LMIs in Control/Matrix and LMI Properties and Tools/Frobenius Norm

Frobenius Norm

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

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

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 $S\in {ℝ}^n$ such that

$\left[\begin{array}{cc}Z& A^T\\ *& 1\end{array}\right]\ge 0,$

$\begin{array}{c}tr(Z)\le {\gamma }^{\text{2}}.\\ \end{array}$

2.There exists $S\in {ℝ}^m$ such that

$\left[\begin{array}{cc}Z& A\\ *& 1\end{array}\right]\ge 0,$

$\begin{array}{c}tr(Z)\le {\gamma }^{\text{2}}.\\ \end{array}$

External Links

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

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