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

Minimum Singular Value of a Complex Matrix The System Consider 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 The matrix 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 : if , where , then: Else if , then: Conclusion: The 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 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. Return to Main Page: