LMIs in Control/Matrix and LMI Properties and Tools/Young’s Relation (Completion of the Squares)

This method is used to solve quadratic equations that can't be factorized.

Matrix inequality

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

${\begin{aligned}\ X^{T}Y+Y^{T}X\leq X^{T}S^{-1}X+Y^{T}SY,\\\end{aligned}}$

which is named Young’s relation or Young’s inequality.

Derivation

Young’s relation can be derived from a completion of the squares as follows.

${\begin{aligned}0\leq (X-SY)^{T}S^{-1}(X-SY)\\0\leq X^{T}S^{-1}X+Y^{T}SY-X^{T}Y-Y^{T}X\\X^{T}Y+Y^{T}X\leq X^{T}S^{-1}X+Y^{T}SY,\end{aligned}}$

which is named Young’s relation.

Reformulation of Young’s Relation

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

${\begin{aligned}\ X^{T}Y+Y^{T}X\leq {\frac {1}{2}}(X+SY)^{T}S^{-1}(X+SY),\\\end{aligned}}$

is a reformulation of Young’s relation.

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)