LMIs in Control/pages/Minimum Singular Value of a Complex Matrix
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LMIs in Control/pages/Minimum Singular Value of a Complex Matrix
Minimum Singular Value of a Complex Matrix
The System
editConsider as well as . A minimum singular value of a matrix is greater than if and only if or , where is the conjugate transpose or Hermitian transpose of the matrix . the inequality used depends on the size of matrix .
The Data
editThe matrix is the only data required.
The LMI: Minimum Singular Value of a Complex Matrix
editThe following LMIs can be constructed depending on the size of :
if , where , then:
Else if , then:
Conclusion:
editThe results from this LMI will give the maximum complex value of the matrix :
This answer can also be proven using the following solution. Note that this solution only works if the matrix is a square, invertible matrix: .
Implementation
edit% 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
editExternal Links
edit- 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.