diff --git a/dcm-simscape.html b/dcm-simscape.html index 834116d..b3c5612 100644 --- a/dcm-simscape.html +++ b/dcm-simscape.html @@ -3,7 +3,7 @@ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> - + DCM - Dynamical Multi-Body Model @@ -39,39 +39,41 @@

Table of Contents

@@ -87,22 +89,22 @@ In this document, a Simscape (.e.g. multi-body) model of the ESRF Double Crystal It is structured as follow:

-
-

1. System Kinematics

+
+

1. System Kinematics

- +

-
-

1.1. Bragg Angle

+
+

1.1. Bragg Angle

%% Tested bragg angles
@@ -118,7 +120,7 @@ dz = d_off./(2*
 
 
-

+

 
jack_motion_bragg_angle.png
 

 
Figure 1: Jack motion as a function of Bragg angle

@@ -136,34 +138,34 @@ dz = d_off./(2*
 

 
-

-
1.2. Kinematics (111 Crystal)

+

+
1.2. Kinematics (111 Crystal)

 

 

 The reference frame is taken at the center of the 111 second crystal.
 

 

 
-

-
1.2.1. Interferometers - 111 Crystal

+

+
1.2.1. Interferometers - 111 Crystal

 

 

 Three interferometers are pointed to the bottom surface of the 111 crystal.
 

 
 

-The position of the measurement points are shown in Figure 2 as well as the origin where the motion of the crystal is computed.
+The position of the measurement points are shown in Figure 2 as well as the origin where the motion of the crystal is computed.
 

 
 
-

+

 
sensor_111_crystal_points.png
 

 
Figure 2: Bottom view of the second crystal 111. Position of the measurement points.

 

 
 

-The inverse kinematics consisting of deriving the interferometer measurements from the motion of the crystal (see Figure 3):
+The inverse kinematics consisting of deriving the interferometer measurements from the motion of the crystal (see Figure 3):
 

 \begin{equation}
 \begin{bmatrix}
@@ -177,7 +179,7 @@ d_z \\ r_y \\ r_x
 \end{equation}
 
 
-

+

 
schematic_sensor_jacobian_inverse_kinematics.png
 

 
Figure 3: Inverse Kinematics - Interferometers

@@ -185,7 +187,7 @@ d_z \\ r_y \\ r_x
 
 
 

-From the Figure 2, the inverse kinematics can be solved as follow (for small motion):
+From the Figure 2, the inverse kinematics can be solved as follow (for small motion):
 

 \begin{equation}
 \bm{J}_{s,111}
@@ -205,7 +207,7 @@ J_s_111 = [1,  0.07, -0.015
- +
@@ -237,7 +239,7 @@ J_s_111 = [1, 0.07, -0.015
Table 1: Sensor Jacobian \(\bm{J}_{s,111}\)

-The forward kinematics is solved by inverting the Jacobian matrix (see Figure 4). +The forward kinematics is solved by inverting the Jacobian matrix (see Figure 4).

\begin{equation} \begin{bmatrix} @@ -251,13 +253,13 @@ x_1 \\ x_2 \\ x_3 \end{equation} -
+

schematic_sensor_jacobian_forward_kinematics.png

Figure 4: Forward Kinematics - Interferometers

- +
@@ -290,15 +292,15 @@ x_1 \\ x_2 \\ x_3 -
-

1.2.2. Piezo - 111 Crystal

+
+

1.2.2. Piezo - 111 Crystal

-The location of the actuators with respect with the center of the 111 second crystal are shown in Figure 5. +The location of the actuators with respect with the center of the 111 second crystal are shown in Figure 5.

-
+

actuator_jacobian_111_points.png

Figure 5: Location of actuators with respect to the center of the 111 second crystal (bottom view)

@@ -319,14 +321,14 @@ d_z \\ r_y \\ r_x \end{equation} -
+

schematic_actuator_jacobian_inverse_kinematics.png

Figure 6: Inverse Kinematics - Actuators

-Based on the geometry in Figure 5, we obtain: +Based on the geometry in Figure 5, we obtain:

\begin{equation} \bm{J}_{a,111} @@ -346,7 +348,7 @@ J_a_111 = [1, 0.14, -0.1525
-
Table 2: Inverse of the sensor Jacobian \(\bm{J}_{s,111}^{-1}\)
+
@@ -392,13 +394,13 @@ d_{u_r} \\ d_{u_h} \\ d_{d} \end{equation} -
+

schematic_actuator_jacobian_forward_kinematics.png

Figure 7: Forward Kinematics - Actuators for 111 crystal

-
Table 3: Actuator Jacobian \(\bm{J}_{a,111}\)
+
@@ -432,8 +434,8 @@ d_{u_r} \\ d_{u_h} \\ d_{d} -
-

1.3. Save Kinematics

+
+

1.3. Save Kinematics

save('mat/dcm_kinematics.mat', 'J_a_111', 'J_s_111')
@@ -443,15 +445,15 @@ d_{u_r} \\ d_{u_h} \\ d_{d}
-
-

2. Open Loop System Identification

+
+

2. Open Loop System Identification

- +

-
-

2.1. Identification

+
+

2.1. Identification

Let’s considered the system \(\bm{G}(s)\) with: @@ -462,11 +464,11 @@ Let’s considered the system \(\bm{G}(s)\) with:

-It is schematically shown in Figure 8. +It is schematically shown in Figure 8.

-
+

schematic_system_inputs_outputs.png

Figure 8: Dynamical system with inputs and outputs

@@ -508,8 +510,8 @@ State-space model with 3 outputs, 3 inputs, and 24 states.
-
-

2.2. Plant in the frame of the fastjacks

+
+

2.2. Plant in the frame of the fastjacks

load('mat/dcm_kinematics.mat');
@@ -517,11 +519,11 @@ State-space model with 3 outputs, 3 inputs, and 24 states.

-Using the forward and inverse kinematics, we can computed the dynamics from piezo forces to axial motion of the 3 fastjacks (see Figure 9). +Using the forward and inverse kinematics, we can computed the dynamics from piezo forces to axial motion of the 3 fastjacks (see Figure 9).

-
+

schematic_jacobian_frame_fastjack.png

Figure 9: Use of Jacobian matrices to obtain the system in the frame of the fastjacks

@@ -542,7 +544,7 @@ The DC gain of the new system shows that the system is well decoupled at low fre
-
Table 4: Inverse of the actuator Jacobian \(\bm{J}_{a,111}^{-1}\)
+
@@ -574,17 +576,17 @@ The DC gain of the new system shows that the system is well decoupled at low fre
Table 5: DC gain of the plant in the frame of the fast jacks \(\bm{G}_{\text{fj}}\)

-The bode plot of \(\bm{G}_{\text{fj}}(s)\) is shown in Figure 10. +The bode plot of \(\bm{G}_{\text{fj}}(s)\) is shown in Figure 10.

-
+

bode_plot_plant_fj.png

Figure 10: Bode plot of the diagonal and off-diagonal elements of the plant in the frame of the fast jacks

-
+

Computing the system in the frame of the fastjack gives good decoupling at low frequency (until the first resonance of the system).

@@ -593,11 +595,11 @@ Computing the system in the frame of the fastjack gives good decoupling at low f
-
-

2.3. Plant in the frame of the crystal

+
+

2.3. Plant in the frame of the crystal

-
+

schematic_jacobian_frame_crystal.png

Figure 11: Use of Jacobian matrices to obtain the system in the frame of the crystal

@@ -652,18 +654,18 @@ The main reason is that, as we map forces to the center of the 111 crystal and n
-
-

3. Active Damping Plant (Strain gauges)

+
+

3. Active Damping Plant (Strain gauges)

- +

In this section, we wish to see whether if strain gauges fixed to the piezoelectric actuator can be used for active damping.

-
-

3.1. Identification

+
+

3.1. Identification

%% Input/Output definition
@@ -671,13 +673,11 @@ clear io; io_i = 1;
 
 %% Inputs
 % Control Input {3x1} [N]
-io(io_i) = linio([mdl, '/u'], 1, 'openinput');  io_i = io_i + 1;
-% % Stepper Displacement {3x1} [m]
-% io(io_i) = linio([mdl, '/d'], 1, 'openinput');  io_i = io_i + 1;
+io(io_i) = linio([mdl, '/control_system'], 1, 'openinput');  io_i = io_i + 1;
 
 %% Outputs
 % Strain Gauges {3x1} [m]
-io(io_i) = linio([mdl, '/sg'], 1, 'openoutput'); io_i = io_i + 1;
+io(io_i) = linio([mdl, '/DCM'], 2, 'openoutput'); io_i = io_i + 1;
@@ -704,33 +704,128 @@ G_sg = linearize(mdl, io); --1.4113e-13 +4.4443e-09 1.0339e-13 3.774e-14 1.0339e-13 --1.4113e-13 +4.4443e-09 3.774e-14 3.7792e-14 3.7792e-14 --7.5585e-14 +4.4444e-09 + + +
+

strain_gauge_plant_bode_plot.png +

+

Figure 12: Bode Plot of the transfer functions from piezoelectric forces to strain gauges measuremed displacements

+
+ +
+

+As the distance between the poles and zeros in Figure 15 is very small, little damping can be actively added using the strain gauges. +This will be confirmed using a Root Locus plot. +

+
-
-

4. Active Damping Plant (Force Sensors)

+
+

3.2. Relative Active Damping

+
+
+
Krad_g1 = eye(3)*s/(s^2/(2*pi*500)^2 + 2*s/(2*pi*500) + 1);
+
+
+ +

+As can be seen in Figure 13, very little damping can be added using relative damping strategy using strain gauges. +

+ + +
+

relative_damping_root_locus.png +

+

Figure 13: Root Locus for the relative damping control

+
+ +
+
Krad = -g*Krad_g1;
+
+
+
+
+ +
+

3.3. Damped Plant

+
+

+The controller is implemented on Simscape, and the damped plant is identified. +

+ +
+
%% Input/Output definition
+clear io; io_i = 1;
+
+%% Inputs
+% Control Input {3x1} [N]
+io(io_i) = linio([mdl, '/control_system'], 1, 'input');  io_i = io_i + 1;
+
+%% Outputs
+% Force Sensor {3x1} [m]
+io(io_i) = linio([mdl, '/DCM'], 1, 'openoutput'); io_i = io_i + 1;
+
+
+ +
+
%% DCM Kinematics
+load('mat/dcm_kinematics.mat');
+
+
+ +
+
%% Identification of the Open Loop plant
+controller.type = 0; % Open Loop
+G_ol = J_a_111*inv(J_s_111)*linearize(mdl, io);
+G_ol.InputName  = {'u_ur',  'u_uh',  'u_d'};
+G_ol.OutputName = {'d_ur',  'd_uh',  'd_d'};
+
+
+ +
+
%% Identification of the damped plant with Relative Active Damping
+controller.type = 2; % RAD
+G_dp = J_a_111*inv(J_s_111)*linearize(mdl, io);
+G_dp.InputName  = {'u_ur',  'u_uh',  'u_d'};
+G_dp.OutputName = {'d_ur',  'd_uh',  'd_d'};
+
+
+ + +
+

comp_damp_undamped_plant_rad_bode_plot.png +

+

Figure 14: Bode plot of both the open-loop plant and the damped plant using relative active damping

+
+
+
+
+ +
+

4. Active Damping Plant (Force Sensors)

- +

Force sensors are added above the piezoelectric actuators. @@ -738,8 +833,8 @@ They can consists of a simple piezoelectric ceramic stack. See for instance fleming10_integ_strain_force_feedb_high.

-
-

4.1. Identification

+
+

4.1. Identification

%% Input/Output definition
@@ -798,16 +893,16 @@ G_fs = linearize(mdl, io);
 
 
 
-

+

 
iff_plant_bode_plot.png
 

-
Figure 12: Bode plot of IFF Plant

+
Figure 15: Bode plot of IFF Plant
-
-

4.2. Controller - Root Locus

+
+

4.2. Controller - Root Locus

Kiff_g1 = eye(3)*1/(1 + s/2/pi/20);
@@ -815,10 +910,10 @@ G_fs = linearize(mdl, io);
-
+

iff_root_locus.png

-

Figure 13: Root Locus plot for the IFF Control strategy

+

Figure 16: Root Locus plot for the IFF Control strategy

@@ -829,8 +924,8 @@ Kiff = g*Kiff_g1;
-
-

4.3. Damped Plant

+
+

4.3. Damped Plant

%% Input/Output definition
@@ -871,13 +966,13 @@ G_dp.OutputName = {'d_ur',  
+

 
comp_damped_undamped_plant_iff_bode_plot.png
 

-
Figure 14: Bode plot of both the open-loop plant and the damped plant using IFF

+
Figure 17: Bode plot of both the open-loop plant and the damped plant using IFF

 

 
-

+

 

 The Integral Force Feedback control strategy is very effective in damping the suspension modes of the DCM.
 

@@ -886,8 +981,8 @@ The Integral Force Feedback control strategy is very effective in damping the su
 

 

 
-

-
4.4. Save

+

+
4.4. Save

 

 

 
save('mat/Kiff.mat', 'Kiff');
@@ -897,18 +992,18 @@ The Integral Force Feedback control strategy is very effective in damping the su
 

 

 
-

-
5. HAC-LAC (IFF) architecture

+

+
5. HAC-LAC (IFF) architecture

 

 

-
+
 

 

 

 

 

 
Author: Dehaeze Thomas

-
Created: 2021-11-30 mar. 11:26

+
Created: 2021-11-30 mar. 11:44

 

 
 
diff --git a/dcm-simscape.org b/dcm-simscape.org
index 88720e2..57b6f9c 100644
--- a/dcm-simscape.org
+++ b/dcm-simscape.org
@@ -738,13 +738,11 @@ clear io; io_i = 1;
 
 %% Inputs
 % Control Input {3x1} [N]
-io(io_i) = linio([mdl, '/u'], 1, 'openinput');  io_i = io_i + 1;
-% % Stepper Displacement {3x1} [m]
-% io(io_i) = linio([mdl, '/d'], 1, 'openinput');  io_i = io_i + 1;
+io(io_i) = linio([mdl, '/control_system'], 1, 'openinput');  io_i = io_i + 1;
 
 %% Outputs
 % Strain Gauges {3x1} [m]
-io(io_i) = linio([mdl, '/sg'], 1, 'openoutput'); io_i = io_i + 1;
+io(io_i) = linio([mdl, '/DCM'], 2, 'openoutput'); io_i = io_i + 1;
 #+end_src
 
 #+begin_src matlab
@@ -762,12 +760,12 @@ dcgain(G_sg)
 #+end_src
 
 #+RESULTS:
-| -1.4113e-13 |  1.0339e-13 |   3.774e-14 |
-|  1.0339e-13 | -1.4113e-13 |   3.774e-14 |
-|  3.7792e-14 |  3.7792e-14 | -7.5585e-14 |
+| 4.4443e-09 | 1.0339e-13 |  3.774e-14 |
+| 1.0339e-13 | 4.4443e-09 |  3.774e-14 |
+| 3.7792e-14 | 3.7792e-14 | 4.4444e-09 |
 
 #+begin_src matlab :exports none
-%% Bode plot for the plant
+%% Bode plot for the plant (strain gauge output)
 figure;
 tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
 
@@ -788,7 +786,8 @@ end
 hold off;
 set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
 ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
-legend('location', 'southwest', 'FontSize', 8, 'NumColumns', 2);
+legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2);
+ylim([1e-14, 1e-7]);
 
 ax2 = nexttile;
 hold on;
@@ -800,12 +799,173 @@ set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
 xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
 hold off;
 yticks(-360:90:360);
-ylim([-180, 180]);
+ylim([-180, 0]);
 
 linkaxes([ax1,ax2],'x');
 xlim([freqs(1), freqs(end)]);
 #+end_src
 
+#+begin_src matlab :tangle no :exports results :results file replace
+exportFig('figs/strain_gauge_plant_bode_plot.pdf', 'width', 'wide', 'height', 'tall');
+#+end_src
+
+#+name: fig:strain_gauge_plant_bode_plot
+#+caption: Bode Plot of the transfer functions from piezoelectric forces to strain gauges measuremed displacements
+#+RESULTS:
+[[file:figs/strain_gauge_plant_bode_plot.png]]
+
+#+begin_important
+As the distance between the poles and zeros in Figure [[fig:iff_plant_bode_plot]] is very small, little damping can be actively added using the strain gauges.
+This will be confirmed using a Root Locus plot.
+#+end_important
+
+** Relative Active Damping
+#+begin_src matlab
+Krad_g1 = eye(3)*s/(s^2/(2*pi*500)^2 + 2*s/(2*pi*500) + 1);
+#+end_src
+
+As can be seen in Figure [[fig:relative_damping_root_locus]], very little damping can be added using relative damping strategy using strain gauges.
+
+#+begin_src matlab :exports none
+%% Root Locus for IFF
+gains = logspace(3, 8, 200);
+
+figure;
+
+hold on;
+plot(real(pole(G_sg)),  imag(pole(G_sg)),  'x', 'color', colors(1,:), ...
+    'DisplayName', '$g = 0$');
+plot(real(tzero(G_sg)), imag(tzero(G_sg)), 'o', 'color', colors(1,:), ...
+    'HandleVisibility', 'off');
+
+for g = gains
+    clpoles = pole(feedback(G_sg, g*Krad_g1, -1));
+    plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:), ...
+        'HandleVisibility', 'off');
+end
+
+% Optimal gain
+g = 2e5;
+clpoles = pole(feedback(G_sg, g*Krad_g1, -1));
+plot(real(clpoles), imag(clpoles), 'x', 'color', colors(2,:), ...
+    'DisplayName', sprintf('$g=%.0e$', g));
+hold off;
+xlim([-6, 0]); ylim([0, 2700]);
+xlabel('Real Part'); ylabel('Imaginary Part');
+legend('location', 'northwest');
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+exportFig('figs/relative_damping_root_locus.pdf', 'width', 'wide', 'height', 'tall');
+#+end_src
+
+#+name: fig:relative_damping_root_locus
+#+caption: Root Locus for the relative damping control
+#+RESULTS:
+[[file:figs/relative_damping_root_locus.png]]
+
+#+begin_src matlab
+Krad = -g*Krad_g1;
+#+end_src
+
+** Damped Plant
+The controller is implemented on Simscape, and the damped plant is identified.
+
+#+begin_src matlab
+%% Input/Output definition
+clear io; io_i = 1;
+
+%% Inputs
+% Control Input {3x1} [N]
+io(io_i) = linio([mdl, '/control_system'], 1, 'input');  io_i = io_i + 1;
+
+%% Outputs
+% Force Sensor {3x1} [m]
+io(io_i) = linio([mdl, '/DCM'], 1, 'openoutput'); io_i = io_i + 1;
+#+end_src
+
+#+begin_src matlab
+%% DCM Kinematics
+load('mat/dcm_kinematics.mat');
+#+end_src
+
+#+begin_src matlab
+%% Identification of the Open Loop plant
+controller.type = 0; % Open Loop
+G_ol = J_a_111*inv(J_s_111)*linearize(mdl, io);
+G_ol.InputName  = {'u_ur',  'u_uh',  'u_d'};
+G_ol.OutputName = {'d_ur',  'd_uh',  'd_d'};
+#+end_src
+
+#+begin_src matlab
+%% Identification of the damped plant with Relative Active Damping
+controller.type = 2; % RAD
+G_dp = J_a_111*inv(J_s_111)*linearize(mdl, io);
+G_dp.InputName  = {'u_ur',  'u_uh',  'u_d'};
+G_dp.OutputName = {'d_ur',  'd_uh',  'd_d'};
+#+end_src
+
+#+begin_src matlab :exports none
+%% Comparison of the damped and undamped plant
+figure;
+tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
+
+ax1 = nexttile([2,1]);
+hold on;
+plot(freqs, abs(squeeze(freqresp(G_ol(1,1), freqs, 'Hz'))), ...
+     'DisplayName', 'd - OL');
+plot(freqs, abs(squeeze(freqresp(G_ol(2,2), freqs, 'Hz'))), ...
+     'DisplayName', 'uh - OL');
+plot(freqs, abs(squeeze(freqresp(G_ol(3,3), freqs, 'Hz'))), ...
+     'DisplayName', 'ur - OL');
+set(gca,'ColorOrderIndex',1)
+plot(freqs, abs(squeeze(freqresp(G_dp(1,1), freqs, 'Hz'))), '--', ...
+     'DisplayName', 'd - IFF');
+plot(freqs, abs(squeeze(freqresp(G_dp(2,2), freqs, 'Hz'))), '--', ...
+     'DisplayName', 'uh - IFF');
+plot(freqs, abs(squeeze(freqresp(G_dp(3,3), freqs, 'Hz'))), '--', ...
+     'DisplayName', 'ur - IFF');
+for i = 1:2
+    for j = i+1:3
+        plot(freqs, abs(squeeze(freqresp(G_dp(i,j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
+             'HandleVisibility', 'off');
+    end
+end
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
+ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]);
+legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2);
+ylim([1e-12, 1e-6]);
+
+ax2 = nexttile;
+hold on;
+plot(freqs, 180/pi*angle(squeeze(freqresp(G_ol(1,1), freqs, 'Hz'))));
+plot(freqs, 180/pi*angle(squeeze(freqresp(G_ol(2,2), freqs, 'Hz'))));
+plot(freqs, 180/pi*angle(squeeze(freqresp(G_ol(3,3), freqs, 'Hz'))));
+set(gca,'ColorOrderIndex',1)
+plot(freqs, 180/pi*angle(squeeze(freqresp(G_dp(1,1), freqs, 'Hz'))), '--');
+plot(freqs, 180/pi*angle(squeeze(freqresp(G_dp(2,2), freqs, 'Hz'))), '--');
+plot(freqs, 180/pi*angle(squeeze(freqresp(G_dp(3,3), freqs, 'Hz'))), '--');
+hold off;
+set(gca, 'XScale', 'log'); set(gca, 'YScale', 'lin');
+xlabel('Frequency [Hz]'); ylabel('Phase [deg]');
+hold off;
+yticks(-360:90:360);
+ylim([-180, 0]);
+
+linkaxes([ax1,ax2],'x');
+xlim([freqs(1), freqs(end)]);
+#+end_src
+
+#+begin_src matlab :tangle no :exports results :results file replace
+exportFig('figs/comp_damp_undamped_plant_rad_bode_plot.pdf', 'width', 'wide', 'height', 'tall');
+#+end_src
+
+#+name: fig:comp_damp_undamped_plant_rad_bode_plot
+#+caption: Bode plot of both the open-loop plant and the damped plant using relative active damping
+#+RESULTS:
+[[file:figs/comp_damp_undamped_plant_rad_bode_plot.png]]
+
 * Active Damping Plant (Force Sensors)
 :PROPERTIES:
 :header-args:matlab+: :tangle matlab/dcm_active_damping_iff.m
diff --git a/dcm-simscape.pdf b/dcm-simscape.pdf
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