# Comparative Study Between Active AMD and ABS Devices by Using \(\mu\)-Synthesis Robust Control
## 2 Materials and Methods
### Fundamentals Concepts Associated with \(\mu\)-Synthesis Control
#### 2.1.1 Uncertainties Representation  
The resilient controller design may have struggled to delineate the uncertainty regions. Based on the model parameter's dependability, we could approximate these ranges for linearly modeled structures based on the dependability of the model parameter. The first step in implementing the synthesis was to delineate the uncertainty. Identifying their origins facilitated their categorization into two types: parametric (real) uncertainties and dynamic (frequency-dependent) uncertainties. The initial category of uncertainties stemmed from fluctuations in the physical system parameters, including temporal constants and natural frequencies, as well as the structural attributes such as mass, stiffness, and damping coefficients. The second category of uncertainty arose from the simplifications in the complex system models, including model order reduction and the omission of high-frequency dynamics. The uncertainties, referred to as frequency-dependent or complex uncertainty, were intrinsic to the model's dynamic nature. Irrespective of the sources of uncertainty, we expressed them in one of two distinct forms: additive or multiplicative, as illustrated in Figure 1, following four combinations. We formulated Equations (1)-(4) to encompass all the identified uncertainty.  
\[k=k_{0}+r_{a}\delta,\ \ |\delta|\leq 1, \tag{1}\]  
\[k=k_{0}(1+r_{m}\delta),\ |\delta|\leq\ 1, \tag{2}\]  
where \(|\delta|=\) norm of the actual parameter deviation.  
\[k_{min}\leq k_{0}\ \leq\ k_{max},\]  
\[k_{0}:=\frac{k_{max}+k_{min}}{2},\ r_{a}:=\frac{k_{max}-k_{min},\ r_{m}:=\frac{ k_{min}+k_{max}}{k_{min}-k_{max}}\]  
\(k_{min}\) and \(k_{max}\) are extreme values of uncertainty.  
Equation (1) expresses the parametric uncertainties in additive form, while Equation (2) expresses parametric uncertainties in multiplicative form.  
Equation (3) expresses dynamic uncertainties in additive form, while Equation (4) expresses them in multiplicative form.  
\[M(s)=M_{0}(s)+W(s)\Delta(s) \tag{3}\]  
\[M(s)=M_{0}(s)(1+W(s)\Delta(s)) \tag{4}\]  
where  
\(M(s)\): represents the global model of the system, which includes uncertainties,  
\(M_{0}(s)\): represents the nominal matrix model devoid of uncertainty,  
\(W(s)\): represents the weighting function that translates the effect of uncertainties,  
\(\Delta(s)\): represents a set of all the stable transfer functions such that \(\Delta eH_{\infty},||\Delta||_{\infty}\leq 1\).  
The block \(\Delta\) corresponds to different uncertainty set formulations of stable functions; three cases of \(\Delta\) are possible:  
* Unstructured uncertainty \[\Delta=\left\{\begin{bmatrix}\Delta_{11}(s)&\Delta_{12}(s)\\ \Delta_{21}(s)&\Delta_{22}(s)\end{bmatrix},\Delta_{ij}\subset H_{\infty}\right\}\] (5)
* Real uncertainty \[\Delta=\left\{\begin{bmatrix}\delta_{1}&0\\ 0&\delta_{2}\end{bmatrix},\delta_{i}\ \in\ \mathbb{R}\right\}\] (6)  
Figure 1: Representation of uncertainties. (**a**) Dynamic uncertainty: additive form. (**b**) Dynamic uncertainty: multiplicative form.  
* Mixed uncertainty \[\Delta=\left\{\begin{array}{ccccc}\begin{bmatrix}\delta_{1}I_{1}&&\\ &\ddots&&\\ &&\delta_{s}I_{rs}&\\ &&&\Delta_{1}&\\ &&&&&\ddots&\\ &&&&&\Delta_{F}\end{bmatrix}\end{array}\right\}\delta_{i}\;\in\;\mathbb{R}, \left\{\begin{array}{c}\Delta_{ij}\subset H_{\infty}\\ \Delta_{ij}\subset H_{\infty}\end{array}\right.\end{array}\] (7) where \(\delta_{j}\) represents the frequency responses of parametric uncertainty; \(\Delta_{i}\) represents the frequency responses of dynamic uncertainty.