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

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

Maximum Singular Value of a Complex Matrix

The System

Consider $A\in {ℝ}^{n\times m}$ as well as $\gamma$. A maximum singular value of a matrix $A$ is less than $\gamma$ if and only if $A{A}^H<{\gamma }^{2}I$, 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 LMI: Maximum Singular Value of a Complex Matrix

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

$\begin{array}{c}\overline{\sigma }(A)<\gamma \\ \left[\begin{array}{cc}\gamma I& A\\ A^H& \gamma I\end{array}\right]& >0\\ \end{array}$

The following LMI is also equivalent:

$\begin{array}{c}\overline{\sigma }(A)<\gamma \\ \left[\begin{array}{cc}\gamma I& A^H\\ A& \gamma I\end{array}\right]& >0\\ \end{array}$

Conclusion:

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

$\begin{array}{c}\overline{\sigma }(A)<\gamma \end{array}$

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.