# LMIs in Control/Matrix and LMI Properties and Tools/Finsler's Lemma

LMIs in Control/Matrix and LMI Properties and Tools/Finsler's Lemma

This method It states equivalent ways to express the positive definiteness of a quadratic form Q constrained by a linear form L. It is equivalent to other lemmas used in optimization and control theory, such as Yakubovich's S-lemma, Finsler's lemma and it is wedely used in Linear Matrix Inequalities

## Theorem

Consider ${\displaystyle \Psi \in \mathbb {S} ^{n},G\in \mathbb {R} ^{n\times m},\mathrm {A} \in \mathbb {R} ^{m\times p},H\in \mathbb {R} ^{n\times p}}$ and ${\displaystyle \sigma \in \mathbb {R} }$ . There exists ${\displaystyle \mathrm {A} }$  such that

${\displaystyle \Psi +G\mathrm {A} H^{T}+H\mathrm {A} ^{T}G^{T}<0,}$

if and only if there exists ${\displaystyle \sigma }$  such that

${\displaystyle \Psi -\sigma GG^{T}<0}$

${\displaystyle \Psi -\sigma HH^{T}<0}$

## Alternative Forms of Finsler's Lemma

Consider ${\displaystyle \Psi \in \mathbb {S} ^{n},Z\in \mathbb {R} ^{p\times n},x\in \mathbb {R} ^{n}}$ and ${\displaystyle \sigma \in \mathbb {R} _{>0}}$ . If there exists ${\displaystyle Z}$  such that

${\displaystyle x^{T}\Psi x,0}$

holds for all ${\displaystyle x}$ ${\displaystyle 0}$  satisfying ${\displaystyle Zx=0}$ , then there exists ${\displaystyle \sigma }$  such that

${\displaystyle \Psi -\sigma Z^{T}Z<0}$

## Modified Finsler's Lemma

Consider ${\displaystyle \Psi \in \mathbb {S} ^{n},G\in \mathbb {R} ^{n\times m},\mathrm {A} \in \mathbb {R} ^{m\times p},H\in \mathbb {R} ^{n\times p}}$ and ${\displaystyle \epsilon \in \mathbb {R} _{>0}}$ , where ${\displaystyle \mathrm {A} ^{T}\mathrm {A} }$  is less that on equal to ${\displaystyle \mathbb {R} }$ , and ${\displaystyle R>0}$ . There exists ${\displaystyle \mathrm {A} }$  such that

${\displaystyle \Psi +G\mathrm {A} H^{T}+H\mathrm {A} ^{T}G^{T}{T}<0,}$

there exists ${\displaystyle \epsilon }$  such that

${\displaystyle \Psi +\epsilon ^{-1}GG^{T}+\epsilon HRH^{T}<0.}$

## Conclusion

In summary, a number of identical methods have been stated above to determine the positive definiteness of LMIs.