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ESRF Double Crystal Monochromator - Dynamical Multi-Body Model

Table of Contents

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

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
\begin{equation} \label{eq:bragg_angle_formula} d_z = \frac{d_{\text{off}}}{2 \cos \theta_b} \end{equation}

This relation is shown in Figure 1.

jack_motion_bragg_angle.png

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 (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 \(\bm{J}_{s,111}\)
1.0 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,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 \(\bm{J}_{s,111}^{-1}\)
0.25 0.5 0.25
7.14 0.0 -7.14
-16.67 33.33 -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 \(\bm{J}_{a,111}\)
1.0 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}

schematic_actuator_jacobian_forward_kinematics.png

Figure 7: Forward Kinematics - Actuators for 111 crystal

Table 4: Inverse of the actuator Jacobian \(\bm{J}_{a,111}^{-1}\)
0.0568 0.4432 0.5
1.7857 1.7857 -3.5714
-4.5455 4.5455 0.0

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('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-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 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-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

noise_budget_dcm_schematic_fast_jack_frame.png

Figure 12: 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 13. It confirms that the sensor noise is low enough to measure the motion errors of the crystal.

open_loop_noise_budget_fast_jack.png

Figure 13: 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

strain_gauge_plant_bode_plot.png

Figure 14: Bode Plot of the transfer functions from piezoelectric forces to strain gauges measuremed displacements

As the distance between the poles and zeros in Figure 17 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 15, very little damping can be added using relative damping strategy using strain gauges.

relative_damping_root_locus.png

Figure 15: 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'};

comp_damp_undamped_plant_rad_bode_plot.png

Figure 16: 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 17. At high frequency, the diagonal terms are constants while the off-diagonal terms have some roll-off.

iff_plant_bode_plot.png

Figure 17: 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);

iff_root_locus.png

Figure 18: 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 9) and the dynamics with IFF (see Figure 19) 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 19.

schematic_jacobian_frame_fastjack_iff.png

Figure 19: 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 20. 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 20: 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. HAC-LAC (IFF) architecture

The HAC-LAC architecture is shown in Figure 21.

schematic_jacobian_frame_fastjack_hac_iff.png

Figure 21: HAC-LAC architecture

6.1. System Identification

Let’s identify the damped plant.

bode_plot_hac_iff_plant.png

Figure 22: Bode Plot of the plant for the High Authority Controller (transfer function from \(\bm{u}^\prime\) to \(\bm{\epsilon}_d\))

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. 

%% 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 23: Bode Plot of the Loop gain for the High Authority Controller

As shown in the Root Locus plot in Figure 24, the closed loop system should be stable.

loci_hac_iff_fast_jack.png

Figure 24: Root Locus for the High Authority Controller

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.

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

isstable(T_hl)

1

And both transmissibilities are compared in Figure 25.

stepper_transmissibility_comp_ol_hac_iff.png

Figure 25: 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.

6.4. 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 26.

close_loop_asd_noise_budget_hac_iff.png

Figure 26: Closed Loop noise budget

Let’s compare the open-loop and close-loop cases (Figure 27).

cps_comp_ol_cl_hac_iff.png

Figure 27: Cumulative Power Spectrum of the open-loop and closed-loop motion error along one fast-jack

Author: Dehaeze Thomas

Created: 2021-11-30 mar. 17:54