LMIs in Control/Matrix and LMI Properties and Tools/Young’s Relation-Based Properties

Young’s Relation-Based Properties

1. Consider $X$,$Y\in \mathbb {R} ^{n\times m}$ and $Z\in \mathbb {S} ^{m}$. The matrix inequality given by

${\begin{aligned}Z+XTY+YTX>0,\end{aligned}}$

is satisfied if and only if there exist $Q\in \mathbb {S} ^{m}$, $P\in \mathbb {S} ^{n}$,$G1\in \mathbb {R} ^{n\times n}$,$G2\in \mathbb {R} ^{n\times m}$,$F\in \mathbb {R} ^{m\times n}$, and $H\in \mathbb {R} ^{m\times m}$, where $Q>0$

and $P>0$, such that

${\begin{bmatrix}P&Y\\*&Q\end{bmatrix}}>0$ and ${\begin{bmatrix}Z+Q+X^{T}PX&F-X^{T}G_{2}&H-X^{T}G_{1}\\*&G_{1}+G_{1}^{T}-P&F^{T}+G_{2}-Y\\*&*&H^{T}+H-Q\end{bmatrix}}$

2. Consider $X$,$Y\in \mathbb {R} ^{n\times n}$ and $W\in \mathbb {S} ^{m}$, where $X$ is full rank and $W>0$. The matrix inequality given by

${\begin{aligned}X^{T}-W>0,\end{aligned}}$

is satisfied if there exist $\lambda \in \mathbb {R} _{>0}$ such that

${\begin{bmatrix}\lambda \mathbf {1} &\lambda \mathbf {1} &\mathbf {0} \\*&\mathbf {X} +\mathbf {X} ^{T}&\mathbf {W} ^{\frac {1}{2}}\\*&*&\lambda \mathbf {1} \end{bmatrix}}>0$

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 List of LMIs by Ryan Caverly and James Forbes. (2.4.1 page 23)