# 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.