ESRF Double Crystal Monochromator - Dynamical Multi-Body Model
+Table of Contents
+-
+
- 1. System Kinematics + + +
- 2. Open Loop System Identification + + +
- 3. Open-Loop Noise Budgeting + + +
- 4. Active Damping Plant (Strain gauges) + + +
- 5. Active Damping Plant (Force Sensors) + + +
- 6. Feedback Control +
- 7. HAC-LAC (IFF) architecture + + +
+
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: +
+-
+
- Section 1: the kinematics of the DCM is presented, and Jacobian matrices which are used to solve the inverse and forward kinematics are computed. +
- Section 2: the system dynamics is identified in the absence of control. +
- Section 3: an open-loop noise budget is performed. +
- Section 4: it is studied whether if the strain gauges fixed to the piezoelectric actuators can be used to actively damp the plant. +
- Section 5: piezoelectric force sensors are added in series with the piezoelectric actuators and are used to actively damp the plant using the Integral Force Feedback (IFF) control strategy. +
- Section 7: the High Authority Control - Low Authority Control (HAC-LAC) strategy is tested on the Simscape model. +
1. System Kinematics
+ + +1.1. Bragg Angle
++There is a simple relation eq:bragg_angle_formula between: +
+-
+
- \(d_{\text{off}}\) is the wanted offset between the incident x-ray and the output x-ray +
- \(\theta_b\) is the bragg angle +
- \(d_z\) is the corresponding distance between the first and second crystals +
+This relation is shown in Figure 1. +
+ + +
+
Figure 1: Jack motion as a function of Bragg angle
++The required jack stroke is approximately 25mm. +
+ +%% Required Jack stroke +ans = 1e3*(dz(end) - dz(1)) ++
+24.963 ++
1.2. Kinematics (311 Crystal)
++The reference frame is taken at the center of the 311 second crystal. +
+1.2.1. Interferometers - 311 secondary Crystal
++Three interferometers are pointed to the bottom surface of the 311 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. +
+ + +
+
Figure 2: Bottom view of the second crystal 311. 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,311} +\begin{bmatrix} +d_z \\ r_y \\ r_x +\end{bmatrix} +\end{equation} + + +
+
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,311} += +\begin{bmatrix} +1 & 0.07 & -0.015 \\ +1 & 0 & 0.015 \\ +1 & -0.07 & -0.015 +\end{bmatrix} +\end{equation} + +%% Sensor Jacobian matrix for 311 crystal +J_s_311 = [1, 0.07, -0.015 + 1, 0, 0.015 + 1, -0.07, -0.015]; ++
| 1.0 | +0.07 | +-0.015 | +
| 1.0 | +0.0 | +0.015 | +
| 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,311}^{-1} +\begin{bmatrix} +x_1 \\ x_2 \\ x_3 +\end{bmatrix} +\end{equation} + + +
+
Figure 4: Forward Kinematics - Interferometers
+| 0.25 | +0.5 | +0.25 | +
| 7.14 | +0.0 | +-7.14 | +
| -16.67 | +33.33 | +-16.67 | +
1.2.2. Interferometers - 311 primary Crystal
++Three interferometers are pointed to the bottom surface of the 311 crystal. +
+ ++The position of the measurement points are shown in Figure 5 as well as the origin where the motion of the crystal is computed. +
+ + +
+
Figure 5: Top view of the primary crystal 311. 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,311} +\begin{bmatrix} +d_z \\ r_y \\ r_x +\end{bmatrix} +\end{equation} + + +
+
Figure 6: Inverse Kinematics - Interferometers
++From the Figure 2, the inverse kinematics can be solved as follow (for small motion): +
+\begin{equation} +\bm{J}_{s,311} += +\begin{bmatrix} +-1 & -0.07 & 0.015 \\ +-1 & 0 & -0.015 \\ +-1 & 0.07 & 0.015 +\end{bmatrix} +\end{equation} + +%% Sensor Jacobian matrix for 311 crystal +J_s_311_1 = [-1, 0.07, -0.015 + -1, 0, 0.015 + -1, -0.07, -0.015]; ++
| -1.0 | +0.07 | +-0.015 | +
| -1.0 | +0.0 | +0.015 | +
| -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,311}^{-1} +\begin{bmatrix} +x_1 \\ x_2 \\ x_3 +\end{bmatrix} +\end{equation} + + +
+
Figure 7: Forward Kinematics - Interferometers
+| -0.25 | +-0.5 | +-0.25 | +
| 7.14 | +0.0 | +-7.14 | +
| -16.67 | +33.33 | +-16.67 | +
1.2.3. Piezo - 311 Crystal
++The location of the actuators with respect with the center of the 311 second crystal are shown in Figure 8. +
+ + +
+
Figure 8: Location of actuators with respect to the center of the 311 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,311} +\begin{bmatrix} +d_z \\ r_y \\ r_x +\end{bmatrix} +\end{equation} + + +
+
Figure 9: Inverse Kinematics - Actuators
++Based on the geometry in Figure 8, we obtain: +
+\begin{equation} +\bm{J}_{a,311} += +\begin{bmatrix} +1 & 0.14 & -0.1525 \\ +1 & 0.14 & 0.0675 \\ +1 & -0.14 & -0.0425 +\end{bmatrix} +\end{equation} + +%% Actuator Jacobian - 311 crystal +J_a_311 = [1, 0.14, -0.1525 + 1, 0.14, 0.0675 + 1, -0.14, -0.0425]; ++
| 1.0 | +0.14 | +-0.1525 | +
| 1.0 | +0.14 | +0.0675 | +
| 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,311}^{-1} +\begin{bmatrix} +d_{u_r} \\ d_{u_h} \\ d_{d} +\end{bmatrix} +\end{equation} + + +
+
Figure 10: Forward Kinematics - Actuators for 311 crystal
+| 0.0568 | +0.4432 | +0.5 | +
| 1.7857 | +1.7857 | +-3.5714 | +
| -4.5455 | +4.5455 | +0.0 | +
1.3. Kinematics (111 Crystal)
++The reference frame is taken at the center of the 111 second crystal. +
+1.3.1. Interferometers - 111 secondary Crystal
++Three interferometers are pointed to the bottom surface of the 111 crystal. +
+ ++The position of the measurement points are shown in Figure 11 as well as the origin where the motion of the crystal is computed. +
+ + +
+
Figure 11: 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 12): +
+\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} + + +
+
Figure 12: Inverse Kinematics - Interferometers
++From the Figure 11, 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]; ++
| 1.0 | +0.07 | +0.015 | +
| 1.0 | +0.0 | +-0.015 | +
| 1.0 | +-0.07 | +0.015 | +
+The forward kinematics is solved by inverting the Jacobian matrix (see Figure 13). +
+\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} + + +
+
Figure 13: Forward Kinematics - Interferometers
+| 0.25 | +0.5 | +0.25 | +
| 7.14 | +0.0 | +-7.14 | +
| 16.67 | +-33.33 | +16.67 | +
1.3.2. Interferometers - 111 primary Crystal
++Three interferometers are pointed to the bottom surface of the 111 crystal. +
+ ++The position of the measurement points are shown in Figure 14 as well as the origin where the motion of the crystal is computed. +
+ + +
+
Figure 14: Top view of the primary crystal 111. Position of the measurement points.
++The inverse kinematics consisting of deriving the interferometer measurements from the motion of the crystal (see Figure 12): +
+\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} + + +
+
Figure 15: Inverse Kinematics - Interferometers
++From the Figure 11, the inverse kinematics can be solved as follow (for small motion): +
+\begin{equation} +\bm{J}_{s,111} += +\begin{bmatrix} +1 & -0.036 & -0.015 \\ +1 & 0 & 0.015 \\ +1 & 0.036 & -0.015 +\end{bmatrix} +\end{equation} + +%% Sensor Jacobian matrix for 111 crystal +J_s_111_1 = [-1, -0.036, -0.015 + -1, 0, 0.015 + -1, 0.036, -0.015]; ++
| -1.0 | +-0.036 | +-0.015 | +
| -1.0 | +0.0 | +0.015 | +
| -1.0 | +0.036 | +-0.015 | +
+The forward kinematics is solved by inverting the Jacobian matrix (see Figure 13). +
+\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} + + +
+
Figure 16: Forward Kinematics - Interferometers
+| -0.25 | +-0.5 | +-0.25 | +
| -13.89 | +0.0 | +13.89 | +
| -16.67 | +33.33 | +-16.67 | +
1.3.3. Piezo - 111 Crystal
++The location of the actuators with respect with the center of the 111 second crystal are shown in Figure 17. +
+ + +
+
Figure 17: 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} + + +
+
Figure 18: Inverse Kinematics - Actuators
++Based on the geometry in Figure 17, we obtain: +
+\begin{equation} +\bm{J}_{a,111} += +\begin{bmatrix} +1 & 0.14 & -0.0675 \\ +1 & 0.14 & 0.1525 \\ +1 & -0.14 & 0.0425 +\end{bmatrix} +\end{equation} + +%% Actuator Jacobian - 111 crystal +J_a_111 = [1, 0.14, -0.0675 + 1, 0.14, 0.1525 + 1, -0.14, 0.0425]; ++
| 1.0 | +0.14 | +-0.1525 | +
| 1.0 | +0.14 | +0.0675 | +
| 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} + + +
+
Figure 19: Forward Kinematics - Actuators for 111 crystal
+| 0.25 | +0.25 | +0.5 | +
| 0.4058 | +3.1656 | +-3.5714 | +
| -4.5455 | +4.5455 | +0.0 | +
1.4. Kinematics (Metrology Frame)
+
+
Figure 20: Top View - Top metrology frame
+%% Sensor Jacobian matrix for 111 crystal +J_m = [1, 0.102, 0 + 1, -0.088, 0.1275 + 1, -0.088, -0.1275]; ++
| 1.0 | +0.102 | +0.0 | +
| 1.0 | +-0.088 | +0.1275 | +
| 1.0 | +-0.088 | +-0.1275 | +
| 0.463 | +0.268 | +0.268 | +
| 5.263 | +-2.632 | +-2.632 | +
| 0.0 | +3.922 | +-3.922 | +
1.5. Verification
+mdl = 'coordinate_transform'; + +%% Input/Output definition +clear io; io_i = 1; + +%% Inputs +io(io_i) = linio([mdl, '/interferometers'], 1, 'openinput'); io_i = io_i + 1; + +%% Outputs +io(io_i) = linio([mdl, '/xtal_111'], 1, 'openoutput'); io_i = io_i + 1; ++
%% Extraction of the dynamics +G_311 = linearize(mdl, io); +G_311 = G_311(:,[1 2 3 7 8 9 13 14 15]); ++
mdl = 'coordinate_transform'; + +%% Input/Output definition +clear io; io_i = 1; + +%% Inputs +io(io_i) = linio([mdl, '/interferometers'], 1, 'openinput'); io_i = io_i + 1; + +%% Outputs +io(io_i) = linio([mdl, '/xtal_111'], 1, 'openoutput'); io_i = io_i + 1; ++
G_111 = linearize(mdl, io);
+G_111 = G_111(:,[4 5 6 10 11 12 13 14 15]);
+
+%% Sensor Jacobian matrix for 1st 111 crystal +J_s_111_1 = [-1, -0.036, -0.015 + -1, 0, 0.015 + -1, 0.036, -0.015]; ++
%% Sensor Jacobian matrix for 2nd 111 crystal +J_s_111_2 = [1, 0.07, 0.015 + 1, 0, -0.015 + 1, -0.07, 0.015]; ++
%% Sensor Jacobian matrix for 111 crystal +J_m = [1, 0.102, 0 + 1, -0.088, 0.1275 + 1, -0.088, -0.1275]; ++
G_111_t = [-inv(J_s_111_1), inv(J_s_111_2), -inv(J_m)] +G_111_t(1,:) = -G_111_t(1,:); ++
| -0.25 | +-0.5 | +-0.25 | +-0.25 | +-0.5 | +-0.25 | +0.463 | +0.268 | +0.268 | +
| 13.889 | +0.0 | +-13.889 | +7.143 | +0.0 | +-7.143 | +-5.263 | +2.632 | +2.632 | +
| 16.667 | +-33.333 | +16.667 | +16.667 | +-33.333 | +16.667 | +0.0 | +-3.922 | +3.922 | +
| -0.25 | +-0.5 | +-0.25 | +-0.25 | +-0.5 | +-0.25 | +0.333 | +0.333 | +0.333 | +
| 13.889 | +0.0 | +-13.889 | +7.143 | +0.0 | +-7.143 | +-5.263 | +2.632 | +2.632 | +
| 16.667 | +-33.333 | +16.667 | +16.667 | +-33.333 | +16.667 | +0.0 | +-3.922 | +3.922 | +
1.6. Save Kinematics
+save('mat/dcm_kinematics.mat', 'J_a_311', 'J_s_311', '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 21. +
+ + +
+
Figure 21: 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('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 22). +
+ + +
+
Figure 22: 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_311*inv(J_s_311)*G; ++
+The DC gain of the new system shows that the system is well decoupled at low frequency. +
+dcgain(G_pz) ++
| 4.4407e-09 | +2.7656e-12 | +1.0132e-12 | +
| 2.7656e-12 | +4.4407e-09 | +1.0132e-12 | +
| 1.0109e-12 | +1.0109e-12 | +4.4424e-09 | +
+The bode plot of \(\bm{G}_{\text{fj}}(s)\) is shown in Figure 23. +
+ +G_pz = diag(1./diag(dcgain(G_pz)))*G_pz; ++
+
Figure 23: 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
+
+
Figure 24: 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-09 | +3.9657e-09 | +7.7944e-09 | +
| 3.9656e-09 | +8.4979e-08 | +-1.5135e-17 | +
| 7.7944e-09 | +-3.9252e-17 | +1.834e-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. Open-Loop Noise Budgeting
+
+
Figure 25: Schematic representation of the control loop in the frame of one fast jack
+3.1. Power Spectral Density of signals
++Interferometer noise: +
+Wn = 6e-11*(1 + s/2/pi/200)/(1 + s/2/pi/60); % m/sqrt(Hz) ++
+Measurement noise: 0.79 [nm,rms] ++ + +
+DAC noise (amplified by the PI voltage amplifier, and converted to newtons): +
+Wdac = tf(3e-8); % V/sqrt(Hz) +Wu = Wdac*22.5*10; % N/sqrt(Hz) ++
+DAC noise: 0.95 [uV,rms] ++ + +
+Disturbances: +
+Wd = 5e-7/(1 + s/2/pi); % m/sqrt(Hz) ++
+Disturbance motion: 0.61 [um,rms] ++ + +
%% Save ASD of noise and disturbances +save('mat/asd_noises_disturbances.mat', 'Wn', 'Wu', 'Wd'); ++
3.2. Open Loop disturbance and measurement noise
++The comparison of the amplitude spectral density of the measurement noise and of the jack parasitic motion is performed in Figure 26. +It confirms that the sensor noise is low enough to measure the motion errors of the crystal. +
+ + +
+
Figure 26: Open Loop noise budgeting
+4. 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. +
+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 +% Strain Gauges {3x1} [m] +io(io_i) = linio([mdl, '/DCM'], 2, 'openoutput'); io_i = io_i + 1; ++
%% Extraction of the dynamics
+G_sg = linearize(mdl, io);
+
+dcgain(G_sg) ++
| 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 | +
+
Figure 27: Bode Plot of the transfer functions from piezoelectric forces to strain gauges measuremed displacements
++As the distance between the poles and zeros in Figure 30 is very small, little damping can be actively added using the strain gauges. +This will be confirmed using a Root Locus plot. +
+ +4.2. 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 28, very little damping can be added using relative damping strategy using strain gauges. +
+ + +
+
Figure 28: Root Locus for the relative damping control
+Krad = -g*Krad_g1; ++
4.3. 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('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'}; ++
+
Figure 29: Bode plot of both the open-loop plant and the damped plant using relative active damping
+5. 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. +
+5.1. 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);
+
++The Bode plot of the identified dynamics is shown in Figure 30. +At high frequency, the diagonal terms are constants while the off-diagonal terms have some roll-off. +
+ + +
+
Figure 30: Bode plot of IFF Plant
+5.2. 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. +
+ +%% Integral Force Feedback Controller +Kiff_g1 = eye(3)*1/(1 + s/2/pi/20); ++
+
Figure 31: Root Locus plot for the IFF Control strategy
+%% Integral Force Feedback Controller with optimal gain +Kiff = g*Kiff_g1; ++
%% Save the IFF controller +save('mat/Kiff.mat', 'Kiff'); ++
5.3. Damped Plant
++Both the Open Loop dynamics (see Figure 22) and the dynamics with IFF (see Figure 32) 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 32. +
+ + +
+
Figure 32: Use of Jacobian matrices to obtain the system in the frame of the fastjacks
++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 33. +It is clear that the Integral Force Feedback control strategy is very effective in damping the resonances of the plant. +
+ + +
+
Figure 33: 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 modes present in the plant. +
+ +6. Feedback Control
+
+
7. HAC-LAC (IFF) architecture
++The HAC-LAC architecture is shown in Figure 34. +
+ + +
+
Figure 34: HAC-LAC architecture
+7.1. System Identification
++Let’s identify the damped plant. +
+ + +
+
Figure 35: Bode Plot of the plant for the High Authority Controller (transfer function from \(\bm{u}^\prime\) to \(\bm{\epsilon}_d\))
+7.2. 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; ++
+
Figure 36: Bode Plot of the Loop gain for the High Authority Controller
++As shown in the Root Locus plot in Figure 37, the closed loop system should be stable. +
+ + +
+
Figure 37: Root Locus for the High Authority Controller
+7.3. 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 38. +
+ + +
+
Figure 38: 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. +
+ +7.4. Close Loop noise budget
++Let’s compute the amplitude spectral density of the jack motion errors due to the sensor noise, the actuator noise and disturbances. +
+ +%% Computation of ASD of contribution of inputs to the closed-loop motion +% Error due to disturbances +asd_d = abs(squeeze(freqresp(Wd*(1/(1 + G_dp(1,1)*Khac(1,1))), f, 'Hz'))); +% Error due to actuator noise +asd_u = abs(squeeze(freqresp(Wu*(G_dp(1,1)/(1 + G_dp(1,1)*Khac(1,1))), f, 'Hz'))); +% Error due to sensor noise +asd_n = abs(squeeze(freqresp(Wn*(G_dp(1,1)*Khac(1,1)/(1 + G_dp(1,1)*Khac(1,1))), f, 'Hz'))); ++
+The closed-loop ASD is then: +
+%% ASD of the closed-loop motion +asd_cl = sqrt(asd_d.^2 + asd_u.^2 + asd_n.^2); ++
+The obtained ASD are shown in Figure 39. +
+ + +
+
Figure 39: Closed Loop noise budget
++Let’s compare the open-loop and close-loop cases (Figure 40). +
+ +
+
Figure 40: Cumulative Power Spectrum of the open-loop and closed-loop motion error along one fast-jack
+
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