LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/Frobenius Norm

Frobenius Norm

Consider $A\in \mathbb {R} ^{n\times m}$ and $\gamma \in \mathbb {R}$_>0

The Frobenius norm of $A$ is $||A||$_F =${\sqrt {tr(A^{T}A)}}={\sqrt {tr(AA^{T})}}$

The Frobenius norm is less than or equal to $\gamma$ if and only if any of the following equivalent conditions are satisfied.

1.There exists $S\in \mathbb {R} ^{n}$ such that

${\begin{bmatrix}Z&A^{T}\\*&1\\\end{bmatrix}}\geq 0,$

${\begin{aligned}\qquad tr(Z)\leq \gamma ^{\text{2}}.\qquad \\\end{aligned}}$

2.There exists $S\in \mathbb {R} ^{m}$ such that

${\begin{bmatrix}Z&A\\*&1\\\end{bmatrix}}\geq 0,$

${\begin{aligned}\qquad tr(Z)\leq \gamma ^{\text{2}}.\qquad \\\end{aligned}}$

External Links

• LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.