From 8292a1006e282725ecbb7c0a027cab9d4e021814 Mon Sep 17 00:00:00 2001 From: Thomas Dehaeze Date: Tue, 30 Nov 2021 11:28:07 +0100 Subject: [PATCH] Export to html --- dcm-simscape.html | 914 ++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 914 insertions(+) create mode 100644 dcm-simscape.html diff --git a/dcm-simscape.html b/dcm-simscape.html new file mode 100644 index 0000000..834116d --- /dev/null +++ b/dcm-simscape.html @@ -0,0 +1,914 @@ + + + + + + +DCM - Dynamical Multi-Body Model + + + + + + + + +
+ UP + | + HOME +
+

DCM - Dynamical Multi-Body Model

+ +
+

This report is also available as a pdf.

+
+ +

+In this document, a Simscape (.e.g. multi-body) model of the ESRF Double Crystal Monochromator (DCM) is presented and used to develop and optimize the control strategy. +

+ +

+It is structured as follow: +

+ + +
+

1. System Kinematics

+
+

+ +

+
+
+

1.1. Bragg Angle

+
+
+
%% Tested bragg angles
+bragg = linspace(5, 80, 1000); % Bragg angle [deg]
+d_off = 10.5e-3; % Wanted offset between x-rays [m]
+
+
+ +
+
%% Vertical Jack motion as a function of Bragg angle
+dz = d_off./(2*cos(bragg*pi/180));
+
+
+ + +
+

jack_motion_bragg_angle.png +

+

Figure 1: Jack motion as a function of Bragg angle

+
+ +
+
%% Required Jack stroke
+ans = 1e3*(dz(end) - dz(1))
+
+
+ +
+24.963
+
+
+
+ +
+

1.2. Kinematics (111 Crystal)

+
+

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

+ + +
+

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): +

+\begin{equation} +\begin{bmatrix} +x_1 \\ x_2 \\ x_3 +\end{bmatrix} += +\bm{J}_{s,111} +\begin{bmatrix} +d_z \\ r_y \\ r_x +\end{bmatrix} +\end{equation} + + +
+

schematic_sensor_jacobian_inverse_kinematics.png +

+

Figure 3: Inverse Kinematics - Interferometers

+
+ + +

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

+\begin{equation} +\bm{J}_{s,111} += +\begin{bmatrix} +1 & 0.07 & -0.015 \\ +1 & 0 & 0.015 \\ +1 & -0.07 & -0.015 +\end{bmatrix} +\end{equation} + +
+
%% Sensor Jacobian matrix for 111 crystal
+J_s_111 = [1,  0.07, -0.015
+           1,  0,     0.015
+           1, -0.07, -0.015];
+
+
+ + + + +++ ++ ++ + + + + + + + + + + + + + + + + + + + +
Table 1: Sensor Jacobian \(\bm{J}_{s,111}\)
1.00.07-0.015
1.00.00.015
1.0-0.07-0.015
+ +

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

+\begin{equation} +\begin{bmatrix} +d_z \\ r_y \\ r_x +\end{bmatrix} += +\bm{J}_{s,111}^{-1} +\begin{bmatrix} +x_1 \\ x_2 \\ x_3 +\end{bmatrix} +\end{equation} + + +
+

schematic_sensor_jacobian_forward_kinematics.png +

+

Figure 4: Forward Kinematics - Interferometers

+
+ + + + +++ ++ ++ + + + + + + + + + + + + + + + + + + + +
Table 2: Inverse of the sensor Jacobian \(\bm{J}_{s,111}^{-1}\)
0.250.50.25
7.140.0-7.14
-16.6733.33-16.67
+
+
+ +
+

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. +

+ + +
+

actuator_jacobian_111_points.png +

+

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

+
+ +

+Inverse Kinematics consist of deriving the axial (z) motion of the 3 actuators from the motion of the crystal’s center. +

+\begin{equation} +\begin{bmatrix} +d_{u_r} \\ d_{u_h} \\ d_{d} +\end{bmatrix} += +\bm{J}_{a,111} +\begin{bmatrix} +d_z \\ r_y \\ r_x +\end{bmatrix} +\end{equation} + + +
+

schematic_actuator_jacobian_inverse_kinematics.png +

+

Figure 6: Inverse Kinematics - Actuators

+
+ +

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

+\begin{equation} +\bm{J}_{a,111} += +\begin{bmatrix} +1 & 0.14 & -0.1525 \\ +1 & 0.14 & 0.0675 \\ +1 & -0.14 & -0.0425 +\end{bmatrix} +\end{equation} + +
+
%% Actuator Jacobian - 111 crystal
+J_a_111 = [1,  0.14, -0.1525
+           1,  0.14,  0.0675
+           1, -0.14, -0.0425];
+
+
+ + + + +++ ++ ++ + + + + + + + + + + + + + + + + + + + +
Table 3: Actuator Jacobian \(\bm{J}_{a,111}\)
1.00.14-0.1525
1.00.140.0675
1.0-0.14-0.0425
+ +

+The forward Kinematics is solved by inverting the Jacobian matrix: +

+\begin{equation} +\begin{bmatrix} +d_z \\ r_y \\ r_x +\end{bmatrix} += +\bm{J}_{a,111}^{-1} +\begin{bmatrix} +d_{u_r} \\ d_{u_h} \\ d_{d} +\end{bmatrix} +\end{equation} + + +
+

schematic_actuator_jacobian_forward_kinematics.png +

+

Figure 7: Forward Kinematics - Actuators for 111 crystal

+
+ + + + +++ ++ ++ + + + + + + + + + + + + + + + + + + + +
Table 4: Inverse of the actuator Jacobian \(\bm{J}_{a,111}^{-1}\)
0.05680.44320.5
1.78571.7857-3.5714
-4.54554.54550.0
+
+
+
+ +
+

1.3. Save Kinematics

+
+
+
save('mat/dcm_kinematics.mat', 'J_a_111', 'J_s_111')
+
+
+
+
+
+ +
+

2. Open Loop System Identification

+
+

+ +

+
+
+

2.1. Identification

+
+

+Let’s considered the system \(\bm{G}(s)\) with: +

+
    +
  • 3 inputs: force applied to the 3 fast jacks
    • +
  • 3 outputs: measured displacement by the 3 interferometers pointing at the 111 second crystal
    • +
+ +

+It is schematically shown in Figure 8. +

+ + +
+

schematic_system_inputs_outputs.png +

+

Figure 8: Dynamical system with inputs and outputs

+
+ +

+The system is identified from the Simscape model. +

+ +
+
%% Input/Output definition
+clear io; io_i = 1;
+
+%% Inputs
+% Control Input {3x1} [N]
+io(io_i) = linio([mdl, '/control_system'], 1, 'openinput');  io_i = io_i + 1;
+
+%% Outputs
+% Interferometers {3x1} [m]
+io(io_i) = linio([mdl, '/DCM'], 1, 'openoutput'); io_i = io_i + 1;
+
+
+ +
+
%% Extraction of the dynamics
+G = linearize(mdl, io);
+
+
+ +
+
size(G)
+
+
+ +
+size(G)
+State-space model with 3 outputs, 3 inputs, and 24 states.
+
+
+
+ +
+

2.2. Plant in the frame of the fastjacks

+
+
+
load('mat/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). +

+ + +
+

schematic_jacobian_frame_fastjack.png +

+

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

+
+ + +
+
%% Compute the system in the frame of the fastjacks
+G_pz = J_a_111*inv(J_s_111)*G;
+
+
+ +

+The DC gain of the new system shows that the system is well decoupled at low frequency. +

+
+
dcgain(G_pz)
+
+
+ + + + +++ ++ ++ + + + + + + + + + + + + + + + + + + + +
Table 5: DC gain of the plant in the frame of the fast jacks \(\bm{G}_{\text{fj}}\)
4.4407e-092.7656e-121.0132e-12
2.7656e-124.4407e-091.0132e-12
1.0109e-121.0109e-124.4424e-09
+ +

+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). +

+ +
+
+
+ +
+

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

+
+ +
+
G_mr = inv(J_s_111)*G*inv(J_a_111');
+
+
+ +
+
dcgain(G_mr)
+
+
+ + + + +++ ++ ++ + + + + + + + + + + + + + + + + + + + +
1.9978e-093.9657e-097.7944e-09
3.9656e-098.4979e-08-1.5135e-17
7.7944e-09-3.9252e-171.834e-07
+ +

+This results in a coupled system. +The main reason is that, as we map forces to the center of the 111 crystal and not at the center of mass/stiffness of the moving part, vertical forces will induce rotation and torques will induce vertical motion. +

+
+
+
+ +
+

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

+
+
+
%% Input/Output definition
+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;
+
+%% Outputs
+% Strain Gauges {3x1} [m]
+io(io_i) = linio([mdl, '/sg'], 1, 'openoutput'); io_i = io_i + 1;
+
+
+ +
+
%% Extraction of the dynamics
+G_sg = linearize(mdl, io);
+
+
+ +
+
dcgain(G_sg)
+
+
+ + + + +++ ++ ++ + + + + + + + + + + + + + + + + + + + +
-1.4113e-131.0339e-133.774e-14
1.0339e-13-1.4113e-133.774e-14
3.7792e-143.7792e-14-7.5585e-14
+
+
+
+ +
+

4. Active Damping Plant (Force Sensors)

+
+

+ +

+

+Force sensors are added above the piezoelectric actuators. +They can consists of a simple piezoelectric ceramic stack. +See for instance fleming10_integ_strain_force_feedb_high. +

+
+
+

4.1. Identification

+
+
+
%% Input/Output definition
+clear io; io_i = 1;
+
+%% Inputs
+% Control Input {3x1} [N]
+io(io_i) = linio([mdl, '/control_system'], 1, 'openinput');  io_i = io_i + 1;
+
+%% Outputs
+% Force Sensor {3x1} [m]
+io(io_i) = linio([mdl, '/DCM'], 3, 'openoutput'); io_i = io_i + 1;
+
+
+ +
+
%% Extraction of the dynamics
+G_fs = linearize(mdl, io);
+
+
+ +
+
dcgain(G_fs)
+
+
+ + + + +++ ++ ++ + + + + + + + + + + + + + + + + + + + +
-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 12: Bode plot of IFF Plant

+
+
+
+ +
+

4.2. Controller - Root Locus

+
+
+
Kiff_g1 = eye(3)*1/(1 + s/2/pi/20);
+
+
+ + +
+

iff_root_locus.png +

+

Figure 13: Root Locus plot for the IFF Control strategy

+
+ +
+
%% Integral Force Feedback Controller
+Kiff = g*Kiff_g1;
+
+
+
+
+ +
+

4.3. Damped Plant

+
+
+
%% 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 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'};
+
+
+ + +
+

comp_damped_undamped_plant_iff_bode_plot.png +

+

Figure 14: 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. +

+ +
+
+
+ +
+

4.4. Save

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

5. HAC-LAC (IFF) architecture

+
+

+ +

+
+
+
+
+

Author: Dehaeze Thomas

+

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

+
+ +