LMIs in Control/pages/Mu Analysis

Mu Synthesis. The technique of $\mu$ synthesis extends the methods of $H\mathrm{\infty }$ 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

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

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 $\bar{\sigma }(DA{D}^{-1})<\gamma$ holds if and only if there exist $X\in {𝕊}^n$ and $\gamma \in {ℝ}_{>0}$, where $X>0$, satisfying:

$\begin{array}{c}{A}^{T}XA-{\gamma }^{2}X<0\end{array}$

Conclusion:

The inequality $\bar{\sigma }(DA{D}^{-1})<\gamma$ 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

External links

• [1]-MATLAB $\mu$-synthesis Implementation
• LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.