#+TITLE: ESRF Double Crystal Monochromator - Dynamical Multi-Body Model :DRAWER: #+LANGUAGE: en #+EMAIL: dehaeze.thomas@gmail.com #+AUTHOR: Dehaeze Thomas #+HTML_LINK_HOME: ../index.html #+HTML_LINK_UP: ../index.html #+HTML_HEAD: #+HTML_HEAD: #+BIND: org-latex-image-default-option "scale=1" #+BIND: org-latex-image-default-width "" #+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 #+PROPERTY: header-args:matlab+ :tangle no #+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}") #+PROPERTY: header-args:latex+ :imagemagick t :fit yes #+PROPERTY: header-args:latex+ :iminoptions -scale 100% -density 150 #+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

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#+end_export #+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:dcm_noise_budget]]: an open-loop noise budget is performed. - Section [[sec:active_damping_strain_gauges]]: it is studied whether if the strain gauges fixed to the piezoelectric actuators can be used to actively damp the plant. - Section [[sec:active_damping_iff]]: piezoelectric force sensors are added in series with the piezoelectric actuators and are used to actively damp the plant using the Integral Force Feedback (IFF) control strategy. - Section [[sec:hac_iff]]: the High Authority Control - Low Authority Control (HAC-LAC) strategy is tested on the Simscape model. * System Kinematics :PROPERTIES: :header-args:matlab+: :tangle matlab/dcm_kinematics.m :END: <> ** 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) <> #+end_src #+begin_src matlab :exports none :results silent :noweb yes <> #+end_src #+begin_src matlab :tangle no :noweb yes <> #+end_src #+begin_src matlab :eval no :noweb yes <> #+end_src #+begin_src matlab :noweb yes <> #+end_src ** Bragg Angle There is a simple relation eqref:eq:bragg_angle_formula between: - $d_{\text{off}}$ is the wanted offset between the incident x-ray and the output x-ray - $\theta_b$ is the bragg angle - $d_z$ is the corresponding distance between the first and second crystals \begin{equation} \label{eq:bragg_angle_formula} d_z = \frac{d_{\text{off}}}{2 \cos \theta_b} \end{equation} #+begin_src matlab :exports none %% 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 :exports none %% Vertical Jack motion as a function of Bragg angle dz = d_off./(2*cos(bragg*pi/180)); #+end_src This relation is shown in Figure [[fig:jack_motion_bragg_angle]]. #+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]] The required jack stroke is approximately 25mm. #+begin_src matlab :results value replace :exports both %% Required Jack stroke ans = 1e3*(dz(end) - dz(1)) #+end_src #+RESULTS: : 24.963 ** Kinematics (311 Crystal) *** Introduction :ignore: The reference frame is taken at the center of the 311 second crystal. *** Interferometers - 311 Crystal Three interferometers are pointed to the bottom surface of the 311 crystal. The position of the measurement points are shown in Figure [[fig:sensor_311_crystal_points]] as well as the origin where the motion of the crystal is computed. #+begin_src latex :file sensor_311_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) {$\mathcal{O}_{311}$}; \end{tikzpicture} #+end_src #+name: fig:sensor_311_crystal_points #+caption: Bottom view of the second crystal 311. Position of the measurement points. #+RESULTS: [[file:figs/sensor_311_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_311]]): \begin{equation} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \bm{J}_{s,311} \begin{bmatrix} d_z \\ r_y \\ r_x \end{bmatrix} \end{equation} #+begin_src latex :file schematic_sensor_jacobian_inverse_kinematics_311.pdf \begin{tikzpicture} % Blocs \node[block] (Js) {$\bm{J}_{s,311}$}; % 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_311 #+caption: Inverse Kinematics - Interferometers #+RESULTS: [[file:figs/schematic_sensor_jacobian_inverse_kinematics_311.png]] From the Figure [[fig:sensor_311_crystal_points]], the inverse kinematics can be solved as follow (for small motion): \begin{equation} \bm{J}_{s,311} = \begin{bmatrix} 1 & 0.07 & -0.015 \\ 1 & 0 & 0.015 \\ 1 & -0.07 & -0.015 \end{bmatrix} \end{equation} #+begin_src matlab %% Sensor Jacobian matrix for 311 crystal J_s_311 = [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_311, {}, {}, ' %.3f '); #+end_src #+name: tab:jacobian_sensor_311 #+caption: Sensor Jacobian $\bm{J}_{s,311}$ #+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_311]]). \begin{equation} \begin{bmatrix} d_z \\ r_y \\ r_x \end{bmatrix} = \bm{J}_{s,311}^{-1} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \end{equation} #+begin_src latex :file schematic_sensor_jacobian_forward_kinematics_311.pdf \begin{tikzpicture} % Blocs \node[block] (Js_inv) {$\bm{J}_{s,311}^{-1}$}; % 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_311 #+caption: Forward Kinematics - Interferometers #+RESULTS: [[file:figs/schematic_sensor_jacobian_forward_kinematics_311.png]] #+begin_src matlab :exports results :results value table replace :tangle no data2orgtable(inv(J_s_311), {}, {}, ' %.2f '); #+end_src #+name: tab:inverse_jacobian_sensor_311 #+caption: Inverse of the sensor Jacobian $\bm{J}_{s,311}^{-1}$ #+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 - 311 Crystal The location of the actuators with respect with the center of the 311 second crystal are shown in Figure [[fig:actuator_jacobian_311_points]]. #+name: fig:actuator_jacobian_311_points #+caption: Location of actuators with respect to the center of the 311 second crystal (bottom view) #+attr_latex: :width \linewidth [[file:figs/actuator_jacobian_311_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,311} \begin{bmatrix} d_z \\ r_y \\ r_x \end{bmatrix} \end{equation} #+begin_src latex :file schematic_actuator_jacobian_inverse_kinematics_311.pdf \begin{tikzpicture} % Blocs \node[block] (Ja) {$\bm{J}_{a,311}$}; % 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_311 #+caption: Inverse Kinematics - Actuators #+RESULTS: [[file:figs/schematic_actuator_jacobian_inverse_kinematics_311.png]] Based on the geometry in Figure [[fig:actuator_jacobian_311_points]], we obtain: \begin{equation} \bm{J}_{a,311} = \begin{bmatrix} 1 & 0.14 & -0.1525 \\ 1 & 0.14 & 0.0675 \\ 1 & -0.14 & -0.0425 \end{bmatrix} \end{equation} #+begin_src matlab %% Actuator Jacobian - 311 crystal J_a_311 = [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_311, {}, {}, ' %.4f '); #+end_src #+name: tab:jacobian_actuator_311 #+caption: Actuator Jacobian $\bm{J}_{a,311}$ #+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,311}^{-1} \begin{bmatrix} d_{u_r} \\ d_{u_h} \\ d_{d} \end{bmatrix} \end{equation} #+begin_src latex :file schematic_actuator_jacobian_forward_kinematics_311.pdf \begin{tikzpicture} % Blocs \node[block] (Ja_inv) {$\bm{J}_{a,311}^{-1}$}; % 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_311 #+caption: Forward Kinematics - Actuators for 311 crystal #+RESULTS: [[file:figs/schematic_actuator_jacobian_forward_kinematics_311.png]] #+begin_src matlab :exports results :results value table replace :tangle no data2orgtable(inv(J_a_311), {}, {}, ' %.4f '); #+end_src #+name: tab:inverse_jacobian_actuator_311 #+caption: Inverse of the actuator Jacobian $\bm{J}_{a,311}^{-1}$ #+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 | ** 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_111]]): \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_111.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_111 #+caption: Inverse Kinematics - Interferometers #+RESULTS: [[file:figs/schematic_sensor_jacobian_inverse_kinematics_111.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_111]]). \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_111.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_111 #+caption: Forward Kinematics - Interferometers #+RESULTS: [[file:figs/schematic_sensor_jacobian_forward_kinematics_111.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_111.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_111 #+caption: Inverse Kinematics - Actuators #+RESULTS: [[file:figs/schematic_actuator_jacobian_inverse_kinematics_111.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_111.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_111 #+caption: Forward Kinematics - Actuators for 111 crystal #+RESULTS: [[file:figs/schematic_actuator_jacobian_forward_kinematics_111.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: | 0.25 | 0.25 | 0.5 | | 0.4058 | 3.1656 | -3.5714 | | -4.5455 | 4.5455 | 0.0 | ** 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 ** Save Kinematics #+begin_src matlab :exports none :tangle no save('matlab/mat/dcm_kinematics.mat', 'J_a_311', 'J_s_311', 'J_a_111', 'J_s_111') #+end_src #+begin_src matlab :eval no save('mat/dcm_kinematics.mat', 'J_a_311', 'J_s_311', 'J_a_111', 'J_s_111') #+end_src * Open Loop System Identification :PROPERTIES: :header-args:matlab+: :tangle matlab/dcm_identification.m :END: <> ** 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) <> #+end_src #+begin_src matlab :exports none :results silent :noweb yes <> #+end_src #+begin_src matlab :tangle no :noweb yes <> #+end_src #+begin_src matlab :eval no :noweb yes <> #+end_src #+begin_src matlab :noweb yes <> #+end_src #+begin_src matlab :noweb yes <> #+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 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: * Open-Loop Noise Budgeting :PROPERTIES: :header-args:matlab+: :tangle matlab/dcm_noise_budget.m :END: <> ** Introduction :ignore: #+begin_src latex :file noise_budget_dcm_schematic_fast_jack_frame.pdf \begin{tikzpicture} % Blocs \node[block] (G) {$G(s)$}; \node[addb, left= of G] (adddu) {}; \node[block, left= of adddu] (K) {$K(s)$}; \node[addb={+}{}{}{}{-}, left= of K] (subL) {}; \node[addb, right= of G] (addd) {}; \node[addb, below right=0.8 and 0.6 of addd] (adddn) {}; % Connections and labels \draw[->] (subL.east) -- (K.west); \draw[->] (K.east) -- (adddu.west); \draw[->] (adddu.east) -- (G.west); \draw[->] (G.east) -- (addd.west); \draw[->] (addd-|adddn)node[branch]{}node[above]{$y_{\text{fj}}$} -- (adddn.north); \draw[->] (adddn.west) -| (subL.south) node[below right]{$y_{\text{fj},m}$}; \draw[<-] (adddu.north) -- ++(0, 1) node[below right]{$d_u$}; \draw[<-] (addd.north) -- ++(0, 1) node[below right]{$d_{\text{fj}}$}; \draw[<-] (adddn.east) -- ++(1, 0)coordinate(dn) node[above left]{$n_{\text{fj}}$}; \draw[->] (addd.east) -- (addd-|dn); \end{tikzpicture} #+end_src #+name: fig:noise_budget_dcm_schematic_fast_jack_frame #+caption: Schematic representation of the control loop in the frame of one fast jack #+RESULTS: [[file:figs/noise_budget_dcm_schematic_fast_jack_frame.png]] ** Matlab Init :noexport:ignore: #+begin_src matlab %% dcm_noise_budget.m % Basic uniaxial noise budgeting #+end_src #+begin_src matlab :tangle no :exports none :results silent :noweb yes :var current_dir=(file-name-directory buffer-file-name) <> #+end_src #+begin_src matlab :exports none :results silent :noweb yes <> #+end_src #+begin_src matlab :tangle no :noweb yes <> #+end_src #+begin_src matlab :eval no :noweb yes <> #+end_src #+begin_src matlab :noweb yes <> %% Frequency vector for noise budget [Hz] f = logspace(-1, 3, 1000); #+end_src ** Power Spectral Density of signals Interferometer noise: #+begin_src matlab Wn = 6e-11*(1 + s/2/pi/200)/(1 + s/2/pi/60); % m/sqrt(Hz) #+end_src #+begin_src matlab :results value replace :exports results :tangle no sprintf('Measurement noise: %.2f [nm,rms]', 1e9*sqrt(trapz(f, abs(squeeze(freqresp(Wn, f, 'Hz'))).^2))); #+end_src #+RESULTS: : Measurement noise: 0.79 [nm,rms] DAC noise (amplified by the PI voltage amplifier, and converted to newtons): #+begin_src matlab Wdac = tf(3e-8); % V/sqrt(Hz) Wu = Wdac*22.5*10; % N/sqrt(Hz) #+end_src #+begin_src matlab :results value replace :exports results :tangle no sprintf('DAC noise: %.2f [uV,rms]', 1e6*sqrt(trapz(f, abs(squeeze(freqresp(Wdac, f, 'Hz'))).^2))); #+end_src #+RESULTS: : DAC noise: 0.95 [uV,rms] Disturbances: #+begin_src matlab Wd = 5e-7/(1 + s/2/pi); % m/sqrt(Hz) #+end_src #+begin_src matlab :results value replace :exports results :tangle no sprintf('Disturbance motion: %.2f [um,rms]', 1e6*sqrt(trapz(f, abs(squeeze(freqresp(Wd, f, 'Hz'))).^2))); #+end_src #+RESULTS: : Disturbance motion: 0.61 [um,rms] #+begin_src matlab :exports none :tangle no save('matlab/mat/asd_noises_disturbances.mat', 'Wn', 'Wu', 'Wd'); #+end_src #+begin_src matlab :eval no %% Save ASD of noise and disturbances save('mat/asd_noises_disturbances.mat', 'Wn', 'Wu', 'Wd'); #+end_src ** Open Loop disturbance and measurement noise The comparison of the amplitude spectral density of the measurement noise and of the jack parasitic motion is performed in Figure [[fig:open_loop_noise_budget_fast_jack]]. It confirms that the sensor noise is low enough to measure the motion errors of the crystal. #+begin_src matlab :exports none %% Bode plot for the plant (strain gauge output) figure; hold on; plot(f, abs(squeeze(freqresp(Wn, f, 'Hz'))), ... 'DisplayName', 'n'); plot(f, abs(squeeze(freqresp(Wd, f, 'Hz'))), ... 'DisplayName', 'd'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); xlabel('Frequency [Hz]'); ylabel('ASD [$m/\sqrt{Hz}$]'); legend('location', 'northeast', 'FontSize', 8, 'NumColumns', 2); xlim([f(1), f(end)]); #+end_src #+begin_src matlab :tangle no :exports results :results file replace exportFig('figs/open_loop_noise_budget_fast_jack.pdf', 'width', 'wide', 'height', 'normal'); #+end_src #+name: fig:open_loop_noise_budget_fast_jack #+caption: Open Loop noise budgeting #+RESULTS: [[file:figs/open_loop_noise_budget_fast_jack.png]] * Active Damping Plant (Strain gauges) :PROPERTIES: :header-args:matlab+: :tangle matlab/dcm_active_damping_strain_gauges.m :END: <> ** 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) <> #+end_src #+begin_src matlab :exports none :results silent :noweb yes <> #+end_src #+begin_src matlab :tangle no :noweb yes <> #+end_src #+begin_src matlab :eval no :noweb yes <> #+end_src #+begin_src matlab :noweb yes <> #+end_src #+begin_src matlab :noweb yes <> #+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: <> ** 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) <> #+end_src #+begin_src matlab :exports none :results silent :noweb yes <> #+end_src #+begin_src matlab :tangle no :noweb yes <> #+end_src #+begin_src matlab :eval no :noweb yes <> #+end_src #+begin_src matlab :noweb yes <> #+end_src #+begin_src matlab :noweb yes <> #+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: <> ** 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) <> #+end_src #+begin_src matlab :exports none :results silent :noweb yes <> #+end_src #+begin_src matlab :tangle no :noweb yes <> #+end_src #+begin_src matlab :eval no :noweb yes <> #+end_src #+begin_src matlab :noweb yes <> #+end_src #+begin_src matlab :noweb yes <> #+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_311*inv(J_s_311)*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]); 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', 'normal'); #+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 ** Close Loop noise budget #+begin_src matlab :exports none %% Load disturbances load('asd_noises_disturbances.mat'); #+end_src Let's compute the amplitude spectral density of the jack motion errors due to the sensor noise, the actuator noise and disturbances. #+begin_src matlab %% Computation of ASD of contribution of inputs to the closed-loop motion % Error due to disturbances asd_d = abs(squeeze(freqresp(Wd*(1/(1 + G_dp(1,1)*Khac(1,1))), f, 'Hz'))); % Error due to actuator noise asd_u = abs(squeeze(freqresp(Wu*(G_dp(1,1)/(1 + G_dp(1,1)*Khac(1,1))), f, 'Hz'))); % Error due to sensor noise asd_n = abs(squeeze(freqresp(Wn*(G_dp(1,1)*Khac(1,1)/(1 + G_dp(1,1)*Khac(1,1))), f, 'Hz'))); #+end_src The closed-loop ASD is then: #+begin_src matlab %% ASD of the closed-loop motion asd_cl = sqrt(asd_d.^2 + asd_u.^2 + asd_n.^2); #+end_src The obtained ASD are shown in Figure [[fig:close_loop_asd_noise_budget_hac_iff]]. #+begin_src matlab :exports none %% Noise Budget (ASD) f = logspace(-1, 3, 1000); figure; hold on; plot(f, asd_n, 'DisplayName', '$n$'); plot(f, asd_u, 'DisplayName', '$d_u$'); plot(f, asd_d, 'DisplayName', '$d$'); plot(f, asd_cl, 'k--', 'DisplayName', '$y$'); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); xlabel('Frequency [Hz]'); ylabel('ASD [$m/\sqrt{Hz}$]'); legend('location', 'southeast', 'FontSize', 8, 'NumColumns', 2); xlim([f(1), f(end)]); ylim([1e-16, 1e-8]); #+end_src #+begin_src matlab :tangle no :exports results :results file replace exportFig('figs/close_loop_asd_noise_budget_hac_iff.pdf', 'width', 'wide', 'height', 'tall'); #+end_src #+name: fig:close_loop_asd_noise_budget_hac_iff #+caption: Closed Loop noise budget #+RESULTS: [[file:figs/close_loop_asd_noise_budget_hac_iff.png]] Let's compare the open-loop and close-loop cases (Figure [[fig:cps_comp_ol_cl_hac_iff]]). #+begin_src matlab :exports none % Amplitude spectral density of the open loop motion errors [m/sqrt(Hz)] asd_ol = abs(squeeze(freqresp(Wd, f, 'Hz'))); #+end_src #+begin_src matlab :exports none % CPS of open-loop motion [m^2] cps_ol = flip(-cumtrapz(flip(f), flip(asd_ol.^2))); % CPS of closed-loop motion [m^2] cps_cl = flip(-cumtrapz(flip(f), flip(asd_cl.^2))); #+end_src #+begin_src matlab :exports none %% Cumulative Power Spectrum - Motion error of fast jack figure; hold on; plot(f, cps_ol, 'DisplayName', sprintf('OL, $\\epsilon_d = %.0f$ [nm,rms]', 1e9*sqrt(cps_ol(1)))); plot(f, cps_cl, 'DisplayName', sprintf('CL, $\\epsilon_d = %.0f$ [nm,rms]', 1e9*sqrt(cps_cl(1)))); hold off; set(gca, 'XScale', 'log'); set(gca, 'YScale', 'log'); xlabel('Frequency [Hz]'); ylabel('CPS [$m^2$]'); legend('location', 'southwest', 'FontSize', 8); xlim([f(1), f(end)]); % ylim([1e-16, 1e-8]); #+end_src #+begin_src matlab :tangle no :exports results :results file replace exportFig('figs/cps_comp_ol_cl_hac_iff.pdf', 'width', 'wide', 'height', 'normal'); #+end_src #+name: fig:cps_comp_ol_cl_hac_iff #+caption: Cumulative Power Spectrum of the open-loop and closed-loop motion error along one fast-jack #+RESULTS: [[file:figs/cps_comp_ol_cl_hac_iff.png]] * 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