# Comparative Study Between Active AMD and ABS Devices by Using \(\mu\)-Synthesis Robust Control
## 2 Materials and Methods
### Criteria for Robustness in Stability and Performance  
The \(\upmu\)-analysis is a procedure that extends the Nyquist stability criterion to robust stability conditions for multi-input multi-output (MIMO) systems. To do this, one has to find the structured singular value \(\mu\) of a system's uncertain transfer matrix, which is written as \(\mu(M(s))\), for any structured complex block \(\Delta\) of uncertainties across frequencies, usually at point \(s=j\omega\).  
In fact, the structured singular value (SSV \(\upmu\)) and the different ways of showing uncertainty that were talked about in the last section are very important for expressing robustness criteria. They provide a special means of measuring the size of any complex matrix \(M\) (n-by-n), thereby establishing a link between uncertainties and the physical system.  
Hence, to define the \(\mu\) value of the transfer matrix of a system \(M\), consider a set of structured complex matrices \(\Delta\), such that  
\[\Delta=\left\{\begin{array}{ccccc}\delta_{1}I_{r1}&&&&\\ &\ddots&&\\ &&\delta_{s}I_{rs}&&\\ &&&\Delta_{1}&\\ &&&&\ddots&\\ &&&&\Delta_{F}\end{array}\right\}\begin{array}{c}\delta_{i}&\in&\mathbb{C}\\ \Delta_{j}&\in&\mathbb{C}^{m_{j}\upmu_{j}}\\ \\ \end{array}\right\}\subset\mathbb{C}^{ntxm} \tag{12}\]  
where \(\sum_{i=1}^{s}r_{i}+\sum_{j=1}^{F}m_{j}=n\).  
Then, \(\mu\) is defined as the following:  
\[\mu_{\Delta}(M)=\frac{1}{\underset{\Delta\in\Delta}{\min}\{\overline{\sigma}( \Delta)\ :\ \det(I-M\Delta)=0\}} \tag{13}\]  
Equation (13) represents a minimization problem that is challenging to resolve both analytically and numerically; hence, upper and lower bounds are calculated in lieu of the \(\upmu\)-value itself. The robustness criteria for stability, resulting from the \(\mu\)-analysis, is articulated through the new definitions based on the SSV \(\upmu\) as follows:  
* **Robustness condition on stability**  
\(M\)(s) is internally stable for any structured block \(\Delta\), \(||\Delta||_{\infty}\leq 1\) and \(\det(I-M(j\omega)\Delta(j\omega))\
eq 0\), if and only if  
\[\mu(M(j\omega)):=\frac{1}{min\{\overline{\sigma}^{\prime}(\Delta(j\omega)): \det(I-M(j\omega)\Delta(j\omega))=0\}}<1,\forall\omega \tag{14}\]  
\(\Delta\in\Delta\)  
where \(\omega\) is the frequency and \(\overline{\sigma}\) is the maximum singular value of a matrix.  
The robust stability requirement is not the sole feature that the controller must guarantee; additional conditions, termed robust performance, should also be evaluated to reflect the degree of degradation linked to specific levels of disturbances and uncertainties. The basic concept is clear: addressing the robust performance issue as a component of the robust stability challenge.  
Consequently, if all the external disturbances affecting a system induce trucking and regulation errors in a weighted matrix, represented by the transfer matrix T, then in a multivariable control system, optimal performance is achieved when the robust stability requirement indicated by the Hoo norm in Equation (15) is satisfied:  
\[||T||_{\infty}:=\underset{\omega\in R}{\max}\overline{\sigma}(T(j\omega))\leq 1 \tag{15}\]  
* **Robust condition on performance**If \(T\) is an uncertain matrix represented as \(T=F_{-}U\left(M,\Delta\right)\), the upper linear fractional transformation (LFT) illustrates the extraction of uncertainty from the system. Consequently, as per Equation (15), a robust performance criterion is established.  
\[||F_{U}(M,\Delta)||_{\infty}:=\ \max_{\omega\in R}\overline{\sigma}(F_{U}(M, \Delta)\ )\leq 1 \tag{16}\]