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#+TITLE : ESRF Double Crystal Monochromator - Dynamical Multi-Body Model
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:DRAWER:
#+LANGUAGE : en
#+EMAIL : dehaeze.thomas@gmail.com
#+AUTHOR : Dehaeze Thomas
#+HTML_LINK_HOME : ../index.html
#+HTML_LINK_UP : ../index.html
#+HTML_HEAD : <link rel="stylesheet" type="text/css" href="https://research.tdehaeze.xyz/css/style.css"/>
#+HTML_HEAD : <script type="text/javascript" src="https://research.tdehaeze.xyz/js/script.js"></script>
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#+LaTeX_CLASS : scrreprt
#+LaTeX_CLASS_OPTIONS : [a4paper, 10pt, DIV=12, parskip=full]
#+LaTeX_HEADER_EXTRA : \input{preamble.tex}
#+LATEX_HEADER_EXTRA : \bibliography{ref}
#+PROPERTY : header-args:matlab :session *MATLAB*
#+PROPERTY : header-args:matlab+ :comments org
#+PROPERTY : header-args:matlab+ :exports both
#+PROPERTY : header-args:matlab+ :results none
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#+PROPERTY : header-args:matlab+ :tangle no
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#+PROPERTY : header-args:matlab+ :eval no-export
#+PROPERTY : header-args:matlab+ :noweb yes
#+PROPERTY : header-args:matlab+ :mkdirp yes
#+PROPERTY : header-args:matlab+ :output-dir figs
#+PROPERTY : header-args:latex :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{$HOME/Cloud/tikz/org/}{config.tex}")
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#+PROPERTY : header-args:latex+ :imoutoptions -quality 100
#+PROPERTY : header-args:latex+ :results file raw replace
#+PROPERTY : header-args:latex+ :buffer no
#+PROPERTY : header-args:latex+ :tangle no
#+PROPERTY : header-args:latex+ :eval no-export
#+PROPERTY : header-args:latex+ :exports results
#+PROPERTY : header-args:latex+ :mkdirp yes
#+PROPERTY : header-args:latex+ :output-dir figs
#+PROPERTY : header-args:latex+ :post pdf2svg(file=*this*, ext="png")
:END:
#+begin_export html
<hr >
<p >This report is also available as a <a href="./dcm-simscape.pdf" >pdf</a >.</p >
<hr >
#+end_export
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#+latex : \clearpage
* Introduction :ignore:
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: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.
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* System Kinematics
:PROPERTIES:
:header-args:matlab+: :tangle matlab/dcm_kinematics.m
:END:
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<<sec:dcm_kinematics >>
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** Introduction :ignore:
** Matlab Init :noexport:ignore:
#+begin_src matlab
%% dcm_kinematics.m
% Computation of the DCM kinematics
#+end_src
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<<matlab-dir >>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init >>
#+end_src
#+begin_src matlab :tangle no :noweb yes
<<m-init-path >>
#+end_src
#+begin_src matlab :eval no :noweb yes
<<m-init-path-tangle >>
#+end_src
#+begin_src matlab :noweb yes
<<m-init-other >>
#+end_src
** Bragg Angle
#+begin_src matlab
%% Tested bragg angles
bragg = linspace(5, 80, 1000); % Bragg angle [deg]
d_off = 10.5e-3; % Wanted offset between x-rays [m]
#+end_src
#+begin_src matlab
%% Vertical Jack motion as a function of Bragg angle
dz = d_off./(2*cos(bragg*pi/180));
#+end_src
#+begin_src matlab :exports none
%% Jack motion as a function of Bragg angle
figure;
plot(bragg, 1e3*dz)
xlabel('Bragg angle [deg]'); ylabel('Jack Motion [mm]');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/jack_motion_bragg_angle.pdf', 'width', 'wide', 'height', 'normal');
#+end_src
#+name : fig:jack_motion_bragg_angle
#+caption : Jack motion as a function of Bragg angle
#+RESULTS :
[[file:figs/jack_motion_bragg_angle.png ]]
#+begin_src matlab :results value replace :exports both
%% Required Jack stroke
ans = 1e3*(dz(end) - dz(1))
#+end_src
#+RESULTS :
: 24.963
** Kinematics (111 Crystal)
*** Introduction :ignore:
The reference frame is taken at the center of the 111 second crystal.
*** 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 [[fig:sensor_111_crystal_points ]] as well as the origin where the motion of the crystal is computed.
#+begin_src latex :file sensor_111_crystal_points.pdf
\begin{tikzpicture}
% Crystal
\draw (-15/2, -3.5/2) rectangle (15/2, 3.5/2);
% Measurement Points
\node[branch] (a1) at (-7, 1.5){};
\node[branch] (a2) at ( 0, -1.5){};
\node[branch] (a3) at ( 7, 1.5){};
% Labels
\node[right] at (a1) {$\mathcal{O}_1 = (-0.07, -0.015)$};
\node[right] at (a2) {$\mathcal{O}_2 = (0, 0.015)$};
\node[left] at (a3) {$\mathcal{O}_3 = ( 0.07, -0.015)$};
% Origin
\draw[->] (0, 0) node[] -- ++(1, 0) node[right]{$x$};
\draw[->] (0, 0) -- ++(0, -1) node[below]{$y$};
\draw[fill, color=black] (0, 0) circle (0.05);
\node[left] at (0,0) {$\mathcal{O}_{111}$};
\end{tikzpicture}
#+end_src
#+name : fig:sensor_111_crystal_points
#+caption : Bottom view of the second crystal 111. Position of the measurement points.
#+RESULTS :
[[file:figs/sensor_111_crystal_points.png ]]
The inverse kinematics consisting of deriving the interferometer measurements from the motion of the crystal (see Figure [[fig:schematic_sensor_jacobian_inverse_kinematics ]]):
\begin{equation}
\begin{bmatrix}
x_1 \\ x_2 \\ x_3
\end{bmatrix}
=
\bm{J}_{s,111}
\begin{bmatrix}
d_z \\ r_y \\ r_x
\end{bmatrix}
\end{equation}
#+begin_src latex :file schematic_sensor_jacobian_inverse_kinematics.pdf
\begin{tikzpicture}
% Blocs
\node[block] (Js) {$\bm{J}_{s,111}$};
% Connections and labels
\draw[->] ($(Js.west)+(-1.5,0)$) node[above right]{$\begin{bmatrix} d_z \\ r_y \\ r_x \end{bmatrix}$} -- (Js.west);
\draw[->] (Js.east) -- ++(1.5, 0) node[above left]{$\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$};
\end{tikzpicture}
#+end_src
#+name : fig:schematic_sensor_jacobian_inverse_kinematics
#+caption : Inverse Kinematics - Interferometers
#+RESULTS :
[[file:figs/schematic_sensor_jacobian_inverse_kinematics.png ]]
From the Figure [[fig:sensor_111_crystal_points ]], 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}
#+begin_src matlab
%% Sensor Jacobian matrix for 111 crystal
J_s_111 = [1, 0.07, -0.015
1, 0, 0.015
1, -0.07, -0.015];
#+end_src
#+begin_src matlab :exports results :results value table replace :tangle no
data2orgtable(J_s_111, {}, {}, ' %.3f ');
#+end_src
#+name : tab:jacobian_sensor_111
#+caption : Sensor Jacobian $\bm{J}_{s,111}$
#+attr_latex : :environment tabularx :width 0.3\linewidth :align ccc
#+attr_latex : :center t :booktabs t
#+RESULTS :
| 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 [[fig:schematic_sensor_jacobian_forward_kinematics ]]).
\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}
#+begin_src latex :file schematic_sensor_jacobian_forward_kinematics.pdf
\begin{tikzpicture}
% Blocs
\node[block] (Js_inv) {$\bm{J}_ {s,111}^{-1}$};
% Connections and labels
\draw[->] ($(Js_inv.west)+(-1.5,0)$) node[above right]{$\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$} -- (Js_inv.west);
\draw[->] (Js_inv.east) -- ++(1.5, 0) node[above left]{$\begin{bmatrix} d_z \\ r_y \\ r_x \end{bmatrix}$};
\end{tikzpicture}
#+end_src
#+name : fig:schematic_sensor_jacobian_forward_kinematics
#+caption : Forward Kinematics - Interferometers
#+RESULTS :
[[file:figs/schematic_sensor_jacobian_forward_kinematics.png ]]
#+begin_src matlab :exports results :results value table replace :tangle no
data2orgtable(inv(J_s_111), {}, {}, ' %.2f ');
#+end_src
#+name : tab:inverse_jacobian_sensor_111
#+caption : Inverse of the sensor Jacobian $\bm{J}_{s,111}^{-1}$
#+attr_latex : :environment tabularx :width 0.3\linewidth :align ccc
#+attr_latex : :center t :booktabs t
#+RESULTS :
| 0.25 | 0.5 | 0.25 |
| 7.14 | 0.0 | -7.14 |
| -16.67 | 33.33 | -16.67 |
*** Piezo - 111 Crystal
The location of the actuators with respect with the center of the 111 second crystal are shown in Figure [[fig:actuator_jacobian_111_points ]].
#+name : fig:actuator_jacobian_111_points
#+caption : Location of actuators with respect to the center of the 111 second crystal (bottom view)
#+attr_latex : :width \linewidth
[[file:figs/actuator_jacobian_111_points.png ]]
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}
#+begin_src latex :file schematic_actuator_jacobian_inverse_kinematics.pdf
\begin{tikzpicture}
% Blocs
\node[block] (Ja) {$\bm{J}_{a,111}$};
% Connections and labels
\draw[->] ($(Ja.west)+(-1.5,0)$) node[above right]{$\begin{bmatrix} d_z \\ r_y \\ r_x \end{bmatrix}$} -- (Ja.west);
\draw[->] (Ja.east) -- ++(1.5, 0) node[above left]{$\begin{bmatrix} d_{u_r} \\ d_ {u_h} \\ d_d \end{bmatrix}$};
\end{tikzpicture}
#+end_src
#+name : fig:schematic_sensor_jacobian_inverse_kinematics
#+caption : Inverse Kinematics - Actuators
#+RESULTS :
[[file:figs/schematic_actuator_jacobian_inverse_kinematics.png ]]
Based on the geometry in Figure [[fig:actuator_jacobian_111_points ]], 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}
#+begin_src matlab
%% Actuator Jacobian - 111 crystal
J_a_111 = [1, 0.14, -0.1525
1, 0.14, 0.0675
1, -0.14, -0.0425];
#+end_src
#+begin_src matlab :exports results :results value table replace :tangle no
data2orgtable(J_a_111, {}, {}, ' %.4f ');
#+end_src
#+name : tab:jacobian_actuator_111
#+caption : Actuator Jacobian $\bm{J}_{a,111}$
#+attr_latex : :environment tabularx :width 0.3\linewidth :align ccc
#+attr_latex : :center t :booktabs t
#+RESULTS :
| 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}
#+begin_src latex :file schematic_actuator_jacobian_forward_kinematics.pdf
\begin{tikzpicture}
% Blocs
\node[block] (Ja_inv) {$\bm{J}_ {a,111}^{-1}$};
% Connections and labels
\draw[->] ($(Ja_inv.west)+(-1.5,0)$) node[above right]{$\begin{bmatrix} d_ {u_r} \\ d_ {u_h} \\ d_d \end{bmatrix}$} -- (Ja_inv.west);
\draw[->] (Ja_inv.east) -- ++(1.5, 0) node[above left]{$\begin{bmatrix} d_z \\ r_y \\ r_x \end{bmatrix}$};
\end{tikzpicture}
#+end_src
#+name : fig:schematic_actuator_jacobian_forward_kinematics
#+caption : Forward Kinematics - Actuators for 111 crystal
#+RESULTS :
[[file:figs/schematic_actuator_jacobian_forward_kinematics.png ]]
#+begin_src matlab :exports results :results value table replace :tangle no
data2orgtable(inv(J_a_111), {}, {}, ' %.4f ');
#+end_src
#+name : tab:inverse_jacobian_actuator_111
#+caption : Inverse of the actuator Jacobian $\bm{J}_{a,111}^{-1}$
#+attr_latex : :environment tabularx :width 0.3\linewidth :align ccc
#+attr_latex : :center t :booktabs t
#+RESULTS :
| 0.0568 | 0.4432 | 0.5 |
| 1.7857 | 1.7857 | -3.5714 |
| -4.5455 | 4.5455 | 0.0 |
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** 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
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** Save Kinematics
#+begin_src matlab :exports none :tangle no
save('matlab/mat/dcm_kinematics.mat', 'J_a_111', 'J_s_111')
#+end_src
#+begin_src matlab :eval no
save('mat/dcm_kinematics.mat', 'J_a_111', 'J_s_111')
#+end_src
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* Open Loop System Identification
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:PROPERTIES:
:header-args:matlab+: :tangle matlab/dcm_identification.m
:END:
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<<sec:open_loop_identification >>
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** Introduction :ignore:
** Matlab Init :noexport:ignore:
#+begin_src matlab
%% dcm_identification.m
% Extraction of system dynamics using Simscape model
#+end_src
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<<matlab-dir >>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init >>
#+end_src
#+begin_src matlab :tangle no :noweb yes
<<m-init-path >>
#+end_src
#+begin_src matlab :eval no :noweb yes
<<m-init-path-tangle >>
#+end_src
#+begin_src matlab :noweb yes
<<m-init-simscape >>
#+end_src
#+begin_src matlab :noweb yes
<<m-init-other >>
#+end_src
** 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 [[fig:schematic_system_inputs_outputs ]].
#+begin_src latex :file schematic_system_inputs_outputs.pdf
\begin{tikzpicture}
% Blocs
\node[block] (G) {$\bm{G}(s)$};
% Connections and labels
\draw[->] ($(G.west)+(-1.5,0)$) node[above right]{$\begin{bmatrix} u_{u_r} \\ u_ {u_h} \\ u_d \end{bmatrix}$} -- (G.west);
\draw[->] (G.east) -- ++(1.5, 0) node[above left]{$\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$};
\end{tikzpicture}
#+end_src
#+name : fig:schematic_system_inputs_outputs
#+caption : Dynamical system with inputs and outputs
#+RESULTS :
[[file:figs/schematic_system_inputs_outputs.png ]]
The system is identified from the Simscape model.
#+begin_src matlab
%% 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;
#+end_src
#+begin_src matlab
%% Extraction of the dynamics
G = linearize(mdl, io);
#+end_src
#+begin_src matlab :exports none
%% Input and Output names
G.InputName = {'u_ur', 'u_uh', 'u_d'};
G.OutputName = {'int_111_1', 'int_111_2', 'int_111_3'};
#+end_src
#+begin_src matlab :results output replace :exports both :tangle no
size(G)
#+end_src
#+RESULTS :
: size(G)
: State-space model with 3 outputs, 3 inputs, and 24 states.
** Plant in the frame of the fastjacks
#+begin_src matlab
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load('dcm_kinematics.mat');
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#+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 ]]).
#+begin_src latex :file schematic_jacobian_frame_fastjack.pdf
\begin{tikzpicture}
% Blocs
\node[block] (G) {$\bm{G}(s)$};
\node[block, right=1.5 of G] (Js) {$\bm{J}_{s}^{-1}$};
\node[block, right=1.5 of Js] (Ja) {$\bm{J}_{a}$};
% Connections and labels
\draw[->] ($(G.west)+(-1.5,0)$) node[above right]{$\begin{bmatrix} u_{u_r} \\ u_ {u_h} \\ u_d \end{bmatrix}$} -- (G.west);
\draw[->] (G.east) -- 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 west) ($(Ja.east)+(0, 1.4)$)}, fill=black!20!white, draw, inner sep=6pt] (system) {};
\node[above] at (system.north) {$\bm{G}_{\text{fj}}(s)$};
\end{scope}
\end{tikzpicture}
#+end_src
#+name : fig:schematic_jacobian_frame_fastjack
#+caption : Use of Jacobian matrices to obtain the system in the frame of the fastjacks
#+RESULTS :
[[file:figs/schematic_jacobian_frame_fastjack.png ]]
#+begin_src matlab
%% Compute the system in the frame of the fastjacks
G_pz = J_a_111*inv(J_s_111)*G;
#+end_src
The DC gain of the new system shows that the system is well decoupled at low frequency.
#+begin_src matlab :results value replace :exports both :tangle no
dcgain(G_pz)
#+end_src
#+name : tab:dc_gain_plan_fj
#+caption : DC gain of the plant in the frame of the fast jacks $\bm{G}_{\text{fj}}$
#+attr_latex : :environment tabularx :width 0.5\linewidth :align ccc
#+attr_latex : :center t :booktabs t
#+RESULTS :
| 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 [[fig:bode_plot_plant_fj ]].
#+begin_src matlab :exports none
%% Bode plot for the plant
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(G_pz(1,1), freqs, 'Hz'))), ...
'DisplayName', 'd');
plot(freqs, abs(squeeze(freqresp(G_pz(2,2), freqs, 'Hz'))), ...
'DisplayName', 'uh');
plot(freqs, abs(squeeze(freqresp(G_pz(3,3), freqs, 'Hz'))), ...
'DisplayName', 'ur');
for i = 1:2
for j = i+1:3
plot(freqs, abs(squeeze(freqresp(G_pz(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', 3);
ylim([1e-13, 1e-6]);
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(G_pz(1,1), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(G_pz(2,2), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(G_pz(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, 180]);
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_plant_fj.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name : fig:bode_plot_plant_fj
#+caption : Bode plot of the diagonal and off-diagonal elements of the plant in the frame of the fast jacks
#+RESULTS :
[[file:figs/bode_plot_plant_fj.png ]]
#+begin_important
Computing the system in the frame of the fastjack gives good decoupling at low frequency (until the first resonance of the system).
#+end_important
** Plant in the frame of the crystal
#+begin_src latex :file schematic_jacobian_frame_crystal.pdf
\begin{tikzpicture}
% Blocs
\node[block] (G) {$\bm{G}(s)$};
\node[block, left=1.5 of G] (Ja) {$\bm{J}_{a}^{-T}$};
\node[block, right=1.5 of G] (Js) {$\bm{J}_{s}^{-1}$};
% Connections and labels
\draw[->] ($(Ja.west)+(-1.5,0)$) node[above right]{$\begin{bmatrix} \mathcal{F}_{z} \\ \mathcal{M}_ {y} \\ \mathcal{M}_{x} \end{bmatrix}$} -- (Ja.west);
\draw[->] (Ja.east) -- node[midway, above]{$\begin{bmatrix} u_{u_r} \\ u_ {u_h} \\ u_d \end{bmatrix}$} (G.west);
\draw[->] (G.east) -- node[midway, above]{$\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$} (Js.west);
\draw[->] (Js.east) -- ++(1.5, 0) node[above left]{$\begin{bmatrix} d_z \\ r_y \\ r_x \end{bmatrix}$};
\begin{scope}[on background layer]
\node[fit={(Ja.south west) ($(Js.east)+(0, 1.4)$)}, fill=black!20!white, draw, inner sep=6pt] (system) {};
\node[above] at (system.north) {$\bm{G}_{\text{cr}}(s)$};
\end{scope}
\end{tikzpicture}
#+end_src
#+name : fig:schematic_jacobian_frame_crystal
#+caption : Use of Jacobian matrices to obtain the system in the frame of the crystal
#+RESULTS :
[[file:figs/schematic_jacobian_frame_crystal.png ]]
#+begin_src matlab
G_mr = inv(J_s_111)*G*inv(J_a_111');
#+end_src
#+begin_src matlab :results value replace :exports both :tangle no
dcgain(G_mr)
#+end_src
#+RESULTS :
| 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 |
#+begin_src matlab :exports none
%% Bode plot for the plant
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(G_mr(1,1), freqs, 'Hz'))), ...
'DisplayName', 'd');
plot(freqs, abs(squeeze(freqresp(G_mr(2,2), freqs, 'Hz'))), ...
'DisplayName', 'uh');
plot(freqs, abs(squeeze(freqresp(G_mr(3,3), freqs, 'Hz'))), ...
'DisplayName', 'ur');
for i = 1:2
for j = i+1:3
plot(freqs, abs(squeeze(freqresp(G_mr(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', 'southwest', 'FontSize', 8, 'NumColumns', 2);
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(G_mr(1,1), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(G_mr(2,2), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(G_mr(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, 180]);
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
#+end_src
#+begin_src matlab :exports none
%% Bode plot for the plant
fig = figure;
tiledlayout(3, 3, 'TileSpacing', 'Compact', 'Padding', 'None');
for i_out = 1:3
for i_in = 1:3
ax = nexttile;
plot(freqs, abs(squeeze(freqresp(G_mr(i_out, i_in), freqs, 'Hz'))));
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
end
end
linkaxes(findall(fig, 'type', 'axes'),'xy');
xlim([freqs(1), freqs(end)]);
#+end_src
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.
** Plant at the center of stiffness :noexport:
Here, we map the piezo forces at the center of stiffness.
Let's first compute the Jacobian:
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* Active Damping Plant (Strain gauges)
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:PROPERTIES:
:header-args:matlab+: :tangle matlab/dcm_active_damping_strain_gauges.m
:END:
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<<sec:active_damping_strain_gauges >>
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** Introduction :ignore:
In this section, we wish to see whether if strain gauges fixed to the piezoelectric actuator can be used for active damping.
** Matlab Init :noexport:ignore:
#+begin_src matlab
%% dcm_active_damping_strain_gauges.m
% Active Damping using relative motion sensors (strain gauges)
#+end_src
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<<matlab-dir >>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init >>
#+end_src
#+begin_src matlab :tangle no :noweb yes
<<m-init-path >>
#+end_src
#+begin_src matlab :eval no :noweb yes
<<m-init-path-tangle >>
#+end_src
#+begin_src matlab :noweb yes
<<m-init-simscape >>
#+end_src
#+begin_src matlab :noweb yes
<<m-init-other >>
#+end_src
** Identification
#+begin_src matlab
%% Input/Output definition
clear io; io_i = 1;
%% Inputs
% Control Input {3x1} [N]
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io(io_i) = linio([mdl, '/control_system'], 1, 'openinput'); io_i = io_i + 1;
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%% Outputs
% Strain Gauges {3x1} [m]
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io(io_i) = linio([mdl, '/DCM'], 2, 'openoutput'); io_i = io_i + 1;
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#+end_src
#+begin_src matlab
%% Extraction of the dynamics
G_sg = linearize(mdl, io);
#+end_src
#+begin_src matlab :exports none
G_sg.InputName = {'u_ur', 'u_uh', 'u_d'};
G_sg.OutputName = {'sg_ur', 'sg_uh', 'sg_d'};
#+end_src
#+begin_src matlab :results value replace :exports both :tangle no
dcgain(G_sg)
#+end_src
#+RESULTS :
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| 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 |
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#+begin_src matlab :exports none
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%% Bode plot for the plant (strain gauge output)
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figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(G_sg(1,1), freqs, 'Hz'))), ...
'DisplayName', 'd');
plot(freqs, abs(squeeze(freqresp(G_sg(2,2), freqs, 'Hz'))), ...
'DisplayName', 'uh');
plot(freqs, abs(squeeze(freqresp(G_sg(3,3), freqs, 'Hz'))), ...
'DisplayName', 'ur');
for i = 1:2
for j = i+1:3
plot(freqs, abs(squeeze(freqresp(G_sg(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',[]);
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legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2);
ylim([1e-14, 1e-7]);
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ax2 = nexttile;
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(G_sg(1,1), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(G_sg(2,2), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(G_sg(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);
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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/strain_gauge_plant_bode_plot.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name : fig:strain_gauge_plant_bode_plot
#+caption : Bode Plot of the transfer functions from piezoelectric forces to strain gauges measuremed displacements
#+RESULTS :
[[file:figs/strain_gauge_plant_bode_plot.png ]]
#+begin_important
As the distance between the poles and zeros in Figure [[fig:iff_plant_bode_plot ]] is very small, little damping can be actively added using the strain gauges.
This will be confirmed using a Root Locus plot.
#+end_important
** Relative Active Damping
#+begin_src matlab
Krad_g1 = eye(3)*s/(s^2/ (2*pi*500)^2 + 2*s/(2*pi*500) + 1);
#+end_src
As can be seen in Figure [[fig:relative_damping_root_locus ]], very little damping can be added using relative damping strategy using strain gauges.
#+begin_src matlab :exports none
%% Root Locus for IFF
gains = logspace(3, 8, 200);
figure;
hold on;
plot(real(pole(G_sg)), imag(pole(G_sg)), 'x', 'color', colors(1,:), ...
'DisplayName', '$g = 0$');
plot(real(tzero(G_sg)), imag(tzero(G_sg)), 'o', 'color', colors(1,:), ...
'HandleVisibility', 'off');
for g = gains
clpoles = pole(feedback(G_sg, g*Krad_g1, -1));
plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:), ...
'HandleVisibility', 'off');
end
% Optimal gain
g = 2e5;
clpoles = pole(feedback(G_sg, g*Krad_g1, -1));
plot(real(clpoles), imag(clpoles), 'x', 'color', colors(2,:), ...
'DisplayName', sprintf('$g=%.0e$', g));
hold off;
xlim([-6, 0]); ylim([0, 2700]);
xlabel('Real Part'); ylabel('Imaginary Part');
legend('location', 'northwest');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/relative_damping_root_locus.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name : fig:relative_damping_root_locus
#+caption : Root Locus for the relative damping control
#+RESULTS :
[[file:figs/relative_damping_root_locus.png ]]
#+begin_src matlab
Krad = -g*Krad_g1;
#+end_src
** Damped Plant
The controller is implemented on Simscape, and the damped plant is identified.
#+begin_src matlab
%% 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
%% DCM Kinematics
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load('dcm_kinematics.mat');
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#+end_src
#+begin_src matlab
%% 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'};
#+end_src
#+begin_src matlab
%% 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'};
#+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_ol(1,1), freqs, 'Hz'))), ...
'DisplayName', 'd - OL');
plot(freqs, abs(squeeze(freqresp(G_ol(2,2), freqs, 'Hz'))), ...
'DisplayName', 'uh - OL');
plot(freqs, abs(squeeze(freqresp(G_ol(3,3), freqs, 'Hz'))), ...
'DisplayName', 'ur - OL');
set(gca,'ColorOrderIndex',1)
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-6]);
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(G_ol(1,1), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(G_ol(2,2), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(G_ol(3,3), freqs, 'Hz'))));
set(gca,'ColorOrderIndex',1)
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]);
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linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
#+end_src
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#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/comp_damp_undamped_plant_rad_bode_plot.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name : fig:comp_damp_undamped_plant_rad_bode_plot
#+caption : Bode plot of both the open-loop plant and the damped plant using relative active damping
#+RESULTS :
[[file:figs/comp_damp_undamped_plant_rad_bode_plot.png ]]
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* Active Damping Plant (Force Sensors)
:PROPERTIES:
:header-args:matlab+: :tangle matlab/dcm_active_damping_iff.m
:END:
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<<sec:active_damping_iff >>
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** Introduction :ignore:
Force sensors are added above the piezoelectric actuators.
They can consists of a simple piezoelectric ceramic stack.
See for instance cite:fleming10_integ_strain_force_feedb_high.
** Matlab Init :noexport:ignore:
#+begin_src matlab
%% dcm_active_damping_iff.m
% Test of Integral Force Feedback Strategy
#+end_src
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<<matlab-dir >>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init >>
#+end_src
#+begin_src matlab :tangle no :noweb yes
<<m-init-path >>
#+end_src
#+begin_src matlab :eval no :noweb yes
<<m-init-path-tangle >>
#+end_src
#+begin_src matlab :noweb yes
<<m-init-simscape >>
#+end_src
#+begin_src matlab :noweb yes
<<m-init-other >>
#+end_src
** Identification
#+begin_src matlab
%% 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;
#+end_src
#+begin_src matlab
%% Extraction of the dynamics
G_fs = linearize(mdl, io);
#+end_src
#+begin_src matlab :exports none
G_fs.InputName = {'u_ur', 'u_uh', 'u_d'};
G_fs.OutputName = {'fs_ur', 'fs_uh', 'fs_d'};
#+end_src
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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.
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#+begin_src matlab :exports none
%% Bode plot for the plant
figure;
tiledlayout(3, 1, 'TileSpacing', 'Compact', 'Padding', 'None');
ax1 = nexttile([2,1]);
hold on;
plot(freqs, abs(squeeze(freqresp(G_fs(1,1), freqs, 'Hz'))), ...
'DisplayName', 'd');
plot(freqs, abs(squeeze(freqresp(G_fs(2,2), freqs, 'Hz'))), ...
'DisplayName', 'uh');
plot(freqs, abs(squeeze(freqresp(G_fs(3,3), freqs, 'Hz'))), ...
'DisplayName', 'ur');
plot(freqs, abs(squeeze(freqresp(G_fs(1,2), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
'DisplayName', 'off-diag');
for i = 1:2
for j = i+1:3
plot(freqs, abs(squeeze(freqresp(G_fs(i,j), freqs, 'Hz'))), 'color', [0, 0, 0, 0.2], ...
'HandleVisibility', 'off');
end
end
hold off;
set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log');
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ylabel('Amplitude'); set(gca, 'XTickLabel',[]);
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legend('location', 'northwest', 'FontSize', 8, 'NumColumns', 2);
ylim([1e-13, 1e-7]);
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(G_fs(1,1), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(G_fs(2,2), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(G_fs(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, 180]);
linkaxes([ax1,ax2],'x');
xlim([freqs(1), freqs(end)]);
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/iff_plant_bode_plot.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name : fig:iff_plant_bode_plot
#+caption : Bode plot of IFF Plant
#+RESULTS :
[[file:figs/iff_plant_bode_plot.png ]]
** Controller - Root Locus
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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.
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#+begin_src matlab
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%% Integral Force Feedback Controller
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Kiff_g1 = eye(3)*1/(1 + s/2/pi/20);
#+end_src
#+begin_src matlab :exports none
%% Root Locus for IFF
gains = logspace(9, 12, 200);
figure;
hold on;
plot(real(pole(G_fs)), imag(pole(G_fs)), 'x', 'color', colors(1,:), ...
'DisplayName', '$g = 0$');
plot(real(tzero(G_fs)), imag(tzero(G_fs)), 'o', 'color', colors(1,:), ...
'HandleVisibility', 'off');
for g = gains
clpoles = pole(feedback(G_fs, g*Kiff_g1, +1));
plot(real(clpoles), imag(clpoles), '.', 'color', colors(1,:), ...
'HandleVisibility', 'off');
end
% Optimal gain
g = 8e10;
clpoles = pole(feedback(G_fs, g*Kiff_g1, +1));
plot(real(clpoles), imag(clpoles), 'x', 'color', colors(2,:), ...
'DisplayName', sprintf('$g=%.0e$', g));
hold off;
axis square;
xlim([-2700, 0]); ylim([0, 2700]);
xlabel('Real Part'); ylabel('Imaginary Part');
legend('location', 'northwest');
#+end_src
#+begin_src matlab :tangle no :exports results :results file replace
exportFig('figs/iff_root_locus.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name : fig:iff_root_locus
#+caption : Root Locus plot for the IFF Control strategy
#+RESULTS :
[[file:figs/iff_root_locus.png ]]
#+begin_src matlab
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%% Integral Force Feedback Controller with optimal gain
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Kiff = g*Kiff_g1;
#+end_src
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#+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
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** Damped Plant
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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
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%% 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
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#+begin_src matlab :exports none
%% Load DCM Kinematics
load('dcm_kinematics.mat');
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#+end_src
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#+begin_src matlab :exports none
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%% 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'};
#+end_src
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#+begin_src matlab :exports none
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%% 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
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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.
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#+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_ol(1,1), freqs, 'Hz'))), ...
'DisplayName', 'd - OL');
plot(freqs, abs(squeeze(freqresp(G_ol(2,2), freqs, 'Hz'))), ...
'DisplayName', 'uh - OL');
plot(freqs, abs(squeeze(freqresp(G_ol(3,3), freqs, 'Hz'))), ...
'DisplayName', 'ur - OL');
set(gca,'ColorOrderIndex',1)
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-6]);
ax2 = nexttile;
hold on;
plot(freqs, 180/pi*angle(squeeze(freqresp(G_ol(1,1), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(G_ol(2,2), freqs, 'Hz'))));
plot(freqs, 180/pi*angle(squeeze(freqresp(G_ol(3,3), freqs, 'Hz'))));
set(gca,'ColorOrderIndex',1)
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/comp_damped_undamped_plant_iff_bode_plot.pdf', 'width', 'wide', 'height', 'tall');
#+end_src
#+name : fig:comp_damped_undamped_plant_iff_bode_plot
#+caption : Bode plot of both the open-loop plant and the damped plant using IFF
#+RESULTS :
[[file:figs/comp_damped_undamped_plant_iff_bode_plot.png ]]
#+begin_important
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The Integral Force Feedback control strategy is very effective in damping the modes present in the plant.
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#+end_important
* HAC-LAC (IFF) architecture
:PROPERTIES:
:header-args:matlab+: :tangle matlab/dcm_hac_iff.m
:END:
2021-11-30 11:29:43 +01:00
<<sec:hac_iff >>
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** Introduction :ignore:
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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 ]]
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** Matlab Init :noexport:ignore:
#+begin_src matlab
%% dcm_hac_iff.m
% Development of the HAC-IFF control strategy
#+end_src
#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
<<matlab-dir >>
#+end_src
#+begin_src matlab :exports none :results silent :noweb yes
<<matlab-init >>
#+end_src
#+begin_src matlab :tangle no :noweb yes
<<m-init-path >>
#+end_src
#+begin_src matlab :eval no :noweb yes
<<m-init-path-tangle >>
#+end_src
#+begin_src matlab :noweb yes
<<m-init-simscape >>
#+end_src
#+begin_src matlab :noweb yes
<<m-init-other >>
#+end_src
2021-11-30 15:17:23 +01:00
** 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
2021-11-30 11:16:48 +01:00
* Helping Functions :noexport:
** Initialize Path
#+NAME : m-init-path
#+BEGIN_SRC matlab
%% Path for functions, data and scripts
addpath('./matlab/mat/ '); % Path for data
addpath('./matlab/ '); % Path for scripts
%% Simscape Model - Nano Hexapod
addpath('./matlab/STEPS/ ')
#+END_SRC
#+NAME : m-init-path-tangle
#+BEGIN_SRC matlab
%% Path for functions, data and scripts
addpath('./mat/ '); % Path for data
%% Simscape Model - Nano Hexapod
addpath('./STEPS/ ')
#+END_SRC
** Initialize Simscape Model
#+NAME : m-init-simscape
#+begin_src matlab
%% Initialize Parameters for Simscape model
controller.type = 0; % Open Loop Control
%% Options for Linearization
options = linearizeOptions;
options.SampleTime = 0;
%% Open Simulink Model
mdl = 'simscape_dcm';
open(mdl)
#+end_src
** Initialize other elements
#+NAME : m-init-other
#+BEGIN_SRC matlab
%% Colors for the figures
colors = colororder;
%% Frequency Vector
freqs = logspace(1, 3, 1000);
#+END_SRC
* Bibliography :ignore:
#+latex : \printbibliography
