## 2 Methodology  
\(\mu\)_-Analysis and \(\mu\)-Synthesis_  
\(\mu\)-Analysis is a method used to analyze the robustness of a control system. It quantifies the extent to which the performance of the system can be degraded in the presence of uncertainties or variations in the plant (the system being controlled). This involves considering uncertainties in the system parameters, dynamics, and disturbances. These uncertainties are typically represented by transfer functions of varying magnitudes and phases. \(\mu\)-Synthesis is an extension of \(\mu\)-analysis, which incorporates a synthesis or design component. This is a systematic method for designing controllers that can handle identified uncertainties and variations, ensuring robust stability and performance. The \(\mu\)-synthesis approach involves finding a controller that minimizes the \(\mu\)-value (uncertainty measure) while satisfying certain performance and stability specifications.  
Every approach to the analysis problem discussed includes implements to assess each controller's performance and facilitate comparisons between controllers. However, a controller that attains a particular performance level in relation to the structured single value \(\mu\) can be synthesized. The (\(D\), \(G\)-\(K\)) iteration procedure was used to perform this synthesis [29; 31; 32]. During this procedure, the task of locating a \(\mu\)-optimal controller K satisfying \(\mu(Fu(F(j\omega),K(j\omega))\leq\beta\) for all \(\omega\) has been altered for the challenge of determining the transfer function matrices \(D(\omega)\in D\) and \(G(\omega)\in G\). Thus,  
\[\sup\overline{\varphi}\bigg{[}\bigg{(}\frac{D(\omega)\left(Fu\left(F(j\omega),K( j\omega)D^{-1}\right)}{\gamma}-\mathrm{j}G(\omega)\right)\quad\left(I+G^{2}( \omega)\right)^{-\frac{1}{2}}\bigg{]}\leq 1,\forall\omega \tag{1}\]  
Unfortunately, this approach does not ensure the discovery of local maxima. Nevertheless, when dealing with intricate perturbations, there is an alternative method called \(D\)-\(K\) iteration (also executable in MATLAB) [29; 33]. This technique integrates \(\mu\)-synthesis and \(\mu\)-analysis and frequently provides satisfactory results. It begins with an initial estimation of \(\mu\), as stated in connection with the scaled singular value:  
\[\mu(\mathrm{N})\leq\underset{D\in D}{\min}\overline{\sigma}(DND^{-1}) \tag{2}\]  
The vision is centered on identifying the controller that minimizes the peak value of its upper bound across different frequencies. In other words, the K-Steps in \(\mu\)-Synthesis are:  
Uncertainty Modeling: Identify and model uncertainties in the system. This may include parametric uncertainties, unmodelled dynamics, and external disturbances.  
\(\mu\)-Analysis: Perform \(\mu\)-analysis to quantify the robustness of the system in the presence of uncertainties. This step helps to identify the critical frequencies and magnitudes wherein robustness is the most challenging.  
Robustness: \(\mu\)-synthesis provides a systematic approach for designing controllers that are robust against uncertainties, ensuring stable and satisfactory performance over a range of operating conditions. The resulting controllers often have a structured form, which makes them more interpretable and easier to implement.  
\[\underset{K}{\min}(\underset{D\in D}{\min}\left\|DN(K)D^{-1}\right\|_{\infty}) \tag{3}\]  
By alternately minimizing the \(\left\|DN(K)D^{-1}\right\|_{\infty}\) with respect to either \(K\) or \(D\) (while keeping the other constant) [27; 29]:  
In the K-step, synthesize an \(H_{\infty}\) controller for the scaled problem, minimizing \(\parallel\left\{\left\|DN(K)D\right\|\right\}^{-1}\parallel_{\infty}\) with a fixed \(D(s)\).  
In the D-step, we determine \(D(j\omega)\) at each frequency to minimize \(\overline{\sigma}(DND^{-1}(j\omega))\) with the fixed transfer function \(N\).  
The magnitude of each element of \(D(j\omega)\) is adjusted to fit a stable and minimum-phase transfer function \(D(s)\) and then returns to the first step.