## 6 Conclusions  
Linear-quadratic regulator (LQR) has been widely applied to structural control as it minimizes a cost function formulated by weighted states and control inputs. However, LQR requires structural states for feedback control which may not be measurable in real application. Therefore, state estimation is indispensable which inevitably takes additional computation time and probably results in time delay during the control application. In addition, the control effectiveness is sensitive to the performance of the state estimator. If the estimated states are not accurate enough and robustly stable, the corresponding control performance become mediocre. In this study, machine learning technology has been applied to learn the behavior of an LQR with optimized weighting matrices such that the control force can be predicted directly from the measurements. Two neural network models have been applied for learning which include a multilayer perceptron (MLP) model and an autoregressive with exogenous inputs (ARX) model.  
In the numerical simulation, the number of acceleration steps for training the MLP model is investigated. Simulation results indicate that 120, 150, and 200 acceleration steps led to stable and satisfactory simulation results. On the other hand, the ARX model takes the previous output as one of the inputs for training. Therefore, the number of inputs becomes much less than that required for the MLP model. Simulation results demonstrate that both MLP and ARX models predict the control force well in the control simulation model. Although the root-mean-square error of the ARX model is larger  
Figure 14: Time histories of the desired and achieved control force of the three controllers.  
Figure 15: Time histories of the structural responses in the experimental validation.  
than that of the MLP model, the elapsed time for training the ARX model is significantly less than that for training the MPL model. As a result, the ARX model is suggested in real application. In the experimental validation, the control force time histories of the MLP and ARX models are close to that of the LQR in both frequency and magnitude. Similar results can be also observed with respect to the structural displacement and acceleration. It demonstrates that the MLP and ARX are able to emulate the LQR controller fairly well in real application. Conclusively, machine-learning based controllers can be well-trained to replace conventional state feedback controllers which generally require an observer or state estimator to obtain the states. Machine-learning based controllers simply collect the measurable response and calculate the corresponding control force similar to the control force generated from a conventional state feedback controller. Accordingly, the control performance of LQR has been successfully emulated and the effect of state estimation error on the seismic control performance of LQR has been reduced. Future works will be focused on inverse training strategy in order to calculate the control force directly from the relationship between the control device and the structural acceleration.  
Conceptualization, resources, original draft preparation, writing, supervision, project administration, and funding acquisition, P.-C.C.; methodology, software, validation, formal analysis, investigation, and data curation, K.-Y.C. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Ministry of Science and Technology, Republic of China (Taiwan), grant number MOST 108-2221-E-011-006-MY2. The APC was funded by the Taiwan Building Technology Center from The Featured Areas Research Center Program within the framework of the Higher Education Sprout Project by the Ministry of Education in Taiwan.  
The experiments were conducted in the small-scale structural laboratory (SSL) of National Center for Research on Earthquake Engineering (NCREE) in Taiwan. The authors would like to thank all the technical support from NCREE.  
The authors declare no conflict of interest.