LMIs in Control/pages/Mixed H2 Hinf optimal state feedback control

Mixed Optimal State Feedback Control The Optimization Problem This Optimization problem involves the same process used on the full-feedback control design; however, instead of optimizing the full output-feedback design of the Optimal output-feedback control design. This is done by defining the 9-matrix plant as such: , , , , , , , , and . Using this type of optimization allows for stacking of optimization LMIs in order to achieve the controller synthesis for both and . The Data The data is dependent on the type the state-space representation of the 9-matrix plant; therefore the following must be known for this LMI to be calculated: , , , , , , , , and . The LMI: Mixed Optimal State Feedback Control There exists the scalars , , along with the matrices , and where: Where is the controller matrix. Conclusion: The results from this LMI give a controller that is a mixed optimization of both an , and optimization. Implementation Hinf/H2 Mixed state-feedback optimization example % Mixed Hinf/H2 state feedback optimization % -- EXAMPLE -- clear; clc; close all; %Given A = [ 1 1 0 1 0 1; -1 -1 -1 0 0 1; 1 0 1 -1 1 1; -1 1 -1 -1 0 0; -1 -1 1 1 1 -1; 0 -1 0 0 -1 -1]; B1 = [ 0 -1 -1; 0 0 0; -1 1 1; -1 0 0; 0 0 1; -1 1 1]; B2 = [ 0 0 0; -1 0 1; -1 1 0; 1 -1 0; -1 0 -1; 0 1 1]; C1 = [ 0 1 0 -1 -1 -1; 0 0 0 -1 0 0; 1 0 0 0 -1 0]; D12= [ 1 1 1; 0 0 0; 0.1 0.2 0.4]; D11= [ 1 2 3; 0 0 0; 0 0 0]; %Error eta = 1E-4; %sizes of matrices numa = size(A,1); %states numb2 = size(B2,2); %actuators numb1 = size(B1,2); %external inputs numc1 = size(C1,1); %regulated outputs %variables gam1= sdpvar(1); gam2= sdpvar(1); Y = sdpvar(numa); Z = sdpvar(numb2,numa,'full'); W = sdpvar(numc1); %Matrix for LMI optimization M1 = Y*A'+A*Y+B2*Z+Z'*B2'+B1*B1'; M2 = [Y (C1*Y+D12*Z)' ; C1*Y+D12*Z W ]; M3 = [Y*A'+A*Y+Z'*B2'+B2*Z B1 Y*C1'+Z'*D12'; B1' -eye(numb1) D11'; C1*Y+D12*Z D11 -gam1*eye(numc1)]; %Constraints Fc = (M1 <= 0); Fc = [Fc; M3 <= 0]; Fc = [Fc; trace(W) <= gam2]; Fc = [Fc; Y >= eta*eye(numa)]; Fc = [Fc; M2 >= zeros(numa+numc1)]; opt = sdpsettings('solver','sedumi'); %Objective function obj = gam1 + gam2; %Optimizing given constraints optimize(Fc,obj,opt); F = value(Z)*inv(value(Y)); %#ok<MINV> fprintf('\n\nState-Feedback controller F matrix') display(F) External Links LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet. Return to Main Page: