# LMIs in Control/pages/Mu Analysis

Mu Synthesis. The technique of $\mu$ synthesis extends the methods of $H\infty$ 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{aligned}{\dot {x}}&=Ax+Bu\\y&=Cx+Du\\\end{aligned}}

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

## Conclusion:

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