Add LQG controller for two-mass system
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Enrico Lumetti
parent
353456519a
commit
5b92f781bc
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%% Plant definition
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% refer to to Sinha, FIGURE 5.1.1
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clear;
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k1 = ureal('k1', 100, 'Percentage', 20);
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k2 = ureal('k2', 500, 'Percentage', 20);
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m = ureal('m', 1, 'Percentage', 1);
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alpha = ureal('alpha', 1, 'Percentage', 5);
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A = [
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0 0 1 0;
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0 0 0 1;
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-(k1+k2)/m k2/m -alpha/m 0;
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k2/m -(k1+k2)/m 0 -alpha/m;
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];
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B = [0 0; 0 0; 1 0; 0 1];
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C = [eye(2) zeros(2)];
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D = zeros(2);
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G = uss(A, B, C, D);
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G_fullstate = uss(A, B, eye(4), zeros(4, 2));
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% multiplicative uncertainty
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w = logspace(0, 3, 40);
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systems = usample(G, 40);
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data = frd(systems, w) ;
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[~, Info] = ucover(data, G.NominalValue, 4, []);
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Im = tf(Info.W1);
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%% Controller Definition
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Gn = G.NominalValue;
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Gn_fullstate = G_fullstate.NominalValue;
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s = tf('s');
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% output sensitivity weight
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Ms = 1.7; % peak sensitivity (~ stability margin)
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omega_b = 5; % open loop bandwidth
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eps_s = 0.01; % low frequency sensitivity upper bound (~ robust tracking)
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Ws = (s/Ms + omega_b)/(s+omega_b*eps_s);
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%%%% loop invertion controller
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% K = inv(Gn);
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% K = K*1/s * 1/(1+10*s) * (1+0.7*s) * 100;
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% K.OutputName = 'u';
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%
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% AP_u = AnalysisPoint('u', 2);
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% Gcl = feedback(G*AP_u*K, eye(2));
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% Gcl.InputName = 'r';
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% Gcl.OutputName = 'y';
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% L = Gn*K;
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%%%% PI diagonal controller
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% D = evalfr(inv(G), 0);
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% K = D*(s^2 + s + 100) / (1+s) / (1+s/10);
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%
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% Gcl = feedback(G*K, eye(2));
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% L = Gn*K;
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%%%% LQI controller
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% state weight + integrators weight
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Q = blkdiag(1e3*eye(4), 1e7*eye(2));
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% input weight
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R = 1e-3*eye(2);
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K = lqi(Gn, Q, R, zeros(6, 2));
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K = ss(K);
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K.InputName = {'x(1)', 'x(2)', 'x(3)', 'x(4)', 'e(1)', 'e(2)'};
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K.OutputName = 'u';
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G_fullstate.InputName = 'u';
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G_fullstate.OutputName = 'x';
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G_feedback = connect(G_fullstate, -K, 'e', 'x', {'u', 'x'});
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% only select first and second output
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G_feedback = G_feedback([1 2], [1 2]);
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G_feedback.OutputName = 'y';
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Gcl = feedback(G_feedback*1/s, eye(2));
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Gcl.InputName = 'r';
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L = ss(G_feedback)*1/s;
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%%%% LQG controller
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% Kalman filter
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G.InputName = 'uin';
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G.OutputName = 'y';
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Sum = sumblk('uin = u + w', 2);
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% G_noise models the plant G with additive input noise w
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G_noise = connect(G, Sum, {'u', 'w'}, 'y');
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% input noise covariance matrix
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Q = 0.1 * eye(2);
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% measurement noise covariance matrix
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R = 0.1 * eye(2); %1 centimeter stdev
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N = zeros(2);
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Kf = kalman(G_noise.NominalValue, Q, R, N, [1 2], [1 2]);
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Klqg = lqgtrack(Kf, K.D, '1dof');
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G_lqg = connect(G_noise, Klqg, {'e1', 'e2', 'w'}, {'y'});
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G_lqg.InputName = {'e(1)', 'e(2)', 'w(1)', 'w(2)'};
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Sum2 = sumblk('e = r - y', 2);
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G_lqg_cl = connect(G_lqg, Sum2, {'r', 'w'}, 'y');
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L_lqg = ss(G_lqg([1 2], [1 2]));
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%L = L_lqg;
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%%%% Sensitivity functions
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S = inv(eye(2)+L);
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T = eye(2) - S;
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if ~isstable(T)
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disp('Nominal closed loop is not stable');
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else
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disp('Nominal closed loop is stable');
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end
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if hinfnorm(S*Ws) <= 1
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disp('Nominal performance specifications on sensitivity are met')
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else
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disp('Nominal performance specifications on sensitivity are not met')
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end
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%% close figures
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close all;
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%% Singular value robustness and performance analysis
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figure('Name', 'Robust stability analysis');
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w = logspace(-1, 10, 100);
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SV = sigma(eye(2) + inv(L), w);
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semilogx(w, 20*log10(min(SV, [], 1))); hold on;
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sigma(Im, w, 'r'); hold off;
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title('Robust stability analysis')
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figure('Name', 'Sensitivity and loop performance');
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% open loop performance specifications
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subplot(2, 2, 1);
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sigma(L);
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title('Open loop specifications: L');
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subplot(2, 2, 2);
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[SV, W] = sigma(S);
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semilogx(W, 20*log10(max(SV, [], 1))); hold on;
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% sensitivity weight
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r = freqresp(1/Ws, W);
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semilogx(W, 20*log10(abs(r(:, :))), 'r');
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hold off;
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title('Open loop specifications: S');
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subplot(2, 2, 3);
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[SV, W] = sigma(T);
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semilogx(W, 20*log10(max(SV, [], 1)));
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title('Open loop specifications: T');
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%% LQG/LTR analysis
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figure('Name', 'LTR robust stability analysis');
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w = logspace(-1, 10, 100);
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SV = sigma(eye(2) + inv(L_lqg), w);
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semilogx(w, 20*log10(min(SV, [], 1))); hold on;
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sigma(Im, w, 'r'); hold off;
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title('Robust stability analysis')
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S_lqg = inv(eye(2)+L_lqg);
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T_lqg = eye(2) - S_lqg;
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figure('Name', 'LTR sensitivity and loop performance');
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% open loop performance specifications
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subplot(2, 2, 1);
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sigma(L_lqg); hold on;
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sigma(L); hold off;
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title('Open loop specifications: L');
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legend('L LTR', 'L');
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subplot(2, 2, 2);
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[SV, W] = sigma(S_lqg);
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semilogx(W, 20*log10(max(SV, [], 1))); hold on;
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% sensitivity weight
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r = freqresp(1/Ws, W);
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semilogx(W, 20*log10(abs(r(:, :))), 'r');
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hold off;
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title('Open loop specifications: S');
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subplot(2, 2, 3);
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[SV, W] = sigma(T_lqg);
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semilogx(W, 20*log10(max(SV, [], 1)));
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title('Open loop specifications: T');
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%% Structured singular value analysis
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opts = robOptions('VaryFrequency','on');
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[stabmarg,wcg,info] = robstab(Gcl,opts);
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figure('Name', 'mu analysis');
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semilogx(info.Frequency,info.Bounds)
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title('Stability Margin vs. Frequency');
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ylabel('Margin');
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xlabel('Frequency');
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legend('Lower bound','Upper bound');
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if stabmarg.LowerBound > 1
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fprintf('Closed loop system is robustly stable: mu=%f-%f at %f rad/s\n', ...
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stabmarg.LowerBound, stabmarg.UpperBound, stabmarg.CriticalFrequency);
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else
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fprintf('Closed loop system is not robustly stable: mu=%f-%f at %f rad/s\n', ...
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stabmarg.LowerBound, stabmarg.UpperBound, stabmarg.CriticalFrequency);
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end
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%% Plant sv and multiplicative uncertainty
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figure('Name', 'Open Loop System');
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subplot(2, 1, 1);
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sigma(G);
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hold on;
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title('Singular values of the system');
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subplot(2, 1, 2);
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sigma(Im);
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title('Multiplicative Uncertainty Magnitude');
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%% Simulation
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t = 0:0.001:3;
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r1 = 0*t + 0.1; % 10 centimeters displacement
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r2 = 0*t;
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% noise
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w = 0.05 .* randn(2, length(t));
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% transfer function from reference to controller output
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G_ur = getIOTransfer(Gcl, 'r', 'u');
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y = lsim(T, [r1; r2], t);
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u = lsim(G_ur, [r1; r2], t);
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realizations = usample(Gcl, 20);
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ninputs = 2;
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y_unc = zeros(ninputs*length(realizations), length(t));
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u_unc = zeros(ninputs*length(realizations), length(t));
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for i=1:length(realizations)
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G_ur_i = getIOTransfer(realizations(:, :, i, 1), 'r', 'u');
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p = lsim(realizations(:,:, i, 1), [r1; r2], t);
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q = lsim(G_ur_i, [r1; r2], t);
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y_unc(ninputs*i, :) = p(:, 1);
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y_unc(ninputs*i+1, :) = p(:, 2);
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u_unc(ninputs*i, :) = q(:, 1);
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u_unc(ninputs*i+1, :) = q(:, 2);
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end
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Twc = usubs(Gcl, wcg);
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Tur = usubs(G_ur, wcg);
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y_wc = lsim(Twc, [r1; r2], t);
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u_wc = lsim(Tur, [r1; r2], t);
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y_noisy = lsim(G_lqg_cl, [r1; r2; w], t);
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%% Output and control action plot
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figure('Name', 'Transient simulation: output');
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subplot(2, 3, 1);
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plot(t, r1, '--b', t, y(:, 1), 'r');
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title('Nominal system output');
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subplot(2, 3, 4);
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plot(t, r2, '--b', t, y(:, 2), 'r');
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subplot(2, 3, 2);
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plot(t, r1, '--b'); hold on;
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for i=1:length(realizations)
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plot(t, y_unc(ninputs*i, :), 'r');
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end
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hold off;
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title('Uncertain system output');
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subplot(2, 3, 5);
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plot(t, r2, '--b'); hold on;
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for i=1:length(realizations)
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plot(t, y_unc(ninputs*i+1, :), 'r');
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end
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hold off;
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subplot(2, 3, 3);
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plot(t, r1, '--b', t, y_wc(:, 1), 'r');
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title('Worst case system output');
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subplot(2, 3, 6);
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plot(t, r2, '--b', t, y_wc(:, 2), 'r');
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figure('Name', 'Transient simulation: control');
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subplot(2, 3, 1);
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plot(t, r1, '--b', t, u(:, 1), 'r');
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title('Nominal system control action');
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subplot(2, 3, 4);
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plot(t, r2, '--b', t, u(:, 2), 'r');
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subplot(2, 3, 2);
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plot(t, r1, '--b'); hold on;
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for i=1:length(realizations)
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plot(t, u_unc(ninputs*i, :), 'r');
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end
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hold off;
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title('Uncertain system control action');
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subplot(2, 3, 5);
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plot(t, r2, '--b'); hold on;
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for i=1:length(realizations)
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plot(t, u_unc(ninputs*i+1, :), 'r');
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end
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hold off;
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subplot(2, 3, 3);
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plot(t, r1, '--b', t, u_wc(:, 1), 'r');
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title('Worst case system control action');
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subplot(2, 3, 6);
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plot(t, r2, '--b', t, u_wc(:, 2), 'r');
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figure('Name', 'Noisy system transient');
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subplot(2, 1, 1);
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plot(t, r1, '--b', t, y_noisy(:, 1), 'r');
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title('Noisy system output');
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subplot(2, 1, 2);
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plot(t, r2, '--b', t, y_noisy(:, 2), 'r');
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