LMIs in Control/Tools/LMI for Matrix Norm Minimization

LMIs in Control/Tools/LMI for Matrix Norm Minimization

The System

$\begin{array}{c}A(x)=A_0+A_1{x}_1+...+A_n{x}_n\end{array}$

Note that $\begin{array}{c}{A}_i,i=1,2,...,n\end{array}$ are symmetric matrices.

The Data

$\begin{array}{c}{A}_0,A_1,...,A_n\text{are given matrices.}\end{array}$

The Optimization Problem

Find

$\begin{array}{c}x=[x_1{x}_2...x_n]\end{array}$

to minimize,

$\begin{array}{c}J(x)=||A(x)|{|}_2\end{array}$

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

$\begin{array}{c}{A}^{T}A-t^{2}I\le 0⟺\left[\begin{array}{cc}-tI& A\\ A^T& -tI\end{array}\right]\le 0\\ \end{array}$

The LMI: Minimization of Maximum Eigenvalue of a Matrix

Mathematical description of the LMI formulation:

$\begin{array}{cc}\text{min}t:& \\ \text{s.t.}\left[\begin{array}{cc}-tI& A(x)\\ A(x{)}^T& -tI\end{array}\right]\le 0\\ \end{array}$

Conclusion:

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

$\begin{array}{c}{x}_i,i=1,2,...,n\text{and}t>0\end{array}$ 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