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

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

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

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

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

Alternative Forms of Finsler's Lemma

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

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

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

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

Modified Finsler's Lemma

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

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

there exists $ϵ$ such that

$\mathrm{\Psi }+{ϵ}^{-1}G{G}^T+ϵHR{H}^T<0.$

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.