# LMIs in Control/pages/LMI for Matrix Norm Minimization

LMIs in Control/pages/LMI for Matrix Norm Minimization

## The System

{\displaystyle {\begin{aligned}A(x)=A_{0}+A_{1}x_{1}+...+A_{n}x_{n}\end{aligned}}}

Note that {\displaystyle {\begin{aligned}A_{i},\quad i=1,2,...,n\end{aligned}}} are symmetric matrices.

## The Data

{\displaystyle {\begin{aligned}A_{0},A_{1},...,A_{n}\quad {\text{are given matrices.}}\end{aligned}}}

## The Optimization Problem

Find

{\displaystyle {\begin{aligned}x=[x_{1}\quad x_{2}...x_{n}]\end{aligned}}}

to minimize,

{\displaystyle {\begin{aligned}J(x)=||A(x)||_{2}\end{aligned}}}

According to Lemma 1.2 in [1] page 11, the following statements are equivalent

{\displaystyle {\begin{aligned}A^{T}A-t^{2}I\leq 0\iff {\begin{bmatrix}-tI&A\\A^{T}&-tI\end{bmatrix}}\leq 0\\\end{aligned}}}

## The LMI: Minimization of Maximum Eigenvalue of a Matrix

Mathematical description of the LMI formulation:

{\displaystyle {\begin{aligned}{\text{min}}\;\quad t:&\\{\text{s.t.}}\quad {\begin{bmatrix}-tI&A(x)\\A(x)^{T}&-tI\end{bmatrix}}\leq 0\\\end{aligned}}}

## Conclusion:

This problem is a slight generalization of the eigenvalue minimization problem for a matrix.

{\displaystyle {\begin{aligned}x_{i},\quad i=1,2,...,n\quad {\text{and}}\quad t>0\end{aligned}}} are parameters to be optimized

## Implementation

A link to Matlab codes for this problem in the Github repository:

## Related LMIs

LMI for Matrix Norm Minimization

LMI for Schur Stabilization