tdehaeze/svd-control
1
0
Fork 0
Code Issues Pull Requests Releases Wiki Activity

Change gravimeter axis

Browse Source
This commit is contained in:
Thomas Dehaeze 2020-12-10 13:26:01 +01:00
parent 754716e4ad
commit 404c78505a
7 changed files with 496 additions and 72 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

BIN
figs/gravimeter_decouple_jacobian.pdf
View File

Binary file not shown.

BIN
figs/gravimeter_decouple_jacobian.png
View File

Binary file not shown.
Side by Side Swipe Overlay

Before

Width:  |  Height:  |  Size: 12 KiB

After

Width:  |  Height:  |  Size: 12 KiB

BIN
figs/gravimeter_model.png
View File

Binary file not shown.
Side by Side Swipe Overlay

Before

Width:  |  Height:  |  Size: 22 KiB

After

Width:  |  Height:  |  Size: 22 KiB

BIN
figs/gravimeter_plant_schematic.pdf
View File

Binary file not shown.

BIN
figs/gravimeter_plant_schematic.png
View File

Binary file not shown.
Side by Side Swipe Overlay

Before

Width:  |  Height:  |  Size: 4.4 KiB

After

Width:  |  Height:  |  Size: 4.6 KiB

478
gravimeter/script.m
View File

@ -20,6 +20,10 @@ open('gravimeter.slx')
% #+caption: Model of the gravimeter
% [[file:figs/gravimeter_model.png]]
% #+name: fig:leg_model
% #+caption: Model of the struts
% [[file:figs/leg_model.png]]
% The parameters used for the simulation are the following:
l = 1.0; % Length of the mass [m]
@ -57,7 +61,7 @@ io(io_i) = linio([mdl, '/Acc_top'], 2, 'openoutput'); io_i = io_i + 1;
G = linearize(mdl, io);
G.InputName = {'F1', 'F2', 'F3'};
G.OutputName = {'Ax1', 'Az1', 'Ax2', 'Az2'};
G.OutputName = {'Ax1', 'Ay1', 'Ax2', 'Ay2'};
@ -125,22 +129,22 @@ end
% The Jacobian corresponding to the sensors and actuators are defined below:
Ja = [1 0 h/2
0 1 -l/2
1 0 -h/2
Ja = [1 0 -h/2
0 1 l/2
1 0 h/2
0 1 0];
Jt = [1 0 ha
0 1 -la
0 1 la];
Jt = [1 0 -ha
0 1 la
0 1 -la];
% And the plant $\bm{G}_x$ is computed:
Gx = pinv(Ja)*G*pinv(Jt');
Gx.InputName = {'Fx', 'Fz', 'My'};
Gx.OutputName = {'Dx', 'Dz', 'Ry'};
Gx.InputName = {'Fx', 'Fy', 'Mz'};
Gx.OutputName = {'Dx', 'Dy', 'Rz'};
size(Gx)
@ -385,6 +389,43 @@ legend('location', 'southwest');
linkaxes([ax1,ax2],'y');
ylim([1e-5, 1e1]);
% #+name: fig:gravimeter_rga
% #+caption: Obtained norm of RGA elements for the SVD decoupled plant and the Jacobian decoupled plant
% #+RESULTS:
% [[file:figs/gravimeter_rga.png]]
% The RGA-number is also a measure of diagonal dominance:
% \begin{equation}
% \text{RGA-number} = \| \Lambda(G) - I \|_\text{sum}
% \end{equation}
% Relative Gain Array for the decoupled plant using SVD:
RGA_svd = zeros(size(Gsvd,1), size(Gsvd,2), length(freqs));
Gsvd_inv = inv(Gsvd);
for f_i = 1:length(freqs)
RGA_svd(:, :, f_i) = abs(evalfr(Gsvd, j*2*pi*freqs(f_i)).*evalfr(Gsvd_inv, j*2*pi*freqs(f_i))');
end
% Relative Gain Array for the decoupled plant using the Jacobian:
RGA_x = zeros(size(Gx,1), size(Gx,2), length(freqs));
Gx_inv = inv(Gx);
for f_i = 1:length(freqs)
RGA_x(:, :, f_i) = abs(evalfr(Gx, j*2*pi*freqs(f_i)).*evalfr(Gx_inv, j*2*pi*freqs(f_i))');
end
RGA_num_svd = squeeze(sum(sum(RGA_svd - eye(3))));
RGA_num_x = squeeze(sum(sum(RGA_x - eye(3))));
figure;
hold on;
plot(freqs, RGA_num_svd)
plot(freqs, RGA_num_x)
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('RGA-Number');
% Obtained Decoupled Plants
% <<sec:gravimeter_decoupled_plant>>
@ -457,7 +498,7 @@ plot(freqs, abs(squeeze(freqresp(Gx(1, 2), freqs, 'Hz'))), 'color', [0,0,0,0.5],
set(gca,'ColorOrderIndex',1)
plot(freqs, abs(squeeze(freqresp(Gx(1, 1), freqs, 'Hz'))), 'DisplayName', '$G_x(1,1) = A_x/F_x$');
plot(freqs, abs(squeeze(freqresp(Gx(2, 2), freqs, 'Hz'))), 'DisplayName', '$G_x(2,2) = A_y/F_y$');
plot(freqs, abs(squeeze(freqresp(Gx(3, 3), freqs, 'Hz'))), 'DisplayName', '$G_x(3,3) = R_y/M_y$');
plot(freqs, abs(squeeze(freqresp(Gx(3, 3), freqs, 'Hz'))), 'DisplayName', '$G_x(3,3) = R_z/M_z$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
@ -605,7 +646,7 @@ plot(freqs, abs(squeeze(freqresp(G_svd(2,2)/s^2, freqs, 'Hz'))), '--');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'YTickLabel',[]); xlabel('Frequency [Hz]');
title('$D_z/D_{w,z}$');
title('$D_y/D_{w,y}$');
ax3 = nexttile;
hold on;
@ -615,8 +656,421 @@ plot(freqs, abs(squeeze(freqresp(G_svd(3,3)/s^2, freqs, 'Hz'))), '--');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'YTickLabel',[]); xlabel('Frequency [Hz]');
title('$R_y/R_{w,y}$');
title('$R_z/R_{w,z}$');
linkaxes([ax1,ax2,ax3],'xy');
xlim([freqs(1), freqs(end)]);
xlim([1e-2, 5e1]); ylim([1e-7, 1e-2]);
% #+name: fig:gravimeter_platform_simscape_cl_transmissibility
% #+caption: Obtained Transmissibility
% #+RESULTS:
% [[file:figs/gravimeter_platform_simscape_cl_transmissibility.png]]
freqs = logspace(-2, 2, 1000);
figure;
hold on;
for out_i = 1:3
for in_i = out_i+1:3
set(gca,'ColorOrderIndex',1)
plot(freqs, abs(squeeze(freqresp(G( out_i,in_i), freqs, 'Hz'))));
set(gca,'ColorOrderIndex',2)
plot(freqs, abs(squeeze(freqresp(G_cen(out_i,in_i), freqs, 'Hz'))));
set(gca,'ColorOrderIndex',3)
plot(freqs, abs(squeeze(freqresp(G_svd(out_i,in_i), freqs, 'Hz'))), '--');
end
end
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Transmissibility'); xlabel('Frequency [Hz]');
% Robustness to a change of actuator position
% Let say we change the position of the actuators:
la = l/2*0.7; % Position of Act. [m]
ha = h/2*0.7; % Position of Act. [m]
%% Name of the Simulink File
mdl = 'gravimeter';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/F1'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/F2'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/F3'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Acc_side'], 1, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Acc_side'], 2, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Acc_top'], 1, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Acc_top'], 2, 'openoutput'); io_i = io_i + 1;
G = linearize(mdl, io);
G.InputName = {'F1', 'F2', 'F3'};
G.OutputName = {'Ax1', 'Ay1', 'Ax2', 'Ay2'};
G_cen_b = feedback(G, pinv(Jt')*K_cen*pinv(Ja));
G_svd_b = feedback(G, inv(V')*K_svd*U_inv(1:3, :));
% The new plant is computed, and the centralized and SVD control architectures are applied using the previsouly computed Jacobian matrices and $U$ and $V$ matrices.
% The closed-loop system are still stable, and their
freqs = logspace(-2, 2, 1000);
figure;
tiledlayout(1, 3, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(G_cen(1,1)/s^2, freqs, 'Hz'))), 'DisplayName', 'Initial');
plot(freqs, abs(squeeze(freqresp(G_cen_b(1,1)/s^2, freqs, 'Hz'))), 'DisplayName', 'Jacobian');
plot(freqs, abs(squeeze(freqresp(G_svd_b(1,1)/s^2, freqs, 'Hz'))), '--', 'DisplayName', 'SVD');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Transmissibility'); xlabel('Frequency [Hz]');
title('$D_x/D_{w,x}$');
legend('location', 'southwest');
ax2 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(G_cen(2,2)/s^2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G_cen_b(2,2)/s^2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G_svd_b(2,2)/s^2, freqs, 'Hz'))), '--');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'YTickLabel',[]); xlabel('Frequency [Hz]');
title('$D_y/D_{w,y}$');
ax3 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(G_cen(3,3)/s^2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G_cen_b(3,3)/s^2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G_svd_b(3,3)/s^2, freqs, 'Hz'))), '--');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'YTickLabel',[]); xlabel('Frequency [Hz]');
title('$R_z/R_{w,z}$');
linkaxes([ax1,ax2,ax3],'xy');
xlim([freqs(1), freqs(end)]);
xlim([1e-2, 5e1]); ylim([1e-7, 3e-4]);
% Decoupling of the mass matrix
% #+name: fig:gravimeter_model_M
% #+caption: Choice of {O} such that the Mass Matrix is Diagonal
% [[file:figs/gravimeter_model_M.png]]
la = l/2; % Position of Act. [m]
ha = h/2; % Position of Act. [m]
%% Name of the Simulink File
mdl = 'gravimeter';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/F1'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/F2'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/F3'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Acc_side'], 1, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Acc_side'], 2, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Acc_top'], 1, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Acc_top'], 2, 'openoutput'); io_i = io_i + 1;
G = linearize(mdl, io);
G.InputName = {'F1', 'F2', 'F3'};
G.OutputName = {'Ax1', 'Ay1', 'Ax2', 'Ay2'};
% Decoupling at the CoM (Mass decoupled)
JMa = [1 0 -h/2
0 1 l/2
1 0 h/2
0 1 0];
JMt = [1 0 -ha
0 1 la
0 1 -la];
GM = pinv(JMa)*G*pinv(JMt');
GM.InputName = {'Fx', 'Fy', 'Mz'};
GM.OutputName = {'Dx', 'Dy', 'Rz'};
figure;
% Magnitude
hold on;
for i_in = 1:3
for i_out = [1:i_in-1, i_in+1:3]
plot(freqs, abs(squeeze(freqresp(GM(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
'HandleVisibility', 'off');
end
end
plot(freqs, abs(squeeze(freqresp(GM(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
'DisplayName', '$G_x(i,j)\ i \neq j$');
set(gca,'ColorOrderIndex',1)
for i_in_out = 1:3
plot(freqs, abs(squeeze(freqresp(GM(i_in_out, i_in_out), freqs, 'Hz'))), 'DisplayName', sprintf('$G_x(%d,%d)$', i_in_out, i_in_out));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
legend('location', 'southeast');
ylim([1e-8, 1e0]);
% Decoupling of the stiffness matrix
% #+name: fig:gravimeter_model_K
% #+caption: Choice of {O} such that the Stiffness Matrix is Diagonal
% [[file:figs/gravimeter_model_K.png]]
% Decoupling at the point where K is diagonal (x = 0, y = -h/2 from the schematic {O} frame):
JKa = [1 0 0
0 1 -l/2
1 0 -h
0 1 0];
JKt = [1 0 0
0 1 -la
0 1 la];
% And the plant $\bm{G}_x$ is computed:
GK = pinv(JKa)*G*pinv(JKt');
GK.InputName = {'Fx', 'Fy', 'Mz'};
GK.OutputName = {'Dx', 'Dy', 'Rz'};
figure;
% Magnitude
hold on;
for i_in = 1:3
for i_out = [1:i_in-1, i_in+1:3]
plot(freqs, abs(squeeze(freqresp(GK(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
'HandleVisibility', 'off');
end
end
plot(freqs, abs(squeeze(freqresp(GK(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
'DisplayName', '$G_x(i,j)\ i \neq j$');
set(gca,'ColorOrderIndex',1)
for i_in_out = 1:3
plot(freqs, abs(squeeze(freqresp(GK(i_in_out, i_in_out), freqs, 'Hz'))), 'DisplayName', sprintf('$G_x(%d,%d)$', i_in_out, i_in_out));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
legend('location', 'southeast');
ylim([1e-8, 1e0]);
% Combined decoupling of the mass and stiffness matrices
% #+name: fig:gravimeter_model_KM
% #+caption: Ideal location of the actuators such that both the mass and stiffness matrices are diagonal
% [[file:figs/gravimeter_model_KM.png]]
% To do so, the actuator position should be modified
la = l/2; % Position of Act. [m]
ha = 0; % Position of Act. [m]
%% Name of the Simulink File
mdl = 'gravimeter';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/F1'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/F2'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/F3'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Acc_side'], 1, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Acc_side'], 2, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Acc_top'], 1, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Acc_top'], 2, 'openoutput'); io_i = io_i + 1;
G = linearize(mdl, io);
G.InputName = {'F1', 'F2', 'F3'};
G.OutputName = {'Ax1', 'Ay1', 'Ax2', 'Ay2'};
JMa = [1 0 -h/2
0 1 l/2
1 0 h/2
0 1 0];
JMt = [1 0 -ha
0 1 la
0 1 -la];
GKM = pinv(JMa)*G*pinv(JMt');
GKM.InputName = {'Fx', 'Fy', 'Mz'};
GKM.OutputName = {'Dx', 'Dy', 'Rz'};
figure;
% Magnitude
hold on;
for i_in = 1:3
for i_out = [1:i_in-1, i_in+1:3]
plot(freqs, abs(squeeze(freqresp(GKM(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
'HandleVisibility', 'off');
end
end
plot(freqs, abs(squeeze(freqresp(GKM(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
'DisplayName', '$G_x(i,j)\ i \neq j$');
set(gca,'ColorOrderIndex',1)
for i_in_out = 1:3
plot(freqs, abs(squeeze(freqresp(GKM(i_in_out, i_in_out), freqs, 'Hz'))), 'DisplayName', sprintf('$G_x(%d,%d)$', i_in_out, i_in_out));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
legend('location', 'southeast');
ylim([1e-8, 1e0]);
% SVD decoupling performances :noexport:
la = l/2; % Position of Act. [m]
ha = 0; % Position of Act. [m]
c = 2e1; % Actuator Damping [N/(m/s)]
%% Name of the Simulink File
mdl = 'gravimeter';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/F1'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/F2'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/F3'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Acc_side'], 1, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Acc_side'], 2, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Acc_top'], 1, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Acc_top'], 2, 'openoutput'); io_i = io_i + 1;
G = linearize(mdl, io);
G.InputName = {'F1', 'F2', 'F3'};
G.OutputName = {'Ax1', 'Ay1', 'Ax2', 'Ay2'};
wc = 2*pi*10; % Decoupling frequency [rad/s]
H1 = evalfr(G, j*wc);
D = pinv(real(H1'*H1));
H1 = pinv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2))));
[U,S,V] = svd(H1);
Gsvd = inv(U)*G*inv(V');
c = 5e2; % Actuator Damping [N/(m/s)]
%% Name of the Simulink File
mdl = 'gravimeter';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/F1'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/F2'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/F3'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Acc_side'], 1, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Acc_side'], 2, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Acc_top'], 1, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Acc_top'], 2, 'openoutput'); io_i = io_i + 1;
G = linearize(mdl, io);
G.InputName = {'F1', 'F2', 'F3'};
G.OutputName = {'Ax1', 'Ay1', 'Ax2', 'Ay2'};
wc = 2*pi*10; % Decoupling frequency [rad/s]
H1 = evalfr(G, j*wc);
D = pinv(real(H1'*H1));
H1 = pinv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2))));
[U,S,V] = svd(H1);
Gsvdd = inv(U)*G*inv(V');
JMa = [1 0 -h/2
0 1 l/2
1 0 h/2
0 1 0];
JMt = [1 0 -ha
0 1 la
0 1 -la];
GM = pinv(JMa)*G*pinv(JMt');
GM.InputName = {'Fx', 'Fy', 'Mz'};
GM.OutputName = {'Dx', 'Dy', 'Rz'};
figure;
% Magnitude
hold on;
for i_in = 1:3
for i_out = [1:i_in-1, i_in+1:3]
plot(freqs, abs(squeeze(freqresp(GM(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
'HandleVisibility', 'off');
end
end
plot(freqs, abs(squeeze(freqresp(GM(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
'DisplayName', '$G_x(i,j)\ i \neq j$');
set(gca,'ColorOrderIndex',1)
for i_in_out = 1:3
plot(freqs, abs(squeeze(freqresp(GM(i_in_out, i_in_out), freqs, 'Hz'))), 'DisplayName', sprintf('$G_x(%d,%d)$', i_in_out, i_in_out));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
legend('location', 'southeast');
ylim([1e-8, 1e0]);
figure;
% Magnitude
hold on;
for i_in = 1:3
for i_out = [1:i_in-1, i_in+1:3]
plot(freqs, abs(squeeze(freqresp(Gsvd(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
'HandleVisibility', 'off');
end
end
plot(freqs, abs(squeeze(freqresp(Gsvd(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
'DisplayName', '$G_x(i,j)\ i \neq j$');
set(gca,'ColorOrderIndex',1)
for i_in_out = 1:3
plot(freqs, abs(squeeze(freqresp(Gsvd(i_in_out, i_in_out), freqs, 'Hz'))), 'DisplayName', sprintf('$G_x(%d,%d)$', i_in_out, i_in_out));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
legend('location', 'southeast');
ylim([1e-8, 1e0]);
figure;
% Magnitude
hold on;
for i_in = 1:3
for i_out = [1:i_in-1, i_in+1:3]
plot(freqs, abs(squeeze(freqresp(Gsvdd(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
'HandleVisibility', 'off');
end
end
plot(freqs, abs(squeeze(freqresp(Gsvdd(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
'DisplayName', '$G_x(i,j)\ i \neq j$');
set(gca,'ColorOrderIndex',1)
for i_in_out = 1:3
plot(freqs, abs(squeeze(freqresp(Gsvdd(i_in_out, i_in_out), freqs, 'Hz'))), 'DisplayName', sprintf('$G_x(%d,%d)$', i_in_out, i_in_out));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
legend('location', 'southeast');
ylim([1e-8, 1e0]);

90
index.org
View File

@ -138,7 +138,7 @@ The parameters used for the simulation are the following:
G = linearize(mdl, io);
G.InputName = {'F1', 'F2', 'F3'};
G.OutputName = {'Ax1', 'Az1', 'Ax2', 'Az2'};
G.OutputName = {'Ax1', 'Ay1', 'Ax2', 'Ay2'};
#+end_src
The inputs and outputs of the plant are shown in Figure [[fig:gravimeter_plant_schematic]].
@ -149,7 +149,7 @@ More precisely there are three inputs (the three actuator forces):
\end{equation}
And 4 outputs (the two 2-DoF accelerometers):
\begin{equation}
\bm{a} = \begin{bmatrix} a_{1x} \\ a_{1z} \\ a_{2x} \\ a_{2z} \end{bmatrix}
\bm{a} = \begin{bmatrix} a_{1x} \\ a_{1y} \\ a_{2x} \\ a_{2y} \end{bmatrix}
\end{equation}
#+begin_src latex :file gravimeter_plant_schematic.pdf :tangle no :exports results
@ -158,7 +158,7 @@ And 4 outputs (the two 2-DoF accelerometers):
% Connections and labels
\draw[<-] (G.west) -- ++(-2.0, 0) node[above right]{$\bm{\tau} = \begin{bmatrix}\tau_1 \\ \tau_2 \\ \tau_2 \end{bmatrix}$};
\draw[->] (G.east) -- ++( 2.0, 0) node[above left]{$\bm{a} = \begin{bmatrix} a_{1x} \\ a_{1z} \\ a_{2x} \\ a_{2z} \end{bmatrix}$};
\draw[->] (G.east) -- ++( 2.0, 0) node[above left]{$\bm{a} = \begin{bmatrix} a_{1x} \\ a_{1y} \\ a_{2x} \\ a_{2y} \end{bmatrix}$};
\end{tikzpicture}
#+end_src
@ -232,12 +232,12 @@ Consider the control architecture shown in Figure [[fig:gravimeter_decouple_jaco
The Jacobian matrix $J_{\tau}$ is used to transform forces applied by the three actuators into forces/torques applied on the gravimeter at its center of mass:
\begin{equation}
\begin{bmatrix} \tau_1 \\ \tau_2 \\ \tau_3 \end{bmatrix} = J_{\tau}^{-T} \begin{bmatrix} F_x \\ F_z \\ M_y \end{bmatrix}
\begin{bmatrix} \tau_1 \\ \tau_2 \\ \tau_3 \end{bmatrix} = J_{\tau}^{-T} \begin{bmatrix} F_x \\ F_y \\ M_z \end{bmatrix}
\end{equation}
The Jacobian matrix $J_{a}$ is used to compute the vertical acceleration, horizontal acceleration and rotational acceleration of the mass with respect to its center of mass:
\begin{equation}
\begin{bmatrix} a_x \\ a_z \\ a_{R_y} \end{bmatrix} = J_{a}^{-1} \begin{bmatrix} a_{x1} \\ a_{z1} \\ a_{x2} \\ a_{z2} \end{bmatrix}
\begin{bmatrix} a_x \\ a_y \\ a_{R_z} \end{bmatrix} = J_{a}^{-1} \begin{bmatrix} a_{x1} \\ a_{y1} \\ a_{x2} \\ a_{y2} \end{bmatrix}
\end{equation}
We thus define a new plant as defined in Figure [[fig:gravimeter_decouple_jacobian]].
@ -252,10 +252,10 @@ $\bm{G}_x(s)$ correspond to the $3 \times 3$transfer function matrix from forces
\node[block, right=0.6 of G] (Ja) {$J_{a}^{-1}$};
% Connections and labels
\draw[<-] (Jt.west) -- ++(-2.5, 0) node[above right]{$\bm{\mathcal{F}} = \begin{bmatrix}F_x \\ F_z \\ M_y \end{bmatrix}$};
\draw[<-] (Jt.west) -- ++(-2.5, 0) node[above right]{$\bm{\mathcal{F}} = \begin{bmatrix}F_x \\ F_y \\ M_z \end{bmatrix}$};
\draw[->] (Jt.east) -- (G.west) node[above left]{$\bm{\tau}$};
\draw[->] (G.east) -- (Ja.west) node[above left]{$\bm{a}$};
\draw[->] (Ja.east) -- ++( 2.6, 0) node[above left]{$\bm{\mathcal{A}} = \begin{bmatrix}a_x \\ a_z \\ a_{R_y} \end{bmatrix}$};
\draw[->] (Ja.east) -- ++( 2.6, 0) node[above left]{$\bm{\mathcal{A}} = \begin{bmatrix}a_x \\ a_y \\ a_{R_z} \end{bmatrix}$};
\begin{scope}[on background layer]
\node[fit={(Jt.south west) (Ja.north east)}, fill=black!10!white, draw, dashed, inner sep=14pt] (Gx) {};
@ -284,8 +284,8 @@ The Jacobian corresponding to the sensors and actuators are defined below:
And the plant $\bm{G}_x$ is computed:
#+begin_src matlab
Gx = pinv(Ja)*G*pinv(Jt');
Gx.InputName = {'Fx', 'Fz', 'My'};
Gx.OutputName = {'Dx', 'Dz', 'Ry'};
Gx.InputName = {'Fx', 'Fy', 'Mz'};
Gx.OutputName = {'Dx', 'Dy', 'Rz'};
#+end_src
#+begin_src matlab :results output replace :exports results
@ -736,7 +736,7 @@ Similarly, the bode plots of the diagonal elements and off-diagonal elements of
set(gca,'ColorOrderIndex',1)
plot(freqs, abs(squeeze(freqresp(Gx(1, 1), freqs, 'Hz'))), 'DisplayName', '$G_x(1,1) = A_x/F_x$');
plot(freqs, abs(squeeze(freqresp(Gx(2, 2), freqs, 'Hz'))), 'DisplayName', '$G_x(2,2) = A_y/F_y$');
plot(freqs, abs(squeeze(freqresp(Gx(3, 3), freqs, 'Hz'))), 'DisplayName', '$G_x(3,3) = R_y/M_y$');
plot(freqs, abs(squeeze(freqresp(Gx(3, 3), freqs, 'Hz'))), 'DisplayName', '$G_x(3,3) = R_z/M_z$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
@ -961,7 +961,7 @@ The obtained transmissibility in Open-loop, for the centralized control as well
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'YTickLabel',[]); xlabel('Frequency [Hz]');
title('$D_z/D_{w,z}$');
title('$D_y/D_{w,y}$');
ax3 = nexttile;
hold on;
@ -971,7 +971,7 @@ The obtained transmissibility in Open-loop, for the centralized control as well
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'YTickLabel',[]); xlabel('Frequency [Hz]');
title('$R_y/R_{w,y}$');
title('$R_z/R_{w,z}$');
linkaxes([ax1,ax2,ax3],'xy');
xlim([freqs(1), freqs(end)]);
@ -1040,7 +1040,7 @@ Let say we change the position of the actuators:
G = linearize(mdl, io);
G.InputName = {'F1', 'F2', 'F3'};
G.OutputName = {'Ax1', 'Az1', 'Ax2', 'Az2'};
G.OutputName = {'Ax1', 'Ay1', 'Ax2', 'Ay2'};
#+end_src
#+begin_src matlab :exports none
@ -1077,7 +1077,7 @@ The closed-loop system are still stable, and their
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'YTickLabel',[]); xlabel('Frequency [Hz]');
title('$D_z/D_{w,z}$');
title('$D_y/D_{w,y}$');
ax3 = nexttile;
hold on;
@ -1087,7 +1087,7 @@ The closed-loop system are still stable, and their
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
set(gca, 'YTickLabel',[]); xlabel('Frequency [Hz]');
title('$R_y/R_{w,y}$');
title('$R_z/R_{w,z}$');
linkaxes([ax1,ax2,ax3],'xy');
xlim([freqs(1), freqs(end)]);
@ -1145,7 +1145,7 @@ To do so, the actuators (springs) should be positioned such that the stiffness m
G = linearize(mdl, io);
G.InputName = {'F1', 'F2', 'F3'};
G.OutputName = {'Ax1', 'Az1', 'Ax2', 'Az2'};
G.OutputName = {'Ax1', 'Ay1', 'Ax2', 'Ay2'};
#+end_src
Decoupling at the CoM (Mass decoupled)
@ -1162,8 +1162,8 @@ Decoupling at the CoM (Mass decoupled)
#+begin_src matlab
GM = pinv(JMa)*G*pinv(JMt');
GM.InputName = {'Fx', 'Fz', 'My'};
GM.OutputName = {'Dx', 'Dz', 'Ry'};
GM.InputName = {'Fx', 'Fy', 'Mz'};
GM.OutputName = {'Dx', 'Dy', 'Rz'};
#+end_src
#+begin_src matlab :exports none
@ -1195,7 +1195,7 @@ Decoupling at the CoM (Mass decoupled)
#+end_src
#+name: fig:jac_decoupling_M
#+caption:
#+caption: Diagonal and off-diagonal elements of the decoupled plant
#+RESULTS:
[[file:figs/jac_decoupling_M.png]]
@ -1220,8 +1220,8 @@ Decoupling at the point where K is diagonal (x = 0, y = -h/2 from the schematic
And the plant $\bm{G}_x$ is computed:
#+begin_src matlab
GK = pinv(JKa)*G*pinv(JKt');
GK.InputName = {'Fx', 'Fz', 'My'};
GK.OutputName = {'Dx', 'Dz', 'Ry'};
GK.InputName = {'Fx', 'Fy', 'Mz'};
GK.OutputName = {'Dx', 'Dy', 'Rz'};
#+end_src
#+begin_src matlab :exports none
@ -1253,7 +1253,7 @@ And the plant $\bm{G}_x$ is computed:
#+end_src
#+name: fig:jac_decoupling_K
#+caption:
#+caption: Diagonal and off-diagonal elements of the decoupled plant
#+RESULTS:
[[file:figs/jac_decoupling_K.png]]
@ -1286,7 +1286,7 @@ To do so, the actuator position should be modified
G = linearize(mdl, io);
G.InputName = {'F1', 'F2', 'F3'};
G.OutputName = {'Ax1', 'Az1', 'Ax2', 'Az2'};
G.OutputName = {'Ax1', 'Ay1', 'Ax2', 'Ay2'};
#+end_src
#+begin_src matlab
@ -1302,8 +1302,8 @@ To do so, the actuator position should be modified
#+begin_src matlab
GKM = pinv(JMa)*G*pinv(JMt');
GKM.InputName = {'Fx', 'Fz', 'My'};
GKM.OutputName = {'Dx', 'Dz', 'Ry'};
GKM.InputName = {'Fx', 'Fy', 'Mz'};
GKM.OutputName = {'Dx', 'Dy', 'Rz'};
#+end_src
#+begin_src matlab :exports none
@ -1335,7 +1335,7 @@ To do so, the actuator position should be modified
#+end_src
#+name: fig:jac_decoupling_KM
#+caption:
#+caption: Diagonal and off-diagonal elements of the decoupled plant
#+RESULTS:
[[file:figs/jac_decoupling_KM.png]]
@ -1373,7 +1373,7 @@ Or it can be decoupled at high frequency if the Jacobians are evaluated at the C
G = linearize(mdl, io);
G.InputName = {'F1', 'F2', 'F3'};
G.OutputName = {'Ax1', 'Az1', 'Ax2', 'Az2'};
G.OutputName = {'Ax1', 'Ay1', 'Ax2', 'Ay2'};
#+end_src
#+begin_src matlab
@ -1405,7 +1405,7 @@ Or it can be decoupled at high frequency if the Jacobians are evaluated at the C
G = linearize(mdl, io);
G.InputName = {'F1', 'F2', 'F3'};
G.OutputName = {'Ax1', 'Az1', 'Ax2', 'Az2'};
G.OutputName = {'Ax1', 'Ay1', 'Ax2', 'Ay2'};
#+end_src
#+begin_src matlab
@ -1430,8 +1430,8 @@ Or it can be decoupled at high frequency if the Jacobians are evaluated at the C
#+begin_src matlab
GM = pinv(JMa)*G*pinv(JMt');
GM.InputName = {'Fx', 'Fz', 'My'};
GM.OutputName = {'Dx', 'Dz', 'Ry'};
GM.InputName = {'Fx', 'Fy', 'Mz'};
GM.OutputName = {'Dx', 'Dy', 'Rz'};
#+end_src
#+begin_src matlab :exports none
@ -1506,36 +1506,6 @@ Or it can be decoupled at high frequency if the Jacobians are evaluated at the C
ylim([1e-8, 1e0]);
#+end_src
** SVD U and V matrices :noexport:
#+begin_src matlab
la = l/2; % Position of Act. [m]
ha = 0; % Position of Act. [m]
#+end_src
#+begin_src matlab
c = 2e1; % Actuator Damping [N/(m/s)]
#+end_src
#+begin_src matlab
%% Name of the Simulink File
mdl = 'gravimeter';
%% Input/Output definition
clear io; io_i = 1;
io(io_i) = linio([mdl, '/F1'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/F2'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/F3'], 1, 'openinput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Acc_side'], 1, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Acc_side'], 2, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Acc_top'], 1, 'openoutput'); io_i = io_i + 1;
io(io_i) = linio([mdl, '/Acc_top'], 2, 'openoutput'); io_i = io_i + 1;
G = linearize(mdl, io);
G.InputName = {'F1', 'F2', 'F3'};
G.OutputName = {'Ax1', 'Az1', 'Ax2', 'Az2'};
#+end_src
* Stewart Platform - Simscape Model
:PROPERTIES:
:header-args:matlab+: :tangle stewart_platform/script.m