# Seismic Response Study of L-Shaped Frame Structure with Magnetorheological Dampers
## 3 Modeling Theory of the L-Shaped Frame Structure
### Semi-Active Control Algorithm  
At present, the classical linear quadratic control algorithm is mostly used to solve for the optimal control force [34]. In this algorithm, the optimal control force \(\mathbf{U}\) is selected through the following performance indicators.  
\[\mathbf{J}=\frac{1}{2}\!\int_{0}^{t}\!\left[\mathbf{Z}^{\mathrm{T}}\mathbf{Q}\mathbf{Z}+\mathbf{U} ^{\mathrm{T}}\mathbf{R}\mathbf{U}\right]\!dt+\frac{1}{2}\mathbf{Z}^{\mathrm{T}}\mathbf{Q}_{0} \mathbf{Z} \tag{21}\]  
where the diagonal positive definite matrix \(\mathbf{R}\), with the dimension of \(\mathrm{m}\times\mathrm{m}\), is the weighting matrix for the control matrix, where \(\mathrm{m}\) is the number of MRDs, the diagonal positive semidefinite matrix \(\mathbf{Q}_{0}\), and the diagonal positive definite matrix \(\mathbf{Q}\) are the weighting matrix for the response of the structure, and \(\mathbf{Z}\) is the state vector of the controlled system. The optimal control question of the system state is to find the optimal control vector in an infinite time interval; the optimal control force can be given as follows:  
\[\mathbf{U}=\mathbf{R}^{-1}\mathbf{B}^{\mathrm{T}}\mathbf{P}\mathbf{Z} \tag{22}\]  
in which the solution of the Riccati matrix algebraic equation \(\mathbf{P}\) is as follows:  
\[-\mathbf{P}\mathbf{A}-\mathbf{A}^{\mathrm{T}}\mathbf{P}+1/2\mathbf{P}\mathbf{B}\mathbf{R}^{-1}\mathbf{B}^{ \mathrm{T}}\mathbf{P}-2\mathbf{Q}=0 \tag{23}\]  
In which, \(\mathbf{A}=\left[\begin{array}{cc}\mathbf{0}&\mathbf{I}\\ -\mathbf{M}^{-1}\mathbf{K}&-\mathbf{M}^{-1}\mathbf{C}\end{array}\right]\), \(\mathbf{B}=\left[\begin{array}{cc}\mathbf{0}\\ -\mathbf{M}^{-1}\mathbf{H}\end{array}\right]\), \(\mathbf{Q}=\alpha\left[\begin{array}{cc}\mathbf{K}&\mathbf{0}\\ \mathbf{0}&\mathbf{M}\end{array}\right]\), \(\mathbf{R}=\beta\mathbf{I}\).  
In which \(\alpha\) and \(\beta\) can be obtained through trial calculation, and \(\mathbf{I}\) is the unit matrix with the dimension of \(\mathrm{m}\times\mathrm{m}\).  
As can be seen in Section 2, the minimum and maximum damping forces of the MRD are \(F_{\mathrm{min}}=3.1\) kN and \(F_{\mathrm{max}}=120\) kN when the current is 0 A and 0.28 A, respectively; therefore, the damping force range of the MRD is 3.1 to 120 kN, as shown in Figure 4. The optimal control force should be regulated when it surpasses the adjustable range of the damping force developed by the MRD, or it is opposite to the relative displacement direction of the two ends of the MRD. In addition, some scholars have found that a real-timecontrolled MRD can trace the required damping force well [35; 36; 37]. Thus, the semi-active control strategy for adjusting the optimal control force is shown in Equation (24).  
\[F_{i}=\left\{\begin{array}{cc}\text{sgn}(u_{i})F_{\text{min}}&\text{when }U_{i} \cdot u_{i}>0\text{ or }|U_{i}|<F_{\text{min}}\\ U_{i}&\text{when }U_{i}\cdot u_{i}<0\text{ and }F_{\text{min}}<|U_{i}|<F_{\text{max}}\\ \text{sgn}(u_{i})F_{\text{max}}&\text{when }U_{i}\cdot u_{i}<0\text{ and }|U_{i}|>F_{\text{max}} \end{array}\right. \tag{24}\]  
where \(U_{i}\) is the optimal control force of the \(i\)th MRD, \(u_{i}\) is the relative displacement at both ends of the \(i\)th MRD, and \(F_{i}\) is the damping force provided by a single MRD on the \(i\)th chevron support [38].