# LMIs in Control/pages/Mu Analysis

Mu Synthesis. The technique of ${\displaystyle \mu }$ synthesis extends the methods of ${\displaystyle H\infty }$ synthesis to design a robust controller for an uncertain plant. You can perform ${\displaystyle \mu }$ synthesis on plants with parameter uncertainty, dynamic uncertainty, or both using the "musyn" command in MATLAB. ${\displaystyle \mu }$ 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

{\displaystyle {\begin{aligned}{\dot {x}}&=Ax+Bu\\y&=Cx+Du\\\end{aligned}}}

where it is assumed that ${\displaystyle D}$ is Invertible.

## The Data

The matrices needed as inputs are only, ${\displaystyle A}$ and ${\displaystyle D}$.

## The LMI: ${\displaystyle \mu }$- Analysis

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

{\displaystyle {\begin{aligned}A^{T}XA-\gamma ^{2}X<0\end{aligned}}}

## Conclusion:

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