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

Young’s Relation-Based Properties

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

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

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

and $P>0$, such that

$\left[\begin{array}{cc}P& Y\\ *& Q\end{array}\right]>0$ and $\left[\begin{array}{ccc}Z+Q+X^{T}PX& F-X^T{G}_2& H-X^T{G}_1\\ *& G_1+G_1^T-P& F^T+G_2-Y\\ *& *& H^T+H-Q\end{array}\right]$

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

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

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

$\left[\begin{array}{ccc}\lambda \mathrm{𝟏}& \lambda \mathrm{𝟏}& \mathrm{𝟎}\\ *& 𝐗+{𝐗}^T& {𝐖}^{\frac{1}{2}}\\ *& *& \lambda \mathrm{𝟏}\end{array}\right]>0$

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)

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