## 3 Adaptation of Particle Swarm Optimization and the Proposed Weighted Control Algorithm
The frequency selectivity of the \(j\)th filter is defined by the second-order transfer function for a bandpass filter [31], expressed as  
\[H_{j}(s)=H_{0}\frac{\omega_{j}}{Q_{0,j}}s\left(s^{2}+\frac{\omega_{j}}{Q_{0,j} }s+\omega_{j}^{2}\right)^{-1}, \tag{4}\]  
Figure 1: Flow chart for PSO-H algorithm. where \(H_{0}\) is set to 1.0 to ensure a unit gain in the filter and \(Q_{0,j}\) is related to the frequency selectivity of the filter, defined as \((2\xi_{j})^{-1}\), such that \(\xi_{j}\) is the damping ratio of the filter and \(\omega_{j}\) is the center frequency of the filter (rad/s). By making these substitutions for \(H_{0}\) and \(Q_{0j}\) in Equation (4), the impulse response of that filter is determined through the inverse Laplace transform, yielding  
\[h_{j}(t)=-2\xi_{j}\omega_{j}\Big{(}\sqrt{1-\xi_{j}^{2}}\Big{)}^{-1}e^{\xi_{j} \omega_{j}t}\sin\left(\omega_{j}\sqrt{1-\xi_{j}^{2}}t-\theta\right), \tag{5}\]  
with  
\[\theta(t)=\tan^{-1}\left(\omega_{j}\sqrt{1-\xi_{j}^{2}}t-\theta\right). \tag{6}\]  
The convolution integral can be used with the input signal, \(y(t)\), and the impulse response of that filter, \(h_{j}(t)\), to obtain the continuous time output of each \(j\)th bandpass filter, \(z_{j}(t)\),  
\[z_{j}(t)=\int_{0}^{t}y(\tau)h_{j}(t-\tau)d\tau. \tag{7}\]  
Decomposing the signal into numerous components via bandpass filters increases the amount of information to manage, which is actually counterproductive to the goal of streamlining the control algorithm and communication. As such, the filtered signals, \(z_{j}(t)\), are passed through a simple peak-picking algorithm and only these peak values are transmitted to the controller in an asynchronous sampling scheme. This allows for the control law to become a weighted sum of these peaks, expressed as  
\[u(t_{c})=\hat{W}P\big{(}k_{p}\big{)}\, \tag{8}\]  
where \(u(t_{c})\) is the calculated control value at control time increment \(t_{c}\), \(\hat{W}\in\mathbb{R}^{1 xn}\) is a control weighting vector, and \(P\big{(}k_{p}\big{)}\in\mathbb{R}^{nx1}\) is the vector of the detected peak values for all filters, given that \(n\) is the total number of filters. This complete process is depicted in \[x_{i}(k+1)=x_{i}(k)+v_{i}(k+1), \tag{2}\]0 For simplicity, the architecture shown in Figure 2 is a single input-single output system but it could easily be extended to a multiple input-multiple output system. There is no empirical method for determining the parameters in the control weighting vector, \(\hat{W}\), as shown in [25], and the PSO is one option for determining these control parameters. In the PSO optimization technique, each particle represents a potential weighting vector, \(\hat{W}\), where \(\hat{W}=[w_{1}\ w_{2}\ \cdots\ w_{n}]\). After completing the optimization, the particle with the smallest fitness function is chosen as the resulting weighting vector. This optimization is completed offline and prior to the control event, thereby minimizing the computational demand on the actuating node during the actual seismic event. \[x_{i}(k+1)=x_{i}(k)+v_{i}(k+1), \tag{2}\]1 Control architecture using the weighted sum of peaks from the filtered system response.