-
-
- 1. System Kinematics +
- 1. Open Loop System Identification
-
-
- 1.1. Bragg Angle -
- 1.2. Kinematics (311 Crystal) - -
- 1.3. Kinematics (111 Crystal) +
- 2. Open-Loop Noise Budgeting -
- 1.4. Kinematics (Metrology Frame) -
- 1.5. Verification -
- 1.6. Save Kinematics +
- 3. Active Damping Plant (Strain gauges) + -
- 2. Open Loop System Identification +
- 4. Active Damping Plant (Force Sensors) -
- 3. Open-Loop Noise Budgeting +
- 5. Feedback Control +
- 6. HAC-LAC (IFF) architecture - -
- 4. Active Damping Plant (Strain gauges) - - -
- 5. Active Damping Plant (Force Sensors) - - -
- 6. Feedback Control -
- 7. HAC-LAC (IFF) architecture -
-
-
- 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. +
- Section 1: the system dynamics is identified in the absence of control. +
- Section 2: an open-loop noise budget is performed. +
- Section 3: it is studied whether if the strain gauges fixed to the piezoelectric actuators can be used to actively damp the plant. +
- Section 4: 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 6: the High Authority Control - Low Authority Control (HAC-LAC) strategy is tested on the Simscape model.
-1. System Kinematics
++1. Open Loop System Identification
- --- -1.1. Bragg Angle
++- -1.1. Identification
--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]; -
--
- -Table 1: Sensor Jacobian \(\bm{J}_{s,311}\) - -- - -- - - - - - - -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
--
-Table 2: Inverse of the sensor Jacobian \(\bm{J}_{s,311}^{-1}\) - -- - -- - - - - - - -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]; -
--
- -Table 3: Sensor Jacobian - Primary Crystal - \(\bm{J}_{s,311}\) - -- - -- - - - - - - --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
--
-Table 4: Inverse of the sensor Jacobian \(\bm{J}_{s,311}^{-1}\) - -- - -- - - - - - - --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]; -
--
- -Table 5: Actuator Jacobian \(\bm{J}_{a,311}\) - -- - -- - - - - - - -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
--
-Table 6: Inverse of the actuator Jacobian \(\bm{J}_{a,311}^{-1}\) - -- - -- - - - - - - -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]; -
--
- -Table 7: Sensor Jacobian \(\bm{J}_{s,111}\) - -- - -- - - - - - - -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
--
-Table 8: Inverse of the sensor Jacobian \(\bm{J}_{s,111}^{-1}\) - -- - -- - - - - - - -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]; -
--
- -Table 9: Sensor Jacobian \(\bm{J}_{s,111}\) - -- - -- - - - - - - --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
--
-Table 10: Inverse of the sensor Jacobian \(\bm{J}_{s,111}^{-1}\) - -- - -- - - - - - - --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]; -
--
- -Table 11: Actuator Jacobian \(\bm{J}_{a,111}\) - -- - -- - - - - - - -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
--
-Table 12: Inverse of the actuator Jacobian \(\bm{J}_{a,111}^{-1}\) - -- - -- - - - - - - -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 +
- 3 outputs: measured displacement by the 3 interferometers pointing at the ring second crystal
-It is schematically shown in Figure 21. +It is schematically shown in Figure 1.
-+
-
Figure 21: Dynamical system with inputs and outputs
+Figure 1: Dynamical system with inputs and outputs
@@ -1413,29 +162,29 @@ State-space model with 3 outputs, 3 inputs, and 24 states.
-2.2. Plant in the frame of the fastjacks
-++1.2. 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). +Using the forward and inverse kinematics, we can computed the dynamics from piezo forces to axial motion of the 3 fastjacks (see Figure 2).
-+-
-
Figure 22: Use of Jacobian matrices to obtain the system in the frame of the fastjacks
+Figure 2: Use of Jacobian matrices to obtain the system in the frame of the fastjacks
@@ -1447,8 +196,8 @@ The DC gain of the new system shows that the system is well decoupled at low fre%% Compute the system in the frame of the fastjacks -G_pz = J_a_311*inv(J_s_311)*G; +G_pz = J_a_h*inv(J_2h_s)*G;
+-
-
Figure 27: Bode Plot of the transfer functions from piezoelectric forces to strain gauges measuremed displacements
+Figure 7: 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. +As the distance between the poles and zeros in Figure 10 is very small, little damping can be actively added using the strain gauges. This will be confirmed using a Root Locus plot.
@@ -1731,23 +480,23 @@ This will be confirmed using a Root Locus plot.-4.2. 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);
-As can be seen in Figure 28, very little damping can be added using relative damping strategy using strain gauges. +As can be seen in Figure 8, very little damping can be added using relative damping strategy using strain gauges.
-+-
-
Figure 28: Root Locus for the relative damping control
+Figure 8: Root Locus for the relative damping control
@@ -1757,9 +506,9 @@ As can be seen in Figure 28, very little damping can b-4.3. Damped Plant
-++-3.3. Damped Plant
+The controller is implemented on Simscape, and the damped plant is identified.
@@ -1787,7 +536,7 @@ 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 = J_a_r*inv(J_s_r)*linearize(mdl, io); G_ol.InputName = {'u_ur', 'u_uh', 'u_d'}; G_ol.OutputName = {'d_ur', 'd_uh', 'd_d'};
@@ -1796,37 +545,37 @@ G_ol.OutputName = {'d_ur',%% 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 = J_a_r*inv(J_s_r)*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
+Figure 9: Bode plot of both the open-loop plant and the damped plant using relative active damping
-5. Active Damping Plant (Force Sensors)
-++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. +See for instance (Fleming and Leang 2010).
-5.1. Identification
-++4.1. Identification
+-%% Input/Output definition clear io; io_i = 1; @@ -1848,22 +597,22 @@ G_fs = linearize(mdl, io);-The Bode plot of the identified dynamics is shown in Figure 30. +The Bode plot of the identified dynamics is shown in Figure 10. At high frequency, the diagonal terms are constants while the off-diagonal terms have some roll-off.
-+
-
Figure 30: Bode plot of IFF Plant
+Figure 10: Bode plot of IFF Plant
-5.2. 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. @@ -1876,10 +625,10 @@ Kiff_g1 = eye(3)*1/ -
+-
-
Figure 31: Root Locus plot for the IFF Control strategy
+Figure 11: Root Locus plot for the IFF Control strategy
@@ -1896,38 +645,38 @@ save('mat/Kiff.mat', 'K-5.3. Damped Plant
-++4.3. Damped Plant
+-Both the Open Loop dynamics (see Figure 22) and the dynamics with IFF (see Figure 32) are identified. +Both the Open Loop dynamics (see Figure 2) and the dynamics with IFF (see Figure 12) 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. +This is schematically represented in Figure 12.
-+
-
Figure 32: Use of Jacobian matrices to obtain the system in the frame of the fastjacks
+Figure 12: 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. +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 13. 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
+Figure 13: 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.
@@ -1937,53 +686,53 @@ The Integral Force Feedback control strategy is very effective in damping the mo-6. Feedback Control
-++5. Feedback Control
+--+
-7. HAC-LAC (IFF) architecture
-++6. HAC-LAC (IFF) architecture
+-The HAC-LAC architecture is shown in Figure 34. +The HAC-LAC architecture is shown in Figure 14.
-+-
-
Figure 34: HAC-LAC architecture
+Figure 14: HAC-LAC architecture
-7.1. System Identification
-++6.1. 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 \(\bm{\epsilon}_d\))
+Figure 15: 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
-++6.2. 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. @@ -2014,28 +763,28 @@ L_hac_lac = G_dp * Khac;
+-
-
Figure 36: Bode Plot of the Loop gain for the High Authority Controller
+Figure 16: 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. +As shown in the Root Locus plot in Figure 17, the closed loop system should be stable.
-+
-
Figure 37: Root Locus for the High Authority Controller
+Figure 17: Root Locus for the High Authority Controller
-7.3. Performances
-++6.3. 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.
@@ -2055,17 +804,17 @@ It is first verified that the closed-loop system is stable:-And both transmissibilities are compared in Figure 38. +And both transmissibilities are compared in Figure 18.
-+-
-
Figure 38: Comparison of the transmissibility of errors from vibrations of the stepper motor between the open-loop case and the hac-iff case.
+Figure 18: 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. @@ -2075,9 +824,9 @@ This reduction is effective inside the bandwidth of the controller.
-7.4. Close Loop noise budget
-++6.4. Close Loop noise budget
+diff --git a/dcm-simscape-model.org b/dcm-simscape-model.org index ee23c61..87a9978 100644 --- a/dcm-simscape-model.org +++ b/dcm-simscape-model.org @@ -54,1232 +54,12 @@ 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 [[sec:dcm_kinematics]]: the kinematics of the DCM is presented, and Jacobian matrices which are used to solve the inverse and forward kinematics are computed. - Section [[sec:open_loop_identification]]: the system dynamics is identified in the absence of control. - Section [[sec:dcm_noise_budget]]: an open-loop noise budget is performed. - Section [[sec:active_damping_strain_gauges]]: it is studied whether if the strain gauges fixed to the piezoelectric actuators can be used to actively damp the plant. - Section [[sec:active_damping_iff]]: 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 [[sec:hac_iff]]: the High Authority Control - Low Authority Control (HAC-LAC) strategy is tested on the Simscape model. -* System Kinematics -:PROPERTIES: -:header-args:matlab+: :tangle matlab/dcm_kinematics.m -:END: -<Let’s compute the amplitude spectral density of the jack motion errors due to the sensor noise, the actuator noise and disturbances.
@@ -2103,24 +852,24 @@ asd_cl = sqrt(asd_d.^2 39. +The obtained ASD are shown in Figure 19. -+@@ -2128,7 +877,7 @@ Let’s compare the open-loop and close-loop cases (Figure -
-
Figure 39: Closed Loop noise budget
+Figure 19: Closed Loop noise budget
-Let’s compare the open-loop and close-loop cases (Figure 40). +Let’s compare the open-loop and close-loop cases (Figure 20).
-+
-
Figure 40: Cumulative Power Spectrum of the open-loop and closed-loop motion error along one fast-jack
+Figure 20: Cumulative Power Spectrum of the open-loop and closed-loop motion error along one fast-jack
Created: 2022-02-15 mar. 14:18
+Created: 2022-06-02 Thu 18:11
> -#+end_src - -#+begin_src matlab :exports none :results silent :noweb yes -< > -#+end_src - -#+begin_src matlab :tangle no :noweb yes -< > -#+end_src - -#+begin_src matlab :eval no :noweb yes -< > -#+end_src - -#+begin_src matlab :noweb yes -< > -#+end_src - -** Bragg Angle -<
