LMIs in Control/LMI Matrix Properties/Minimization of Maximum Eigenvalue of a Matrix

LMIs in Control/LMI Matrix Properties/Minimization of Maximum Eigenvalue of a Matrix

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,

$\begin{array}{c}J(x)={\lambda }_{max}(A(x))\end{array}$

According to Lemma 1.1 in [1] page 10, the following statements are equivalent

$\begin{array}{c}{\lambda }_{max}(A(x))\le t⟺A(x)-tI\le 0\end{array}$

The LMI: The Lyapunov Inequality

Title and mathematical description of the LMI formulation.

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

Conclusion:

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

Implementation

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

https://github.com/asalimil/LMI-for-Minimizing-the-Maximum-Eigenvalue-of-Matrix/blob/master/README.md

Related LMIs

LMI for Matrix Norm Minimization

External Links

A list of references documenting and validating the LMI.

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