User:Asalimil/sandbox

LMI for Minimizing the Maximum Eigenvalue of a Linear Matrix Function

The System


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

The Data


\begin{align} A_{0}, A_{1}, ..., A_{n} \end{align} $$

The Optimization Problem
Find $$ \begin{align} x = [x_{1} \quad x_{2} \quad ... \quad x_{n}] \end{align} $$

to minimize : $$ \begin{align} J(x) = \lambda_{max}(A(x)) \end{align} $$

$$ \begin{align} \text{min} \quad t \end{align} $$

$$ \begin{align} \text{s.t.} \quad A(x) - tI \leq 0 \\ \quad \quad t > 0 \end{align} $$

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

Using Lemma 1.1,

$$ \begin{align} \lambda_{max}(A(x)) \iff \quad A(x) - tI \leq 0 \end{align} $$

Implementation
A link to CodeOcean or other online implementation of the LMI

Related LMIs
LMI for Matrix norm minimization

LMI for Schur stabilization