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

Maximum Singular Value of a Complex Matrix The System 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 Data The matrix is the only data required. The Optimization Problem The LMI: Maximum Singular Value of a Complex Matrix Using the Shur complement procedure, the following LMIs can be constructed: The following LMI is also equivalent: Conclusion: The results from this LMI will give the maximum complex value of the matrix : Implementation Maximum Singular Value Example % 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 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. Return to Main Page: