LMIs in Control/LMI Matrix Properties/Maximum Singular Value of a Complex Matrix

Maximum Singular Value of a Complex Matrix

The System

Consider $A\in \mathbb {R} ^{n\times m}$ as well as $\gamma$ . A maximum singular value of a matrix $A$ is less than $\gamma$ if and only if $AA^{H}<\gamma ^{2}I$ , where $A^{H}$ is the conjugate transpose or Hermitian transpose of the matrix $A$ .

The Data

The matrix $A$ is the only data required.

The Optimization Problem

The LMI: Maximum Singular Value of a Complex Matrix

Using the Shur complement procedure, the following LMIs can be constructed:

{\begin{aligned}{\bar {\sigma }}(A)<\gamma \\{\begin{bmatrix}\gamma I&A\\A^{H}&\gamma I\end{bmatrix}}&>0\\\end{aligned}}

The following LMI is also equivalent:

{\begin{aligned}{\bar {\sigma }}(A)<\gamma \\{\begin{bmatrix}\gamma I&A^{H}\\A&\gamma I\end{bmatrix}}&>0\\\end{aligned}}

Conclusion:

The results from this LMI will give the maximum complex value of the matrix $A$ :

{\begin{aligned}{\bar {\sigma }}(A)<\gamma \end{aligned}}

Implementation

A link to CodeOcean or other online implementation of the LMI

Related LMIs

[\\ Minimum singular value of a complex matrix]

External Links

LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.