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

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

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

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

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

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

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

there exists $$\epsilon$$ such that

$$ \Psi + \epsilon^{-1} G G^{T} + \epsilon H R H^{T} < 0.$$

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