LMIs in Control/pages/LMI for Matrix Norm Minimization

LMIs in Control/pages/LMI for Matrix Norm Minimization

The System

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

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

The Data

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

The Optimization Problem

Find

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

to minimize,

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

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

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

The LMI: Minimization of Maximum Eigenvalue of a Matrix

Mathematical description of the LMI formulation:

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

Conclusion:

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

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

Implementation

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

https://github.com/asalimil/LMI-for-Matrix-Norm-Minimization

Related LMIs

LMI for Matrix Norm Minimization

LMI for Schur Stabilization

External Links

A list of references documenting and validating the LMI.

• [1] - LMI in Control Systems Analysis, Design and Applications