# Comparative Study Between Active AMD and ABS Devices by Using \(\mu\)-Synthesis Robust Control
## 2 Materials and Methods
### Mathematical Representation of the Structural Model  
We model the structural system using established geometry, material properties, and boundary conditions. This representation of the structure constitutes the nominal model.  
Given the configuration of a linear building structure subject to horizontal seismic excitation in Figure 3, we can formulate the equation of motion governing this system as follows:  
\[M\ddot{y}(t)+C\ \dot{y}(t)+\dot{K}y(t)=\ Hf(t)+E\ddot{u}_{g}(t) \tag{21}\]  
The matrices of mass (see Equation (22)), damping (see Equation (23)), and stiffness (see Equation (24)) represent the mass, stiffness, and damping of the ith storey, respectively, where n is the number of structure stores.  
\[M=\begin{bmatrix}m_{1}&\cdots&0\\ \vdots&\ddots&\vdots\\ 0&\cdots&m_{n}\end{bmatrix} \tag{22}\]  
Figure 3: AMD system attached to the top floor.  
\[C=\begin{bmatrix}c_{1}+c_{2}&-c_{2}&0&\cdots&0\\ -c_{2}&c_{2}+c_{3}&-c_{3}&\cdots&\vdots\\ 0&-c_{3}&\ddots&-c_{n-1}&0\\ \vdots&\cdots&-c_{n-1}&c_{n-1+}c_{n}&-c_{n}\\ 0&\cdots&0&-c_{n}&c_{n}\end{bmatrix} \tag{23}\]  
\[K=\begin{bmatrix}k_{1}+k_{2}&-k_{2}&0&\cdots&0\\ -k_{2}&k_{2}+k_{3}&-k_{3}&\cdots&\vdots\\ 0&-k_{3}&\ddots&-k_{n-1}&0\\ \vdots&\cdots&-k_{n-1}&k_{n-1+}k_{n}&-k_{n}\\ 0&\cdots&0&-k_{n}&k_{n}\end{bmatrix} \tag{24}\]  
where \(y(t),\;\dot{y}(t)\;\) and \(\ddot{y}(t)\) are, respectively, the displacement, velocity, and acceleration vectors; \(f(t)\) is the force provided by the actuator depending on the control law, and \(g(t)\) is the ground acceleration. \(H\;\in\mathbb{R}^{n\times n}\;\) is a vector that gives the location of the force while \(E\) characterizes the impact of the external forces, namely the seismic excitation.  
We use the state space representation to re-express Equation (21) in a more suitable form, allowing the new system model to preserve the temporal notions of its physical properties.  
Moreover, we can model measurements of the structural system using the system states and exogenous inputs, such as noise, disturbance, and control inputs.  
Therefore, we can substitute \(x(t)=\begin{bmatrix}y(t)\;\dot{y}(t)\end{bmatrix}\) in Equation (21) to create the following state vector:  
\[\left\{\begin{array}{l}\dot{x}(t)=Ax(t)+Bu(t)+Bw\;w(t)\\ y_{n}=Cx(t)+Du(t)+Dw\;w(t)+n\end{array}\right. \tag{25}\]  
The system matrices \(A\), \(B\), and \(B_{w}\) are defined in the following way:  
\[A=\begin{bmatrix}0&I\\ -M^{-1}K&-M^{-1}C\end{bmatrix} \tag{26}\]  
\(A\) is the system matrix that encapsulates the relationships between the states and their derivatives. Matrix \(A\) converts the second-order linear differential equation (refer to Equation (21)) into a first-order differential equation (refer to Equation (25)). Equation (25) converts the unknown variables from Equation (21) into six unknowns.  
\[B=\begin{bmatrix}0\\ M^{-1}\;H\end{bmatrix} \tag{27}\]  
\[B_{w}=-\begin{bmatrix}0\\ M^{-1}E\end{bmatrix} \tag{28}\]  
The desired output variable, \(y_{n}\), forms the basis for \(C\), \(D\), and \(D_{w}\), which are weighting matrices. \(B\) and \(B_{w}\) represent the disturbance and control input weighting vectors, respectively.