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More advanced analysis

This commit is contained in:
Enrico Lumetti 2021-08-27 21:51:00 +02:00
parent 89e068e794
commit 0137222ca5
1 changed files with 107 additions and 50 deletions
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157
robust_control/two_mass_system.m
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@ -35,11 +35,21 @@ Gn_fullstate = G_fullstate.NominalValue;
s = tf('s'); s = tf('s');
% output sensitivity weight
Ms = 1.7; % peak sensitivity (~ stability margin)
omega_b = 5; % open loop bandwidth
eps_s = 0.01; % low frequency sensitivity upper bound (~ robust tracking)
Ws = (s/Ms + omega_b)/(s+omega_b*eps_s);
%%%% loop invertion controller %%%% loop invertion controller
% K = inv(Gn); % K = inv(Gn);
% K = K*1/s * 1/(1+10*s) * (1+0.7*s) * 100; % K = K*1/s * 1/(1+10*s) * (1+0.7*s) * 100;
% K.OutputName = 'u';
% %
% Gcl = feedback(G*K, eye(2)); % AP_u = AnalysisPoint('u', 2);
% Gcl = feedback(G*AP_u*K, eye(2));
% Gcl.InputName = 'r';
% Gcl.OutputName = 'y';
% L = Gn*K; % L = Gn*K;
%%%% PI diagonal controller %%%% PI diagonal controller
@ -51,22 +61,23 @@ s = tf('s');
%%%% LQI controller %%%% LQI controller
% state weight + integrators weight % state weight + integrators weight
Q = blkdiag(1e3*eye(4), 1e9*eye(2)); Q = blkdiag(1e3*eye(4), 1e7*eye(2));
% input weight % input weight
R = 1e-6*eye(2); R = 1e-8*eye(2);
K = -lqi(Gn, Q, R, zeros(6, 2)); K = -lqi(Gn, Q, R, zeros(6, 2));
K = ss(K); K = ss(K);
K.InputName = {'x(1)', 'x(2)', 'x(3)', 'x(4)', 'e(1)', 'e(2)'}; K.InputName = {'x(1)', 'x(2)', 'x(3)', 'x(4)', 'e(1)', 'e(2)'};
K.OutputName = 'u'; K.OutputName = 'u';
G_fullstate.InputName = 'u'; G_fullstate.InputName = 'u';
G_fullstate.OutputName = 'x'; G_fullstate.OutputName = 'x';
G_feedback = connect(G_fullstate, K, "e", "x"); G_feedback = connect(G_fullstate, K, 'e', 'x', 'u');
% only select first and second output % only select first and second output
G_feedback = G_feedback([1 2], [1 2]); G_feedback = G_feedback([1 2], [1 2]);
G_feedback.OutputName = 'y'; G_feedback.OutputName = 'y';
Gcl = feedback(G_feedback*1/s, eye(2)); Gcl = feedback(G_feedback*1/s, eye(2));
L = G_feedback.NominalValue*1/s; Gcl.InputName = 'r';
L = ss(G_feedback)*1/s;
%%%% Sensitivity functions %%%% Sensitivity functions
S = inv(eye(2)+L); S = inv(eye(2)+L);
@ -78,55 +89,33 @@ else
disp('Nominal closed loop is stable'); disp('Nominal closed loop is stable');
end end
%% Simulation
t = 0:0.001:1;
u1 = 0*t + 1;
u2 = 0*t;
y = lsim(T, [u1; u2], t);
realizations = usample(Gcl, 20);
ninputs = 2;
y_unc = zeros(ninputs*length(realizations), length(t));
for i=1:length(realizations)
p = lsim(realizations(:,:, i, 1), [u1; u2], t);
y_unc(ninputs*i, :) = p(:, 1);
y_unc(ninputs*i+1, :) = p(:, 2);
end
%% close figures %% close figures
close all; close all;
%% Plant sv and multiplicative uncertainty
figure('Name', 'Open Loop System');
subplot(2, 1, 1);
sigma(G);
hold on;
title('Singular values of the system');
subplot(2, 1, 2);
sigma(Im);
title('Multiplicative Uncertainty Magnitude');
%% Singular value robustness and performance analysis %% Singular value robustness and performance analysis
figure('Name', 'Loop Analysis'); figure('Name', 'Robust stability analysis');
subplot(2, 2, 1);
w = logspace(-1, 10, 100); w = logspace(-1, 10, 100);
SV = sigma(eye(2) + inv(L), w); SV = sigma(eye(2) + inv(L), w);
semilogx(w, 20*log10(min(SV, [], 1))); hold on; semilogx(w, 20*log10(min(SV, [], 1))); hold on;
sigma(Im, w, 'r'); hold off; sigma(Im, w, 'r'); hold off;
title('Robust stability analysis') title('Robust stability analysis')
figure('Name', 'Sensitivity and loop performance');
% open loop performance specifications % open loop performance specifications
subplot(2, 2, 2); subplot(2, 2, 1);
sigma(L); sigma(L);
title('Open loop specifications: L'); title('Open loop specifications: L');
subplot(2, 2, 3); subplot(2, 2, 2);
[SV, W] = sigma(S); [SV, W] = sigma(S);
semilogx(W, 20*log10(max(SV, [], 1))); semilogx(W, 20*log10(max(SV, [], 1))); hold on;
% sensitivity weight
r = freqresp(1/Ws, W);
semilogx(W, 20*log10(abs(r(:, :))), 'r');
hold off;
title('Open loop specifications: S'); title('Open loop specifications: S');
subplot(2, 2, 4); subplot(2, 2, 3);
[SV, W] = sigma(T); [SV, W] = sigma(T);
semilogx(W, 20*log10(max(SV, [], 1))); semilogx(W, 20*log10(max(SV, [], 1)));
title('Open loop specifications: T'); title('Open loop specifications: T');
@ -134,6 +123,7 @@ title('Open loop specifications: T');
%% Structured singular value analysis %% Structured singular value analysis
opts = robOptions('VaryFrequency','on'); opts = robOptions('VaryFrequency','on');
[stabmarg,wcg,info] = robstab(Gcl,opts); [stabmarg,wcg,info] = robstab(Gcl,opts);
figure('Name', 'mu analysis');
semilogx(info.Frequency,info.Bounds) semilogx(info.Frequency,info.Bounds)
title('Stability Margin vs. Frequency'); title('Stability Margin vs. Frequency');
ylabel('Margin'); ylabel('Margin');
@ -148,35 +138,102 @@ else
stabmarg.LowerBound, stabmarg.UpperBound, stabmarg.CriticalFrequency); stabmarg.LowerBound, stabmarg.UpperBound, stabmarg.CriticalFrequency);
end end
Twc = usubs(Gcl, wcg); %% Plant sv and multiplicative uncertainty
y_wc = lsim(Twc, [u1; u2], t); figure('Name', 'Open Loop System');
subplot(2, 1, 1);
sigma(G);
hold on;
title('Singular values of the system');
%% Transient plot subplot(2, 1, 2);
figure('Name', 'Transient simulation'); sigma(Im);
title('Multiplicative Uncertainty Magnitude');
%% Simulation
t = 0:0.001:3;
r1 = 0*t + 1;
r2 = 0*t;
% transfer function from reference to controller output
G_ur = getIOTransfer(Gcl, 'r', 'u');
y = lsim(T, [r1; r2], t);
u = lsim(G_ur, [r1; r2], t);
realizations = usample(Gcl, 20);
ninputs = 2;
y_unc = zeros(ninputs*length(realizations), length(t));
u_unc = zeros(ninputs*length(realizations), length(t));
for i=1:length(realizations)
G_ur_i = getIOTransfer(realizations(:, :, i, 1), 'r', 'u');
p = lsim(realizations(:,:, i, 1), [r1; r2], t);
q = lsim(G_ur_i, [r1; r2], t);
y_unc(ninputs*i, :) = p(:, 1);
y_unc(ninputs*i+1, :) = p(:, 2);
u_unc(ninputs*i, :) = q(:, 1);
u_unc(ninputs*i+1, :) = q(:, 2);
end
Twc = usubs(Gcl, wcg);
Tur = usubs(G_ur, wcg);
y_wc = lsim(Twc, [r1; r2], t);
u_wc = lsim(Tur, [r1; r2], t);
%% Output and control action plot
figure('Name', 'Transient simulation: output');
subplot(2, 3, 1); subplot(2, 3, 1);
plot(t, u1, '--b', t, y(:, 1), 'r'); plot(t, r1, '--b', t, y(:, 1), 'r');
title('Nominal system response'); title('Nominal system output');
subplot(2, 3, 4); subplot(2, 3, 4);
plot(t, u2, '--b', t, y(:, 2), 'r'); plot(t, r2, '--b', t, y(:, 2), 'r');
subplot(2, 3, 2); subplot(2, 3, 2);
plot(t, u1, '--b'); hold on; plot(t, r1, '--b'); hold on;
for i=1:length(realizations) for i=1:length(realizations)
plot(t, y_unc(ninputs*i, :), 'r'); plot(t, y_unc(ninputs*i, :), 'r');
end end
hold off; hold off;
title('Uncertain system response'); title('Uncertain system output');
subplot(2, 3, 5); subplot(2, 3, 5);
plot(t, u2, '--b'); hold on; plot(t, r2, '--b'); hold on;
for i=1:length(realizations) for i=1:length(realizations)
plot(t, y_unc(ninputs*i+1, :), 'r'); plot(t, y_unc(ninputs*i+1, :), 'r');
end end
hold off; hold off;
subplot(2, 3, 3); subplot(2, 3, 3);
plot(t, u1, '--b', t, y_wc(:, 1), 'r'); plot(t, r1, '--b', t, y_wc(:, 1), 'r');
title('Worst case system response'); title('Worst case system output');
subplot(2, 3, 6); subplot(2, 3, 6);
plot(t, u2, '--b', t, y_wc(:, 2), 'r'); plot(t, r2, '--b', t, y_wc(:, 2), 'r');
figure('Name', 'Transient simulation: control');
subplot(2, 3, 1);
plot(t, r1, '--b', t, u(:, 1), 'r');
title('Nominal system control action');
subplot(2, 3, 4);
plot(t, r2, '--b', t, u(:, 2), 'r');
subplot(2, 3, 2);
plot(t, r1, '--b'); hold on;
for i=1:length(realizations)
plot(t, u_unc(ninputs*i, :), 'r');
end
hold off;
title('Uncertain system control action');
subplot(2, 3, 5);
plot(t, r2, '--b'); hold on;
for i=1:length(realizations)
plot(t, u_unc(ninputs*i+1, :), 'r');
end
hold off;
subplot(2, 3, 3);
plot(t, r1, '--b', t, u_wc(:, 1), 'r');
title('Worst case system control action');
subplot(2, 3, 6);
plot(t, r2, '--b', t, u_wc(:, 2), 'r');