## 2 The Analysis of Structures with ATMD
Since the velocity of the structure directly affects the kinetic energy of the structure excited by the earthquake and the ATMD was positioned on the top story, where the responses are generally maximum, the error signal was chosen as the top story velocity of the structure (\(\dot{x}_{\rm N}\)) in the study. Here, velocity is measured via sensor and velocity feedback control is performed. In the optimum tuning of PID-controlled ATMD, six different design variables are considered to reduce the value of the objective function, which is taken as the displacement of the top story. The maximum absolute value solution is considered for extremum solution, and the maximum response under different excitations is considered in the optimization process. The objective function is formulized as Equation (10). In this study, the minimization of maximum top story displacement is chosen as the optimization objective. Also, other critical responses can be considered in structural control systems. The different optimum design strategies were reviewed by [PERSON] et al. [31]. The performance criteria used as objectives may be related to energy perspective results [32], inter-story drifts, accelerations [33] or multi-objectives of several of them [34]. \[f(x)=max(|x_{\rm N}|) \tag{10}\]  
The objective function is constrained with a function (g\({}_{1}\)) to consider the limitation of the movement of ATMD. It is formulated as Equation (11). \[g_{1}=\frac{max(|x_{\rm d}-x_{\rm N}|)_{\rm with\,ATMD}}{max(|x_{\rm N}|)_{ \rm without\,ATMD}}<st_{\rm max} \tag{11}\]  
\(st_{\rm max}\) is a user-defined value for the stroke limitation of ATMD. In the constraint, a normalization is performed with respect to the uncontrolled structure. The design variables are the physical parameters of mass damper and the gains of the controller. The ranges of physical parameters can be defined according to application. The mass of ATMD must be limited with respect to the axial force capacity of the structure, and it is generally defined with a ratio respect to the total mass of the structure. The stiffness of ATMD (k\({}_{\rm d}\)) is tuned by optimizing the period of ATMD (T\({}_{\rm atmd}\)), as formulated as Equation (12). The value of T\({}_{\rm atmd}\) must be close to the critical period of the structure. A range around that value can be chosen. \[T_{\rm atmd}=2\pi\sqrt{\frac{m_{\rm d}}{k_{\rm d}}} \tag{12}\]The damping of ATMD is generally defined with damping ratio (\(\xi_{\rm d}\)), given as Equation (13). For that reason, \(\xi_{\rm d}\) is taken as a design variable. \[\xi_{\rm d}=\frac{c_{\rm d}}{2m_{\rm d}\sqrt{\frac{k_{\rm d}}{m_{\rm d}}}}x100 \tag{13}\]  
The PID controller gains can be optimum in any value. This situation may increase optimization process time and some values will have a negative effect on structure. In some combinations of controller gains, a stability problem that resulted in a resonance state of the structure can be seen. The numerical integration methods may give non-defined solutions, and the optimization process may be interrupted. For that reason, trial and error may be used for finding a range, but the optimum value may be out of the range that is found with trial and error. To circumvent these pitfalls, a Matlab Simulink block diagram comparison check is performed on the displacement of a ATMD-controlled structure for each time-lag; if the value exceeds the value of the maximum displacement of the structure without control, the simulation is ended. In addition, to prevent the stability problem, it is saved from the computation time of the optimization process. The optimization iterations continue by generating new values.