LMIs in Control/pages/Switched systems Hinf Optimization

Switched Systems Optimization The Optimization Problem This Optimization problem involves the use of the State-feedback plant design, with the difference of optimizing the system with a system matrix which "switches" in properties during optimization. This is similar to considering the system with variable uncertainty; an example of this would be polytopic uncertainty in the matrix . which ever matrix is switching states, there must be an optimization for both cases using the same variables for both. This is first done by defining the 9-matrix plant as such: , , , , , , , , and . Using this type of optimization allows for stacking different LMI matrix states in order to achieve the controller synthesis for . 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 . What must also be considered is which matrix or matrices will be "switched" during optimization. This can be denoted as . The LMI: Switched Systems Optimization There exists a scalar , along with the matrices and where: Where is the controller matrix. This also assumes that the only switching matrix is ; however, other matrices can be switched in states in order for more robustness in the controller. Conclusion: The results from this LMI gives a controller gain that is an optimization of for a switched system optimization. Implementation Hinf switched state-feedback optimization example % Switched System Hinf example % -- 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]; B21= [ 0 0 0; -1 0 1; -1 1 0; 1 -1 0; -1 0 -1; 0 1 1]; B22= [ 0 0 0; -1 0 1; -1 1 0; 1 1 0; 1 0 1; 0 -3 -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(B21,2); %actuators numb1 = size(B1,2); %external inputs numc1 = size(C1,1); %regulated outputs %variables gam = sdpvar(1); Y = sdpvar(numa); Z = sdpvar(numb2,numa,'full'); %Matrix for LMI optimization M1 = [Y*A'+A*Y+Z'*B21'+B21*Z B1 Y*C1'+Z'*D12'; B1' -gam*eye(numb1) D11'; C1*Y+D12*Z D11 -gam*eye(numc1)]; M2 = [Y*A'+A*Y+Z'*B22'+B22*Z B1 Y*C1'+Z'*D12'; B1' -gam*eye(numb1) D11'; C1*Y+D12*Z D11 -gam*eye(numc1)]; %Constraints Fc = (M1 <= 0); Fc = [Fc; M2 <= 0]; Fc = [Fc; Y >= eta*eye(numa)]; opt = sdpsettings('solver','sedumi'); %Objective function obj = gam; %Optimizing given constraints optimize(Fc,obj,opt); %Displays output Hinf gain fprintf('\n\nHinf for Hinf optimal state-feedback problem is: ') display(value(gam)) 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: