LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/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 $\Psi \in \mathbb {S} ^{n},G\in \mathbb {R} ^{n\times m},\mathrm {A} \in \mathbb {R} ^{m\times p},H\in \mathbb {R} ^{n\times p}$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

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

there exists $\epsilon$ such that

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

Conclusion

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

External Links

A list of references documenting and validating the LMI.

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
• LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
• LMI Properties and Applications in Systems, Stability, and Control Theory - A downloadable book on LMIs by Ryan James Caverly and James Richard Forbes.