# 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 $M$  that depends affinely on the variable $x$ , subject to the LMI constraint $F(x)$  > 0. This problem can be reformulated as the GEVP.

## The Optimization Problem:

The GEVP can be formulated as follows:

minimize $\gamma$

subject to $F(x)$  > 0;

$\mu$ >0;

$\mu I$ < $M(x)$  < $\gamma \mu I$ .

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

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

## The LMI:

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

miminize$\gamma$

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

## Conclusion:

The LMI is feasible.