LMIs in Control/Click here to continue/LMIs in system and stability Theory/Peak-to-Peak Norm

Peak-to-peak norm performance of a system The System Considering the following system: Where is the state signal, is the input signal, and is the output. When given an initial condition , the system can be defined to map the output and input signals for the peak-to-peak performance. The Data The matrices , , , and are the only data sets required for this optimization problem. The Optimization Problem Consider a continuous-time LTI system, , given that: , , , and , , , and . Given that the matrix is Hurwitz,The peak-to-peak norm of is given as: The LMI: Peak-to-Peak norm There exists a matrix and , , , where the following constraints are used: Since this optimization has in the constraints, this does make this optimization bi-linear. attempting to solve this LMI is not feasible unless some type of substitute is implemented to the variables . Conclusion: The results from this LMI will give the peak to peak norm of the system: Implementation Peak-to-Peak Norm Example % Peak-to-Peak Norm % -- EXAMPLE -- %Clears all variables clear; clc; close all; %Example Matrices A = [ 1 1 0 1 0 1; -1 0 -1 0 0 1; 1 0 0 -1 1 1; -1 1 -1 0 0 0; -1 -1 1 1 -1 -1; 0 -1 0 0 -1 0]; B = [ 0 -1 -1; 0 0 0; -1 -1 1; -1 0 0; 0 0 1; -1 1 1]; C = [ 0 1 0 -1 -1 -1; 0 0 0 -1 0 0; 1 0 0 0 -1 0]; D = [ 0 1 1; 0 0 0; 1 1 1]; %SDPVAR variables gam = sdpvar(1); eps = sdpvar(1); up = sdpvar(1); %SDPVAR MATRIX P = sdpvar(size(A,1),size(A,1),'symmetric'); %Constraint matrices M1 = [A'*P+P*A+gam*P P*B ; B'*P -eps*eye(3)]; M2 = [gam*P zeros(6,3) C' ; zeros(3,6) (up-eps)*eye(3) D' ; C D up*eye(3)]; %Constraints Fc = (P >= 0); Fc = [Fc; gam >= 0]; Fc = [Fc; eps >= 0]; Fc = [Fc; M1 <= 0]; Fc = [Fc; M2 >= 0]; %Objective function obj = up; %Settings for YALMIP opt = sdpsettings('solver','sedumi'); %Optimization optimize(Fc,obj,opt) fprintf('\nRepresentation of what occurs when attempting to solve \n') fprintf('problem without considering bilinearity\n\n') fprintf('setting gamma to a certain value\n eg: 0.5') gam = 0.5; eps = sdpvar(1); up = sdpvar(1); %SDPVAR MATRIX P = sdpvar(size(A,1),size(A,1),'symmetric'); %Constraint matrices M1 = [A'*P+P*A+gam*P P*B ; B'*P -eps*eye(3)]; M2 = [gam*P zeros(6,3) C' ; zeros(3,6) (up-eps)*eye(3) D' ; C D up*eye(3)]; %Constraints Fc = (P >= 0); Fc = [Fc; eps >= 0]; Fc = [Fc; M1 <= 0]; Fc = [Fc; M2 >= 0]; %Objective function obj = up; %Optimization optimize(Fc,obj,opt) fprintf('mu value: ') disp(value(up)) External Links 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. Linear Matrix Inequalities in Control - A downloadable book on LMIs by Carsten Scherer and Siep Weiland Return to Main Page: