# 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.