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

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


Maximum Singular Value of a Complex Matrix


The SystemEdit

Consider   as well as  . A maximum singular value of a matrix   is less than   if and only if  , where   is the conjugate transpose or Hermitian transpose of the matrix  .

The DataEdit

The matrix   is the only data required.

The Optimization ProblemEdit

The LMI: Maximum Singular Value of a Complex MatrixEdit

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

 

The following LMI is also equivalent:

 

Conclusion:Edit

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

 

ImplementationEdit

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

%Clears all variables
clear; clc; close all;

%SDPVAR variables
gam1 = sdpvar(1);
gam2 = sdpvar(1);

%Example Matrix A
A = rand(9,6)+rand(9,6)*1i;

%Constraint Matrix for LMI optimization
M1 = [gam1*eye(9) A; A' gam1*eye(6)];

%Equivalent counter Matrix
M2 = [gam2*eye(6) A';A gam2*eye(9)]; 

%Constraints
Fc1 = (M1 >= 0);
Fc2 = (M2 >= 0);

%Objective function
obj1=gam1;
obj2=gam2;

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

%Optimization
optimize(Fc1,obj1,opt)
optimize(Fc2,obj2,opt)

%Displays output
fprintf('\nValue of Max singular value (using first method): ')
disp(value(gam1))

fprintf('\nValue of Max singular value (using second method): ')
disp(value(gam2))

fprintf('\nMATLAB verified output: ')
disp(norm(norm(A)))

Related LMIsEdit

External LinksEdit

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