# Comparative Study Between Active AMD and ABS Devices by Using \(\mu\)-Synthesis Robust Control
## 2 Materials and Methods
### \(\mu\)-Synthesis and D-K Iteration Algorithm  
The \(\mu\)-synthesis technique aims to develop a controller \(K\) for a feedback uncertain system (\(G\), \(\Delta\)), as illustrated in Figure 2, by integrating \(H\infty\) control design with diagonal scaling methods from the SSV \(\mu\) framework. The \(\Delta\) block illustrates both parametric and non-parametric uncertainty. These uncertainties are termed \"structured uncertainties\" and are constrained by the \(H\infty\) norm of uncertain gains. These gains illustrate the impact of various alterations on one or more inputs or outputs of a system. The controller's design process will specifically target the minimization of the structured singular value \(\mu\) of a complex cost matrix. The structured singular value \(\mu\) serves as an effective instrument for evaluating the system's resilience via the \(\mu\) analysis technique, addressing both stability and performance, as demonstrated in the preceding section. Figure 2 represents the \(\mu\)-synthesis control configuration with the following input-output structure:  
* Inputs: \(w\) (disturbances), \(u\) (control inputs);
* Outputs: \(z\) (performance variables), \(y\) (measured outputs). \(\Delta\) represents the uncertainty block, \(M=F_{U}(G,\Delta)\) represents the generalized uncertain plant, \(K\) is the controller and, \(F_{L}(M,K)\) represents the interconnections of \(M\), and \(K\) forms the closed-loop system. The control objective is often formulated in terms of minimizing the \(H\infty\) norm of the transfer function from \(w\) to \(z\), ensuring that the system can handle worst-case disturbances. \[\mu_{\Delta}(M)\leq\min_{D\in D}\overline{\sigma}\Big{(}DMD^{-1}\Big{)} \tag{17}\]  
The \(\mu\)-synthesis approach poses the problem of finding a matrix \(D\) for a complex matrix \(M=\mathcal{F}_{\ell}(G,K)\), as shown in Figure 2, with  
\[\mathcal{D}=\left\{ \begin{array}{ccccc}\begin{bmatrix}d_{1}I_{r1}&&&&\\ &\ddots&&\\ &&d_{s}\ I_{rs}&&\\ &&&\mathcal{D}_{1}&&\\ &&&&\ddots&\\ &&&&\mathcal{D}_{F}\end{bmatrix}&\begin{bmatrix}D_{i}\in\mathbb{C}^{\Delta_{ i}\times\Delta_{i}}\\ D_{i}=D_{i}^{*}>0\\ d_{j}\in\mathbb{R}\\ d_{j}>0\\ \end{bmatrix},\\ \\ \end{array} \right. \tag{18}\]  
where \(M\) is the uncertain system and \(K\) is the controller to be designed. The \(\mu\)-synthesis problem consists of designing a nominally stabilizing controller \(K\) and solving the following minimization problem, which represents the \(\mu\) upper bound over frequencies. \[\min_{K}\min_{\omega^{\prime}}\min_{D(j\omega)\in D}\overline{\sigma}D\omega \mathcal{F}_{\ell}(G,K)(j\omega)D^{-1} \tag{19}\]  
This problem is equivalent to a minimizing scaled H\(\infty\) norm, which can be expressed as  
\[\min_{K}\min_{D,D^{-1}\varepsilon\mathcal{R}H_{\infty},D(j\omega)\in D}||D \mathcal{F}_{\ell}(M,K)D^{-1}||_{\infty}\leq 1 \tag{20}\]  
Equation (20) yields the cost function, which is a non-convex minimization problem involving \(D\) and \(K\). The synthesis is executed using an algorithm known as \(D\)-\(K\) iteration to find a local minimum. Once the controller \(K\) is fixed and constitutes a convex problem, the \(\mu\)-analysis procedure can be derived. However, if \(D\) is fixed, the problem becomes convex, and we are in front of an \(H\infty\) optimal controller design. If the minimized \(\mu\) value is less than one, the resulting controller \(K\) is a robust stabilizing controller. For the numerical computation of a \(\mu\)-controller, an efficient algorithm called \(D\)-\(K\) iteration, summarized in seven steps, is executed, as follows:  
Step 1: Hoo synthesis is used to find a controller that minimizes the nominal system's closed-loop gain;  
Step 2: The system conducts a robustness analysis to gauge the closed-loop system's robust Hoo performance.