# Earthquake Hazard Mitigation for Uncertain Building Systems Based on Adaptive Synergetic Control
## 3 Adaptive Synergetic Control Design
### Adaptive Synergetic Control  
Due to earthquake, uncertainty in the coefficients of a vibrated building structure will appear. For this study, two uncertainties are considered; one concerns the coefficient ofdamping viscosity \(B\), and the other uncertainty is due to change in the stiffness coefficient \(K\) of the building system [39; 40]. This can be expressed as  
\[\hat{B}=B+\widetilde{B},\ \hat{K}=K+\widetilde{K} \tag{17}\]  
where \(\hat{B}\) and \(\hat{K}\) are the estimated values of the nominal viscous damping coefficient \(B\) and the nominal stiffness coefficient \(K\), respectively. The adaptive synergetic controller is designed on the basis of Lyapunov stability analysis to develop the control and adaptive laws for estimating uncertain coefficients. The objective of the proposed adaptive scheme is to reduce the vibration effect on an MR damper-supported building structure due to earthquake.  
**Assumption:** The stability analysis is based on the assumption that the actual values of the stiffness and viscosity coefficients are stationary or vary slowly with time.  
The candidate Lyapunov function can be defined as  
\[V=\frac{1}{2}\psi\ (e)^{2}+\frac{1}{2}\ \gamma_{1}\ \widetilde{B}^{2}+\frac{1}{2} \ \gamma_{2}\ \widetilde{K}^{2} \tag{18}\]  
where \(\gamma_{1}\), \(\gamma_{2}\) denote the adaptation gains. The time derivative of Equation (18) is given by  
\[\dot{V}=\psi(e)\ \big{(}c.\dot{e}+\widetilde{e}\big{)}+\gamma_{1}\ \widetilde{B}. \widetilde{B}+\gamma_{2}\ \widetilde{K}.\ \hat{K} \tag{19}\]  
Substituting Equation (10) into Equation (19), we obtain  
\[\dot{V}=\psi(e)\ \bigg{(}c\ \dot{e}+\bigg{(}-\frac{B_{s}}{M_{s}}\ x_{2}- \frac{K_{s}}{M}x_{1}+\frac{1}{M_{s}}v+\widetilde{x}_{g}\bigg{)}\bigg{)}+ \gamma_{1}\ \widetilde{B}\ \dot{\hat{B}}+\gamma_{2}\ \widetilde{K}\ \dot{\hat{K}} \tag{20}\]  
The control law \(v\) can be determined on the basis of Equation (16) using the chosen estimated values of  
\[v=\hat{B}_{s}\ x_{2}+\hat{K}_{s}\ x_{1}-M_{s}\ \ddot{x}_{g}-M_{s}\ c\ \dot{e}- \frac{M_{s}}{T}\psi(e) \tag{21}\]  
Using Equation (21), the expression for \(\dot{V}\) becomes  
\[\dot{V}=\psi(e)\bigg{(}-\frac{B_{s}}{M_{s}}\ x_{2}-\frac{K_{s}}{M_{s}}x_{1}+ \frac{\hat{B}_{s}}{M_{s}}x_{2}+\frac{\hat{K}_{s}}{M_{s}}x_{1}-\frac{\psi(e)}{ T}\bigg{)}+\gamma_{1}\ \widetilde{B}\ \dot{\hat{B}}+\gamma_{2}\ \widetilde{K}\ \dot{\hat{K}} \tag{22}\]  
Using Equation (17), we have  
\[\dot{V}=-\psi^{2}(e)+\frac{\widetilde{B}}{M_{s}}x_{2}\ \psi(e)+\frac{ \widetilde{K}}{M_{s}}x_{1}\ \psi(e)+\gamma_{1}\ \widetilde{B}\ \dot{\hat{B}}+\gamma_{2}\ \widetilde{K}\ \dot{\hat{K}} \tag{23}\]  
or  
\[\dot{V}=-\psi^{2}(e)+\widetilde{B}\ \bigg{(}\gamma_{1}\dot{\hat{B}}+\psi(e) \frac{x_{2}}{M_{s}}\bigg{)}+\widetilde{B}\ \bigg{(}\gamma_{2}\ \dot{\hat{K}}+\psi(e)\frac{x_{1}}{M_{s}}\bigg{)} \tag{24}\]  
To guarantee \(\dot{V}<0\), the last two terms must be evaluated as zeros; that is,  
\[\gamma_{1}\dot{\hat{B}}+\psi(e)\frac{x_{2}}{M_{s}}=0 \tag{25}\]  
\[\gamma_{2}\ \dot{\hat{K}}+\psi(e)\frac{x_{1}}{M_{s}}=0 \tag{26}\]  
This produces the adaptive synergetic laws of an adaptive-controlled structure subjected to earthquake:  
\[\dot{\hat{B}}=-\frac{x_{2}}{\gamma_{1}M_{s}}\psi(e) \tag{27}\]\[\dot{\hat{K}}=-\frac{x_{1}}{\gamma_{2}\ M_{s}}\psi(e) \tag{28}\]  
On the basis of the adaptive laws given by Equations (27) and (28), \(\dot{V}\) is negative definite, and this ensures the asymptotic stability of the earthquakes one-story structure assisted by the MR damper and controlled by the adaptive synergetic controller.