## 2 Semi-Active Control Theory
### Semi-Active Control Algorithm  
Semi-active control algorithms are the core of structural semi-active control. Summarizing the more commonly used semi-active control algorithms, there are two main types: (1) switching type, such as simple bang-bang control algorithm, optimal bang-bang control algorithm, and maximum switching control algorithm. (2) Continuity, such as bounded bang-bang control algorithm, bounded sliding control algorithm, and bounded Hrovat optimal control algorithm. The following is a brief description of the three algorithms for comparative analysis in this paper.  
1. Simple bang-bang control algorithm The output force of the MR damper can be expressed as \[f_{d}=c_{d}\dot{x}_{d}+f_{dw}sgn\left(\dot{x}_{d}\right)\] (8) where \(\dot{x}_{d}\) is the relative velocity of the MR damper; \(c_{d}\) is the viscous damping coefficient of the MR damper, and \(f_{dw}\) is adjustable Coulomb damping force of the MR damper.  
Then, the simple bang-bang algorithm (semictrl1) can be expressed as \[u_{d}(t)=\left\{\begin{array}{l}f_{dmax}\left(x\dot{x}\right)>0\\ f_{dmin}\left(x\dot{x}\right)\leq 0\end{array}\right.\] (9) where \(x=x_{d}\) is the relative displacement of the MR damper, and this control algorithm indicates the maximum damping force (\(f_{dmax}\)) that can be achieved by applying the damper when the MR damper location is vibrating away from the equilibrium point and the minimum damping force (\(f_{dmin}\)) otherwise.
2. Optimal bang-bang control algorithm The optimal bang-bang algorithm (semictrl2) can be expressed as \[u_{d}(t)=\left\{\begin{array}{l}f_{dmax}\left(u\dot{x}\right)<0\\ f_{dmin}\left(u\dot{x}\right)\geq 0\end{array}\right.\] (10)  
The equation represents the maximum damping force that can be achieved by applying the damper when the optimal control force is in the opposite direction of the vibration at the location of the MR damper. Otherwise, the minimum damping force is applied. The algorithm exactly reflects the fact that the MR damper applies only the force that prevents the structure from moving but not the force that drives the structure.
3. Bounded Hrovat control algorithm The bounded Hrovat algorithm (semictrl3) can be expressed as \[u_{d}(t)=\left\{\begin{array}{l}f_{dmax}\left(u\dot{x}<0\ and\ |\ u\right|>u_{dmax}\right)\\ |u|sgn\left(\dot{x}\right)\left(u\dot{x}<0\ and\ |\ u\right|<u_{dmax}\right)\\ f_{dmin}\left(u\dot{x}\geq 0\right)\end{array}\right.\] (11)where \(u_{dmax}=c_{d}\big{|}\,\dot{x}\big{|}\,+f_{dmax}\) is the maximum damping force that can be achieved by the MR damper relative to the active optimal control force at the moment \(u(t)\). Based on the optimal bang-bang control algorithm, the Hrovat semi-active control force (\(\left|\,u\,|sgn(\dot{x})\right.\)) is added to the range of Coulomb damping force that can be realized by MR damper, which is the middle part of the equation.  
It should be noted that the relationship between the semi-active control force \(u_{is}(t)\) and the semi-active damping force \(f_{id}(t)\) is \(u(t)=-f(t)\). The damping force of the MR damper is \(f_{id}(t)=c_{id}y_{is}(t)+f_{idy}sgn[y_{is}(t)]\), where \(c_{id}\) is the viscous damping coefficient of the magnetroheological damper, \(f_{idy}\) is the adjustable Coulomb damping force of the MR damper, and \(y_{is}(t)\) is the relative velocity of the bridge junction connected by the two segments of the MR damper--that is, the relative motion velocity of the \(i\)-th MR piston and cylinder block.