LMIs in Control/Matrix and LMI Properties and Tools/Continuous Time/Structured Singular Value

LMIs in Control/Matrix and LMI Properties and Tools/Continuous Time/Structured Singular Value

The LMI can be used to find a $\mathrm{\Theta }$ that belongs to the set of scalings $P\mathrm{\Theta }$. Minimizing $\gamma$ allows to minimize the squared norm of $\mathrm{\Theta }M{\mathrm{\Theta }}^{-}1$.

The System

$\begin{array}{ccc}M\text{ with transfer function }\hat{M}(s)=C(sI-A{)}^{-1}B+D,& & \hat{M}\in H_{\mathrm{\infty }}\end{array}$

The Data

The matrices $A\in R^{n\times n},B\in R^{n\times m},C\in R^{o\times n},D\in R^{o\times m}$.

The Optimization Problem

$\begin{array}{c}\text{There exists }\mathrm{\Theta }\in \mathrm{\Theta }\text{ such that }||\mathrm{\Theta }M{\mathrm{\Theta }}^{-1}|{|}^2<\gamma .\end{array}$

The LMI:

$\begin{array}{cc}\text{Find}& X>0:\\ \left[\begin{array}{cc}{A}^{T}X+XA& XB\\ B^{T}X& -\mathrm{\Theta }\end{array}\right]+{\gamma }^{-2}\left[\begin{array}{c}{C}^T\\ D^T\end{array}\right]\mathrm{\Theta }\left[\begin{array}{cc}C& D\end{array}\right]<0\\ \end{array}$

Conclusion:

$\text{The optimization problem and the LMI are equivalent. }\gamma \text{ must be optimized using bisection.}$

Implementation

https://github.com/mcavorsi/LMI

Related LMIs

Eigenvalue Problem

External Links

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.