From c56ce7e3c8b9f3ece1df6fe6d6673113906f283b Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Tue, 30 Nov 2021 15:17:23 +0100 Subject: [PATCH] Add HAC-IFF strategy and analysis --- dcm-simscape.html | 463 +++--- dcm-simscape.org | 444 +++++- dcm-simscape.pdf | Bin 1104973 -> 1498370 bytes figs/bode_plot_hac_iff_plant.pdf | Bin 0 -> 105870 bytes figs/bode_plot_hac_iff_plant.png | Bin 0 -> 105745 bytes figs/bode_plot_iff_plant.pdf | 1288 +++++++++++++++++ figs/bode_plot_iff_plant.png | Bin 0 -> 102872 bytes figs/hac_iff_loop_gain_bode_plot.pdf | Bin 0 -> 89693 bytes figs/hac_iff_loop_gain_bode_plot.png | Bin 0 -> 98098 bytes figs/iff_plant_bode_plot.pdf | Bin 101951 -> 101718 bytes figs/iff_plant_bode_plot.png | Bin 119627 -> 118991 bytes figs/loci_hac_iff_fast_jack.pdf | Bin 0 -> 76713 bytes figs/loci_hac_iff_fast_jack.png | Bin 0 -> 34693 bytes ...ematic_jacobian_frame_fastjack_hac_iff.pdf | Bin 0 -> 97705 bytes ...ematic_jacobian_frame_fastjack_hac_iff.png | Bin 0 -> 26281 bytes ...ematic_jacobian_frame_fastjack_hac_iff.svg | 696 +++++++++ .../schematic_jacobian_frame_fastjack_iff.pdf | Bin 0 -> 72099 bytes .../schematic_jacobian_frame_fastjack_iff.png | Bin 0 -> 23598 bytes .../schematic_jacobian_frame_fastjack_iff.svg | 459 ++++++ ...epper_transmissibility_comp_ol_hac_iff.pdf | Bin 0 -> 79161 bytes ...epper_transmissibility_comp_ol_hac_iff.png | Bin 0 -> 115267 bytes matlab/simscape_dcm.slx | Bin 56339 -> 57776 bytes 22 files changed, 3129 insertions(+), 221 deletions(-) create mode 100644 figs/bode_plot_hac_iff_plant.pdf create mode 100644 figs/bode_plot_hac_iff_plant.png create mode 100644 figs/bode_plot_iff_plant.pdf create mode 100644 figs/bode_plot_iff_plant.png create mode 100644 figs/hac_iff_loop_gain_bode_plot.pdf create mode 100644 figs/hac_iff_loop_gain_bode_plot.png create mode 100644 figs/loci_hac_iff_fast_jack.pdf create mode 100644 figs/loci_hac_iff_fast_jack.png create mode 100644 figs/schematic_jacobian_frame_fastjack_hac_iff.pdf create mode 100644 figs/schematic_jacobian_frame_fastjack_hac_iff.png create mode 100644 figs/schematic_jacobian_frame_fastjack_hac_iff.svg create mode 100644 figs/schematic_jacobian_frame_fastjack_iff.pdf create mode 100644 figs/schematic_jacobian_frame_fastjack_iff.png create mode 100644 figs/schematic_jacobian_frame_fastjack_iff.svg create mode 100644 figs/stepper_transmissibility_comp_ol_hac_iff.pdf create mode 100644 figs/stepper_transmissibility_comp_ol_hac_iff.png diff --git a/dcm-simscape.html b/dcm-simscape.html index b3c5612..a999d8f 100644 --- a/dcm-simscape.html +++ b/dcm-simscape.html @@ -3,9 +3,9 @@ "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> - + -DCM - Dynamical Multi-Body Model +ESRF Double Crystal Monochromator - Dynamical Multi-Body Model @@ -34,46 +34,51 @@ | HOME
-

DCM - Dynamical Multi-Body Model

+

ESRF Double Crystal Monochromator - Dynamical Multi-Body Model

@@ -89,22 +94,23 @@ 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
@@ -120,7 +126,7 @@ dz = d_off./(2*
 
 
-
+

jack_motion_bragg_angle.png

Figure 1: Jack motion as a function of Bragg angle

@@ -138,34 +144,34 @@ 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} @@ -179,7 +185,7 @@ d_z \\ r_y \\ r_x \end{equation} -
+

schematic_sensor_jacobian_inverse_kinematics.png

Figure 3: Inverse Kinematics - Interferometers

@@ -187,7 +193,7 @@ 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} @@ -207,7 +213,7 @@ J_s_111 = [1, 0.07, -0.015
- +
@@ -239,7 +245,7 @@ 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} @@ -253,13 +259,13 @@ x_1 \\ x_2 \\ x_3 \end{equation} -
+

schematic_sensor_jacobian_forward_kinematics.png

Figure 4: Forward Kinematics - Interferometers

- +
@@ -292,15 +298,15 @@ 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)

@@ -321,14 +327,14 @@ 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} @@ -348,7 +354,7 @@ J_a_111 = [1, 0.14, -0.1525
-
Table 2: Inverse of the sensor Jacobian \(\bm{J}_{s,111}^{-1}\)
+
@@ -394,13 +400,13 @@ 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 \(\bm{J}_{a,111}\)
+
@@ -434,8 +440,8 @@ 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')
@@ -445,15 +451,15 @@ 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: @@ -464,11 +470,11 @@ 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

@@ -510,20 +516,20 @@ 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');
+
load('dcm_kinematics.mat');
 

-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

@@ -544,7 +550,7 @@ The DC gain of the new system shows that the system is well decoupled at low fre
-
Table 4: Inverse of the actuator Jacobian \(\bm{J}_{a,111}^{-1}\)
+
@@ -576,17 +582,17 @@ 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).

@@ -595,11 +601,11 @@ 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

@@ -654,18 +660,18 @@ 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
@@ -724,15 +730,15 @@ G_sg = linearize(mdl, io);
 
 
 
-
+

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. +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.

@@ -740,8 +746,8 @@ This will be confirmed using a Root Locus plot.
-
-

3.2. Relative Active Damping

+
+

3.2. Relative Active Damping

Krad_g1 = eye(3)*s/(s^2/(2*pi*500)^2 + 2*s/(2*pi*500) + 1);
@@ -749,11 +755,11 @@ This will be confirmed using a Root Locus plot.
 

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

@@ -766,8 +772,8 @@ As can be seen in Figure 13, very little damping can b
-
-

3.3. Damped Plant

+
+

3.3. Damped Plant

The controller is implemented on Simscape, and the damped plant is identified. @@ -789,7 +795,7 @@ io(io_i) = linio([mdl, '/DCM'], 1,

%% DCM Kinematics
-load('mat/dcm_kinematics.mat');
+load('dcm_kinematics.mat');
 
@@ -812,7 +818,7 @@ G_dp.OutputName = {'d_ur', +

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

@@ -821,11 +827,11 @@ G_dp.OutputName = {'d_ur', -

4. Active Damping Plant (Force Sensors)

+
+

4. Active Damping Plant (Force Sensors)

- +

Force sensors are added above the piezoelectric actuators. @@ -833,8 +839,8 @@ 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
@@ -856,44 +862,13 @@ G_fs = linearize(mdl, io);
 
-
-
dcgain(G_fs)
-
-
- - +

+The Bode plot of the identified dynamics is shown in Figure 15. +At high frequency, the diagonal terms are constants while the off-diagonal terms have some roll-off. +

--- -- -- - - - - - - - - - - - - - - - - - - - -
-1.4113e-131.0339e-133.774e-14
1.0339e-13-1.4113e-133.774e-14
3.7792e-143.7792e-14-7.5585e-14
- - -
+

iff_plant_bode_plot.png

Figure 15: Bode plot of IFF Plant

@@ -901,109 +876,213 @@ G_fs = linearize(mdl, io);
-
-

4.2. Controller - Root Locus

+
+

4.2. Controller - Root Locus

+

+We want to have integral action around the resonances of the system, but we do not want to integrate at low frequency. +Therefore, we can use a low pass filter. +

+
-
Kiff_g1 = eye(3)*1/(1 + s/2/pi/20);
+
%% Integral Force Feedback Controller
+Kiff_g1 = eye(3)*1/(1 + s/2/pi/20);
 
-
+

iff_root_locus.png

Figure 16: Root Locus plot for the IFF Control strategy

-
%% Integral Force Feedback Controller
+
%% Integral Force Feedback Controller with optimal gain
 Kiff = g*Kiff_g1;
 
+ +
+
%% Save the IFF controller
+save('mat/Kiff.mat', 'Kiff');
+
+
-
-

4.3. Damped Plant

+
+

4.3. Damped Plant

-
-
%% Input/Output definition
-clear io; io_i = 1;
+

+Both the Open Loop dynamics (see Figure 9) and the dynamics with IFF (see Figure 17) are identified. +

-%% Inputs -% Control Input {3x1} [N] -io(io_i) = linio([mdl, '/control_system'], 1, 'input'); io_i = io_i + 1; +

+We are here interested in the dynamics from \(\bm{u}^\prime = [u_{u_r}^\prime,\ u_{u_h}^\prime,\ u_d^\prime]\) (input of the damped plant) to \(\bm{d}_{\text{fj}} = [d_{u_r},\ d_{u_h},\ d_d]\) (motion of the crystal expressed in the frame of the fast-jacks). +This is schematically represented in Figure 17. +

-%% Outputs -% Force Sensor {3x1} [m] -io(io_i) = linio([mdl, '/DCM'], 1, 'openoutput'); io_i = io_i + 1; -
+ +
+

schematic_jacobian_frame_fastjack_iff.png +

+

Figure 17: Use of Jacobian matrices to obtain the system in the frame of the fastjacks

-
-
%% 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 IFF
-controller.type = 1; % IFF
-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'};
-
-
+

+The dynamics from \(\bm{u}\) to \(\bm{d}_{\text{fj}}\) (open-loop dynamics) and from \(\bm{u}^\prime\) to \(\bm{d}_{\text{fs}}\) are compared in Figure 18. +It is clear that the Integral Force Feedback control strategy is very effective in damping the resonances of the plant. +

-
+

comp_damped_undamped_plant_iff_bode_plot.png

-

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

+

Figure 18: 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. +The Integral Force Feedback control strategy is very effective in damping the modes present in the plant.

-
-
-
- -
-

4.4. Save

-
-
-
save('mat/Kiff.mat', 'Kiff');
-
-
-

5. HAC-LAC (IFF) architecture

+
+

5. HAC-LAC (IFF) architecture

- +

+

+The HAC-LAC architecture is shown in Figure 19. +

+ + +
+

schematic_jacobian_frame_fastjack_hac_iff.png +

+

Figure 19: HAC-LAC architecture

+
+
+
+

5.1. System Identification

+
+

+Let’s identify the damped plant. +

+ + +
+

bode_plot_hac_iff_plant.png +

+

Figure 20: Bode Plot of the plant for the High Authority Controller (transfer function from \(\bm{u}^\prime\) to \(\bm{\epsilon}_d\))

+
+
+
+ +
+

5.2. High Authority Controller

+
+

+Let’s design a controller with a bandwidth of 100Hz. +As the plant is well decoupled and well approximated by a constant at low frequency, the high authority controller can easily be designed with SISO loop shaping. +

+ +
+
%% Controller design
+wc = 2*pi*100; % Wanted crossover frequency [rad/s]
+a = 2; % Lead parameter
+
+Khac = diag(1./diag(abs(evalfr(G_dp, 1j*wc)))) * ... % Diagonal controller
+        wc/s * ... % Integrator
+        1/(sqrt(a))*(1 + s/(wc/sqrt(a)))/(1 + s/(wc*sqrt(a))) * ... % Lead
+        1/(s^2/(4*wc)^2 + 2*s/(4*wc) + 1); % Low pass filter
+
+
+ +
+
%% Save the HAC controller
+save('mat/Khac_iff.mat', 'Khac');
+
+
+ +
+
%% Loop Gain
+L_hac_lac = G_dp * Khac;
+
+
+ + +
+

hac_iff_loop_gain_bode_plot.png +

+

Figure 21: Bode Plot of the Loop gain for the High Authority Controller

+
+ +

+As shown in the Root Locus plot in Figure 22, the closed loop system should be stable. +

+ + +
+

loci_hac_iff_fast_jack.png +

+

Figure 22: Root Locus for the High Authority Controller

+
+
+
+ +
+

5.3. Performances

+
+

+In order to estimate the performances of the HAC-IFF control strategy, the transfer function from motion errors of the stepper motors to the motion error of the crystal is identified both in open loop and with the HAC-IFF strategy. +

+ +

+It is first verified that the closed-loop system is stable: +

+ +
+
isstable(T_hl)
+
+
+ +
+1
+
+ + +

+And both transmissibilities are compared in Figure 23. +

+ + +
+

stepper_transmissibility_comp_ol_hac_iff.png +

+

Figure 23: Comparison of the transmissibility of errors from vibrations of the stepper motor between the open-loop case and the hac-iff case.

+
+ +
+

+The HAC-IFF control strategy can effectively reduce the transmissibility of the motion errors of the stepper motors. +This reduction is effective inside the bandwidth of the controller. +

+ +
+

Author: Dehaeze Thomas

-

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

+

Created: 2021-11-30 mar. 15:17

diff --git a/dcm-simscape.org b/dcm-simscape.org index 57b6f9c..870c31e 100644 --- a/dcm-simscape.org +++ b/dcm-simscape.org @@ -1,4 +1,4 @@ -#+TITLE: DCM - Dynamical Multi-Body Model +#+TITLE: ESRF Double Crystal Monochromator - Dynamical Multi-Body Model :DRAWER: #+LANGUAGE: en #+EMAIL: dehaeze.thomas@gmail.com @@ -65,6 +65,7 @@ It is structured as follow: :header-args:matlab+: :tangle matlab/dcm_kinematics.m :END: <> + ** Introduction :ignore: ** Matlab Init :noexport:ignore: @@ -378,6 +379,13 @@ data2orgtable(inv(J_a_111), {}, {}, ' %.4f '); | 1.7857 | 1.7857 | -3.5714 | | -4.5455 | 4.5455 | 0.0 | +** TODO Inputs and Outputs :noexport: + +Disturbances: +- Motion errors of the stepper motor +- Vibrations from the outside +- Vibrations from the cooling system directly applied on the crystals + ** Save Kinematics #+begin_src matlab :exports none :tangle no save('matlab/mat/dcm_kinematics.mat', 'J_a_111', 'J_s_111') @@ -482,7 +490,7 @@ size(G) ** Plant in the frame of the fastjacks #+begin_src matlab -load('mat/dcm_kinematics.mat'); +load('dcm_kinematics.mat'); #+end_src Using the forward and inverse kinematics, we can computed the dynamics from piezo forces to axial motion of the 3 fastjacks (see Figure [[fig:schematic_jacobian_frame_fastjack]]). @@ -886,7 +894,7 @@ io(io_i) = linio([mdl, '/DCM'], 1, 'openoutput'); io_i = io_i + 1; #+begin_src matlab %% DCM Kinematics -load('mat/dcm_kinematics.mat'); +load('dcm_kinematics.mat'); #+end_src #+begin_src matlab @@ -1030,14 +1038,8 @@ G_fs.InputName = {'u_ur', 'u_uh', 'u_d'}; G_fs.OutputName = {'fs_ur', 'fs_uh', 'fs_d'}; #+end_src -#+begin_src matlab :results value replace :exports both :tangle no -dcgain(G_fs) -#+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 | +The Bode plot of the identified dynamics is shown in Figure [[fig:iff_plant_bode_plot]]. +At high frequency, the diagonal terms are constants while the off-diagonal terms have some roll-off. #+begin_src matlab :exports none %% Bode plot for the plant @@ -1062,7 +1064,7 @@ for i = 1:2 end hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); -ylabel('Amplitude [m/N]'); set(gca, 'XTickLabel',[]); +ylabel('Amplitude'); set(gca, 'XTickLabel',[]); legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 2); ylim([1e-13, 1e-7]); @@ -1092,7 +1094,11 @@ exportFig('figs/iff_plant_bode_plot.pdf', 'width', 'wide', 'height', 'tall'); [[file:figs/iff_plant_bode_plot.png]] ** Controller - Root Locus +We want to have integral action around the resonances of the system, but we do not want to integrate at low frequency. +Therefore, we can use a low pass filter. + #+begin_src matlab +%% Integral Force Feedback Controller Kiff_g1 = eye(3)*1/(1 + s/2/pi/20); #+end_src @@ -1136,12 +1142,61 @@ exportFig('figs/iff_root_locus.pdf', 'width', 'wide', 'height', 'tall'); [[file:figs/iff_root_locus.png]] #+begin_src matlab -%% Integral Force Feedback Controller +%% Integral Force Feedback Controller with optimal gain Kiff = g*Kiff_g1; #+end_src +#+begin_src matlab :exports none :tangle no +save('matlab/mat/Kiff.mat', 'Kiff'); +#+end_src + +#+begin_src matlab :eval no +%% Save the IFF controller +save('mat/Kiff.mat', 'Kiff'); +#+end_src + ** Damped Plant -#+begin_src matlab +Both the Open Loop dynamics (see Figure [[fig:schematic_jacobian_frame_fastjack]]) and the dynamics with IFF (see Figure [[fig:schematic_jacobian_frame_fastjack_iff]]) are identified. + +We are here interested in the dynamics from $\bm{u}^\prime = [u_{u_r}^\prime,\ u_{u_h}^\prime,\ u_d^\prime]$ (input of the damped plant) to $\bm{d}_{\text{fj}} = [d_{u_r},\ d_{u_h},\ d_d]$ (motion of the crystal expressed in the frame of the fast-jacks). +This is schematically represented in Figure [[fig:schematic_jacobian_frame_fastjack_iff]]. + +#+begin_src latex :file schematic_jacobian_frame_fastjack_iff.pdf +\begin{tikzpicture} + % Blocs + \node[block={4.0cm}{3.0cm}] (G) {$\bm{G}(s)$}; + \coordinate[] (inputF) at ($(G.south west)!0.5!(G.north west)$); + \coordinate[] (outputF) at ($(G.south east)!0.8!(G.north east)$); + \coordinate[] (outputL) at ($(G.south east)!0.2!(G.north east)$); + + \node[block, right=1.5 of outputL] (Js) {$\bm{J}_{s}^{-1}$}; + \node[block, right=1.5 of Js] (Ja) {$\bm{J}_{a}$}; + \node[addb, left= of G] (addF) {}; + \node[block, above=0.5 of G] (Kiff) {$\bm{K}_{\text{IFF}}(s)$}; + + % Connections and labels + \draw[->] ($(addF.west)+(-1.5,0)$) node[above right]{$\begin{bmatrix} u_{u_r}^\prime \\ u_{u_h}^\prime \\ u_d^\prime \end{bmatrix}$} -- (addF.west); + \draw[->] (addF.east) -- node[miday, above]{$\begin{bmatrix} u_{u_r} \\ u_{u_h} \\ u_d \end{bmatrix}$} (inputF); + \draw[->] (outputF) -- ++(2.0, 0) node[above left]{$\begin{bmatrix} \tau_{u_r} \\ \tau_{u_h} \\ \tau_d \end{bmatrix}$}; + \draw[->] ($(outputF) + (0.6, 0)$)node[branch]{} |- (Kiff.east); + \draw[->] (Kiff.west) -| (addF.north); + \draw[->] (outputL) -- node[midway, above]{$\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$} (Js.west); + \draw[->] (Js.east) -- node[midway, above]{$\begin{bmatrix} d_z \\ r_y \\ r_x \end{bmatrix}$} (Ja.west); + \draw[->] (Ja.east) -- ++(1.5, 0) node[above left]{$\begin{bmatrix} d_{u_r} \\ d_{u_h} \\ d_{d} \end{bmatrix}$}; + + \begin{scope}[on background layer] + \node[fit={(G.south -| addF.west) (Ja.east |- Kiff.north)}, fill=black!20!white, draw, inner sep=6pt] (system) {}; + \node[above] at (system.north) {$\bm{G}_{\text{fj,IFF}}(s)$}; + \end{scope} +\end{tikzpicture} +#+end_src + +#+name: fig:schematic_jacobian_frame_fastjack_iff +#+caption: Use of Jacobian matrices to obtain the system in the frame of the fastjacks +#+RESULTS: +[[file:figs/schematic_jacobian_frame_fastjack_iff.png]] + +#+begin_src matlab :exports none %% Input/Output definition clear io; io_i = 1; @@ -1154,12 +1209,12 @@ io(io_i) = linio([mdl, '/control_system'], 1, 'input'); io_i = io_i + 1; 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'); +#+begin_src matlab :exports none +%% Load DCM Kinematics +load('dcm_kinematics.mat'); #+end_src -#+begin_src matlab +#+begin_src matlab :exports none %% Identification of the Open Loop plant controller.type = 0; % Open Loop G_ol = J_a_111*inv(J_s_111)*linearize(mdl, io); @@ -1167,7 +1222,7 @@ G_ol.InputName = {'u_ur', 'u_uh', 'u_d'}; G_ol.OutputName = {'d_ur', 'd_uh', 'd_d'}; #+end_src -#+begin_src matlab +#+begin_src matlab :exports none %% Identification of the damped plant with IFF controller.type = 1; % IFF G_dp = J_a_111*inv(J_s_111)*linearize(mdl, io); @@ -1175,6 +1230,9 @@ G_dp.InputName = {'u_ur', 'u_uh', 'u_d'}; G_dp.OutputName = {'d_ur', 'd_uh', 'd_d'}; #+end_src +The dynamics from $\bm{u}$ to $\bm{d}_{\text{fj}}$ (open-loop dynamics) and from $\bm{u}^\prime$ to $\bm{d}_{\text{fs}}$ are compared in Figure [[fig:comp_damped_undamped_plant_iff_bode_plot]]. +It is clear that the Integral Force Feedback control strategy is very effective in damping the resonances of the plant. + #+begin_src matlab :exports none %% Comparison of the damped and undamped plant figure; @@ -1237,24 +1295,67 @@ exportFig('figs/comp_damped_undamped_plant_iff_bode_plot.pdf', 'width', 'wide', [[file:figs/comp_damped_undamped_plant_iff_bode_plot.png]] #+begin_important -The Integral Force Feedback control strategy is very effective in damping the suspension modes of the DCM. +The Integral Force Feedback control strategy is very effective in damping the modes present in the plant. #+end_important -** Save -#+begin_src matlab :exports none :tangle no -save('matlab/mat/Kiff.mat', 'Kiff'); -#+end_src - -#+begin_src matlab :eval no -save('mat/Kiff.mat', 'Kiff'); -#+end_src - * HAC-LAC (IFF) architecture :PROPERTIES: :header-args:matlab+: :tangle matlab/dcm_hac_iff.m :END: <> ** Introduction :ignore: + +The HAC-LAC architecture is shown in Figure [[fig:schematic_jacobian_frame_fastjack_hac_iff]]. + +#+begin_src latex :file schematic_jacobian_frame_fastjack_hac_iff.pdf +\begin{tikzpicture} + % Blocs + \node[block={4.0cm}{3.0cm}] (G) {$\bm{G}(s)$}; + \coordinate[] (inputF) at ($(G.south west)!0.5!(G.north west)$); + \coordinate[] (outputF) at ($(G.south east)!0.8!(G.north east)$); + \coordinate[] (outputL) at ($(G.south east)!0.2!(G.north east)$); + + \node[block, right=1.3 of outputL] (Js) {$\bm{J}_{s}^{-1}$}; + + \node[addb, left= of G] (addF) {}; + \node[block, above=0.5 of G] (Kiff) {$\bm{K}_{\text{IFF}}(s)$}; + + \node[block, left=1.3 of addF] (Khac) {$\bm{K}_{\text{HAC}}(s)$}; + \node[block, left=1.3 of Khac] (Ja) {$\bm{J}_{a}$}; + \node[addb={+}{}{}{}{-}, left=1.3 of Ja] (subL) {}; + \coordinate[left=1.3 of subL] (mux); + \node[block] (bragg2d) at ($(mux) + (-1.2, 0.8)$) {$\frac{d_{\text{off}}}{2 \cos \theta_b}$}; + \coordinate (rx) at ($(mux) + (0, -0.8)$); + + + % Connections and labels + \draw[->] (Khac.east) -- node[midway, above]{$\begin{bmatrix} u_{u_r}^\prime \\ u_{u_h}^\prime \\ u_d^\prime \end{bmatrix}$} (addF.west); + \draw[->] (addF.east) -- node[midway, above]{$\begin{bmatrix} u_{u_r} \\ u_{u_h} \\ u_d \end{bmatrix}$} (inputF); + \draw[->] (outputF) -- ++(2.0, 0) node[above left]{$\begin{bmatrix} \tau_{u_r} \\ \tau_{u_h} \\ \tau_d \end{bmatrix}$}; + \draw[->] ($(outputF) + (0.6, 0)$)node[branch]{} |- (Kiff.east); + \draw[->] (Kiff.west) -| (addF.north); + \draw[->] (outputL) -- node[midway, above]{$\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$} (Js.west); + \draw[->] (Js.east) -- ++(1.0, 0); + + + \draw[line width=4pt] ($(mux) + (0, -1.0)$) -- ($(mux) + (0, 1.0)$); + \draw[->] (mux) -- node[midway, above]{$\begin{bmatrix} r_{d_z} \\ r_{r_y} \\ r_{r_x} \end{bmatrix}$} (subL.west); + \draw[->] (subL.east) -- node[midway, above]{$\begin{bmatrix} \epsilon_{d_z} \\ \epsilon_{r_y} \\ \epsilon_{r_x} \end{bmatrix}$} (Ja.west); + \draw[->] (Ja.east) -- node[midway, above]{$\begin{bmatrix} \epsilon_{d_{u_r}} \\ \epsilon_{d_{u_h}} \\ \epsilon_{d_d} \end{bmatrix}$} (Khac.west); + + \draw[->] ($(Js.east) + (0.6, 0)$)node[branch]{}node[above]{$\begin{bmatrix} d_z \\ r_y \\ r_x \end{bmatrix}$} -- ++(0, -1.0) -| (subL.south); + \draw[->] ($(bragg2d.west) + (-0.6, 0)$)coordinate(ref_input)node[above right]{$\theta_b$} -- (bragg2d.west); + \draw[->] (bragg2d.east) --node[midway, above]{$r_{d_z}$} (bragg2d-|mux); + \draw[->] (ref_input|-mux)node[above right]{$r_{r_y}$} -- (mux); + \draw[->] (ref_input|-rx) node[above right]{$r_{r_x}$} -- (rx-|mux); +\end{tikzpicture} +#+end_src + +#+name: fig:schematic_jacobian_frame_fastjack_hac_iff +#+caption: HAC-LAC architecture +#+RESULTS: +[[file:figs/schematic_jacobian_frame_fastjack_hac_iff.png]] + ** Matlab Init :noexport:ignore: #+begin_src matlab %% dcm_hac_iff.m @@ -1285,6 +1386,291 @@ save('mat/Kiff.mat', 'Kiff'); <> #+end_src +** System Identification +Let's identify the damped plant. + +#+begin_src matlab :exports none +%% 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 :exports none +%% Load DCM Kinematics and IFF controller +load('dcm_kinematics.mat'); +load('Kiff.mat'); +#+end_src + +#+begin_src matlab :exports none +%% Identification of the damped plant with IFF +controller.type = 1; % IFF +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_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-8]); + +ax2 = nexttile; +hold on; +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/bode_plot_hac_iff_plant.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:bode_plot_hac_iff_plant +#+caption: Bode Plot of the plant for the High Authority Controller (transfer function from $\bm{u}^\prime$ to $\bm{\epsilon}_d$) +#+RESULTS: +[[file:figs/bode_plot_hac_iff_plant.png]] + +** High Authority Controller +Let's design a controller with a bandwidth of 100Hz. +As the plant is well decoupled and well approximated by a constant at low frequency, the high authority controller can easily be designed with SISO loop shaping. + +#+begin_src matlab +%% Controller design +wc = 2*pi*100; % Wanted crossover frequency [rad/s] +a = 2; % Lead parameter + +Khac = diag(1./diag(abs(evalfr(G_dp, 1j*wc)))) * ... % Diagonal controller + wc/s * ... % Integrator + 1/(sqrt(a))*(1 + s/(wc/sqrt(a)))/(1 + s/(wc*sqrt(a))) * ... % Lead + 1/(s^2/(4*wc)^2 + 2*s/(4*wc) + 1); % Low pass filter +#+end_src + +#+begin_src matlab :exports none :tangle no +save('matlab/mat/Khac_iff.mat', 'Khac'); +#+end_src + +#+begin_src matlab :eval no +%% Save the HAC controller +save('mat/Khac_iff.mat', 'Khac'); +#+end_src + +#+begin_src matlab +%% Loop Gain +L_hac_lac = G_dp * Khac; +#+end_src + +#+begin_src matlab :exports none +%% Bode Plot of the Loop Gain +figure; +tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None'); + +ax1 = nexttile([2,1]); +hold on; +plot(freqs, abs(squeeze(freqresp(L_hac_lac(1,1), freqs, 'Hz'))), '-', ... + 'DisplayName', 'd'); +plot(freqs, abs(squeeze(freqresp(L_hac_lac(2,2), freqs, 'Hz'))), '-', ... + 'DisplayName', 'uh'); +plot(freqs, abs(squeeze(freqresp(L_hac_lac(3,3), freqs, 'Hz'))), '-', ... + 'DisplayName', 'ur'); +for i = 1:2 + for j = i+1:3 + plot(freqs, abs(squeeze(freqresp(L_hac_lac(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('Loop Gain'); set(gca, 'XTickLabel',[]); +legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2); +ylim([1e-2, 1e1]); + +ax2 = nexttile; +hold on; +plot(freqs, 180/pi*angle(squeeze(freqresp(L_hac_lac(1,1), freqs, 'Hz'))), '-'); +plot(freqs, 180/pi*angle(squeeze(freqresp(L_hac_lac(2,2), freqs, 'Hz'))), '-'); +plot(freqs, 180/pi*angle(squeeze(freqresp(L_hac_lac(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/hac_iff_loop_gain_bode_plot.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:hac_iff_loop_gain_bode_plot +#+caption: Bode Plot of the Loop gain for the High Authority Controller +#+RESULTS: +[[file:figs/hac_iff_loop_gain_bode_plot.png]] + +#+begin_src matlab :exports none +%% Compute the Eigenvalues of the loop gain +Ldet = zeros(3, length(freqs)); + +Lmimo = squeeze(freqresp(L_hac_lac, freqs, 'Hz')); +for i_f = 1:length(freqs) + Ldet(:, i_f) = eig(squeeze(Lmimo(:,:,i_f))); +end +#+end_src + +As shown in the Root Locus plot in Figure [[fig:loci_hac_iff_fast_jack]], the closed loop system should be stable. + +#+begin_src matlab :exports none +%% Plot of the eigenvalues of L in the complex plane +figure; +hold on; +for i = 1:3 + plot(real(squeeze(Ldet(i,:))), imag(squeeze(Ldet(i,:))), 'k.'); + plot(real(squeeze(Ldet(i,:))), -imag(squeeze(Ldet(i,:))), 'k.'); +end +plot(-1, 0, 'kx', 'HandleVisibility', 'off'); +hold off; +set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'lin'); +xlabel('Real'); ylabel('Imag'); +xlim([-3, 1]); ylim([-2, 2]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/loci_hac_iff_fast_jack.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:loci_hac_iff_fast_jack +#+caption: Root Locus for the High Authority Controller +#+RESULTS: +[[file:figs/loci_hac_iff_fast_jack.png]] + +** Performances +In order to estimate the performances of the HAC-IFF control strategy, the transfer function from motion errors of the stepper motors to the motion error of the crystal is identified both in open loop and with the HAC-IFF strategy. + +#+begin_src matlab :exports none +%% Input/Output definition +clear io; io_i = 1; + +%% Inputs +% Jack Motion Erros {3x1} [m] +io(io_i) = linio([mdl, '/stepper_errors'], 1, 'input'); io_i = io_i + 1; + +%% Outputs +% Interferometer Output {3x1} [m] +io(io_i) = linio([mdl, '/DCM'], 1, 'output'); io_i = io_i + 1; +#+end_src + +#+begin_src matlab :exports none +%% Identification of the transmissibility of errors in open-loop +controller.type = 0; % Open Loop +T_ol = inv(J_s_111)*linearize(mdl, io)*J_a_111; +T_ol.InputName = {'e_dz', 'e_ry', 'e_rx'}; +T_ol.OutputName = {'dx', 'ry', 'rx'}; +#+end_src + +#+begin_src matlab :exports none +%% Load DCM Kinematics and IFF controller +load('dcm_kinematics.mat'); +load('Kiff.mat'); +#+end_src + +#+begin_src matlab :exports none +%% Identification of the transmissibility of errors with HAC-IFF +controller.type = 3; % IFF +T_hl = inv(J_s_111)*linearize(mdl, io)*J_a_111; +T_hl.InputName = {'e_dz', 'e_ry', 'e_rx'}; +T_hl.OutputName = {'dx', 'ry', 'rx'}; +#+end_src + +It is first verified that the closed-loop system is stable: + +#+begin_src matlab :results value replace :exports both :tangle no +isstable(T_hl) +#+end_src + +#+RESULTS: +: 1 + +And both transmissibilities are compared in Figure [[fig:stepper_transmissibility_comp_ol_hac_iff]]. + +#+begin_src matlab :exports none +%% Transmissibility of stepper errors +f = logspace(0, 3, 1000); + +figure; +hold on; +plot(f, abs(squeeze(freqresp(T_ol(1,1), f, 'Hz'))), '-', ... + 'DisplayName', '$d_z$ - OL'); +plot(f, abs(squeeze(freqresp(T_ol(2,2), f, 'Hz'))), '-', ... + 'DisplayName', '$r_y$ - OL'); +plot(f, abs(squeeze(freqresp(T_ol(3,3), f, 'Hz'))), '-', ... + 'DisplayName', '$r_x$ - OL'); +set(gca,'ColorOrderIndex',1) +plot(f, abs(squeeze(freqresp(T_hl(1,1), f, 'Hz'))), '--', ... + 'DisplayName', '$d_z$ - HAC-IFF'); +plot(f, abs(squeeze(freqresp(T_hl(2,2), f, 'Hz'))), '--', ... + 'DisplayName', '$r_y$ - HAC-IFF'); +plot(f, abs(squeeze(freqresp(T_hl(3,3), f, 'Hz'))), '--', ... + 'DisplayName', '$r_x$ - HAC-IFF'); +hold off; +set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); +xlabel('Frequency [Hz]'); ylabel('Stepper transmissibility'); +legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2); +ylim([1e-2, 1e2]); +xlim([f(1), f(end)]); +#+end_src + +#+begin_src matlab :tangle no :exports results :results file replace +exportFig('figs/stepper_transmissibility_comp_ol_hac_iff.pdf', 'width', 'wide', 'height', 'tall'); +#+end_src + +#+name: fig:stepper_transmissibility_comp_ol_hac_iff +#+caption: Comparison of the transmissibility of errors from vibrations of the stepper motor between the open-loop case and the hac-iff case. +#+RESULTS: +[[file:figs/stepper_transmissibility_comp_ol_hac_iff.png]] + +#+begin_important +The HAC-IFF control strategy can effectively reduce the transmissibility of the motion errors of the stepper motors. +This reduction is effective inside the bandwidth of the controller. +#+end_important + * Helping Functions :noexport: ** Initialize Path #+NAME: m-init-path diff --git a/dcm-simscape.pdf b/dcm-simscape.pdf index 595ee5c07e23c2ab99283090fe465afd37fb16dd..80ee8959acf1dc80da5682e4575e76ece5e0eab8 100644 GIT binary patch delta 665193 zcmZsjbzD>b8}{93knR+uks54+Au1&$-QA5y3n(}ckd%grw3H|fQj*dwf`l~EA<`h> zbI`Bs_j;b^kNNWEv+cgl;lstb-sc|uHPX-P3`-61z``Q&q+o93%czD&t;`g%UiaD= zCGjcvY5HC_DJFi)vVJcn3MpcpkaSB`{4q=&!lqHp?|Wmx?FZ>kEHD$n&Y;n+1#?3l ziI$fwWM`bBy+b!?mTDw08Uwt_eV$Hh!|BRt$0TkaF;uWjE`C7 z6cC@h93uXWY5k3b^dXJIk2cEs^e9i{_$|LEZHBp}`8uwG5jaHBXH}jvl|oik 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