# Machine-Learning Based Optimal Seismic Control of Structure with Active Mass Damper

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Active Mass Damper

_{1}, c

_{1}, and k

_{1}represent the mass, damping coefficient, and stiffness of the primary structure; m

_{2}is the mass of the AMD mass block; u is the control force; and f is the equivalent external force. Numerous studies have been focused on controller synthesis and analysis for computing desired force for AMD installed on a structure for the past decades. Various control algorithms for AMD have been proposed and validated such as the complete-feedback control algorithm which incorporates displacement, velocity and acceleration measurements as the feedback signal [2]; the linear-quadratic regulator (LQR) which requires displacement and velocity feedback [3]; and the linear-quadratic Gaussian (LQG) controller with acceleration feedback control [4,5]. Meanwhile, the neural network has been proposed and become an alternative to replace the control algorithm for structural control [6]. Chen et al. used the back-propagation algorithm proposed by Rumelhart et al. [7] for training neural network in structural control [8]. Bani-Hani and Ghaboussi applied the neural network control algorithm to a nonlinear three degrees-of-freedom steel frame model [9]. Kim et al. proposed an optimal control algorithm using neural networks through minimization of the cost function [10]. Hung et al. used neural networks with a training algorithm for active pulse structural control without the trial-and-error selection of the learning rate [11]. Cho et al. applied a neural network control for nonlinear bridge systems with earthquake excitation [12]. Lin et al. proposed a smart active control framework which utilized fiber Bragg grating sensors and neural networks [13]. In addition, the neural network controller has been applied to reduce the wind-induced vibration of a tall building through an active tuned mass damper [14]. An AMD controller used neural network based on optimal control method for a cable-stayed bridge structure [15]. Most of these studies have been focused on numerical simulation without experimental validation. Recently, Chen and Yang [16] proposed a neural network with modified Newton method for structural control with an AMD and validated the control performance through shake table testing. Experimental results have shown that the neural network controller is effective in damping control applications. However, the mass ratio of the AMD used in the experiment was more than 20% which is not realistic in real implementation for buildings. Practical experimental studies regarding to the AMD seismic control performance achieved by using machine learning are considered limited.

#### 1.2. Motivation

## 2. Modal LQR with Optimized Weighting Matrices

#### 2.1. Modal LQR

**M**,

**C**, and

**K**represent the mass, damping, and stiffness matrix of the building, respectively;

**x**(t) is the displacement vector; u(t) is the control force;

**Λ**is the location vector of the control force;

**l**is a vector of order N with each element equal to unity; and ${\ddot{x}}_{g}(t)$ is the ground acceleration. It is noted that proportional damping of structures has been frequently assumed in structural dynamic analyses. As a result, if the building is proportional damping system, the equation of motion can be expressed in modal configuration by letting $x(t)=\Phi q(t)$. Hence, the equation of motion in modal space becomes:

**Φ**is the modal matrix which contains all the mode shapes;

**q**(t) is the modal displacement vector; $\overline{C}=diag\left(\begin{array}{ccc}2{\zeta}_{1}{\omega}_{1}& \cdots & 2{\zeta}_{N}{\omega}_{N}\end{array}\right)$ is a modal damping matrix in which the parameters ζ

_{i}and ω

_{i}represent the damping ratio and natural frequency of the i-th mode, respectively; and

**Ω**

^{2}is the modal natural frequency matrix and can be expressed as ${\Omega}^{2}=diag\left(\begin{array}{ccc}{\omega}_{1}^{2}& \cdots & {\omega}_{N}^{2}\end{array}\right)$. The state-space formulation of the building in modal space can be represented as:

**A**, the control force distribution matrix

_{m}**B**, and the disturbance location matrix

_{m}**E**in modal configuration become

_{m}**y**(t) is selected as the modal absolute acceleration of each mode and the matrices

_{m}**C**and

_{m}**D**become:

_{m}**Q**and

**R**determined by users. In the modal LQR design, the modal absolute acceleration is taken as the state to be regulated. Therefore, the quadratic cost function of LQR for structural control is defined as:

**Q**depends on the number of selected modes to be controlled. The weighting R is a scalar since there is merely one control force imposed on the top of the structure. The state feedback gain matrix

**K**can be obtained by solving the Ricatti equation:

_{m}**P**is the solution of the Ricatti equation:

**K**(t). In order to convert the control input from the modal space to the configuration space, the gain matrix

_{m}z_{m}**K**needs to be converted into

_{m}**K**through:

_{g}**K**can be used to calculate the control force for the structure by u(t) = −

_{g}**K**(t) where

_{g}z**z**is the states in the configuration space which contains the displacement and velocity vectors, i.e., $\dot{z}(t)={\left[\begin{array}{cc}x(t)& \dot{x}(t)\end{array}\right]}^{T}$.

#### 2.2. Symbiotic Organisms Search

**x**. In the mutualism phase, two organisms are randomly selected and the position of each organism is updated by following the equation:

_{best}**x**and

_{i}’**x**are the updated position of the i-th and j-th organisms, respectively; and BF

_{j}’_{1}and BF

_{2}are the benefit factors of the i-th and j-th organisms, respectively which are randomly assigned as either 1 or 2 at each iteration step. The benefit factors are utilized to simulate whether the two organisms partially or fully benefit from the other. Since BF

_{1}and BF

_{2}are randomly assigned, tuning is not required in the mutualism phase. The commensalism phase is performed after the mutualism phase is completed. Similar to the mutualism phase, two organisms are randomly selected and one of them intends to benefit from the other by:

## 3. Neural Network Models

#### 3.1. Multilayer Perceptron Model

#### 3.2. Autoregressive Exogenous with Exogenous Inputs Model

## 4. Numerical Study: A 10-Story Shear Building

#### 4.1. Modal LQR with Optimized Weighting Matrices

^{2}/m and 2000 kN/m, respectively. The undamped natural frequencies of the first five modes of the structure are shown in Table 1. An active mass damper was assumed to be installed on the top floor of the structural model with a force capacity of ±50 kN, which was roughly equal to 5% structural weight. The damping ratio of the structural model was assumed 2% for all vibration modes. Figure 6 depicts the schematic of the 10-story shear building with an AMD at the top.

^{2}/m, 9.14 kN-s

^{2}/m, and 3.09 kN-s

^{2}/m, respectively, the accumulated effective modal mass of the first three modes were 97.02 kN-s

^{2}/m which was larger than 95% of the structural weight (95 kN-s

^{2}/m). As a result, the first three modes were adopted as the control modes in the numerical study. MATLAB/Simulink, provided by The Mathworks Inc., Natick, MA, USA was adopted to conduct the numerical simulation of the 10-story building with AMD optimized by SOS with respect to the objective function:

_{mi}represents the modal acceleration of the i-th mode; and n

_{m}is the number of control modes. Actually, the objective function is the SRSS of maximum acceleration of the selected modes. For the optimization of weighting matrices, the weighting R was remained a fixed value of 100 while 10

^{10}and 0 were selected as the upper and lower bounds value of each element in the weighting marix

**Q**. A band-limited white noise (BLWN) with a peak ground acceleration (PGA) of 2.4 m/s

^{2}, a bandwidth from 0 Hz to 15 Hz, and a duration of 140 s was used to excite the 10-story shear building. The root-mean-square (RMS) of the BLWN was 0.5526 m/s

^{2}. The bandwidth was considered sufficient to cover the frequency component of most historical earthquake. It is worth noting that saturation with 10% of the actuator capacity of AMD was used in the optimization because the generated force of the LQR could reach the force capacity of the AMD actuator without the 10% saturation. A number of 100 organisms and 35 iterations were adopted for SOS optimization. Figure 7 depicts the convergence curve with respect to the objective function. It shows that the optimal value was reached at the 11th iteration. Note that the initial values in the weighting matrix

**Q**were randomly assigned; however, the corresponding LQR still achieved effective control performance. After the first iteration, the objective function value was significantly reduced. Finally, the corresponding optimized state feedback gain matrix

**K**was obtained as:

_{g}#### 4.2. Training and Validating of MLP Model

**K**shown in Equation (13) was used in the time history analysis of the 10-story shear building with an AMD at the roof subjected to a BLWN ground acceleration. The corresponding absolute acceleration of each floor was taken as the input of the MLP model and the generated control force was the desired output to be predicted by the MLP model. In the training process, the MLP model with one input layer, three hidden layers, and one output layer was structured. Each hidden layer contained 300 neurons. The output layer had merely one neuron representing the control force at the current predicting step. Since the control force at the current step was related to previous various steps of structural response, the number of neurons in the input layer was investigated first in the study. Note that the acceleration and control force from the time history analysis were normalized to a peak value of ±1.0 before the training process of MLP was conducted. The objective of training the MLP aims to optimize the weights of the model; therefore, a loss function was required for training. In this study, the mean square error (MSE) between the predicted and desired control force was adopted as the loss function. The adaptive moment estimation (Adam) optimizer [19] which computed adaptive learning rates for each parameter was used to train the model in addition to the characteristics of adaptive gradient descent. Since the calculation of Adam included offset correction, the weight updating of the MLP model was sustained within a specific range. As a result, the update of parameters was relatively smooth and the calculation efficiency was exceptional.

_{g}_{p}[k] are the desired and the predicted control force at the kth step, respectively. It is realized that better learning leads to smaller RMSE. After training, the MLP model was then implemented in the control simulation model as shown in Figure 8 using MATLAB/Simulink. The input of the MLP was the acceleration response of each floor with the current and previous steps, and the output was the calculated control force imposed on the 10-story shear building. Note that the training data of acceleration and control force for MLP were normalized to a peak value of ±1.0; therefore, the acceleration responses of the building were scaled by a gain before they were input to the MLP model. Similarly, the calculated control force of the MLP model was also scaled before it was imposed on the shear building. Table 2 shows the training results considering various number of acceleration steps in the input layer. It can be found that the RMSE between the desired and predicted control force is smaller than 2.0% for all cases, indicating that the MLP model is able to learn the control force generated from an LQR with optimized weighting matrices through the corresponding controlled acceleration responses. Thus, it was expected that the implementation of the MLP model as depicted in Figure 8 should be able to generate similar control force to the LQR control force. The ode5 solver using the Dormand–Prince formula was adopted for the control simulation in Simulink. Table 2 also shows the RMSE between the desired and predicted control force in the control simulation model. All the simulations were computed using a sampling rate of 200 Hz. However, the control simulation model was divergent and became inexecutable for the cases of 10, 50, and 100 acceleration steps due to error propagation in the control loop. On the other hand, the cases of 120, 150, and 200 acceleration steps all led to stable and satisfactory simulation results. Among the three successful cases, the computational effort of the 120 steps was the least. Therefore, it was suggested that a minimum number of 120 steps was needed as the input of the MLP model. Accordingly, a minimum number of 1210 neurons was required in the input layer for the MLP model.

#### 4.3. Training and Validating of ARX Model

_{p}[k] are the desired and the predicted control force at the k-th step, respectively; and n

_{a}and n

_{f}are the number of acceleration and control force delay steps for training in ARX model. In the series-parallel architecture, the predicted control force u

_{p}[k] is calculated from the current and previous steps of the absolute acceleration at each floor as well as the previous steps of the desired control force. In the parallel architecture, the predicted control force u

_{p}[k] is performed from the present and past values of the absolute acceleration at each floor as well as the past predicted control force. In this study, the series-parallel architecture was adopted for training while the parallel architecture was implemented in the control simulation model of MATLAB/Simulink. In the training process, the ARX model with one input layer, two hidden layers, and one output layer was structured. The first and second hidden layer contained 100 and 50 neurons, respectively. In the output layer, there was simply one neuron representing the control force at the current step. The number of acceleration and control force delay steps in the training process were selected one, i.e., n

_{a}= n

_{f}= 1. Accordingly, the dimension of the ARX neural network model was 21 × 100 × 50 × 1. Similar to the MLP training, the training data of acceleration and control force were normalized to a peak value of ±1.0, and the MSE between the predicted and desired control force was used as the loss function and Adam was applied as the optimizer. The number of epochs was 5000. The RMSE between the desired and predicted control force was 0.25%.

#### 4.4. Comparison of Seismic Control Performance

_{i}(t) is the relative displacement of the i-th floor during the excitation; and x

^{max}represents the maximum displacement of the uncontrolled shear building. The second performance index is the maximum normalized inter-story drift which can be represented as:

_{i}(t) is the inter-story drift of the i-th floor during the excitation; h

_{i}is the story height of the i-th floor; and d

_{n}

^{max}represents the maximum normalized inter-story drift of the uncontrolled shear building. The third performance index is the normalized peak absolute acceleration which can be expressed as:

^{2}were adopted as the ground excitation to the 10-story shear building with an AMD. The AMD control force was generated by using the LQR state feedback control with optimized weighting matrices using SOS (LQR-SOS), the MLP models with 120, 150, and 200 input acceleration steps (MLP-120, MLP-150, and MLP-200), and the ARX model. As mentioned previously the force capacity of the AMD was assumed ±50 kN. The AMD controlled 10-story shear building was subjected to the fourteen earthquakes. Table 3, Table 4, Table 5, Table 6 and Table 7 list the performance indices of each simulation control case. Table 8 compares the average seismic performance of the 10-story shear building subjected to the fourteen earthquakes and the average RMSE of the fourteen excitation cases between the control force generated from LQR-SOS and the neural network controllers. It can be found that the three MLP models emulated the LQR-SOS controller extraordinarily. The seismic responses of the LQR-SOS, and the three MLP models are nearly identical in terms of the four performance indices. The RMSE between the control force generated by the LQR-SOS and the three MLP models is less than 2.0% for each case. On the other hand, the ARX model emulated the LQR-SOS fairly well. The seismic responses of the LQR-SOS, and ARX model are close to each other with respect to the four performance indices. Meanwhile, the RMSE between the LQR-SOS control force and the ARX control force is less than 5.0% for nearly all cases. Generally speaking, the RMSE of the ARX model is approximately two times larger than that of the MPL model.

## 5. Experimental Validation

#### 5.1. Experimental Setup

^{®}Controller FT-100 digital controller, manufactured by MTS Systems Corporation, Eden Prairie, MN, USA was used to control the actuator using a proportional-integral-derivative control algorithm. An analog controller converted voltage into current was used to control the torque of the motor. Three high-resolution MEMS accelerometers manufactured by Silicon Designs, Inc., Kirkland, WA, USA were installed on the bottom rigid plate connected to the shake table, the roof of specimen, and the mass block to measure the absolute acceleration. Each accelerometer was able to measure a range of acceleration of ±20 m/s

^{2}. Note the control force of AMD was obtained by multiplying the mass of the mass block by the absolute acceleration measured from the accelerometer. Meanwhile, a Temposonics GH linear-position sensor, manufactured by MTS Systems Corporation, Eden Prairie, MN, USA was installed between the roof and the platen of shake table to measure the relative displacement of the specimen.

#### 5.2. Experimental Results

^{2}, a bandwidth from 0 to 15 Hz, and a duration of 140 s was used to excite the identified structural model with a time step of 0.005 sec. Accordingly, the weightings Q and R in the cost function shown in Equation (6) were optimized and the optimal value of Q and R were 1.8357 and 100, respectively in the experimental validation. Accordingly, the state feedback gain

**K**was obtained as [−0.2393, 0.1329]. Afterwards, the structural model was subjected to the band-limited white noise excitation again and the control force and structural acceleration were obtained. Then, MLP with 150 steps of acceleration (MLP-150) and the ARX controllers were trained offline using these data.

_{g}^{2}in the experimental validation. The LQR-SOS, MLP-150 and ARX were implemented in the experimental validation. A Kalman filter was designed based on the identified structural model in order to obtain the structural states for LQR state feedback control. Since both the MLP and ARX required the acceleration measured from the specimen to calculate the control force, the noise in the acceleration could result in high-frequency control force and enlarge the acceleration response. Therefore, a 4th-order elliptic low-pass filter with a cutoff frequency of 15 Hz was implemented for the measured acceleration. Table 9 shows the experimental results of the three controllers with respect to the performance indices. Note that the specimen used in the validation was a SDOF shear building. Therefore, the performance indices J

_{1}and J

_{2}were identical. From Table 9, it is found the results of MLP-150 and ARX in the experimental validation are inferior to the numerical simulation results considering the effectiveness of LQR emulation. This is due to the fact that the structural states and acceleration were permanently accurate and precise in the numerical simulation, the error of the Kalman filter and effect of the low-pass filter could not be observed. Nevertheless, the overall performance of the trained MLP-150 and ARX models is considered fairly well comparing with the LQR-SOS. The performance indices of the three controllers in average are approximately within the same level. Figure 14 and Figure 15 illustrate the time histories of control force and structural response of the specimen, respectively when the specimen was subjected to El Centro earthquake. From Figure 14, it is investigated that there exists tracking error between the desired and achieved control force in the three control cases. The achieved control force is frequently smaller than the desired one. However, the tracking performance is considered acceptable considering the friction between the guide screw and the mass block. In addition, the control force time histories of the three control cases are similar in both frequency and magnitude. Similar trend can be also observed in the structural responses as depicted in Figure 15. It demonstrates that the MLP and ARX are able to emulate the LQR controller reasonably successful in the experimental validation.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Blocked diagram of the training process. LQR: linear-quadratic regulator; ML: machine learning.

**Figure 9.**Architectures of the exogenous inputs (ARX) neural network: (

**a**) series-parallel, and (

**b**) parallel.

**Figure 13.**Flowchart of the design and training procedure for the experimental validation. BLWN: band-limited white noise.

1st Mode | 2nd Mode | 3rd Mode | 4th Mode | 5th Mode |
---|---|---|---|---|

0.336 | 1.002 | 1.644 | 2.250 | 2.807 |

**Table 2.**Training results considering various acceleration steps for the multilayer perceptron (MLP).

Number of Acceleration Steps | ||||||
---|---|---|---|---|---|---|

10 | 50 | 100 | 120 | 150 | 200 | |

Training RMSE (%) | 1.95 | 0.40 | 0.62 | 0.68 | 0.89 | 0.87 |

Simulation RMSE (%) | N.A. | N.A. | N.A. | 1.77 | 1.82 | 1.70 |

**Table 3.**Seismic control performance of the LQR (linear-quadratic regulator) -SOS (symbiotic organisms search).

Earthquakes | J_{1} | J_{2} | J_{3} | J_{4} |
---|---|---|---|---|

El Centro | 0.434 | 0.523 | 0.803 | 0.020 |

Chichi | 0.472 | 0.814 | 0.690 | 0.016 |

Kobe | 0.531 | 0.584 | 0.645 | 0.029 |

Meinong | 0.686 | 0.747 | 0.655 | 0.049 |

Northridge | 0.553 | 0.655 | 0.895 | 0.025 |

Kumamoto | 0.543 | 0.718 | 0.648 | 0.020 |

Capemendocino | 0.557 | 0.677 | 0.684 | 0.028 |

Parkfield | 0.702 | 0.797 | 0.732 | 0.022 |

Chuetsu Oki | 0.605 | 0.549 | 0.674 | 0.030 |

Montenegro | 0.778 | 0.812 | 0.701 | 0.030 |

Imperial Valley | 0.553 | 0.531 | 0.606 | 0.040 |

Taipei | 0.501 | 0.487 | 0.462 | 0.047 |

El Mayor | 0.584 | 0.618 | 0.569 | 0.030 |

Darfield | 0.846 | 0.896 | 0.744 | 0.033 |

Average | 0.596 | 0.672 | 0.679 | 0.030 |

Earthquakes | J_{1} | J_{2} | J_{3} | J_{4} | RMSE |
---|---|---|---|---|---|

El Centro | 0.433 | 0.523 | 0.801 | 0.020 | 0.018 |

Chichi | 0.471 | 0.813 | 0.690 | 0.016 | 0.019 |

Kobe | 0.531 | 0.582 | 0.643 | 0.030 | 0.017 |

Meinong | 0.686 | 0.747 | 0.656 | 0.049 | 0.011 |

Northridge | 0.553 | 0.655 | 0.890 | 0.025 | 0.009 |

Kumamoto | 0.543 | 0.720 | 0.645 | 0.020 | 0.016 |

Capemendocino | 0.558 | 0.677 | 0.686 | 0.028 | 0.016 |

Parkfield | 0.702 | 0.791 | 0.731 | 0.022 | 0.015 |

Chuetsu Oki | 0.604 | 0.549 | 0.670 | 0.030 | 0.015 |

Montenegro | 0.779 | 0.812 | 0.699 | 0.030 | 0.012 |

Imperial Valley | 0.552 | 0.531 | 0.597 | 0.040 | 0.010 |

Taipei | 0.501 | 0.485 | 0.450 | 0.047 | 0.008 |

El Mayor | 0.583 | 0.615 | 0.572 | 0.030 | 0.013 |

Darfield | 0.847 | 0.897 | 0.739 | 0.033 | 0.011 |

Average | 0.596 | 0.671 | 0.676 | 0.030 | 0.014 |

Earthquakes | J_{1} | J_{2} | J_{3} | J_{4} | RMSE |
---|---|---|---|---|---|

El Centro | 0.433 | 0.523 | 0.801 | 0.020 | 0.013 |

Chichi | 0.471 | 0.813 | 0.690 | 0.016 | 0.015 |

Kobe | 0.532 | 0.583 | 0.644 | 0.029 | 0.014 |

Meinong | 0.687 | 0.747 | 0.655 | 0.049 | 0.010 |

Northridge | 0.553 | 0.655 | 0.891 | 0.025 | 0.008 |

Kumamoto | 0.543 | 0.720 | 0.646 | 0.020 | 0.011 |

Capemendocino | 0.559 | 0.677 | 0.687 | 0.028 | 0.013 |

Parkfield | 0.703 | 0.790 | 0.731 | 0.022 | 0.015 |

Chuetsu Oki | 0.605 | 0.550 | 0.669 | 0.030 | 0.011 |

Montenegro | 0.779 | 0.812 | 0.699 | 0.030 | 0.011 |

Imperial Valley | 0.553 | 0.531 | 0.600 | 0.040 | 0.008 |

Taipei | 0.501 | 0.487 | 0.453 | 0.047 | 0.007 |

El Mayor | 0.583 | 0.615 | 0.573 | 0.030 | 0.011 |

Darfield | 0.847 | 0.897 | 0.736 | 0.033 | 0.009 |

Average | 0.596 | 0.671 | 0.677 | 0.030 | 0.011 |

Earthquakes | J_{1} | J_{2} | J_{3} | J_{4} | RMSE |
---|---|---|---|---|---|

El Centro | 0.434 | 0.524 | 0.803 | 0.020 | 0.013 |

Chichi | 0.471 | 0.813 | 0.690 | 0.016 | 0.015 |

Kobe | 0.532 | 0.583 | 0.645 | 0.029 | 0.016 |

Meinong | 0.687 | 0.747 | 0.656 | 0.049 | 0.011 |

Northridge | 0.554 | 0.655 | 0.893 | 0.025 | 0.010 |

Kumamoto | 0.543 | 0.720 | 0.645 | 0.020 | 0.011 |

Capemendocino | 0.559 | 0.677 | 0.688 | 0.028 | 0.013 |

Parkfield | 0.703 | 0.791 | 0.731 | 0.022 | 0.016 |

Chuetsu Oki | 0.605 | 0.550 | 0.671 | 0.030 | 0.011 |

Montenegro | 0.779 | 0.812 | 0.699 | 0.029 | 0.012 |

Imperial Valley | 0.553 | 0.531 | 0.601 | 0.040 | 0.008 |

Taipei | 0.502 | 0.488 | 0.455 | 0.047 | 0.008 |

El Mayor | 0.583 | 0.617 | 0.573 | 0.030 | 0.012 |

Darfield | 0.847 | 0.897 | 0.738 | 0.033 | 0.010 |

Average | 0.596 | 0.672 | 0.678 | 0.030 | 0.012 |

Earthquakes | J_{1} | J_{2} | J_{3} | J_{4} | RMSE |
---|---|---|---|---|---|

El Centro | 0.431 | 0.522 | 0.799 | 0.020 | 0.045 |

Chichi | 0.472 | 0.809 | 0.686 | 0.016 | 0.054 |

Kobe | 0.527 | 0.576 | 0.632 | 0.030 | 0.049 |

Meinong | 0.685 | 0.747 | 0.654 | 0.048 | 0.032 |

Northridge | 0.553 | 0.655 | 0.894 | 0.025 | 0.037 |

Kumamoto | 0.541 | 0.722 | 0.647 | 0.020 | 0.039 |

Capemendocino | 0.557 | 0.677 | 0.682 | 0.028 | 0.043 |

Parkfield | 0.696 | 0.770 | 0.733 | 0.022 | 0.045 |

Chuetsu Oki | 0.602 | 0.549 | 0.662 | 0.030 | 0.031 |

Montenegro | 0.779 | 0.809 | 0.696 | 0.029 | 0.040 |

Imperial Valley | 0.549 | 0.531 | 0.577 | 0.039 | 0.026 |

Taipei | 0.500 | 0.483 | 0.438 | 0.046 | 0.027 |

El Mayor | 0.579 | 0.611 | 0.556 | 0.029 | 0.039 |

Darfield | 0.845 | 0.899 | 0.714 | 0.032 | 0.034 |

Average | 0.594 | 0.669 | 0.669 | 0.030 | 0.039 |

Controller | J_{1} | J_{2} | J_{3} | J_{4} | RMSE |
---|---|---|---|---|---|

LQR-SOS | 0.596 | 0.672 | 0.679 | 0.030 | 0 |

MLP-120 | 0.596 | 0.671 | 0.676 | 0.030 | 0.014 |

MLP-150 | 0.596 | 0.671 | 0.677 | 0.030 | 0.011 |

MLP-200 | 0.596 | 0.672 | 0.678 | 0.030 | 0.012 |

ARX | 0.594 | 0.669 | 0.669 | 0.030 | 0.039 |

Controller | LQR-SOS | MLP-150 | ARX | ||||||
---|---|---|---|---|---|---|---|---|---|

Earthquakes | J_{1}/J_{2} | J_{3} | J_{4} | J_{1}/J_{2} | J_{3} | J_{4} | J_{1}/J_{2} | J_{3} | J_{4} |

El Centro | 0.473 | 0.455 | 0.036 | 0.431 | 0.437 | 0.033 | 0.456 | 0.544 | 0.036 |

Chichi | 0.668 | 0.690 | 0.021 | 0.672 | 0.752 | 0.026 | 0.693 | 0.728 | 0.027 |

Kobe | 0.447 | 0.438 | 0.056 | 0.519 | 0.567 | 0.042 | 0.525 | 0.515 | 0.036 |

Northridge | 0.899 | 0.885 | 0.030 | 0.776 | 0.841 | 0.032 | 0.843 | 0.832 | 0.038 |

Parkfield | 0.449 | 0.414 | 0.039 | 0.389 | 0.446 | 0.036 | 0.441 | 0.501 | 0.031 |

Montenegro | 0.671 | 0.641 | 0.043 | 0.613 | 0.616 | 0.035 | 0.635 | 0.642 | 0.034 |

Meinong | 0.637 | 0.616 | 0.048 | 0.644 | 0.621 | 0.033 | 0.647 | 0.679 | 0.045 |

Capemendocino | 0.743 | 0.685 | 0.041 | 0.764 | 0.839 | 0.051 | 0.790 | 0.787 | 0.041 |

Average | 0.623 | 0.603 | 0.039 | 0.601 | 0.640 | 0.036 | 0.629 | 0.653 | 0.036 |

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## Share and Cite

**MDPI and ACS Style**

Chen, P.-C.; Chien, K.-Y. Machine-Learning Based Optimal Seismic Control of Structure with Active Mass Damper. *Appl. Sci.* **2020**, *10*, 5342.
https://doi.org/10.3390/app10155342

**AMA Style**

Chen P-C, Chien K-Y. Machine-Learning Based Optimal Seismic Control of Structure with Active Mass Damper. *Applied Sciences*. 2020; 10(15):5342.
https://doi.org/10.3390/app10155342

**Chicago/Turabian Style**

Chen, Pei-Ching, and Kai-Yi Chien. 2020. "Machine-Learning Based Optimal Seismic Control of Structure with Active Mass Damper" *Applied Sciences* 10, no. 15: 5342.
https://doi.org/10.3390/app10155342