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Clean simscape models

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This commit is contained in:
Thomas Dehaeze
2020-11-25 09:40:17 +01:00
parent 9b29c73fff
commit a1581cb873
17 changed files with 11319 additions and 10289 deletions
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15
gravimeter/align.m
View File

@@ -1,15 +0,0 @@
function [A] = align(V)
%A!ALIGN(V) returns a constat matrix A which is the real alignment of the
%INVERSE of the complex input matrix V
%from Mohit slides
if (nargin ==0) || (nargin > 1)
disp('usage: mat_inv_real = align(mat)')
return
end
D = pinv(real(V'*V));
A = D*real(V'*diag(exp(1i * angle(diag(V*D*V.'))/2)));
end

34
gravimeter/pzmap_testCL.m
View File

@@ -1,34 +0,0 @@
function [] = pzmap_testCL(system,H,gain,feedin,feedout)
% evaluate and plot the pole-zero map for the closed loop system for
% different values of the gain
[~, n] = size(gain);
[m1, n1, ~] = size(H);
[~,n2] = size(feedin);
figure
for i = 1:n
% if n1 == n2
system_CL = feedback(system,gain(i)*H,feedin,feedout);
[P,Z] = pzmap(system_CL);
plot(real(P(:)),imag(P(:)),'x',real(Z(:)),imag(Z(:)),'o');hold on
xlabel('Real axis (s^{-1})');ylabel('Imaginary Axis (s^{-1})');
% clear P Z
% else
% system_CL = feedback(system,gain(i)*H(:,1+(i-1)*m1:m1+(i-1)*m1),feedin,feedout);
%
% [P,Z] = pzmap(system_CL);
% plot(real(P(:)),imag(P(:)),'x',real(Z(:)),imag(Z(:)),'o');hold on
% xlabel('Real axis (s^{-1})');ylabel('Imaginary Axis (s^{-1})');
% clear P Z
% end
end
str = {strcat('gain = ' , num2str(gain(1)))}; % at the end of first loop, z being loop output
str = [str , strcat('gain = ' , num2str(gain(1)))]; % after 2nd loop
for i = 2:n
str = [str , strcat('gain = ' , num2str(gain(i)))]; % after 2nd loop
str = [str , strcat('gain = ' , num2str(gain(i)))]; % after 2nd loop
end
legend(str{:})
end

327
gravimeter/script.m
View File

@@ -4,13 +4,23 @@ clear; close all; clc;
%% Intialize Laplace variable
s = zpk('s');
% Simscape Model - Parameters
freqs = logspace(-1, 2, 1000);
% Gravimeter Model - Parameters
% <<sec:gravimeter_model>>
open('gravimeter.slx')
% Parameters
% The model of the gravimeter is schematically shown in Figure [[fig:gravimeter_model]].
% #+name: fig:gravimeter_model
% #+caption: Model of the gravimeter
% [[file:figs/gravimeter_model.png]]
% The parameters used for the simulation are the following:
l = 1.0; % Length of the mass [m]
h = 1.7; % Height of the mass [m]
@@ -22,13 +32,15 @@ m = 400; % Mass [kg]
I = 115; % Inertia [kg m^2]
k = 15e3; % Actuator Stiffness [N/m]
c = 0.03; % Actuator Damping [N/(m/s)]
c = 2e1; % Actuator Damping [N/(m/s)]
deq = 0.2; % Length of the actuators [m]
g = 0; % Gravity [m/s2]
% System Identification - Without Gravity
% System Identification
% <<sec:gravimeter_identification>>
%% Name of the Simulink File
mdl = 'gravimeter';
@@ -54,32 +66,21 @@ G.OutputName = {'Ax1', 'Az1', 'Ax2', 'Az2'};
% #+RESULTS:
% [[file:figs/gravimeter_plant_schematic.png]]
% \begin{equation}
% \bm{a} = \begin{bmatrix} a_{1x} \\ a_{1z} \\ a_{2x} \\ a_{2z} \end{bmatrix}
% \end{equation}
% \begin{equation}
% \bm{\tau} = \begin{bmatrix}\tau_1 \\ \tau_2 \\ \tau_2 \end{bmatrix}
% \end{equation}
% We can check the poles of the plant:
pole(G)
% #+RESULTS:
% #+begin_example
% -0.000183495485977108 + 13.546056874877i
% -0.000183495485977108 - 13.546056874877i
% -7.49842878906757e-05 + 8.65934902322567i
% -7.49842878906757e-05 - 8.65934902322567i
% -1.33171230256362e-05 + 3.64924169037897i
% -1.33171230256362e-05 - 3.64924169037897i
% #+end_example
% | -0.12243+13.551i |
% | -0.12243-13.551i |
% | -0.05+8.6601i |
% | -0.05-8.6601i |
% | -0.0088785+3.6493i |
% | -0.0088785-3.6493i |
% The plant as 6 states as expected (2 translations + 1 rotation)
% As expected, the plant as 6 states (2 translations + 1 rotation)
size(G)
@@ -91,8 +92,6 @@ size(G)
% The bode plot of all elements of the plant are shown in Figure [[fig:open_loop_tf]].
freqs = logspace(-1, 2, 1000);
figure;
tiledlayout(4, 3, 'TileSpacing', 'None', 'Padding', 'None');
@@ -124,7 +123,7 @@ end
% #+RESULTS:
% [[file:figs/gravimeter_decouple_jacobian.png]]
% The jacobian corresponding to the sensors and actuators are defined below.
% The Jacobian corresponding to the sensors and actuators are defined below:
Ja = [1 0 h/2
0 1 -l/2
@@ -135,17 +134,25 @@ 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'};
size(Gx)
% #+RESULTS:
% : size(Gx)
% : State-space model with 3 outputs, 3 inputs, and 6 states.
% The diagonal and off-diagonal elements of $G_x$ are shown in Figure [[fig:gravimeter_jacobian_plant]].
freqs = logspace(-1, 2, 1000);
figure;
% Magnitude
@@ -168,10 +175,12 @@ xlabel('Frequency [Hz]'); ylabel('Magnitude');
legend('location', 'southeast');
ylim([1e-8, 1e0]);
% Real Approximation of $G$ at the decoupling frequency
% <<sec:gravimeter_real_approx>>
% Decoupling using the SVD
% <<sec:gravimeter_svd_decoupling>>
% Let's compute a real approximation of the complex matrix $H_1$ which corresponds to the the transfer function $G_u(j\omega_c)$ from forces applied by the actuators to the measured acceleration of the top platform evaluated at the frequency $\omega_c$.
% In order to decouple the plant using the SVD, first a real approximation of the plant transfer function matrix as the crossover frequency is required.
% Let's compute a real approximation of the complex matrix $H_1$ which corresponds to the the transfer function $G(j\omega_c)$ from forces applied by the actuators to the measured acceleration of the top platform evaluated at the frequency $\omega_c$.
wc = 2*pi*10; % Decoupling frequency [rad/s]
@@ -182,16 +191,23 @@ H1 = evalfr(G, j*wc);
% The real approximation is computed as follows:
D = pinv(real(H1'*H1));
H1 = inv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2))));
H1 = pinv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2))));
% SVD Decoupling
% <<sec:gravimeter_svd_decoupling>>
% First, the Singular Value Decomposition of $H_1$ is performed:
% #+caption: Real approximate of $G$ at the decoupling frequency $\omega_c$
% #+RESULTS:
% | 0.0092 | -0.0039 | 0.0039 |
% | -0.0039 | 0.0048 | 0.00028 |
% | -0.004 | 0.0038 | -0.0038 |
% | 8.4e-09 | 0.0025 | 0.0025 |
% Now, the Singular Value Decomposition of $H_1$ is performed:
% \[ H_1 = U \Sigma V^H \]
[U,~,V] = svd(H1);
[U,S,V] = svd(H1);
@@ -201,18 +217,27 @@ H1 = inv(D*real(H1'*diag(exp(j*angle(diag(H1*D*H1.'))/2))));
% [[file:figs/gravimeter_decouple_svd.png]]
% The decoupled plant is then:
% \[ G_{SVD}(s) = U^{-1} G_u(s) V^{-H} \]
% \[ \bm{G}_{SVD}(s) = U^{-1} \bm{G}(s) V^{-H} \]
Gsvd = inv(U)*G*inv(V');
size(Gsvd)
% #+RESULTS:
% : size(Gsvd)
% : State-space model with 4 outputs, 3 inputs, and 6 states.
% The 4th output (corresponding to the null singular value) is discarded, and we only keep the $3 \times 3$ plant:
Gsvd = Gsvd(1:3, 1:3);
% The diagonal and off-diagonal elements of the "SVD" plant are shown in Figure [[fig:gravimeter_svd_plant]].
freqs = logspace(-1, 2, 1000);
figure;
% Magnitude
@@ -232,25 +257,22 @@ end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); ylabel('Magnitude');
legend('location', 'southeast', 'FontSize', 8);
legend('location', 'southwest', 'FontSize', 8);
ylim([1e-8, 1e0]);
% TODO Verification of the decoupling using the "Gershgorin Radii"
% <<sec:comp_decoupling>>
% Verification of the decoupling using the "Gershgorin Radii"
% <<sec:gravimeter_gershgorin_radii>>
% The "Gershgorin Radii" is computed for the coupled plant $G(s)$, for the "Jacobian plant" $G_x(s)$ and the "SVD Decoupled Plant" $G_{SVD}(s)$:
% The "Gershgorin Radii" of a matrix $S$ is defined by:
% \[ \zeta_i(j\omega) = \frac{\sum\limits_{j\neq i}|S_{ij}(j\omega)|}{|S_{ii}(j\omega)|} \]
% This is computed over the following frequencies.
freqs = logspace(-2, 2, 1000); % [Hz]
% Gershgorin Radii for the coupled plant:
Gr_coupled = zeros(length(freqs), size(Gu,2));
H = abs(squeeze(freqresp(Gu, freqs, 'Hz')));
for out_i = 1:size(Gu,2)
Gr_coupled = zeros(length(freqs), size(G,2));
H = abs(squeeze(freqresp(G, freqs, 'Hz')));
for out_i = 1:size(G,2)
Gr_coupled(:, out_i) = squeeze((sum(H(out_i,:,:)) - H(out_i,out_i,:))./H(out_i, out_i, :));
end
@@ -273,7 +295,7 @@ hold on;
plot(freqs, Gr_coupled(:,1), 'DisplayName', 'Coupled');
plot(freqs, Gr_decoupled(:,1), 'DisplayName', 'SVD');
plot(freqs, Gr_jacobian(:,1), 'DisplayName', 'Jacobian');
for in_i = 2:6
for in_i = 2:3
set(gca,'ColorOrderIndex',1)
plot(freqs, Gr_coupled(:,in_i), 'HandleVisibility', 'off');
set(gca,'ColorOrderIndex',2)
@@ -284,25 +306,99 @@ end
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
hold off;
xlabel('Frequency (Hz)'); ylabel('Gershgorin Radii')
legend('location', 'northwest');
ylim([1e-3, 1e3]);
legend('location', 'southwest');
ylim([1e-4, 1e2]);
% TODO Obtained Decoupled Plants
% Verification of the decoupling using the "Relative Gain Array"
% <<sec:gravimeter_rga>>
% The relative gain array (RGA) is defined as:
% \begin{equation}
% \Lambda\big(G(s)\big) = G(s) \times \big( G(s)^{-1} \big)^T
% \end{equation}
% where $\times$ denotes an element by element multiplication and $G(s)$ is an $n \times n$ square transfer matrix.
% The obtained RGA elements are shown in Figure [[fig:gravimeter_rga]].
% Relative Gain Array for the decoupled plant using SVD:
RGA_svd = zeros(length(freqs), size(Gsvd,1), size(Gsvd,2));
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(length(freqs), size(Gx,1), size(Gx,2));
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
figure;
tiledlayout(1, 2, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile;
hold on;
for i_in = 1:3
for i_out = [1:i_in-1, i_in+1:3]
plot(freqs, RGA_svd(:, i_out, i_in), '--', 'color', [0 0 0 0.2], ...
'HandleVisibility', 'off');
end
end
plot(freqs, RGA_svd(:, 1, 2), '--', 'color', [0 0 0 0.2], ...
'DisplayName', '$RGA_{SVD}(i,j),\ i \neq j$');
plot(freqs, RGA_svd(:, 1, 1), 'k-', ...
'DisplayName', '$RGA_{SVD}(i,i)$');
for ch_i = 1:3
plot(freqs, RGA_svd(:, ch_i, ch_i), 'k-', ...
'HandleVisibility', 'off');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Magnitude'); xlabel('Frequency [Hz]');
legend('location', 'southwest');
ax2 = nexttile;
hold on;
for i_in = 1:3
for i_out = [1:i_in-1, i_in+1:3]
plot(freqs, RGA_x(:, i_out, i_in), '--', 'color', [0 0 0 0.2], ...
'HandleVisibility', 'off');
end
end
plot(freqs, RGA_x(:, 1, 2), '--', 'color', [0 0 0 0.2], ...
'DisplayName', '$RGA_{X}(i,j),\ i \neq j$');
plot(freqs, RGA_x(:, 1, 1), 'k-', ...
'DisplayName', '$RGA_{X}(i,i)$');
for ch_i = 1:3
plot(freqs, RGA_x(:, ch_i, ch_i), 'k-', ...
'HandleVisibility', 'off');
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
xlabel('Frequency [Hz]'); set(gca, 'YTickLabel',[]);
legend('location', 'southwest');
linkaxes([ax1,ax2],'y');
ylim([1e-5, 1e1]);
% Obtained Decoupled Plants
% <<sec:gravimeter_decoupled_plant>>
% The bode plot of the diagonal and off-diagonal elements of $G_{SVD}$ are shown in Figure [[fig:simscape_model_decoupled_plant_svd]].
% The bode plot of the diagonal and off-diagonal elements of $G_{SVD}$ are shown in Figure [[fig:gravimeter_decoupled_plant_svd]].
freqs = logspace(-1, 2, 1000);
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
% Magnitude
ax1 = nexttile([2, 1]);
hold on;
for i_in = 1:6
for i_out = [1:i_in-1, i_in+1:6]
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
@@ -310,20 +406,20 @@ end
plot(freqs, abs(squeeze(freqresp(Gsvd(1, 2), freqs, 'Hz'))), 'color', [0,0,0,0.5], ...
'DisplayName', '$G_{SVD}(i,j),\ i \neq j$');
set(gca,'ColorOrderIndex',1)
for ch_i = 1:6
for ch_i = 1:3
plot(freqs, abs(squeeze(freqresp(Gsvd(ch_i, ch_i), freqs, 'Hz'))), ...
'DisplayName', sprintf('$G_{SVD}(%i,%i)$', ch_i, ch_i));
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
legend('location', 'northwest');
ylim([1e-1, 1e5])
legend('location', 'southwest');
ylim([1e-8, 1e0])
% Phase
ax2 = nexttile;
hold on;
for ch_i = 1:6
for ch_i = 1:3
plot(freqs, 180/pi*angle(squeeze(freqresp(Gsvd(ch_i, ch_i), freqs, 'Hz'))));
end
hold off;
@@ -336,24 +432,22 @@ linkaxes([ax1,ax2],'x');
% #+name: fig:simscape_model_decoupled_plant_svd
% #+name: fig:gravimeter_decoupled_plant_svd
% #+caption: Decoupled Plant using SVD
% #+RESULTS:
% [[file:figs/simscape_model_decoupled_plant_svd.png]]
% [[file:figs/gravimeter_decoupled_plant_svd.png]]
% Similarly, the bode plots of the diagonal elements and off-diagonal elements of the decoupled plant $G_x(s)$ using the Jacobian are shown in Figure [[fig:simscape_model_decoupled_plant_jacobian]].
% Similarly, the bode plots of the diagonal elements and off-diagonal elements of the decoupled plant $G_x(s)$ using the Jacobian are shown in Figure [[fig:gravimeter_decoupled_plant_jacobian]].
freqs = logspace(-1, 2, 1000);
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
% Magnitude
ax1 = nexttile([2, 1]);
hold on;
for i_in = 1:6
for i_out = [1:i_in-1, i_in+1:6]
for i_in = 1:3
for i_out = [1:i_in-1, i_in+1:3]
plot(freqs, abs(squeeze(freqresp(Gx(i_out, i_in), freqs, 'Hz'))), 'color', [0,0,0,0.2], ...
'HandleVisibility', 'off');
end
@@ -361,41 +455,35 @@ end
plot(freqs, abs(squeeze(freqresp(Gx(1, 2), freqs, 'Hz'))), 'color', [0,0,0,0.5], ...
'DisplayName', '$G_x(i,j),\ i \neq j$');
set(gca,'ColorOrderIndex',1)
plot(freqs, abs(squeeze(freqresp(Gx('Ax', 'Fx'), freqs, 'Hz'))), 'DisplayName', '$G_x(1,1) = A_x/F_x$');
plot(freqs, abs(squeeze(freqresp(Gx('Ay', 'Fy'), freqs, 'Hz'))), 'DisplayName', '$G_x(2,2) = A_y/F_y$');
plot(freqs, abs(squeeze(freqresp(Gx('Az', 'Fz'), freqs, 'Hz'))), 'DisplayName', '$G_x(3,3) = A_z/F_z$');
plot(freqs, abs(squeeze(freqresp(Gx('Arx', 'Mx'), freqs, 'Hz'))), 'DisplayName', '$G_x(4,4) = A_{R_x}/M_x$');
plot(freqs, abs(squeeze(freqresp(Gx('Ary', 'My'), freqs, 'Hz'))), 'DisplayName', '$G_x(5,5) = A_{R_y}/M_y$');
plot(freqs, abs(squeeze(freqresp(Gx('Arz', 'Mz'), freqs, 'Hz'))), 'DisplayName', '$G_x(6,6) = A_{R_z}/M_z$');
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', '$G_x(3,3) = R_y/M_y$');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('Magnitude'); set(gca, 'XTickLabel',[]);
legend('location', 'northwest');
ylim([1e-2, 2e6])
legend('location', 'southwest');
ylim([1e-8, 1e0])
% Phase
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Ax', 'Fx'), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Ay', 'Fy'), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Az', 'Fz'), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Arx', 'Mx'), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Ary', 'My'), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx('Arz', 'Mz'), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx(1, 1), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx(2, 2), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(Gx(3, 3), freqs, 'Hz'))));
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
ylabel('Phase [deg]'); xlabel('Frequency [Hz]');
ylim([0, 180]);
ylim([-180, 180]);
yticks([0:45:360]);
linkaxes([ax1,ax2],'x');
% #+name: fig:svd_control
% #+name: fig:svd_control_gravimeter
% #+caption: Control Diagram for the SVD control
% #+RESULTS:
% [[file:figs/svd_control.png]]
% [[file:figs/svd_control_gravimeter.png]]
% We choose the controller to be a low pass filter:
@@ -404,24 +492,23 @@ linkaxes([ax1,ax2],'x');
% $G_0$ is tuned such that the crossover frequency corresponding to the diagonal terms of the loop gain is equal to $\omega_c$
wc = 2*pi*80; % Crossover Frequency [rad/s]
wc = 2*pi*10; % Crossover Frequency [rad/s]
w0 = 2*pi*0.1; % Controller Pole [rad/s]
K_cen = diag(1./diag(abs(evalfr(Gx, j*wc))))*(1/abs(evalfr(1/(1 + s/w0), j*wc)))/(1 + s/w0);
L_cen = K_cen*Gx;
G_cen = feedback(G, pinv(J')*K_cen, [7:12], [1:6]);
G_cen = feedback(G, pinv(Jt')*K_cen*pinv(Ja));
K_svd = diag(1./diag(abs(evalfr(Gsvd, j*wc))))*(1/abs(evalfr(1/(1 + s/w0), j*wc)))/(1 + s/w0);
L_svd = K_svd*Gsvd;
G_svd = feedback(G, inv(V')*K_svd*inv(U), [7:12], [1:6]);
U_inv = inv(U);
G_svd = feedback(G, inv(V')*K_svd*U_inv(1:3, :));
% The obtained diagonal elements of the loop gains are shown in Figure [[fig:gravimeter_comp_loop_gain_diagonal]].
freqs = logspace(-1, 2, 1000);
figure;
tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
@@ -429,7 +516,7 @@ tiledlayout(3, 1, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile([2, 1]);
hold on;
plot(freqs, abs(squeeze(freqresp(L_svd(1, 1), freqs, 'Hz'))), 'DisplayName', '$L_{SVD}(i,i)$');
for i_in_out = 2:6
for i_in_out = 2:3
set(gca,'ColorOrderIndex',1)
plot(freqs, abs(squeeze(freqresp(L_svd(i_in_out, i_in_out), freqs, 'Hz'))), 'HandleVisibility', 'off');
end
@@ -437,7 +524,7 @@ end
set(gca,'ColorOrderIndex',2)
plot(freqs, abs(squeeze(freqresp(L_cen(1, 1), freqs, 'Hz'))), ...
'DisplayName', '$L_{J}(i,i)$');
for i_in_out = 2:6
for i_in_out = 2:3
set(gca,'ColorOrderIndex',2)
plot(freqs, abs(squeeze(freqresp(L_cen(i_in_out, i_in_out), freqs, 'Hz'))), 'HandleVisibility', 'off');
end
@@ -450,12 +537,12 @@ ylim([5e-2, 2e3])
% Phase
ax2 = nexttile;
hold on;
for i_in_out = 1:6
for i_in_out = 1:3
set(gca,'ColorOrderIndex',1)
plot(freqs, 180/pi*angle(squeeze(freqresp(L_svd(i_in_out, i_in_out), freqs, 'Hz'))));
end
set(gca,'ColorOrderIndex',2)
for i_in_out = 1:6
for i_in_out = 1:3
set(gca,'ColorOrderIndex',2)
plot(freqs, 180/pi*angle(squeeze(freqresp(L_cen(i_in_out, i_in_out), freqs, 'Hz'))));
end
@@ -467,7 +554,7 @@ yticks([-180:90:360]);
linkaxes([ax1,ax2],'x');
% TODO Closed-Loop system Performances
% Closed-Loop system Performances
% <<sec:gravimeter_closed_loop_results>>
% Let's first verify the stability of the closed-loop systems:
@@ -497,53 +584,39 @@ isstable(G_svd)
freqs = logspace(-2, 2, 1000);
figure;
tiledlayout(2, 2, 'TileSpacing', 'None', 'Padding', 'None');
tiledlayout(1, 3, 'TileSpacing', 'None', 'Padding', 'None');
ax1 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'Ax', 'Dwx')/s^2, freqs, 'Hz'))), 'DisplayName', 'Open-Loop');
plot(freqs, abs(squeeze(freqresp(G_cen('Ax', 'Dwx')/s^2, freqs, 'Hz'))), 'DisplayName', 'Centralized');
plot(freqs, abs(squeeze(freqresp(G_svd('Ax', 'Dwx')/s^2, freqs, 'Hz'))), '--', 'DisplayName', 'SVD');
set(gca,'ColorOrderIndex',1)
plot(freqs, abs(squeeze(freqresp(G( 'Ay', 'Dwy')/s^2, freqs, 'Hz'))), 'HandleVisibility', 'off');
plot(freqs, abs(squeeze(freqresp(G_cen('Ay', 'Dwy')/s^2, freqs, 'Hz'))), 'HandleVisibility', 'off');
plot(freqs, abs(squeeze(freqresp(G_svd('Ay', 'Dwy')/s^2, freqs, 'Hz'))), '--', 'HandleVisibility', 'off');
plot(freqs, abs(squeeze(freqresp(G( 1,1)/s^2, freqs, 'Hz'))), 'DisplayName', 'Open-Loop');
plot(freqs, abs(squeeze(freqresp(G_cen(1,1)/s^2, freqs, 'Hz'))), 'DisplayName', 'Centralized');
plot(freqs, abs(squeeze(freqresp(G_svd(1,1)/s^2, freqs, 'Hz'))), '--', 'DisplayName', 'SVD');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('$D_x/D_{w,x}$, $D_y/D_{w, y}$'); set(gca, 'XTickLabel',[]);
ylabel('Transmissibility'); xlabel('Frequency [Hz]');
title('$D_x/D_{w,x}$');
legend('location', 'southwest');
ax2 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'Az', 'Dwz')/s^2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G_cen('Az', 'Dwz')/s^2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G_svd('Az', 'Dwz')/s^2, freqs, 'Hz'))), '--');
plot(freqs, abs(squeeze(freqresp(G( 2,2)/s^2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G_cen(2,2)/s^2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G_svd(2,2)/s^2, freqs, 'Hz'))), '--');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('$D_z/D_{w,z}$'); set(gca, 'XTickLabel',[]);
set(gca, 'YTickLabel',[]); xlabel('Frequency [Hz]');
title('$D_z/D_{w,z}$');
ax3 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'Arx', 'Rwx')/s^2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G_cen('Arx', 'Rwx')/s^2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G_svd('Arx', 'Rwx')/s^2, freqs, 'Hz'))), '--');
set(gca,'ColorOrderIndex',1)
plot(freqs, abs(squeeze(freqresp(G( 'Ary', 'Rwy')/s^2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G_cen('Ary', 'Rwy')/s^2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G_svd('Ary', 'Rwy')/s^2, freqs, 'Hz'))), '--');
plot(freqs, abs(squeeze(freqresp(G( 3,3)/s^2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G_cen(3,3)/s^2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G_svd(3,3)/s^2, freqs, 'Hz'))), '--');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('$R_x/R_{w,x}$, $R_y/R_{w,y}$'); xlabel('Frequency [Hz]');
set(gca, 'YTickLabel',[]); xlabel('Frequency [Hz]');
title('$R_y/R_{w,y}$');
ax4 = nexttile;
hold on;
plot(freqs, abs(squeeze(freqresp(G( 'Arz', 'Rwz')/s^2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G_cen('Arz', 'Rwz')/s^2, freqs, 'Hz'))));
plot(freqs, abs(squeeze(freqresp(G_svd('Arz', 'Rwz')/s^2, freqs, 'Hz'))), '--');
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
ylabel('$R_z/R_{w,z}$'); xlabel('Frequency [Hz]');
linkaxes([ax1,ax2,ax3,ax4],'xy');
linkaxes([ax1,ax2,ax3],'xy');
xlim([freqs(1), freqs(end)]);
ylim([1e-3, 1e2]);
xlim([1e-2, 5e1]); ylim([1e-7, 1e-2]);