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.

References

• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.