View source for LMi for Minimizing Eigenvalues of a Matrix


{{FULLPAGENAME}}

== '''The System''' ==

:<math>

\begin{align}
A(x) = A_{0}+A_{1}x_{1}+ ... + A_{n}x_{n}
\end{align}</math>

Note that <math> \begin{align} A_{i}, \quad i=1, 2, ..., n \end{align}</math> are symmetric matrices.

== '''The Data''' ==

:<math>

\begin{align}
A_{0}, A_{1}, ..., A_{n} \quad \text{are given matrices.}
\end{align}</math>

== '''The Optimization Problem''' ==

Find 

<math>\begin{align} x = [x_{1} \quad x_{2} ... x_{n}] \end{align}</math>

to minimize,

<math>\begin{align} J(x) = \lambda_{max}(A(x)) \end{align}</math>

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

<math>\begin{align} \lambda_{max}(A(x)) \leq t \iff A(x)-tI \leq 0 \end{align}</math>

== '''The LMI:''' The Lyapunov Inequality ==

Title and mathematical description of the LMI formulation.

::<math>
\begin{align}
\text{min} \; \quad t:&\\
    \text{s.t.} \quad \begin{bmatrix} A(x)-tI \end{bmatrix}& \leq 0
\end{align}</math>

== '''Conclusion:''' ==

<math>\begin{align}  x_{i}, \quad i =1, 2, ... , n \quad \text{and} \quad t > 0 \end{align}</math> 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

LMI for Schur Stabilization

== External Links ==

A list of references documenting and validating the LMI.

* [https://www.crcpress.com/LMIs-in-Control-Systems-Analysis-Design-and-Applications/Duan-Yu/p/book/9781466582996] - LMI in Control Systems Analysis, Design and Applications

== '''Return to Main Page:''' ==

{{BookCat}}