (i.e., displacement and velocity) of all floors are measurable or estimates them using techniques such as the Kalman filter. The LQR uses the algebraic Ricatti equation to minimize the cost function  
Figure 3:0  
subject to the full state feedback control law, expressed as \(u=\) -\(K\), where \(\overline{y}e\ \mathbb{R}^{1\times 2m}=\left[y\,\dot{y}\right]\) is a vector of the inter-story displacement and velocity of all the floors of the structure and \(K\in\mathbb{R}^{p\times 2m}\) is the resulting constant feedback gain matrix, given \(m\) states (or floors) and \(p\) control forces. This minimization is subject to two parameters: \(Q\in\mathbb{R}^{2m\times 2m}\), which applies a weight to the cost of the structural response and \(R\ e\ \mathbb{R}^{p\times p}\), which applies a weight to the cost of the control effort. In the algorithm, \(Q\) and \(R\) are chosen using the commonly accepted Bryson's Rule [38] that establishes these values as proportional to the inverse of the square of the maximum acceptable displacement and control force, respectively. For this application, the non-zero entries of \(Q\) are \(10^{10}\) and the non-zero entries of \(R\) are \(10^{-5.4}\). Based on limitations described in [7], it is assumed that the control sampling frequency is limited to 40 Hz, which subjects the control effort to realistic experimental constraints.The results from this controller are shown in Tables 2-4 for the El Centro, Kobe, and Loma Prieta earthquakes. Figure 3:1 Seismic signals used as base excitation in the simulation in time (**a**) and frequency (**b**) domains. Abbreviations: SE, southeast; NS, north–south; and JMA, Japan Meteorological Agency. Figure 3:2 Inter-story drift response of the fifth floor when subject to the El Centro earthquake, shown on a full-time scale (**a**) and sub-section of time (**b**). The weighted-sum control algorithm has similar control effectiveness as the LQR algorithm when considering the El Centro earthquake for all cost functions (Table 2). However, the LQR controller is much more adaptable to other earthquakes as demonstrated by its continued effectiveness at mitigating inter-story displacement and acceleration for both the Kobe and Loma Prieta earthquakes. The weighted-sum control algorithm is able to reduce the displacement measures (i.e., \(J_{1}\) and \(J_{2}\)) when the structure is subject