## 2 Methodology
### Linear Quadratic Regulator (LQR) Feedback Control  
Equation (14) cannot be solved directly because the number of equations (12) is less than the number of unknown variables (13) in the case of base isolation, i.e., 12 \(\times\) response outputs \(Z(t)\) and 1 \(\times\) control force \(u(t)\). Thus, one more equation in this case is needed to solve the control problem. This required equation is referred to as the feedback control law. There are three control outlines through which the feedback control law can be implemented [59; 60; 61] to the smart structure model, i.e., open-loop feedback control, closed-loop feedback control, and open-closed loop feedback controls. For open-loop feedback control, the control force is determined by feedback of excitation at the base, such as sinusoidal motion or earthquake ground motion. The input information for its control law is only the base excitation data acquired with the help of accelerometers, etc. The obtained information is then used to calculate the required control force. For closed-loop feedback control, the control force is determined by the feedback of structure responses at each or some degrees-of-freedom. The input information for its control law consists of the structure responses such as velocities or relative displacements with the help of sensors. The acquired information is then used to calculate the required control force. The open-closed-loop scheme is a combination of both the closed-loop and open-loop control schemes. This can obtain information on both the ground motions and the structural responses. The control force is dependent on ground motion, displacement, and velocity responses [62]. Utilizing one of the above-discussed control schemes, Equation (14) can be solved mathematically. For a structure with closed-loop feedback control, the control force matrix is determined by using the measurements of response values and feeding them back to the equation. In this way, the feedback law will be given as  
\[\{u(t)\}\ =-[G]\{Z(t)\} \tag{19}\]  
In the case of the MRE base-isolated model, u will be a single value of force applied at the isolation level. \(G\) is a 1 \(\times\) 12 matrix of feedback gain. Hence, the key parameter to make Equation (14) mathematically solvable is gain matrix \(G\). In the current study, the gain matrix is obtained by using the linear quadratic regulator (LQR) algorithm and is discussed ahead. Once \(G\) is obtained, the closed-loop system takes the form as expressed below  
\[\{Z(t)\}=\ [A_{c}]\{Z(t)\}\ +\ [B_{r}]\ddot{x}_{g}(t) \tag{20}\]  
where \(A_{c}\) is the closed-loop plant matrix obtained through a substitution operation of Equation (19) into Equation (14), and expressed as  
\[[A_{c}]\ =\ [A]\ -\ [B_{u}][G] \tag{21}\]  
The linear quadratic regulator (LQR) algorithm for structural control is an optimal control algorithm. The key objective of LQR is to minimize the following quadratic cost function \(J\) with respect to the control force input \(u(t)\). \[J=\int_{0}^{t}\Bigl{[}z^{T}(t)Q\ z(t)\ +\ u^{T}(t)\ R\ u(t)\Bigr{]}dt \tag{22}\]  
In regulator-type algorithm problems, the system is idealized to be in equilibrium, and the LQR control algorithm serves the purpose of maintaining that equilibrium, even though the system is subjected to turbulences or of minimizing the response of the system under consideration from any sorts of disturbances [6]. The parameters \(Q\) and \(R\) in the quadratic cost function \(J\) are used as design parameters to penalize the state variables and the control signals, respectively. The larger these values, the greater the penalizing of these signals. Choosing a large value for \(R\) means the system is intended to become stabilized with less (weighted) force. This is usually known as an expensive control strategy. On the other hand, choosing a small value for \(R\) means the control force will not be penalized.