1725 lines
55 KiB
Org Mode
1725 lines
55 KiB
Org Mode
#+TITLE: ESRF Double Crystal Monochromator - Dynamical Multi-Body Model
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:DRAWER:
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#+LANGUAGE: en
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#+EMAIL: dehaeze.thomas@gmail.com
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#+AUTHOR: Dehaeze Thomas
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#+HTML_LINK_HOME: ../index.html
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#+HTML_LINK_UP: ../index.html
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#+HTML_HEAD: <link rel="stylesheet" type="text/css" href="https://research.tdehaeze.xyz/css/style.css"/>
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#+HTML_HEAD: <script type="text/javascript" src="https://research.tdehaeze.xyz/js/script.js"></script>
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#+BIND: org-latex-image-default-option "scale=1"
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#+BIND: org-latex-image-default-width ""
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#+LaTeX_CLASS: scrreprt
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#+LaTeX_CLASS_OPTIONS: [a4paper, 10pt, DIV=12, parskip=full]
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#+LaTeX_HEADER_EXTRA: \input{preamble.tex}
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#+LATEX_HEADER_EXTRA: \bibliography{ref}
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#+PROPERTY: header-args:matlab :session *MATLAB*
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#+PROPERTY: header-args:matlab+ :comments org
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#+PROPERTY: header-args:matlab+ :exports both
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#+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
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#+PROPERTY: header-args:matlab+ :noweb yes
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#+PROPERTY: header-args:matlab+ :mkdirp yes
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#+PROPERTY: header-args:matlab+ :output-dir figs
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#+PROPERTY: header-args:latex :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{$HOME/Cloud/tikz/org/}{config.tex}")
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#+PROPERTY: header-args:latex+ :imagemagick t :fit yes
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#+PROPERTY: header-args:latex+ :iminoptions -scale 100% -density 150
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#+PROPERTY: header-args:latex+ :imoutoptions -quality 100
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#+PROPERTY: header-args:latex+ :results file raw replace
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#+PROPERTY: header-args:latex+ :buffer no
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#+PROPERTY: header-args:latex+ :tangle no
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#+PROPERTY: header-args:latex+ :eval no-export
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#+PROPERTY: header-args:latex+ :exports results
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#+PROPERTY: header-args:latex+ :mkdirp yes
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#+PROPERTY: header-args:latex+ :output-dir figs
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#+PROPERTY: header-args:latex+ :post pdf2svg(file=*this*, ext="png")
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:END:
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#+begin_export html
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<hr>
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<p>This report is also available as a <a href="./dcm-simscape.pdf">pdf</a>.</p>
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<hr>
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#+end_export
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#+latex: \clearpage
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* Introduction :ignore:
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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.
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It is structured as follow:
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- 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.
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- Section [[sec:open_loop_identification]]: the system dynamics is identified in the absence of control.
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- 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.
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- 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.
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- 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
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:PROPERTIES:
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:header-args:matlab+: :tangle matlab/dcm_kinematics.m
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:END:
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<<sec:dcm_kinematics>>
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** Introduction :ignore:
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** Matlab Init :noexport:ignore:
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#+begin_src matlab
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%% dcm_kinematics.m
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% Computation of the DCM kinematics
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#+end_src
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#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
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<<matlab-dir>>
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#+end_src
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#+begin_src matlab :exports none :results silent :noweb yes
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<<matlab-init>>
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#+end_src
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#+begin_src matlab :tangle no :noweb yes
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<<m-init-path>>
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#+end_src
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#+begin_src matlab :eval no :noweb yes
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<<m-init-path-tangle>>
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#+end_src
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#+begin_src matlab :noweb yes
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<<m-init-other>>
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#+end_src
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** Bragg Angle
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#+begin_src matlab
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%% Tested bragg angles
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bragg = linspace(5, 80, 1000); % Bragg angle [deg]
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d_off = 10.5e-3; % Wanted offset between x-rays [m]
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#+end_src
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#+begin_src matlab
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%% Vertical Jack motion as a function of Bragg angle
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dz = d_off./(2*cos(bragg*pi/180));
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#+end_src
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#+begin_src matlab :exports none
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%% Jack motion as a function of Bragg angle
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figure;
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plot(bragg, 1e3*dz)
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xlabel('Bragg angle [deg]'); ylabel('Jack Motion [mm]');
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#+end_src
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#+begin_src matlab :tangle no :exports results :results file replace
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exportFig('figs/jack_motion_bragg_angle.pdf', 'width', 'wide', 'height', 'normal');
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#+end_src
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#+name: fig:jack_motion_bragg_angle
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#+caption: Jack motion as a function of Bragg angle
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#+RESULTS:
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[[file:figs/jack_motion_bragg_angle.png]]
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#+begin_src matlab :results value replace :exports both
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%% Required Jack stroke
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ans = 1e3*(dz(end) - dz(1))
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#+end_src
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#+RESULTS:
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: 24.963
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** Kinematics (111 Crystal)
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*** Introduction :ignore:
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The reference frame is taken at the center of the 111 second crystal.
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*** Interferometers - 111 Crystal
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Three interferometers are pointed to the bottom surface of the 111 crystal.
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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.
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#+begin_src latex :file sensor_111_crystal_points.pdf
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\begin{tikzpicture}
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% Crystal
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\draw (-15/2, -3.5/2) rectangle (15/2, 3.5/2);
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% Measurement Points
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\node[branch] (a1) at (-7, 1.5){};
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\node[branch] (a2) at ( 0, -1.5){};
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\node[branch] (a3) at ( 7, 1.5){};
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% Labels
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\node[right] at (a1) {$\mathcal{O}_1 = (-0.07, -0.015)$};
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\node[right] at (a2) {$\mathcal{O}_2 = (0, 0.015)$};
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\node[left] at (a3) {$\mathcal{O}_3 = ( 0.07, -0.015)$};
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% Origin
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\draw[->] (0, 0) node[] -- ++(1, 0) node[right]{$x$};
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\draw[->] (0, 0) -- ++(0, -1) node[below]{$y$};
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\draw[fill, color=black] (0, 0) circle (0.05);
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\node[left] at (0,0) {$\mathcal{O}_{111}$};
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\end{tikzpicture}
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#+end_src
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#+name: fig:sensor_111_crystal_points
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#+caption: Bottom view of the second crystal 111. Position of the measurement points.
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#+RESULTS:
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[[file:figs/sensor_111_crystal_points.png]]
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The inverse kinematics consisting of deriving the interferometer measurements from the motion of the crystal (see Figure [[fig:schematic_sensor_jacobian_inverse_kinematics]]):
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\begin{equation}
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\begin{bmatrix}
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x_1 \\ x_2 \\ x_3
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\end{bmatrix}
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=
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\bm{J}_{s,111}
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\begin{bmatrix}
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d_z \\ r_y \\ r_x
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\end{bmatrix}
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\end{equation}
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#+begin_src latex :file schematic_sensor_jacobian_inverse_kinematics.pdf
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\begin{tikzpicture}
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% Blocs
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\node[block] (Js) {$\bm{J}_{s,111}$};
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% Connections and labels
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\draw[->] ($(Js.west)+(-1.5,0)$) node[above right]{$\begin{bmatrix} d_z \\ r_y \\ r_x \end{bmatrix}$} -- (Js.west);
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\draw[->] (Js.east) -- ++(1.5, 0) node[above left]{$\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$};
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\end{tikzpicture}
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#+end_src
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#+name: fig:schematic_sensor_jacobian_inverse_kinematics
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#+caption: Inverse Kinematics - Interferometers
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#+RESULTS:
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[[file:figs/schematic_sensor_jacobian_inverse_kinematics.png]]
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From the Figure [[fig:sensor_111_crystal_points]], the inverse kinematics can be solved as follow (for small motion):
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\begin{equation}
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\bm{J}_{s,111}
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=
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\begin{bmatrix}
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1 & 0.07 & -0.015 \\
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1 & 0 & 0.015 \\
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1 & -0.07 & -0.015
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\end{bmatrix}
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\end{equation}
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#+begin_src matlab
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%% Sensor Jacobian matrix for 111 crystal
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J_s_111 = [1, 0.07, -0.015
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1, 0, 0.015
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1, -0.07, -0.015];
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#+end_src
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#+begin_src matlab :exports results :results value table replace :tangle no
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data2orgtable(J_s_111, {}, {}, ' %.3f ');
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#+end_src
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#+name: tab:jacobian_sensor_111
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#+caption: Sensor Jacobian $\bm{J}_{s,111}$
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#+attr_latex: :environment tabularx :width 0.3\linewidth :align ccc
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#+attr_latex: :center t :booktabs t
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#+RESULTS:
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| 1.0 | 0.07 | -0.015 |
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| 1.0 | 0.0 | 0.015 |
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| 1.0 | -0.07 | -0.015 |
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The forward kinematics is solved by inverting the Jacobian matrix (see Figure [[fig:schematic_sensor_jacobian_forward_kinematics]]).
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\begin{equation}
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\begin{bmatrix}
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d_z \\ r_y \\ r_x
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\end{bmatrix}
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=
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\bm{J}_{s,111}^{-1}
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\begin{bmatrix}
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x_1 \\ x_2 \\ x_3
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\end{bmatrix}
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\end{equation}
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#+begin_src latex :file schematic_sensor_jacobian_forward_kinematics.pdf
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\begin{tikzpicture}
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% Blocs
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\node[block] (Js_inv) {$\bm{J}_{s,111}^{-1}$};
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% Connections and labels
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\draw[->] ($(Js_inv.west)+(-1.5,0)$) node[above right]{$\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix}$} -- (Js_inv.west);
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\draw[->] (Js_inv.east) -- ++(1.5, 0) node[above left]{$\begin{bmatrix} d_z \\ r_y \\ r_x \end{bmatrix}$};
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\end{tikzpicture}
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#+end_src
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#+name: fig:schematic_sensor_jacobian_forward_kinematics
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#+caption: Forward Kinematics - Interferometers
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#+RESULTS:
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[[file:figs/schematic_sensor_jacobian_forward_kinematics.png]]
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#+begin_src matlab :exports results :results value table replace :tangle no
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data2orgtable(inv(J_s_111), {}, {}, ' %.2f ');
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#+end_src
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#+name: tab:inverse_jacobian_sensor_111
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#+caption: Inverse of the sensor Jacobian $\bm{J}_{s,111}^{-1}$
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#+attr_latex: :environment tabularx :width 0.3\linewidth :align ccc
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#+attr_latex: :center t :booktabs t
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#+RESULTS:
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| 0.25 | 0.5 | 0.25 |
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| 7.14 | 0.0 | -7.14 |
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| -16.67 | 33.33 | -16.67 |
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*** Piezo - 111 Crystal
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The location of the actuators with respect with the center of the 111 second crystal are shown in Figure [[fig:actuator_jacobian_111_points]].
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#+name: fig:actuator_jacobian_111_points
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#+caption: Location of actuators with respect to the center of the 111 second crystal (bottom view)
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#+attr_latex: :width \linewidth
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[[file:figs/actuator_jacobian_111_points.png]]
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Inverse Kinematics consist of deriving the axial (z) motion of the 3 actuators from the motion of the crystal's center.
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\begin{equation}
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\begin{bmatrix}
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d_{u_r} \\ d_{u_h} \\ d_{d}
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\end{bmatrix}
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=
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\bm{J}_{a,111}
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\begin{bmatrix}
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d_z \\ r_y \\ r_x
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\end{bmatrix}
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\end{equation}
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#+begin_src latex :file schematic_actuator_jacobian_inverse_kinematics.pdf
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\begin{tikzpicture}
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% Blocs
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\node[block] (Ja) {$\bm{J}_{a,111}$};
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% Connections and labels
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\draw[->] ($(Ja.west)+(-1.5,0)$) node[above right]{$\begin{bmatrix} d_z \\ r_y \\ r_x \end{bmatrix}$} -- (Ja.west);
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\draw[->] (Ja.east) -- ++(1.5, 0) node[above left]{$\begin{bmatrix} d_{u_r} \\ d_{u_h} \\ d_d \end{bmatrix}$};
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\end{tikzpicture}
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#+end_src
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#+name: fig:schematic_sensor_jacobian_inverse_kinematics
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#+caption: Inverse Kinematics - Actuators
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#+RESULTS:
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[[file:figs/schematic_actuator_jacobian_inverse_kinematics.png]]
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Based on the geometry in Figure [[fig:actuator_jacobian_111_points]], we obtain:
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\begin{equation}
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\bm{J}_{a,111}
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=
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\begin{bmatrix}
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1 & 0.14 & -0.1525 \\
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1 & 0.14 & 0.0675 \\
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1 & -0.14 & -0.0425
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\end{bmatrix}
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\end{equation}
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#+begin_src matlab
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%% Actuator Jacobian - 111 crystal
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J_a_111 = [1, 0.14, -0.1525
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1, 0.14, 0.0675
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1, -0.14, -0.0425];
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#+end_src
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#+begin_src matlab :exports results :results value table replace :tangle no
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data2orgtable(J_a_111, {}, {}, ' %.4f ');
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#+end_src
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#+name: tab:jacobian_actuator_111
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#+caption: Actuator Jacobian $\bm{J}_{a,111}$
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#+attr_latex: :environment tabularx :width 0.3\linewidth :align ccc
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#+attr_latex: :center t :booktabs t
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#+RESULTS:
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| 1.0 | 0.14 | -0.1525 |
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| 1.0 | 0.14 | 0.0675 |
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| 1.0 | -0.14 | -0.0425 |
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The forward Kinematics is solved by inverting the Jacobian matrix:
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\begin{equation}
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\begin{bmatrix}
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d_z \\ r_y \\ r_x
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\end{bmatrix}
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=
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\bm{J}_{a,111}^{-1}
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\begin{bmatrix}
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d_{u_r} \\ d_{u_h} \\ d_{d}
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\end{bmatrix}
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\end{equation}
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#+begin_src latex :file schematic_actuator_jacobian_forward_kinematics.pdf
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\begin{tikzpicture}
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% Blocs
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\node[block] (Ja_inv) {$\bm{J}_{a,111}^{-1}$};
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% Connections and labels
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\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);
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\draw[->] (Ja_inv.east) -- ++(1.5, 0) node[above left]{$\begin{bmatrix} d_z \\ r_y \\ r_x \end{bmatrix}$};
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\end{tikzpicture}
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#+end_src
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#+name: fig:schematic_actuator_jacobian_forward_kinematics
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#+caption: Forward Kinematics - Actuators for 111 crystal
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#+RESULTS:
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[[file:figs/schematic_actuator_jacobian_forward_kinematics.png]]
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#+begin_src matlab :exports results :results value table replace :tangle no
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data2orgtable(inv(J_a_111), {}, {}, ' %.4f ');
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#+end_src
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#+name: tab:inverse_jacobian_actuator_111
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#+caption: Inverse of the actuator Jacobian $\bm{J}_{a,111}^{-1}$
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#+attr_latex: :environment tabularx :width 0.3\linewidth :align ccc
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#+attr_latex: :center t :booktabs t
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#+RESULTS:
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| 0.0568 | 0.4432 | 0.5 |
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| 1.7857 | 1.7857 | -3.5714 |
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| -4.5455 | 4.5455 | 0.0 |
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** TODO Inputs and Outputs :noexport:
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Disturbances:
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- Motion errors of the stepper motor
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- Vibrations from the outside
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- Vibrations from the cooling system directly applied on the crystals
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** Save Kinematics
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#+begin_src matlab :exports none :tangle no
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save('matlab/mat/dcm_kinematics.mat', 'J_a_111', 'J_s_111')
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#+end_src
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#+begin_src matlab :eval no
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save('mat/dcm_kinematics.mat', 'J_a_111', 'J_s_111')
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#+end_src
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* Open Loop System Identification
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:PROPERTIES:
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:header-args:matlab+: :tangle matlab/dcm_identification.m
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:END:
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<<sec:open_loop_identification>>
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** Introduction :ignore:
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** Matlab Init :noexport:ignore:
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#+begin_src matlab
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%% dcm_identification.m
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% Extraction of system dynamics using Simscape model
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#+end_src
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#+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name)
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<<matlab-dir>>
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#+end_src
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#+begin_src matlab :exports none :results silent :noweb yes
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<<matlab-init>>
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#+end_src
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#+begin_src matlab :tangle no :noweb yes
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<<m-init-path>>
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#+end_src
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#+begin_src matlab :eval no :noweb yes
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<<m-init-path-tangle>>
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#+end_src
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#+begin_src matlab :noweb yes
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<<m-init-simscape>>
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#+end_src
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#+begin_src matlab :noweb yes
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<<m-init-other>>
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#+end_src
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** Identification
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Let's considered the system $\bm{G}(s)$ with:
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- 3 inputs: force applied to the 3 fast jacks
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- 3 outputs: measured displacement by the 3 interferometers pointing at the 111 second crystal
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It is schematically shown in Figure [[fig:schematic_system_inputs_outputs]].
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|
|
#+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
|
|
load('dcm_kinematics.mat');
|
|
#+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:
|
|
|
|
* Active Damping Plant (Strain gauges)
|
|
:PROPERTIES:
|
|
:header-args:matlab+: :tangle matlab/dcm_active_damping_strain_gauges.m
|
|
:END:
|
|
<<sec:active_damping_strain_gauges>>
|
|
** 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]
|
|
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;
|
|
#+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:
|
|
| 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 |
|
|
|
|
#+begin_src matlab :exports none
|
|
%% Bode plot for the plant (strain gauge output)
|
|
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',[]);
|
|
legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2);
|
|
ylim([1e-14, 1e-7]);
|
|
|
|
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);
|
|
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
|
|
load('dcm_kinematics.mat');
|
|
#+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]);
|
|
|
|
linkaxes([ax1,ax2],'x');
|
|
xlim([freqs(1), freqs(end)]);
|
|
#+end_src
|
|
|
|
#+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]]
|
|
|
|
* Active Damping Plant (Force Sensors)
|
|
:PROPERTIES:
|
|
:header-args:matlab+: :tangle matlab/dcm_active_damping_iff.m
|
|
:END:
|
|
<<sec:active_damping_iff>>
|
|
** 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
|
|
|
|
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.
|
|
|
|
#+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');
|
|
ylabel('Amplitude'); set(gca, 'XTickLabel',[]);
|
|
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
|
|
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.
|
|
|
|
#+begin_src matlab
|
|
%% Integral Force Feedback Controller
|
|
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
|
|
%% Integral Force Feedback Controller with optimal gain
|
|
Kiff = g*Kiff_g1;
|
|
#+end_src
|
|
|
|
#+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
|
|
|
|
** Damped Plant
|
|
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
|
|
%% 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
|
|
load('dcm_kinematics.mat');
|
|
#+end_src
|
|
|
|
#+begin_src matlab :exports none
|
|
%% 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 :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
|
|
|
|
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.
|
|
|
|
#+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
|
|
The Integral Force Feedback control strategy is very effective in damping the modes present in the plant.
|
|
#+end_important
|
|
|
|
* HAC-LAC (IFF) architecture
|
|
:PROPERTIES:
|
|
:header-args:matlab+: :tangle matlab/dcm_hac_iff.m
|
|
:END:
|
|
<<sec:hac_iff>>
|
|
** Introduction :ignore:
|
|
|
|
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={3.0cm}{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=1.2 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=1.2 of addF] (Khac) {$\bm{K}_{\text{HAC}}(s)$};
|
|
\node[block, left=1.2 of Khac] (Ja) {$\bm{J}_{a}$};
|
|
\node[addb={+}{}{}{}{-}, left=1.0 of Ja] (subL) {};
|
|
|
|
|
|
% 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[->] ($(subL.west) + (-0.8, 0)$) -- 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);
|
|
\end{tikzpicture}
|
|
#+end_src
|
|
|
|
#+name: fig:schematic_jacobian_frame_fastjack_hac_iff
|
|
#+caption: HAC-LAC architecture
|
|
#+attr_latex: :width \linewidth
|
|
#+RESULTS:
|
|
[[file:figs/schematic_jacobian_frame_fastjack_hac_iff.png]]
|
|
|
|
** 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
|
|
|
|
** 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;
|
|
% Angle used to draw the circles
|
|
theta = linspace(0, 2*pi, 100);
|
|
% Unit circle
|
|
plot(cos(theta), sin(theta), '--');
|
|
% Circle for module margin
|
|
plot(-1 + min(min(abs(Ldet + 1)))*cos(theta), min(min(abs(Ldet + 1)))*sin(theta), '--');
|
|
|
|
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
|
|
% Unstable Point
|
|
plot(-1, 0, 'kx', 'HandleVisibility', 'off');
|
|
hold off;
|
|
set(gca, 'XScale', 'lin'); set(gca, 'YScale', 'lin');
|
|
xlabel('Real'); ylabel('Imag');
|
|
axis square;
|
|
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', 'normal', 'height', 'normal');
|
|
#+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]);
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xlim([f(1), f(end)]);
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#+end_src
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|
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#+begin_src matlab :tangle no :exports results :results file replace
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exportFig('figs/stepper_transmissibility_comp_ol_hac_iff.pdf', 'width', 'wide', 'height', 'normal');
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|
#+end_src
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|
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#+name: fig:stepper_transmissibility_comp_ol_hac_iff
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#+caption: Comparison of the transmissibility of errors from vibrations of the stepper motor between the open-loop case and the hac-iff case.
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|
#+RESULTS:
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|
[[file:figs/stepper_transmissibility_comp_ol_hac_iff.png]]
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|
|
|
#+begin_important
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|
The HAC-IFF control strategy can effectively reduce the transmissibility of the motion errors of the stepper motors.
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|
This reduction is effective inside the bandwidth of the controller.
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|
#+end_important
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|
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|
* Helping Functions :noexport:
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|
** Initialize Path
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|
#+NAME: m-init-path
|
|
#+BEGIN_SRC matlab
|
|
%% Path for functions, data and scripts
|
|
addpath('./matlab/mat/'); % Path for data
|
|
addpath('./matlab/'); % Path for scripts
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|
|
|
%% 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
|