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

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

if and only if there exists $\sigma$ such that

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

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

Alternative Forms of Finsler's Lemma

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

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

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

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

Modified Finsler's Lemma

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