User:Margav06/sandbox/Click here to continue/LMIs in system and stability Theory/mu-Analysis

Mu Synthesis. The technique of $$ \mu $$ synthesis extends the methods of $$ H\infin $$ synthesis to design a robust controller for an uncertain plant. You can perform $$ \mu $$ synthesis on plants with parameter uncertainty, dynamic uncertainty, or both using the "musyn" command in MATLAB. $$ \mu $$ analysis is an extremely powerful multivariable technique which has been applied to many problems in the almost every industry including Aerospace, process industry etc.

The System:
Consider the continuous-time generalized LTI plant with minimal states-space realization



\begin{align} \dot x &=Ax + Bu\\ y &=Cx + Du\\ \end{align}$$

where it is assumed that $$ D $$ is Invertible.

The Data
The matrices needed as inputs are only, $$ A $$ and $$ D $$.

The LMI: $$ \mu $$- Analysis
The inequality $$ \overline{\sigma}(DAD^{-1})< \gamma $$ holds if and only if there exist $$ X \in \mathbb{S}^n $$ and $$ \gamma \in \mathbb{R}_{>0}$$, where $$ X>0 $$, satisfying:



\begin{align}

A^TXA - \gamma^2X <0 \end{align}$$

Conclusion:
The inequality $$ \overline{\sigma}(DAD^{-1}) < \gamma $$ holds for $$ D= X^{1/2}, $$ where X satisfies the above Inequality.

Implementation

 * https://github.com/Ricky-10/coding107/blob/master/Mu%20Analysis - $$ \mu $$-Analysis