# LMIs in Control/pages/LMI for Minimizing Condition Number of Positive Definite Matrix

LMIs in Control/pages/LMI for Minimizing Condition Number of Positive Definite Matrix

## The System:

A related problem is minimizing the condition number of a positive-defnite matrix ${\displaystyle M}$ that depends affinely on the variable ${\displaystyle x}$, subject to the LMI constraint ${\displaystyle F(x)}$ > 0. This problem can be reformulated as the GEVP.

## The Optimization Problem:

The GEVP can be formulated as follows:

minimize ${\displaystyle \gamma }$

subject to ${\displaystyle F(x)}$ > 0;

${\displaystyle \mu }$>0;

${\displaystyle \mu I}$< ${\displaystyle M(x)}$ < ${\displaystyle \gamma \mu I}$.

We can reformulate this GEVP as an EVP as follows. Suppose,

${\displaystyle M(x)}$= ${\displaystyle M_{0}}$ +${\displaystyle \sum _{n=1}^{m}}$${\displaystyle x_{i}M_{i}}$ , ${\displaystyle F(x)}$= ${\displaystyle F_{0}}$+ ${\displaystyle \sum _{n=1}^{m}}$${\displaystyle x_{i}F_{i}}$

## The LMI:

Defining the new variables ${\displaystyle \nu }$=${\displaystyle 1/\mu }$ , ${\displaystyle {\tilde {x}}}$=${\displaystyle x/\mu }$ we can express the previous formulation as the EVP with variables ${\displaystyle {\tilde {x}},\nu }$ and ${\displaystyle \gamma }$:

miminize${\displaystyle \gamma }$

subject to ${\displaystyle \nu F_{0}}$+ ${\displaystyle \sum _{n=1}^{m}}$${\displaystyle x_{i}F_{i}}$ >0; ${\displaystyle I}$ < ${\displaystyle \nu M_{0}}$+ ${\displaystyle \sum _{n=1}^{m}}$${\displaystyle x_{i}M_{i}}$ < ${\displaystyle \gamma I}$

## Conclusion:

The LMI is feasible.