LMIs in Control/pages/Minimum Singular Value of a Complex Matrix

LMIs in Control/pages/Minimum Singular Value of a Complex Matrix

Minimum Singular Value of a Complex Matrix

The System

Consider $A\in \mathbb {C} ^{n\times m}$ as well as $\gamma$. A minimum singular value of a matrix $A$ is greater than $\gamma$ if and only if $AA^{H}>\gamma ^{2}I$ or $A^{H}A>\gamma ^{2}I$, where $A^{H}$ is the conjugate transpose or Hermitian transpose of the matrix $A$. the inequality used depends on the size of matrix $A$.

The Data

The matrix $A$ is the only data required.

The LMI: Minimum Singular Value of a Complex Matrix

The following LMIs can be constructed depending on the size of $A$:

if $A\in \mathbb {C} ^{n\times m}$, where $n\leq m$, then:

${\begin{aligned}{\bar {\sigma }}(A)>\gamma \\AA^{H}>\gamma ^{2}I\end{aligned}}$

Else if $n\geq m$, then:

${\begin{aligned}{\bar {\sigma }}(A)>\gamma \\A^{H}A>\gamma ^{2}I\end{aligned}}$

Conclusion:

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

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

This answer can also be proven using the following solution. Note that this solution only works if the matrix $A$ is a square, invertible matrix: $\sigma _{min}=1/||A^{-1}||_{2}$.

Implementation

• Minimum Singular Value Example

% Minimum Singular Value of Complex Matrix
% -- EXAMPLE --

%Clears all variables
clear; clc; close all;

%SDPVAR variables
gam = sdpvar(1);

%Example Matrix A
A = rand(6,6)+rand(6,6)*1i;

%Constraints
Fc = ( A'*A >= gam*eye(6));

%Objective function
obj=-gam;

%options
opt = sdpsettings('solver','sedumi');

%Optimization
optimize(Fc,obj,opt)

%Displays output
fprintf('\nValue of Min singular value: ')
disp(value(sqrt(gam)))

fprintf('\nMATLAB verified output: ')
disp(1/norm(norm(A^(-1))))

Related LMIs

• Maximum 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.